1st INTERNATIONAL EURASIAN CONFERENCE ON MATHEMATICAL SCIENCES AND APPLICATIONS

PROCEEDING BOOK

SEPTEMBER 03-07, 2012 PRISHTINE KOSOVO

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

FOREWORD

The “1st International Eurasian Conference on Mathematical Sciences and Applications (IECMSA)’’ jointly organized by Bilecik University, Kocaeli University, Sakarya University and Prishtine University will be hold on 03-07 September 2012 in Prishtine, KOSOVO.

The aim of this conference is to bring the math society all over the Europa and Asia working in the new trends of applications and progresses in mathematical science in Prishtine, KOSOVO, the country in which the cemetery of the Soltan Murad Hudavendigar of Ottoman Empire, the symbol of justice. With this conference we also aimed to promote the publicity of Republic of Kosovo.

The organization scheme of this conference was formed by Organizing, Scientific and Honorary committees. The Scientific committee is formed from different University worldwide. Seven worldwide distinguished speakers are invited to the conference.

Although we have announced the conference within the limited time span we have taken a lot of applications all over the world. These big numbers of applications have given us opportunity to choose the best ones to reach the higher scientific level.

The first three days are organized for the scientific discussions and talks and the remaining two days for the social program. In the social program, there will be trips to historical and touristic place of Kosovo.

Within the social program there will be a Albanian and Turkish cultural program. We do believe that with these cultural programs we will give the opportunity to the participants to appreciate the Albanian and Turkish Culture to a certain extent.

The talks that are going to be delivered cover a wide range of mathematics which is one of the goals of this organization. Among the topics that are going to be presented are mostly on pure

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

and applied mathematics, engineering applications, statistics, algebra, computer mathematics, geometry, etc. We aimed to promote the interdisciplinary collaborations.

We would like to thank to Bilecik University, Kocaeli University, Sakarya University and Prishtine University for their invaluable supports. We would also like to thank to TİKA for their support.

We also thank to the invited speaker, who share their scientific knowledge with us, to the Organizing and Scientific committee. Last but not least we thank to all presenters and participants. We do believe and hope that each of them will get the maximum benefit from the conference.

We hope to see you in the second, third,... International Eurasian Conference on Mathematical Sciences and Applications in different countries and cities of Europe and Asia.

Sincerely yours,

Prof. Dr. Murat TOSUN, Chair

On behalf of the Organizing Committee

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

HONORARY COMMITTEE

Professor Misir MARDANOV (Azerbaijan Minister of Education, TWMS President)

Professor Baktyzhan ZHUMAGULAV (Kazakhstan Minister of Education, TWMS Honorary

President)

Professor Sezer Sener KOMSUOGLU (Kocaeli University Rectorate)

Professor Muzaffer ELMAS (Sakarya University Rectorate)

Professor Rugova MUJE (Prishtine University Rectorate)

Professor Azmi OZCAN (Bilecik Seyh EdebaliUniversity Rectorate)

Professor H. Hilmi HACISALIHOGLU (TWMS Honorary President)

Professor Ilyas OZTURK (Sakarya University Faculty of Arts and Sciences Dean)

Professor Fikret ALIYEV (TWMS Vice President)

Professor Amanbek JAINAKOV (TWMS Vice President)

Professor Nargozy DANAEV (TWMS Vice President)

Professor Etibar PANAHOV (TWMS Vice President)

Professor Mahmut ERGUT (TWMS Member of Steering Board)

Professor Murat TOSUN (TWMS Member of Steering Board)

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

SCIENTIFIC COMMITTEE

Professor H.Hilmi HACISALIHOGLU (Bilecik Seyh Edebali University)

Professor Fikret ALIYEV (Baku State University)

Professor Eberhard MALKOWSKY (Giessen University)

Professor Ravi P. AGARWAL (Texas University)

Professor Mohammad MURSALEEN (Aligarh Muslim University)

Professor Ekrem SAVAS (Istanbul Commerce University)

Professor Amanbek JAINAKOV (National Academy of Sciences of Kyrgyz Republic)

Professor Kazim ILARSLAN (Kirikkale University)

Professor Selim JUSUFI (Prishtine University)

Professor Sergio AMAT (Polytecnic University of Cartagena)

Professor Cihan OZGUR (Balikesir University)

Professor Sui Sun CHENG (Tsing Hua University)

Professor Levent KULA (Ahi Evran University)

Professor Ibrahim OKUR (Sakarya University)

Professor Rexhep GJERGJI (Prishtine University)

Professor Arif SALIMOV (Atatürk University)

Professor Florim ISUFI (Prishtine University)

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Professor Etibar PANAHOV (Firat University)

Professor Danaev NARGOZY (El-Farabi National Kazakh University)

Professor Varga KALANTAROV (Koc University)

Professor Anatolij DVURECENSKIJ (Slovak Academy of Sciences)

Professor Yusif GASİMOV (Baku State University)

Professor Irfan SIAP (Yildiz Tecnical University)

Professor Qendrim GASHI (Prishtine University)

Professor Nuri KURUOGLU (Bahcesehir University)

Professor Halis AYGUN (Kocaeli University)

Professor Muhib LOHAJ (Prishtine University)

Professor Mahmut ERGUT (Firat Universtiy)

Prof. Dr. Arjun K. GUPTA (Bowling Green State University)

Professor Fevzi BERISHA (Prishtine University)

Professor Nihal YILMAZ OZGUR (Balikesir University)

Professor Ali AKHMADOV (Baku State University)

Professor Ramadan ZEJNULLAHU (Prishtine University)

Professor Sadik KELES (Inonu University)

Professor Yusuf YAYLI (Ankara University)

Professor Faton BERISHA (Prishtine University)

Professor Kadri ARSLAN (Uludag University)

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Professor Qamil HAXHIBEQIRI (Prishtine University)

Professor Bayram SAHIN (Inonu University)

Professor Harry MILLER (International University of Sarajevo)

Professor Cengizhan MURATHAN (Uludag University)

Professor Soltan ALIYEV (National Academy of Sciences of Azarbaijan Republic)

Professor Fahir Talay AKYILDIZ (Gaziantep University)

Professor Omer AKIN (University of Economics and Technology)

Professor Naim BRAHA (Prishtine University)

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ORGANIZING COMMITTEE

Professor H.Hilmi HACISALIHOGLU (Bilecik Seyh Edebali University)

Professor Murat TOSUN (Sakarya University)

Professor Munir EFENDIJA (Pristhine University)

Professor Kazim ILARSLAN (Kirikkale University)

Professor Cihan OZGUR (Balıkesir University)

Professor Levent KULA (Ahi Evran University)

Professor Halis AYGUN (Kocaeli University)

Professor Avdulla ALIJA (Prishtine University)

Professor Gamar MAMMADOVA (Baku State University)

Professor Nargiz SAFAROVA (Baku State University)

Professor Agim GASHI (Prishtine University)

Professor Latifa AGAMALIEVA (Baku State University)

Professor Naser TRONI (Prishtine University)

Professor Musaj PACARIZI (Prishtine University)

Professor Mimoza DUSHI (Prishtine University)

Professor Naim SYLA (Prishtine University)

Associate Professor Mehmet Ali GUNGOR (Sakarya University)

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Associate Professor Hatice Gül INCE ILARSLAN (Gazi University)

Associate Professor Kamile SANLI KULA (Ahi Evran University)

Associate Professor Soley ERSOY (Sakarya University)

Assistant Professor Emrah Evren KARA (Bilecik Seyh Edebali University)

Assistant Professor Mahmut AKYIGIT (Sakarya University)

Assistant Professor Ergin JABLE (Prishtine University)

Assistant Professor Ercan MASAL (Sakarya University)

Assistant Professor Ayse Zeynep AZAK (Sakarya University)

Assistant Professor Murat SARDUVAN (Sakarya University)

Assistant Professor Mehmet GUNER (Sakarya University)

Assistant Professor Melek MASAL (Sakarya University)

Doctor Bujar FEJZULLAHU (Prishtine University)

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SECRETARIAT

Professor Murat TOSUN (Sakarya University)

Lecturer Muhsin CELIK (Sakarya University)

Lecturer Dogan UNAL (Sakarya University)

Research Assistant Hidayet Huda KOSAL (Sakarya University)

Research Assistant Hatice PARLATICI (Sakarya University)

Research Assistant Tulay SOYFIDAN (Erzincan University)

Research Assistant Valbon BYTYQI (Prishtine University)

Research Assistant Fisnik ALIAJ (Prishtine University)

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CHAPTERS

INVITED TALKS 1-8

ALGEBRA 9-38

ANALYSIS 39-76

APPLIED MATHEMATICS 77-146

DISCRETE MATHEMATICS 147-162

GEOMETRY 163-218

TOPOLOGY 219-246

STATISTICS 247-260

MATHEMATICS EDUCATION 261-280

THE OTHER AREAS 281-328

POSTER 329-372

PARTICIPIANTS 373-379

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CONTENTS

INVITED TALKS

Variable-range Approximate Systems: a unified approach to categories of fuzzy 1 topology and theories of fuzzy rough sets Alexander Sostak

Minimax Solution of the Problem of The Choice of Optimum Modes for Gas-Lift Process 2 F. A. Aliev, N. Ismayilov

Dimension of Fractals 3 H. H. Hacısalihoğlu

The Borwein-Ditor Theorem and My Oxford Moment 4 H. I. Miller

Bijective S-boxes and Self Dual Codes 5 P. Sole

Index of a Subfactor in a Real Factor 6 Sh. A. Ayupov, A. A. Rakhimov

ALGEBRA

Transfer of Orbital Integrals and Division Algebras 9 A. Aydoğdu, R. Öztürk, E. Özkan, Y. Z. Flicker

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras 10 A. Koç, S. Esin, İ. Güloğlu, M. Kanuni

Inequalities Involving vn( !) p 11 A. Sh. Shabani, V. Krasniqi

Zbërthimi në vlera singular I një matrice dhe zbatime 12 A. Vrapi, F. Hoxha

Structure of Intuitionistic Fuzzy Sets in Γ−Semihyperrings 13 B. A. Ersoy, B. Davvaz

-Vague Module 14 D. Bayrak, S. Yamak 푻푳 ∆-Primitive Elements of Free Metabelian Lie Algebras 15 D. Ersalan, Z. G. Esmerligil

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A Note on Direct Product of Fuzzy Modules over Fuzzy Rings 17 E. Yetkin

On Almost Prime Elements in Lattice Modules 18 E. Yetkin, Z. Kılıç, Ü. Tekir

On Multiplication Lattice Modules 19 F. Çallıalp, Ü. Tekir

Comultiplication Lattice Modules 20 F. Çallıalp, Ü. Tekir

On Closed Submodules of a Finite Rank Free Module over a Complete DVR 21 H. Garminia, G. Rahmaputri, P. Astuti

Commutative and Normal Groups on Soft Sets 22 İ. Şimşek, N. Çağman, K. Kaygısız, İ. Deli

On topological Structure on Ternary Semihyperrings 23 K. Hila, A. Sh. Shabani

N-dimensional (α, β)-fuzzy ideals of hemirings 24 M. Aslam, S. Abdullah

Test Rank of The Lie Algebra F/,[] RF′ 25 N. Ş. Öğüşlü, N. Ekici

Pullback for Simplicial Lie Algebras 27 Ö. G. Alansal

Induced Inner Product on a Quotient Tensor Product Space 28 P. Astuti

Normality and Quadraticity for Special Ample Line Bundles on Toric Varieties Arising from 29 Root Systems Q. R. Gashi, T. Schedler

The Square Terms in Generalized Fibonacci Sequence 30 R. Keskin, Z. Şiar

Generalized Fuzzy Bi-hyperideals in Semihypergroups 31 S. Abdullah, K. Hila

Some Relations on Extending Modules 32 S. Doğruöz

Generalized (,)fg Derivations on Lattices 33 Ş. Ceran, M. Aşcı

Algebraic Hyperstructures of Soft Sets Associated to n-ary Polygroups 34 Ş. Yılmaz, O. Kazancı

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Irreducible, Prime and Maximal Idealistics Soft WS-Algebra 35 U. Acar, A. Harmancı

Some Regular Elements, Idempotents and Right Units of Complete Semigroups of Binary 36 Relations Defined by Semilattices of the Class Lower Incomplete Nets Y. Diasamidze, A. Erdoğan, N. Aydın

Parafree Lie Algebras with Certain Properties 38 Z. Velioğlu, N. Ekici

ANALYSIS

On Some New Paranormed Sequence Spaces of Non-Absolute Type 39 A. A. Karaaslan, V. Karakaya

40 On saturation Order of Functions in the Space L p ππ p ∞<<− )1(),( for Holder Method of

Fourier Series Summation A. J. Aliyeva, J. M. Aliyev

On the Fine spectrum of Some Generalized Difference Operators 41 A. M. Akhmedov, S. R. El-Shabrawy

Fixed Point Theorems in p − Summable Symmetric n − Cone Normed Sequence Spaces 42 A. Sahiner, T. Yigit

Matrix Inequality Hardy’s Type 43 B. Sherali

Asymptotics of Modified Jacobi Polynomials in a Neighborhood of Endpoints of the Interval 44 of Orthogonality B. Xh. Fejzullahu

Compact Operators on the Fibonacci Difference Sequence Spaces ( ) and ( ) 45 E. E. Kara, M. Başarır, M. Mursaleen 퓵풑 푭� 퓵∞ 푭� On Some New Paranormed B-Difference Sequence Spaces Derived by Generalized Weighted 46 Mean Matrix E. E. Kara, S. Demiriz

Spectrum of Matrix Operators on a Difference Sequence Space of Weighted Means 47 E. Erdoğan, V. Karakaya

On Matrix Transformations Between Sequence Spaces 48 E. Savaş

Approximation of Functions by Algebraic Polynomials in Jacobi Weighted Lp Spaces 50 F. Berisha, S. Makolli

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Shpejtësia e Tentimit Në Zero e Koeficientëve Furie Të Funksioneve Me Derivat Të wely-it 52 F. Berisha

Approximating Classes of Functions Defined by a Generalised Modulus of Smoothness 54 F. M. Berisha, N. S. Berisha

Starlikeness and Convexity of Generalized Struve Functions 55 H. Orhan, N. Yağmur

An Extremum Problem in the Metric Space 57 H.S. Akhundov, M.A. Sadygov

On the Boundedness of Riesz-Bessel Transforms associated with the B-GSO 58 I. Ekincioglu, C. Keskin

On Privalov Type Estimates of High order Riesz-Bessel Transformations Generated by a 60 Bessel Generalized Shift Operator

I. Ekincioglu, C. Keskin

Some Structural Properties of Vector Valued Orlicz Sequence Space Generilazed by an Infinite 62 Matrix M. Candan Almost Convergence and Generalized Double Sequential Band Matrix 63 M. Candan

On Pointwise and Uniform λ-Statistical Convergence of Order α of Sequences of Functions 64 M. Et, M. Çınar, M. Karakaş

Some Results on dist -formulas, Toeplitz Operators and Berezin Symbols 65 M. Gürdal, M.T. Karaev, S. Saltan, U. Yamancı

Quasi-Diagonal Operators 67 M. Lohaj, S. Lohaj

Radon-Nikodym Property for Vector Valued Measures 68 M. Öztürk, C. Çevik

On mt-Convexity 69 M. Tunç, H. Yıldırım

On New Inequalities for h-Convex Functions via Riemann-Liouville Fractional Integration 70 M. Tunç

Two New Definitions on Convexity and Related Inequalities 72 M. Tunç

On One Class of Sequences and a Convolution Operator in This Class 73 N. Ibadov

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Möbius Transformations and Circles on the Complex Plane 75 N. Y. Özgür

APPLIED MATHEMATICS

Some Properties of a Family of Analytic Functions 77 A. Altın

Numerical Approximation for Boundary Value Problem by a New Collocation Approach 78 A. B. Koç, A. Kurnaz

Statistical Integral on Banach Space 79 A. Caushi, A. Tato

Dynamical Problem for the Pre-Stressed Bi-Layered Plate-Strip with Finite Length under the 81 Action of Arbitrary Time-Harmonic Forces Resting on a Rigid Foundation A. Daşdemir, M. Eröz

On the Solution of Vibration (Eigenvalue) Problem of Ceramic Cylindrical Shells with FG 82 Coatings A. Deniz, A. H. Sofiyev

Finite Difference/Spectral Approximations for the Time Fractional Higher Order Partial 84 Differential Equations A. Kablan, F. T. Akyıldız

Anew Metaheuristic Algorithm Based Chemical Process: Atom Algorithm 85 A. Karcı

The Measurement of the Effect of the Pre Crisis and Post Crisis Economic Dynamics 87 (Parameters) over the Stock Index with the 2-factor Cross Classification Model of Variance

Analysis Models and Turkey Example A. Mazmanoğlu Exact Solutions of the Zakharov Equations by Using the First Integral Method 89 A. Öğün Ünal

About Existence of the Local and Global on Time Solutions of the Initial and Initial-Boundary 90 Value Problems for One-Dimensional Nonlinear Boltzmann’s Moment System Equations A. Sakabekov, Y. Auzhani

Cubic B-Spline Finite Element Method for the Numerical Solution of Fisher’s Equation 91 A. Şahin, H. Bilgil

The Problem of Constructing a Step Function, the Least Fluctuating around a Given Function 93 B. Bayraktar

Matrix Analogues of Some Properties for Bessel Functions 95 B. Çekim, A. Altın

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A General Solution Procedure for Fractionally Damped Beams 96 D. Dönmez Demir, N. Bildik

Generalizations of Hölder’s Inequalities on Time Scales 97 D. Uçar, A. Deniz

Some Specials Results and Applications of Pseudo – Measure Theory 98 D. Valera

Numerical Solution for Two-Point Fuzzy Boundary Value Problem 100 E. Can, C. Köroğlu, A. Golayoğlu Fatullayev

Modal Characteristics of Optical Waveguides with Cardioid-Shaped Cross Sections 101 E. Eroğlu, H. Altural

Approximate Solution of a Model Describing Biological Species Living Together by Taylor 102 Collocation Method E. Gökmen, M. Sezer

Laguerre-Krall Orthogonal Polynomials: Outer Relative Asymptotics and an Electrostatic 103 Model for Their Zeros. E. J. Huertas, F. Marcellán, H. Pijeira

On the Inverse Problems for Singular Differential Operators 104 E. S. Panakhov, M. Sat

A Problem with Mixed Boundary Conditions for an Elastic Strip When There Are Friction 105 Forces in the Contact Area E. Yusufoğlu , H. Oğuz Numerical Solution of a Crack Problem with the Help of Gauss Quadrature Formulas 106 E. Yusufoğlu, İ. Turhan

Method for the Solution of the Optimal Control Problem for the Linear Descriptor Systems 107 F.A. Aliev, N.I. Velieva , Y.S. Gasimov, L.F. Agamalieva

Iterative Algorithms for the Solution of the Discrete Optimal Regulator Problem in Stationary 108 and Periodic Cases F.A. Aliev, N.A. Safarova, N. I. Velieva

Polynomial Presentation of the Binary Multidimensional Nonlinear Modular Dynamic Systems 109 F.G Feyziev, G.H Mammadova, Sh.T. Mammadov

A Kantorovich Type of Szasz Operators Including Brenke Type Polynomials 110 F. Taşdelen, R. Aktaş , A. Altın

The Energy Levels, Binding Energies, and Dipole Moment in a Gaas/Alas Spherical Quantum 111 Dot as Function of the Donor Position H. Akbas, P. Bulut, C. Dane

Steady Stokes Flow in a Sectorial Cavity: Lids Moving in the Same Direction 112 H. Bilgil, A. Şahin

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A Crack Identification Method for Bridge Type Structures under Vehicular Load Using 114 Wavelet Transform and Particle Swarm Optimization H. Gökdağ

Green’s Functions for Elasto-hydrodynamic Model of 1D Quasicrystals 115 H. K. Akmaz

An Efficient Scheme for Designing Substitution-boxes Based on TD-ERCS Chaotic Sequence 116 I. Hussain, M. A. Gondal

Image Analysis with Differential Geometry Applications 118 K. Hanbay, M. F. Talu

Transferring of Energy via Special Functions; A Special Case: “Airy Function” 120 K. Köklü, E. Eroğlu, N. Eren

To properties of solutions of one nonlinear problem of heat onductivity in a heterogenic media 121 M. Aripov, M. Khodjimurodova

About Extremum Problems with Constraints 122 M. A. Sadygov

Koçak’s Acceleration Method Smoothly Gears Up Iterative Solvers 123 M. Ç. Koçak

The Stress Field Problem for a Pre-Stressed Plate-Strip with Finite Length under the Action of 124 Arbitrary Time-Harmonic Forces M. Eröz

The Influence of Initial Stresses on a Pre-stressed Orthotropic Plate-strip with Finite Length 125 Resting on a Rigid Foundation M. Eröz

On Solving a Type of Fractional Volterra-Integro Differential Equations Using Homotopy 126 Analysis Method M. F. Karaaslan, M. Kurulay

About one Problem of Solution Stabilization of the Loaded Heat Equation 127 M.M.Amangaliyeva, M.T.Jenaliyev, K.B.Imanberdiyev and M.I.Ramazanov

On the Best Approximate (P,Q)-Orthogonal Symmetric and Skew-Symmetric Solution of the 128 Matrix Equation AXB= C M. Sarduvan, S. Şimşek, H. Özdemir

A Fractional Order Nonlinear Dynamical Model 129 N. Özalp, İ. Koca

Some Approximation Results for Lupaş Operators 130 O. Doğru, K. Kanat

Asymptotic of Solution Functions of a Sturm-Liouville-Type Problem and the Green Function 131 O. Kuzu, M. Kadakal

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Some Possible Fuzzy Solutions for Second Order Fuzzy Initial Value Problems Involving 133 Forcing Terms Ö. Akın, T. Khaniyev, Ö. Oruç, I. B. Türkşen

Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq 135 Equations of Sobolev Type Ö. F. Gözükızıl, Ş. Akçağıl

Explicit Solutions for Fractional Schrödinger Equation 136 R. Yılmazer, E. Bas

Sharp Weighted Rellich Type Inequalities for Generalized Greiner Vector Fields 137 S. Ahmetolan, İ. Kombe

Review of special relativity mathematics 138 S. Klinaku

Numerical Simulations of Coupled Sine-Gordon Equations by Reduced Differential Transform 139 Method S. Servi, A. B. Koç, Y. Keskin

On Removable Sets of Solutions for Degenerate Linear Parabolic Equations 141 T.S. Gadjiev, S.A. Aliev, O.S. Aliyev

Calculation of Shear Stresses Occurred on Elastic Soils under Rectangular Foundation with 142 Analytical and Numerical Methods U. Dağdeviren, Z. Gündüz

On a Shape Design Problem for the Eigenfrequency of the Membrane with Fixed Boundary 144 Lenth Y. S. Gasimov, L. I. Amirova

Reduced Differential Transform Method for Solving High-Dimensional PDEs 145 Y. Keskin, A. Betül Koç, S. Servi

DISCRETE MATHEMATICS

Connected Dominating Sets in Unit Disk Graphs 147 D. A. Mojdeh, S. Kordi

Perfect Codes in the Euclidean Metric 149 F. Temiz, V. Şiap

On One Dimensional 2r+1-Cyclic Rule Cellular Automata 150 İ. Şiap, H. Akın, M. E. Köroğlu

Image Representation as Skip Graph 152 M. Aksu, M. Canayaz, A. Karcı

On Generalized Gaussian Fibonacci Numbers and Matrix Methods 154 M. Aşcı

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Content Based Image Retrieval via Graph Similarity 155 M. Canayaz, M. Baykara, M. Aksu, T. Özseven, A. Karcı

Perfect Mannheim, Lipschitz and Hurwitz Weight Codes 157 M. Güzeltepe

Codes over Hurwitz Integers 158 M. Güzeltepe

A Study on Minimal Codewords of the Codes Associated with the Symmetric 160 v,( k, λ) − Designs S. Çalkavur, E. Balkanay

On MacWilliams Identity for M-spotty Weight Enumerators of Linear Codes over Frobenius 162 Rings V. Şiap, M. Özen

GEOMETRY

On Hypercomplex Structures 163 A. A. Salimov

On the Energy and the Angle of a Frenet Vector Fields 164 A. Altın

The Gauss Equations of Timelike Surface Invaryant under One Parameter Group of Lorentzian 165 Isometry A. İnalcık, S. Ersoy

On Inclined Curves in Euclıdean n -Space 166 A. Şenol, E. Zıplar, Y. Yaylı, H. H. Hacısalihoğlu

On Three-Dimensional Normal Almost Contact Metric Manifolds Admitting Quarter 167 Symmetric Non-Metric Connection A. Yıldız, A. Çetinkaya

Weingarten and Linear Weingarten Type Tubes with Darboux Frame in E3 168 A. Z. Azak, M. Tosun

On Semiparallelity of Wintgen Ideal Surfaces 169 B. Bulca, K. Arslan

Umbrella Motion in R3 170 B. Karakaş, Ş. Baydaş

Umbrella Surfaces in R3 as an Orbit Surfaces 171 B. Karakaş, Ş. Baydaş

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Anti-Invariant Riemannian Submersions from Almost Hermitian Manifolds:Curvatures 172 B. Şahin

B.-Y. Chen Inequalities for Submanifolds of a Conformally Flat Manifold 174 C. Özgür

On the Two Parameter Homothetic Motions in the Complex Plane 175 D. Ünal, M. Çelik, M. A. Güngör e-Curvature Functions in 4-Dimensional Lorentzian Space 176 E. İyigün

f -Eikonal Helix Submanifolds and f -Eikonal Helix Curves 177 E. Zıplar, A. Şenol, Y. Yaylı

Fractals of Finite Area 179 F. Çilingir

Half-Lightlike Hypersurfaces with Planar Normal Sections in 180 F. E. Erdoğan, R. Güneş, B. Şahin ퟒ 푹ퟐ On Weyl Manifolds with Semi-Symmetric Reccurent Metric Connection 181 F. Özdemir

Euler’s Formula and De-Moivre’s Formula for Dual Hyperbolic Quaternions 182 H. H. Kösal, M. Akyiğit, M. Tosun

On the Geometry of the Sasaki semi-Riemann Manifold on the Tangent Sphere Bundle of the 183 de-Sitter Space İ. Ayhan

Notes on Curves in Sasakian 3-manifolds 184 İ. Gök, Y. Yaylı

On Null Rectifying Curves in 185 K. İlarslan ퟒ 푹ퟐ On Special Curves in Sol Space 186 L. Sizer, H. Başeğmez, M. Altınok, L. Kula

Fibonacci Split Quaternions 187 M. Akyiğit, H. H. Kösal, M. Tosun

A Note on Fibonacci Generalized Quaternions 188 M. Akyiğit, H. H. Kösal, M. Tosun

Slant Helices Generated by Plane Curves and Applications 189 M. Altınok, L. Kula

On the Two Parameter Lorentzian Homothetic Motions 190 M. Çelik, D. Ünal, M. A. Güngör

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On ξ -Conformally and ξ -Projecitvely Flat Lorentzian Sasakian Manifolds with Tanaka- 191 Webster Connection M. Erdoğan, J. Alo, G. Yilmaz

3 Special Curves in Three Dimensional Finsler Manifold F 193 M. Ergüt, M. Külahcı

Some Characterizations on Biharmonic General Helices in Finsler Manifold F3 194 M. Ergüt, M. Y. Yılmaz

Inextensible Flows of Curves in Three Dimensional Finsler Manifold F3 195 M. Ergüt, A. O. Öğrenmiş

Inverse Surfaces of Tangent, Principal Normal and Bi-normal Surfaces of a Space Curve in 196 Euclidean 3-Space M. Ergüt, M. E. Aydın

On the First Fundamental Theorem for Dual Orthogonal Group O(2, D) 197 M. İncesu, O. Gürsoy, D. Khadjıev

On the B-Manifold Defined by Algebra of Plural Numbers 198 M. İşcan, A. Mağden

CMC-Surfaces in Hyperbolic Space Hn 200 M. Mak, B. Karlığa

Torsion Tensors of Pure Π−Connections 202 N. Cengiz

Screen Semi-Invariant Half-Lightlike Submanifolds 203 O. Bahadır

n A Note on Inextensible Flows of Curves in E 204 Ö. G. Yıldız, M. Tosun, S. Ö. Karakuş

A Generalization of Some Well-Known Distances and Related Isometries 205 R. Kaya

Bobillier Formula for One Parameter Motions in the Complex Plane 206 S. Ersoy, N. Bayrak

Arc-Lenghts, Curvature and Natural Lifts of Spherical Indicatrix of Timelike Mannheim Curve 208 with Spacelike Binormıal Spacelike Curve Partner S. Şenyurt, S. Demet

Arc-Lenghts, Curvature and Natural Lifts of Spherical Indicatrix of Timelike Bertrand Curve 209 Partner S. Şenyurt, Ö. F. Çalışkan

Lightlike Hypersurfaces of an ( )-Para Sasakian Manifold 210 S. Y. Perktaş, E. Kılıç, M. M. Tripathi 휺

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On the Ruled Surfaces Whose Frames are the Bishop Frame in the Euclidean 3-Space 211 Ş. Kılıçoğlu, H. H. Hacısalihoğlu

Constant Angles Surfaces in Hyperbolic Space 212 T. Mert, B. Karlığa

On the Quaternionic Curves According to Parallel Transport Frame 214 T. Soyfidan, H. Parlatıcı, M. A. Güngör

4 On the Quaternionic Rectifying Curves in Semi-Euclidean Space 215 2 T. Soyfidan, M. A. Güngör

Conformal Triangles in Hyperbolic and Spherical Space 216 Ü. Tokeşer, B. Karlığa

Framed-complex Submersions 217 Y. Gündüzalp

Affine Differential Invariants of a Pair of Curves 218 Y. Sağıroğlu

TOPOLOGY

On Multimetric Spaces 219 A.A. Borubaev

On τ − sequential Spaces 220 A. A. Borubaev, B. A. Boljiev

On τ − metric Spaces 222 A.A. Borubaev, K. Ishmakhametov

On Some Types of Strongly Continuous Multifunctions 223 A. Açıkgöz, S. Göktepe

Simplicial Weak (4,2)-chain Complexes of Simplicial Complexes 224 A. Ahmeti

Actions and Coverings of Topological Groupoids 226 A. F. Özcan

On τ − ultracompact and τ − bounded Spaces. 227 B. A. Boljiev

Separation Axioms on Soft Topological Spaces 228 B. P. Varol, H. Aygün

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Completeness Properties of Di-Uniform Texture Spaces 229 F. Yıldız

Topological Internal Groupoids and Their Coverings 230 H. F. Akız, N. Alemdar, T. Şahan, O. Mucuk

Some Properties of Polynomials G and N, and Tables of Knots and Links 232 İ. Altıntaş

On Uniformly Paracompact Mappings 234 K. B. Emenovich

Proper C − Tameness and Proper Deformation Dimension 236 M. Efendija

The Product of Shape Fibrations 237 Q. Haxhibeqiri

Rough Topology on Covering-Based Rough Sets 239 S. Akduman, E. Z.Yıldız, A. Z. Özcelik, S. Narlı

Simplicial Homotopy in Digital Images 240 S. Öztunç, N. Bildik, A. Mutlu

Semi-Compactness in Ditopological Texture Spaces 241 Ş. Dost

Some Generalized Closed Sets and Continuous Functions in Ideal Topological Spaces 242 Ü. Karabıyık

Extension of Shi’s Quasi-Uniformity to the Fuzzy Soft Sets 245 V. Çetkin, H. Aygün

STATISTICS

Weight Function Efficiency Against Outliers in Nonlinear Regression 247 A. Pekgör, A. Genç

Transmuted Exponentiated Exponential Distribution 248 F. Merovci

Comparisons of Imputation Methods for Missing Data in Mixture Discriminant Analysis 250 M. Erişoğlu, İ. K. Sarı

Determining Number of Clusters with Analytic Hierarchy Process 251 M. Erişoğlu

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Analysis of Divorces Using Survival Models 252 N. Ata, T. P. Tacoğlu A Study on an Additive Decomposition of the Blues in a Partitioned Linear Model and Its Two 254 Small Models N. Güler

Properties of Systems with Nonidentical Components 256 N. Torrado

Bayesian Approach of Structural Equation Models with Conjugate Priors for Discrete 257 Distributions S. Şehribanoglu, H. Okut

L-Moments Estimations for Mixture of Two Weibull Distributions 259 Ü. Erişoğlu, M. Erişoğlu

MATHEMATICS EDUCATION

Analysis of the Effect of Using Interactive Learning Objects on Critical Thinking Skills of 261 Math Teacher Candidates A. Ş. Özdemir, A. B. Yaprakdal

The Relationship between 7th Grade Students Visual Literacy Competencies and Their 263 Academic Achievement A. Ş. Özdemir, S. Göktepe 264 The Effects of History of Mathematics on Mathematics Education D. Demiroğlu, E. Aydın, A. Delice

The Correlation Between Achievement Test Scores and Creative Thinking Skills of Non-gifted 266 Students E. Altıntaş, A. Ş. Özdemir, A. Kerpiç

The Changes in the Cognitive and Affective Properties of the Pre-Service Elementary 267 Mathematics Teachers from the Beginning of the Teacher Training Program to Graduation E. Masal, M. Masal, E. Akgün, M. Takunyacı

Open Source Softwares in Mathematics Education 269 E. Özüsağlam, G. Biçer, A. Atalay, P. Poşpoş

Sequences and Series: What are These Concepts Through the Eyes of Pre Service Secondary 270 Mathematics Teachers? G. Yazgan-Sağ, Z. Argün

The Concept of Proof is Examined the Size of Teachers 272 İ. Yavuz, M. Taştepe

Analysis of the Elementary 8th Grade Students’ Mathematical Ability Levels from the 273 Perspective of PISA M. Altun, N. Aydın, R. Akkaya, D. Uzel

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Proof Schemes Used by Mathematics Teacher Candidates 274 S. Şengül, P. Güner

Preservice Elementary Mathematics Teachers’ Proving Skills 275 Y. Kıymaz, B. K. Doruk, T. Horzum

The Conceptual Methodical Approach in Training as a Mean of Increase of Degree of Social 277 Protection of Disabled Children Z. K. Yuldashev, M. A. Yuldasheva

THE OTHER AREAS

Quartic B-Spline Collocation Method for Numerical Solutions of the Klein-Gordon Equation 281 A. Boz, İ. Dağ

Vehicle Crashworthiness Optimization Using Differential Evolution Approach 282 A. R. Yıldız

Optimization of Vehicle Components Using Artificial Bee Colony Algorithm 285 A.R. Yıldız

Modeling of Flux Decline at Nanofiltration Membrane by Artificial Neural Networks 287 B. Eren

Reduction of Dimensionality in the Problem of Diffraction From a Half-Plane 288 B. Kamishi, R. Bejtullahu

Estimation of Lake Sapanca Daily Water Level Using Artificial Intelligent Techniques 289 E. Doğan, S. Demir

Seismic Q Estimation Using Artifical Neural Network E. Yıldırım, R. Saatçılar, G. Horasan, S. Ergintav 290

A Real-Time Virtual Sculpting Application with a Haptic Device 291 G. Cit, K. Ayar, S. Serttaş, C. Öz

Pre-Service Student Teachers’ Point, Line and Plane Concept Knowledges in Geometry and 293 Used Multiply Representation G. Tuluk Self-Dual Codes from Self-Dual Codes of Smaller Lengths and Recursive Algorithm 295 H. Topcu, H.Aktas

Theoretical Calculation of the Ground-State Magnetic Moments in the Odd-Mass Nuclei 297 H. Yakut, E. Tabar, E. Guliyev, O. Örnek

Performans Ranking of Turkish Insurance Companies : the Anp Application 298 İ. Akhisar

Relations on FP-Soft Sets Applied to Decision Making Problems 299 İ. Deli, N. Çağman, İ. Şimşek

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Assessing the Relationship Between Educational Performance and Attitudes of Turkish 300 Students İ. Demir, S. Kılıç

Comparing Turkey’s Domestic Debt Stock Increment with Linear Regression, Ridge 301 Regression and Principle Component Regression İ. Demir, G. Altunel, E. Cene

The Adomian Decomposition Method for Solving Nonlocal Boundary Value Problems for 303 First-Order Linear Hyperbolic Equation L. Bougoffa

Genetic Algorithm and Financial Optimisation 304 M. A. Ünal

Uniform Time Controllability of Affine Control Systems on Semisimple Lie Groups 305 M. Kule

The Effect of Instruction States Designed According to Van Hiele Geometrical Thinking 306 Levels on the Geometrical Success M. Terzi, Ş. Mirasyedioğlu

On a Problem of Thermal Convection with Unset Flow Rate 309 N.T. Danaev, B.S. Darybaev, B.A. Urmashev

Solution of Spherical Triangles in Geodesy with the Equations Used in Spherical Trigonometry 310 N. Ersoy, E. Yavuz, R. G. Hoşbaş

Application of Incomplete Cylindrical Functions in the Diffraction of Gaussian Beam from a 311 Half-Plane R. Bejtullahu, B. Kamishi, Z. Tolaj, F. Aliaj

Asynchronous Motor with Finite Element Method Nonlinear Analysis 312 S. A. Korkmaz, H. Kürüm Relationship Between Classical and Quantum Physic 314 S. Ahmetaj, S. Kabashi, S. Bekteshi

Modeling of gas flow through a rectangular channel (variable leak valve) 314 S. Avdiaj, N. Syla and F. Aliaj

Modelling Kosovo’s Power System and Scenarios for Sustainable Development 315 S. Bekteshi, S. Kabashi, S. Ahmetaj The Specific Exponential Stability of Solutions of Linear Homogeneous Volterra Integro- 317 differential Equation of Sixth Order S. Iskandarov

Rendering Virtual Welding Seam Form 318 S. Serttaş, K. Ayar, C. Öz, G. Cit

The Relationships Among Interest Rate, Exchange Rate and Stock Price: A BEKK – 321 MGARCH Approach S. Türkyılmaz, B. Yıldız

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The Causality Relationship Among Inflation, Output Growth and Their Uncertainties: 323 Evidence For Turkey S. Türkyılmaz, M. Balıbey

Bounds on the Largest Eigenvalue of the Distance Signless Laplacian of Connected Graphs 325 Ş. B. Bozkurt, D. Bozkurt

An Historical Overview of Visual Mathematical Arts 326 Z. Tez

Processing of Non-determined Results of Observations with Interval Interpolation Polynoms 328 Z. Yuldashev, A. Ibragimov, S. Tadjibaev

POSTER

The Generalized Incomplete Pochhammer Symbols and Their Applications to Exponential 329 Functions A. Çetinkaya, O. Kıymaz

A Solution of the Radial Schrödinger Equation for the Potential Family 331 AB 2 V() r = −+Cr + Dr r2 r A. Güleroğlu , C. Dane, H. Akbaş

Modeling of Concentrators Influence on Stress Condition of Elastic-plastic Structures 332 A.M. Polatov

The Special Solution of Schrödinger Equation with by Symmetries 333 C. Dane, K. Kasapoğlu, H. Akbaş

An Analysis of Blood flow in the Human Descending Aorta with Different Reynolds Number 334 D. Pandır, Y. Pandır

Ideals and Their Characterizatios in Factor Γ - Near – Rings 335 E. Domi, I. Braja

Numerical Solution of Some Dynamic Problems in Nanomechanics 336 E. Hazar, M.K. Cerrahoğlu

Inclined Curves in the Lightlike Cone 337 E. S. Yakıcı, İ. Gök, F. N. Ekmekci, Y. Yaylı

A Study on The Spherical Null Curves and Bertrand Curves in Minkowski 4 Spacetime 338 G. Güner, N. Ekmekci, Y. Sağıroğlu

Reconstruction in Education 339 H. Bozyokuş

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State Estimation in Induction Motors via Block-Pulse Functions 340 H. Kızmaz, S. Aksoy

Geodesics on the Tangent Sphere Bundle of the 3-sphere 342 İ. Ayhan

One-Sided Similar Ideals in Γ-Semigroups and Some Properties of the Principal Quasi-Ideals 343 I. Braja, E. Domi

A Different Characterization of U1(ZC8) 344 K. Arı , M. Görgülü

The Dependence Described by Copulas in the Reinsurance Treaties 345 K. Haxhi, O. Zacaj

346 Algebraic Hyperstructure of Soft Sets Associated to Ternary Semihypergroups K. Hila, S. Sadiku, K. Naka

Solution of Time-Independent One Dimensional Schrödinger Equation Using Symmetries 347 K. Kasapoğlu, C. Dane, H. Akbaş

Parameter Estimation Based on Type-II Fuzzy Logic 348 K. Ş. Kula, T. E. Dalkılıç

Intrinsic Geometry of the NLS Equation According to Bishop Frame in Euclidean 349 3-Space M. Ergüt, H. Öztekin, S. Aykurt

On the Mcshane Integral on the Riesz space 350 M. Shkëmbi, I. Temaj

Developing Attitude Scale toward the Geometric Objects for the Preservice Teachers 352 N. Gürefe, A. Kan

Modeling the RC Electric Circuit with Finite Element Method 353 N. Syla, F. Aliaj, S. Avdiaj

Extended Caputo Fractional Derivative and Its Applications 354 O. Kıymaz, A. Çetinkaya, Y. Sökmen

Non-Symmetric Divisor Problem over the Ring of Gaussian Integers 356 O. Savastru

Harmonic Curvature Function of Special Curves in Lie Groups 357 O. Z. Okuyucu, İ. Gök, Y. Yaylı, F. N. Ekmekci

Applications to Markov Chains: Introducing a BONUS-MALUS Model for the MTPL 358 Portfolio in Albania O. Zacaj, K. Haxhi

Fractional Calculus Operator for a Singular Sturm-Liouville Equation 360 R. Yılmazer, Ö. Öztürk

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State Variables Computation Technique for Linear Systems by Using Block-Pulse Functions 361 S. Aksoy, H. Kızmaz

On Fuzzy Soft Γ-ring 363 S. Onar, B. A. Ersoy

Some Examples of Crossed Modules 364 T. S. Kuzpınarı

Oscillation Criteria for Second Order Nonlinear Differential Equations 365 X. Beqiri, E. Koci

Solutions of the Nonlinear Differential Equations by Use of Symmetric Fibonacci Functions 367 Y. A. Tandoğan, Y. Pandır

A New Approach to the Trial Equation Method 368 Y. Gürefe, Y. Pandır, E. Mısırlı

Elliptic Function Solutions of a Nonlinear Partial Differential Equation 369 Y. Pandır, Y. Gürefe, E. Mısırlı

On Parallel Surfaces in Minkowski 3-Space 370 Y. Ünlütürk, E. Özüsağlam

Computation of Energies of Atoms Over Integer and Non Integer Quantum Numbers 371 Y. Yakar, B. Çakır, A. Özmen

Participiants 373

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INVITED TALKS

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Variable-range approximate systems: a unified approach to categories of fuzzy topology and theories of fuzzy rough sets Alexander Sostak University of Latvia, Riga,Latvia, [email protected]

Abstract. In our paper A. Sostak, Towards the theory of  -approximate systems: Fundamentals and examples, Fuzzy Sets and Syst. 161 (2010), 2440-2461. the concept of an  -approximate system where  is a fixed complete lattice was introduced and basic properties of the category of M-approximate systems were studied. We regard the concept of an  approximate system and the corresponding category as the framework for a unified approach to various categories related to (fuzzy) (bi)toplogical spaces and to (fuzzy) rough sets. Although the attempts to study the relations between fuzzy topological space and fuzzy rough sets and to introduce a context allowing to give a unified view on these notions were undertaken also by other authors, the approach presented in our paper is essentially different. In this work we continue the research of  -approximate systems. However, as different from our previous work here we consider the case of a variable range  , that is allow to change lattice M. In particular this allows to include also the category of  -topological spaces with varied lattices  and  in the scope of our research.

The author gratefully acknowledge a partial financial support by the ESF research project 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008. 1

1 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Minimax Solution of The Problem of The Choice of Optimum Modes for Gas-Lift Process Aliev F. A. (1) and Ismayilov N. (2) (1) Baku State University, Baku, Azerbaijan, [email protected]

Abstract. As it is known [1], gas-lift method for oil production plays an important role after fountain process. In spite of this way is used for a long time, till now there are no exact mathematical models of the corresponding control problem [2]. Really, in [2] mathematical model of operation of the gas-lift where the initial motion of fluids is described by the system of ordinary differential equations is suggested. However, in [3] it is demonstrated that the motion of fluids is described by the partial differential equation of hyperbolic type. Here we try to solve the general problem (optimal modes and their stabilization), reducing it to the linear quadratic problem of optimization by means of the method of straight lines [5]. Introducing the small parameter an asymptotical method is proposed to the solution of the choice of program trajectory and control. Basically, it is supposed that the bulk of debit should be fixed. However, in practice the problem consists in a finding of minimal bulk of gas to provide maximal production of oil [1]. This is a mathematical statement of “minimax problem” [6]. By the aggregating of the positive weighting matrix coefficients in control and of the negative weighting coefficients, in debit it is possible to construct the quality criterion and to formulate the “minimax problem”. The purpose of the present work is study of this problem. Keywords. minimax problem, Euler-Lagrange equation AMS 2010. 15A24

References

[1] Mirzadzhanzade A.H., Ahmetov I.M., Khasaev A.M., etc. Oil Production Technology and Equipment. Under the editorship of A.H.Mirzadzhanzade, Moscow, Nedra, 382p., 1986.

[2] Eikrem G.O., Aamo O.M. and Foss B.A. Stabilization of Gas-Distribution Instability in Single-Point Dual Gas Lift Wells, SPE Production & Operations, vol. 21, N2, pp.1-20, 2006.

[3] Aliev F.A., Ilyasov M. Kh, Dzhamalbekov M. A. Modeling the operation of a gas-lift well, Reports NAS Azerbaijan, N.4, pp.30-41, 2008.

[5] Aliev F.A., Ilyasov M. H, Nuriev N.B. Problem of mathematical modeling, optimization and gas-lift controls, Reports NAS Azerbaijan, №2, pp.43-57, 2009.

[6] Demyanov V.F, Malozemov V.N. The introducing to the minimax. M.:Nauka, 368p., 1972.

[7] Başar T., Bernhard P. I. H ∞ - optimal control and related minimax design problems. A dynamic game approach. Birkhauser, Boston, Basel, Berlin. 1995.

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Dimension of Fractals H. Hilmi Hacısalihoğlu Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]

Abstract. MANDELBROT has been generalized the concept of fractal dimension which was given by F.HAUSDOFF in 1919. This concept essentially was given by KOLMOGOROV in 1958, as the capacity of a geometric figure. The concept of dimensional has so much different mathematical meanings. The most common one of these is well known as the topologic dimension. In regular geometry the dimension is an entire number. But for the fractals, dimension is a rational number or a negative number also. For example L. E. BROWER (1882-1926), F. HAUSDORFF (1886-1942), A. S. BERICOWICH (1891-1970) and A. N. KOLMOGOROV (1903-1987) used the second kind numbers for the fractals. This means that these dimensions were not the entire numbers.

A definition for fractal dimension was given by B. MANDELROT as

log n d = lim a→0 1 log a where d denotes the dimension and a denotes the criterion for distance and n denotes the numbers a which gives the length af the fractal curve.

This definition gives us that

d d −1 1 1 n =  and na. =  . a a

We calculated some fractal dimensions as examples.

Ex1. The dimension of Waclaw Siepinsky triangle as

log 3 d = =1,5850. log 2

Ex2. The dimension of the curve of KOCH as

log 4 d = =1,26. log 3

3 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Borwein-Ditor Theorem and My Oxford Moment Harry I. Miller International University of Sarajeevo, Bosnia and Herzegovina, [email protected]

Abstract. In 1978 Borwein and Ditor published a paper answering a real analysis question of Erdos. Since then a series of authors have made contributions to this area. I will review the work of these authors and in particular present the results from my joint paper (with A.J. Ostaszewski) that just appeared in the JMA.

4 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Bijective S-boxes and Self Dual Codes Patrick Sole University of Nice Sophia Antipolis, Nice, France, [email protected]

Abstract. We introduce a new class of rate one half binary codes: complementary information set codes. A binary linear code of length 2n and dimension n is called complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this work we investigate this new class of codes: we give optimal or best known CIS codes of length < 132. We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths ≤ 12 by the building up construction. Some nonlinear S-boxes are constructed by using Z4-codes, based on the notion of dual distance of an unrestricted code. (joint work with Carlet, Gaborit, Kim).

5 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Index of a Subfactor in a Real Factor Sh.A.Ayupov (1) and A.A.Rakhimov (2) (1) National University of Uzbekistan, Tashkent, Uzbekistan, [email protected] (2) Tashkent Institute of Automobile and Road Construction, Tashkent, Uzbekistan, [email protected]

Abstract. Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. A weakly closed *-subalgebra M containing the identity operator 1 in B(H) is called a W*- algebra. A real *-subalgebra R of B(H) is called a real W*-algebra if it is closed in the weak operator topology and R ∩ iR={0}. A real W*-algebra R is called a real factor if its center consists of the elements { λ ⋅1, λ ∈R}. Let R be a real W*-algebra, and let Q be a real W*-subalgebra of R. The set of normal faithful semi-finite operator-valued weights from R to Q is denoted by P(R,Q). We have the following results.

Theorem 1. P(R,Q) ≠ ∅ ⇔ P(R+iR,Q+iQ) ≠ ∅

Theorem 2. If R' and Q' are the commutants of R and Q, respectively, then

P(R,Q) ≠ ∅ ⇔ P(R',Q') ≠ ∅ , and there exists the canonical order-reversing bijection (denoted by Φ : E →E −1 ) between P(R,Q) and P(Q',R').

Let now R be a σ -finite real factor and let Q be a real subfactor of R. We recall that a linear positive mapping E:R →Q is called a conditional expectation if it satisfies the following conditions: E(1)=1, E(E(x)y)=E(x)E(y)=E(xE(y)) and E(x)*E(x)≤ E(x*x), for all x,y∈R. We fix a normal conditional expectation E from R onto Q. Since E is an operator-valued weight we get E −1 ∈P(Q', R'). For any unitary u∈R', we have u E −1 (1)u*= E −1 (u1u*)= E −1 (1). It is obvious that E(1)=1, but in general we have E −1 (1) ≠ 1. Since R is a real factor the element E −1 (1) is a scalar (possibly + ∞ ).

6 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Definition. The number E −1 (1) is called the index of the subfactor Q in the real factor R, and denoted by [R:Q]. By this definition it is clear that [R:Q]=[R+iR:Q+iQ]. Thus, we obtain the following real version of the index theorem.

Theorem 3. Let R be a σ -finite real factor and let Q be a real subfactor of R. Then one has either [R:Q]=4cos 2 π /q, for some integer q ≥ 3 or [R:Q] ≥ 4.

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8 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

ALGEBRA

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Transfer of Orbital Integrals and Division Algebras Ali Aydoğdu (1), Rukiye Öztürk (1), Engin Özkan (2) and Yuval Z.Flicker (3) (1) Ataturk University, Turkey, [email protected]; [email protected] (2) Erzincan University, Erzincan, Turkey, [email protected] (3) The Ohio State University, Columbus, USA, [email protected]

Abstract Let Fu be a local non-archimedean field, Gu the multiplicative group of a division algebra Du central of rank n over Fu , and Guu′ = GL(, n F ) There is an embedding of the set of conjugacy classes γ in Gu as the set of elliptic conjugacy classes γ ′ in Gu′ defined by ppγγ= ′ here pγ is the characteristic polynomial of γ and pγ ′ is that of γ ′ . In a fundamental but unpublished work [2] ( see also [1] ) of the late 1970's, Deligne and Kazhdan proved:

Theorem 1. There is a bijection from the set of equivalence classes of irreducible Gu modules

π u to the set of equivalence classes of irreducible square-integrable Gu′ -modules π u′ defined n−1 by the character relation χγ′ = −1 χγ for every regular in γ in G with image ππuu′ ()()() u

γ ′ in G′ . Here χπ ′ denotes the character of π ′ and χπ that of π u u u u u We shall briefly discuss the history of this important theorem, give a simple, global proof of it, which does not use matching of orbital integrals, and rather deduce this matching from the correspondence. ∞ Theorem 2. For every function fu in the Hecke algebra CGcu() there is a function fu′ in the Hecke algebra CG∞ ′ and for every f ′ in CG∞ ′ there is f in CG∞ whose cu() u cu() u cu() orbital integrals match Φ=Φ()()γγ,,ffuu′′ for all regular γ in Gu and γ ′ in Gu′ with

ppγγ= ′ Keywords. division algebra, automorphic representations, trace formula, L-functions AMS 2010. 11R52, 11F70, 11F72,11S40

References [1] J. Bernstein, P. Deligne, D. Kazhdan, M.- F. Vigneras, Representations des groupes reductifs sur un corps local, Hermann, Paris 1984.

[2] P. Deligne, D. Kazhdan, On representations of local division algebras, unpublished manuscript.

9 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras A. Koç (1), S. Esin (2), İ. Güloğlu (2) and M. Kanuni (3) (1) Istanbul Kultur University, Istanbul, Turkey, [email protected] (2) Dogus University, Istanbul, Turkey, [email protected], [email protected] (3) Bogazici University, Istanbul, Turkey, [email protected]

Abstract. Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with κ(A)=n.

Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices. Keywords. Finite dimensional semisimple algebra, Leavitt path algebra, Truncated trees, Line graphs. AMS 2010. 16D70, 05C60.

References

[1] Abrams, G., Pino, A., Molina, M.S., Finite-dimensional Leavitt path algebras, Journal of Pure Appl. Algebra 209, 753 – 762, 2007.

[2] Lam, T.Y., A First Course In Noncommutative Rings, Springer-Verlag 2001.

[3] Abrams, G., Pino, A., The Leavitt path algebra of a graph, Journal of Algebra 293, 2, 319- 334, 2005.

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Inequalities Involving vn( !) p Armend Sh. Shabani (1) and Valmir Krasniqi (2) (1) University of Prishtina, Prishtina, Kosovo, [email protected] (2) University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. We establish some inequalities involving vnp ( !) , the greatest power of prime p in factorization of n!. Also, we give some partial solutions and comments to the problems appeared in [2]. Keywords. Factorial function, Prime number, Inequality AMS 2010. 05A10, 11A41, 26D15, 26D20.

References

[1] M.B. Nathanson, Elementary Methods in Number Theory, Springer 2000.

[2] M. Hassani, Equations and Inequalities involving vnp ( !) , J. Ineq. Pure Appl. Math., 6(2) (2005), Article 29.

[3] A.M. Robert, A Course in p-adic Analysis, Springer, 2000.

11 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Zbërthimi në vlera singular I një matrice dhe zbatime

Alba Vrapi and Fatmir Hoxha

Abstract. The singular value decomposition of matrices stands as one of the most important concepts in mathematics, because of its variety of applications in mathematics , statistics, biology and many other areas of science. In this thesis , we present the singular value decomposition and its relation to the spectral decomposition . We also investigate the singular value decomposition of a matrix together with some of its applications.

Key words. Vector space,vector subspace,rank,base,matrix,eigen values, eigen vectors, singular value, matrix rates.

12 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Structure of Intuitionistic Fuzzy Sets in Γ−Semihyperrings Bayram Ali Ersoy (1) and Bijan Davvaz (2) (1) Yildiz Technical University, Istanbul, Turkey [email protected] (2) Yazd University, Yazd, Iran, [email protected], [email protected]

Abstract. As we know, intuitionistic fuzzy sets are extensions of the standard fuzzy sets. Now, in this paper, the basic definitions and properties of intuitionistic fuzzy Γ−hyperideals of a Γ−semihyperring are introduced. A few examples are presented. In particular, some characterizations of Artinian and Noetherian Γ−semihyperring are given.

13 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

-Vague Module

Dilek Bayrak푻푳 (1) and Sultan Yamak (2) (1) Karadeniz Technical University, Trabzon, Turkey, [email protected] (2) Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. Although the general theory of vague groups and vague rings has been investigated, the concept of TL-vague module has not been examined yet. In this study, the concept of TL-vague module is defined and some related basic properties are obtained. Keywords. -vague group, -vague ring, -vague module.

AMS 2010. 푇퐿03B52, 20N25. 푇퐿 푇퐿

References

[1] Demirci, M. ve Recasens, J., Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets Syst., 144, 441-458, 2004.

[2] Demirci, M., Vague groups, J. Math. Anal. Appl., 230, 142–156, 1999.

[3] Sezer, S., Vague groups and generalized vague subgroups on the basis of ([0,1], , ),

Information Sciences, 174, 123-142, 2005. ≤ ⋀

[4] Mordeson, J. N. ve Bhutani, K. R., Vague groups and Vague Field, J. Fuzzy Math. 15, No. 4, 927-944, 2007.

14 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

∆-Primitive Elements of Free Metabelian Lie Algebras Dilek Ersalan (1) and Zerrin Gül Esmerligil (2) (1) Cukurova University, Adana, Turkey, [email protected] (2) Cukurova University, Adana, Turkey, [email protected]

Abstract. Let L2 be a free metabelian Lie algebra of rank two. We obtain a characterization of ∆-primitive elements of L2. In particular, we prove that every edomorphism Φ of L2 is an automorphism if and only if Φ takes a ∆-primitive element of L2 to another ∆-primitive element. Also we show that in a free metabelian Lie algebra of odd rank there are no ∆-primitive elements. Keywords. Primitive element, Free Metabelian Lie algebra AMS 2010. 17B01, 17B40.

References

[1] Drensky, V.; Automorphisms of relatively free algebras, Commun. In Algebra 18 No.12, 4323-4351, 1990.

[2] Fox, R.H.; Free differential calculus I.Derivation in free group ring, Ann.of Math.2, No.57, 547-560, 1953.

[3] Mikhalev, A.A., Zolotykh, A.A.; The rank of element of a free Lie (p)superalgebra, Doklady Akad. Nauk 334(1994), No.6, 690-693; Translation in Russian Acad. Sci. Dokl. Math. 49 no.1,189-193, 1994.

[4] Mikhalev, A.A., Zolotykh, A.A.; Rank and primitivity of elements of free color Lie (p)superalgebras, Int.J. of Algebra and Comp., 4,No.4, 617-655, 1994.

[5] Mikhalev, A.A., Yu, J.T.; Primitive, almost primitive, test and Δ-primitive elements of free algebras with the Nielsen-Schreier Property, J. of Algebra, 228,603-623, 2000.

[6] Shmelkin, A.L., Wreath products of Lie algebras and their application in the theory of groups(Russian), Trudy Moskov. Math. Obshch., 29(1973) 247-260.Tanslation:Trans. Moscow Math. Soc. 29,239-252, 1973.

15 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[7] Shpilrain, V., On generators of L/R² Lie algebras, Proc. Amer. Math. Soc. 119, No.4, 1039-1043,1993.

[8] Shpilrain, V., Yu, J.T.; On generators of polynomial algebras in two commuting or non- commuting variables, J. Pure Appl. Algebra 132, 309-315, 1998.

[9] Shpilrain, V.; Generalized primitive elements of a free group, Arch. Math. 71, 270-278, 1998.

16 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Note on Direct Product of Fuzzy Modules over Fuzzy Rings Ece Yetkin Marmara University, Istanbul, Turkey, [email protected]

Abstract. In this study, a new kind of fuzzy module over a fuzzy ring is defined and then direct product of fuzzy modules over fuzzy rings is introduced. Some of their basic properties are presented analogous to ordinary module theory. Keywords. Fuzzy ring, fuzzy module AMS 2010. 08A72, 03E72, 20K25.

References

[1] Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl., 35, 512-517, 1971.

[2] Ersoy, B.A., A generalization of cartesian product of fuzzy subgroups and ideals, Pakistan Journal of Applied Sciences, 3, 100-102, 2003.

[3] Malik, D.S., Mordeson, J.N., Fuzzy Commutative Algebra, World Scientific Publishing, 1998.

[4] Malik, D.S., Mordeson, J.N, Fuzzy relations on rings and groups, Fuzzy Sets and Systems, 43, 117-123, 1991.

[5] Aktaş, H., Çağman, N., A type of fuzzy ring, Arch. Math. Logic, 46, 165-177, 2007.

[7] Yuan, X., Lee, E.S., Fuzzy group based on fuzzy binary operation, Comput. Math. App., 47, 631-641, 2004.

[8] Liu, W.J., Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8, 133- 139, 1982.

17 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Almost Prime Elements in Lattice Modules Ece Yetkin (1), Zeliha Kılıç (2) and Ünsal Tekir (3) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected] (3) Marmara University, Istanbul, Turkey, [email protected]

Abstract. In this study, we introduce the concept of almost prime elements in lattice modules over a multiplicative lattice as a new generalization of prime and weakly primary elements. Some characterizations of this type of elements are given in lattice modules. Keywords. Almost prime element, lattice module, multiplicative lattice. AMS 2010. 03G10, 06B99.

References

[1] Atani, S. E., Ferzalipour, F., On weakly primary ideals, Georgian Mathematical Journal, 12, 423-429, 2005.

[2] Bataineh, M., Kuhail, S., Generalizations of primary ideals and submodules, Int. J. Contemp. Math. Sciences, 6, 811-824, 2011.

[3] Callialp, F., Jayaram, C., Tekir, U., Weakly prime elements in multiplicative lattices, Communications in Algebra, 40, 1-16, 2012.

[4] Callialp, F., Tekir, U., Multiplication lattice modules, Iranian Journal of Science and Technology, A4, 309-313, 2011.

[5] Jabbar, A. K., Ahmed, C. A., On almost primary ideals, International Journal of Algebra, 5, 627-636, 2011.

[6] Jayaram, C., Johnson, E.W., s-prime elements in multiplicative lattices, Periodica Mathematica Hungarica, 31, 201-208, 1995.

[7] Jayaram, C., Johnson, E.W., Strong compact elements in multiplicative lattices, Czechoslovak Mathematical Journal, 47, 105-112, 1997.

[8] Jayaram, C., Laskerian lattices, Czechoslovak Mathematical Journal, 53, 351-363, 2003.

18 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Multiplication Lattice Modules

Fethi Çallıalp (1) and Ünsal Tekir (2) (1) Dogus University, Istanbul,Turkey, [email protected] (2) Marmara University, Istanbul,Turkey,[email protected]

Abstract. This study concerns with the further investigation of multiplicaiton lattice modules especially faithful and principally generated multiplication lattice modules over a multiplicaitive lattice Multiplication lattice modules have been studied in [5]. In this work we continue to the characterization of multiplication lattice modules, in particular, the proofs of Cohen theorem and Nakayama lemma are given in multiplication lattice modules. Afterwards, a small element is defined in a lattice module and a new type of lattice module is characterized called hollow lattice module by means of small elements. Several results concerning with these elements and hollow lattice modules are considered.

Keywords. Multiplicative lattices, Lattices modules, prime elements.

AMS 2000. Primary 16F10; Secondary 16F05.

References

[1] C.Jayaram and E.W.Johnson. (1995) : Some results on almost principal element lattices, Period.Math.Hungar, 31; 33 - 42:

[2] C.Jayaram and E.W.Johnson.(1997): Dedekind lattices, Acta. Sci. Math.(Szeged); 63; 367-378:

[3] C.Jayaram and E.W.Johnson, Strong compact elements in multiplicative lattices. (1997) : Czechoslovak Math. J, 47 (122) ; 105 – 112:

[4] D. Scott Culhan. (2005) : Associated Primes and Primal Decomposition in modules and Lattice modules, and their duals, Thesis, University of California Riverside.

[5] Fethi Çallıalp and Ünsal Tekir. (2011):Multiplication lattice modules, Iranian Journal of Science & Technology,no.4, 309 - 313:

[6] Z.A.El-Bast and P.F.Smith. (1985) : Multiplication modules, Comm. Algebra 16; 755 - 799:

19 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Comultiplication Lattice Modules

Fethi Çallıalp (1) and Ünsal Tekir (2) (1) Dogus University, Istanbul,Turkey, [email protected] (2) Marmara University, Istanbul,Turkey,[email protected]

Abstract. Let L be a multiplicative lattice and M be a lattice L-module. M is said to be a comultiplication L-module if for every element N of M there exists an element such that = (0 : ). This study is devoted to investigate some properties of comultiplicaiton푎 ∈ 퐿 lattice푁 modules.푀 푀 In푎 this work, K-lattice is defined and a sufficient condition is given to say that every element of a comultiplication K-lattice module has a primary decomposition. Furthermore, we introduce the definition of a second element in comultiplication lattice module and the relation between prime element and second element is given. Afterwards, coprime element is defined and some related results are proved. Additionally, large and small elements are defined and various properties concerning the relation between small elements and large elements are obtained.

AMS 2000. Primary 16F10; Secondary 16F05.

Keywords . Multiplicative lattices, Lattices modules, comultiplication lattice modules, coprime elements.

References.

[1] C.Jayaram. (2003) : Laskerian lattices, CzechoslovakMath. J, 53 (128) ; 351- 363: [2] D. Scott Culhan. (2005) : Associated Primes and Primal Decomposition in modules and Lattice modules,and their duals, Thesis, University of California Riverside. [3] Eaman A.Al-Khouja. (2003) : Maximal Elements and Prime elements in Lattice Modules, Damascus University for Basic Sciences 19, 9 -20: [4] H. Ansari-Toroghy, F. Farshadifar, On Multiplication and comultiplication modules,Acta Mathematica Scientia 2011,31B(2),694 - 700: [5] H. Ansari-Toroghy, F. Farshadifar, On comultiplication modules, Korean Ann. Math. 25 (2008), no. 1-2, 57 - 66: [6] F.Callialp, U.Tekir, Multiplication lattice modules, Iranian Journal of Science & Technology (2011);no.4, 309 - 313:

20 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Closed Submodules of a Finite Rank Free Module over a Complete DVR Hanni Garminia (1), Gantina Rahmaputri (2) and Pudji Astuti (3) (1) Institut Teknologi Bandung, Bandung, Indonesia, [email protected], (2) Institut Teknologi Bandung, Bandung, Indonesia, [email protected] (3) Institut Teknologi Bandung, Bandung, Indonesia, [email protected]

Abstract. Fuhrmann [1] introduced the complete notion of a subspace of formal series in term of an algebraic condition connected to certain projection. In this paper we show that this algebraic condition can be adapted to obtain a characterization of closed submodules of a finite rank free module over a complete discrete valuation domain (DVR). Keywords. Complete discrete valuation domain, closed submodule, exact sequence, p-adic topology, projection.

References

[1] P.A. Fuhrmann, A Study of Behaviors, Linear Algebra and Its Appl, 351-352, 303-380, 2002.

21 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Commutative and Normal Groups on Soft Sets İrfan Şimşek (1), Naim Çağman (2), Kenan Kaygısız (3) and İrfan Deli (4) (1) Gaziosmanpasa University, Tokat, Turkey, [email protected] (2) Gaziosmanpasa University, Tokat, Turkey, [email protected] (3) Gaziosmanpasa University, Tokat, Turkey, [email protected] (4) University of 7 Aralik, Kilis, Turkey, [email protected]

Abstract. Soft set theory introduced by Molodtsov in 1999 in order to deal with uncertainties. In this work, we define commutative and normalizer groups on the soft sets. We then investigate their related properties by inspiring the group theory. Keywords. Soft sets, normalizer, soft groups, soft int-groups, commutative soft int- groups, normal soft int-group. AMS 2010. 20D15, 20D35, 20F18

References

[1] Ali, M.I., Feng, F., Liu, X., Min W.K., and Shabir, M., On some new operations in soft set theory, Computers and Mathematics with Applications 57, 1547-1553, 2009.

[2] Çağman N. and Enginoğlu, S., Soft set theory and uni-int decision making, European Journal of Operational Research 207, 848-855, 2010.

[3] Molodtsov, D.A., Soft set theory-first results, Computers and Mathematics with Applications 37, 19-31, 1999.

[4] Akgül, M., Some properties of soft groups, J. Math. Anal. Appl. 133, 93-100. 29, 1988.

[5] Aktaş, H. and Çağman, N., Soft sets and soft groups, Inform. Sci. 177, 2726-2735 2007.

[6] Kaygısız, K., On soft int-groups, Ann. Fuzzy Math. Inform., in press.

[7] Kaygısız, K., Normal soft int-groups, submitted.

22 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On topological Structure on Ternary Semihyperrings Kostaq Hila (1) and Armend Sh. Shabani (2) (1) University of Gjirokastra, Gjirokastra, Albania, [email protected] (2) University of Prishtina, Prishtinë, Kosova, [email protected]

Abstract. This paper deals with the topological structure of a class of algebraic hyperstructures called ternary semihyperrings, which are a generalization of semirings and ternary semirings. In this paper we introduce a class of hyperideals in ternary semihyperrings called prime k-hyperideal. Considering and investigating properties of the collection A of all proper prime k-hyperideals of a ternary semihyperring H, we construct a topology τ A on A by means of closure operator defined in terms of intersection and inclusion relation among these hyperideals of ternary semihyperring H. The topological space (A, τ A ) is called the structure space of the ternary semihyperring H. We study a several principal topological axioms and properties in this structure space of ternary semihyperring such as separation axioms, compactness etc. Keywords. Ternary semihyperring; Prime k-hyperideal; k-Noetherian ternary semihyperring; Hull-Kernel topology; Structure space. AMS 2010. 16Y99, 20N20, 17A30.

References

[1] R. Ameri and H. Hedayati, On k-hyperideals of semihyperrings, J. Discrete Math. Sci. Cryptogr.10 (2007), 41-54.

[2] M. R. Adhikari, M. K. Das, Structure spaces of semirings, Bull. Cal. Math. Soc. 86 (1994), 313-317.

[3] A.P. Pojidaev, Enveloping algebras of Fillipov algebras, Comm. Algebra 31 (2003) 883- 900.

[4] A. Cayley, Comb. Math. J. 4(1) (1845).

[5] P. Bonansinga and P. Corsini, On semihypergroup and hypergroup homomorphisms, Boll. Un. Mat. Ital. B (6) 1(2) (1982), 717-727.

[6] G. L. Booth and N. J. Groendwald, On uniformly strongly prime Γ - rings, Bull. Austral.Math. Soc. Vol 37 (1988), 437-445.

[7] S. Chattopadhyay and S. Kar, On structure space of Γ - semigroups, Acta Univ. Palacki.Olomuc., Fac. rer. nat., Mathematica 47 (2008), 37-46.

[8] P. Corsini, Prolegomena of hypergroup theory, Second edition, Aviani editor, 1993.

23 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

N-dimensional (α, β)-fuzzy ideals of hemirings Muhammad Aslam (1) and Saleem Abdullah (2) (1) Quaid-I-Azam University, Islamabad, Pakistan, [email protected] (2) Quaid-I-Azam University, Islamabad, Pakistan, [email protected]

Abstract. In this paper, we introduce the notion of N-dimensional (α, β)-fuzzy ideals in hemirings by using N-dimensional fuzzy point and N-dimensional fuzzy set,. Moreover we introduced N-dimensional (α, β)-fuzzy h-ideals in hemirings. We give some characterization Theorems of N-dimensional (α, β)-fuzzy h-ideals in hemirings. N-dimensional (α, β)-fuzzy ideals in hemirings is most generalization of N-dimensional fuzzy ideals in hemirings and (α, β)-fuzzy ideals in hemirings. We also define N-dimensional prime (α, β)-fuzzy ideals in hemirings. Keywords. Hemiring. N-dimensional fuzzy set, N-dimensional (α, β)-fuzzy h-ideals

24 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

' Test Rank of the Lie Algebra F/, RF

Nazar Şahin Öğüşlü (1) and Naime Ekici (2) (1) Cukurova University, Adana, Turkey, [email protected] (2) Cukurova University, Adana, Turkey, [email protected]

Abstract. Let F be a free Lie algebra of rank n ≥ 2 and R be a fully invariant ideal ' of F . We show that the test rank of the Lie algebra F/, RF is equal to 1 when n is even and less than or equal to 2 when n is odd. Keywords. Free Lie algebras, Test sets, Test rank. AMS 2010. 17B01, 17B40.

References

[1] Esmerligil, Z., Ekici, N., Test sets and test rank of a free metabelian Liealgebra. Comm. Algebra, 31, 11, 5581-5589, 2003.

[2] Esmerligil, Z., Kahyalar, D., Ekici, E., Test rank of F/R’ Lie algebras. Internat. J. Algebra Comput., 16, 4, 817-825, 2006.

[3] Fox, R. H., Free differential calculus. I. Derivations in free group rings. Ann. Math. 2, 57, 547-560, 1995.

[4] Mikhalev, A. A., Yu, J. T., Primitive, almost primitive, test and Δ-primitive elements of free algebras with the Nielsen-Schreier property, J. Algebra, 228, 603-623, 2000.

[5] Mikhalev, A. A., Umirbaev, U. U., Yu, J. T., Generic, almost primitive and test elements of free Lie algebras, Proc. Amer. Math. Soc., 130, 5, 1303-1310, 2001.

[6] Shpilrain, V., On generators of L/ R2 Lie algebras, Proc. Amer. Math. Soc., 119, 4, 1039- 1043, 1993.

[7] Shpilrain, V., On the rank of an element of a free Lie algebra, Proc. Amer. Math. Soc., 123, 5, 1303-1307, 1995.

25 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[8] Temizyürek, A., Ekici, E., A particular test element of a free solvable Lie algebra of rank two, Rocky Mountain J. Math., 37, 4, 1315-1326, 2007.

[9] Timoshenko, E. I., Test sets in free metabelian Lie algebras, Siberian Math. J., 43, 6, 1135-1140, 2002.

[10] Timoshenko, E. I., Shevelin, M. A., Computing the test rank of a free solvable Lie algebra, Siberian Math. J., 49, 6, 1131-1135, 2008.

[11] Umirbaev, U. U., Approximation of free Lie algebras relative to inclusion, Monoids, Rings and Algebras, Uchen. Zap. Tartu Univ., 878, 142-145, 1990.

[12] Umirbaev, U. U., Partial derivatives and endomorphisms of some relatively free Lie algebras, Sibirsk. Mat. Zh., 42, 3, 720-723, 2001.

[13] Yagzhev, A. V., Endomorphisms of free algebras, Sibirsk. Math. Zh.21, 181-192; English Transl. Siberian Math. J., 21, 133-141, 1980.

26 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Pullback for Simplicial Lie algebras Özgün Gürmen Alansal Dumlupinar University, Kutahya, Turkey, [email protected]

Abstract. Pullback simplicial object had been defined by Glenn [4]. In this work we consider this object for lie algebra case. Keywords. Simplicial algebra, Pullback object, Crossed module. AMS 2010. 13G05, 18G50, 20J05.

References

[1] Brown, R., Topology, Ellis Horwood Series in Mathematics and its applications Ellis Horwood, Ltd. p. 460. 1988.

[2] Brown, R., Higgins, P., On the Connection between the Second Relative Homotopy Groups of some Related Spaces, Proc. LondonMath. Soc.,3, 36, 193-212. 1978.

[3] Curtis, E. B.,Simplicial Homotopy Theory, Adv in Math 6, 107-209 1971.

[4] Glenn P.G., Realization of Cohomology Classes in arbitrary exact categories, Journal of pure and applied Algebra 25, 33-105, 1982.

27 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Induced Inner Product on a Quotient Tensor Product Space Pudji Astuti Institut Teknologi Bandung, Bandung, Indonesia, [email protected]

Abstract. The concept of generalized n-inner product was introduced by Trenčevsky and Malčeski [6] as a generalization of an n-inner product by Misiak [3]. Following this intro-duction, a number of results concerning the structure and properties of generalized n- inner product spaces can be found in literature (see [1], [2] and [4]). A standard generalized n-inner product is a generalized n-inner product induced by an inner product [6]. In this presentation we show that one can develop an inner product on a quotient tensor product space [5] induced by any standard generalized n-inner product. As a result, the study of any alternating multilinear functional on a standard generalized n-inner product space can be lifted to the study of the corresponding linear functional on the induced inner product space. Keywords. Inner products, generalized n-inner products, tensor products. AMS 2010. 46C05, 15A69.

References

[1] R. Chugh and S. Lather, On Generalized n-inner product spaces, Navi Sad J. Math., 41(2011) 73-80.

[2] R. Chugh and Sushma, Some results in Generalized n-inner product spaces, Internat. Math. Forum, 4(2009) 1013-1020.

[3] A. Misiak, n-inner product spaces, Math. Nachr, 140 (1989) 200-319.

[4] B.S. Reddy, Some Properties in Generalized n-inner product spaces, Int. Journal of Math. Analysis, 4(2010) 2229-2234.

[5] S. Roman, Advanced Linear Algebra, third ed., Springer, 2008.

[6] K. Trenčevsky and R. Malčeski, On generalized n-inner product and the corres-ponding Cauchy-Schwarz inequality, J. Inequal. Pure and Appl. Math., 7 Art. 57 (2006).

28 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Normality and Quadraticity for Special Ample Line Bundles on Toric Varieties Arising from Root Systems Qëndrim R. Gashi (1) and Travis Schedler (2) (1) University of Prishtina, Prishtina, Kosovo, [email protected] (2) MIT, Cambridge (MA), USA, [email protected]

Abstract. We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semigroup algebras are quadratic AMS 2010. 00B05, 14M25, 05E15

29 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Square Terms in Generalized Fibonacci Sequence

Refik Keskin (1) and Zafer Şiar (2) (1) Sakarya University, Sakarya, TURKEY, [email protected] (2) Bilecik Seyh Edebali University, Bilecik, TURKEY, [email protected]

Abstract. Let P and Q be nonzero integers. Generalized Fibonacci sequence {}U n and

Lucas sequence {}Vn are defined by U 0 = ,0 U1 = 1 and U n+1 PU n += QU n−1 for n ≥ 1 and

V0 = ,2 1 = PV and Vn+1 PVn += QVn−1 for n ≥ 1, respectively. In this paper, we assume that 2 2 2 2 Q = 1. We solve the equations = mn xUU , n = 2 m xUU , n = 3 m xUU and n = 6 m xUU 2 when P is even and ≠ 2mn . Moreover, we solve the equations n = 3 m xUU and 2 n = 6 m xUU when P is odd.

Keywords. Generalized Fibonacci numbers; Generalized Lucas numbers; Congruences. AMS 2010. 11B37, 11B39.

References

[1] R. Keskin and Z. Yosma, Some new identities concerning generalized Fibonacci and Lucas numbers (Accepted for publication).

[2] M. Mignotte and A. Pethö, Sur les carres dans certanies suites de Lucas, Journal de Theorie des nombers de Bordeaux, 5 no. 2 (1993), 333--341.

[3] K. Nakamula and A. Pethö, Squares in binary recurrence sequences, Number Theory: Diophantine, computational and algebraic aspects; proceedings of the international conference, de Gruyter, 1998, 409--421.

[4] P. Ribenboim and W. L. McDaniel, On Lucas sequence terms of the form kx², Number Theory: proceedings of the Turku symposium on Number Theory in memory of Kustaa Inkeri (Turku, 1999), de Gruyter, Berlin, 2001, 293--303.

30 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Generalized Fuzzy Bi-hyperideals in Semihypergroups Saleem Abdullah (1) and Kostaq Hila (2) (1) Quaid-i-Azam University, Islamabad, Pakistan, [email protected] (2) Gijrokastera University, Gijrokastera, Albania, [email protected]

Abstract. In this paper we introduced interval valued ()αβ, -fuzzy bi-hyperideal in semihypergroups which is a generalization of fuzzy hyperideals and ()αβ, -fuzzy bi- hyperideal in semihypergroups, where αβ,∈{ ∈ ,,qqq ∈∨ , ∈∧ } with α ≠∈∧q . So we can easily construct twelve different types of interval valued fuzzy hyperideals in semihypergroups. Using the idea of a quasi-coincidence of an interval valued fuzzy point with a interval valued fuzzy set, the concept of an ()αβ, -fuzzy bi-hyperideal in semihypergroups is introduced and some interesting characterizations theorems are obtained. A special attention in given to ()∈∈∨, q -fuzzy bi-hyperideals. We prove that for a semihypergroups S . Let A be an interval valued fuzzy subset of S . Then, A is an interval valued (,∈∈∨q ) -fuzzy bi-hyperideal of if and only if A is a bi-hyperideal t  of S for all tD ∈ (0,1] , where A:= { x ∈ S | x ∈∨ qA }. t t Keywords. Fuzzy algebra; semihypergroups; quasi-coincidence; (,αβ )-fuzzy bi- hyperideal, (,∈∈∨q )-fuzzy bi-hyperideals. AMS 2010. 53A40, 20M15.

References [1] L. A. Zadeh, Fuzzy Sets, Inform. and Control, 8 (1965) 338-353.

[2] B. Davvaz, Fuzzy hyperideals in semihypergroups, Italian J. Pure and Appl. Math. no. 8, (2000), 67-74.

[3] B. Davvaz, Strong Regularity and Fuzzy Strong Regularity in Semihypergroups., Korean J.Comput. & Appl. Math. Vol. 7 ,No 1, (2000), 205-213.

[4] Tosun, M., Kuruoğlu, N., On B. Davvaz: ( , q)-fuzzy subnearrings and ideals, Soft Comput, 10:(2006),206-211. ∈ ∈∨ [5] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer- Verlag, New-York, 1979.

31 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Relations on Extending Modules Semra Doğruöz Adnan Menderes University, Aydin, Turkey, [email protected]

Abstract. In the development of extending module theory, behaviour of closed and complement submodules take an important place. In this work all recent generalizations of extending modules has been investigated through closed submodules and give some relations among them for future researchers in this field. Keywords. Closed submodule, essential submodule, extending module, module classes, torsion theory. AMS 2010. 16S90,16D80,13D30.

References

[1] Doğruöz S., Extending modules relative to module classes, PhD. Thesis, Glasgow University, 1997.

[2] Doğruöz S., Classes of extending modules associated with a torsion theory, East-West Journal of Math. Vol. 8 Number 2, (2006), 163-180.

[3] Doğruöz S., Extending modules relative to a torsion theory, Czechoslovak Mathematical Journal, 58 (133), (2008), 381-393.

[4] Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R., Extending Modules, Pitman Research Notes in Mathematics series, (Longman, Harlow), 1994.

[5] Harmanci, A. and Smith, P.F., Relative injectivity and module classes, Comm. in Algebra, 20 (9) (1992), 2471-2511.

[6] Mohamed, S. and Müller, B.J, Continuous and Discrete Modules, London Mathematical Soc. Lecture Note series 147, Cambridge University Press, 1990.

[7] Doğruöz S. and Ürün Ö., On extending modules:A survey, Surveys in Mathematics and Mathematical Sciences, FJMS, to appear 2012.

32 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Generalized (,)fg Derivations on Lattices Şahin Ceran (1) and Mustafa Aşcı (2) (1) Pamukkale University, Denizli, Turkey, [email protected] (2) Pamukkale University, Denizli, Turkey, [email protected]

Abstract. In this paper as a generalization of f-derivation and (f,g)-derivation of a lattice on a lattice we introduce the notion of generalized (f,g)-derivation of a lattice. We give interesting results about this derivation. Keywords. Lattice, Derivation, (f,g)-derivation AMS 2010. 06B35, 06B99, 16B70

References

[1] Asci, M.; Ceran, Ş. Generalized symmetric bi-(σ,τ)-derivations on prime near rings. Algebras Groups Geom. 24 (2007), no. 3, 291--301.

[2] Ceran, Ş.; Aşci, M. Symmetric bi-(σ,τ) derivations of prime and semi prime gamma rings. Bull. Korean Math. Soc. 43 (2006), no. 1, 9--16.

[3] Ceran, Ş., Aşci, M. "On traces of Symmetrıc Bi- -Derivations on Prime Rings" Algebras, Groups and Geometries 26, (2009), no:2, 203-214.

[4] Çeven, Y. Öztürk, M. A. On f-derivations of lattices. Bull. Korean Math. Soc. 45 (2008), no. 4, 701--707.

[5] Çeven, Y. Symmetric bi derivations of Lattices, Quaestiones Mathematicae, 32(2009), 1- 5.

[6] G. Birkhoof, Lattice Theory, American Mathematical Society, New York, 1940.

[7] X. L. Xin, T. Y. Li, and J. H. Lu, On derivations of lattices, Inform. Sci. 178 (2008), no. 2, 307-316.

[8] Y. B. Jun and X. L. Xin, On derivations of BCI-algebras, Inform. Sci. 159 (2004), no. 3-4, 167-176.

33 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Algebraic Hyperstructures of Soft Sets Associated to n-ary Polygroups Şerife Yılmaz (1) and Osman Kazancı (2) (1) Karadeniz Technical University, Trabzon, Turkey, [email protected] (2) Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. This paper concerns a relationship between soft sets and n-ary polygroups. First we consider the notion of an n-ary polygroup as a generalization of a polygroup. Then we introduce the notion of a soft n-ary subpolygroup of an n-ary poygroup which is an extended notion of an n-ary subpolygroup in an n-ary polygroup. We gave some examples and investigate several related properties. Keywords. n-ary polygroup, soft set, soft n-ary subpolygroup. AMS 2010. 20N25, 20N20, 20N15.

References

[1] Wang, J., Yin, M., Gu, W., Soft Polygroups, Computers and Mathematics with Applications 62, 3529–3537, 2011.

[2] Davvaz, B., Vougiouklis, T., N-ary hypergroups, Iranian Journal of Science and Technology Transection A 30, 165-174, 2006.

[3] Yamak, S., Kazancı, O., Davvaz, B., Soft hyperstructure, Computers and Mathematics with Applications 62, 797-803, 2011.

34 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Irreducible, Prime and Maximal Idealistics Soft WS-algebra Ummahan Acar (1) and Abdullah Harmancı (2) (1) Mugla Sitki Kocman University, Mugla, Turkey, [email protected] (2) Hacettepe University, Ankara, Turkey, [email protected]

Abstract. In this work, the notions of irreducible idealistic soft WS-algebra , maximal idealistic soft WS-algebra and prime idealistic soft WS-algebra over WS-algebra X are introduced and several examples are given to illustrate. Relations between prime idealistic, irreducible idealistic and maximal idealistic soft WS-algebras are investigated. Keywords. idealistic soft WS-algebra, Prime idealistic soft WS-algebra.

References

[1] H. Aktas, N. Çağman, Soft sets and soft Groups , Information Science, 177 (2007), 2726- 2735.

[2] Y. B. Jun, Soft BCK/BCI-algebra, Comput. Math. App. 56 (2008), 1408-1413.

[3] Y. B. Jun, C.H. Park, Application of Soft sets in ideal theory of BCK-algebra, Information Sciences, 178 (2008), 2466-2475.

[4] Y. B. Jun, H.S. Kim, E.H.Roh, Ideal theory of subtraction algebras, Sci. math. Jpn. online-e, (2004)397-402

[5] Y. B. Jun, K. H. Kim, Prime and irreducible ideals in Subtraction algebras, International Math. Forum, 3(2008)10, 457-462

[6] Y.H. Kim, C.H.S. Kim, Subtraction algebras and BCK-Algebras, Math. Bohemica,128(2003)1,21-24

[7] J. Meng, Y.B. Jun, BCK-algebras , Kyungmoon Sa Co., Soul, 1994.

[8] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. App. 45 (2003), 555- 562.

[9] D.Molodtsov, Soft set theory-first result, Comput. Math. App. 37 (1999),19-31.

[10] C.H. Park, Y.B. Jun and M.A. Öztürk, Soft WS-Algberas, Commun. Korean Math. Soc. 23 (2008), No. 3, pp. 313-324.

35 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Regular Elements, Idempotents and Right Units of Complete Semigroups of Binary Relations Defined by Semilattices of The Class Lower Incomplete Nets Yasha Diasamidze (1), Ali Erdoğan (2) and Neşet Aydın (3) (1) Sh. Rustaveli State University, Batumi, Georgia, [email protected] (2) Hacettepe University, Ankara, Turkey, [email protected] (3) Canakkale Onsekiz Mart University, Canakkale, Turkey, [email protected]

Abstract. Let X be an arbitrary nonempty set, D be an X − semilattice of unions, i.e. such a nonempty set of subsets of the set X that is closed with respect to the set − theoretic operations of unification of elements from D , f be an arbitrary mapping of the set X in the semilattice D . To each such a mapping f we put into correspondence a binary relation α f on the set X that satisfies the condition α =(){}()x × fx . The set of all such α ()fX: → D is f  f xX∈ denoted by BDX () . It is easy to prove that BDX () is a semigroup with respect to the operation of multiplication of binary relations, which is called a complete semigroup of binary relations defined by an X − semilattice of unions D .

Definition Let Nmm = {}0,1,2,..., ()m ≥ 1 be some subset of the set of all natural numbers.

A subsemilattice Q=⊆∈{ Tij Xi| Ns , j ∈ N k } \{} T00 of the complete X − semilattice of unions D is called lower incomplete net which contains two T sk subsets QT1= {} 10,..., Ts 0 , QT2= {} 01,..., T 0k and

T − sk1 Tsk −1 satisfies the following conditions:     T3k  Tsk−−11  T a) TT⊂⊂... T and TT⊂ ⊂⊂... T;   s3 10 20s 0 01 02 0k   T   T T2k 31k − s−13 T   s2 b) QQ12∩=∅;   T1k T21k −  Ts−12 T  T s1  33  c TT≠ pq,,≠ i j T   ) pq ij if ()() ; T0k 11k −   Ts−11 T  T T  s0  23 32  T   T d) the elements of the sets Q1 and Q2 are pairwise 01k −  T T T  s−10  13 22 31    T T T noncomparable; 03 12 21 T30

T T11 T20 ′ ′ 02 e) TTij∪= i′′ j T pq , if p= max{} ii , and q= max{} jj , . T 01 T10 Fig.1 Note that the diagram of the given X − semilattice of unions Q is shown in Fig. 1.

In this paper, we investigate such a regular elements α and idempotents of the complete semigroup of binary relations BDX () defined by semilattices of the class lower incomplete

36 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

nets, for which VD(),α = Q. Also we investigate right units of the semigroup BQX () . For the case where X is a finite set we derive formulas be means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup.. Keywords. Semigroups, binary relation, regular element, idempotents, right units. AMS 2010. Primary 20M30, 20M10, Secondary 20M15.

References

[1] Ya. I. Diasamidze, Complete semigroups of binary relations. Journal of Mathematical Sciences,Plenum Publ. Corp., New York 117(2003), No. 4, 4271–4319.

[2] Diasamidze Ya., Makharadze Sh., Complete semigroups of binary relations. Sputnik +, Moscow, 2010, 1- 657 (in Russian).

[3] Diasamidze Ya., Makharadze Sh., Complete Semigroups of Binary Relations Defined by X − Semilattices of Unions. Journal of Mathematical Sciences, Plenum Publ. Cor., New York, Vol. 166, No. 5, 2010, 615-633.

37 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Parafree Lie Algebras with Certain Properties Zehra Velioğlu (1) and Naime Ekici (2) (1) Cukurova University, Adana, Turkey, [email protected] (2) Cukurova University, Adana, Turkey, [email protected]

Abstract. We prove that the union of the free Lie algebras of rank two is parafree and we employ this result to construct some parafree Lie algebras with certain properties. Keywords. Parafree Lie algebras, Free Lie algebras, Directed system. AMS 2010. 17B99, 17B01.

References

[1] Baumslag, G., Groups with the same lower central sequence as a relatively free group I. The groups, Trans. Amer. Math. Soc., 129, 308-321, 1967.

[2] Baumslag, G., Groups with the same lower central sequence as a relatively free group. II Properties, Trans. Amer. Math. Soc., 142, 507-538, 1969.

[3] Baumslag, G., Parafree groups, Progress in Math., Vol. 248, 1-14, 2005.

[4] Baumslag, G., And Stammbach, U., On the inverse limit of free nilpotent groups, Comment. Math. Helvetici, 52, 219-233, 1977.

[5] Baumslag, G., and Cleary, S., Parafree one-relator groups, J. Of groups theory, 2005.

[6] Baumslag, G., and Cleary, S. And Havas, G., Experimenting with infinite group, Experimental Math., 13, 495-502, 2004.

[7] Baur, H., Parafreie Liealgebren und homologie, Diss. Eth Nr., 6126, 1978.

[8] Baur, H., A note on parafree Lie algebras, Commun. in Alg., Vol.8, 10, 953-960,1980.

[9] Knus, M.A. and Stammbach, U., Anwendungen der homologietheorie der Liealgebren auf zentralreihren unda uf prasentierungen, Comment. Math. Helvetici, 42, 297-306, 1967.

[10] Morandi, J., Patrick, Mathematical Notes, sierra.nmsu.edu/morandi/, 2012.

38 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

ANALYSIS

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Some New Paranormed Sequence Spaces of Non-Absolute Type Arife Aysun Karaaslan (1) and Vatan Karakaya (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. For a sequence = xx k )( , we denote the difference sequence by

∞ ∞ −=∆ xxx kk −1)( . Let = uu )( nn =0 and = vv )( kk =0 be the sequences of real numbers such that

un ≠ ,0 vk ≠ 0 for all k Ν∈ . The sequence spaces λ∆ { k ∈∈== λ pxVwxxpvu )()(:)(),,( } where

∞ λ ∈{}∞ ,, ccl 0 and the matrix = vV )( knnk =0, is obtained by multiplying weighted means matrix and difference matrix ]1[ . In this study, we introduce some new generalized sequence spaces related to the spaces ∞ ( ), pcpl )( and 0 pc )( . Furthermore, we investigate some topological properties, and we also give some inclusion relations among these spaces. In addition, we compute the , βα −− and γ − duals of these spaces, and characterize certain matrix transformations on these sequence spaces. Keywords. Paranormed sequence space, , βα −− and γ − duals, matrix transformation.

References [1] Mursaleen, M., Karakaya, V., Polat, H., Şimşek, N., Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Computers and Mathematics with Applications, Vol. 62, Issue 2, pp. 814-820, 2011.

[2] Karakaya V, Noman A.K., Polat H., On paranormed λ -sequence spaces of non-absolute type, Mathematical and Computer Modelling, Vol. 54, Issue 5, pp. 1473-1480, (2011)

[3] K.-G. Gross-Erdman, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl. 180(1993)223-238

39 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On saturation Order of Functions in the Space L p ππ p ∞<<− )1(),( for Holder Method of Fourier Series Summation A. J. Aliyeva (1) and J. M. Aliyev (2) (1) Azerbaijan Technical University, Baku, Azerbaijan (2) Institute of Applied Mathematics, BSU, Baku, Azerbaijan, [email protected]

Abstract. In the paper, we prove theorems on saturation order in the space

L p − ππ ),( for Holder method of Fourier series summation. Applying the Holder method to the Fourier series, we get

n r a0 r n xH )( += ∑ nk (ah k coskx + bk sin kx n = ,...)2,1,0() , 2 k=1

r r where the multipliers hnk are determined by the formulae, hn0 = ,1

−kn − )1( p C p r = k −1 −kn = nk nCh n−1 ∑ r , nk ),...,2,1( . p=0 kpp +++ )1)(1( Theorem. Holder’s summation method of order r ≥ 0 is saturated with approximation of saturation order

 lg nr−1  r − =   n xfxH )()( O  . (1) Lp − ππ ),(  n  since the functions satisfying (1) should be sought in the class of functions satisfying the

 lgnr−1  =   condition n )( OfE  , where n fE )( Lp is the best approximation.  n  By proving we are limited with the case r ≥ 1, though the theorem is true for any r > 0 Corollary. This statement is more general than A.Kh. Turetskiys theorem [1]. Similar results have been also obtained in [2]. Keywords. Saturation, Summation. AMS 2010. 26A15.

References [1] Turetsky, A.Kh. On saturation classes in space C . Transaction of AN SSSR, ser. Matem., 25(1961). [2]. Aliyeva, A.J., Aliyev, J.M. On saturation classes of Fourier series. Proceedings of Az.TU, Fundamental Sciences, N1, 2012.

40 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the Fine spectrum of Some Generalized Difference Operators Ali M. Akhmedov (1) and Saad R. El-Shabrawy (1) (1) Baku State University, Baku, Azerbaijan, [email protected]

Abstract. The spectrum and fine spectrum of the difference operator and generalized difference operators over sequence spaces have been examined by several authors. This work is divided into two parts. In the first part, we mainly review several recent results concerning the fine spectrum of the difference operator ( , ), the generalized difference operator ( ) , and the operator , over some sequence퐵 푟spaces.푠 In the second part, we discuss the 퐵operator푟 푠 , with some new∆푎 푏 conditions and then determine the fine spectrum of the operator , over∆ the푎 푏 sequence spaces and . Also, some examples are given that may be helpful to ∆provide푎 푏 the support for the obtained푐0 results.푐 Keywords. Spectrum of an operator, Generalized di¤erence operator, Sequence spaces. AMS 2010. 47A10, 47B37.

References

[1] Akhmedov A.M., El-Shabrawy S.R. On the fine spectrum of the operator , over the sequence space . Comput. Math. Appl. 61, 2994-3002, 2011. ∆푎 푏 푐 [2] Altay B., Basar F. On the fine spectrum of the difference operator , on and .

Inform. Sci. 168, 217-224, 2004. ∆푎 푏 푐0 푐

[3] Furkan H., Bilgiç H., Altay B. On the .ne spectrum of the operator ( , ) over and .

Comput. Math. Appl. 53, 989-998, 2007. 퐵 푟 푠 푐0 푐

41 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Fixed Point Theorems in p − Summable Symmetric n − Cone Normed Sequence Spaces Ahmet Şahiner (1) and Tuba Yiğit(2) (1) Suleyman Demirel University, Isparta, Turkey, [email protected] (2) Suleyman Demirel University, Isparta, Turkey, [email protected]

Abstract. In this study fixed point theorems and related concepts in p − summable symmetric n − cone normed sequence spaces are investigated. Keywords. Symmetric n-normed space, Symmetric n-cone Banach space, Fixed point theorems. AMS 2010. 46A45, 47H10.

References − c [1] A. Sahiner, Fixed point theorems in symmetric 2 cone Banach space ( l p .,., p ), Journal of Nonlinear Analysis and Optimization: Theory and Applications, 2012 (in press).

[2] H. Gunawan, The space of p -summable sequences and its natural n -norm, Bull. Aust. Math. Soc., 64, 1, 137-147, 2001.

[3] H. Gunawan, Mashadi, On finite-dimensinal 2 -normed spaces, Soochow J. Math. 27, 321-329, 2001.

[4] S. Gahler, 2 -Metrische raume und ihre topologische struktur, Math. Nachr., 26, 115-148, 1963.

[5] Z. Lewandowska, Generalized 2 -normed spaces, Slupskie Prace Mathematyczno- Fizyczne 1, 33-40, 2001.

42 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Matrix Inequality Hardy’s Type Bilal Sherali Informatics and Mechanics MES RK, Almaty, Kasakhstan, [email protected]

∞ ∞ ∞ ∞ Abstract. Let = {}ww i i=1, = {}vv i i=1, = {}uu i i=1 and = {}rr i i=1 are consequence of ∞ nonnegative numbers. Let = {}ff i i=1 are arbitrary consequence of real numbers. In this paper the correctness is established of the following inequalities:

≤+ (1) sup uk() Afkk Csup vii f sup wk() Rf , 1≤k <∞ 11≤kk <∞ ≤ <∞ k k f ≥ ,0 where ()Af k = ∑ ki fa i , ()Rf k = ∑ fr ii , k ≥ .1 i=1 i=1

If wk = ,0 k ≥∀ ,1 then (1) will have look

≤ ≥ (2) sup uk() Afk Csup vkk f , fk ,0 11≤kk <∞ ≤ <∞

k ≥ .1 If also vi = ,0 i ≥∀ ,1 then (1) will rewritten in the following form

i i sup i ∑ ij j ≤ sup ∑ frwCfau jji , f j ≥ ,0 1 i ∞<≤ j =1 1 i ∞<≤ j =1 (3) For case ,1 qp ∞<< this problem was considered in [].1 Estimate of matrix operator. We consider the inequality (2), when the matrix operator A look as k ()Af k = ∑ ki fa i , k ≥ .1 i=1 (4) ∞ ∞ Let aij ≥ 0 and exist sequences = {}cc i i=1, c > 0 и = {}bb i i=1, b > 0 such that

aik akj aij c j +≈ bi under jki ≥≥≥ .1 ck bk (5)

Theorem. Let the matrix ()aij , aij ≥ 0 satisfies to a condition (5). Then the inequality (2) for the operator (4) fairly if and only if , k k 1 −1 1 −1 = max{}MMM 21 ∞< ,, M1 = sup sup i ik ∑ vcau jj , M 2 = sup sup ii ∑ kj vabu j , k 1 ck ik ∞<≤≥ j=1 k 1 bk ik ∞<≤≥ j=1 at that ≈ CM , where C the best const in (2).

References [1] Oynarov R., Shalginbayev S. H. Weight additive estimate of the class of matrix operators // Izv. MON RK AN. Series physical. - a mat., 2004, No. 1.(In Russian).

43 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Asymptotics of Modified Jacobi Polynomials in a Neighborhood of Endpoints of the Interval of Orthogonality Bujar Xh. Fejzullahu University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. For the classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) the asymptotic results on and away from the interval of orthogonality, as well as at its end-points, can be derived using multiple properties that these orthogonal polynomials satisfy: the differential equation, the Rodrigues formula, integral representation, etc. In a general situation the problem is much more difficult. However, many asymptotic results were found for polynomials that are orthogonal with respect to various classes of weights. For instance, in [3] (see also [4]) authors deduced the more refined Bernstein-Szegö asymptotic formula for the modified Jacobi polynomials. This formula gives a good description of the corresponding polynomial on compact sets interior to the interval of orthogonality. At the end-points of the interval of orthogonality, the Bernstein-Szegö formula becomes unsuitable. In this contribution the ladder operator approach has been applied together with the result of Aptekarev [1] to deduce the complete characterizations of orthogonal polynomials

α β with respect to the modified Jacobi weight ω h •+•−•=• ,)1()1)(()( where βα > 0, and h real analytic and strictly positive on [−1,1], near end-points of the interval of orthogonality [-1,1], so-called Mehler-Heine formulas ([2]). Keywords. Orthogonal polynomials, ladder operator, Mehler-Heine formula. AMS 2010. 33C47, 42C05.

References

[1] A. I. Aptekarev, Asymptotics of orthogonal polynomials in a neighborhood of endpoints of the interval of orthogonality, Russian Acad. Sci. Sb. Math. 76, 35-50, 1993. [2] B. Xh. Fejzullahu, Mehler-Heine formulas for orthogonal polynomials with respect to the modified Jacobi weight, submitted to publication. [3] A. Foulquié Moreno, A. Martínez-Finkelshtein, V. L. Sousa, Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1], Constr. Approx. 33, 219-263, 2011. [4] A. B. J. Kuijlaars, K. T-R McLaughlin, W. Van Assche, M. Vanlessen, The Riemann- Hilbert approach to strong asymptotics for orthogonal polynomials on [−1, 1], Adv. Math. 188, 337-398, 2004.

44 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Compact Operators on The Fibonacci Difference Sequence Spaces ( ) and ( )

E. Evren Kara (1), M. Başarır (2) and M. Mursaleen (3)퓵 풑 푭� 퓵∞ 푭� (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Aligarh Muslim University, Aligarh, India, [email protected]

Abstract. In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces ( ) and ( ) to be compact, where 1 < .

Keywords. Fibonacciℓ푝 퐹Numbers� ℓ∞, Sequence퐹� Spaces, Compact≤ Operators, 푝 ∞ Hausdorff Measure of Noncompactness AMS 2010. 11B39, 46A45, 46B50

References

[1] M. Mursaleen, A.K. Noman, Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Inequal. Appl. (Preprint).

[2] M. Başarır, E.E. Kara, On compact operators on the Riesz ( )-difference sequence 푚 spaces, Iran. J. Sci. Technol. 35(A4) (201) 279--285. 퐵

[3] E.E. Kara, M. Başarır, On some Euler ( ) difference sequence spaces and compact 푚 operators, J. Math. anal. Appl. 379 (2011) 499-511.퐵

[4] M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal. TMA, 73 (8) (2010) 2541--2557.

[5] M. Et, On some difference sequence spaces, Turkish J. Math.7 (1993) 18-24.

45 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Some New Paranormed B − Difference Sequence Spaces Derived by Generalized Weighted Mean Matrix Emrah Evren Kara (1) and Serkan Demiriz (2) (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Gaziosmanpasa University, Tokat, Turkey, [email protected]

Abstract. The paranormed sequence spaces  ∞ ()p , cp() and cp0 () were introduced and studied by Maddox [1]. In the present paper, we introduce the paranormed sequence spaces c0 () uvpB,; , , cuvpB(),; , and  ∞ ()uvpB,; , of non-absolute type which are derived by generalized weighted mean matrix and B − difference matrix and is proved that the spaces

c0 () uvpB,; , , cuvpB(),; , and  ∞ ()uvpB,; , are paranorm isomorphic to the spaces cp0 () , cp()

and  ∞ ()p ; respectively. Furthermore, the αβ−−, and γ − duals of the spaces c0 () uvpB,; , ,

cuvpB(),; , and  ∞ ()uvpB,; , are computed and the bases of the spaces c0 () uvpB,; , and

cuvpB(),; , are constructed. Finally, the matrix mappings from the sequence spaces XuvpB(,; , ) to a given sequence space λ and from the sequence space λ to the sequence space

XuvpB(,; , ) are characterized, where X∈{} cc0 ,, ∞ .

Keywords. Paranormed Sequence Spaces, Generalized Weighted Mean Matrix, B − Difference Matrix, Matrix Transformations AMS 2010. 40C05, 40A05

References

[1] Maddox I. J., Paranormed sequence spaces generated by infinite matrices, Proc. Comb. Phil. Soc., 64, 335-340, 1968.

[2] Başar, F., Çakmak, A. F., Domain of the triple band matrix on some Maddox’s spaces, Ann. Funct. Anal. 3 (1), 32-48, 2012.

[3] Aydın, C., Başar, F., Some new paranormed sequence spaces, Inform. Sci. 160, 27-40, 2004.

[4] Lascarides, C. G., Maddox, I.J., Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc., 68, 99-104, 1970.

[5] Başarır, M., Kara, E.E., On the B − difference Sequence space derived by Generalized weighted mean and compact operators, J. Math. Anal. Appl., 391, 67-81, 2012.

46 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Spectrum of Matrix Operators on a Difference Sequence Space of Weighted Means Ezgi Erdoğan (1) and Vatan Karakaya (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. For a sequence xx ()k , we denote the difference sequence by

  x() xxkk  1 . Let uu ()nn0 and vv ()kk0 be the sequences of positive real numbers such that un  0 , vk  0 for all k   . The difference sequences of weighted means

(,,uv ) are defined as (uv , , ) { x  ( xk ): W ( x )  }, where    p and the matrix

Ww ()nk is obtained by multiplying weighted mean matrix and difference matrix [1].

In this paper, we investigated the spectrum of this matrix over the sequence space  p . The fine spectrum of the Cesáro operator of order one on the sequence space  p studied by

González [2], where 1 < p < 1. Also, weighted mean matrices of operators on  p have been investigated by Cartlidge [3]. Karakaya et al. obtained the fine spectrum of the second order difference operator over the sequence spaces  p and bvp [4]. Keywords. Spectrum, sequence space, difference operator.

References

[1] Mursaleen, M., Karakaya, V., Polat, H., Şimşek, N., Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Computers and Mathematics with Applications, Vol. 62, Issue 2, pp. 814-820, 2011.

[2] González, M., The fine spectrum of the Cesáro operator in  p (1p  ) Arch. Math., 44, 355-358, 1985.

[3] Cartlidge, P.J., Weighted mean matrices as operators on  p , Ph. D. Dissertation, Indiana University, 1978.

[4] Karakaya, V., Dzh. Manafov, M. and Şimşek, N., On the fine spectrum of the second order difference operator over the sequence spaces  p and bvp , Mathematical and Computer Modelling, Vol. 55, Issue 3, pp. 426-436, 2012.

47 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Matrix Transformations Between Sequence Spaces Ekrem Savaş Istanbul Commerce University, Istanbul, Turkey, [email protected]

λ λ Abstract. The main object of this paper is to study Vp() and Vp() (the σ σ 0

λ definitions are given below) and characterize certain matrices in Vpσ ().

If pm is real such that pm > 0 and sup pm <∞, we define

λ p = m = Vpσ ( ) x : lim tmn, () x 0, uniformly in n 0 { } m→∞ and

λ p V( p )= x : lim t() x −= le m 0, for somel , uniformly inn . σ { mn, } m→∞

λλλλ In particular if pp= > 0 for all m , we have Vp()= V and Vp()= V. In m σσ00σσ

λ λ Theorem 4, we prove that Vp() and Vp() are complete linear topological spaces. σ 0 σ

Theorem 7 characterizes the matrices in the class ()c0 (),() pVσ p . In Theorem 8 we 0

λ determine the matrix in the class cp(),. V ()σ

References

[1] S. Banach, Theorie des Operations Linearies, Warszawa, 1932.

[2] G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc. 2(2) (1983), 61-70.

[3] V. Karakaya, θσ − sumable sequences and some matrix transformations, Tamkang J. Math. 35(4)(2004), 313-320.

[4] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta. Math. 80(1948), 167-190.

[5] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. 18(1967), 345-355.

48 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[6] I. J. Maddox and J. W. Roles, Absolute convexity in certain topological linear spaces, Proc. Cambridge Philos. Soc. 66(1969), 541-545.

[7] I. J. Maddox, Elements of functional analysis, Camb. Univ. Press (1970).

[8] M. Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford, 34(1983), 77-86.

[9] M. Mursaleen, A. M. Jarrah and S. A. Mohiuddine, Bounded linear poperators for some new matrix transformations, Iran. J. Sci. Technol. Trans. A Sci. 33 (2009), no. 2, 169-177.

[10] S. Nanda, Infinite matrices and almost convergence, Journal of the Indian Math. Soc. 40(1976), 173-184.

[11] E. Savaş, On infinite matrices and lacunary s- convergence, Appl. Math. Comp., 218(3), (2011), 1036-1040.

[12] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36(1972), 104-110.

49 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Approximation of Functions by Algebraic Polynomials in Jacobi Weighted Lp Spaces Fevzi Berisha (1) and Shkumbin Makolli (2) (1) FNA- Universiteti i Prishtinës, [email protected] (2) Shkumbin Makolli, FIM- Universiteti i Prishtinës,[email protected]

Abstract. The idea of approximation in the past, as well as in present time, has shown to be the most efficient and rational manner in achieving certain goals in every field of science as well as in practice. The core topic in the Mathematical Analysis are functions of different kind. The complexity of these functions has inspired many authors who have worked in replacing these functions with more simple functions. The simplest functions are polynomials, that is why we have many published works regarding approximation of functions with algebraic or trigonometric polynomials. While approximating certain apparatus with a simpler apparatus it is evident that an approximation error will appear. Therefore, it is of great interest to minimize the approximation error. The approximation error will be evaluated in accordance with the certain approximation apparatus which will be used. If the approximation apparatus is “good”, then the approximation error is very small, respectively, it tends to zero. In this work we will consider approximation of non-periodic functions with algebraic polynomials. We will prove the Jackson type inequality for non-periodic functions. We observe that in

Lp []0, 2π spaces, for p ≥1 , the Jackson’s theorem is proved by the creation of direct trigonometric polynomials in accordance with the given function f . Specifics of Lp []0, 2π spaces for 01<

Lp ()() ab,,1≤ p ≤∞ , a different manner will be used to prove statements. Keywords algebraic polynomials, Jacobi weight, approximation. AMS 2010. 41A10, 41AM17.

50 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

References

[1] Storozhenko E.A., Osvald P- Teorema Dzheksona v prostranstva LRpk(),0<< p 1- Dok. AN SSR, 1976, 229, N=3,, 554-557.

[2] Storozhenko E.A., Krotov V.G, Osvald P- Pramije i obratnie teoremi tipa Dzheksona v prostranstva Lpp ,0<< 1- Mat. Zbor. 1975, 98, 140, 395-415.

[3] Ivanov V.I. - Pramije i obratnie teoremi teorii priblizhenija v metrike Lpp ,0<< 1- Matem. Zametki, 1975, 18, N=5, 641-658.

[4] Storozhenko E.A.,Priblizhenije algebraiqeskimi mnogoqlenami funkcii klasa Lpp ,0<< 1- Izv. AN. SSSR ,ser mtem. 1977, 41,N=3,652-662.

[5] Timan A.F.- Teorija priblizhenija funkcii dejstvitelnogo peremenogo M.1950

[6] Brudnij J.A.-Priblizhenije funkciji algebraiqeskimi mnogoqlenami-Izv. AN. SSSR ,ser mtem. 1968, 32,N=4,780-787.

[7] Nguen Xuon Ky- On wighted polinomial approximation with a wighted ()()α 11 +− xx β in

L2 space. Act. Math. Sci. hungarica, t-27 (1-2). 1976, 101-107. [8] Muharrem Berisha, Fevzi Berisha – Nga teoria e përafrimeve të funksioneve, Akademia e Shkencave dhe Arteve të Kosovës, 2010 .

51 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Shpejtësia e Tentimit Në Zero e Koeficientëve Furie Të Funksioneve Me Derivat Të wely-it Fevzi Berisha FNA- Universiteti i Prishtinës, [email protected]

Abstract. Problemi i paraqitjes së funksionit 2π - periodik me anë të serisë Furie nuk është trivial. Qysh në vitin 1876, Xy Bua Rejmond ka formuar funksionin e vazhdueshëm me serinë Furie, që divergjon në pikat e veçanta. Më vitin 1925, A.N. Kolmogorov ka konstruktuar funksionin e shumueshëm, seria Furie e së cilës divergjon në secilën pikë. Pyetja se a ekziston funksion i vazhdueshëm që ka serinë Furie kudo (pothuajse kudo) divergjente një kohë të gjatë ka mbetur pa përgjigje. Vetëm në vitin 1966, L. Karleson ka vërtetuar se seria Furie e secilit funksion 2π - periodik, konvergjon te ky funksion pothuajse kudo ( ky ka

π ∈ 2 vërtetuar se konstatimi vlen edhe për funksionin fL1 , për të cilin integrali ∫ f është i −π fundmë). Me këtë rast do ti shqyrtojmë koeficientët Furie të funksioneve që kanë derivat të Wely-it dhe

r lidhjen e tyre me klasën N pθ . Pohimet e vërtetuar paraqesin vlerësimin e koeficienteve Furie në formë të shumës kështu që drejtpërdrejtë edhe nuk mund të vërejmë shpejtësinë e tentimit në zero të koeficienteve Furie

r të funksioneve nga klasa N pθ e që njëherit lidhet edhe me ekzistencën e derivatit të Wely-it. Prandaj që të vërejmë më thjesht shpejtësinë e tentimit në zero të koeficienteve Furie paraqesim edhe vlerësimet në formë të një koeficienti.

r Keywords. Koeficienët Furie, derivati i Wely-it , klasa N pθ . AMS 2010. 42A16

References

[1] Kokiashvilli V. M. - Ob ocenka najluqshih pribilizhenii i modulei glatkostli v razliqnih Lebegovski prostranstvah periodiqeskih funkcii s preobrazovanim rijadom Furije. Soobshq. AN Gruz. SSR 1964, 35 No.7 s.3-8 [2] Kokoashvili V. M. - O priblizhenii periodiçeskih funkcii, Tr Tbilis, Matemth int-ta 1968, 34, s. 51-81.

52 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[3] Sagher Y. - An application of interpollation theory to Fourier series - Studio. Math 1972 V. 71 No.2 pp 169-181. [4] Rodin V. A . - Neravenstva Xheksona i Nikolskogo dlja trigonometriçeskih polinomov v simetriçeskom prostranstvo. V. Kn. Trudi 7 - Zimnei Shkolli. Drogobiq, 1974, M 1976. s 133-139. [5] Koljada V. I. , Teorema vlozhenja i neravenstva razni metrik dlja najluqshih priblizheni. Mat. Sb. , 1977, 102 (144) No 2 s. 195-215. [6] Kokiashvili V. M. -Ob odnom funkcionalinom prostranstve i koefficientah Furie. Soobshq. AN Gruz. SSR, 1964, 35 No 3 S. 523-530. [7] M.K.Potapov,M.Berisha-Moduli gladkosti i koeficienti Fourie periodiqeskih funkcii odnogo peremenogo, Pub.De.Ins.Mat,Beograd,26(40),1979. [8] E.S.Smailov-Teoremi vlozhenija dja funksionalnih prostranstva s ortogonalnim bazisom,SSR,Alma-Ata, 1976. [9] F.Berisha-Procena pomoqu najbole aproksimacije Fourierovih koeficienata funkcija klas

r N pθ , Bul.pun.shkenc i FSHMN, Prishtinë ,1986.

r [10]F.Berisha-Mbi koeficientët Fourie të funksioneve nga klasa N pθ , Bul.pun.shkenc.i FSHMN,Prishtinë ,1996.

53 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Approximating Classes of Functions Defined by a Generalised Modulus of Smoothness Faton M. Berisha (1) and Nimete S. Berisha (2) (1) University of Prishtina, Prishtina, Kosovo, [email protected] (2) University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. In the paper, we use an operator of generalised translation in order to define a generalised modulus of smoothness. By its means, we define certain classes of functions, and we give their structural and constructive characteristics. Specifically, we prove some direct and inverse types theorems in approximation theory for best approximations by algebraical polynomials and the generalised modulus of smoothness. Keywords. Generalised modulus of smoothness, structural characteristics, constructive characteristics. AMS 2010. 41A10, 41A17, 41A27.

References

[1] M. K. Potapov, The operators of generalized translation in the approximation theory, Proc. II Math. Conf. Pristina, 1997, 27--36.

[2] M. K. Potapov, F. M. Berisha, Approximation of classes of functions defined by a generalized $k$th modulus of smoothness, East J. Approx. 4 (1998), no. 2, 217--241.

[3] M. K. Potapov, On the approximation of functions, characterized by a family of nonsymmetric generalized shift operators, by algebraic polynomials, Dokl. Akad. Nauk 373 (2000), no. 4, 456—458.

[4] M. K. Potapov, F. M. Berisha, On a relation between best approximations by algebraic polynomials and the $r$th generalized modulus of smoothness, J. Math. Sci. (N. Y.) 155 (2008), no. 1, 153—169.

54 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Starlikeness and Convexity of Generalized Struve Functions Halit Orhan (1) and Nihat Yağmur(2) (1) Ataturk University, Erzurum, Turkey, [email protected] (2) Erzincan University, Erzincan, Turkey, [email protected]

Abstract. In this work, we give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk. Keywords. Analytic function, Struve functions AMS 2010. 30C45, 33C10

References

[1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann.of Math. 17 (1915), 12-22.

[2] A. Baricz, Applications of the admissible functions method for some differential equations, Pure Math. Appl. 13 (2002), 433-440.

[3] A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73/1-2 (2008), 155-178.

[4] R. K. Brown, Univalence of Bessel functions, Proc. Amer. Math. Soc. 11 (1960), 278-283.

[5] P. L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.

[6] L. Fejer, Untersuchungen Äuber Potenzreihen mit mehrfach monotoner Koeffizientenfolge, Acta Litterarum ac Scientiarum 8 (1936), 89-115.

[7] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469- 474.

[8] W. Kaplan, Close-to-convex schlicht functions, Mich. Math. J. 1 (1952), 169-185.

[9] E. Kreyszig and J. Todd, The radius of univalence of Bessel functions, Illinois J. Math. 4 (1960), 143-149.

55 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[10] S. S. Miller and P. T. Mocanu, Differential subordinations and inequalities in the complex plane, J. Differential Equations 67 (1987), 199-211.

[11] I. R. Nezhmetdinov and S. Ponnusamy, New coefficient conditions for the starlikeness of analytic functions and their applications, Houston J. Math. 31(2) (2005), 587-604.

[12] K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. 2 (1935),129-155.

[13] S. Ozaki, On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daigaku 2(1935), 167-188.

[14] S. Ponnusamy and M. Vuorinen, Univalence and convexity properties for confluent hypergeometric functions, Complex Variables Theory and Appl. 36 (1998), 73-97.

[15] S. Ponnusamy and M. Vuorinen, Univalence and convexity properties of Gaussian hypergeometric functions, Rocky Mountain J. Math. 31 (2001), 327-353.

[16] V. Selinger, Geometric properties of normalized Bessel functions, Pure Math. Appl. 6(1995), 273-277.

[17] S. E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), 310-340.

[18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press,Cambridge, 1962.

56 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

An Extremum Problem in the Metric Space H.S. Akhundov (1) and M.A. Sadygov (2) (1) Institute of Applied Mathematics, BSU, Baku, Azerbaijan, [email protected]

Abstract. We studied unconditional extremum problem in metric space, obtained from conditional extremum problem by using notion of covering operators. Obtained necessary optimality conditions for considered problem. In the work Clark [1] the exact penalty function is constructed for the extremum problems with constraints using the distance functions in the class of Lipschits functions. In [2] Lipschits functions (),,, δϑβα and − (),,, δϑβαϕ are defined in the point of Banach space, some their properties are studied, and extremum problems with constraints are investigated. Using the distance function some theorems are obtained on the exact penalty. Also high order necessary and sufficient conditions are derived within constraints. Our aim in this paper is solving unconditionally extremum problem in metric space. Let ,YX be Banach spaces, G open set in X and : → YGF , operator, the point

x0 be a local minimum of the function f on the set { =∈= 0 },)()(: fxFxFGxM satisfies to

Lipschts condition neigberhood of the point 0 , Fx is strictly differentiable at the point x0 and ′ 0 )( = YXxF . Then there exists a number L > 0 such, that

m * xf 0 +∂∈ LB ′ xF 0 )()(0 ,

* here B - is unit sphere. Keywords. Banach space, strictly differentiable.

AMS 2010. 46N10, 54E35.

References

[1] Clark, F. Optimization and Nonsmooth Analysis. M.: Nauka, 1988, 280.

[2] Sadygov, M.A. Investigation of Nonsmooth Optimization Problems. Baku, 2002, 125.

57 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the Boundedness of Riesz-Bessel Transforms associated with the B-GSO I. Ekincioğlu (1) and C. Keskin (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Dumlupinar University, Kutahya, Turkey, [email protected]

Abstract. Riesz Bessel transforms RBxj related to the Bessel operators are studied in this paper. Since Bessel generalized shift operator (B-GSO) is translation operator corresponding to the Bessel operator, we construct a family of RBxjby using Bessel generalized shift operator depends on Bessel operator. Finally, we analyse weighted inequalities involving RBxj. Keywords. Riesz-Bessel Singular Integral, Bessel Generalized Shift Operator,Bessel operator. AMS 2010. Primary 42B37,42B20; Secondary 47G10,47B37.

References

[1] I.A.Aliyev, On Riesz transformations generated by generalized shift operator,Izv. Acad. Sci. Azerb, 1,p.7-13 1987.

[2] I.A. Aliyev and A.D. Gadjiev, Weighted estimates for multidimensional Singular Integrals Generated by the Generalized shift Operator, Rus. Acad. Sci. Sb. Math,77(1) 37-55, 1994.

[3] R. Courant and H. Hilbert, Methods of Mathematical Physics,Volume 2,Interscience Publ., New York, 1962.

[4] M. Cheng, On a Theorem of Nicolesco and Generalized Laplace Operators, Proc.Amer. Math. Soc., 2, p.77-86, 1951.

[5] I.Ekincioglu, H.Y³ld³r³m and O.Ak³n, The Mean Value Theorem For Laplacean-Bessel Equation, Invited Lecture Delivered at the seventh International Colloquium on Di®erential Equations, Plovdiv, Bulgaria, Vol.II, Academic Publications, PP. 29-37, 1996.Preprint submitted to Elsevier 3 July 2012

58 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[6] I. Ekincioglu, and I.K. Ozk³n , On Higher Order Riesz Transformations Generated By Generalized Shift Operator, Turkish Journal of Mathematics, Vol.21, p.51-60, 1997.

[7] I.Ekincioglu and S.Solak, On High Order Riesz-Bessel Transformations Generated By Generalized Shift Operator,, Journal of Ins. Math. And Comp.Vol. 13, (2), p 227-233, 2000.

[8] A.D. Gadjiev and E.V. Guliyev, Two weighted inequality for singular integrals in Lebesgue spaces associated with the Laplace-Bessel di®erential operator, Proc.A. Razmadze Math. Inst., 138, 1-15, 2005

[9] I.A. Gray, A Treatise on the Theory of Bessel Functions, Mac. London, 1931.

[10] V.S. Guliev, Sobolev Theorems for B-Riesz Potentials, Dokl. RAN 358(4), 450-451, 1987 (Russian)

[11] I.A. Kipriyanov, Singular Elliptic Boundary Value Problems, Nauka, Moscow 1997, (Russian).

[12] I.A. Kipriyanov, Boundary Value Problems for Elliptic Partial Di®erential Operators, Soviet Math.Dokl.II, p.1416-1419 1970.

[13] I.A. Kipriyanov, Fourier Bessel Transformations and Imbedding Theorems, Trudy Math. Inst. Steklov 89, 130-213, 1967 (Russian)

[14] I.A. Kipriyanov and M.I. Klyuchantsev, On Singular Integrals Generated by The Generalized Shift Operator, II. Sib. Mat. Zh., 11, 1060-1083, 1970 (Russian)

[15] M.I. Klyuchantsev, On Singular Integrals Generated by The Generalized Shift Operator, I. Sib. Mat. Zh., 11, 810-8221, 1970 (Russian)

[16] N.I. Kipriyanova, The Mean Value Formula for Singular Di®erential Operator With Order Second,Di®erential Equations, Vol.121, No:11, 1985.

[17] B.M. Levitan, Bessel Function Expansions in Series and Fourier Integrals, Volume 6, Ushki Math. Nauk, (Russian) No: 2. p. 102-143, 1951.

[18] B.M. Levitan , The Theory of Generalized Translation Operators, Nauka, Moscow, (Russian), 1973.

59 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Privalov Type Estimates of High Order Riesz-Bessel Transformations Generated by a Bessel Generalized Shift Operator I. Ekincioğlu (1) and C. Keskin (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Dumlupinar University, Kutahya, Turkey, [email protected]

Abstract. We obtain an estimate for the certain singular integral operators, Riesz- Bessel transformations generated by Bessel-generalized shift operator which is called of I.I. Pri-valov type. Keywords. Riesz-Bessel transformations, Bessel Generalized Shift Operator. AMS 2010. Primary 47G10; Secondary 45E10,47B37.

References

[1] Agmon, S., Douglis, A., Nirenberg, L., Estimates of solutions of elliptic equations near the boundary, Comm. Pure and Appl. Math., 12, 623-727, 1959.

[2] Aliev, I.A., Gadzhiev, A.D., Weighted estimates for multidimensional singular integrals generated by the generalized shift operator, Mat. Sb., 183, no.9, 45-66, 1992, translation in Russian Acad. Sci. Math. 77, no.1, 37-55, 1994.

[3] Calderon, A.P.,Zygmund, A., On singular integrals, Amer. J. Math., 78, 289-309, 1956.

[4] Ekincioglu, I. and Serbetci, A., On weighted estimates of high order Riesz-Bessel transformations generated by the generalized shift operator, Acta Mathematica Sinica, 21, no.1, 53-64, 2005.

[5] Lyakhov, L.N., On a class of spherical functions and singular pseudodi®erential operators, Dokl. Akad. Nauk. 272, no.4, 781-784, 1983,translation in Soviet Math.Dokl. 28, no. 2, 431-434, 1983.Preprint submitted to Elsevier 3 July 2012

[6] Lyakhov,L.N., Spherical weighted harmonic functions and singular pseudodi®erential operators, Di®erents. Uravn. 21, no. 6, 1020-1032,1985; translation in Di®. Equ. 21, no. 6, 693-703, 1985.

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[7] Kipriyanov, I.,A., Klyuchantsev, M.I., On singular integrals generated by the generalized shift operator II, Sibirsk. Mat. Zh., 11,1060-1083, 1970; translation in Siberian Math. J., 11, 787-804, 1970.

[8] Klyuchantsev, M.I., On singular integrals generated by the generalized shift operator I, Sibirsk. Math. Zh. 11, 810-821, 1970; translation in Siberian Math. J. 11, 612-620, 1970.

[9] Mihlin, S.G., Multidimensional singular integrals and integral equations,Fizmatgiz, Moscow, 1962; english transl. Pergamon Press, NY, 1965.

61 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Structural Properties of Vector Valued Orlicz Sequence Space Generilazed by an Infinite Matrix Murat Candan Inonu University, Malatya, Turkey, [email protected]

Abstract. In this paper, we introduce the vector valued sequence space

( k ,,,, spMXAF ) and study the closed subspace of it. Here, F is a normal sequence algebra with absolutely monotone norm . F and having a Schauder base ()ek , where

ek = (  ),,0,1,0,,0 with 1 in the k − th place; A is a nonnegative matrix; X k is seminormed space over the complex field with seminorm qk for each ∈ INk ; M is an Orlicz function;

= ()pp k be any sequence of strictly positive real numbers and s be any non-negative real number. We examine various algebraic and topological properties of this space and also investigate some inclusion relations on it. Keywords. Orlicz function, Orlicz sequence space, vector valued sequence space, paranormed space. AMS 2010. 46A45, 40D25, 46A35.

62 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Almost Convergence and Generalized Double Sequential Band Matrix Murat Candan Inonu University, Malatya, Turkey, [email protected]

Abstract. The class f of almost convergent sequences was introduced by G.G. Lorentz, using the idea of the Banach limits [A contribution to the theory of divergent ~ ~ sequences, Acta Math. 80(1948), 167--190]. Let 0 ()Bf and ()Bf be the domain of the double ~ ~ ~ sequential band matrix (), srB in the sequence spaces f0 and f . In the present paper, the ~ β − and γ − duals of the space ()Bf are determined. Additionally, we give some inclusion ~ ~ ~ theorems concerning with the spaces 0 ()Bf and ()Bf . Furthermore, the classes (Bf )(, µ ) and ~ µ , (Bf )() of infinite matrices are characterized, and the characterizations of some other classes are also given as an application of those main results, where µ is any given sequence space. Keywords. Almost convergence, matrix domain of a sequence space, generalized difference matrix, β − and γ − duals and matrix transformations. AMS 2010. 46A45, 40C05

63 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Pointwise and Uniform λ-Statistical Convergence of Order α of Sequences of Functions Mikail Et (1), Muhammed Çınar (2) and Murat Karakaş (3) (1) Firat University, Elazıg, Turkey, [email protected] (2) Mus Alparslan University, Mus, Turkey, [email protected] (3) Bitlis Eren University, Bitlis, Turkey, [email protected]

Abstract. In this study, we introduce the notion of pointwise and uniform λ-statistical convergence of order α of sequences of real-valued functions. Furthermore, we introduce the concept of λ-statistically Cauchy sequence and study statistical analogue of convergence and Cauchy criterion for sequences of real-valued functions. Also some relations between ( )- 훼 statistical convergence of order α and strong ( )summability of order β are given.푆휆 Also푓 훽 some relations between the spaces ( , )and푆 휆 푓( ) are examined. 훼 훼 Keywords: Statistical convergence,푤푝 푓 휆 sequences푆휆 푓 of functions, Cesàro summability. AMS 2010: 40A05, 40C05, 46A45

References

[1] Çolak, R., Statistical convergence of order α, Modern Methods in Analysis and Its Applications, NewDelhi, India: Anamaya Pub, 2010: 121--129.

[2] Connor, J. S., The Statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 47-63.

[3] Fridy, J., On statistical convergence, Analysis 5 (1985) 301-313.

[4] Mursaleen, M., λ-statistical convergence, Math. Slovaca, 50(1) (2000), 111 -115.

[5] Gökhan, A., Güngör, M.On pointwise statistical convergence, Indian J. appl. Math. 33(9) 2002 1379-1384

[6] Güngör, M., Gökhan, A., On uniform statistical convergence. Int. J. Pure Appl. Math. 19 (2005), no. 1, 17--24.

[7] Šalát, T., On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.

[8] Duman, O.; Orhan, C. μ-statistically convergent function sequences. Czechoslovak Math. J. 54(129) (2004), no. 2, 413--422.

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Some Results on dist -formulas, Toeplitz Operators and Berezin Symbols M. Gürdal (1), M.T. Karaev (2), S. Saltan (3) and U. Yamancı (4) (1) Suleyman Demirel University, Isparta, Turkey, [email protected] (2) Suleyman Demirel University, Isparta, Turkey, [email protected] (3) Suleyman Demirel University, Isparta, Turkey, [email protected] (4) Suleyman Demirel University, Isparta, Turkey, [email protected]

∞ Abstract. The distance from the nonconstant function ϕ in L T)( to the set Fconst of all

2 constant functions is estimated in terms of Hankel operators on the Hardy space H D)( over

∞ the unit disk { CD zz <∈= 1: }. More detailly is discussed a partial case ϕ ∈ H . More precisely,

ϕ ≥ ∗ in this case we prove that dist Fconst sup),( HH θϕ . We give a sufficient condition ensuring the θ Σ∈ )(

ϕ = ϕ equality dist Fconst ),( ∞ . Moreover, we investigate the maximal numerical range and maximal Berezin set for some Toeplitz operators. Also we present some other results related with applications of Berezin symbols, in particular, we characterize normal operators in tems of reproducing kernels and Berezin symbols, and prove an inequality for the Berezin number

2 of operator on H D).( Keywords. Berezin symbol, Hardy space, Hankel operator, Toeplitz operator, Maximal numerical range, Berezin number AMS 2010. 47B35, 47B10.

References

[1] Davidson, K., The distance to the analytic Toeplitz operators, Ill. J. Math., 39, 265-273, 1987.

[2] Douglas, R.G., Banach algebra techniques in operator theory, Academic Press, New York, 1972.

[3] Fisher, S., Exposed points in spaces of bounded analytic functions, Duke Math. J., 36, 479-489, 1969.

[4] Garnett, J.B., Bounded analytic functions, Academic Press, New York-London, 1981.

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[5] Hoffman, K., Banach spaces of analytic functions, Prentice Hall, England.Cliffs, NJ, 1962.

[6] Kadison, R.V., Derivations of operator algebras, Ann. of Math., 83, 280-293, 1966.

[7] Karaev, M.T. and Gürdal, M., On the Berezin symbols and Toeplitz operators, Extracta Mathematicae, 25, 83-102, 2010.

[8] Mustafaev, G.S., Distance estimation in the space of Toeplitz operators, Izv. Akad. Nauk Azerbaidzhan, SSR.Ser.Fiz.-Tekhn.Mat.Nauk, 6, 10-14, 1985.

[9] Mustafaev, G.S. and Shulman, V.S., Estimates for the norms of inner derivations in some operator algebras, Math. Notes, 45, 337-341, 1989.

[10] Nikolski, N.K., Treatise on the shift operator, Heidelberg, 1986.

[11] Nordgren, E. and Rosenthal, P., Boundary values of Berezin symbols, Operator Theory: Advances and Applications, 73, 362-368, 1994.

[12] Sakai, S., Derivations of W ∗ -algebras, Ann. of Math., 83, 273-279, 1966.

[13] Sakai, S., Derivations of simple C ∗ -algebras, J. Funct. Anal., 2, 202-206, 1968.

[14] Stampfli, J.G., The norm of a derivation, Pacif. J. Math., 33, 737-747, 1970.

[15] Sz.-Nagy, B. and Foias, C., Harmonic analysis of operators on Hilbert space, North Holland, New York, 1970.

[16] Zhu, K., Operator Theory in Function Spaces, Marcel Delcker, Ins., 1990.

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Quasi-Diagonal Operators Muhib Lohaj (1) and Shqipe Lohaj (1) (1) Department of Mathematics Avenue "Mother Theresa " 5, Prishtine, 10000, Kosova [email protected] , [email protected]

Abstract. Let be a separable complex Hilbert space and let denote the algebra of all bounded linear operators on : If is a quasi-normal Fredholm operator we prove that if and only if . We also show that if is quasi-normal and is quasi-diagonal with respect to any sequence in such that , strongly, then ; where is a normal operator and is a compact operator. Keywords. quasi-diagonal operators. AMS 2010. 47Bxx, 47B20.

References

[1] D.A. Herrero, Approximation of Hilbert space operators, I. Research Notes in Math., Vol.72(London-Boston-Melbourne: Pitman Books Ltd.,1982).

[2] P.R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton 1967.

[3] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970) 887-933.

[4] Muhib R. Lohaj, Necessary conditions for quasidiagonality of some special nilpotent opera-

tors, Rad. Mat. 10(2001),209-217. [5] G.R.Luecke, A note on quasidiagonal and quasitriangular operators, Pacific. J . Math. 56 (1975), 179-185.

[6] Carl M. Pearcy, Some recent developments in operator theory, Conference Board Math. Sci.

Vol 36, 1978. [7] R.A. Smucker, Quasidiagonal and quasitriangular operators, Disertation, Indiana Univ.1973.

[8] R.A. Smucker, Quasidiagonal weighted shifts, Pacific. J. Math. 98 (1982), 173-182.

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Radon-Nikodym Property for Vector Valued Measures Mine Öztürk (1) and Cüneyt Çevik (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. We explain tensor integrable functions and give some properties of them. We obtain the classical Radon-Nikodym property extends for vector valued measures. Keywords. Radon-Nikodym property, tensor integrable function, vector measure. AMS 2010. 53A40, 28B05.

References

[1] Chakraborty, N.D., Basu, S., On some properties of the space of tensor integrable functions, Anal. Math. 33 (1), 1-16, 2007.

[2] Diestel, J.; Uhl, J.J., Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R.I., 1977.

[3] Ryan, R.A., Introduction to tensor products of Banach spaces, Springer-Verlag, London, 2002.

[4] Stefánsson, G.F., Integration in vector spaces, Illinois J. Math. 45 (3), 925-938, 2001.

[5] Turan, B., Çevik, C., On the ideal centre of the space of vector valued integrable functions, Positivity 13 (2), 427-433, 2009.

[6] Turan, B., Çevik, C., On the ideal centre of L1(v) for a vector measure v, Rev. Roumaine Math. Pures Appl. 51 (2), 257-264, 2006.

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On mt-Convexity Mevlüt Tunç (1) and Hüseyin Yıldırım (2) (1) University of Kilis 7 Aralik, Kilis, Turkey, [email protected] (2) University of Kahramanmaras Sutcu Imam, Kahramanmaras, Turkey, [email protected]

Abstract. In this paper, one new class of convex functions which is called MT-convex functions is given. We also establish some Hadamard-type inequalities. Keywords. Convexity, AM-GM inequality, Similarly ordered. AMS 2010. 26D15

References

[1] J.L.W. V. Jensen, Sur les fonctions convexes et les inéqualités entre les valeurs moyennes, Acta Mathematica 30 (1906), 175-193.

[2] B. Bollobás, Linear Analysis, an introductory course (Cambridge Univ. Press 1990).

[3] B.G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6 (E), 2003.

[4] H.J. Skala, On the characterization of certain similarly ordered super-additive functionals, Proceedings of the American Mathematical Society, 126 (5) (1998), 1349-1353.

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On New Inequalities for h-Convex Functions via Riemann-Liouville Fractional Integration Mevlüt Tunç University of Kilis 7 Aralik, Kilis, Turkey, [email protected]

Abstract. In this paper, some new inequalities of the Hermite-Hadamard type for h- convex functions via Riemann-Liouville fractional integral are given. Keywords. Riemann-Liouville fractional integral, h-convex function, Hadamard’s inequality. AMS 2010. 26D15

References

[1] Alzer, H.: A superadditive property of Hadamard.s gamma function, Abh. Math. Semin. Univ. Hambg., 79, 11-23 (2009).

[2] Belarbi, S. and Dahmani, Z.: On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3), Art. 86 (2009).

[3] Bombardelli, M. and Varosanec, S.: Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities, Computers and Mathematics with Applications, 58, 1869-1877 (2009)

[4] Burai, P. and Házy, A.: On approximately h-convex functions, Journal of Convex Analysis, 18 (2) (2011).

[5] Dahmani, Z.: New inequalities in fractional integrals, International Journal of Nonlinear Science, 9(4), 493-497 (2010).

[6] Dahmani, Z.: On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1(1), 51-58 (2010).

[7] Dahmani, Z., Tabharit, L. and Taf, S.: Some fractional integral inequalities, Nonl. Sci. Lett. A., 1(2), 155-160 (2010).

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[8] Dahmani, Z., Tabharit, L. and Taf, S.: New generalizations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3), 93-99 (2010).

[9] Dragomir, S.S. and Agarwal, R. P.: Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11 (5), 91-95 (1998).

[10] Özdemir, M.E., Kavurmacı, H. and Avcı, M.: New inequalities of Ostrowski type for mappings whose derivatives are (α,m)-convex via fractional integrals, RGMIA Research Report Collection, 15, Article 10, 8 pp (2012).

[11] Özdemir, M.E., Kavurmacı, H. and Yıldız, Ç.: Fractional integral inequalities via s- convex functions, arXiv:1201.4915v1 [math.CA] 24 Jan 2012.

[12] Sarıkaya, M. Z., Sağlam, A. and Yıldırım, H.: On some Hadamard-type inequalities for h-convex functions. Journal of Math. Ineq., 2 (3), 335-341 (2008).

[13] Sarıkaya, M. Z., Set, E. and Özdemir, M.E.: On some new inequalities of Hadamard type involving h-convex functions, Acta Math. Univ. Com., Vol. LXXIX, 2, 265-272 (2010).

[14] Sarıkaya, M. Z., Set, E., Yaldiz, H. and Başak, N.: Hermite-Hadamard.s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling,In Press.

[15] Set, E.: New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., In Press, Corrected Proof, 29 December 2011.

[16] Varo.anec, S.: On h-convexity, J. Math. Anal. Appl., 326, 303-311 (2007).

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Two New Definitions on Convexity and Related Inequalities Mevlüt Tunç University of Kilis 7 Aralik, Kilis, Turkey, [email protected]

Abstract. We have made some new definitions using the inequalities of Young and Nesbitt. And we have given some features of these new definitions. After, we established new Hadamard type inequalities for convex functions in the Young and Nesbitt sense. Keywords. Hadamard’s, Young’s, Nesbitt’s, Pachpatte’s inequality. AMS 2010. 26D15

References

[1] B.G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll., 6 (E), 2003.

[2] S.S. Dragomir, R.P. Agarwal and N.S. Barnett, Inequalities for beta and gamma functions via some classical and new integral inequalities, J. Inequal. & Appl., 5 (2000), 103-165.

[3] R.B. Manfrino, J.A.G. Ortega and R.V. Delgado, Inequalities A Mathematical Olympiad Approach, Birkhäuser, Basel-Boston-Berlin, 2009.

72 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On One Class of Sequences and a Convolution Operator in This Class Nadir Ibadov Ganja State University, Ganja, Azerbaijan, [email protected]

* Abstract. In this paper, a space of sequences A ,1 ∞ and its adjoint space A ,1 ∞ are built.

* Also, there is formed a biunique correspondence between the spaces A ,1 ∞ and ()CH * through

Mellin transform, where C is a complex plane, * CC −= }0{ , and ()CH * is a space of nearly integer functions. Also, there are considered homogeneous convolution equations in the space of sequences A ,1 ∞ , and an approximation problem for a homogeneous convolution equation in this space is solved. Generally, we have got the following results. We have proved the following Lemma. Lemma 1. The space A is a banach space. In the set = AA , we can consider ,1 σ ,1 ∞  ,1 σ σ >0 a topology of inductive limit A ,1 ∞ = limind A ,1 σ . σ >0 Furter in the paper, we have had the following theorem. Theorem 1. A is a space ( M )*. Thereby, there exists an intersection * = AA * , ,1 ∞ ,1 ∞  ,1 σ σ >0 where we can introduce a topology of projective limit (it follows from the property of the

* * * * space ( LN ) and A ,1 ∞ = lim pr A ,1 σ . The space A ,1 ∞ is adjoint to the space A ,1 ∞ . σ >0

α Theorem 2.2. The following is true: 1) For ∈∀ Cz * . The sequences ∈ Az ,1 ∞ , where

∞ ˆ α α ≥ 0 . 2) The Mellin transform + () = ∑ α zbzF of the functional ∈ AF ,1 ∞ accomplishes a α =0

* biunique correspondence between the spaces A ,1 ∞ and ()CH * .

* Definition 1. Convolution operator generated by the functional ∈ AF ,1 ∞ is an operator

M F functioning by the following rules: F []{}α α∈Z == ,{aFaaM α +m }()∈Zm == {cc m } ∈Zm , where

* ∞ {m }∈Zm == , {aFcc α +m }()α∈Z и {}α ∈= Aaa ,1 ∞ . The functional ∈= AbF ,1 defines a

+∞ convolution operator in the form of F [][]b ∑ αα +m mabaMaM ±±±=⋅== ,...,2,1,0, α −∞=

73 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Keywords. convolution operator, Mellin transform, homogeneous convolution equation AMS 2010: 46-XX Functional Analysis, 44A35 Convolution, 46A45 Sequence spaces, 46B45 Banach sequence spaces

References

[1] Kolmogorov A.N., Fomin S.B. The Elements of the Theory of Functions and Functional Analysis., Nauka M., 1976, 544 pp. / Колмогоров А.Н. Фомин С.Б. Элементы теории функций и функционального анализа. Наука, М., 1976, 544 стр.

[2] Robertson A., Robertson I., Topological Vector Spaces. Mir, M., 1967, 258 pp. / Робертсон А., Робертсон И., Топологические векторные пространства. Мир, М., 1967, 258 стр.

[3] Jose Sebastian-da-Silva About some classes of locally-convex spaces important in Applications. – Mathematics, 1957, 1, № 1, pp.60-77 / Жозе Себаштьян-и-Силва. О некоторых классах локально выпуклых пространств, важных в приложениях.- Сб. Математика 1957, 1, № 1, с.60-77.

[4] Leontiev A.F. Integer Functions. Exponent Series. Nauka, M., 1983, 176 pp. / Леонтьев А.Ф. Целые функции, Ряды экспоненты. Наука, М.,1983, 176 стр.

[5] Ibadov N.V. About one convolution operator., Book of IV congress of the Turkic world mathematical society., 1-3 July 2011, Baku, Azerbaijan, p.87.

* [6] Ibadov N.V. Biunique correspondence between the spaces A ,1 ∞ and ()CH * through Mellin transformation, Proceedings of conference on New Problems in Mathematics. Theory and Practice, 23-25 September 2011, Gandja, Azerbaijan, p.12-18

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Möbius Transformations and Circles on the Complex Plane Nihal Yılmaz Özgür Balikesir University, Balikesir, Turkey, [email protected]

Abstract. A Möbius transformation (also called a fractional linear transformation, projective linear transformation, or a bilinear transformation) of the complex plane is a rational function of the form az+ b Tz()= cz+ d where the coefficients a, b, c, d are complex numbers satisfying ad−≠ bc 0 . It is well-known that Möbius transformations map circles to circles (where straight lines are considered to be circles through ∞). This is their most basic geometric property. In this study, we consider the circles corresponding to any norm function on the complex plane and their images under the Möbius transformations. Keywords. Möbius transformations, preservation of circles. AMS 2010. 30C35.

References

[1] Beardon, A. F., Algebra and Geometry, Cambridge University Press, Cambridge, 2005.

[2] Jones, G. A., Singerman, D., Complex functions. An algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987.

[3] Yılmaz Özgür, N., On Some Mapping Properties of Möbius Transformations, Aust. J. Math. Anal. Appl., 6, Art. 13, 8 pp., 2010.

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APPLIED MATHEMATICS

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Properties of a Family of Analytic Functions Abdullah Altın Ankara University, Ankara, Turkey, [email protected]

Abstract. In this paper, we define a class of analytic functions and we give miscellaneous properties of these functions as generating function, Rodrigues formula and recurrence relation. We derive various classes of bilinear and bilateral generating functions for these functions. We also show that some particular cases of these functions reduce to well- known polynomials. Keywords. Rodrigues formula, recurrence relation, generating function, bilinear and bilateral generating function, Hermite polynomial. AMS 2010. 42C05.

References

[1] Altın, A., Aktaş, R., A generating function and some recurrence relations for a family of polynomials, Proceedings of the 12 th WSEAS International Conference on Applied Mathematics, 118-121, 2007.

[2] Altın, A., Aktaş, R., A class of polynomials in two variables, Mathematica Moravica, 14, 1, 1-14, 2010.

[3] Altın, A., Erkuş, E., On a multivariable extension of the Lagrange-Hermite polynomials, Integral Transform. Spec. Funct. 17, 239-244, 2006.

[4] Aktaş, R., Şahin, R., Altın, A., On a multivariable extension of Humbert polynomials, Appl. Math. Comp, 218, 662-666, 2011.

[5] Erkuş, E., Srivastava, H.M., A unified presentation of some families of multivariable polyomials, Integral Transforms and Special Functions, 17, 267-273, 2006.

[6] Rainville, E.D., Special Functions, The Macmillan Company, New York, 1960.

[7] Srivastava, H.M., Manocha, H.L., A Treatise on Generating Functions, Halsted Press, New York, 1984.

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Numerical Approximation for Boundary Value Problem By A New Collocation Approach Ayşe Betül Koç (1) and Aydın Kurnaz (2) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. In this work, a new approximate method for solving boundary value problem is presented. The method is based on the polynomial collocation methods. The solution is obtained in terms of special functions. Numerical results and conclusions is presented. Keywords. Collocation methods, boundary value problem. AMS 2010. 53A40, 20M15.

References [1] Fox L., Chebyshev Methods for ordinary differential equations, The Computer Journal, 1962,, 318-331.

[2] Fox L. , Parker I.B., Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968.

[3] Scraton R. E., The solution of linear differential equations in Chebyshev series, The Computer Journal, 1965, 57-61.

[4] N.K. Basu, On double Chebyshev series approximation, SIAM J. Numer. Anal., 10(1973), 496–505.

[5] Sezer M., Doğan S.,Chebyshev series solutions of Fredholm integral equations, Internaitonal Journal of Mathematical Education inScience andTechnology,27(1996),649-657.

[6] Akyüz A., Sezer M., A Chebyshev collocation method for the solution of linear integro- differential equations, 72(1998), 491-507.

[7] Mason J. C., Handscomb D. C., Chebyshev polynomials, CRCPress, BocaRaton, 2003.

[8] Akyüz A., Sezer M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Applied Mathematics and Computation, 144(2003), 237-247.

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Statistical Integral on Banach Space Anita Caushi (1) and Agron Tato (2) (1) Polytechnic University of Tirana, Albania, [email protected] (2) Polytechnic University of Tirana, Albania, [email protected]

Abstract. The idea of statistical convergence was given by Zygmund (1935) in the first edition of his monograph. The concept of statistical convergence was formally introduced by Steinhaus and Fast(1951) and later statistical convergence was reintroduced by Schoenberg(1959). Although statistical convergence was introduced over nearly the last fifty years, it has become an active area of research in recent years. The main idea of the statistical convergence of a sequence x is that majority of elements from x converge and we don’t care what is going on with other elements. Concept of statistical convergence in many works is directly connected with convergence of such statistical characteristics as mean and standard deviation and at the same time with measurement and computation.

The main object of this presentation is to introduce a new concept of the statistical integration on Banach space. We propose one type of Bohner integration in context of this convergence. We used the Lebesgue measure and measurable functions in usual meaning and extend the property of the statistical Cauchy sequence of functions on Banach space and profit a number of prepositions about the representative sequences of function. We prove a kind of Egorov theorem by concept of equi-statistical convergence. The main results are construction of the statistical integral and prove usual properties of integration and show that ones are generalization of the Bohner integral. We also proposed some convergence theorems. Key words. statistical convergence, statistical pointweis convergence, statistical integration. AMS 2010. 40A05.

References

[1] J. Connor, M. Ganichev, and V. Kadets, “A characterization of Banach spaces ëith separable duals via ëeak statistical convergence,” Journal of Mathematical Analysis and Applications, vol. 244,no. 1, pp. 251–261, 1989.

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[2] H. Fast, “Sur la convergence statistique,” ColloquiumMathematicum, vol. 2, pp. 241–244, 1951.

[3] J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.

[4] J. A. Fridy, “Statistical limit points,” Proceedings of the American Mathematical Society, vol. 118, no. 4, pp. 1187–1192, 1993.

[5] Gökhan A., Güngör M., On pointëise statistical convergence, Indian Journal of pure and application mathematics, 33(9) : 1379-1384, 2002.

[6] I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, no. 5, pp. 361–375, 1959.

[7] Schwabik, S., Guoju, Y., Topics in banach space integration, Series in Analysis vol. 10. World Scientific Publishing Co. Singapore 2005.

[8] H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” ColloquiumMathematicum, vol. 2, pp. 73–74, 1951.

[9] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979.

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Dynamical Problem for the Pre-stressed Bi-layered Plate-strip with Finite Length Under the Action of Arbitrary Time-Harmonic Forces Resting on a Rigid Foundation Ahmet Daşdemir (1) and Mustafa Eröz (2) (1) Aksaray University, Aksaray, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. According to the principle of the three-dimensional linearized theory of elastic waves in the initially stressed bodies (TLTEWISB), a pre-stressed bi-layered plate- strip under the action of an arbitrary inclined force resting on a rigid foundation is studied. It is assumed that the force applied to upper free surface of the plate-strip is time-harmonic and the materials used are linearly elastic, homogenous and isotropic. By employing FEM modeling the governing system of partial differential equations of motion is approximately solved. The different dependencies of the problem such as the length and initial stress of the materials are numerically investigated. Particularly the effect of angle is analyzed. It is observed that the results obtained according to this angle converge to the ones in the previous studies. Keywords. Fem Modelling, Wave Propagations, initial stress, wave dispersion, layered material, time-harmonic stress field. AMS 2010. 74B10, 74S05.

References

[1] Guz, A. N., Elastic Waves in a Body Initial Stresses, I. General Theory, Naukova Dumka, Kiev, (in Russian) 1986.

[2] Guz, A. N., Elastic Waves in a Body Initial Stresses, II. Propagation Laws, Naukova Dumka, Kiev, (in Russian) 1986.

[3] Guz, A. N., Elastic Waves in a Body with Initial (Residual) Stresses, A.S.K., Kiev, (in Russian) 2004.

[4] Guz, A. N., Elastic Waves in a Body with Initial (Residual) Stresses, International Applied Mechanics, 38, 1, 23-59, 2002.

[5] Akbarov, S. D., Recent investigations on Dynamic Problems for an Elastic Body with Initial (Residual) Stresses (Review), International Applied Mechanics, 43, 12, 1305-1324, 2007.

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On the Solution of Vibration (Eigenvalue) Problem of Ceramic Cylindrical Shells with FG Coatings A. Deniz (1) and A.H. Sofiyev (2) (1) Usak University, Usak, Turkey, [email protected] (2) Suleyman Demirel University, Isparta, Turkey, [email protected]

Abstract. The applications of sandwich structures in aerospace industry, shipbuilding and civil engineering offer numerous advantages related to their low weight-to-strength and weight-to-stiffness ratios and high durability [1,2]. Modern aircraft fuselage design requires better sound and vibration isolation. Optimal design of cylindrical sandwich shells with minimum noise and vibration transmission is a great interest to aerospace industry. The vibration or eigenvalue problem of cylindrical sandwich shells with FG coatings has not been studied thoroughly. In this study, the solution of free vibration or eigenvalue problem of ceramic cylindrical shells with functionally graded (FG) coatings is presented. Functionally graded material (FGM) refers to a heterogeneous composite material with gradient compositional variation of the constituents from one surface of the material to the other which results in continuously varying material properties[1-4]. The fundamental relations and basic equations of ceramic cylindrical shells with FG coatings are derived. By using Galerkin method to the basic equations, the formula for the frequency parameter of the cylindrical shell with FG coatings is obtained. The influences of sandwich shell characteristics and the volume fraction distribution of FG coatings on the frequency parameter are discussed. Keywords: Functionally graded coatings, eigenvalue problem, frequency parameters AMS 2010. 74E30, 34B09, 74H45.

References:

[1] Liew, K.M., Yang, J. and Wu, Y.F. Nonlinear Vibration of a Coating-FGM-Substrate Cylindrical Panel Subjected to a Temperature Gradient. Computer Meth. Appl. Mech. Eng. 195(9-12), 1007-1026, 2006.

[2] Pitakthapanaphong, S., Busso, E.P., Self-Consistent Elasto-Plastic Stress Solutions for Functionally Graded Material Systems Subjected to Thermal Transients, Journal of Mechanics and Physics of Solids, 50, 695–716, 2002.

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[3] Y. Fu, J. Wang, Y. Mao, Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment. Applied Mathematical Modelling, in Press, 2012.

[4] Sofiyev A.H., Deniz A., Akcay İ,H., Yusufoğlu E. The vibration and stability of a three-layered conical shell containing a FGM layer subjected to axial compressive load. Acta Mechanica 183, 129-144, 2006.

83 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Finite Difference/Spectral Approximations for the Time Fractional Higher Order Partial Differential Equations A. Kablan and F. Talay Akyıldız University of Gaziantep, Gaziantep, Turkey

Abtsract. In this report, we consider time fractional higher order partial differential equations which arise from viscoelastic fluid flow. The main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time-fractional diffusion equation. The proposed method is based on a finite difference scheme in time and Legendre spectral methods in space. Stability and convergence of the method are rigorously established. We prove that the full discretization is unconditionally stable. Numerical experiments are carried out to support the theoretical claims. Keywords. Fractional order, Higher order Partial differential equations, Spectral method, Stability analysis. AMS 2010. 26A33, 65N06, 65N30

84 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Anew Metaheuristic Algorithm Based on Chemical Process : Atom Algorithm

Ali Karcı Inonu University, Malatya, Turkey, [email protected]

Abstract. Most of the metaheuristic algorithms are based on the natural processes. Some of them are inspired by physical processes; some of them are inspired by biological processes; some of them are inspired by social processes; some of them are based on chemical processes; some of them are based on Social-Biological processes; some of them are inspired by biological-geography process; some of them are based on music, and finally some of them are hybrid. In this paper, we proposed anew metaheuristic algorithm based on chemical element making compound process. In order to make compound some elements give electrons, and some take electrons. The constructed bond is called ionic bond. In another case, at least two elements use at least one electron in common, and the constructed bond is called covalent bond. Each solution in the proposed algorithm is called atom, and each parameter of solution is called electron. The proposed algorithm can be sketched as follow

Algorithm : Atom Algorithm – A2

1- Create random Atom Set (A0) 2- Compute individual effect of each electron for each atom. 3- i←0 4- Do the following until stopping criteria met

a. Apply ionic operator to Ai. // B←IonicOperator(Ai-1) b. Compute effects of electrons in RCR.

c. Apply covalent operator to Ai. //

Ai+1←CovalentOperator(B) d. Compute objective function value for each atom. e. i←i+1

References

[1] Akay, B., Karaboğa, D.,(2012), “A Modified Artificial Bee Colony Algorithm for Real- Parameter Optimization”, Information Sciences, Article in Press.

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[2] Alataş, B.(2011), ”ACROA: Artificial Chemical Reaction Optimization Algorithm for Global Optimization”, Expert Systems with Applications, 38, 13170-13180.

[3] Baykaşoğlu, A., Özbakır, L., Tapkan, P., (2007), “Artificial bee colony algorithm and its application to generalized assignement problem”, In F.T.S. Chan & M.K. Tiwari (Eds). Chapter 8 of Swarm Intelligence: Focyus on ant and particle swarm optimization, pp. 532, Itech Education and Publishing.

[4] Birbil, S.I., Fang, S.C., (2003), “An electromagnetism-like mechanism for global optimization”, Journal of Global Optimization, 25, 263-282.

[5] Chu, S.C., Tsai, P.W., Pan, J.S., (2006), “Cat swarm optimization”, LNCS, 4099, 854-858.

[6] Erol, O.K., Eksin, İ., (2006), “A new optimization method: Big bang-big crunch”, Advances in Engineering Software, 37, 106-111.

[7] Karci, A., (2007), “Theory of saplings growing-up algorithm”, LNCS, 4431, 450-460.

[8] Kennedy, J., Eberhart, R.C., (1995), “Particle swarm optimization”, In proceedings of IEEE international conference on neural Networks, Australia, 4, 1942-1948.

[9] Lamberti, L., Pappalettere, C., (2007), “Weight optimization of skeletal structures with multi-point simulated annealing”, Computer Modeling in Engineering and Sciences, 18, 183- 221.

[10] Maziezzo, F., Colorni, (1996), “Ant system : optimization by a colony of cooperating agents“, Transactions on Systems, Man and Cybernetics – Part B : Cybernetics,26, 29-41.

[11] Yang, X.S., (2009), “Firefly algorithms for multimodal optimization”, In stochastic algorithms: Foundations and applications, SAGA, 2009. LNCS, 5792, 169-178.

86 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Measurement of the Effect of the Pre Crisis and Post Crisis Economic Dynamics (Parameters) over the Stock Index with the 2-factor Cross Classification Model of Variance Analysis Models and Turkey Example Adnan Mazmanoğlu Istanbul Aydin University, Istanbul, Turkey, [email protected]

Abstract.1The fact that the financial fluctuations is random in economy, which is one of the most important fields of social sciences, makes it difficult for us to make a decision about future. Because, in each case there is an order which is not observed at first glance, and also there is one or more factors which leads this case. During an experiment or research, it is not usually possible to examine all units which form a mass. The mass is infinite or numerous. In this case, by selecting part of the units, a decision and generalizations can be made about the mass. We have thought, with the consideration of Turkish Stock Index which is very popular and in the position of being the engine of national economy, the case of the measurement of the effect which causes the changes on it, with the pre world economic crisis and the post world economic crisis economic Dynamics (parameters). At first glance, we see that a solution must be made with qualitative variables Variance Analysis (V.A.) method. To answer the question of how this can be made, we can say that in variance analysis, like other statistical methods, the decisions can’t be separated from concepts like chance, sampling. We wanted to investigate how crisis period affect (factor) affected the pre crisis Dynamics and levels of GSMH growth rate, the proportion of the monetary shares of the foreigners in stock, central bank nightly interest rate and capital rate of post crisis economic Dynamics affected the stock index. As well as there can be some uninspectable factors beside these four, we want to test which factor is the most effective of the inspectable factors with a certain probability threshold. We will add average rates of 10 observations about 5(five) different average economic indicator rates and 2(two) different circuits to our project. We have thought of applying 2-way Cross Classification model to our example. In the sample matris which will be constituted in our application we considered one observation case in each cell. Because we can examine if there is an interaction between the factor groups when the observation number

87 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

concerning every composition is equal to or greater than 1. Our model which will be formed from qualitative variables won’t have a single solution. If we think this model like 2-Way Linear Model or 2-Way Cross Classification Model, our model will be like y +++= εβαµ ij ji ij (1). Here, if we call first factor A, and second factor B,

α i = Shows the effect of ith level of factor A,

β j = Shows the effect of jth level of factor B.

µ: Shows the general average,

ε ij : Shows the random values (error term)

yij : Shows the effect on the ith row (crisis periods) and jth column (financial dynamics). Here, the a levels of α − factor and b levels of β − factor can be indiced like below: i = 1, 2, 3, ...... , a j = 1, 2, 3, ...... , b In our example, it will be a=2; b = 5. The term (a,b) shows the number of observations in the cells. The matrixial expression of our model in equation (1) will be:

Y = X β + ' XXe )( = X β + ε . The normal equations ′ )( = ′YXbXX or ( ′ )βˆ = ′YXXX which corresponds to these equations, can be written. The βˆ vector of ′XX matrix which consists from the values of the qualitative variables can only be solved with g-inverse. We will try to find the best estimator and the single solution vector from these solutions. Here βˆ is the vector of the unknown with ˆ β ′ = { ,, βαµ cr }, (where r: rows, c: columns). We shall say that it consists from three sub-vectors. That is, in the form of ˆ β ′ = { ,,,,,, ββββααµ 432121 }. Keywords . The analysis of variance, linear models, models not full rank, the 2-way crossed classification, without interaction, factorial designs

88 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Exact Solutions of the Zakharov Equations by Using the First Integral Method Arzu Öğün Ünal Ankara University, Ankara, Turkey, [email protected]

Abstract. Nonlinear evolution equations (such as KdV, Burgers, Bousinesq, etc.) are widely used to describe nonlinear phenomena in physics fields like the fluid mechanics, plasma physics, optics. In recent years various techniques have been developed to obtain exact solutions of nonlinear evolution equations such as Bäcklund transformation method, Painlevé method, inverse scattering method, Hirota's bilinear method, tanh method and the first integral method. The first integral method used in the theory of commutative algebra was first proposed by Feng to solve the Burgers Korteweg-de Vries equation [1]. Recently, many authors has applied this method to various types of nonlinear problems [1-5]. In this work, we use the first integral method to find the exact solutions of the Zakharov equations. Keywords. First integral method; Traveling wave solutions; Zakharov equations. AMS 2010. 35C07, 35Q55.

References

[1] Z.S. Feng, The first-integral method to the Burgers--KdV equation, J. Phys. A 35 (2002) 343--350.

[2] Z.S. Feng, X.H Wang, The first integral method to the two-dimensional Burgers-KdV equation, Phys. Lett. A 308 (2003) 173 - 178.

[3] Ke, Yun-Quan; Yu, Jun, The first integral method to study a class of reaction-diffusion equations, Commun. Theor. Phys. (Beijing) 43 (2005) no. 4 597--600.

[4] A. H. A. Ali,K. R. Raslan, The first integral method for solving a system of nonlinear partial differential equations, Int. J. Nonlinear Sci. 5 (2008) no. 2 111--119.

[5] K. R. Raslan, The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynam. 53 (2008) no. 4 281--286.

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About Existence of the Local and Global on Time Solutions of the Initial and Initial- Boundary Value Problems for one-Dimensional Nonlinear Boltzmann’s Moment System Equations A.Sakabekov (1) and Y.Auzhani (2) (1) Kazakh-British Technical University, Almaty, Kazakhstan, [email protected] (2) Kazakh-British Technical University, Almaty, Kazakhstan, [email protected]

Abstract. Boltzmann’s moment system equations are intermediate between Boltzmann’s equation and hydro-dynamical approximation [1]. Boltzmann’s moment system equations represent not studied class of the nonlinear partial differential equations. Studying the different problems for a Boltzmann’s moment system equations represent important and actual problem of the gas dynamic. We prove the existence and uniqueness of the local solution of the initial and boundary value problems for one-dimensional nonlinear Boltzmann’s moment system equations in arbitrary approximation with Vladimirov-Marshak generalized condition in the space of functions that are continuous on time and square summable on x. As well known [2] for nonlinear Boltzmann’s equation mass conservation law, momentum conservation law, energy conservation law and Boltzmann’s H-theorem take place. The fulfillment of the mass conservation law and Boltzmann’s H-theorem analogue for one-dimensional nonlinear moment system equations of Boltzmann in second approximation is proven. We have proved the existence of the global on time solutions of the Caushy value problem and initial-boundary value problem for one-dimensional nonlinear Boltzmann’s moment system equations in second approximation in the function space that are continuous by time and summable by space variables. Keywords. Boltzmann’s moment system equations.

References

[1] Sakabekov A., Initial and boundary value problem for Boltzmann’s moment system equations, Almaty, Gylym, 2002, 276p.

[2] Cercignani C., Theory and application of Boltzmann’s equation, Milano, Italy, 1975.

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Cubic B-Spline Finite Element Method for the Numerical Solution of Fisher’s Equation Ali Şahin (1) and Halis Bilgil (2) (1) Aksaray University, Aksaray, Turkey, [email protected] (2) Aksaray University, Aksaray, Turkey, [email protected]

Abstract. In this study, we present a B-spline finite element scheme for the numerical solution of Fisher’s equation which is a nonlinear reaction-diffusion equation describing the relation between the diffusion and the nonlinear multiplication of a species. In computations, the solution domain is partitioned into uniform mesh and the cubic B-spline collocation method is used for the numerical purpose. The method yields stable accurate solutions. Obtained results are acceptable and in unison with some earlier studies. Keywords. Fisher’s equation, collocation method, B-spline function AMS 2010. 35K57, 65L60, 65D07

References

[1] Fisher, R.A., The wave of advance of advantageous genes, Ann. Eugen., 7: 355-369, 1936.

[2] Kolmogoroff, A., Petrovsky, I. and Piscounoff, N., E´ tude de l’e´quation de la diffusion avec croisance de la quantite´ de matie´re et son application a´ un proble´me biologique, Bull. Univ, Etat. Moscou Ser. Int. A, 1: 1-25, 1937.

[3] Gazdag, J. and Canosa, J., Numerical solution of Fisher’s equation, J. Appl. Prob., 11: 445-457, 1974.

[4] Prenter, P.M., Splines and Variational Methods, Wiley, New York, NY, 1975.

[5] Tang, S. and Weber, R.O., Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method, J. Austral. Math. Soc. Ser. B, 33: 27-38, 1991.

[6] Carey, G.F. and Shen, Y.,Least-Squares finite element approximation of Fisher’s reaction- diffusion equation, Numer. Methods Partial Differential Eq., 11: 175-186, 1995.

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[7] Qiu, Y. and Sloan, D.M., Numerical solution of Fisher’s equation using amoving mesh method, J. Comput. Phys. 146: 726–746, 1998.

[8] Zhao, S. and Wei, G.W., Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation, SIAM J. Sci. Comput., 25: 127–147, 2003.

[9] Olmos, D. and Shizgal, B.D., A pseudospectral method of solution of Fisher’s equation, J. Comput. Appl. Math., 193: 219–242, 2006.

[10] Dag, I., Sahin, A., Korkmaz, A., Numerical Investigation of the Solution of Fisher’s Equation via the B-Spline Galerkin Method, Numer Methods Partial Differential Eq., 26: 1483–1503, 2010.

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The Problem of Constructing a Step Function, the Least Fluctuating Around a Given Function Bahtiyar Bayraktar Uludag University, Bursa, Turkey, [email protected]

Abstract. It is well known that step functions are often used in regulation and optimal control. Nevertheless, as far as we know, this problem was not raised previously. Therefore, the task of building a step function, that in the integral sense the least fluctuates around given function is the main point of the article. This is a multiextremal task. In particular, such problems mostly arise in the projection of piping systems, namely in problems of choosing of a working schedule for a pumping station. In this paper a mathematical model of the task is clearly formulated. And for the solution of the problem an effective method of shrinking neighborhoods was built. A software algorithm was developed as well. Keywords. calculus of variations, optimal control, step functions, pipelines, pumping stations, regulating tanks.

References [1] K.Vairavamoorthy, M. Ali, Optimal Design of Water Distribution Systems Using Generic Algorithms, Comput.-Aided Civ. Infrastruct. Eng.15, 374-382, 2000.

[2] I.Sârbu, F.Kamlar, Optimization of looped water supply Networks, Periodica Polytechnica Ser. Mech. Eng. 46 (1), 75-90, 2002.

[3] E. Keedwell, Hhu Soon-Thiam, A hybrid genetic algorithm for the design of water distribution networks, Engineering Applications of Artificial Intelligence 18, 461- 472, 2005.

[4] Z. W.Geem,. Harmony Search Optimisation to the Pump-Included Water Distribution Network Design. Civil Engineering and Environmental Systems 26 (3), 211- 221, 2009.

[5] P. F. Boulos, Z.Wu, C. H. Orr, M. Moore, P. Hsiung, D.Thomas. Optimal Pump Operation of Water Distribution Systems Using Genetic Algorithms, http://www.rbfconsulting.com/papers/genetic_algo.pdf, 2008.

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[6] I. Sârbu, Analysis of Looped Water Distribution Networks. 4. international conference on Hyro-Science and Engineering, Seul, Korea (ICHE 2000), 2000.

[7] N.N Abramov. Theory and a design procedure of systems of giving and water distribution. - TH.: Stroyizdat, 1983.

[8] S. N. Karambirov, Perfection of methods of calculation of systems of giving and distribution of water in the conditions of multimodiness and the incomplete initial information: the dissertation... Dr. Sci. Tech.: 05.23.16, 05.23.04. - Moscow, 2005.346 with: silt. RGB ODES, 71 06-5/213.

[9] B. R. Bayraktarov and V. Ch. Kudaev, Problem of Definition of The Regulating Volume of Tanks And Water Towers of Pipeline Systems. Academy of Sciences of Russia, Izvestiya Kabardino – Balkarskogo Nauchnogo Centra RAN № 6 (32), 123-129, 2009

94 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Matrix Analogues of Some Properties for Bessel Functions Bayram Çekim (1) and Abdullah Altın (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected]

Abstract. Some relations for Bessel matrix functions of the first kind for the case where no eigenvalue of a matrix is an negative integer are obtained. Then, two new integral representations of these matrix functions are derived. Furthermore, new connections between Bessel matrix functions and Laguerre matrix polynomials are given. Keywords. Bessel matrix function, Jordan block, Gamma matrix function, Beta matrix function, Laguerre matrix polynomials. AMS 2010. 33C10, 33C45, 15A60.

References

[1] Dunford, N. and Schwartz, J., Linear operators, Interscience, New- York, 1987.

[2] Jódar, L., Company, R. and Navarro, E., Solving explicitly the Bessel matrix differential equation, without increasing problem dimension, Congr. Numer., 92, 261-276, 1993.

[3] Jódar, L., Company, R. and Navarro, E., Bessel matrix functions: explicit solution of coupled Bessel type equations, Util. Math., 46, 29-141, 1994.

[4] Jódar, L., Company, R. and Navarro, E., Laguerre matrix polynomials and systems of second-order differential equations, Appl. Numer. Math., 15, 53-63, 1994.

[5] Jódar, L. and Cortés J. C., Some properties of Gamma and Beta matrix functions, Appl. Math. Lett. 11, 1 , 89-93, 1998.

[6] Rainville, E. D., Special Functions, Chelsea Pub., New- York, 1973.

95 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A General Solution Procedure for Fractionally Damped Beams Duygu Dönmez Demir (1) and Necdet Bildik (2) (1) Celal Bayar University, Manisa, Turkey, [email protected] (2) Celal Bayar University, Manisa, Turkey, [email protected]

Abstract. Linear vibrations of a general model of the beams and rods with the fractional derivatives are considered. The model consists of the arbitrary linear operators with spatial and time derivatives. The equation of motion is solved by the method of multiple scales. We introduce effects on dynamical analysis of the beams and rods of damping term modelled with fractional derivative. The approximate solution gives us the amplitude and the phase modulation equations obtained in terms of the operators. Steady-state solutions and their stability are discussed. The solution procedure is applied to an engineering problem. Keywords. Perturbation method, Fractional derivative, Method of the Multiple Scales. AMS 2010. 35Q74, 65M12.

References [1] Rossikhin, Y. A., Shitikova, M. V., Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results, Applied Mechanics Reviews, 63/010801, 1-52, 2010.

[2] Rossikhin, Y. A., Shitikova, M. V., Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids, Applied Mechanics Reviews, 50(1), 15–67, 1997.

[3] Boyacı, H., Pakdemirli, M., A Comparison of Different Versions of the Method of Multiple Scales for Partial Differential Equations, Journal of Sound and Vibration, 204(4),595-607, 1997.

[4] Nayfeh, A. H., Introduction to Perturbation Techniques, A Wiley Interscience, John Wiley & Sons , New York, 1981.

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Generalizations of Hölder’s Inequalities on Time Scales Deniz Uçar (1) and Ali Deniz (2) (1) Usak University, Usak, Turkey, [email protected] (2) Usak University, Usak, Turkey, [email protected]

Abstract. Hölder's inequalities and their extensions have received considerable attention in the theory of differential and difference equations. In this paper we establish some new generalizations and refinements of Hölder’s inequality and some related inequalities on time scales. We also show that many existing inequalities related to the Hölder’s inequality are special cases of the inequalities presented on time scales. Keywords. Hölder’s inequalities, time scales, integral inequalities. AMS 2012. 26B25, 26D15, 26E70.

References [1] Qiang, H., Hu, Z., Generalizations of Hölder’s and some related inequalities, Comp. and Math. with Appl., 61, 392-396, 2011.

[2] Yang, X., A generalization of Hölder inequality, J. Math. Anal. Appl. 196, 328-330, 2000.

[3] Masjed-Jamei, M., A functional generalization of the Cauchy-Schwarz inequality and some subclasses, Appl. Math. Lett. 22, 1335-1339, 2009.

[4] Callebaut, D.K., Generalization of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 12, 491-494, 1965.

[5] Bohner, M. and Peterson, A., Dynamic equations on time scales, An introduction with applications, Birkhauser, Boston, 2001.

[6] Wu, S., Debnanth, L., Generalizations of Aczel’s inequality and Popoviciu’s inequality, Indian J. Pure Appl. Math. 36 (2), 49-62, 2005.

[7] Wu, S., A new sharpened and generalized version of Hölder’s inequality and its applications, Appl. Math. Comput. 197, 708-714, 2008.

[8] Agarwall, R., Bohner, M., Peterson, A., Inequalities on time scales: A survey, Math. Inequalities and Applications, 4, 4, 535-557, 2001.

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Some Specials Results and Applications of Pseudo – Measure Theory Dhurata Valera University “Aleksandër Xhuvani”, Elbasan, Albania, [email protected]

Abstract. Many authors have presented the advantages of the pseudo – analysis and ëe ëant to stress here some of them. Pseudo- analysis is based on the semiring structure on the real interval [a,b] [ , + ] and the pseudo – operations extend to the whole extended real line [ , + ]. There⊂ −∞ are considered∞ extensions of operations and non-commutative and non−∞-associative∞ cases. The obtained results are applied ⨁through⨀ the pseudo-linear superposition principle on some nonlinear partial differential equations. Are presented some parts of mathematicals analysis in analogy with the classical mathematical analysis as measure theory, integration, integral operators, convolution, Laplase transform, etc. There are also many applications in fields as nonlinear differential and difference equations, economy, game theory etc..Some basic properties of the pseudo-integral of set-valued functions have been shown. Further research of this combination of set-valued functions theory and pseudo- analysis will be directed to the problems of convergence for sequences of set-valued pseudo- integrals and possible applications. The results serve as a basis for the further investigations of fuzzy measure and theory and integrals. Keywords. Pseudo- analysis, Pseudo- additive measure, Pseudo – additions, Pseudo- multiplications, Pseudo-integral, Set-valued functions. AMS 2010. 53A40, 20M15.

References

[1] Pap, E., Pseudo-analysis and its applications, Tatra Mountains Math. Publ. 12 (1997),1- 12.

[2] Benvenuti,P., Mesiar, R., and Vivona, D., Monotone Set Functions-Based Integrals, in: E. Pap (Ed.), Handbook of Measure Theory, Volume II.Elsevier, Amsterdam, 2002, pp. 1329- 1379.

[3] Klement, E.,P., Mesiar, R., and Pap, E., Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.

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[4] Pap, E., Pseudo-analysis approach to nonlinear partial differential equations , Acta Polytechnica Hungarica 5 (2008), 31-45.

[5] Zhang, D., Guo, C., Integrals of set-valued functions for -decomposable measures,

Fuzzy Sets and Systems 78 (1996) 341-346. ⨁

[6] Ichihashi, H.-Tanka, H.- Asai, K.: Fuzzy integrals based on Pseudo – additions and multiplications, J.Math. Anal. Appl. 130 (1988), 354-364

[7] Pap, E., Vivona, D., Non-commutative and non-associative pseudo-analysis and its applications on nonlinear partial differential equations, J.Math. Anal. Appl. 246/2 (2000) 390-408.

[8] Pap, E., Pseudo-additive measures and their applications, in: E. Pap (Ed.), Handbook of Measure Theory, Volume II. Elsevier, Amsterdam, 2002,pp. 1403-1465.

99 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Numerical Solution For Two-Point Fuzzy Boundary Value Problem Emine Can (1), Canan Köroğlu (2) and Afet Golayoğlu Fatullayev (3) (1) Kocaeli University, Kocaeli, Turkey , [email protected] (2) Hacettepe University, Ankara, Turkey, [email protected] (3) Baskent University, Ankara, Turkey, [email protected]

Abstract. In this work we solve numerically two-point boundary value problem for second order fuzzy differential equation in the form  ′′ = ′ tytytGty ))(),(,()(  (1)  lyy )(,)0( == λγ =∈ α = α = λλλγγγ where lTt ],0[ and α α αα ],,[][],,[][ are fuzzy numbers. The problem (1) is reformulated to four different system of crisp boundary value problems, when the fuzzy derivative is considered as a generalization of the H- derivative [1]. So, problem (1) has four different solutions. Numerical solution of each system is obtained by applying the shooting method. Numerical examples to illustrate the method are presented. Keywords. Boundary value problem, Second order fuzzy differential equations, generalized differentiability. AMS. 65L10, 34A07, 65L12.

References

[1] Y.Cholco-Cano, H. Roman-Flores, On new solution of fuzzy differential equations, Chaos, Solutions & Fractals 38, 112-119, 2008.

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Modal Characteristics of Optical Waveguides with Cardioid-Shaped Cross Sections Emre Eroğlu (1) and Hayriye Altural (2) (1) Kirklareli University, Kirklareli, Turkey, [email protected] (2) Kirklareli University, Kirklareli, Turkey, [email protected]

Abstract. Optical waveguides have important role in optical telecommunications and integrated optic devices and thus optical waveguide research has made tremendous advances during the past four decades [1]. The basic building blocks of these devices that can switch, modulate, combine, multiplex and demultiplex optical signals in realizing optical systems are integrated optical waveguides [2]. Optical waveguides with circular symmetry are very common and widely applied in photonic devices [3]. In this study, a simple analytical method [4] that consists of solution of Helmholtz equation is used to analyze wave propagation in cardioid-core optical waveguides. Keywords. Cardioid-shaped waveguide, partial differential equation, dispersion curves. AMS 2010. 42.82.Et, . 02.30.Jr, 42.50.Nn

References

[1] Kumar, N., Ojha, S.P., Toward modal dispersion characteristics of a new unconventional optical waveguide with a core cross-section of plano-concave lens shape, Optics, in press.

[2] Du, C-H., Chiou, Y-P., Higher-order full-vectorial finite-difference analysis of waveguiding structures with circular symmetry, IEEE Photonic Technology Letters, 24, 894- 896, 2012.

[3] Altural, H., Study of effective index method for wave propagation in integrated optic devices (in Turkish), Erciyes University, Graduate School of Natural and Applied Sciences M.S. Thesis, 2005.

[4] Pipes, L.A., Harvill, L.R., Applied mathematics for engineers and physicists, McGraw- Hill, New York, 1970.

101 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Approximate Solution of a Model Describing Biological Species Living Together by Taylor Collocation Method Elçin Gökmen (1) and Mehmet Sezer (2) (1) Mugla Sitki Kocman University, Mugla, Turkey, [email protected] (2) Mugla Sitki Kocman University, Mugla, Turkey, [email protected]

Abstract. In this study, a numerical method is presented to obtain approximate solutions for the system of nonlinear integro-differential equations derived from considering biological species living together. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points [4-6]. Also, to illustrate the pertinent features of the method examples are presented and results are compared with the other methods [1-3]. All numerical computations have been performed on the computer algebraic system Maple 9 Keywords. System of nonlinear integro-differential equations, Taylor polynomials and series, Collocation points, Biological species.

References

[1] Yousefi,S. A., Numerical Solution of a Model Describing Biological Species Living Together by Using Legendre Multiwavelet Method, International Journal of Nonlinear Science, 11, 109-113, 2011.

[2] Babolian E., Biazar J., Solving the problem of biological species living together by Adomian decomposition method, Applied Mathematics and Computation, 129, 339-343, 2002.

[3] Shakeri F., Dehghan M., Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling, 48, 685-699, 2008.

[4] Yalçınbaş S., Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations, Applied Mathematics and Computation, 127, 195-206, 2002.

[5] Gülsu M., Sezer M., Taylor collocation method for solution of systems of high-order linear Fredholm- Volterra integro-differential equations, 83, 429-448, 2006.

[6] Bülbül B., Gülsu M., Sezer M., A new Taylor collocation method for non-linear Fredholm-Volterra integro-differential equations, 26, 1006-1020, 2010.

102 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Laguerre-Krall Orthogonal Polynomials: Outer Relative Asymptotics and an Electrostatic Model for Their Zeros E.J. Huertas (1), F. Marcellán (2) and H. Pijeira (3) (1) University Carlos III of Madrid, Madrid, Spain, [email protected] (2) University Carlos III of Madrid, Madrid, Spain, [email protected] (3) University Carlos III of Madrid, Madrid, Spain, [email protected]

Abstract. Our contribution is based on the work [4], and it generalizes some results in [1], [2] and [3]. We study the outer relative asymptotics and the behavior of the zeros of the sequences of orthogonal polynomials associated with the Laguerre measure perturbed by the simultaneous addition of m mass points located in the negative real semiaxis. The outline of the talk is the following. First, we introduce the representation of the perturbed orthogonal polynomial sequences in terms of the classical ones and we analyze the outer relative asymptotics for these Laguerre-Krall polynomials. Second, we provide an electrostatic model for the zeros of these polynomials as equilibrium points in a logarithmic potential interaction under the action of an external field. Keywords. Orthogonal polynomials, outer relative asymptotics, electrostatic interpretation. AMS 2010. 33C45, 33C47.

References

[1] H. Dueñas, E. J. Huertas, and F. Marcellán, Analytic properties of Laguerre-type orthogonal polynomials, Integral Transforms Spec. Funct. 22 (2012), 107-122.

[2] B. Xh. Fejzullahu and R. Xh. Zejnullahu, Orthogonal polynomials with respect to the Laguerre measure perturbed by the canonical transformations, Integral Transforms Spec. Funct. 17 (2010), 569–580.

[3] E. J. Huertas, F. Marcellán, and F. R. Rafaeli, Zeros of orthogonal polynomials generated by canonical perturbations of measures, Appl. Math. Comput. 218 (2012), 7109–7127.

[4] E. J. Huertas, F. Marcellán, and H. Pijeira, An Electrostatic Model for Zeros of Laguerre Polynomials, submitted to the Proceedings of the American Mathematical Society.

103 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the Inverse Problems for Singular Differential Operators Etibar S. Panakhov (1) and Murat Sat (2) (1) Firat University, Elazig, Turkey, [email protected] (2) Erzincan University, Erzincan, Turkey, [email protected]

Abstract. In this study, it will be talked about Hochstadt theorem, Hochstadt- Lieberman theorem and inverse problem according to two spectrum for singular differential operators such as hydrogen atom equation, Sturm-Liouville operator with Coulomb potential and Bessel operator. Keywords. Spectrum, Coulomb potential, operator. AMS 2010. 34A55, 34B24.

References

[1] Hochstadt, H., The inverse Sturm-Liouville problem, Comm. Pure Appl. Math., 26, 715- 729, 1973.

[2] Hochstadt, H., Lieberman B., An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34, 676-680, 1978.

[3] Panakhov, E.S., The definition of differential operator with peculiarity in zero on two spectrum, VINITY, 4407-80, 1-16, 1980.

104 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A problem with Mixed Boundary Conditions for an Elastic Strip When There are Friction Forces in the Contact Area Elçin Yusufoğlu (1) and Hüseyin Oğuz (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Dumlupinar University, Kutahya, Turkey, [email protected]

Abstract. The contact interaction between the absolutely rigid punch and the elastic strip is investigated. By using the Fourier integral transform, the problem is reduced to a Cauchy type singular integral equation with unknown pressure at the contact area. The solution of last equation is built by the method of discrete vortices. The effects of geometrical and mechanical parameters of the materials on various subjects of interest are discussed and shown graphically and tabular form. Keywords. Fourier Intregral Transform, Singular Integral Equation, Method of Discrete Vortices AMS 2010. 45E05, 41A55.

References

[1] N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leiden, 1953.

[2] Xiaoqing Jin, Leon M. Keer, Qian Wang, A practical method for singular integral equations of the second kind, Engineering Fracture Mechanics 75, 1005-1014, 2008.

[3] Belocerkovskiy S.M. and Lifanov I.K., Numerical methods in Singular integral Equations, Nauka, Moskow,1985.

[4] Ufliand I. S. Integral Transforms in the Problem of the Theory of Elastisity, Leningrad, Nauka, 1967.

[5] Aleksandrov, V. M. and Kovalenko, Ye. V., Problems of Continuum Mechanics with Mixed Boundary Conditions, Nauka, Moscow, 1986.

[6] Chebakov M. I. Asymptotic solution of contact problems for a relatively thick elastic layer when there are friction forces in the contact area, Journal of Applied Mathematics and Mechanics 69, 296-304, 2005.

105 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Numerical Solution of a Crack Problem with the help of Gauss Quadrature Formulas Elçin Yusufoğlu (1) and İlkem Turhan (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Dumlupinar University, Kutahya, Turkey, [email protected]

Abstract. In this paper, a problem of a symmetrically located crack in an orthotropic strip is studied under plane strain conditions. The solution is obtained by reducing the problem to Cauchy type singular integral equation and by using Gauss Quadrature formula. The effect of relative thickness and mechanical properties of strip on stress intensity factors of Mode I is examined under different loading conditions. The results are compared with the results obtained other methods. Keywords. Gauss Quadrature, crack, theory of elasticity. AMS 2010. 41A55, 45E05.

References [1] N.I. Muskheleshvili, Singular Integral Equations, Edited by J.R.M. Rodok, Noordhoff International publishing Leyden, 1997. [2] Aleksandrov V.M., Smetanin B.I., Sobol B.V.: Thin Stress Concentrators in Elastic Solids, Nauka, Moskow, 1993. [3] Aleksandrov V.M.: Two problems with mixed boundary conditions for an elastic orthotropic strip. J. Appl. Math. Mech. 70: 128-138, 2006. [4] Erbaş B, Yusufoğlu E, Kaplunov J. A plane contact problem for an elastic orthotropic strip. J. Engrg. Math. 70: 399-409, 2011. [5] Xiaoqing J., Leon M.K., Qian W., A practial method for singular integral equations of the second kind, Engineering Fracture Mechanics, 75, 1005-1014, 2008.

106 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Method for the Solution of the Optimal Control Problem for the Linear Descriptor Systems∗ F.A. Aliev (1), N.I. Velieva (1), Y.S. Gasimov (1) and L.F. Agamalieva (1) (1) Baku State University, Baku, Azerbaijan, [email protected], [email protected], [email protected]

Abstract. A numerical algorithm is proposed for the solution of the linear quadratic optimal control regulator problem, when movement of the object is described by the descriptor system. On the base of Levin-Athans method [1] algorithm is shown that the proposed algorithm requares solution of two generalized algebraic Lyapunov equation that is solved by signium function and orthogonal projection methods. Initial value is chosen from the asymptotical stability condition of the closed-loop system. The obtained results are illustrated by examples. Let the movement of a system be described by the system of linear differential equations with constant coefficients Ex(t) Fx(t)+= Gu(t); Ex( 0 = X) 0 (1) y = Cx(t) It needs to minimize the functional ∞ = ∫(xJ ′Qx + ′ )dtRuu (2) 0 subject to tu = Ky t)()( . (3) Here tx )( is a coordinate vector, ()ty − vector of measurements, tu )( – control vector, = QQCGFE ′ ≥ ,0,,,, = RR ′ > 0 are constant matrices. This problem is reduced to the solution of two generalized Lyapunov equations that is solved by the signium function and orthogonal projection methods. Keywords. iterative method optimal control problem discrete-time systems AMS 2010. 15A24.

References [1] Levine, W.S., Athans M. On the determination of the optimal constant output feedback gains for linear multivariable systems. IEEE Trans. Autom. Control, V.AC-15, N1, 44-48, 1970.

∗ The work was supported by Science Development Foundation of the Republic of Azerbaijan № EIF-2011- 1(3)-82/25/1

107 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Iterative Algorithms for the Solution of the Discrete Optimal Regulator Problem in Stationary and Periodic Cases∗ F.A. Aliev (1), N.A. Safarova (1) and N.I.Velieva (1) (1) BSU, Baku, Azerbaijan, [email protected], [email protected], [email protected]

Abstract. In the work, the discrete stationary and periodic optimal regulator problems are considered. Iterative algorithms are proposed to the solution of these problems. Let the movement of the system be described by the following equations

Γ+Ψ=+ )0(),()()1( 0 ixxiuixix == ,...,2,1,0 (1) and it needs to find a feedback chain ()()= iKxiu (2) that gives minimum to the functional ∞ =∑( ′ ixJ Qx + ′ iui Ru i ),)()()()( (3) i=0 where all matrices have proper dimensions. There exist various methods [1, 2] to find the solution of corresponding algebraic Riccati and Lyapunov equations. The advantage of the proposed here algorithm is that it doesn’t solve these equations in each step. We solve the Riccati equation once and find the initial regulator. Using this we solve once the Lyapunov equation. Using them we develop an iterative algorithm that continues while the exactness criteria will not satisfied. The results are illustrated by examples. Keywords.Iterative algorithms, optimal regulator problem, Lyapunov equation. AMS 2010. 93B40, 93B52.

References

[1] Aliev, F.A.. Solution Methods for the Applied Optimization Problems for the Dynamic Systems.Elm, Baku, 1989 (in Russian).

[2] Bordyug, B.A., Larin, V.B., Timoshenko, A.G..The Control Problem with Biped Apparatus.NaukaDumka, Kiev, 1985 (in Russian).

∗The work is supported by Science Development Foundation of the Republic of Azerbaijan Grant N. EIF-2011-1(3-82)/25/1-M-29.

108 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Polynomial Presentation of the Binary Multidimensional Nonlinear Modular Dynamic Systems F. G. Feyziev (1), G.H. Mammadova (2) and Sh.T. Mammadov (3) (1) Sumgait State University, Sumgait, Azerbaijan, [email protected], (2) Baku State University, Baku, Azerbaijan, [email protected] (3) Azerbaijan National Academy of Sciences, Baku

Abstract. The analytical presentation of binary multidimensional nonlinear modular dynamic systems (BMNMDS) with fixed memory n0 given by the following functional relationship is considered in the work.

= ][{][ 0 ≤≤− nmnnmxGny },GF ),2( (1)

r k где nx ∈GF (),2][ ny ∈GF )2(][ , {...} = ( 1 2 GGGG k {...},...,{...}{...}, ).

GF  GF ××= GF )2(...)2()2( .  

Theorem. Let n0 be a fixed memory of the vector-functional ][{ 0 ≤≤− nmnnmxG }. Then BMNMDS (1) can be represented by the following two-dimensional analogy of Volterra polynomial.

()0 +1 rn η = − β α = α ny ][ ∑ ∑ ∑ ,ηα nh 1 ][ ∏ ∏  nnu 1  )],,([ GF k,,...,1),2( (2) i=0 η Φ∈() i1 Γ∈ () in ,η ∈Q()η β =1 where ηα nh 1, ∈{}1,0][ , 1 ()in ηη Φ∈Γ∈ ()i ,,, (0 +∈ 1,...,1 ){}rni ,

Φ i == ()1,...,{)( r 1 r i ηηηηηη j 0 =+∈=++ ,...,1)},1(,...,0{,... rjn },

Q ααη ∈= ,...,1{{)( r}, ηα ≠ 0},

Γ i η 1 =×= 1 nnn 1  η n1  1  η ),(...)1,(0)),(),...,1,(()({),( ≤<<≤ nn 0}. ∈Q()η The problem of finding the unknown coefficients of the polynomial (2) by known input and output sequences of this polynomial is also considered. Keywords. Volterra polynomial, multidimensional nonlinear modular dynamic systems. AMS 2010. 37Nxx

109 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Kantorovich Type of Szasz Operators Including Brenke Type Polynomials Fatma Taşdelen (1), Rabia Aktaş (2) and Abdullah Altın (3) (1) Ankara University, Ankara, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected] (3) Ankara University, Ankara, Turkey, [email protected]

Abstract. In this paper, we give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity and Peetre's K-functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya type result is given for these operators including Gould-Hopper polynomials. Keywords. Szasz operator, Modulus of continuity, Rate of convergence, Brenke type polynomials, Gould-Hopper polynomials, Voronovskaya type theorem AMS 2010. 41A25, 41A36.

References

[1] Butzer, P. L., On the extensions of Bernstein polynomials to the infinite interval, Proc. Amer. Soc., 5, 547-553, 1954.

[2] Ditzian, Z., Totik, V., Moduli of Smoothness, Springer-Verlag, New York, 1987.

[3] Gould, H. W., Hopper, A. T., Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29, 51-63, 1962.

[4] Ismail, M.E.H., On a generalization of Szász operators, Mathematica (Cluj), 39, 259-267, 1974.

[5] Jakimovski, A., Leviatan, D., Generalized Szasz operators for the approximation in the infinite interval, Mathematica (Cluj) 11, 97-103, 1969.

[6] Szasz, O., Generalization of S. Bernstein's polynomials to n the infinite interval, J. Research Nat. Bur. Standards, 45, 239-245, 1950.

[7] Varma, S., Sucu, S., İçöz, G., Generalization of Szasz operators involving Brenke type polynomials, Comput. Math. Appl., 64 (2), 121-127, 2012.

110 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Energy Levels, Binding Energies, and Dipole Moment in a GaAs/AlAs Spherical Quantum Dot as Function of the Donor Position H. Akbas (1), P. Bulut (2) and C. Dane (3) (1) Trakya University, Edirne, Turkey, [email protected] (2) Namik Kemal University , Tekirdağ, Turkey, [email protected] (3) Trakya University, Edirne, Turkey, [email protected]

Abstract. Off-center impurity in a spherical quantum dot(SQD) are theoretically studied by the variational method within the effective mass approximation. The energy levels, binding energies [1], and electric dipole moment of the transition from 1s state to the 2p state [2], [3] are computed for a GaAs/AlAs SQD as function of the donor position. Such theoretical studies will lead to better understanding of the properties of SQDs. Keywords. Quantum dot, donor impurity, dipole moment.

References

[1] Chuu D.S.,C.M.Hsaio, and Mei W.N., Hydrogenic impurity states in quantum dots and quantum wires., Phys. Rev., 46,3898-3904, 1992.

[2] Yilmaz S., Şafak H., Oscillator strengths for the intersubband transitions in a CdS-SiO2 quantum dot with hydrogenic impurity, Physica E,36, 40-44, 2007.

[3] Boichuk V.I.,Bilynskyi I.V, Leshko R.Y., Turyanska L.M.,The effect of the polarization charges on the optical properties of a spherical quantum dot with an off-central hydrogenic impurity., Physica E,44, 476-482, 2011.

111 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Steady Stokes Flow in a Sectorial Cavity: Lids Moving in the Same Direction Halis Bilgil (1) and Ali Şahin (2) (1) Aksaray University, Aksaray, Turkey, [email protected] (2) Aksaray University, Aksaray, Turkey, [email protected]

Abstract. This work analyses the Stokes flow generated in a sectorial driven cavity formed a pair of curved stationary side-walls capped by straight translating lids. The flow is governed by two physical control parameters: the cavity aspect ratio,

(where and are the radii of the inner and outer curved side-walls, respectively) and the ratio of the upper to the lower lid speed. A boundary value problem is formulated, which is solved analytically for the streamfunction, ψ, expressed as an infinite series of Papkovich– Faddle eigenfunctions. The solution of streamfunction is then expanded about any stagnation point to reveal changes in the local flow structure as A and S are varied. At critical values of A and S stagnation point bifurcations arise and the local flow topology is transformed. For the case of lids moving in the same direction (i.e. ), the various flow transformations are tracked as A is decreased and hence the means is identified by which new eddies appear and become fully developed. The key results are shown in an (S, A) control space diagram. It is shown that for , the number of eddies increases from 2 to 4 via the development of corner eddies. Keywords. Stokes flow, eddy, eigenfunction, flow structure, bifurcation, stagnation point, biorthogonality AMS 2010. 35Q30, 76D05, 76D07

References

[1] Khuri, S. A., Biorthogonal series solution of stokes flowproblems in sectorial regions. Siam.,J. Appl. Math. 56, (1), 19–39, 1996.

112 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[2] Gürcan, F., Gaskell, P. H., Savage, M. D. and M. Wilson, Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Proc. of The˙Instn. Mech. Eng. Part-C: J. Mec. Eng. Sci, Vol.217, No:3, 353-364, 2003.

[3] Gaskell, P. H.,Savage, M. D., Wilson, M., Flow structures in a half-filled annulus between rotating co-axial cylinders. Journal of FluidMechanics 337, 263–282, 1997.

[4] Bilgil, H., Bifurcations of Solution Actions of 2-D Navier Stokes Equation in The Sectorial Cavities, PhD Thesis, University of Erciyes, 2011.

113 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Crack Identification Method for Bridge Type Structures Under Vehicular Load Using Wavelet Transform and Particle Swarm Optimization Hakan Gökdağ Bursa Technical University, Bursa, Turkey, [email protected]

Abstract. In this work a crack identification method for bridge structures carrying moving vehicle is proposed. The bridge is modeled as an Euler-Bernoulli beam, and open cracks are assumed to exist on several points of the beam. Half car model is adopted for the vehicle. Then, coupled equations of the beam-vehicle system [1] are solved using Newmark Beta method, so that dynamic responses of the beam are obtained. Using these and the measured displacements, an objective function is derived. The unknown parameters, i.e. crack locations and depths, are determined by solving this optimization problem. To this end, a robust evolutionary algorithm, i.e. the particle swarm optimization (PSO) [2], was employed. To enhance the performance of the method, the measured displacements are denoised using wavelet transform (WT) [3]. It was observed that by the proposed method it is possible to determine small cracks with depth ratio of 0.1 [4] even if 5% noise interference. Keywords. Moving vehicle, damage detection, particle swarm optimization, wavelet transform.

References

[1] Nguyen, K. V., Tran, H. T., Multi-cracks detection of a beam-like structure based on the on-vehicle vibration signal and wavelet analysis, J. of Sound and Vib., 329, 4455-4465, 2010.

[2] Trelea, I.C., The particle swarm optimization algorithm: convergence analysis and parameter selection, Information Processing Letters, 85, 317-325, 2003.

[3] Gökdağ, H., Wavelet-based damage detection method for a beam-type structure carrying moving mass, Structural Engineering and Mechanics, 38(1), 81-97, 2011.

[4] Zhu, X.Q., Law, S.S., Wavelet-based crack identification of bridge beam from operational deflection time history, Int. J. of Solids and Structures, 43, 2299-2317, 2006.

114 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Green's Functions for Elasto-hydrodynamic Model of 1D Quasicrystals Hakan K. Akmaz Cankiri Karatekin University, Cankiri, Turkey, [email protected]

Abstract. The Green's functions for dynamic system of elasticity for 1D quasicrystals is considered [1]. In literature, there are different arguments for dynamic models of quasicrystals. In this study, the elasto-hydrodynamic model based on the argument of Lubensky et al is considered [2]. According to this model, the phonon fields describe wave propagation and phason fields are diffusive. A method based on integral transformations is suggested for the construction of Green's functions. Keywords. 1D Quasicrystals, elasto-hydrodynamics, Green's function AMS 2010. 52C23, 35E05

References [1] Fan, T.Y., Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Science Press, Beijing and Springer-Verlag, Berlin Heidelberg, 2011.

[2] Lubensky T.C., Ramaswamy S., Toner J., Hydrodynamics of icosahedral quasicrystals, Phys. Rev. B, 32(11), 7444–7452, 1985.

115 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

An Efficient Scheme for designing Substitution-boxes based on TD-ERCS Chaotic Sequence Iqtadar Hussain and Muhammad Asif Gondal (1) National University of Computer and Emerging Sciences Islamabad, Pakistan, [email protected] (2) National University of Computer and Emerging Sciences Islamabad, Pakistan, [email protected]

Abstract. You A substitution box (S-box) plays a central role in cryptographic algorithms. In this paper, an efficient method for designing S-boxes based on chaotic maps is proposed. The proposed method is based on the TD-ERC chaotic maps. The S-box so constructed has very optimal nonlinearity, bit independence criterion, strict avalanche criterion, differential and linear approximation probabilities. The proposed S-box is more secure against differential and linear cryptanalysis compared to recently proposed chaotic S- boxes.

Keywords. S-box, Chaos, LP, DP, Nonlinearity, BIC . References [1] J. Daemen and V. Rijmen. The Design of Rijndael- AES: The Advanced Encryption Standard, Springer-Verlag, 2002.

[2] E. Biham and A. Shamir, “Differential Cryptanalysis of DES-like Cryptosystems,” Journal of Cryptology, vol. 4, no. 1, 1991, pp. 3-72.

[3] M. Matsui, “Linear Cryptanalysis Method of DES Cipher,” Advances in Cryptology, Proc. Eurocrypt’93, LNCS 765, 1994, pp. 386-397.

[4] K. Nyberg, “Differentially Uniform Mappings for Cryptography,” Advances in Cryptology, Proc. Eurocrypt’93, LNCS 765, 1994, pp. 55-64.

[5] G. Jakimoski and L. Kocarev, “Chaos and Cryptography: Block Encryption Ciphers Based on Chaotic Maps,” IEEE Trans. Circuits Syst. I, vol. 48, no. 2, 2001, pp. 163-169.

[6] S. Li, G. Chen, and X. Mou, “On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps,” Int’l Journal of Bifurcation and Chaos, vol. 15, no. 10, 2005, pp. 3119-3151.

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[7] H-O. Peitegen, H. Jürgens, and D. Saupe, Chaos and Fractals-New Frontiers of Science, second Edition, Springer-Verlag, 2004.

[8] G. Alvarez and S. Li, “Some Basic Cryptographic Requirements for Chaos-Based cryptosystems,” Int’l Journal of Bifurcation and Chaos, vol. 16, no. 8, 2006, pp. 2129-2151.

[9] G. Chen, Y. Chen, and X. Liao, “An Extended Method for Obtaining S-Boxes Based on Three-Dimensional Chaotic Baker Maps,” Chaos, Solitons and Fractals, vol. 31, 2007, pp. 571-579.

[10] G. Tang, X. Liao, Y. Chen, "A novel method for designing S-boxes based on chaotic maps," Chaos, Solitons and Fractals, vol. 23, 2005, pp. 413-419.

[11] M. Asim, V. Jeoti, " Efficient and Simple Method for Designing Chaotic S-Boxes," ETRI Journal, vol. 30, 2008, pp. 170-172.

[12] T. Kohda, A. Tsuneda, "Statistics of chaotic binary sequences," IEEE Trans Inform Theory, vol. 43, 1997, pp.104–120.

[13] L. Sheng, K. Sun, C. Li, "Study of a discrete chaotic system based on tangent-delay for elliptic reflecting cavity and its properties," ACTA PHYSICA SINICA, vol. 53, 2004, pp.2871-2876.

[14] L. Sheng, L, L. Cao, K. Sun, "Pseudo-random number generator based on TD-ERCS chaos and its statistic characteristics analysis," ACTA PHYSICA SINICA, vol. 54, 2005, pp.4031-4037.

117 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Image Analysis with Differential Geometry Applications Kazım Hanbay (1) and M. Fatih Talu (2) (1) Bingol University, Bingol, Turkey, [email protected] (2) Inonu University, Malatya, Turkey, [email protected]

Abstract. Differential geometry is an area that defines and analyzes the shape using the derivative calculations. The biggest admission in differential geometry is that the differential of the surface of curve or any shape can be calculated in any point. Especially, there is no global coordinate system in the differential geometry calculations of 3-d shapes. All calculations are based on the tangent space. For images, a coordinate system that defines image density can be used. However, differential geometry calculations of a function analysis and a data having multiple dimensions are different. At this point, calculation formats are different but concepts are the same.

In this work, using differential geometry calculus, the clarification of basic information in the image and emphasize important features in new image obtained are aimed. Maximum, minimum and saddle points of a function as mathematicly is determined by looking at its first and second order conditions. If the curve, surface or any shape have non-linear structure and first order condition does not provide any information about data, the Hessian matrix can be used at this point. So the Hessian matrix has been used in optimization problems that have large scale and non-linear structure inside methods with Newton type [1]. The Hessian matrix is a square and symetric matrix, and consists of second-order partial derivatives of the function. The information about maximum, minimum and saddle points of the function have been obtained by looking the minor determinant of Hessian matrix. To obtain more meaningful information in revealing of fundamental directions in data, the eigen vector and values of Hessian matrix have been calculated. So the eigen vector and values of Hessian matrix represent gradient direction of curve [2]. Therefore, the eigen vector and values of Hessian matrix is named as fundamental directions and these vectors are perpendicular to each other. If these facts known as mathematically are applied to an input image formed as a matrix, an output matrix that are able to represent better fundamental aspects of image can be obtained [3].

118 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

In this study, highlighting of fundamental and important features in gray and color images by analyzing their Hessian and eigen information is aimed. Firstly, calculating Hessian matrix of image, then calculating eigen matrix of Hessian, finally analysing eigen matrix by using differential geometry, image enhancement and highlighting important pixel groups are provided.

References

[1] Gill, Phillip E. and Walter Murray. 1974. “Newton-Type Methods for Unconstrained and Linearly Constrained Optimization.” Mathematical Programming 7:311–50.

[2] W. Murray, Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions, Journal of Optimization Theory and Applications Volume 7, Number 3, 189-196, 1971.

[3] Ruan Lakemond · Sridha Sridharan · Clinton Fookes, Hessian-Based Affine Adaptation of Salient Local Image Features, J Math Imaging Vision, pages: 44:150–167, 2012.

119 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Transferring of Energy via Special Functions; A Special Case: “Airy Function” Kevser Köklü (1), Emre Eroğlu (2) and Nevra Eren (3) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Kirklareli University, Kirklareli, Turkey, [email protected] (3) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. In this study, a problem for electromagnetic fields produced by a time dependent source function in a waveguide with perfect electric conductor surfaces is considered by a direct analytical time-domain method called Evolutionary Approach to Electromagnetics (EAE)[1]. The previous works are revisited for energy and surplus of energy of time-domain modes in the waveguides [2]. A complete set of TE and TM waveguide modes is obtained in time-domain, directly. Every field component of the modes is product of two factors: First one is a vector function of transverse waveguide coordinates which corresponds to a modal basis problem. It is specified via well studied Dirichlet and Neumann boundary eigenvalue problems. Physically, these vector functions are distributions of the modal force lines in the waveguide cross-section. The second one is a scalar function corresponds to a time-dependent modal amplitude problem. This is obtained as the solution of Klein-Gordon equation depend on the waveguide’s longitudinal coordinate and time [3]. Consequently, the problem of time-domain signal propagation in the waveguide is solved analytically in compliance with a causality principle. The graphical results are shown for the cases when the energy for the waveguide time-domain waveguide modes are represented via the Airy function [4]. Keywords. Time domain waveguide mode; Klein-Gordon equation; Airy function; Energy.

References [1] Tretyakov, O.A., Evolutionary waveguide equations, Sov.J.Comm.Tech.Electron. Vol. 35, No. 2, 7-17, 1990. [2] Eroglu, E., Aksoy, S. and Tretyakov, O.A., Surplus of energy for time-domain waveguide modes, Energy Education Science and Technology Part A: Energy Science and Research 2012 Volume (issues) 29(1): 495-506 [3] Aksoy, S. And Tretyakov, O.A., Evolution equations for analytical study of digital signals in waveguides, Journal of Electromagnetic Waves and Applications, Vol. 17, No. 12, 1665- 1682, 2003. [4] Tretyakov, O.A. and Akgün, O., Progress in Elektromagnetics Research, PİER, 105, 171- 191, 2010.

120 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications To properties of solutions of one nonlinear problem of heat conductivity in a heterogenic media Aripov М. (1), Khodjimurodova M. (2) (1) The National University of Uzbekistan, Tashkent, Uzbekistan, [email protected] (2) The National University of Uzbekistan, Tashkent, Uzbekistan, [email protected]

Abstract. The problem Cauchy for the equation of nonlinear heat conductivity a presence of an convective transfer absorption is investigated. It is proved, that if the coefficient of heat conductivity includes, time-dependent function that occurs localization of the solution even an absence of absorption. The Kalashnikov A.S , Knerr B.F , Kershner R type estimation of a weak solution of the problem Cauchy and an estimation of a free boundary are received. As it has been shown in [1] environments with the coefficient of heat conductivity depending on temperature under the sedate law, temperature waves when thermal indignation extends with final speed, forming front (free boundary) temperature wave can take place. By consideration of processes of heat conductivity and in nonlinear environments at presence, a convective transfer which speed depends on time and volume and heat absorption in the assumption, that capacity of thermal "stocks" is sedate function at some value of parametres arise localisation of solutions even at absence of absorption. The mathematical model, describing these processes, can be written down in the form of a problem p2 N u  um  u  u   (())()x k t  f t u  (1) ti1  xi  xi  x i N u(0, x ) u0 ( x )  0, x  R (2)

Where и- the temperature of environment,  —Laplasian operator, (.)  grad x (.) ,   0 - m const, k( t ) 0 — generalized coefficient of heat conductivity, x -coefficient of heterogeneity, f( t ) 0 — coefficient, which described of volume of absorption with p 2,  1.

In [2] has been shown, that volume absorption of heat in some cases leads to effect of spatial localization of thermal indignations. In these cases the stop of front of a temperature wave and thermal indignantly takes place in the environment only on final distance. Key words: heterogenic media, conductivity, absorption

References:

[1] А. С. Калашников, О характере распространения возмущений в процессах, описываемых квазилинейными вырождающимися параболическими уравнениями. Труды семинара им. И. Г. Петровского, 1 (1975).

[2] А. С. Калашников, О дифференциальных свойствах обобщенных решений уравнений типа нестационарной фильтрации, Вестник МГУ, мат., 1 (1974).

[3] Арипов М. Методы эталонных уравнений для решения нелинейных краевых задач. Ташкент. Фан. 1988. стр. 137.

[4] Арипов М. Asymptotes of the Solutions of the Non-Newton Polytrophic Filtration Equation. ZAMM, Berlin 2000, suppl.3, 767-768.

121 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

About Extremum Problems with Constraints M.A. Sadygov BSU, Baku, Azerbaijan, [email protected]

Abstract. In the work, we define ωδνβα ),,,,( - Lipschitz functions at a point in a metric space. We also study a number of their properties and consider extremum problems with constraints. Using distance functions, we obtain a number of theorems concerning the exact penalty. We obtain necessary extremum conditions under some constraints in a Banach space. In this work, we define a subdifferential of the first order and study some of its properties. It should be noted that the introduced definitions of subdifferentials are more natural in a one-dimensional space. In work [1,2] we have defined class − δνβαϕ ),,,( -Lipschitz functions at a point in a Banach space and using that function we have obtained high order sufficient and necessary conditions of extremum for extremal problem with constraints. We also study a number of their properties and using distance functions, we obtain a number of theorems concerning the exact penalty. Let )d,X( be a metric space, G , C⊂X, f: X→R, α>0, ν>0, β≥αν, δ>0 and

ω : + → RR + , where ω = 0)0( , + = +∞),0[R . Let us assume δ { ∈= yxdXyxB ),(:),( ≤ δ}. The function f is called a ωδνβα ),,,,( -Lipschitz function with the constant K at the point x if f satisfies the following condition  −ανβ   −ανβ  f(x)f(y) ≤− K y)d(x, ν )xd(x, + y)d(x, α + ω ))xd(x,()     for y,x ∈ δ),x(B . If ≡ω 0)t( , then the function f is called a δνβα ),,,( - Lipschitz function with the constant K at the point x . The function f is called a ωνβα ),,,( -Lipschitz function with the constant K at the point ∈Gx with respect to the set G if f satisfies the following condition  −ανβ   −ανβ  f(x)f(y) ≤− K y)d(x, ν )xd(x, + y)d(x, α + ω ))xd(x,()     for , ∈ Gyx . If ω t ≡ 0)( , then the function f is called a νβα ),,( - Lipschitz function with the constant K at the point ∈Gx with respect to the set G .

Keywords. Extremal problem, Lipschitz function, Metric space, Banach space, Subdifferential. AMS 2010. 46N10, 54E35.

References

[1] Sadygov, M.A. Analysis of nonsmooth optimization problems. Baku, 2002.

[2] Sadygov, M.A. Investigation of the first and second order subdifferentials of nonsmooth functions. Baku, Elm, 2007.

122 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Koçak’s Acceleration Method Smoothly Gears up Iterative Solvers Mehmet Çetin Koçak Ankara University, Ankara, Turkey, [email protected]

Abstract. Many mathematical applications involve the solution of an equation of the form x= g(x) by a repetitive scheme xk1+ = g k = g(x k )where k is the iteration count. Let z be the target fixed-point of g. Koçak’s method generates a superior solver gK via the fixed- point preserving transformation gK =−− (g m x ) / (1 m) where m= w g′′k +− (1 w ) g (z) . The previous two versions [1,2] employed constant w values. Using higher derivatives of g, the new version adjusts w at each step so as to zero {}g,g,′KK ′′  . Let n denote the convergence 1 1 order of g. If n1= , then lim w = and gK is of third-order. If n1> , then lim w = and xz→ 2 xz→ n

gK is of order n+1. This gK smoothly flies the process to z from a starting point where other solvers fail. The benefits amply compensate the cost of extra derivatives. Keywords. Acceleration, iterative solvers, nonlinear equations. AMS 2010. 26A18, 65B99.

References

[1] Koçak, M.Ç., 2008, Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order, The Proceedings of the Twelfth International Congress on Computational and Applied Mathematics (ICCAM2006), 10-14 July 2006, Leuven, Belgium. Goovaerts, M.J., Vandewalle, S., Van Daele, M., Wuytack, L. (eds) J. Comput.Appl.Math., 218 (2): 350-363. DOI:10.1016/j.cam.2007.02.036.

[2] Koçak, M.Ç., 2009, Acceleration of iterative methods, in Reports of the Third Congress of the World Mathematical Society of Turkic Countries, Ed. Zhumagulov, B.T., Almaty, June 30-July 4, 2009, Kazakhstan. ISBN 978-601-240-063-2.

123 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Stress Field Problem for a Pre-Stressed Plate-Strip with Finite Length under the Action of Arbitrary Time-Harmonic Forces Mustafa ERÖZ Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In the framework of the three-dimensional linearized theory of elastodynamics the finite element modelling of the stress field problem for the pre-stressed plate-strip with finite length resting on a rigid foundation under the action of inclined linearly located time-harmonic forces is developed. The numerical results involving the normal stress acting on the interface plane of the plate-strip and the rigid foundation are presented. Moreover, the dependencies between this stress, the frequency of the arbitrary inclined linearly located external force and the initial stretching of the plate-strip are analyzed. Keywords. Initial stress; Time harmonic load; Forced vibration; Finite element method. AMS 2010. 35A15, 65N30.

References [1] S.D. Akbarov, A. Yildiz, M. Eröz, FEM modeling of the time-harmonic dynamical stress field problem for a pre-stressed plate-strip resting on a rigid foundation, Appl. Math. Model. 35 (2011) 952-964.

[2] S.D. Akbarov, A. Yildiz, M. Eröz, Forced vibration of the pre-stressed bilayered plate- strip with finite length resting on a rigid foundation, Appl. Math. Model. 35 (2011) 250-256.

[3] S.D. Akbarov, C. Guler, On the stress field in a half-plane covered by the pre-stretched layer under the action of arbitrary linearly located time-harmonic forces. Applied Mathematical Modelling, 31 (2007) 2375-2390.

[4] Ya S. Uflyand, Integral Transformations in the Theory of Elasticity, Nauka, Moscow- Leningrad, 1963.

124 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Influence of Initial Stresses on a Pre-stressed Orthotropic Plate-strip with Finite Length Resting on a Rigid Foundation Mustafa ERÖZ Sakarya University, Sakarya, Turkey, [email protected]

Abstract. The influence of initial stress on a pre-stressed orthotropic plate-strip with finite length resting on a rigid half plane is investigated by utilizing Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies. The material of the plate- strip is assumed to be orthotropic. Finite element modeling is developed for the considered boundary-value problem. Numerical results which concern the influence of the initial stress and the finiteness of the length of the plate-strip are presented. Moreover, the numerical results obtained for the various values of the problem parameters are presented and discussed. Also, it is established that, the initial stretching of the plate-strip causes to decrease the stress on the interface plane between the plate-strip and the half plane. Keywords. Orthotropic material; Initial stress; Stress distribution; Finite element method; Forced vibration; Time-harmonic load. AMS 2010. 35A15, 65N30.

References [1] S.D. Akbarov, A. Yildiz, M. Eröz, FEM modeling of the time-harmonic dynamical stress field problem for a pre-stressed plate-strip resting on a rigid foundation, Appl. Math. Model. 35 (2011) 952-964.

[2] S.D. Akbarov, A. Yildiz, M. Eröz, Forced vibration of the pre-stressed bilayered plate- strip with finite length resting on a rigid foundation, Appl. Math. Model. 35 (2011) 250-256.

[3] M. Eröz, The stress field problem for a pre-stressed plate-strip with finite length under the action of arbitrary time-harmonic forces, Appl. Math. Model. (2012), doi:10.1016/j.apm.2011.12.058

[4] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, Holden Day, San Francisco, 1963.

125 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Solving a Type of Fractional Volterra-Integro Differential Equations Using Homotopy Analysis Method Mehmet Fatih Karaaslan (1) and Muhammet Kurulay (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. In this work, homotopy analysis method is used to solve fractional integro differential equation of Volterra type. We will find an approximate solution of our model differential equation. Here, the fractional derivatives are described in Caputo sense. The homotopy analysis method in differential equation for integer order is extended to derive numerical solutions of the fractional differential equations. Keywords. Fractional derivative, Fractional Volterra-Integro differential equation, Homotopy analysis method

126 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

About one Problem of Solution Stabilization of the Loaded Heat Equation M.M.Amangaliyeva, M.T.Jenaliyev, K.B.Imanberdiyev and M.I.Ramazanov Institute of Mathematics, Informatics and Mechanics, Almaty, Kazakhstan, [email protected]

Abstract. Statement is given a problem of solution stabilization of the loaded heat equation with help of the boundary conditions. It is proved the solvability theorem of the boundary value problem. It is developed algorithm of approximate construction of boundary controls.

Statement of the problem. To find such boundary controls 1 2 Ltutu 2 ∞∈ ),,0()(),( that solution txy ),( boundary value problem:

t xx α },{,0),0(),(),( ∈=⋅+− Qtxtytxytxy , (1)

π =− 1 π = 2 tutytuty )(),2/(),(),2/( , (2)

= 0 xyxy )()0,( , (3) tents to the zero at t ∞→ as follows [1]:

−σ txy ||),(|| ≤ eC 0t , (4) L2 − ππ 0)2/,2/( where txQ <<−= ππ tx > }0,2/2/|,{ , α ∈ С , C , σ 00 – given positive numbers,

0 Lx 2 −∈ ππ )2/,2/()(y – given function. Equation (1) is named loaded [2]. The problem (1)-(4) is a inverse problem of finding of functions

{ 1 2 tututxy )(),(),,( }, satisfying the boundary value problem (1)-(3) and additional condition (4). The functions { 1 2 tututxy )(),(),,( } define with help of the spectrum methods. Keywords. Loaded heat operator, boundary control, stabilization, spectrum problem. AMS 2010. 35B35; 35K20; 35R10.

References

[1] Fursikov, A.P., Stabilizability of quasilinear parabolic equation by feedback boundary control (Russian), Mat. Sb. 192, no.4, 115-160, 2001.

[2] Jenaliyev, M.T., and Ramazanov, M.I., The loaded equations as perturbations of the differential equations (Russian), Gylym, Almaty, 2010.

127 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On The Best Approximate (P,Q)-Orthogonal Symmetric and Skew-Symmetric Solution of The Matrix Equation AXB= C Murat Sarduvan (1), Sinem Şimşek (2) and Halim Özdemir (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Kirklareli University, Kirklareli, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. Suppose that the matrix equation AXB= C with unknown matrix X is given, where A , B , and C are known matrices of suitable size. The matrix nearness problem is considered over the ()PQ, -orthogonal symmetric and ()PQ, -orthogonal skew-symmetric solutions of the matrix equation AXB= C . The implicit forms of the best approximate solutions of the problems considered are established. Moreover, two numerical examples and a comparative table, depending on the examples chosen from literature, are given. Keywords. Best approximate solution, Frobenius norm, Matrix equations, Spectral decomposition, Matrix nearness problem, The minimum residual problem. AMS 2010. 15A06, 15A24, 65F35.

References [1] Horn, R.A., Johnson, C.R., Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. [2] Golub, G.H., Van Load, C.F., Matrix Computatios (third ed.), The Johns Hopkins University Press, Batimore and London, 1996. [3] Chu, K.E., Singular value and generalized singular value decompositions and the solutions of linear matrix equations, Linear Algebra and its Applications, 88, 83-98, 1987. [4] Peng, Z.Y., An iterative method for the least squares symmetric solution of the matrix equation AXB= C , Applied Mathematics and Computation, 170, 711-723, 2005. [5] Peng, X.Y., Hu, X.Y., and Zhang, L., An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB= C , Applied Mathematics and Computation, 160, 763-777, 2005. [6] Zhao, L.L., Chen, G.L., and Liu, Q.B., Least squares (P,Q)--orthogonal symmetric solution of the matrix equation and its optimal approximation, Electronic Journal of Linear Algebra, 20, 537-551, 2010.

128 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Fractional Order Nonlinear Dynamical Model Nuri Özalp (1) and İlknur Koca (2) (1) Ankara University, Ankara, Turkey, [email protected] (2) Gaziantep University, Gaziantep, Turkey, [email protected]

Abstract. In this work, we introduce a fractional order nonlinear dynamical model of love. The proposed model describes the dynamic behavior of a love triangle under different structures. We give a detailed analysis for the asymptotic stability of mutual apathy and positive fixed points. Finally, our results are validated by numerical simulations. Keywords. Fractional model, Initial value problem, Stability, Numerical Example. AMS 2010. 34A08, 70K75.

References

[1] Ahmed, E., El-Sayed, A.M.A. and El-Saka, H. A. A. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, JMAA 325, 542- 553, (2007).

[2] Podlubny, I. Fractional Differential Equations, (Academic Press, 1999).

[3] Matignon, D. Stability results for fractional differential equations with applications to control processing, in: Computational Eng. in Sys. Appl. 2 (Lille, France, 1996), p 963.

[4] Sprott, J.C. Dynamical Models of Love, Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 8, No. 3, July, (2004).

[5] E. Demirci, N. Özalp, A method for solving differential equations of fractional order, Journal of Computational and Applied Mathematics 236 (2012) 2754-2762.

129 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Approximation Results for Lupaş Operators Ogün Doğru (1) and Kadir Kanat (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. In this study, a Kantorovich type generalization of Lupaş operators is introduced. Then the Korovkin type statistical approximation properties of these operators are investigated. Finally, the rates of statistical convergence of this modification are also studied. Keywords. Bernstein operators, Modulus of continuity, q − Bernstein operators, The Korovkin type approximation theorem. AMS 2010. 41A25, 41A35, 41A36.

References

[1] Lupaş, A., A q-analogue of the Bernstein operator, in: Seminar on Numerical and Statistical Calculus (Cluj-Napoca), Univ. ‘‘Babeş -Bolyai’’, 9, 85–92, 1987.

[2] Phillips, G. M., Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4, 511–518, 1997.

[3] Bernstein, S. N., Demonstration du theoreme de weierstrass fondee sur la calcul des probabilities, Commun. Soc. Math. Charkow. 13, 1–2, 1912.

[4] Dalmanoğlu, Ö., Doğru, O., On statistical approximation properties of Kantorovich type q-Bernstein operators, Math. Comput. Modelling 52, 4, 760–771, 2010.

[5] Gauchman, H., Integral inequalities in q-Calculus, Comput. Math. Appl. 47, 281–300, 2004.

[6] Gadjiev, A. D., Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129–138, 2002.

130 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Asymptotic of Solution Functions of a Sturm-Liouville-Type Problem and the Green Function Okan Kuzu (1) and Mahir Kadakal (2) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In this study we have created Hilbert Space of boundary value problem consisting of the Sturm Liouville equation + ( ) = ′′ with boundary conditions 휏푢 ≔ −푢 푞 푥 푢 휆푢 (0) = (0) ′ [ ( ) 푢 ( )] =−휆푢 ( ) ( ) ′ ′ consisting eigenvalue parameter.− 훽1푢 휋 −We 훽 2have푢 휋 shown휆�훽 �symmetric1푢 휋 − 훽� 2of푢 appropriate휋 � operator to the problem. We have obtained asymptotic of the solution functions and asymtotic of the characteristic function by using them. Moreover, we have examined Green function and asymtotic expansion of eigenvalues. Here qx() is a continuous function defined in 0,π [] interval. Keywords. Boundary value problem, Sturm-Liouville Theory, Differential operator, eigenvalue, eigenfunction, asymptotic behavior, Green’s Function. AMS 2010. 34B24

References [1]. Birkhoff, G. D. On The Asymptotic Character of The Solution of The Certain Linear Differential Equations Containing Parameter, Trans. Amer. Math. Soc., Vol.9 1908, pp. 219 – 231.

[2]. Birkhoff, G. D. Boundary Value and Expantion Problems of Ordinary Linear Differential Equations, Trans. Amer. Math. Soc., Vol. 9 1908, pp. 373 – 395.

[3]. Boyce, W. E.; Diprima, R. C. Elementary Differential Equations and Boundary Value Problems, John Willey and Sons, New York, 1977, pp. 544-554.

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[4]. Fulton, C. T. Two-point Poundary Value Problems with Eigenvalue Parameter Contained in The Boundary Condition, Proceedings of the Royal Society of Edinburgh. Section A 77, 1977, p. 293-308.

[5]. Hinton, D. B. An Expansion Theorem for Eigenvalue Problem with Eigenvalue Parameter in The Boundary Condition, Quart. J. Math. Oxford, vol. 30, No: 2, 1979, 33-42.

[6]. Kerimov, N. B.; Mamedov, Kh. K. On a Boundary Value Problem with a Spectral Parameter in The Boundary Conditions, Sibirsk. Math. J. 40, No:2, 1999, 281-290.

[7]. Levitan, B. M.; Sarqsyan, I.S. Sturm-Liouville ve Direkt Operatörler, Moskova, Nauka, (Rusca), 1988.

[8]. Naimark, M. A. Linear Differantial Operators, Ungar, New York, 1967.

[9]. Schneider, A. A Note Eigenvalue Problems with Eigenvalue Parameter in The Boundary Conditions, Math. Z. 136, 1974, 163-167.

[10]. Shkalikov, A. A. Boundary Value Problems for Ordinary Differential Equations with a Parameter in Boundary Conditions, Trudy., Sem., Imeny, I.G.Petrovsgo, 9, 1983, 190-229.

[11]. Titchmarsh, E. C. Eigenfunction Expansions Associated with Second Order Differential Equations, 2nd end, Oxford Univ. Pres, London, 1962.

[12]. Walter, J. Regular Eigenvalue Problems with Eigenvalue Parameter in The Boundary Conditions, Math. Z., 133, 1973, 301-312.

[13]. Zayed, E.M.E. and Ibrahim, S.F.M., Regular Eigenvalue Problem with Eigenparameter in the Boundary Conditions, Bull. Cal. Math. Soc. 84 379-393, 1992.

132 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Possible Fuzzy Solutions for Second Order Fuzzy Initial Value Problems Involving Forcing Terms Ö. Akın (1), T. Khaniyev (2), Ö. Oruç (3) and I. B. Türkşen (4) (1) TOBB Economics and Technology University, Ankara, Turkey, [email protected] (2) TOBB Economics and Technology University, Ankara, Turkey, [email protected] (3) Inonu University, Malatya, Turkey, [email protected] (4) TOBB Economics and Technology University, Ankara, Turkey, [email protected]

Abstract. In this study, we state a fuzzy initial value problem of the second order fuzzy differential equations. Here we investigate problems with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions where fuzzy terms are fuzzy number valued. We propose an algorithm based on alpha-cut of a fuzzy set. Finally we present some examples by using our proposed algorithm. Keywords. Alpha-cut operation, Fuzzy number, Fuzzy initial value, Fuzzy initial value problem, Fuzzy differential equations. AMS 2010. 53A40, 20M15.

References

[1] Ö.Akın, T.Khaniyev, Ö.Oruç, and B.Türk¸sen. An algorithm for the solution of second order fuzzy initial value problems. Expert Systems with Applications, 2012. doi:10.1016/j.eswa.2012.05.052.

[2] S. E. Amrahov and I.N.Askerzade. A novel approach to definition of fuzzy functions. Commun. Fac. Sci.Univ.Ank.,Series A1, 59(1):1–7, 2010.

[3] B. Bede and S.G. Gal. Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equation. Fuzzy Sets and Systems, 151:581– 599, 2005.

[4] B. Bede, I.J. Rudas, and A.L. Bencsik. First order linear fuzzy differential equations under generalized differentiability. Information Sciences, 177:1648–1662, 2007.

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[5] S.G Gal. Approximation theory in fuzzy setting, in: G.A. Anastassiou (Ed.), Handbook of Analytic-Computational Methods in Applied Mathematics. Chapman & Hall/CRC Press, 2000.

[6] N. Gasilov, ¸ S. E. Amrahov, and A. G. Fatullayev. A geometric approach to solve fuzzy linear systems of differential equations. Appl. Math. Inf. Sci., 5, no. 3:484–499, 2011.

[7] E. Hüllermeier. An approach to modelling and simulation of uncertain dynamical systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 5, no. 2:117–137, 1997.

[8] O. Kaleva. Fuzzy differential equations. Fuzzy Sets and Systems, 24:301–317, 1987.

[9] A. Kandel and W. J. Byatt. Fuzzy differential equations. In in Proceedings of the International Conference on Cybernetics and Society, pp. 1213-1216, Tokyo, Japan,, 1978.

[10] A. Khastan, F. Bahrami, and K. Ivaz. New results on multiple solutions for nth-order fuzzy differential equations under generalized differentiability. Boundary Value Problems, Volume 2009:13 pages, 2009.

[11] M. Misukoshi, Y.Chalco-Cano, H.Román-Flores, and R.C.Bassanezi. Fuzzy differential equations and the extension principle. Information Sciences, 177:3627–3635, 2007.

[12] M. Oberguggenberger and S. Pittschmann. Differential equations with fuzzy parameters. Mathematical and Computer Modelling of Dynamical Systems, 5:181–202, 1999.

[13] M. Puri and D. Ralescu. Differential and fuzzy functions. Journal of Mathematical Analysis and Applications, 91:552–558, 1983.

[14] C. Wu and Z. Gong. On henstock integral of fuzzy-number-valued functions i. Fuzzy Sets and Systems, 120:523–532, 2001.

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Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type Ömer Faruk Gözükızıl (1) and Şamil Akçağıl (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]

Abstract. By using tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations namely the Benney- Luke equation and the higher-order improved Boussinesq equation. We show that the tanh- coth method is a useful, reliable and concise method to solve these types of equations. Keywords. Sobolev type equation, The tanh--coth method, travelling wave solution, the Benney-Luke equation, the higher-order improved Boussinesq equation.

References

[1] Carroll, R. W. and Showalter, R. E., Singular and Degenerate Cauchy Problem, Academic Press, New York, San Francisco, London(1976).

[2] Sobolev, S. L., Some new problems in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 3-50 .

[3] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat. 18:1 (1954), 3-50

[4] S. A. Gabov, New problems of the mathematical theory of waves, Fizmatlit, Moscow 1998.

[5] M. O. Korpusov, Yu. D. Pletner, and A. G. Sveshnikov, "Unsteady waves in anisotropic dispersive media", Zh. Vychisl. Mat. Mat. Fiz. 39 (1999), 1006-1022.

[6] M. O. Korpusov and A. G. Sveshnikov, Three-dimensional nonlinear evolutionary pseudoparabolic equations in math. Phy. Zh. Vychisl. Mat. Mat. Fiz. 43 (2003), 1835-1869

[7] P. I. Naumkin and I. A. Shishmarev, Nonlinear nonlocal equations in the theory of waves, Translations of MathematicalMonographs, no. 133, Amer. Math. Soc., Providence, RI 1994.

[8] E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, The Cauchy problem for an equation of Sobolev type with power non-linearity, Izv. RAN. Ser. Mat., 69:1 (2005), 61-114

[9] R. L. Pego, J. R. Quintero , Two dimensional solitary waves for a Benney-Luke equation, Physica D: Nonlinear Phenomena, 132 (1999), 476-496.

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Explicit Solutions for Fractional Schrödinger Equation Reşat Yılmazer (1) and Erdal Bas (2) (1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this study, we consider the Schrödinger equation on α - dimensional fractional space with a columb potential depending on a parameter. Using fractional calculus operator, we obtain explicit solution of the following second order linear ordinary differential equation dd2φφ hz+() nz ++ν h s + νφ n()() z =0 h ≠ 0,ν ∈ , dz2 dz which has a solution of the form

−sh φ ()()z= K hz e−nz h , ν −1 where K is an arbitrary constant. Keywords. Fractional calculus; Schrödinger equation; Radial equation; Explicit solution. AMS 2010. 26A33, 34A08

References

[1] Eid, R. Muslih, S.I. Baleanu, D. Rabei, E., On Fractional Schrödinger Equation in α- dimensional Fractional Space, Nonlinear Anal. RWA 10, 1299-1304, 2009.

[2] Tu, S.-T., Chyan, D.-K. Srivastava, H.M., Some Families of Ordinary and Partial Fractional Differintegral Equations, Integral Transform. Spec. Funct. 11, 291-302, 2001.

[3] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications Mathematics in Science and Enginering, vol. 198, Academic Press, New York, London, Tokyo and Toronto, 1999.

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Sharp Weighted Rellich Type Inequalities for Generalized Greiner Vector Fields Semra Ahmetolan (1) and İsmail Kombe (2) (1) Istanbul Technical University, Istanbul, Turkey, [email protected] (2) Istanbul Commerce University, Istanbul, Turkey, [email protected]

Abstract. In this work, sharp weighted Rellich type inequalities are obtained for generalized Greiner vector fields:

= + 2 | | , and = + 2 | | , = 1,2, … , . 휕 2푘−2 휕 휕 2푘−2 휕 푗 푗 푗 푗 where |푋| = (휕푥| 푗 | +푘푦| | 푧) / and휕푙 1. 푌Here휕푦 a푗 generic푘푥 point푧 in휕푙 푗 , 1, is푛 defined by 2 2 1 2 2푛+1 = ( ,푧 ) = (푥, , ) 푦 where푘 ≥ , , = + 1 ℝ. Furthermore,푛 ≥ we present 2푛+1 푛 several휔 푧 weighted푙 푥 푦Rellich푙 ∈ ℝ type inequalities푥 푦 with ∈ ℝ remainder푧 푥 terms.√− 푦 Keywords. Rellich type inequalities, sharp constant, Greiner vector fields. AMS 2010. 22E30, 43A80, 26D10.

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Review of special relativity mathematics

Shukri Klinaku University of Prishtina, Prishtina, Kosova, [email protected]

Abstract. The second postulate of special relativity states that the velocity of light is constant in any reference system in relative motion. This postulate is necessary but not sufficient condition for the derivation of Lorentz transformations. The special relativity, based on this postulate, has sacrificed the homogeneous and absolute time. More than that, to generate Lorentz transformations, based on this postulate, special relativity sacrifices a few simple mathematical concepts. In this paper we will attempt to prove that the preservation of the postulate about the velocity of light brings in question the validity of mathematical rule for vector addition even for velocities; that the derivation of these transformations violates the solving rules for systems with two linear equations; that the Lorentz transformations contain only one transformation and not two as special relativity claimed; that the special relativity’s velocity addition formula is invalid. In the end, three equations that express the consequences of special relativity will be reviewed.

Keywords. Lorentz transformation, velocity of light, system of linear equations. AMS 2010. 83A05.

References [1] Klinaku, Sh., The explanation of Michelson’s experiment, AIP Conf. Proc.; Volume 1316, 2010 [2] Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics, John Wiley&Sons, USA, 2004. [3] Einstein, A., Relativity: The special and general theory, London, 1920 [4] Born, M., Die Relativitätstheorie Einsteins, Sechste Auflage, Springer-Verlag, Berlin – Heidelberg - New York, 2001. [5] Lämmerzahl, C., Test Theories for Lorentz Invariance, Springer-Verlag Berlin Heidelberg 2006 [6] Landau, L. D., Lifschitz, E. M., Lehrbuch der Teoretischen Physik I, Akademie Verlag, Berlin, 1977.

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Numerical Simulations of Coupled Sine-Gordon Equations by Reduced Differential Transform Method Sema Servi (1), Ayşe Betül Koç (2), Yıldıray Keskin (3) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected] (3) Selcuk University, Konya, Turkey, [email protected]

Abstract. In this paper, a general framework of the reduced differential transform method (RDTM) is presented for solving coupled sine-Gordon equation. Coupled sine- Gordon equation have special importance in nonlinear wave processes. the comparsion of the results obtained by variational iteration method (VIM) and RDTM shows that RDTM is a powerful method for the solution of linear and nonlinear systems of PDEs. Keywords. Reduced differential transform method, Variational iteration method, coupled KdV, coupled Burgers equations. AMS 2010. 35M11, 74G15

References [1] L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhauser, Boston, 1997.

[2] K. Al-Khaled, Numerical approximations for population growth models, Appl. Math. Comput.,160 (2005), 865-873.

[3] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burgers’ and coupled Burgers’ equations, Journal of Computational and Applied Mathematics 181 (2005) 245–251.

[4] El-Wakil, S.A., Abulwafa, E.M. and Abdou, M.A., An Improved variational iteration method for solving coupled KdV and Boussinesq-like B(m,n) equations. Chaos

Solitons Fractals, in press.

[5] K.R. Khusnutdinova, D.E. Pelinovsky, On the exchange of energy in coupled Klein– Gordon equations, Wave Motion 38 (2003) 1–10.

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[6] S. Nakagiri, J. Ha, Coupled sine-Gordon equations as nonlinear second order evolution equations. Taiwanese J. Math. 5 (2001) 297-315.

[7] R.S. Jonson, Shallow water waves in a viscous fluid-the undular bore, Phys. Fluids 15 (1972) 1693–1699.

[8] S.S. Ray, A numerical solution of the coupled sine-Gordon equation using the modified decomposition method. Appl. Math. Comput. 175 (2006) 1046-1054.

[9] B. Batiha, M. S. M. Noorani, I. Hashim, Approximate analytical solution of the coupled sine-Gordon equation using the variational iteration method, Physica Scripta 76

(2007) 445-448.

[10] Batiha, B., M. S. M. Noorani, I. Hashim Numerical solution of sine-Gordon equation by variational iteration method, Physics Letters A 370 (2007) 437-440.

[11] Abdul-Majid Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Computers & Mathematics with Applications,54 (7-8) 2007, 895-902.

[12] A. Sadighi, D.D. Ganji; and B. Ganjavi, Traveling Wave Solutions of the Sine- Gordon and the Coupled Sine-Gordon Equations Using the Homotopy-Perturbation Method, Transaction B: Mechanical Engineering, 16 (2009) 189-195.

[13] A. Sami Bataineh, M. S. M. Noorani, I. Hashim, Approximate analytical solutions of systems of PDEs by homotopy analysis method, Computers & Mathematics with Applications, 55(12) 2008, 2913-2923.

[14] F. Ayaz, Solutions of the system of differential equations by differential transform method, Appl. Math. Comput., 147 (2004) 547-567.

[15] Y. Keskin, G. Oturanc, Reduced Differential Transform Method for Partial Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (6) (2009) 741-749.

[16] Y. Keskin, G. Oturanc, Reduced Differential Transform Method for fractional partial differential equations, Nonlinear Science Letters A 1(1) (2010) 61-72

[17] Y. Keskin, Oturanç, G., “Reduced differential transform method for generalized KdV Equations”, Mathematical and Computational Applications, 2010, 15 (3), 382-393.

[18] Y. Keskin, Ph.D. Thesis, University of Selcuk.

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On Removable Sets of Solutions for Degenerate Linear Parabolic Equations T. S. Gadjiev (1), S. A. Aliev (1) and O. S. Aliyev (1) (1) Institute of Mathematical and Mechanics of NAS of Azerbaijan, Baku, [email protected]

Abstract. The sufficient conditions of removability of sets for mixed value problems are obtained for linear elliptic equations in nondicergent form.

Let D be a bounded domain in Euclidean space n+1 ,3, ∂≥ DnE its boundary. The boundary ∂D is nonsmooth surface. Consider in D the following equation n n −= ()()()+ + uxcuxbuxauhu , (1) t ∑ ij ji ∑ xixx i ji =1, i=1 where ()ij ()xa is a real symmetric matrix ij ()xa are Lipschitzian in D and

n 2 −1 2 ()ξγω ≤ ∑ ij ()()xax ji ≤ Ex n γξξωγξξ ∈∈ ( 1,0,; ], (2) ji =1,

i () ; 00 () =≤≤−≤ nixcbbxb ,,1;0 . (3) where b0 ≥ 0 is a constant, ω()x > 0 is a weight function. The following theorems have been proved.

Theorem 1. Let D be some nonsmooth domain in En , ⊂ DE be a compact set. The coefficients L satisfy the conditions (2)-(3). In order that the compact E be removable λ relative to Dirichlet problem for equation (1) in the space β ()DC it is sufficient that

n 2+− λ Η ()Em = 0 (6) γ where mΗ is a Hausdorff measure of order γ .

Theorem 2. Let D be some nonsmooth domain in En , ⊂ DE be a compact set. In order that the compact E be removable relative to Neumann problem for equation (1) in the space λ β ()DC it is sufficient that

n 2+− λ Η ()Em = 0 . (7) The analogous analytical result is proved for the mixed boundary value problem.

141 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Calculation of Shear Stresses Occurred on Elastic Soils under Rectangular Foundation with Analytical and Numerical Methods Uğur Dağdeviren (1) and Zeki Gündüz (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. Soils are subjected to normal and shear stresses due to their own weight and external loads applied by buildings. Stresses applied by structures to the soil under and around the structure are not constant and varies along the depth. Knowledge of the stresses distribution has a great importance in many problems solution and projects design of geotechnical engineering. The actual stress values resulting from external loads depend on the magnitude of the applied load, the foundation dimensions and soil properties. However, to do in the real stress- strain analysis is very difficult due to the heterogeneous and anisotropic structures of the soil. Therefore, in practice, stress increment in soils can be usually determined by the acceptance of semi-infinite, weightless, isotropic, homogeneous soil and elastic half-space medium. In accordance with these assumptions, Boussinesq (1885) solved the stresses occurred in soil due to a single load acting perpendicular to plane, in a cartesian coordinate system. However, the stress distributions obtained for a single load is not realistic in many civil engineering problem since the structural loads are transferred to a soil through the foundations. Therefore, by integrating the solutions developed for the single load through the loaded area, it is analytically possible to solve stress distributions in soil due to the different foundation geometries (circular, rectangular, trapezoidal, etc.). In this study, the exact analytical solution was carried out in order to determine shear stresses on horizontal planes due to uniformly loaded rectangular foundation within the elastic soil. For practical uses, it is required to make the analytical solution to determine shear stresses, which will occur under uniformly loaded rectangular foundation, in simple and convenient form. Therefore, the obtained shear stresses were converted to a simple table form. Also, the problem is solved with a numerical integration method, and a computer program is prepared. As a result, the largest shear stresses, which will occur under the different sizes of rectangular foundations, are examined throughout the depth. Findings show that the largest stress under uniformly loaded rectangular foundations will occur at the bottom (z = 0 depth)

142 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

and at the middle of the corner plane of a foundation. This result conforms with the tilting problem of structures because of liquefaction and the loss of bearing capacity during earthquakes. Except rectangular foundations close to strip foundation, the results show that the effect of shear stresses because of structures are reduced to a negligible level after the depth ratio is z/B> 2.5. Especially for the determination of shear stresses acting on the soil prior to dynamic loading, analytical and numerical solutions obtained in the study will be used as a practical method. Keywords. Shear stress, rectangular foundation, elastic soil, analytical solution, numerical solution.

References [1] Boussinesq, J., “Application des Potentials a L’Etude de L’Equilibre et due Mouvement des Solides Elastiques”, Gauthier-Villars, Paris, 1885.

[2] Hyodo, M., Yamamoto, Y., Sugiyama, M., “Undrained Cyclic Shear Behaviour of Normally Consolidated Clay Subjected to Initial Static Shear Stress”, Soils and Foundations, Vol. 34, No. 4, p.1-11, 1994.

[3] Algın, H.M., “Stresses From Linearly Distributed Pressures Over Rectangular Areas”, Int. J. Numer. Anal. Meth. Geomech., Vol. 24, p.681-692, 2000.

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On a Shape Design Problem for the Eigenfrequency of the Membrane with Fixed Boundary Lenth* Yusif S. Gasimov (1) and Leyla I. Amirova (1) (1) Baku State University, Baku, Azerbaijan, [email protected]

Abstract. In the work the following eigenvalue problem is considered =∆− λ ∈ Dxuu ,, (1)

u = ,0 ∈ Sx D , (2)

n where ∆ - Laplace operator, D -convex bounded domain from R , S D - its boundary. Let

n 2 { 0 ,: D ∈∈∈= CSKDEDK } ,

n where K 0 the set of all convex bounded domains from R with S D = 2π , SD -denotes the surface area of D . ∗ The problem is: For n = 2 to find a domain ∈ KD that solves the problem λ1 ()D → min . Indeed this is a problem of finding of the shape that makes minimal the first eigenfrequency of the membrane under across vibrations [1]. For the considered problem the following condition of optimality is obtained

∗ 2 ∗ 2 ∇ () ( )()∇≤ ( ) ∗ ( )()dsxnPxudsxnPxu , (3) ∫ 1 D ∫ 1 D S ∗ S ∗ D D

∗ where 1 ()xu is the eigenvibration of the membrane corresponding to the eigenfrequency

λ1()D . It is shown that the unit ball B satisfies to this condition. As is shown in [2] λ()D is quazi convex with respect to D . So, (3) is also sufficient condition for the considered problem. The considered problem is solved for arbitrary n . Keywords. shape design, eigenfrequency, membrane. AMS 2010. 31B20, 49N45

References [1] Gould S.H. Variational Methods for Eigenvalue Problems, Oxford University Press, 328p., 1966. [2] Gasimov Y.S. On some properties of the eigenvalues by the variation of the domain, Mathematical Physics, Analysis, Geometry, V.10, N.2, , pp.249- 255, 2003.

* The work was supported by Science Development Foundation of the Republic of Azerbaijan № EIF-2011- 1(3)-82/25/1

144 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Reduced Differential Transform Method for Solving High-Dimensional PDEs Yıldıray Keskin (1), Ayşe Betül Koç (2) and Sema Servi (3) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. In this paper, reduced differential transform method was applied to a few high-dimensional PDEs with initial and boundary conditions. The proposed procedure solved these problems quite satisfactorily even if it has discontinuous boundary conditions. Using the differential transform, PDEs with n-variables can be transformed into algebraic equations and the resulting algebraic equations can be solved by simple manipulations. It shows that the proposed transformation is a feasible tool to solve linear or nonlinear PDEs and the resulting algebraic equations can be less complicated to manipulate compared with the integral transform methods. Keywords. Reduced differential transform method, high-dimensional PDEs. AMS 2010. 35G40, 74G15.

References [1] Zhou JK. Differential transform and its applications for electrical circuits. Wuhan. China: Huazhong University Press; 1986.

[2] Chen CK. Solving partial differential equations by two-dimensional differential transform. Appl Math Comput 1999;106:171-179

[3] F. Ayaz, On the two-dimensional di erential transform method, Appl. Math. Comput. 143, 361 (2003). ff [4] A. Kurnaz, G. Oturan¸c and M. E. Kiris, n-Dimensional di erential transformation method for solving PDEs, Int. J. Comput. Math. 82 (2005) 369. ff [5] Y. Keskin, G. Oturanc, Reduced Differential Transform Method for Partial Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (6) (2009) 741-749.

[6] Y. Keskin, G. Oturanc, Reduced Differential Transform Method for fractional partial differential equations, Nonlinear Science Letters A 1(1) (2010) 61-72

[7] Y. Keskin, Oturanç, G., “Reduced differential transform method for generalized KdV Equations”, Mathematical and Computational Applications, 2010, 15 (3), 382-393.

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146 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

DISCRETE MATHEMATICS

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Connected Dominating Sets in Unit Disk Graphs D. A. Mojdeh (1) and S. Kordi (2) (1) University of Tafresh, Tafresh, Iran, [email protected] (2) University of Tafresh, Tafresh, Iran

Abstract. A subset of vertices in a graph is called a dominating set if every vertex is either in the subset or adjacent to a vertex in the subset. A dominating set is connected if it induces a connected subgraph. A subset of vertices in a graph is independent if no two vertices are connected by an edge. Many constructions for approximating the minimum connected dominating set are based on the construction of a maximal independent set. Let

| | and be the size of a maximum independent set and the size of a minimum connected dominating set in the same graph respectively. In [Theoretical Computer Science 352 (2006) 1-7] Wu et al by showing that

, they have really shown the relation between

and plays an important role in establishing the performance ratio of those approximation algorithms. They have also conjectured "the neighbor area of a -star subgraph in a unit disk graph contains at most independent vertices". In this paper we show that for all unit disk graphs and improve the conjecture. Keywords. Connected dominating set; independent set; unit disk graph. AMS 2010. 05C69.

References

[1] H. Eriksson, MBone: The Multicast backbone, Comm ACM, 37, (1994), 54-60.

[2] W.K. Hale, Frequency assignment: Theory and Applications, Proc. IEEE, Vol. 68, (1980), 1497-1514.

[3] K. Kammerlander, C 900 an Advanced Mobile Radio Telephone system with optimum frequency utilization, IEEE Tran. selected Area in Communication, Vol. 2, (1984), 538-597.

[4] D.B. West, Introduction to Graph Theory (Second Edition). Prentice Hall USA (2001).

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[5] D.W. Wong and Y.S. Kuo, A study of two geometric Location Problems, Inf. Proc. Letter, Vol. 28, No. 6, (1988), 281-286.

[6] W.Wu, H. Du, X. Jia, Y. Li, Scott C.H. Huang, Minimum connected dominating sets and maximal independent sets in unit disk graphs, Theoretical Computer Science, 352, (2006), 1-7

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Perfect Codes in the Euclidean Metric Fatih Temiz (1) and Vedat Şiap (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. One of the most important and investigated recently topics in coding theory is the topic of perfect codes. The study of nontrivial perfect codes is important for several interdisciplines. In this work, we propose bounds for error correcting codes of some particular

Euclidean weights over  p ( p ≥ 5, a prime). Besides, we study these bounds to determine whether there exist perfect linear codes or not. Keywords. Perfect codes, Euclidean weight, Linear Codes AMS 2010. 94B60

References

[1] Jain S., Nam K.B., Lee K.S., On some perfect codes with respect to Lee metric, Linear Algebra Appl., 405, 104-120, 2005.

[2] Solov'eva F., On perfect binary codes, Discrete Appl. Math., 156, 1488-1498, 2008.

[3] Heden O., Roos C., The non-existence of some perfect codes over non-prime power alphabets, Discrete Math., 311, 1344-1348, 2011.

[4] Siap, I., Ozen, M., Siap, V., On the existence of perfect linear codes over $\mathbb{Z}_{2^l}$, To appear in The Arabian Journal for Science and Enginnering.

149 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On One Dimensional 2r+1-Cyclic Rule Cellular Automata* İrfan Şiap (1), Hasan Akın (2) and M. Emin Köroğlu (3) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Zirve University, Gaziantep, Turkey, [email protected] (3) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. Cellular automata (CA) models the behavior of discrete dynamical systems which have found a wide range of applications in science. On the other hand, error correcting codes are used for detecting and correcting errors occurred in digital data transmission. In this work, we make use of the theory of error correcting codes in studying a special family of one dimensional cellular automata. Here, we introduce a new family of one dimensional cellular automata named 2r+1-cyclic cellular automata (2r+1-CCA) over primitive fields. As a special case with r=2 the structure and reversibility problem over respectively binary fields is studied by del Rel et al. in [1] and primitive fields is studied by Cinkir et al. in [2] and Siap et al. in [3]. The approach of studying the algebraic structure and their reversibility property for this general case is generalized from [3]. First, we state the definition of 2r+1-CCA, next we determine their rule matrix. After presenting some necessary definitions and theorems from the theory of error correcting code we establish the connection between the generator matrices of cyclic codes and the rule matrix of 2r+1-CCA. One of the important feature of CA is their reversibility problem [4,5]. Here, by taking advantage of this connection, we are able to determine the reversibility problem of 2r+1- CCA. Finally, we present an application of reversible 2r+1-CCA to error correcting codes by extending the method first introduced by Chowdhury et al. in [6] for binary cases. We conclude by observing the advantage of 2r+1- CCA if used in encoding and decoding. Keywords. Cellular Automata, 2r+1 -Cyclic Cellular Automata, Reversibility, Error Correcting Codes. AMS 2010. Primary37B15; Secondary 37A35, 37B40, 94B40.

(*) The work is partially supported by The Scientific and Technological Research Council of Turkey-TUBITAK (Project No: 110T713).

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References

[1] A. Martin del Rey, G. Rodriguez Sanchez, Reversibility of linear cellular automata, Applied Mathematics and Computation, Vol. 217, 21, 1, 8360–836 (2011).

[2] Z. Cinkir, H. Akın, I. Siap, Reversibility of 1D Cellular automata with periodic boundary over finite fields Zp, Journal of Statistical Physics, 143 (4) 807-823 (2011).

[3] Irfan Siap, and Hasan Akin, On One Dimensional Penta-Cyclic Rule Cellular Automata, 2nd World Conference on Information Technology, Kemer, Antalya, Turkey, 23-27 November 2011.

[4] E. Czeizler, On the size of inverse neighborhoods for one-dimensional reversible cellular automata. Theor. Comput. Sci. 273-284 (2004).

[5] G. Manzini, L. Margara, Invertible linear cellular automata over: Algorithmic and dynamical aspects, J. Comput. Syst. Sci. 56 60-67 (1998).

[6] D.R. Chowdhury, S. Basu, I.S. Gupta, and P.P. Chaudhuri, Design of CAECC-Cellular Automata Based Error Correcting Code, IEEE Trans. Computers , 43 759-764 (1994).

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Image Representation as Skip Graph Mustafa Aksu (1), Murat Canayaz (2), Ali Karcı (3) (1) Sutcu Imam University, Kahramanmaras, Turkey, [email protected] (2)Yuzuncu Yil University, Van, Turkey, [email protected] (3) Inonu University, Malatya, Turkey, [email protected]

Abstract. In this study, new approaches based on skip lists and skip graphs presented for image representation and segmentation procedure. As a term, graph is commonly used instead of image processing in similar studies. However, in our study it was shown that Skip Graph can be used in image processing, which is derived from Skip List data structure. Skip List data structure uses nodes and trees like graphs to build Skip Graphs. In Skip Graph, data structure can be presented in different levels. Moreover, nodes can be added, subtracted, deleted or joined on available data. Considering these features of Skip Graph structure , it can be used in image processing applications as well. It provides a variety of advantages in terms of adding or subtracting objects to images, or even dividing images into levels and convenience to access different levels. Furthermore, cleaning the image off dust, noise and removing color tones from the image can be achieved in this method. Keywords. Skip graph, image representation, skip list, segmentation, graph theory.

References

[1] Aspnes J., Shah G., Skip graphs, in Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 384–393, 2003.

[2] Denis N. G., Christos A. F., Automatic synthesis of mathematical models using graph theory for optimisation of thermal energy systems, Volume 48, Issue 11, Pages 2818–2826. November 2007.

[3] Huang Q. et al, A robust graph-based segmentation method for breast tumors in ultrasound images,Volume 52, Issue 2 Pages 266–275, February 2012.

[4] Jianbo S., Jitendra M., Normalized Cuts and Image Segmentation, IEEE Transactions on PAMI, Vol. 22, No. 8, Aug. 2000.

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[5] Pugh W. ,Skip Lists: A Probabilistic Alternative to Balanced Trees ,Communications of the ACM, 33(6):668–676, June 1990.

[6] Tremeau A., Colantoni P., Region Adjacency Graph Applied to Color image segmentation,Universite Jean Monnet, Institut d’lngenierie de la Vision – IEEE Trans Image Process.;9(4):735-44, 2000.

[7] Zhang Y., Cheng X., Medical Image Segmentation Based on Watershed and Graph Theory , 3rd International Congress on Image and Signal Processing, 2010.

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On Generalized Gaussian Fibonacci Numbers and Matrix Methods Mustafa Aşcı Pamukkale University, Denizli, Turkey, [email protected]

Abstract. Many authors define certain generalizations of the usual Fibonacci, Pell and Lucas numbers by matrix methods and then obtain the Binet formulas and combinatorial representations of the generalizations of these number sequence. In this article firstly we define and study the generalized Gaussian Fibonacci numbers and then find the matrix representation of positively and negatively subscripted terms of Gaussian Fibonacci numbers and prove some theorems by these matrix representations. We also find the sums of generalized Gaussian Fibonacci numbers by matrix method. Keywords. Gaussian Fibonacci, Gaussian Pell, Matrix Method. AMS 2010. 11BXX

References

[1] Berzsenyi, G. Gaussian Fibonacci numbers. Fibonacci Quart. 15 (1977), no. 3, 233--236.

[2] Asci M., Gurel E. Bivariate Gaussian Fibonacci and Lucas Polynomials Accepted.

[3] Asci M., Gurel E. Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers Submitted.

[4] Good, I. J. Complex Fibonacci and Lucas numbers, continued fractions, and the square root of the golden ratio. Fibonacci Quart. 31 (1993), no. 1, 7--20.

[5] G, Berzsenyi, Gaussian Fibonacci Numbers. The Fibonacci Quarterly 15 (1977):233-236.

[6] Harman, C. J. Complex Fibonacci numbers. Fibonacci Quart. 19 (1981), no. 1, 82--86.

[7] Horadam, A. F. A Generalized Fibonacci Sequence, American Math. Monthly 68 (1961):455~459,

[8] Horadam, A. F. Complex Fibonacci Numbers and Fibonacci Quaternions. American Math. Monthly 70 (1963):289-291.

154 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Content Based Image Retrieval via Graph Similarity Murat Canayaz (1), Muhammet Baykara (2), Mustafa Aksu (3), Turgut Özseven (4) and Ali Karcı (5) (1) Yuzuncu Yil University, Van, Turkey, [email protected] (2) Firat University, Elazıg, Turkey, [email protected] (3 Sutcu Imam University , Kahramanmaras, Turkey, [email protected] (4) Gazi Osmanpasa University, Tokat, Turkey, [email protected] (5) Inonu University, Malatya, Turkey, [email protected]

Abstract. The need of finding the intended image in a database or a collection is considered as necessity by a great numbers of people working on medicine, architecture, fashion or publishing. This study aims to develop a graph based model which can be used in the applications of image retrieval providing the picture we need among the images increasing continuously in this digital age. Principally, the representation of image as graph is realized and choosing the intended image from database is accomplished with the help of the similarity of graph. Especially, in this study which is transacted on the medical data, the original database taken from hospitals is used and an approach which can be useful for the medical field is endeavoured to improve. Keywords. Image Retrieval, Graph Similarity

References

[1] Baeza-Yates R.,Valiente G, An Image Similarity Measure based on Graph Matching, String Processing and Information Retrieval, Proceedings. Seventh International Symposium on, Page(s): 28 2000.

[2] Denis N. G., Christos A. F., Automatic synthesis of mathematical models using graph theory for optimisation of thermal energy systems, Volume 48, Issue 11, Pages 2818–2826. November 2007.

[3] Huang Q. et al, A robust graph-based segmentation method for breast tumors in ultrasound images,Volume 52, Issue 2 Pages 266–275, February 2012.

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[4] Jianbo S., Jitendra M., Normalized Cuts and Image Segmentation, IEEE Transactions on PAMI, Vol. 22, No. 8, Aug. 2000.

[6] Tremeau A., Colantoni P., Region Adjacency Graph Applied to Color image segmentation,Universite Jean Monnet, Institut d’lngenierie de la Vision – IEEE Trans Image Process.;9(4):735-44, 2000.

[7] Zhang Y., Cheng X., Medical Image Segmentation Based on Watershed and Graph Theory , 3rd International Congress on Image and Signal Processing, 2010.

156 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Perfect Mannheim, Lipschitz and Hurwitz Weight Codes Murat Güzeltepe Sakarya University, Sakarya, Turkey, [email protected]

Abstract. If a code attains an upper bound (the sphere-packing bound) in a given metric, then it is called a perfect code. Perfect codes have always drawn the attention of coding theorists and mathematicians since they play an important role in coding theory for theoretical and practical reasons. All perfect codes with respect to Hamming metric over finite fields are known [1-3] For non-field alphabets only trivial codes are known and by similar methods it was proved in [4] Perfect codes have been investigated not only with respect to Hamming metric but also other metrics, for example Lee metric. Lee metric was introduced in [5]. Some perfect codes with respect to Lee metric were discovered in [3]. In this paper, upper bounds on codes over Gaussian integers, Lipschitz integers and Hurwitz integers with respect to Mannheim metric, Lipschitz and Hurwitz metric are given. Keywords. Block codes, Lipschitz distance, Mannheim distance, perfect code. AMS 2010. 94B05, 94B15.

References

[1] J. H. van Lint, "Nonexistence theorems for perfect error-correcting codes," in Computers in Algebra and Number Theory, vol. IV, SIAM-AMS Proceedings, 1971.

[2] A. Tietäväinen, "On the nonexistence of perfect codes over finite fields," SIAM J. Appl. Math., vol. 24, pp. 88-96, 1973.

[3] S. Jain, K. Nam and K. Lee, "On some perfect codes with respect to Lee metric," Linear Algebra and Appl., vol. 405, pp.104-120, 2005.

[4] C. Martinez, R. Beivide and E. Gabidulin, “Perfect Codes from Cayley Graphs over Lipschitz Integers". IEEE Trans. Inf. Theory, Vol. 55, No. 8, Aug. 2009.

[5] C.Y. Lee, Some properties of non-binary error correcting codes, IEEE Trans. Inform. Theory, vol. 4, pp. 77–82, 1958.

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Codes over Hurwitz Integers Murat Güzeltepe Sakarya University, Sakarya, Turkey, [email protected]

Abstract. Hamming and Lee distances have been revealed to be inappropriate metrics to deal with quadrature amplitude modulation (QAM) signal sets and other related constellations [1]. To solve this problem, different authors have constructed new error- correcting codes over fields or rings. For example, Huber discovered a new way to construct codes for two-dimensional signals in terms of Gaussian integers, i.e., the integral points on the complex plane [2]. His original idea is to regard a finite field as a residue field of the Gaussian integer ring modulo a Gaussian prime and, by Euclidean division, to get a unique element of minimal norm in each residue class, which represents each element of a finite field. Therefore, each element of a finite field can be represented by a Gaussian integer with the minimal Galois norm in the residue class; and the set of the selected Gaussian integers is called a constellation. Since the Galois norm of integral points on the complex plane coincides with the Euclidean metric, Huber's constellation is of minimal energy. Moreover, Huber introduced the Mannheim weight by means of the Manhattan metric of the constellation, and obtained linear codes which are of one Mannheim error-correcting capability. In [3], Huber developed his wonderful idea further to the Eisenstein integers, i.e., the algebraic integers of the cyclotomic field generated by the sixth roots of unity. Although Huber's work constitutes a relevant contribution, unfortunately the Mannheim distance is not a true metric as was proved in [5]. Later, T. P. da Nobrega Neto et al. in [4] discussed the algebraic integer rings of quadratic fields which are Euclidean norm, and proposed a new class of linear codes. In [4], codes over the ring []i of Gaussian integers and codes over the ring Ap []ρ of Eisenstein- Jacobi integers were presented. The metric used in [4] is inspired by Mannheim metric. In this study, we obtain new classes of linear codes over Hurwitz integers equipped with a new metric. We refer to the metric as Hurwitz metric. Also, we define decoding algorithms for these codes when up to two coordinates of a transmitted code vector are affected by error of arbitrary Hurwitz weight. The interest in the codes with respect to Hurwitz metric is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, whereas the Hamming metric and the Lee metric is not appropriate for this purpose.

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Keywords. Block codes, Cyclic codes, Syndrome decoding, Hurwitz metric. AMS 2010. 94B05, 94B15, 94B35, 94B60.

References

[1] S. W. Golomb and L. R. Welch, "Perfect codes in the Lee metric and the packing of polyominoes," SIAM J. Applied Mathematics, vol. 18, No 2, pp.302-317, 1970.

[2] K. Huber., "Codes Over Gaussian integers," IEEE Trans. Inform.Theory, vol. 40, pp. 207- 216, Jan. 1994.

[3] K. Huber., "Codes Over Eisenstein-Jacobi integers," AMS. Contemp. Math., vol. 158, pp.165-179, 2004.

[4] T. P. da N. Neto, J. C. Interlando, M. O. Favareto, M. Elia and R. Palazzo Jr., "Lattice constellation and codes from quadratic number fields," IEEE Trans. Inform. Theory, vol. 47, No. 4, pp. 1514-1527, May. 2001.

[5] C. Martinez, R. Beivide and E. Gabidulin, "Perfect Codes from Cayley Graphs over Lipschitz Integers," IEEE Trans. Inf. Theory, Vol. 55, No. 8, pp. 3552-3562, Aug. 2009.

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A Study on Minimal Codewords of the Codes Associated with the Symmetric v,( k, λ) − Designs Selda Çalkavur (1) and Erol Balkanay (2) (1) Kocaeli University, Kocaeli, Turkey, [email protected] (2) Istanbul Kultur University, Istanbul, Turkey, [email protected]

Abstract. It is important that a secret key, passwords, informations of the plan of a secret place or an important formula of a product or i.e. must be kept secret. For the secret sharing problem the main problem is to divide the secret into pieces instead of storing the whole. The secret sharing schemes were introduced by Blakley [2] and Shamir [10] in 1979. Since then, many constructions have been proposed. In a secret sharing scheme, to know whether of codewords are minimal is an important problem. So, in this paper, we prove two theorems about minimal codewords. These theorems are given below. Theorem. Let C be the binary code of a symmetric v,( k, λ) − design. If k + λ)(2 w < , max λ then all nonzero codewords in a dual code C ⊥ are minimal.

In this theorem, the wmax is maximum nonzero weight in the code C . Theorem. Let A be the incidence matrix of the symmetric v,( k, λ) − design. All the nonzero codewords in the dual code C ⊥ of the binary code C which is generated by the rows of the matrix 1     D =   A   1   + kv )(2 are minimal if w < . max k In this theorem, the matrix A is the incidence matrix of a − v,(2 k, λ) design.

Keywords. Linear code, the code of a symmetric design, secret sharing scheme, minimal codeword. AMS 2010. 14G50, 94A60, 94C30.

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References [1] Ashikmin, A. and Barg, A. “Minimal Vectors in Linear Codes”, IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 2010-2017, Sep. 1998.

[2] Blakley, G. R. “Safeguarding Cryptographic Keys”, in Proc. 1979 National Computer Conf., New York, Jun. 1979, pp. 313-317.

[3] Chaudhuri, D. R. (Editor) Coding Theory and Design Theory, “Designs, Intersection Numbers and Codes, Shrikhande, M. S.” Part II, Design Theory with 32 Illustrations pp. 304- 317, Springer-Verlag.

[4] Çalkavur, S. “Symmetric Designs, Codes and a Study on Secret Sharing Schemes”, Doctoral Thesis pp. 59-63. İstanbul: İstanbul Kültür University, July 2010.

[5] Ding, C. and Yuan, J. “Covering and Secret Sharing with Linear Codes” in Discrete Mathematics and Theoretical Computer Science (Lecture Notes in Computer Science). Berlin, Germany : Springer-Verlag, 2003, vol. 2731, pp. 11-25.

[6] Hill, Raymond. “A First Course in Coding Theory”, Oxford: Oxford University, 1986.

[7] Lander, E. S. “Symmetric Designs an Algebraic Approach”, Cambridge: Cambridge University, 1983.

[8] Massey, J. L. “Minimal Codewords and Secret Sharing”, in Proc. 6th Joint Swedish- Russian Workshop on Information Theory, Mölle, Sweden, pp. 276-279, Aug. 1993.

[9] Özadam, H. “Constructing of Secret Sharing Schemes Using Linear Codes”, IAM 589 Term Project. Ankara: Middle East Technical University Institute of Applied Mathematics.

[10] Shamir, A. “How to Share a Secret”, Commun. Assoc. Comp. Mach., vol. 22, pp. 612- 613, 1979.

[11] Yuan, J. and Ding, C. “Secret Sharing Schemes from Two Weight Codes”, in Proc. R. C. Bose Centenary Symp. Discrete Mathematic and Applications, Koklata, India, Dec. 2002.

[12] Yuan, J., Ding, C., Senior Member, IEEE, “Secret Sharing Schemes from Three Classes of Linear Codes”, IEEE Trans. on Inf. Theory, vol. 52, no. 1, pp. 206-212, Jan. 2006.

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On MacWilliams Identity for M-spotty Weight Enumerators of Linear Codes over Frobenius Rings Vedat Siap (1) and Mehmet Ozen (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. A new class of byte error control codes called spotty byte error control codes has been specifically designed to fit the large capacity memory systems that use high- density random access memory chips with input/output data of 8, 16, and 32 bits [1-6]. In this work, we present a MacWilliams type identity for spotty byte error control codes with the m- spotty metric over finite commutative Frobenius rings. Keywords. MacWilliams identity, weight enumerator, m-spotty weight. AMS 2010. 94B05, 94B60.

References

[1] Umanesan, G., Fujiwara, E., A class of random multiple bits in a byte error correcting and single byte error detecting codes, IEEE Trans. Comput., 52 (7), 835-847, 2003.

[2] Umanesan, G., Fujiwara, E., A class of codes for correcting single spotty byte error, IEICE Trans. Fundam., E86-A, 704-714, 2003.

[3] Suzuki, K., Kashiyama, T. and Fujiwara, E., A general class of m-spotty byte error control codes, IEICE Trans. Fundam., E90-A (7), 1418-1427, 2007.

[4] Suzuki, K., Fujiwara, E., MacWilliams identity for m-spotty weight enumerator, IEICE Trans. Fundam., E93-A (2), 526-531, 2010.

[5] Ozen, M., Siap, V., The MacWilliams identity for m-spotty weight enumerators of linear codes over finite fields, Comput. Math. Appl., 61 (4), 1000-1004, 2011.

[6] Ozen, M., Siap, V., The MacWilliams identity for m-spotty Rosenbloom-Tsfasman weight enumerator, J. Franklin Inst., http://dx.doi.org/10.1016/j.jfranklin.2012.06.002, 2012.

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GEOMETRY

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Hypercomplex Structures A.A. Salimov Ataturk University, Erzurum, Turkey, [email protected]

Abstract. A hypercomplex algebra is a real associative algebra with unit 1. A poly- affinor structure on a manifold is a family of endomorphism fields i.e. tensor fields of type (1,1) ) If poly-affinor structure is an algebra (under the natural operations) isomorphic to a hypercomplex algebra, the poly-affinor structure is called hypercomplex. In this paper we define some tensor operators which are applied to pure tensor fields. Using these operators we study some properties of integrable commutative hypercomplex structures endowed with a holomorphic torsion-free pure connection whose curvature tensor satisfy the purity condition with respect to the covariantly constant structure affinors. Keywords. Pure tensors and connections, Holomorphic tensors and connections, Tachibana operator, Norden metric, Regular representations. AMS 2010. 53C15, 53B05, 15A69, 16G60, 32A10.

163 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the Energy and the Angle of a Frenet Vector Fields Ayşe Altın Hacettepe University, Ankara, Turkey, [email protected]

Abstract. In this paper we compute the energy of a Frenet vector field and angle between Frenet vectors for a givien curve C in n-dimensional Euclidien space We observe that the energy and angle may be expressed in the curvature functions of C Keywords. Energy, Frenet vector field Sasaki metric. AMS 2010. 53A04, 53C04.

References

[1] Altın, A.: The Angles Between Serre-Frenet Frames of the Curves in Euclidian Space. The Journal of the Indian Academy of Mathematics. 1999, Vol.21, No.1, p.59-63.

[2] Chacón, P. M.; Naveira, A. M.; Weston, J. M.: On the Energy of Distributions, with Application to the Quarternionic Hopf Fibration. Monatshefte fûr Mathematik.2001, 133. p.281-294.

[3] Chacón, P.M.;Naveira, A.M.: Corrected Energy of Distributions on Riemannian Manifold. Osaka Journal Mathematics. 2004, Vol.41, p.97-105.

[4] Gerretsen,J.C.H.: Lectures on Tensor Calculus and Differential Geometry, Noordhoff, Groningen,1962.

[5] Higuchi,A., Kay, B.S., Wood, C. M.: The energy of unit vector fields on the 3-sphere. Journal of Geometry and Physics. 2001, Vol.37, 137–155.

[6] O'Neill, B.:Elementary Diffrential Geometry, Academic Press Inc.,1966.

[7] Wood, C. M.: On the Energy of a Unit Vector Field. Geometrae Dedicata. 1997, 64, p. 319-330.

164 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Gauss Equations of Timelike Surface Invaryant under One Parameter Group of Lorentzian Isometry Abdullah İnalcık (1) and Soley Ersoy (2) (1) Artvin Coruh University, Artvin, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this study, we obtain some fundamental theorems in order to apply them to timelike surfaces invaryant under one parameter Lorentzian group of isometries. In this manner, we get the Gauss equations of timelike surfaces in class of helicoidal surface, generalized cylinders and surfaces of revolution and also in subclass of maximal, Delaunay and isothermic surfaces. Moreover, we invistigate the solutions of these differential equations and find the second fundamental forms of these timelike surfaces by these solutions.

Keywords. Lorentz metric, Global surface theory, Gaussian curvature, Gauss equation, mean curvature, isothermic, deformation. AMS 2010. 53B30, 53C45.

References

[1] McNERTNEY, L.V., One-parameter families of surface with constant curvature in Lorentzian 3-space, PhD. Thesis, Brown University, 1980. [2] MIRA, P., PASTOR, J. A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math., 140(2003), 315-334. [3] LOPEZ, R., DEMİR, E., Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, 2010, arXiv:1006.2345v2.

n [4] MAGID, M. A., Lorentzian isothermic surfaces in  j . Rocky Mountain J. Math. Volume 35, Number 2 (2005), 627-640. [5] O’NEILL, B., Semi-Riemannian Geometry, Academic Press, New York 1983. [6] INALCIK, A., ERSOY, E., Non-null helicoidal surface as non-null Bonnet surfaces, (submitted). [7] STEPHANIDIS, NK., Differential Geometry, Thessaloniki, Greece, 1987 [8] ROUSSOS, I. M., Surface in E3 invaryant under a one parameter group of isometries of E3 , An. Acad. Bras. Ci, 72(2000), 125-151. [9] MANHART, F., Affine surface with planar affine normals in 3-dimensional 3 Minkowski space 1 , Mathematica Pannonica, 17(2006), 69-82. [10] CHEN, W., LI, H., On the classification of the timelike Bonnet surfaces, Geometry and Topology of Submanifolds, 10, Chern, S. Chen, W., Shelton Street, Covent Garden, London, 18-31, 1999.

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On Inclined Curves in Euclidean n -Space Ali Şenol (1), Evren Zıplar (2),Yusuf Yaylı (3) and H. H. Hacısalihoğlu (4) (1) Cankiri Karatekin University, Cankiri, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected] (3) Ankara University, Ankara, Turkey, [email protected] (4) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]

Abstract. We give a characterization for inclined curves in n -dimensional Euclidean space E n with the Theorem “ A curve is an inclined curve in E n if and only if

n−2 2 ≠ ∑ Hi =constant and Hn−2 0 , where {HH12, ,..., Hn− 2} be the harmonic curvature functions i=1 of the curve”. Moreover, we specify a new characterization fort he inclined curves in E n−1 . Keywords. Inclined curve, Harmonic curvature, Frenet frame. AMS 2010. 14H45,14H50,53A04

References

[1] Camcı, Ç., İlarslan, K., Kula, L., Hacısalihoğlu, H.H., Harmonic curvatures and

n generalized helices in E , Chaos,Solitons and Fractals 40, 2590-2596, 2009.

[2] Di Scala, A.J., Ruiz-Hernandez, G., Higher codimensional euclidean helix submanifolds, Kodai Math. J. 33, 192-210, 2010.

[3] Özdamar, E., Hacısalihoğlu, H.H., A characterization of inclined curves in euclidean n - space, Comm.Fac.Sci.Univ. Ankara 3,15-23, 1975.

[4] Scarr, G., Helical tensgrity as a structural mechanism in human anatomy, International journal of osteopathic medicine 14, 24-32,2011.

166 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Three-dimensional Normal Almost Contact Metric Manifolds Admitting Quarter Symmetric Non-metric Connection Ahmet Yıldız (1) and Azime Çetinkaya (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Dumlupinar University, Kutahya, Turkey, azzimece@hotmail

Abstract. In this paper we study Ricci solitons and gradient Ricci solitons on a three- dimensional normal almost contact metric manifolds admitting quarter symmetric non- metric connection. At first we prove on a three-dimensional normal almost contact metric manifold given with quarter symmetric non-metric connection, Ricci soliton with a potential vector field V collinear with the characteristic vector field ξ has constant scalar curvature provided α,β=constant. Also we investigate gradient Ricci solitons for a three-dimensional normal almost contact metric manifolds admitting quarter symmetric non-metric connection. Finally we study a three-dimensional normal almost contact metric manifolds admitting Ricci solitons, which satisfies R.S=0, P.S=0 and Z.S=0 with quarter symmetric non-metric connection. Keywords. Ricci soliton, gradient ricci soliton, quarter symmetric non-metric connection, normal almost contact metric manifold. AMS. 53D15, 53B05.

References

[1] Golab S., On semi-symmetric and quarter-symmetric linear connections, Tensor N.S., 29, 249-254, 1975.

[2] Hamilton R.S., The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, American Math. Soc., 237-262, 1988.

[3] Oubina J.A., New classes of almost contact metric structures, Pulb.Math.Debrecen, 32(1985), 187-193, 1979.

[4] Sengupta J. And Biswas B., Quarter-symmetric non-metric connection on a Sasakian manifold, Bull. Cal. Math. Soc., 95, 169-176, 2003.

167 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Weingarten and Linear Weingarten Type Tubes with Darboux Frame in E3 Ayşe Zeynep Azak (1) and Murat Tosun (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this study, we found that the tube surface with Darboux frame was also Weingarten surface in the Euclidean 3-space E3 . Then we get the necessary and sufficient conditions for being ()()HK,II ,, KKII types Weingarten and linear tube surface according to

Darboux frame. Here , KK II and H are Gauss curvature, second Gauss curvature and mean curvature of tube surface, respectively. Key Words. Tube, Weingarten surface, Darboux frame

AMS 2010. 53A04, 53A05, 53C40

References

[1] Blair, D. E., Koufogiorgos, Th., Ruled Surfaces with Vanishing Second Gaussian Curvature, Monatsh. Math., 113, 177-181, 1992.

[2] Doğan, F., Yaylı, Y., Tubes with Darboux Frame, Int. J. Contemp. Math. Sciences, 7, 16, 751-758, 2012.

[3] Kühnel, W., Ruled W-surfaces, Arch. Math., 62, p.p. 475-480, 1994.

[4] Kühnel, W., Steller, M., On Closed Weingarten Surfaces, Monatshefte für Mathematik, 146, 2, 113-126, 2005.

[5] L´opez, R., Special Weingarten Surfaces Foliated by Circles, Monatsh.Math.,154, 4, 289- 302, 2008.

[6] O’Neill, B., Elementary Differential Geometry, Academic Press Inc., New York, 1966.

[7] Ro, J. S., Yoon, D. W., Tubes of Weingarten Types in a Euclidean 3-Space, Journal of Chungcheong mathematical Society, 22, 3, 359-366, 2009.

168 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Semiparallelity of Wintgen Ideal Surfaces Betül Bulca (1) and Kadri Arslan (2) (1) Uludag University, Bursa, Turkey, [email protected] (2) Uludag University, Bursa, Turkey, [email protected]

Abstract. Wintgen ideal surfaces in E form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously,⁴ Wintgen ideal surfaces satisfy the 2+d equality K+KN=‖H‖². In the present study we consider Wintgen ideal surfaces in E . We have shown that a Wintgen ideal surfaces in E2+d satisfying the semiparallelity condition R(X,Y) h=0 are normally flat. We give some examples of these type of surfaces in E .

Keywords.⋅ Wintgen ideal surfaces, Semiparallel surface, Ellipse of curvature⁴. AMS 2010. 53C40, 53C42.

References

[1] Chen, B-Y. Geometry of Submanifols,. Dekker, New York(1973).

[2] Chen, B.-Y. Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvature, Ann.Global Anal. Geom., 38 (2010), 257-265.

[3] DeSmet, P.J.,Dillen, F.,Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory.

[4] Decu, S., Petrovi´c-Torgašev, M., Verstraelen, L.: On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds. Tamkang J. Math. 41(2) (2010) (in print).

[5] Ge J. and Tang Z. A proof of the DDVV conjecture and its equality case, arXiv:0801.0650v2 [math.DG] 11 Jan 2008.

[6] Özgür, C. Arslan. and Murathan, C. On a Class of Surfaces in Euclidean Spaces,Commun. Fac. Sci. Univ. Ank. SeriesA1, 52(2002), 47-54.

[7] Deprez, J. Semi-parallel surfaces in Euclidean space, Journal of Geometry, 25(1985), 192- 200.

169 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Umbrella Motion in R3 Bülent Karakaş (1) and Şenay Baydaş (2) (1) Yuzuncu Yil University, Van, Türkiye, [email protected] (2) Yuzuncu Yil University, Van, Türkiye, [email protected]

Abstract. In this article we define umbrella motion in R3 as two parametric motion and study some properties of this motion. In addition we give Matlab applications. Keywords. Motion, umbrella, two parametric motion. AMS 2010. 70E17, 83C10, 70B10

References

[1] Baydas, S., Karakas, B., Modelling of the 3R motion at non-parallel planes, Journal of Informatics and Mathematical Sciences, 4, 1, 85-92, 2012.

[2] Davydov, A. A., Whitney umbrella and slow-motion bifurcations of relaxaiıon-type equations, Journal of Mathematical Sciences, 126, 4, 1251-1258, 2005.

[3] Karakas, B., Baydas, S., A Non-Rigid Symmetric Motion With One Center: NRS Motion, YYU, BAPB, 2011.

[4] Kuruoğlu, N., Hacısalihoğlu, H. H., On the Lie group of umbrella matrices. Communications De La Faculte Des Sciences De L’Universite D’Ankara, 32, 131-144, 1983.

[5] Tagare, H. D., O’Shea, D., Groisser, D., Non-Rigid Shape Comparison of Plane Curves in Images, Journal of Mathematical Imaging and Vision, 16, 57-68, 2002.

170 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Umbrella Surfaces in R3 as an Orbit Surfaces Bülent Karakaş (1) and Şenay Baydaş(2) (1) Yuzuncu Yil University, Van, Turkey, [email protected] (2) Yuzuncu Yil University, Van, Turkey, [email protected]

Abstract. In this study we define umbrella surfaces as an orbit surfaces of umbrella motions. Matlab applications and some examples are given. Keywords. Umbrella, surface, motion, two parametric motion. AMS 2010. 97G70, 53A05.

References

[1] Baydas, S., Karakas, B., Modelling of the 3R motion at non-parallel planes, Journal of Informatics and Mathematical Sciences, 4, 1, 85-92, 2012.

[2] Darafsheh, M.R., Farjami, Y., Ashrafi, A.R., Hamadanian, M., Full non-rigid group of Sponge and Pina, Journal of Mathematical Chemistry, 41, 3, 315-326, 2007.

[3] Davydov A. A., “Whitney umbrella and slow-motion bifurcations of relaxaiıon-type equations, Journal of Mathematical Sciences, 126, 4, 1251-1258, 2005.

[4] Getino, J., Gonzalez, A. B., Escapa, A., The rotation of a non-rigid, nonsymmetrical earth II: Free nutations and dissipative effects. Celestial Mechanics and Dynamical Astronomy, 76, 1-21, 2000.

[5] Karakaş, B., Baydaş, S., A Non-Rigid Symmetric Motion With One Center: NRS Motion, YYU, BAPB, 2011.

[6] Kuruoğlu, N., Hacısalihoğlu, H. H., On the Lie group of umbrella matrices. De La Faculte Des Sciences De L’Universite D’Ankara, 32, 131-144, 1983.

[7] Wang, Q., Ai, H., Xu, G., Learning-Based Tracking of Complex Non-Rigid Motion. J. Comput. Sci. Technol., 19 , 4, 489-500, 2004.

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Anti-invariant Riemannian Submersions from Almost Hermitian Manifolds:Curvatures Bayram Şahin Inonu University, Malatya, Turkey, [email protected]

Abstract. We investigate curvature relations between the total manifold and the base manifold of an anti-invariant Riemannian submersion from an almost Hermitian manifold.

References

[1].Altafini, C. Redundant robotic chains on Riemannian submersions IEEE Transactions on Robotics and Automation, (2004), 20(2), 335-340.

[2]. Baird, P. and Wood, J. C. Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, No. 29, Oxford University Press, The Clarendon Press, Oxford, 2003.

[3]. Chinea, D. Almost contact metric submersions. Rend. Circ. Mat. Palermo, (1985), 34(1),89-104.

[4].Escobales, R. H. Jr. Riemannian submersions from complex projective space. J. Differential Geom., (1978), 13(1), 93-107.

[5]. Falcitelli, M., Ianus, S., Pastore, A. M., Riemannian Submersions and Related Topics. World Scientific, River Edge, NJ, 2004.

[6]. Gray, A. Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech., (1967), 16, 715-737.

[7]. Ianus, S., Mazzocco, R., Vilcu, G. E., Riemannian submersions from quaternionic manifolds. Acta Appl. Math., (2008), 104(1), 83-89.

[8].Marrero, J. C., Rocha, J. Locally conformal K\"{a}hler submersions. Geom. Dedicata, (1994), 52(3), 271-289.

[9]. O'Neill, B. The fundamental equations of a submersion, Mich. Math. J., (1966), 13, 458- 469.

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[10]. Ponge, R., Reckziegel, H. Twisted products in pseudo-Riemannian geometry. Geom. Dedicata, (1993), 48(1), 15-25.

[11]. Şahin, B., Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math. 8(3), (2010), 437-447.

[12]. Watson, B. Almost Hermitian submersions. J. Differential Geometry, (1976), 11(1), 147-165.

[13]. Yano, K. and Kon, M. Structures on Manifolds, World Scientific, Singapore, 1984.

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B.-Y. Chen Inequalities for Submanifolds of a Conformally Flat Manifold Cihan Özgür Balikesir University, Balikesir, Turkey, [email protected]

Abstract. In this study, we prove B.-Y. Chen inequalities for submanifolds of a conformally flat manifold. Some relations between the mean curvature, scalar and sectional curvatures of the ambient space are found. Some generalizations of the previously proven results are obtained. Keywords. Conformally flat manifold, B.-Y. Chen inequality. AMS 2010. 53C40, 53C42

References

[1] Chen, B.-Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math. (Basel), 60, 568-578, 1993.

[2] Chen, B.-Y., Strings of Riemannian invariants, inequalities, ideal immersions and their applications, The Third Pacific Rim Geometry Conference (Seoul, 1996), 7--60, Monogr. Geom. Topology, 25, Int. Press, Cambridge, MA, 1998.

[3] Chen, B.-Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasg. Math. J., 41, 33-41, 1999.

[4] Chen, B.-Y., δ-invariants, Inequalities of Submanifolds and Their Applications, in Topics in Differential Geometry, Eds. A. Mihai, I. Mihai, R. Miron, Editura Academiei Romane, Bucuresti, 29-156, 2008.

[5] Chen, B.-Y., Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.) 26, 105-127, 2000.

[6] Chen, B.-Y., Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.

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On the Two Parameter Homothetic Motions in the Complex Plane Doğan Ünal (1), Muhsin Çelik (2) and Mehmet Ali Güngör (3) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. One and two parameter planar motions investigated in a detailed manner [1]. Moreover, the relations between the complex velocities one and two parameter motions in the complex plane were provided by [1]. In this study, two parameter planar and planar homothetic motions are defined in the complex plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Keywords. Two parameter motions, homothetic motions, complex plane. AMS 2010. 53A17.

References

[1] H.R. Müller, Kinematik Dersleri. Ankara Üniversitesi Fen Fakültesi Yayınları, Um.96- Mat No:2, 1963.

[2] O. Bottema, B. Roth, Theoretical Kinematics. North Holland Publ. Com., 1979.

[3] A. Karger, J. Novak, Space Kinematics And Lie Groups. Breach Science Publishers S.A. Switzerland, 1985.

[4] M.K. Karacan, İki Parametreli Hareketlerin Kinematik Uygulamaları. Doktora Tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü, 2004.

[5] M. Düldül, Kompleks Düzlemde 1-Parametreli Hareketler ve Holditch Teoremi. Yüksek Lisans Tezi, Ondokuz Mayıs Üniversitesi Fen Bilimleri Enstitüsü, 2000.

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e-Curvature Functions in 4-Dimensional Lorentzian Space Esen İyigün Uludag University, Bursa, Turkey, [email protected]

Abstract. In this study; we define e-curvature functions in 4-dimensional Lorentzian space.We find a relation between Frenet formulas and e-curvature functions, and also results related to ccr-curves, harmonic curvatures and e-curvature functions. Keywords. e-curvature functions, harmonic curvatures, constant curvature ratios. AMS 2010. 53C40, 53C42.

References

[1] O'Neill, B., Semi-Riemannian geometry with applications to relativity. Academic Pres, New-York, (1983).

[2] İyigün, E., Arslan, K., On harmonic curvatures of curves in Lorentzian n-space. Commun. Fac. Sci.Univ. Ank. Series A1, V.54, No(1) , pp.29-34, (2005).

[3] Ekmekçi, N., Hacisalihoglu, H.H., İlarslan, K., Harmonic curvatures in Lorentzian space. Bull. Malaysian Math. Sc. Soc. (Second Series) 23(2000), 173-179.

[4] Petrovic-Torgasev, M., Sucurovic, E., W-curves in Minkowski Space-Time. Novi Sad J. Math.,Vol. 32, No.2, (2002), 55-65.

[5] Yılmaz, S., Özyılmaz, E., Turgut, M., The Differential Geometry Of The Curves In Minkowski Space-Time II. International Journal of Computational and Mathematical Sciences, 3 (2) (2009).

176 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

f -Eikonal Helix Submanifolds and f -Eikonal Helix Curves Evren Zıplar (1) , Ali Şenol (2) and Yusuf Yaylı (3) (1), (3) Ankara University, Ankara, Turkey, [email protected] (2) Cankiri Karatekin University, Cankiri, Turkey, [email protected]

Abstract. Let M ⊂ n be a Riemannian helix submanifold with respect to the unit direction d ∈ n and fM: →  be a eikonal function. We say that M is a f -eikonal helix submanifold if for each qM∈ the angle between ∇f and d is constant. Let M ⊂ n be a Riemannian submanifold and α : IM→ be a curve with unit tangent T . Let fM: →  be a eikonal function. We say that α is a f -eikonal helix curve if the angle between ∇f and T is constant along the curve α . ∇f will be called as the axis of the f -eikonal helix curve. The aim of this article is to give that the relations between f -eikonal helix submanifolds and f - eikonal helix curves, and to investigate f -eikonal helix curves on Riemannian manifolds. Keywords. Helix submanifold, Eikonal function, Helix line. AMS 2010. 53A04, 53B25,53C40,53C50

References [1] Barros, M., General helices and a Theorem of Lancret, Proceedings of the American Mathematical Society 125, 5, 1503-1509, 1997.

[2] Cermelli, P., Di Scala, A.J., Constant angle surfaces in liquid crystals, Philos. Mag. 87, 1871-1888, 2007.

[3] Di Scala, A.J., Ruiz-Hernandez, G., Higher codimensional euclidean helix submanifolds, Kodai Math. J. 33, 192-210, 2010.

[4] Dillen, F., Fastenakels, J., Van der Verken, J., Vrancken, L., Constant angle surfaces in S 2 ×  , Monatsh. Math. 152, 89-96, 2007.

[5] Dillen, F., Munteanu, M.I., Constant angle surfaces in H 2 ×  , Bull. Braz. Math. Soc. 40, 85-97, 2009.

[6] Ghomi, M., Shadows and convexity of surfaces, Ann. Math. 155, 281-293,2002.

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[7] Hicks, N.J., Notes on differential geometry, Van Nostrand Reinhold Company, London, 1974.

[8] Hitzelberger, P., Lytchak, A., Spaces with many afine functions, Proceedings of The American Mathematical Society 135, 2263-2271, 2007.

[9] Innami, N., Splitting Theorems of Riemannian manifolds, Compositio Mathematica 47, 237-247, 1982.

[10] Ruiz-Hernandez, G., Helix, shadow boundary and minimal submanifolds, Illinois J.Math 52, 1385-1397, 2008.

[11] Sakai, T., On Riemannian manifolds Admitting a function whose gradient is of constant norm, Kodai Math.J. 19, 39-51, 1996.

[12] Şenol, A., Yaylı, Y., LC helices in space forms, Chaos Solitions&Fractals 42, 2115- 2119,2009.

178 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Fractals of Finite Area Figen Çilingir Cankaya University, Ankara, Turkey, [email protected]

Abstract. When the relaxed Newton’s method is applied to the func tion, F(z) = P(z)Exp[Q(z)], where P and Q are polynomials, the basin of attraction of a root of F has a finite area if the degree of Q exceeds 2. In this presentation, it will be shown that the beautiful images of these areas which are the fractals of the basins of attraction of roots of complex functions. These basins are obtained under condations for the polynomial P, if the degree of Q is less than 3. The striking point about the basins of attraction for the roots of the function F approximated by the relaxed Newton’s method is that they are called Julia sets of the rational function of NF_h(z), where h (0,1]. Especially, the Julia sets on the fractal image will be described. ∈ Keywords. Relaxed Newton’s method, Julia sets, rational iteration.

References. [1] Beardon, A. Iteration of Rational Functions. Sprringer-Verlag, 1991.

[2] Çilingir, F. The Dynamics of Relaxed Newton’s Method on the Ex- ponential function and its Fractals, Thesis,Georg-August-Universiteat, Go ̈ttingen, 2002.

[3] Çilingir, F. Finiteness of the Area of Basins of Attraction of Re- laxed Newton Method for Certain Holomorphic Functions, Interna- tional Journal of Biffurcation and Chaos, Vol. 14(12), 2004, pp. 4177- 4190.

[4] Çilingir, F. On infinite area for complex exponential function. Chaos, Solitons and Fractals Vol. 22, 2004, pp. 1189-1198.

[5] Haruta, M. Newton’s method on the complex exponential function. Trans. Amer. Math. Soc. Vol. 351(6), 1999, pp. 2499–2513.

[6] Milnor, J. ”Dynamics in One Complex Variable”, Vieweg, 1999.

179 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Half-lightlike Hypersurfaces with Planar Normal Sections in ퟒ Feyza Esra Erdoğan (1), Rıfat Güneş (2) and Bayram Şahin (3)푹 ퟐ (1) Adiyaman University, Adiyaman, Turkey, [email protected] (2) Inonu University, Malatya, Turkey, [email protected] (3) Inonu University, Malatya, Turkey, [email protected]

Abstract. A half-lightlike hypersurface M in is said to have planar normal sections 4 if normal sections of M are planar curves. In the present푅2 paper we investigate necessary and sufficient conditions for a half-lightlike hypersurface in to have degenerate and non- 4 degenerate planar normal sections, respectively. 푅2 Keywords. Half-Lightlike Hypersurface, Planar Curve, Degenerate Planar Normal Section, Non-degenerate Planar Normal Section. AMS 2010. 53C50.

References

[1] B. O'Neill, Semi Riemannian Geometry with Applications to Relativity (Academic Press,1983)

[2] K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Academic Press, 2010.

[3] B-Y. Chen, Classification of Surfaces with Planar Normal Sections, Journal of Geometry Vol.20, (1983).

[4] Y. H. Ki, Surfaces in a Pseudo-Euclidean Space With Planar Normal Sections, Journal of Geometry, Vol.35, (1989).

180 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Weyl Manifolds with Semi-Symmetric Reccurent Metric Connection Fatma Özdemir Istanbul Technical University, Istanbul, Turkey, [email protected]

Abstract. Semi-symmetric metric connectin manifolds are widely examined in literature [1-5]. In this work, we study Weyl manifolds with semi-symmetric recurrent metric connection. We obtain a condition for such spaces to be isotropic and projective flat. Keywords. Weyl manifold, semi symmetric connection, reccurent metric connection. AMS 2010. 53A30, 53A40.

References

[1] Canfes, E.Ö., Özdeğer, A., Some applications of prolonged covariant differentiation in weyl spaces, Journal of Geometry., Vol.60, 7-16, 1997.

[2] Eisenhart, L.P., Non-Riemannian geometry, New York, American Mathematical Society, 1927.

[3] Özdemir, F., Yıldırım Çivi G., On conformally recurrent Kahlerian-Weyl spaces, Topology and Its Applications., 153, 477-484, 2005.

[4] Yano., K., On semi-symmetric metric connection, Type, Rev. Roumania Math. Prues Appl., 15,1579-1586, 1970.

[5] Liang, Y., On semi-symmetric and reccurent metric connection, Tensor,N.S., 55, 134-138, 1988.

181 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Euler’s Formula and De-Moivre’s Formula for Dual Hyperbolic Quaternions Hidayet Hüda Kösal (1) Mahmut Akyiğit (2) and Murat Tosun (3) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this paper, we define dual hyperbolic quaternions. Moreover, Euler’s formula and De Moivre’s formula are given for dual hyperbolic quaternions. Also, it is shown that there is only one quaternion satisfying qpn = for any nZ∈ . Key Words. Hyperbolic quaternion, Hyperbolic numbers, Free module, Euler’s formula, De Moivre’s formula.

References

[1] C. Muses, Applied Hypernumbers: Computational concept. Appl. Math. Comput., 3, (1976), 211-216.

[2] S. Demir, M. Tanışlı, N. Candemir, Hyperbolic Quaternions Formulation of Electromacnetism. Adv. Appl. Clifford Algebras, 20, (2010), 547-563.

[3] K. Carmody, Circular and Hyperbolic Quaternions, Octanions and Sedenions-Further Result. Appl. Math. And Comput., Volume 84, (1997), 27-47.

[4] E. Cho., De Moivre’s Formula for Quaternions. Appl. Math. Lett., 11 (6), (1998), 33-35.

[5] H. H. Hacısalihoğlu, Hareket Geometrisi ve Kuaterniyonlar Teorisi. Gazi Ünv. Publishing, 1983.

[6] M. Özdemir, The roots of a split quaternion. Applied Mathematics Letters, Volume 22, (2009) 258-263.

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On the Geometry of the Sasaki semi-Riemann Manifold on the Tangent Sphere Bundle of the de-Sitter Space İsmet Ayhan Pamukkale University, Denizli, Turkey, [email protected]

Abstract. The Sasaki semi-Riemann metric g S on the unit tangent sphere bundle

2 2 ST 11 of the unit 2-sphere S1 in Minkowski Space-Time called 2-dimensional de-Sitter space

2 is obtained by using the parametric representation of the S1 . The coefficients of the Levi- Civita connection of the Sasaki semi-Riemann manifold on tangent sphere bundle of the de-

2 S Sitter space 11 gST ),( are found. Moreover, the differential equations of geodesics of

2 S 11 gST ),( are calculated. Finally, the coefficients of the Riemann curvature tensor of

2 S 11 gST ),( are obtained. Keywords. The tangent bundle of the de-Sitter space, Sasaki semi-Riemann metric. AMS 2010. 55R25, 53C25.

References [1] Klingenberg, W., and Sasaki, S., On the tangent sphere bundle of a 2-sphere. Tohuku Math. Journ. 27, 49-56, 1975.

[2] Nagy, P.T., On the tangent sphere bundle of a Riemannian 2-manifold. Tohuku Math. Journ. 29, 203-208, 1977.

[3] O’Neill, B. Semi-Riemannian Geometry with applications to relativity. Academic Press, New York, 1997 [4] Sasaki, S., Geodesic on the tangent sphere bundles over spaces forms. Journ. Für die reine und angewandte math.288, 106-120, 1976.

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Notes on Curves in Sasakian 3-manifolds İsmail Gök (1) and Yusuf Yaylı (2) (1) Ankara University, Ankara, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected]

Absract. In this paper, we study some special curves in 3 -dimensional Sasakian manifolds. Cho et.al ]7[ study Bertrand-Lancret-de Saint Venant type problems for slant curves in Sasakian 3 -manifolds. We consider a curve whose normal vector field makes a constant angle with the characteristic vector field ξ and give some characterizations. Keywords. Sasakian manifold, helices, slant helices. AMS 2010. 53C25, 53B25

References ]1[ C. Baikoussis and D. E. Blair, Finite type integral submanifold of the contact manifold , Bull. Math. Acad. Sinica , 19, (1991), 327-350.

]2[ C. Baikoussis and D. E. Blair, On Legendre curves in contact 3 -manifolds, Geom. Dedicata, 49, (1994), 135-142.

]3[ D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer, Berlin, Hiedelberg, New York, 1976.

]4[ C. Özgür and ¸S. Güvenç, On some types of slant curves in contact pseudo-Hermitian 3-manifolds, Annales Polonici Math., in press.

]5[ C. Camci and H.H. Hacisalihoglu, Finite type curve in 3 -dimensional Sasakian Manifold, Bull. Korean Math. Soc. 47 (2010), No. 6, pp. 1163-1170, DOI 10.4134/BKMS.2010.47.6.1163.

]6[ C. Camci; Y. Yayli; H. H. Hacisalihoglu, On the characterization of spherical curves in 3 -dimensional Sasakian spaces, J. Math. Anal. Appl., 342(2008), no. 2, 1151-1159. MR2445265 (2009k:53097).

]7[ J. T. Cho; J.-I. Inoguchi; J.-E. Lee, On slant curves in Sasakian 3 -manifolds, Bull. Austral. Math. Soc., 74(2006), no. 3, 359-367. MR2273746 (2007g:53059)

]8[ J. T. Cho; J.-E. Lee, Slant curves in contact pseudo-Hermitian 3 -manifolds, Bull. Aust. Math. Soc., 78(2008), no. 3, 383-396. MR2472274 (2009m:53135)

184 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Null Rectifying Curves in * Kazım İlarslan Kirikkale University, Kirikkale, Turkey, [email protected]

Abstract. In this paper we give the necessary and sufficient conditions for a null curves in 4-dimensional semi-Euclidean space with indeks 2 to be rectifying curves in terms of their curvature functions. Finally, we give some examples of the null rectifying curves in

. Keywords. Null curve, curvature functions, rectifying curve, semi-Euclidean space. AMS 2010. 53C50, 53C40 *This paper was supported by Project No. TUBITAK-210T151-2012-Kırıkkale-Turkey.

References

[1] Bonnor, W. B., Null curves in a Minkowski space-time, Tensor 20 (1969), 229-242.

[2] Chen, B. Y., When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110 (2003), 147-152.

[3] Chen, B. Y. and Dillen F., Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Academia Sinica 33, No. 2 (2005) 77-90.

[4] Duggal, K. L. and Jin, D. H., Null curves and hypersurfaces of semi-Riemannian manifolds, World Scientific, Singapore, 2007.

[5] İlarslan, K., Nešović, E. and Petrović-Torgašev, M., Some characterizations of rectifying curves in the Minkowski 3-space. Novi Sad J. Math. 33 (2003), no. 2, 23–32.

[6] İlarslan, K. and Nešović, E., On rectifying curves as centrodes and extremal curves in the Minkowski 3-space. Novi Sad J. Math. 37 (2007), no. 1, 53–64

[7] İlarslan, K. and Nešović, E., Some charactherizations of null, pseudo null and partially null rectifying curves in Minkowski space-time, Taiwanese J. Math., 12(5) (2008), 1035-1044

[8] İlarslan, K., Altın Erdem, H. and Kılıç, N., 4-Boyutlu, 2-İndeksli Yarı-Öklidyen Uzayda Rektifiyen Eğriler, (in Turkish, unpublished research project: TUBITAK-210T151) Kırıkkale, 2012.

185 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Special Curves in Sol Space Lütfü Sizer (1), Hülya Başeğmez (2), Mesut Altınok (3) and Levent Kula (4) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected] (3) Ahi Evran University, Kirsehir, Turkey, [email protected] (4) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In this paper, we consider Sol Space, one model space of Thurston’s 3- dimensional geometries. We characterize some special curves in Sol Space. Keywords. Special Curves, Sol Geometry. AMS 2010. 53A35, 14H50.

References [1] Ou, Y. and Wang, Z, Biharmonic maps into Sol and Nil spaces, http://arxiv.org/abs/math/0612329v1, 2006.

[2] Bölcskei, A. and Brigitta, S., Frenet formulas and geodesics in Sol Geometry, Beitrage Algebra Geometry, 48, No.2, 411-421, 2007.

[3] López, R. and Munteanu, M. I., Minimal translation surfaces in Sol, http://arxiv.org/abs/1010.1085, 2010.

[4] López, R. and Munteanu, M. I., Surfaces with constant mean curvature in Sol geometry, Differential geometry and its Applications, 29, 238-245, 2011.

[5] Bölcskei, A. and Szilagyi, B., Visualization of curves and Spheres in Sol Geometry, KoG 10, 27-32, 2006.

This work is supported by Ahi Evran University BAP Project with Project number FBA-11- 23.

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Fibonacci Split Quaternions Mahmut Akyiğit (1), Hidayet Hüda Kösal (2) and Murat Tosun (3) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this paper, Split Fibonacci quaternions, Split Lucas quaternions and generalized Split Fibonacci quaternions are examined. The relations between these quaternions are given. Keywords. Split quaternion, Fibonacci Split Quaternion, Lucas Split Quaternion AMS 2010. 11R52 51B20

References

[1] A. F. Horadam, “A Generalized Fibonacci Sequence”, American Math. Monthly, 68, pp.455-459, 1961.

[2] A. F. Horadam, “Complex Fibonacci Numbers and Fibonacci Quaternions”, Amer. Math. Monthly, 70, pp. 289-291, 1963.

[3] M. R. Iyer, “A Note On Fibonacci Quaternion”, The Fib. Quarterly, 3,225-229, 1969.

[4] M. N. S. Swamy, “On Generalized Fibonacci Quaternions”, The Fib. Quarterly, 5, pp.547-550, 1973.

[5] T. Koshy, “Fibonacci and Lucas Numbers with Applications”, A Wiley- Interscience publication, U.S.A., 2001.

[6] L. Kula, Bölünmüş Kuaterniyonlar ve Geometrik Uygulamaları, Ph.D Thesis, Ankara University, Institute of Science, Ankara, 2003.

[7] S. Vajda, “Fibonacci and Lucas Numbers and the Golden Section”, Ellis Horwood Limited Publ., England, 1989.

[8] M.R. Iyer “Some Results on Fibonacci Quaternions” The Fib. Quarterly, 7 (2), pp.201- 210, 1969.

187 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Note on Fibonacci Generalized Quaternions Mahmut Akyiğit (1), Hidayet Hüda Kösal (2) and Murat Tosun (3) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this paper, firstly we define Fibonacci generalized quaternions. Also, the properties of Fibonacci generalized quaternions and some sums formulas this quaternions are investigated. Keywords. Generalized quaternion, Fibonacci generalized, Lucas Generalized quaternion AMS 2010. 11R52 51B20

References

[1] Julia Tumasova Upton “The Hidden Subgroup Problem For Generalized Quaternions”, The University of Alabama, the Department of Mathematics, Phd. Thesis, Tuscaloosa, Alabama, 2009.

[2] R.Graves, Life of Sir William Rowan Hamilton. Ayer Co Pub, 1975.

[4] Pottman H., Wallner J., Computational Line Geometry. Springer-Verlag Berlin Heidelberg New York, 2000.

[6] A. F. Horadam, “A Generalized Fibonacci Sequence”, American Math. Monthly, 68, pp.455-459, 1961.

[7] A. F. Horadam, “Complex Fibonacci Numbers and Fibonacci Quaternions”, Amer. Math. Monthly, 70, pp. 289-291, 1963.

[8] M. R. Iyer, “A Note On Fibonacci Quaternion”, The Fib. Quarterly, 3,225-229, 1969.

[9] M. N. S. Swamy, “On Generalized Fibonacci Quaternions”, The Fib. Quarterly, 5, pp.547-550, 1973.

188 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Slant Helices Generated by Plane Curves and Applications Mesut Altınok (1) and Levent Kula (2) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In this paper, we investigate the relationship between the plane curves and slant helices in . Moreover, we show how could be obtained to a slant helix from a plane 3 curve. Finally, weℝ give some slant helix examples generated by plane curves in Euclidean 3- space. Keywords. Plane curve, Slant helix. AMS 2010. 53A04, 14H50.

References

[1] Izumiya, S. and Takeuchi, N., Generic properties of helices and Bertrand curves, Journal of Geometry, 74, 97-109, 2002.

[2] Kula, L. and Yayli, Y., On slant helix and its spherical indicatrix. Applied Mathematics and Computation, 169, 600-607, 2005.

[3] Kula, L., Ekmekçi, N., Yayli, Y. and İlarslan, K., Characterizations of slant helices in Euclidean 3-space. Turkish J. Math., 34, 261-274, 2010.

[4] Lawrence, J. D., A catalog of special plane curves, Dover Publications, 1972.

[5] Altınok, M., The characterization of some special space curves with plane curves, Master thesis, Ahi Evran University, Graduate School of Natural and Applied Sciences, 2011.

189 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the Two Parameter Lorentzian Homothetic Motions Muhsin Çelik (1), Doğan Ünal (2) and Mehmet Ali Güngör (3) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. One and two parameter planar motions are investigated in a detailed manner [1]. Moreover, two parameter motions in three dimensional spaces are defined [2] and [3]. In this study, sliding velocity, pole lines, Hodograph and acceleration poles of two parameter Lorentzian homothetic motions at ( , ) positions are obtained. By defining two parameter

Lorentzian homothetic motion along∀ 휆a curve휇 in Lorentzian space , the theorems related to 3 this motion and characterizations of trajectory surface are given. ℝ1 Keywords. Two Parameter Motions, Homothetic Motions, Lorentzian Plane. AMS 2010. 53A17.

References

[1] H.R. Müller, Kinematik Dersleri. Ankara Üniversitesi Fen Fakültesi Yayınları, Um.96- Mat No:2, 1963.

[2] O. Bottema, B. Roth, Theoretical Kinematics. North Holland publ. Com., 1979.

[3] A. Karger, J. Novák, Space Kinematics And Lie Groups. Breach Science Publishers S.A. Switzerland, 1985.

[4] M.K. Karacan, İki Paramatreli Hareketlerin Kinematik Uygulamaları, Ph.D Thesis, Ankara University, Institute of Science, Ankara, 2004.

[5] B.O'Neill, Semi-Riemannian Geometry with Aplications to Relativity, Acedemic Press, London, 1963.

[6] L.Kula, Bölünmüş Kuaterniyonlar ve Geometrik Uygulamalar, Ph.D Thesis, Ankara University, Institute of Science, Ankara, 2004.

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On ξ -Conformally and ξ -Projectively Flat Lorentzian Sasakian Manifolds with Tanaka-Webster Connection Mehmet Erdoğan (1), Jeta Alo (2) and Gülşen Yilmaz (3) (1) Yeni Yuzyil University, Istanbul, Turkey, [email protected] (2) Beykent University Istanbul, Turkey, [email protected] (3) Yeni Yuzyil University, Istanbul, Turkey, [email protected]

Abstract. In this work, the Tanaka-Webster connection on a Lorentzian Sasakian manifold is defined and the notions ξ -conformally flat and ξ -projectively flat structures on a Lorentzian Sasakian manifold are introduced. After that, it is proved that if any Lorentzian Sasakian manifold with Tanaka-Webster connection is an η- Einstein manifold, then the

Tanaka-Webster connection ∇ˆ is ξ-conformally flat if the Lorentzian connection ∇ satisfies the condition R(X, Y )ξ = 0 and also it is ξ-projectively flat if and only if the Tanaka-Webster connection ∇ˆ is ξ-projectively flat. Furthermore, we give some structure theorems on various curvature tensors of Lorentzian Sasakian manifold with respect to the Tanaka-Webster connection. Keywords. Lorentzian Sasakian manifold, Tanaka-Webster connection, ξ-conformally flat, ξ -projectively flat. AMS 2010. 53C15, 53C25, 53D10

References

[1] Belkhelfa, M., Hirica, I.E., Rosca, R., Verstraelen, L., On Legendre curves in Riemannian and Lorentzian Sasaki spaces, Soochow Jour. Math., 28, No. 1, January(2002), 81-91.

[2] Blair, D.E., Contact manifolds in Riemannian Geometry, Lecture Notes in Math., 509 (1976), Springer-Verlag, Berlin-Heidelberg-New-York.

[3] Cabrerizo, J.L., Fernandez, L., Fernandez, M. and Zhen, G., The structure of a class of K- contact manifolds, Acta math.Hungar., 82(4) (1999), 331-340.

[4] De, U.C. and Biswas, S., A note on ξ-conformally flat contact manifolds, Bull. Malaysian Math. Sci. Soc., 29(1)(2006), 51-57.

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[5] Ikawa, T., Erdogan, M., Sasakian manifolds with Lorentzian metric, Kyungpook Math. J., 35 (1996), 517-526.

[6] Mishra, R.S., Structure on differentiable manifold and their applications, Chandrama Prakasana, Allahabad, India, 1984.

[7] Mondal, A.K., De, U.C., Some properties of a quarter-symmetric metric connection on a Sasakian manifold, Bull. Math.Analysis and Appl., Vol.1, Issue 3 (2009), 99-108.

[8] Shaikh, A.A. and Biswas, S., On LP-Sasakian manifolds, Bull. Malaysian Math. Sci. Soc., 27(2004), 17-26.

[9] Takahashi, T., Sasakian manifold with pseudo Riemannian metric, Tohoku Math J., 21 (1969), 271-290.

[10] Tanaka, N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan Jour. Math., 2 (1976), 131-190.

[11] Webster, S.M., Pseudohermitian structures on a real hypersurface, J. Differ. Geom., 13 (1978), 25-41.

[12] Weyl, H., Reine infinitesimalgeometrie, Math.Z., 2(3-4) (1918), 384-411.

[13] Weyl, H., Zur infinitesimalgeometrie, Einordnung der projektiven und der konformen Auffassung, Gottingen Nachrichten., (1921), 99-112.

[14] Yano, K. and Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom., 2(1968), 161-184.

[15] Zhen, G., Cabrerizo, J.L., Fernandez, L.M. and Fernandez, M., On ξ-conformally flat contact metric manifolds, Indian J. Pure Appl. Math., 28(6) (1997), 725-734.

192 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Special Curves in Three Dimensional Finsler Manifold F3

Mahmut Ergüt (1) and Mihriban Külahcı (2)

(1) Firat University, Elazig, Turkey, [email protected], (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this paper, we study AW(k)-type curves in three dimensional Finsler manifold and we give some characterizations related to these curves.

Keywords. AW(k)-type curve, Frenet formulas.

AMS 2010. 53A04, 53C40

References

[1] Bejancu, A. and Farran H.R., Geometry of Pseudo-Finsler Submanifolds, Kluwer Academic Publishers,Dordrecht, Boston, London, 2000.

[2] Arslan, K., Özgür, C., Curves and Surfaces of AW(k)-type, in: Geometry and Topology of Submanifolds, IX (Valencienne/ Lyan/ Leuven, 1997), World. Sci. Publishing, River Edge, NJ, 1999, pp. 21-26.

[3] Özgür, C., Gezgin, F., On Some Curves of AW(k)-type, Differ. Geo. Dyn. Syst. 7(2005), 74-80.

[4] Külahcı, M., Bektaş, M., Ergüt, M., Curves of AW(k)-type in 3-dimensional null cone, Physics Letters A 371 (2007), 275-277.

[5] Külahcı, M., Bektaş, M., Ergüt, M., On Harmonic Curvatures of Null Curves of the AW(k)-type in Lorentzian Space, Z. Naturforsch. 63a (2008), 248-252.

[6] Külahcı, M., Bektaş, M., Ergüt, M., On Harmonic Curvatures of a Frenet Curve in Lorentzian Space, Chaos, Solitons and Fractals 41(2009), 1668-1675.

[7] Külahcı, M., Ergüt, M., Bertrand Curves of AW(k)-type in Lorentzian Space, Nonlinear Analysis 70 (2009), 1725-1731.

[8] Yıldırım Yılmaz, M., Bektaş, M., Bertrand Curves on Finsler Manifolds, International Journal of Physical and Mathematical Sciences (2011), 5-10.

193 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Characterizations on Biharmonic General Helices in Finsler Manifold F3

Mahmut Ergüt (1) and Münevver Yıldırım Yılmaz

(1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this paper, we focus on the biharmonic general helices of 3 dimensional Finsler manifold and obtain some characterizations for helices of this type.

Keywords. Finsler manifold, biharmonic helices.

AMS 2010. 58J60.

Reference

[1] Bejancu, A. and Farran H.R., Geometry of Pseudo-Finsler Submanifolds, Kluwer Academic Publishers,Dordrecht, Boston, London, 2000.

[2] Yaz, N., Ekmekçi, N., Biharmonic General Helices and Submanifolds In An Indefinite- Riemannian Manifold, Tensor, N.S., Vol. 64 (2003).

[3] Yıldırım Yılmaz, M., Bektaş, M., Bertrand Curves on Finsler Manifolds, International Journal of Physical and Mathematical Sciences (2011), 5-10.

[4] Külahcı, M., Bektaş, M., Ergüt, M., On Harmonic Curvatures of Null Curves of the AW(k)-type in Lorentzian Space, Z. Naturforsch. 63a (2008), 248-252.

[5] Külahcı, M., Bektaş, M., Ergüt, M., On Harmonic Curvatures of a Frenet Curve in Lorentzian Space, Chaos, Solitons and Fractals 41(2009), 1668-1675.

194 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Inextensible Flows of Curves in Three Dimensional Finsler Manifold F3

Mahmut Ergüt (1) and Alper Osman Öğrenmiş (2)

(1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this paper after a short description of Finsler Manifolds, we investigate inextensible flows of a curve with respect to the Frenet frame of the 3-dimensional Finsler manifold F3. Some conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature and torsions.

Keywords. Inextensible Flows, Finsler Manifold AMS 2010. 53B40, 58B20

References

[1] Bejancu, A. and Farran H.R., Geometry of Pseudo-Finsler Submanifolds, Kluwer Academic Publishers,Dordrecht, Boston, London, 2000.

[2] Brandt, E.H., Finslerian quantum field theory, Nonlinear Analysis, 63, 5-7, e119-e130, 2005.

[3] Kwon, D.Y., Park, F.C., Evolution of inelastic plane curves, Appl. Math. Lett. 12, 115-119, 1999.

[4] Kwon, D.Y., Park, F.C. and Chi, D.P., Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18, 1156-1162, 2005.

[5] Latifi, D., and Razavi, A., Inextensible Flows of Curves in Minkowskian Space, Adv. Studies Theor. Phys. 2(16), 761-768, 2008.

[6] Öğrenmiş, A.O. and Yeneroğlu, M., Inextensible Curves in the Galilean Space, International Journal of the Physical Sciences, 5(9), 1424-1427, 2010.

[7] Öğrenmiş, A.O.,Yeneroğlu, M. and Külahcı, M., Inelastic Admissible Curves in the Pseudo – Galilean Space G31, Int. J. Open Problems Compt. Math., Vol. 4, No. 3,199-207, 2011.

[8] Solange F.R. and Portugal, R.:FINSLER-acomputer algebra package for Finsler geometries, Nonlinear Analysis, No.47(9), 6121-6134, 2001.

[9] Yıldırım Yılmaz, M., Bektaş, M., Bertrand Curves on Finsler Manifolds, International Journal of Physical and Mathematical Sciences, 5-10 2011.

195 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Inverse Surfaces of Tangent, Principal Normal and Bi-normal Surfaces of a Space Curve in Euclidean 3-Space Mahmut Ergüt (1) and Muhittin Evren Aydın (2) (1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this study, we consider inverse surfaces of tangent, principal normal and bi-normal surfaces of a space curve in Euclidean 3-Space E³ with respect to the sphere ( ).

We give the geometric properties about these surfaces and also obtain various results.. 푆푐 푟 Keywords. Inversion, Inverse surface, Developable surface, Christoffel symbols. AMS 2010. 11A25, 53A04, 53A05.

References

[1] A. Gray: Modern differential geometry of curves and surfaces with mathematica. CRC Press LLC, 1998.

[2] E. Ozyilmaz, Y. Yayli: On the closed space-like developable ruled surface, Hadronic J. 23 4, 439-456, 2000.

[3] P. Alegre, K. Arslan, A. Carriazo, C. Murathan and G. Öztürk: Some Special Types of Developable Ruled Surface, Hacettepe Journal of Mathematics and Statistics, 39 3, 319 – 325, 2010.

[4] S. Izumiya, N. Takeuchi: New Special Curves and Developable Surfaces, Turk J Math 28, 153-163, 2004.

196 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the First Fundamental Theorem for Dual Orthogonal Group O(2, D) Muhsin İncesu (1) , Osman Gürsoy (2) and Djavvat Khadjıev (3) (1) Mus Alparslan University, Mus, Turkey, [email protected] (2) Maltepe University, Istanbul, Turkey, [email protected] (3) Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. Let D be set of dual numbers then in D2 we investigated the first fundamental theorem of dual vectors for dual orthogonal group O(2, D). Then we compared obtaining results to real space R4 for O(4,R). Keywords. Dual Invariants, first fundamental theorem. AMS 2010. 22D20 , 13A50, 14L24.

References

[1]Weyl, H. The Classical Groups, Their Invariants and Representations, 2nd ed., with suppl., Princeton University Press, Princeton, 1946.

[2] Hacısalihoğlu, H. H., Hareket geometrisi ve kuaterniyonlar teorisi, Gazi Üniversitesi, Fen- Edebiyat Fakultesi Yayinlari 2, 1983.

[3] Khadjiev, Dj.,Some Questions in The Theory of Vevtor Invariants, Math.USSR – Sbornic,1 ,3, 383-396 , 1967.

[4] Khadjiev, Dj.,An Application of Invariant Theory to the Differential Geometry of Curves, Fan, Tahkent, 1988, (in Russian)

[4] Incesu, M., The Complete System of Point Invariants in the Similarity Geometry, Ph. D. Thesis, Karadeniz Technical University, 2008.

197 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On the B-manifold Defined by Algebra of Plural Numbers Murat İşcan (1) and Abdullah Mağden (2) (1) Ataturk University, , Erzurum, Turkey, [email protected] (2) Ataturk University, Erzurum, Turkey, [email protected]

Abstract. In this paper we investigated a plural-holomorphic B-manifold. Firstly, we proved that an almost B-manifold is plural-holomorphic B-manifold if and only if the almost tangent structure is parallel with respect to the Levi-Civita connection ∇ . Also, we proved that the almost tangent structure ϕ on almost B-manifold is integrable if φϕ g = 0 . Finally, we give an example for plural holomorphic B-manifold. Keywords. B-manifold, pure tensor, plural-holomorphic tensor, complete lift. AMS 2010. 53C15, 53C25, 53C55

References

[1] N. Cengiz and A.A. Salimov, Complete lifts of derivations to tensor bundles, Bol.Soc.Mat. Mexicana 8(3) (2002), 75-82.

[2] G.T. Ganchev and A. V. Borisov, Note on the almost complex manifolds with a Norden metric, C.R. Acad. Bulgarie Sci. 39 (5) (1986), 31-34.

[3] K. Gribachev, D. Mekerov and G. Djelepov, Generalized B-manifolds, C. R. Acad. Bulgare Sci. 38 (3) (1985), 299-302.

[4] K. Gribachev, D. Mekerov and G. Djelepov, On the geometry of almost B-manifolds, C. R. Acad. Bulgare Sci. 38 (5) (1985), 563-566.

[5] G.I. Kruchkovich, Hypercomplex structure on a manifold I, Tr. Sem. Vect. Tens. Anal., Moscow Univ. 16 (1972), 174-201.

[6] D. Mekerov, On some classes of almost B-manifolds, C. R. Acad. Bulgare Sci. 38 (5) (1985),559-561.

[7] A.P. Norden, On a certain class of four-dimensional A-spaces, Izv. Vuzov. Mat. no.4 (1960), 145-157.

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[8] A.A. Salimov, Generalized Yano-Ako operator and the complete lift of tensor fields, Tensor (N.S.) 55, no.2 (1994), 142-146.

[9] A.A. Salimov and A. Magden, Complete lift of tensor fields on a pure cross-section in the tensor bundle, Note di Matematica 18 (1) (1998), 27-37.

[10] A.A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology and its Application 154 (2007), 925-933.

[11] A.P. Shirokov, Spaces over algebras and their applications, Journal of Mathematical Sciences 108 (2) (2002), 232-248.

[12] G. Thompson and U. Schwardmann, Almost tangent and cotangent structures in the large, Trans. Amer. Math. Soc. 327 (1) (1991), 313-328.

[13] V.V. Vishnevskii, Structures of projective spaces generated by affinors, Izv. Vuzov. Mat. 6 (1969), 35-46.

199 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

CMC-Surfaces in Hyperbolic Space Hn Mahmut Mak (1) and Baki Karliğa (2) (1) Ahi Evran University, Kırsehir, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. In this study, CMC-surfaces are obtained with special method in H n

34 n (n ≥ 3). We also examine the geometric invariants of the surfaces in HH, and H . Finally, we show that these surfaces are de-Sitter minimal but they are not hyperbolic minimal. Keywords. Hyperbolic Space, Minimal Surface, de-Sitter Mean Curvature, Hyperbolic Mean Curvature AMS 2010. 53A35, 53C42

References

[1] Izumiya, S., Pei, D., Romero Fuster, M.C., Takahashi, M., The horospherical geometry of submanifolds in hyperbolic space, J. London Math. Soc., 2(71), 779-800, 2005.

[2] Izumiya, S., Pei,D., Romero Fuster, M.C., The horospherical geometry of surfaces in hyperbolic 4-space, Israel Journal of Math., 154(1), 361-379, 2006.

[3] Izumiya, S., Horospherical flat surfaces in Hyperbolic 3-Space, J. Math. Soc. Japan, 62(3), 789-849, 2010.

[4] Izumiya, S., Pei, D., Sano, T., Singularities of Hyperbolic Gauss Maps, Proceedings of London Mathematical Society, 86, 485-512, 2003.

[5] Aminov, Yu., The Geometry of Submanifolds, Overseas Publishers Association, Singapore, 2001.

[6] Ratcliffe, J., G., Foundations of Hyperbolic Manifolds 2. ed., Springer-Verlag, Graduate Texts in Mathematics 149, New York, 2006.

[7] O’Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.

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[8] Kythe, Prem K., Partial differential equations and Mathematica, CRC Press, USA, 1997.

201 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Torsion Tensors of Pure Π−connections Nejmi Cengiz Ataturk University, Erzurum, Turkey, [email protected]

Abstract. Let S be a torsion tensor field of pure Π−connection ∇ . We have following results:

1. The Π−connection ∇ is pure if and only if the torsion tensor of ∇ is pure.

2. The pure torsion tensor field of the Π−connection ∇ is a real model of the ∗ hypercomplex torsion tensor of hypercomplex connection ∇ .

3. A torsion-free Π−connection ∇ is always pure.

∗ ∗ σ uu 4. If ∇ is a torsion-free Π−connection, then ∇ with components Γ= wvτ, wveσ is a torsion-free connection.

Keywords. Torsion tensör, Pur tensors and connections. AMS 2010. 53C15; 53B05; 15A69; 16G60; 32A10.

202 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Screen Semi-Invariant Half-Lightlike Submanifolds Oğuzhan Bahadır Hitit University, Corum, Turkey, [email protected]

Abstract. In this paper, we study half-lightlike submanifold of a semi-Riemannian product manifold. We introduce a class half-lightlike submanifolds of called screen semi- invariant half-lightlike submanifolds, screen invaryant and radical anti-invariant half-lightlike submanifolds. We consider half-lightlike submanifold which is determined by the product structure. We give some equivalent conditions for integrability of distributions of semi- Riemannian manifold and some results. Keywords. Half-lightlike submanifold, Product manifolds, Screen semi-invariant. AMS 2010. 53C15, 53C25, 53C40.

References

[1] Duggal, K. L. and Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer, Dordrecht, 1996.

[2] Duggal, K. L. and Sahin, B., Diferential Geometry of Lightlike Submanifolds,

Birkhauser Verlag AG, Basel-Boston-Berlin, 2010.

[3] Duggal, K.L. and Jin, D.H.: Null Curves and Hypersurfaces of Semi-Riemannian manifolds, World Scienti_c Publishing Co. Pte. Ltd., 2007.

[4] Atceken, M. and Kilic, E., Semi-Invariant lightlike submanifolds of a semi-Riemannian product manifold, Kodai Math. J., Vol. 30, No. 3, (2007), pp.361-378.

[5] Kilic, E. and Sahin, B., Radical anti-invariant lightlike submanifolds of a semi- Riemannian product manifold, Turkish J. Math., 32, (2008), 429-449.

[6] Massamba, F., Killing and geodesic lightlike hypersurfaces of indefinite Sasakian manifolds, Turk. J. Math. 32, (2008), 325-347.

[7] Massamba, F., Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilineer forms, Diferential Geometry-Dynamical Systems, Vol.10, (2008), 226- 234.

[8] Schouten, J. A., Ricci calculus, Springer, 1954.

203 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Note on Inextensible Flows of Curves in E n Önder Gökmen Yıldız (1), Murat Tosun (2) and Sıddıka Ö. Karakuş (3) (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]

Abstract. In this paper, we investigate the general formulation for inextensible flows of curves in E n . The necessary and sufficient conditions for inextensible curve flow are expressed as a partial differential equation involving the curvatures. Keywords. Curvature flows, inextensible, Euclidean n-space.. AMS 2010. 53C44, 53a04, 53A05, 53A35.

References

[1] . H. Hacisalihoğlu, Differential Geometry, University of İnönü Press, Malatya, (1983).

[2] D. Y. Kwon, F.C. Park, D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18 (2005) 1156-1162.

[3] D. Y. Kwon, F.C. Park, D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18 (2005) 1156-1162.

[4] D.J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract. 50, 33-- 38 (1991).

[5] O. G. Yildiz, S. Ersoy, M. Masal, A Note on Inextensible Flows of Curves on Oriented Surface, arXiv:1106.2012v1.

[6] O. G. Yildiz, Minkowski Uzayında Yüzey Üzerinde Eğrilerin elastik Olmayan Hareketleri, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Yüksek Lisans Tezi, (2011).

204 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Generalization of Some Well-Known Distances and Related Isometries Rüstem Kaya Eskisehir Osmangazi University, Eskisehir, Türkiye, [email protected]

Abstract. This talk is based on our joint paper [1] with the same title. In that paper, we define a family of distance functions in the real plane, m-distance function, which includes the taxicab. Chinese checker, maximum, and alpha distance functions as special cases, and we show that the m-distance function determines a metric. Then we give some properties of the m-distance, and determine isometries of the plane with respect to the m-distance. Finally, we extend the m-distance function to three and n-dimensional spaces, and show that each extended distance function determines a metric. We also give some properties of the m- distance in three-dimensional space. Keywords. Metric, m-distance, taxicab distance, Chinese checker distance, maximum distance, alpha distance, minimum distance set, isometry. AMS 2010. 51K05, 51K99, 51F99, 51N99.

References

[1] H. B. Çolakoğlu- R. Kaya, A generalization of some well-known distances and related isometries, Mathematical Communications, Math. Commun. 16(2011), 21-35.

[2] Ö. Gelisgen, R. Kaya, On Alpha Distance in Three Dimensional Space, Appl. Sci. 8.(2006), 656-69.

[3] Ö. Gelisgen, R. Kaya, The Taxicab Space Group, Acta Math. Hungar. 122(2009), 187- 200.

[4] S. Tian, A Generalization of Chinese Checker Distance and Taxicab Distance, Missouri J. Math. Sci. 17(2005), 35-40.

205 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Bobillier Formula for One Parameter Motions in the Complex Plane Soley Ersoy (1) and Nurten Bayrak (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. In this study, the complex number forms of the Euler Savary formula for the radius of curvature of the trajectory of a point in the moving complex plane during one parameter planar motion are taken into consideration and using the geometrical interpretation of the Euler Savary formula, Bobillier formula is established for one parameter planar motions in the complex plane. Moreover, a direct way is chosen to obtain Bobillier formula without using the Euler Savary formula in the complex plane. As a consequence, the Euler Savary given in the complex plane will appear as a particular case of Bobillier formula.

Keywords. Euler Savary formula, Bobillier formula, planar motion, complex plane AMS 2010. 53A40, 20M15.

References

[1] Sandor, G. N., Erdman, A. G., Hunt, L., and Raghavacharyulu, E., 1982, New Complex Number Forms of the Euler-Savary Equation in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact, ASME J. Mech. Des., 104, pp. 227– 232.

[2] Sandor, G. N., Erdman, A. G., Hunt, L., and Raghavacharyulu, E., 1982, New Complex- Number Forms of the Cubic of Stationary Curvature in a Computer- Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact, ASME J. Mech. Des., 104, pp. 233–238.

[3] Tutar, A., and Duldul, M., 2001, On the Moving Coordinate System on the Complex Plane and Pole Points, Bull. Pure Appl. Sci. Sec. E, Math. Stat., 20(1), pp. 1–6.

[4] Masal, M., Tosun, M., and Pirdal, A. Z., 2010, Euler Savary Formula for the One Parameter Motions in the Complex Plane C, Int. J. Phys. Sci., 5(1), pp. 6–10.

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[5] Sandor, G. N., Arthur, G. E., and Raghavacharyulu, E., 1985, Double Valued Solutions of the Euler-Savary Equation and Its Counterpart in Bobillier’s Construction, Mech. Mach. Theory, 20(2), pp. 145–178.

[6] Sandor, G. N., Xu, Y., and Weng, T.-C., 1990, A Graphical Method for Solving the Euler- Savary Equation, Mech. Mach. Theory, 25(2), pp. 141–147.

[7] Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North Holland Publishing Company, Amsterdam/New York/Oxford.

[8] Garnier, R., 1956, Cours de cinematique, Gauthier-Villar, Paris, p. 38.

[9] Dijskman, E. A., 1976, Motion Geometry of Mechanism, Cambridge University Press, Cambridge.

[10] Fayet, M., 1988, Une Nouvelle Formule Relative Aux Courbures Dans un Mouvement Plan, Mech. Mach. Theory, 23(2), pp. 135–139.

[11] Fayet, M., 2002, Bobillier Formula as a Fundamental Law in Planar Motion, Z. Angew. Math. Mech., 82(3), pp. 207–210.

[12] Waldron, K. J., and Kinzel, G. L., 1999, Kinematics, Dynamics, and Design of Machinery, John Wiley and Sons, Inc., New York.

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Arc-Lenghts, Curvature and Natural Lifts of Spherical Indicatrix of Timelike Mannheim Curve with Spacelike Binormıal Spacelike Curve Partner Süleyman Şenyurt (1) and Selma Demet (1) (1) Ordu University, Ordu, Turkey [email protected], [email protected]

Abstract. In this study, the Mannheim curvature is taken as timelike and the partner curvature by spacelike binormal is considered as a spacelike, the global indicator curvatures of the partner curvatures, ()()TN∗∗, ,()B∗ and the links between the geodesic curvatures and the arc lengths of the fixed centrode curvature ()C∗ have been found according to the IL3

2 2 Lorentz space and S1 Lorentz sphere or H0 Hyperbolic sphere. In addition, what kind of curvature the Mannheim curvature must be is explained by examining the requirement that fort he Geodesic spray, the natural lifts of the global indicators of the partner curvature need to be integral curvatures. Keywords. Mannheim eğrisi, Geodezik eğrilik, Geodezik spray, Tabii lift. AMS 2010. 53A04, 53B30

References

[]1 Çalışkan M., Sivridağ A.İ. , Hacısalihoğlu H. H., Some Characterizations For The Natural Lift Curves And The Geodesic Sprays, Ankara Üniv., Fen Fak.,Commun., Cilt 33, pp.235-242 ,1984.

3 []2 Orbay K. and Kasap E., On Mannheim partner curves in E , International Journal of Physical Sciences Vol. 4 (5), pp. 261-264, May 2009. []3 Liu, H. and Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, Volume 88, Numbers 1-2, March 2008, pp. 120-126(7), March 2008. []4 Ergun E. and Calışkan M., On the M -Integral Curves and M -Geodesic Sprays In Minkowski 3-Space, Int.J.Contemp. Math. Sci. Vol.6., no.39, pp. 1935-1939, 2011. []5 Şenyurt S. and Bektaş Ö., Timelike-Spacelike Mannheim Partner Curves in E3 , International Journal of the Physical Sciences, Vol.7(1),pp 100-106, 2 January, 2012. []6 Şenyurt S., Natural Lifts and The Geodesic Sprays for the Spherical Indicatrices of the Mannheim Partner Curves in E3 , International Journal of the Physical Sciences, Vol.(7),no.16, pp. 2414-2421, 2012.

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Arc-Lenghts, Curvature and Natural Lifts of Spherical Indicatrix of Timelike Bertrand Curve Partner Süleyman Şenyurt (1) and Ömer Faruk Çalışkan (1) (1) Ordu University, Ordu, Turkey [email protected], [email protected]

Abstract. In this study, when a pair of timelike Bertrand curve, ()αα, ∗ is given, the geodesic curves and the arc lengths of the curvatures, TN∗∗, , B∗ and C∗ which are ()() () () generated over the Lorentz sphere or the hyperbolic sphere by the Frenet vectors, {}TNB∗,, ∗∗

∗ and the unit Darboux vector, C have been calculated. The connections of the global indicator curvatures among the Geodesic curvatures have been found, then for the Geodesic spray, the condition of which the natural lifts of these curvatures must be curvatures of an integral and some important results have been obtained. Keywords. Lorentz space, bertrand curves couple, tabii lift, geodesic spray AMS 2010. 53A04, 53B30

References

[]1 Balgetir, H. Bektaş, M. and Ergut, M., Bertrand Curves For Nonnull Curves in 3- Dimensional Lorentzian Space, Hadronıc Journal, 27, 229-236, (2004). []2 Çalışkan M., Sivridağ A.İ., Hacısalihoğlu H. H., “ Some Characterizations For The Natural Lift Curves And The Geodesic Sprays” Ankara Üniv., Fen Fak., Communications, Cilt 33, 235-242 (1984 []3 Ersoy S. and Inalcık A., On the Generalized Timelike Bertrand Curves in 5-Dimensional Lorentzian Space, Dıfferential Geometry-Dynamıcal Systems, Vol.13, 2011, pp. 77-88. []4 O’neill, B., Semi Riemann Geometry, Academic Press, New York, London, 468p.,1983.

[]5 Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Springer-Verlag New York, Inc., New York, 736., 1994. []6 Sivridağ A.İ., Çalışkan M., “ On The M-Integral Curves And M-Geodesic Sprays ” Erciyes Üniv., Fen Bilimleri Dergisi, Cilt 7, Sayı 1-2, 1283-1287 (1991). []7 Uğurlu, H.H., On the Geometry of Timelike Surfaces, Commun. Fac. Sci. Ank. Series A1 V.46.pp. 211-223., 1997. []8 Woestijne, V.D.I., Minimal Surfaces of the 3-diemensional Minkowski space. Proc. Congres “Geometrie differentielle et aplications” Avignon (30 May 1988), Wold Scientific Publishing. Singapore. 344-369, 1990.

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Lightlike Hypersurfaces of an -Para Sasakian Manifold Selcen Yüksel Perktaş (1), Erol Kılıç (2) and Mukut Mani Tripathi (3) (1) Adiyaman University, Adiyaman, Turkey, [email protected] (2) Inonu University, Malatya, Turkey, [email protected] (3) Banaras Hindu University, Varanasi, India, [email protected]

Abstract. In the present paper we study lightlike hypersurfaces of an -almost paracontact metric manifold which are tangent to the structure vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively and give examples. Integrability conditions for the distributions involved in the definition of a screen semi-invariant lightlike hypersurface are investigated when the ambient manifold is an -para Sasakian Manifold. Keywords. -Almost Paracontact Metric Manifold, -Para Sasakian Manifold, Lightlike Hypersurface, Invariant Lightlike Hypersurface, Screen Semi-Invariant Lightlike Hypersurface. AMS 2010. 53C25, 53C40, 53C50.

References

[1] Duggal, K.L. and Bejancu, A., Lightlike Submanifolds of Semi-Riemannian Manifolds and Its Applications, Kluwer, Dordrecht, 1996.

[2] Duggal, K.L. and Jin, D.H., Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scienti.c Publishing Co. Pvt. Ltd., 2007.

[3] Duggal, K.L. and Şahin, B., Differential geometry of lightlike submanifolds, Birkhäuser, 2010.

[4] Tripathi, M.M., Kılıç, E., Perktaş, S.Y. and Keleş, S., Indefinite Almost Paracontact Metric Manifolds, Int. J. Math. Math. Sci., vol.2010, Art. Id.846195, pp. 19, 2010.

[5] Perktaş, S.Y., Kılıç, E., Tripathi, M.M. and Keleş, S. 3-Dimensional -Para Sasakian Manifolds, International Journal of Pure and Applied Mathematics, vol. 77, No.4, 2012.

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On the Ruled Surfaces whose Frames are the Bishop Frame in the Euclidean 3-space Şeyda Kılıçoğlu (1) and H.Hilmi Hacısalihoğlu (2) (1) Baskent University, Ankara, Turkey, [email protected] (2) Bilecik Seyh Edebali University, Ankara, Turkey, [email protected]

Abstract. In this study we investigate the differential geometric elements (such as normal vector field N, shape operator S, curvatures K and H, the fundamental forms I, II, III) of a ruled surface according to its Bishop frames in the Euclidean 3-space. Keywords. Shape operator, fundamental forms, Bishop frame. AMS 2010. 53A05, 53A04.

References

[1] Andrew, J. H., Ma, H., Parallel Transport Approach To Curve Framing, Indiana University, Techreports- TR425, January 11,1995.

[2] Andrew, J. H., Ma, H., Quaternion Frame Approach to Streamline Visualization, Ieee Transactions On Visualization And Computer Graphics, Vol. I , No. 2, June 1995.

[3] Bishop, L. R., There is more than one way to frame a curve, Amer. Math. Monthly, Volume 82, Issue 3, 246-251,1975.

[4] Bukcu, B. , Karacan, M. K. Special Bishop Motion and Bishop Darboux Rotation Axis of space curve, Journal of Dynamical Systems and Geometric Theories. 6(1),27-34, 2008.

[5] Körpınar, T. , Baş, S. , On Characterization Of B-Focal curves In E³ , Bol. Soc. Paran. Mat.31 (1); 175-178, 2013.

[6] Springerlink. Encyclopedia of Mathematics, Springer-Verlag, Berlin Heidelberg New York, 2002.

[7] Shifrin,T., Differential Geometry: A First Course in Curves and Surfaces. University of Georgia, Preliminary Version 2008.

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Constant Angles Surfaces in Hyperbolic Space Tuğba Mert (1) and Baki Karlığa (2) (1) Cumhuriyet University, Sivas, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. In this paper we study and classify constant angle surface in Hyperbolic space H³. Inparticular, we acquired parametrizations of constant timelike and spaecelike angles surfaces with spacelike axis. A constant angle surface in Hyperbolic space is a spacelike surface whose unit normal vector field makes a constant angle with a fixed spacelike vector in R .In a word,a constant angle surface in Hyperbolic space H³ is a surface whose tangent planes₁⁴ make a constant angle with a fixed vector field on R . Keywords. Constant angle surfaces , Hyperbolic space , Helix. ₁⁴ AMS 2010. 53A40, 20M15.

References

[1] R. Lopez, M.I. Munteanu, Constant angle surfaces in Minkowski space, Bulletin of the Belgian Math. So. Simon Stevin, Vo.18 (2011) 2,271-286

[2] S.Izumıya, K.Sajı, M.Takahashı, Horospherical flat surfaces in Hyperbolic 3-space, J.Math.Soc.Japan, Vol.87 (2010), 789-849

[3] S.Izumıya, D.Peı, M.D.C.R. Fuster, The horospherical geometry of surfaces ın hyperbolic 4-spaces,Israel Journal of Mathematıcs, Vol.154 (2006), 361-379

[4] C.Thas, A gauss map on hypersurfaces of submanıfolds ın Euclıdean spaces, J.Korean Math.Soc., Vol.16 (1979) No.1

[5] S.Izumıya, D.Peı, T.Sano, Sıngularities of hyperbolic gauss map, London Math.Soc. Vol.3 (2003), 485-512

[6] M.I.Munteanu, A.I.Nistor, A new approach on constant angle surfaces in E³ , Turk T.Math. Vol.33 (2009), 169-178

[7] C.Takızawa, K.Tsukada , Horocyclic surfaces in hyperbolic 3-space, Kyushu J.Math. Vol.63 (2009), 269-284

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[8] S.Izumıya, M.D.C.R. Fuster, The horospherical Gauss-Bonnet type theorem in hyperbolic space, J.Math.Soc.Japan, Vol.58 (2006) 965-984

[9] B. O'Neill, Semi-Riemannıan Geometry with applıcations to relativity, Academic Press, New York, 1983

[10] W.Fenchel , Elementary Geometry in Hyperbolic Space, Walter de Gruyter , New York , 1989

[11] J.G.Ratcliffe, Foundations of Hyperbolic Manifolds, Springer

[12] P.Cermelli, A.J. Di Scala, Constant angle surfaces in liquid crystals, Phylos. Magazine,Vol.87 (2007),1871-1888

[13] A.J. Di Scala, G. Ruiz-Hernandez, Helix submanifolds of Euclidean space, Monatsh. Math. DOI 10.1007 / s00605-008-0031-9

[14] G.Ruiz-Hernandez, Helix, shadow boundary and minimal submanifolds, Illinois J. Math. Vol.52 (2008) 1385-1397

[15] F. Dillen, J. Fastenakels, J. Van der Veken, L. Vrancken, Constant angle surfaces in S²×R , Monaths. Math. , Vol.152 (2007), 89-96

[16] F. Dillen and M. I. Munteanu, Constant angle surfaces in H²×R , Bull. Braz. Math. soc. , Vol.40 (2009) 85-97

[17] J. Fastenakels, M. I: Munteanu , J. Van der Veken, Constant angle surfaces in the Heisenberg group, Acta Math. Sinica (English Series) Vol.27 (2011), 747-756

[18] R.Lopez, Differantial Geometry of Curves and Surfaces in Lorentz-Minkowski space, arXiv: 0810.3351 (2008)

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On the Quaternionic Curves According to Parallel Transport Frame Tülay Soyfidan (1), Hatice Parlatıcı (2) and Mehmet Ali Güngör(3) (1) Erzincan University, Erzincan, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this paper, we studied parallel transport frame for a quaternionic curve in  3 and  4 . Moreover, using this parallel transport frame, we defined inclined curves and harmonic curvatures for the quaternionic curves in  4 . Keywords. Quaternionic parallel transport frame, Quaternionic inclined curves, Euclidean space. AMS 2010. 11R52, 14H45, 53A04.

References

[1] K. Bharathi and M. Nagaraj, Quaternion valued function of a Real variable Serret-Frenet Formulae, Indian J. Pure Appl. Math. 18, 6, (1987), 507-511.

[2] L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, Vol. 82, Issue 3, (1975), 246-251.

[3] F. Gökçelik, İ. Gök, F. N. Ekmekci, Y. Yaylı, On inclined curves according to parallel

4 transport frame in  , X. Geometri Sempozyumu, Burhaniye-Balıkesir, 13-16 Haziran 2012.

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On The Quaternionic Rectifying Curves in Semi-Euclidean Space 4  2 Tülay Soyfidan (1) and Mehmet Ali Güngör (2) (1) Erzincan University, Erzincan, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this study, semi-real spatial quaternionic rectifying curves in semi-

3 Euclidean space in 1 are defined and the some characterizations are obtained for these

4 curves. Moreover, semi-real quaternionic rectifying curves are investigated in  2 and the characterizations of these curves are given. Keywords. Semi-quaternionic rectifying curves, Semi-real quaternion, Semi- Euclidean space. AMS 2010. 11R52, 53A04, 53C50.

References

[1] K. Bharathi and M. Nagaraj, Quaternion valued function of a Real variable Serret-Frenet Formulae, Indian J. Pure Appl. Math. 18, 6, (1987), 507-511.

[2] A. C. Çöken and A. Tuna, On the quaternionic inclined curves in the semi-Euclidean

4 space  2 , Applied Mathematics and Computation, 155, (2004), 373-389.

[3] K. İlarslan, E. Nešović, M. Petrović-Torgašev, Some characterizations of rectifying curves in the Minkowski 3-space, Novı Sad J. Math. Vol. 33, No. 2, (2003), 23-32.

[4] M. A. Güngör, M. Tosun, Some characterizations of quaternionic rectifying curves, Differential Geometry-Dynamical System, Vol.13, (2011), pp. 89-100.

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Conformal Triangles in Hyperbolic and Spherical Space Ümit Toşeker (1) and Baki Karlığa (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. In this presentation, conformal simplices and conformity in Euclidian space which were considered in the joint paper of Igor RIVIN and Daryl COOPER are investigated [4]. After this investigation, conformity conditions in spherical and hyperbolic spaces are obtained. Keywords. Simplex, Conformal simplex, Hyperbolic and Spherical conformal triangle. AMS 2010. 30F45, 52A55, 97G60.

References

[1] B. Karliga, Edge matrix of hyperbolic simplices, Geom. Dedicata, 109, 1-6, 2004.

[2] B. Karliga and A.T. Yakut, Vertex angles of a simplex in hyperbolic space, Geom. Dedicata, 120, 49-58, 2006.

[3] J.G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag, Berlin, 1994.

[4] I. Rivin and D. Cooper, Combinatorial scalar curvature and rigidity of ball packings, Math. Res. Lett., 3, 1, 51-60, 1996.

[5] A.T. Yakut, Hiperbolik Uzayda Simplekslerin Tepe Açıları, Doktora Tezi, Gazi Uni., 2004.

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Framed-complex Submersions Yılmaz Gündüzalp Dicle University, Diyarbakir, Turkey, [email protected]

Abstract. In this study, we introduce the concept of framed-complex submersions from a framed metric manifold onto an almost Hermitian manifold. We provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the framed metric structure of the total manifold. Moreover, we obtain various properties of the O’Neill’s tensors for such submersions and find the integrability of the horizontal distribution. Keywords. Framed metric manifold, almost Hermitian manifold, Riemannian submersion, framed-complex submersion. AMS 2010. 53C15, 53C12, 53C40.

References

[1] Chinea, D., Almost contact metric submersions, Rend. Circ. Mat. Palermo, II Ser., 34, 89- 104, 1985.

[2] Falcitelli, M., Ianus, S. and Pastore, A. M., Riemannian Submersions and Related Topics, World Scientific, 2004.

[3] O‘Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13, 459 469, 1966.

[4] Terlizzi, L. D., On invariant submanifolds of C-and S-manifolds, Acta Math. Hungar., 85, 229-239, 1999.

[5] Watson, B., Almost Hermitian submersions, J. Diff. Geom., 11, 147-165, 1976.

[6] Yano, K., On a structure defined by a tensor field f satisfying 3 ff =+ ,0 Tensor, 14, 99- 109, 1963.

[7] Yano, K., Kon, M., Structures on manifolds, World Scientific, 1984.

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Affine Differential Invariants of a Pair of Curves Yasemin Sağıroğlu Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. Let = ( , ) be the group of all transformations in as ( ) = 푛 + such that 퐺 푆퐴푓푓( , )푛 and푅 . The system of generators for the푅 differential퐹 푥 푛 fields푔푥 푏of all -invariant푔 ∈ 푆퐴푓푓 differential푛 푅 rational푏 ∈ 푅 functions of a pair of curves is found for the group (퐺, ). The conditions for -equivalence of a pair of curves is obtained in terms of affine푆퐴푓푓 differential푛 푅 invariants. Finally,퐺 it is shown that the generator system should be minimal. Keywords. Differential invariant, affine group, equivalence of curves. AMS 2010. 53A35, 53A55.

References

[1] Gardner, R.B., Wilkens, G.R., The fundamental theorems of curves and hypersurfaces in centro-affine geometry, Bull. Belg. Math. Soc. Simon Stevin, 3, 379-401, 1997.

[2] Khadjiev, Dj., Pekşen, Ö., The complete system of global integral and differential invariants for equi-affine curves, Diff. Geom. and its Appl., 20, 167-175, 2004.

[3] Sağıroğlu, Y., The equivalence of curves in SL(n,R) and its application to ruled surfaces, Applied Mathematics and Computation, 218, 1019-1024, 2011.

[4] Sağıroğlu, Y., Pekşen, Ö., The equivalence of centro-equiaffine curves, Turk. J. Math., 34, 95-104, 2010.

[5] Weyl, H., The Classical Groups-Their Invariants and Representations, Princeton University press, Princeton NJ, 1946.

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TOPOLOGY

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications On multimetric spaces Altay Asylkanovich Borubaev Kyrgyz National University named after J.Balasagyn, Bishkek, Kyrghyzstan, [email protected]

Abstract. Here it is offered a new notion of topological space, i.e. multimetric space. Let

τ τ ++=[0,, ∞)() = −∞ ,, ∞ τ be any cardinal number,  + and  be Tychonoff product of τ

τ τ many of  + and  respectively. In spaces  + and  in natural way (componentwisely) are defined addition, product and product on scalar operations and also partial ordering. Let X τ be nonempty

τ set. A mapping ρτ : XX×→ + is called multimetric and a pair ()X , ρτ as multimetric space if the following well-known axioms are fulfilled:

τ 1. ρθτ (,xy )=⇔= x y, where θ ∈  + with all components equal to 0.

2. ρρττ(,xy )= (,) yx and

3. ρτ(,xy )≤+ ρρ ττ (,) xz (, zy ) for all xyz,,,∈ X.

Let ()X ,σ be topological space. It is said that this space is multimetrizable provided there exists some multimetric ρτ such that a subset O belongs to σ iff for any xO∈ one can find an elementary neighborhood UU,..., it follows that yO∈ . X ,τ can be called as ρ − metrizable εε1 n () τ space too and we agree to use ()X , ρτ instead of ()X ,σ . The following theorem demonstrates the wideness of multimetric spaces.

THEOREM 1. Uniform space ()XU, is ρτ − metrizable iff ωτ()U = , where ω ()U means uniform weight of U . In natural way are defined Cauchy filter in any multimetric space ()X ,σ and µ - completness for any cardinal ℵ≤0 µτ ≤ .

DEFINITION 1. A mapping fX:,()()ρρττ→ X , is called contracting if

ρττ( f()()() x,, f y≤⋅λρ xy) for all xy, ∈ X and some λ from ()0,1 .

THEOREM 2. Let fX:,()()ρρττ→ X , be a contracting mapping and if

()X , ρτ is sequentially complete (ℵ0 -complete) then f has a unique fixed point. THEOREM 2 is a generalizaation of Banach fixed point theorem. It has been characterized images and preimages of multimetric spaces under open, factor, compactly-open and perfect mappings and also generalized Khan-Banach theorem on extending of linear continuous functionals.

219 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On τ − sequential spaces Altay Asylkanovich Borubaev (1) and Buras Asanbekovich Boljiev (2) Kyrgyz National University named after J.Balasagyn, Bishkek, Kyrghyzstan [email protected], [email protected]

Abstract. Here we study a new notion of topological space, i.e. τ − sequential space.The idea of τ − sequential space belongs to A.A.Borubaev. Let τ be any cardinal, τ 1 may be finite. Let T=, nN ∈ . On T τ we consider the following partial order ≥ : for n

αβ, ∈T τ with αα= and ββ= we set βα provided αβ≥ for any i . Any ()i ()i  ii

τ τ collection {}xTα :α ∈ is called as τ − sequence and τ − sequence {}xTα :α ∈ is said to

τ converge to some x if for any neighborhood Ox of x there exists α ∈T such that xOβ ∈ x for all βα and this fact is denoted as {}xxα → .

DEFINITION 1. Topological space ()X ,σ is called τ − sequential if for any

∈ τ − ⊂ nonclosed subset A one can find some point x[] AA\ and some sequence {}xAα converging to xx(){}α → x. PROPOSITION 1. Let fX: → Y be a factor mapping of τ − sequential space onto topological space Y . Then Y is τ − sequential space. THEOREM 1. Let τ be any finite cardinal. Then each τ − sequential space is sequential one. A standard τ − sequence T τ is defined as a set τ with added to it point θ with all 0 T ττ τ coordinates equal to 0. Thus TT0 = ∪{}θ . All points in T0 are declaired to be open subsets

θ θ α ∈ τ except A base of neighborhood of form the following collection {}OTα : where =β ∈ τ βα OTα { :  } DEFINITION 2. Any topological sum of standard τ − sequences is called a standard τ − sequential space. THEOREM 2. Topological space ()X ,σ is τ − sequential iff it is an image of

220 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

some τ − sequential space under factor mapping. In case of τ =1we obtain well-known COROLLARY. Topological space is sequential iff it is an image of metrizable space under factor mapping. Now we study a special case of τ − sequential space, i.e. Freshet-Urysohn space. DEFINITION 3. Topological space ()X ,σ is called τ − Freshet-Urysohn if for

∈ τ − ⊂ any nonclosed subset A and for any point x[] AA\ there is some sequence {}xAα , converging to xx(){}α → x. THEOREM 3. There is a τ − sequential space which is not τ − Freshet-Urysohn one. PROPOSITION 2. Pseudo-open image of τ − Freshet-Urysohn space is τ − Freshet-Urysohn spaces.

DEFINITION 4. Let ()X ,σ be topological space and AY⊂ , then A is called

τ − sequentially open if for any aA∈ and any τ − sequence {}xaα → if follows that for some α0 xα ∈ A for all αα 0 .

DEFINITION 5. A subset A is called sequentially closed if from {}xxα → and

{}xAα ⊂ it follows that xA∈ .

THEOREM 4. Topological space ()X ,σ is τ − sequential iff every sequentially open (closed) subset is open (closed) one. THEOREM 5. Topological space ()X ,σ is τ − Freshet-Urysohn one iff it is an image of some standard τ − sequential space under pseudo-open mapping. As a corollary in case of τ = ℵ we obtain well-known result of Archangelskii 0 A.V.

221 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On τ − metric Spaces A.A. Borubaev (1) and K. Ishmakhametov (2)

Abstract. τ − metric spaces as generalizations of metric spaces have been recently introduced and studied by A. A. Borubaev. This paper considers some more problems corcerning such spaces.

τ Definition (A. A. Borubaev). Let X be set, R =[0, ∞ ), τ be a cardinal, and R be a + +

τ product of τ copies of R . A map ρ : XX×→ R satisfying the condition that + τ +

τ ρθ(,xy )= if and only if xy= , where θ =(0,0,....,0) ∈ R , is called o −−τ metric. In τ +

this case the pair ()X , ρτ is referrred to as o −−τ metric space. Let X be topological space with the weak base (in the sense of A. V. Archangelsky) and gXχ() is the character of it. It is said that the space X satisfies the weak first τ − countability axiom if gXχτ()≤ . Theorem 1. A topological space X is an 0 −−τ metrizable if and only if gXχτ()≤ . Theorem 2. A topological space X satisfies the weak first τ − countability axiom if and only if it is an image of a τ − metric space under a strong factorial mapping [1].

References

[1] A. A. Borubaev. Images of metric spaces under factorial mappings with additional conditions, Proceedings of young scientists, issue IV, Frunze (1975), 14-20.

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On Some Types of Strongly Continuous Multifunctions Ahu Açıkgöz (1) and Seda Göktepe (2) (1) Balikesir University, Balikesir, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. Continuity of functions is one of the most important and basic topics in the theory of classical point set topology and several branches of mathematics. A good number of continuous functions have been extended to the setting of multifunctions. Multifunctions have many applications in mathematical programming, probability, fixed-point theorems and economics [1] [2]. The purpose of this paper is to define upper (lower) strongly θ-β*g- continuous multifunctions and to get several properties of these continuities. Some properties of these multifunctions are studied and some characterizations of them are obtained. We describe new forms of continuity of multifunctions by using the concepts of β*g-θ-closed sets. Keywords. β*g-θ-closed, strongly continuity, multifunction. AMS 2010. 54A05, 54C05,54C08,54C10,54C60

References

[1] Dentcheva ,D., Regular castaing representations of multifunctions with applications to stochastics programming, Sıam Publications, 10, 732-749, 2000.

[2] Yalla M.V.V.S, Fennell E.C., Application of multifunction generator projection systems, IEEE Transactions on Power Delivery ,14,4,1285-1294,1999.

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Simplicial Weak (4,2)-chain Complexes of Simplicial Complexes Ajet Ahmeti University of Prishtina, Kosovo, [email protected]

Abstract. In this paper we give construction simplicial weak (4,2)-chain complex of topological space X , based on simplekses in a simplicial complex whose realization is the topological spaces X, denoted by wS(K) . Then, by appling the functors described in [13] , we obtain three homology groups denoted by HCn,2 , HCn,+ , HCn,* .

References

[1] G. Čupona: Vector valued semigroups; Semigroup Forum, Vol. 26, (1983), 65-74.

[2] G. Čupona, D. Dimovski: On a class of vector valued groups, Proc. of Conf. Alg. and Logic, Zagreb, (1984), 29-373.

[3] G. Čupona, D. Dimovski, A. Samardziski: Fully commutative vector valued groups, Prilozi, MANU, Skopje, 1987, 5-17.

[4] G. Čupona, N. Celakoski, S. Markovski, D. Dimovski: Vector valued groupoids, semigroups and groups, Vector Val. Sem. and Groups, MANU, Skopje, 1988, 1-79.

[5] Dold A.: Lectures on algebraic topology, Springer-Verlag, Berlin, Heidelberg, New York, 1972.

[6] D. Dimovski: Some existence conditions for vector valued groups, God. zbor. na Mat. Fak., 33-34, Skopje, 1982-83, 99-103.

[7] D. Dimovski: Examples of vector valued groups, Prilozi, MANU, VI 2, Skopje, 1985, 105-144.

[8] D. Dimovski: Free (n+1,n)-groups, Vector Val. Sem and Groups, MANU, Skopje, 1988, 103-122.

[9] D. Dimovski: Groups with unique product structures, Jour. of Algebra, V. 146, No.1, 1992, 205-209.

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[10] D. Dimovski, S. Ilič: Commutative (2m,m)-groups. Mac. acad. of sci. and arts. Skopje 1988, 79-90.

[11] D. Dimovski, S. Ilič: Free commutative (2m,m)-groups, Mat. Bilten, 13, 1989, Skopje, 25-34.

[12] D. Dimovski, B. Janeva, S. Ilič: Free (n,m)-groups, Communications in Algebra, 19 (3), 1991, 965-979.

[13] D. Dimovski, A.Ahmeti : (4,2)-Homology groups, MANU, Skopje, 2000, 33-46

[14] A.Ahmeti, D.Dimovski : Cubical weak (4,2)-chain complexes for topological spaces, Mathematica Macedonica Vol.5., 2007

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Actions and Coverings of Topological Groupoids A. Fatih Özcan Inonu University, Malatya, Turkey, [email protected]

Abstract. Let R be a topological group-groupoid. We define a category TGGdCov(R) of coverings of R and a category TGGdOp(R) of actions of R on topological groups and then prove the equivalence of these categories. Further, if R is topological ring-groupoid then we define a category TRGdCov(R) of coverings of R and a category TRGdOp(R) of actions of R on topological rings and then prove the equivalence of these categories. Key Words. Topological coverings, topological group-groupoids, topological ring- groupoids, actions

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On τ − ultracompact and τ − bounded spaces. Buras Asanbekovich Boljiev Kyrgyz Technicall University named after I.Razzakova, Bishkek, Kyrghyzstan, [email protected]

Abstract. In paper [4] V.Saks proved that ℵ0 -bounders and ultracompactness are equivalent in the class of regular space and due to this fact Saks set the following problem: does there exist separable ultracompact noncompact space? In papers [1], [2] one can find positive answer to the problem. Here we generalize this results on case of any regular cardinal. Let τ be any regular cardinal and X be discrete space of power τ . As usually β X is Stone-Cech compactification of X and β X is considered as the set of all ultrafilters on X . Each point p∈ β XX\ is free ultrafilter and just as in [3] we define the notions of p − limit point and p − converging τ − sequence. DEFINITION 1. Topological space is called τ − ultracompact if any τ − sequence has p − limit point for any p∈ β XX\ . DEFINITION 2. Topological space is called τ − bounded if closure of any subset A − of power no more than τ is compact. THEOREM 1. The notion of τ − ultracompactness and τ − boundedness are equivalent in the class of regular spaces. σ τ − σ σ For any topological space ()Y, is defined so called leader τ of ()Y, with σστ ⊇ .

THEOREM 2. Let ()Y,σ be compact space with tY() >τ and dY() =τ . σ τ − Then ()Y, τ is ultracompact noncompact space. From theorem 1 and 2 in case τ = ℵ we obtain results of V.Saks and the author. 0 References [1] Boljiev B.A. Ob odnom classe prostranstv, soderjashem sekvensialnye prostranstva. KGU. Frunze. 1987. [2] Boljiev B.A. O sekvensialnosti i compactnosti po ultafiltram. Bakinskaya mejdunarodnaya topologicheskaya konferensiya. Thesises. Baku. 1987. Chast 2. p. 48. [3] Bernstein A.R. A new kind of compactness for topological spaces. Fund.Math. 1970. V- 66. p. 185-193. [4] Saks V. Ultrafilters invariant on topological spaces.- Trans. Amer. Math. Soc.-1973.V- 241. P. 79-97.

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Separation Axioms on Soft Topological Spaces Banu Pazar Varol (1) and Halis Aygün (2) (1) Kocaeli University, Kocaeli, Turkey, [email protected] (2) Kocaeli University, Kocaeli, Turkey, [email protected]

Abstract. The concept of soft set theory has been initiated by Molodtsov [4] in 1999 as a general mathematical tool for modeling uncertainties. By a soft set we mean a pair ( , ) where is a set interpreted as the set of parameters and the mapping : ( ) is referred퐹 퐸 to as the퐸 soft structure on . After the introduction of the notion퐹 퐸of → soft ℘ 푋 sets several researchers improved this theory푋 in different mathematical concepts. In particular, the concepts of fuzzy soft sets [3], fuzzy soft groups [1], soft topological spaces [6], soft compactness [2], soft neighborhood structures [5], etc., were considered. In the present work, we introduce some new concepts in soft topological space such as convergence of sequences, homeomorphism and investigate the relations between these concepts and soft separation axioms, particularly soft Hausdorff axiom, in soft topological space. Keywords. Soft set, soft topology, soft continuity, soft separation axioms, soft compactness. AMS 2010. 06D72, 54A40.

References

[1] Aygünoğu, A., Aygün, H., Introduction to fuzzy soft groups, Comput. Math. Appl., 58, 1279-1286, 2009.

[2] Aygünoğu, A., Aygün, H., Some notes on soft topological spaces, Neural Comput & Applic., DOI 10.1007/ s00521-011-0722-3.

[3] Maji, P.K., Biswas, R., Roy, A.R., Fuzzy soft sets, J. Fuzzy Math. 9(3), 589-602, 2001.

[4] Molodtsov, D., Soft set theory-First results, Comput. Math. Appl., 37 (4\5), 19-31, 1999.

[5] Pazar Varol, B., Aygün, H., Shostak, A.P., Categories related to topology viewed as soft sets, Proceedings of the 7th Conferences of the European Society for Fuzzy Logic and Technology (EUSFLAT 2011), 1-1, 883-890, 2011.

[6] Shabir, M., Naz, M., On soft topological spaces, Comput. Math. Appl 61, 1786-1799, 2011.

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Completeness Properties of Di-Uniform Texture Spaces Filiz Yıldız Hacettepe University, Ankara, Turkey, [email protected]

Abstract. The notion of texture was introduced as an appropriate setting for the development of complement-free mathematical concepts and it was shown that the ditopological texture spaces provide a unified setting for the study of topology, bitopology and fuzzy topology. For some relations of the ditopological texture spaces with the topological and bitopological cases, the basic references are [1], [2] and [3]. Following that, a suitable uniformity theory in textures was constructed under the name di- uniformity, and this represented a generalization of classical quasi uniform spaces. Therefore, in the previous studies, by defining regular Cauchy difilter in texture, the concept of dicompleteness [2] has been introduced in di-uniform texture spaces as a natural counterpart of “completeness” in the sense of quasi-uniformity. Correspondingly, in this work, • It will be especially interested in categorical aspects of the connections between di- uniformity and classical uniformity, quasi-uniformity. • Some useful results obtained by constructing isomorphic categories and using the connections between quasi uniformity and di-uniformity will be presented insofar as completeness is concerned. Keywords. Texture, Ditopology, Di-uniformity, Quasi-uniformity, Regular Cauchy difilter, Isomorphic Categories, Comleteneses, Dicompleteness AMS 2010. 54E15, 54C30, 54A05, 54E55, 54B30, 03E20

References

[1] Yıldız, F., Connections Between Real Compactifications in Various Categories, submitted, 2012.

[2] Yıldız, F., Brown, L. M., Dicompleteness and Real Dicompactness of Di- topological Texture Spaces, Topology and Its Applications, 158, (15), 1976--1989, 2011.

[3] Yıldız , F. , Brown, L. M., Categories of dicompact bi-T2 texture spaces and a Banach- Stone theorem, Quaestiones Mathematicae, 30, 167--192, 2007.

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Topological Internal Groupoids and Their Coverings Hürmet Fulya Akız (1), Nazmiye Alemdar (2), Tunçar Şahan (3) and Osman Mucuk (4) (1) Bozok University, Yozgat, Turkey, [email protected] (2) Erciyes University, Yozgat, Turkey, [email protected] (3) Erciyes University, Yozgat, Turkey, [email protected] (4) Erciyes University, Yozgat, Turkey, [email protected]

Abstract.. By a covering group of a topological group X we mean a covering map ~ ~ p: → XX such that X is a topological group and p is also a morphism of groups. In this way we obtain a category TGpCov/X of covering groups of X (see for example \cite[3]). If

X is a topological group the fundamental groupoid π1X becomes a group-groupoid. So we have an analogous category GpGdCov/π1X of group-groupoid coverings of π1X . It is proved [1] that for a topological group X the categories TGpCov/X and GpGdCov/π1X are equivalent.

On the other hand if A is a group and group-groupoid G acts on A wia → OA G then we get a category GpGdAct(G) of group-groupoid actions of G. Brown and Mucuk in [1] proved that the categories GpGdCov/G and GpGdAct(G) are equivalent. The object of this paper is to show that these two results can be generalised to a wide class of algebraic structures, which include groups, rings (without 1), Lie algebras, Jordan algebras, and many others, as they have topology structures. Let C be a category of same type of groups with operations handled in [11] and X is a topological group with operations in C. We will prove that the category TGpOpCov/X of coverings of topological groups with operations X in C is equivalent to the category

TIntGdCov/π1X of coverings of topological internal groupoid π1X .We also prove that if G is an internal category in the category of topological groups with operations, i.e. G is a topological internal groupoid, the category TIntGdCov/G of topological internal groupoid \- coverings of G is equivalent to the category TIntGdAct/G of topological internal groupoid actions of G. Keywords. group with operations, internal groupoids

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References

[1] Brown, R. and Mucuk, O., Covering groups of non-connected topological groups revisited, Math. Proc. Camb. Phill. Soc. 115 (1994) 97-110.

[2] Brown, R. and Mucuk, O., Monodromy groupoid of a Lie Groupoid, Cah. Top. G\'eom. Diff. Cat. 36 (1995) 345-370.

[3] Chevalley, C., Theory of Lie groups, Princeton University Press, 1946.

[4] Higgins, P.J., Groups with multiple operators, Proc. London Math. Soc. (3) 6 (1956) 366- 416.

[5] Brown, R., Spencer, B. C. , G-groupoids, Crossed Modules and the Fundemental Grupoid of a Topological Group, Proc. Konink. Nederl. Akad. Wetensch., 296-302, 1976.

[6] Mucuk, O., Covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid, PhD Thesis, University of Wales, 1993.

[7] Mucuk O., Coverings and ring-groupoids, Geor. Math. J., 5 (1998) 475-482

[8] Olver, P.J., Non-associatibe local Lie groups, J. Lie Theory 6 (1996), 23-51.

[9] Whitehead, J.H.C., Combinatorial Homotopy II.Bull. Amer. Math. Soc. 55 (1949), 453 496

[10] Taylor, R.L., Covering groups of non-connected topological groups, Proc. Amer. Math. Soc., 5 (1954) 753-768.

[11] Orzech, G., Obstruction theory in algebraic categories I and II, J. Pure. Appl. Algebra 2 (1972) 287-314 and 315-340.

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Some Properties of Polynomials G and N, and Tables of Knots and Links İsmet Altıntaş Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In [1], we have constructed a polynomial invariant of regular isotopy, GL , for oriented knot and link diagrams L. From GL by multiplying it by normalizing factor, we obtained an ambient isotopy invariant, N L , for oriented knots and links. In this paper, we give some properties of these polynomials. We also calculate the polynomials GL and N L of the knots through nine crossings and the two-component links through eight crossing. Keywords. G-polynomial, N-polinomial, Jones Polynomial, regular isotopy, ambient isotopy, tables of knots and links. AMS 2010. 57M25

References

[1] Altintas, I., An oriented state model for the Jones polynomial and its applications alternating links, Applied Mathematics and Computation, 194,, 168-178, 2007.

[2] Jones, V.F.R., A new knot polynomial and Von Neuman algebras, Notices. Amer. Math. Soc., 1985.

[3] Jones, V.F.R., A new knot polynomial and Von Neuman algebras, Bul. Amer. Math. Soc. 12, 103-111, 1985.

[4] Jones, V.F.R., Hecke algebra representations of braid groups and link polynomial, Ann. Math., 126, 335-388, 1987.

[5] Kauffman, L.H., State models and the Jones polynomial, Topology, 26, 395-407, 1987.

[6] Kauffman, L.H., New invariants in the theory of knots, Amer. Math. Monthly vol. 95, 195- 242, 1988.

[7] Kauffman, L.H., An invariant of regular isotopy, Trans. Amer. Math. Soc. 318, 417-471, 1990.

[8] Kauffman, L.H., Knot and physics, Worıd Scientific, 1991, (second edition 1993).

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[9] Tait, P. G., On Knots I,II,III., Scientific Papers Vol. I, Cambridge University Press, London, 273-347, 1898.

[10] Kirkman, T.P., The enumeration, description and construction of knots with fewer than 10 crossings, Trans.R.Soc. Edinb., 32, 281-309, 1865.

[11] Little, C.N., Non-alternate  knots, Trans.R.Soc. Edinb., 35, 663-664,1889.

[12] Murasugi, K., Knot theory and its applications, translated by Kurpito, B., Birkhause, Boston, 199).

[13] Rolfsen, D., Knot and Links, Mathematics Lectures Series No. 7 Publish or Perish Press, 1976.

[14] Kauffman, L.H., On knots, Princeton University Pres, Princeton, New Jersey, 1987.

[15] www.knotplot.com

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On Uniformly Paracompact Mappings Kanetov Bekbolot Emenovich Kyrgyz National University named after J.Balasagyn, Bishkek, Kyrghyzstan

Abstract. It has become popular in the theory of uniformly continuous mappings to extend some properties if uniform spaces on mappings. A space can be considered as spacial case a mapping when the space is identified with mapping of it into a point. So one meets an idea of extending on mappings some notions and statements for spaces allowing obtain generalizations of some results. In this paper the notions of uniformly U - paracompact, uniformly strongly U -paracompact and uniformly weakly u − paracompact spaces have been extended on mappings and studied. DEFINITION 1. Uniformly continuous mappings f:,()() XU→ YV , of uniform space

()XU, onto uniform space ()YV, is called uniformly U -paracompact mapping if for any

α ∈U one can find β ∈V and locally finite uniform cover γ ∈U such that f −1βγ∧  α. Uniform space is called uniformly U -paracompact if in any uniform cover one can inscribe locally finite uniform cover.

Let f:,()() XU→ YV , be uniformly continuous mapping and if ()XU, is U -paracompact space then f is uniformly U -paracompact mappings.

Let f:,()() XU→ YV , is uniformly U -paracompact mapping and Yy= {} then ()XU, is uniformly U -paracompact space. Composition of two uniformly U -paracompact mappings is uniformly U -paracompact mappings. Let f:,()() XU→ YV , be uniformly continues mappings from ()XU, onto ()YV, and

MU, be a subspace of XU, . If f is uniformly U -paracompact mapping then ()M () restricted mapping f:, MU→ YV , is uniformly U -paracompact mapping too. MM()() THEOREM 1. If f:,()() XU→ YV , is uniformly U -paracompact mapping and

()YV, is uniformly U -paracompact space then ()XU, is uniformly U paracompact space then ()XU, is uniformly U -paracompact space.

DEFINITION 2. Uniformly continuous mappings f:,()() XU→ YV , of

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uniform space ()XU, onto uniform space ()YV, is called uniformly strongly (weakly) U - paracompact mapping if for any α ∈U there are β ∈V and star (pointly) finite uniform cover γ ∈U such that f −1βγ∧  α. Uniform space is called uniformly strongly (weakly) U -paracompact if in any uniform cover one can inscribe star (pointly) finite uniform cover. Let f:,()() XU→ YV , be uniformly continuous mapping. If ()XU, is uniformly strongly (weakly) U -paracompact space then f is uniformly strongly (weakly) U -paracompact mappings. THEOREM 2. If f:,()() XU→ YV , uniformly strongly (weakly)U - paracompact mapping and ()YV, is uniformly strongly (weakly) U -paracompact space ()XU, is uniformly strongly (weakly) U -paracompact space.

THEOREM 3. Let f:,()() XU→ YV , be paracompact mappings of ()XU, onto ()YV, . Then the following statements are equivalent:

1. f is uniformly strongly U -paracompact.

2. f is uniformly U -paracompact. 3. f is uniformly weakly U -paracompact.

235 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Proper C − Tameness and Proper Deformation Dimension Minir Efendija University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. Theory of proper shape, for locally compact metric spaces, was introduced by Ball and Sher [1]. Cerin defined some geometric properties at infinity for locally compact metric spaces, as are: C − tameness at infinity [3], C − movability at infinity, C − triviality at infinity, C − calmness at infinity, e.t.c. We introduce proper versions of Cerin’s C − tameness and Dydak’s deformation dimension and show that they are invariants of proper quasi-domination [6]. i.e. proper shape invariants. Moreover, for a special class of C , it is proved equivalence of those notions. Keywords. Proper homotopy, proper shape, ANR' s , absolute proper shape retract ()APSR .

AMS 2010. 55P55, 55P57, 54C55.

References

[1] Ball, B. J, Sher, R.B, Theory of proper shape for locally compact metric spaces, Fund. Math. 86(1074), 163-192.

[2] Ball, B, Alternative approaches to proper shape theory, Studies in Topology, edited by N. M. Stavrakas, K. R. Allen, Academic Press, New, York, 1-27,1975.

[3] Ball, B, Proper shape retracts, Fund. Math. 89, 177-189, 1975.

[4] Borsuk, K, Theory of shape, Monogr. Matem. 59, Polish Scientific Publishers, Warszawa, 1977.

[5] Cerin, Locally compact spaces C − tame at infinity, Publications de I’institut mathematique, tome 22 (36) 1977, 49-59.

[6] Efendija, M, Prave kvazi-relacije i prava C − pitomost, Kërkime (Reaserches), Acad. of Sc. And Arts of Kosovo, Book 4, 11-16, 1986.

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The Product of Shape Fibrations Qamil Haxhibeqiri University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. The notion of shape fibration for maps between metric compact was introduced by S. Mardešić and T. B. Rushing in [4] and [5]. In [3] S. Mardešić has extented this notion to maps of arbitrary topological spaces. The author has estabilished some further properties of shape fibrations in the noncompact case (see e.g. [1], [2]). The main result of this paper is the following Theorem 1: Let pE: → B, pE′′: → B ′ be maps of compact Hausdorff spaces. Then ppEE×′′: × →× BB ′ is a shape fibration if and only if p and p′ are shape fibrations. Our proof is designed so that if the following Proposition 1: Let E and B be compact Hausdorff spaces. Then for every normal covering  of EB× there are normal covering  of E and an open covering  of B such that ×=×{:,V WV ∈ W ∈  } is a normal covering of EB× which refines  . holds for E arbitrary topological space then the above Theorem 1 on product of shape fibrations remains true also in the case when E , E′ are arbitrary topological spaces. Thus, answer in the Question: Let E , E′ be arbitrary topological spaces and B , B′ compact Hausdorff spaces. Is true that ppEE×′′: × →× BB ′ is shape fibration if and only if pE: → B and pE′′: → B ′ are shape fibrations? is equivalent to the answer in the following Question: Does the Proposition 1 above holds when E is an arbitrary topologicial space? In order to obtain our main result (Theorem 1), we have shown the following result about resolutions of product spaces:

Theorem 2: Let qE=()()qλ : E →= Eqλ ,, λλ′ Λ and rB=()()rµ : B →= Brµ ,, µµ′ M are morphisms of pro-Cpt such that E and B are compact ANR −systems. Then

qr× =()qrλ,: µ EB × →EB × =( Eλ × Bq µ,, λλ′′ × r µµ Λ× M) is a resolution of EB× if and only if q and r are resolutions of E and B , respectively.

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Keywords. Shape fibration, resolution, approximate homotopy lifting property AMS 2010. 55 P 55, 54 B 25, 54 B 10.

References

[1] Q. Haxhibeqiri, Shape fibrations for topologicial spaces, Glas. Mat. 17 (37) (1982), pp. 381-401. [2] Q. Haxhibeqiri, The exact sequence of a shape fibration, Glas. Mat. 18 (38) (1983) pp. 157-177.

[3] S. Mardešić, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), pp. 53-78.

[4] S. Mardešić and T. B. Rushing, Shape fibrations I, Gen. Top. And Appl. 9(1978), pp. 193- 215.

[5] S. Mardešić and T. B. Rushing, Shape fibrations II, Gen. Top. And Appl. 9(1979), pp. 283-298.

[6] T. Watanabe, Approximative shape theory, Mimeographed Notes, Univ. Of Yamaguchi, 1982.

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Rough Topology on Covering-Based Rough Sets Setenay Akduman(1), Eylem Zeliha Yıldız(2), Ahmet Zemci Özcelik(3) and Serkan Narlı(4) (1) Dokuz Eylul University, Izmir, Turkey, [email protected] (2) Dokuz Eylul University, Izmir, Turkey, [email protected] (3) Dokuz Eylul University, Izmir, Turkey, [email protected] (4) Dokuz Eylul University, Izmir, Turkey, [email protected]

Abstract A classical rough set theory was given by Pawlak [2]. Since then rough set theory has been investigated by many researchers. Rough set theory can be viewed as an approach to vagueness. Many scientific fields find vagueness concepts interesting; therefore the rough set theory has important applications in fields such as data analysis and information systems. There are many works on topological and algebraic approaches to rough sets. For example, the union and intersection operations of classical rough sets were introduced by Pomykala [3]. Classical rough sets have been defined on partitions which is a special case of the formalism of coverings. This allows us to study covering-based rough sets. In this paper, we consider covering-based rough sets as an extension of classical rough sets studied by Bryniarski [1]. We also construct a rough topology on a given covering-based rough set. In order to do that, union and intersection operations will be extended to covering- based rough sets. In the future, our aim is to examine applications of rough topology on covering-based rough sets. This work leads us to study some properties of covering-based rough topology. Keywords. Rough Sets, Partition , Covering . AMS 2010. 54B99, 03E99

References

[1] Bryniarski, E., A calculus of rough sets of the first order, Bulletin of the Polish academy of sciences., 16, 71- 77, 1989.

[2] Pawlak, Z., Rough sets: power set hierarchy, ICS PAS Reports., 470, 1982.

[3] Pomykala, J., Pomykala, J., The stone algebra of rough sets, Bulletin of the Polish Academy of Sciences., 36(7-8), 495- 508, 1988.

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Simplicial Homotopy in Digital Images Simge Öztunç (1), Necdet Bildik (2) and Ali Mutlu (3) (1) Celal Bayar University, Manisa, TURKEY, [email protected] (2) Celal Bayar University, Manisa, TURKEY, [email protected] (3) Celal Bayar University, Manisa, TURKEY, [email protected]

Abstract. In this paper we introduced the Kan Complexes in digital images. Then we defined the path and path component for simplicial sets in digital images and we invastigated homotopy properties of simplicial sets in digital images by using simplicial identities. Keywords. Digital Image, Adjacency Relation, Simplicial Map, Digital Homotopy, Simplicial Homotopy. AMS 2010. 55N35, 68R10, 68U05, 68U10.

References

[1] L. Boxer, “Digitally continuous functions,” Pattern Recognition Letters, Vol. 15, pp. 833–839, 1994.

[2] L. Boxer, “Properties of Digital Homotopy,” Journal of Mathematical Imaging and Vision, Vol. 22, pp. 19–26, 2005.

[3] G. Freidman, An Elemantary Illustrated Introduction to Simplicial Set, 2011

[4] I. Karaca, L. Boxer and A Öztel, Topological Invariants in digital images, Jour. of Mathematical Sciences: Advances and Applications, 11(2011) No 2, 109-140

[5] I. Karaca and L. Boxer, Some Properties of digital covering spaces, Journal of Mathematical Imaging and Vision, 37(2010), 17-26.

[6] I. Karaca and L. Boxer., The classification of Digital Covering spaces, Journal of Mathematical Imaging and Vision, 32(2008), 23-29.

[7] A. Mutlu, B. Mutlu, S. Öztunç, On Digital Homotopy Theory of Digital Images, Research Journal of Algebra, Vol 2, No:6, 147-154, 2012.

[8] A. Rosenfeld, “Continuous functions on digital pictures” Pattern Recognition Letters, Vol. 4, 1986, pp. 177–184.

[9] S. Öztunç, A. Mutlu, Structure of Simplicial Sets in Digital Image, International Sympozyum on Applied Analysis and Algebra, 2012

240 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Semi-Compactness in Ditopological Texture Spaces Şenol Dost Hacettepe University, Ankara, Turkey, [email protected]

Abstract. By a texturing [2] of a set S we mean a subset δ of the power set P(S) which is a point separating complete, completely distiributive lattice with respect to inclusion which contains S and ∅ , and for which arbitrary meets coincide with intersections and finite joins coincide with unions. The pair ()S,δ is then called a texture space, or simply texture.

By a ditopology [1] ()τκ, on ()S,δ we mean a family τδ⊆ of open sets containing S and

∅ and closed under finite meets and arbitrary joins, and a general unrelated family κδ⊆ of closed sets containing S and ∅ and closed under finite joins and arbitrary meets. Difunctions [3] arise often in the study of textures and ditopological texture spaces. A difunction is a direlation ()fF, satisfying certain additional conditions.

A ditopology is a "topology" for which there is no a priori relation between the open and closed sets. As a result of this, dual properties such as compactness-cocompactness [1] occur often in the theory of ditopological texture spaces. In this study, we introduce the notion of semi-compactness in ditopological texture spaces. Result include preservation under surjective semi-(co)continuous difunctions. On the other hand new concepts such as semi-stability and semi-costability and their properties are given in ditopological texture spaces. Keywords. Ditopological texture space, Semi-compactness, Semi-cocompactness, Semi-stability, Semi-costability. AMS 2010. 54A05, 54D20, 54D30.

References

[1] Brown, L. M., Diker, M., Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems, 98, 217{224, 1998.

[2] Brown, L. M., Ertürk, R., Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy Sets and Systems, 110 (2), 227{236, 2000.

[3] Brown L. M., Ertürk R., Dost Ş., Ditopological texture spaces and fuzzy topology, I. Basic Concepts, Fuzzy Sets and Systems, 147 (2), 171-199, 2004.

241 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Generalized Closed Sets and Continuous Functions in Ideal Topological Spaces Ümit Karabıyık Necmettin Erbakan University, Konya, Turkey, [email protected]

Abstract. In this paper, definitions of some closed sets, g-closed[10]sets and rg- closed[13] sets in general topology have been given. Ig-closed, rIg-closed, τ * -closed [9], *- perfect[7], O* -set, sets corresponding to the sets in topological spaces and connections between them have been analyzed by using local functions and definitions of ideal topological spaces. Also by means of these analyzed sets, Ig-continuous, rIg-continuous, Ic-continuous, perfectly rg-continuous [20], perfectly rIg-continuous, strongly I-continuous, strongly rg- continuous [20], strongly rIg-continuous functions has been defined and connections between them have been examined and finally new properties between these functions have been found. Keywords. Ideal topological space, Ig-closed set, rIg-closed set, O* - set, continuous function.

References

[1] Arya, S.P. and Gupta, R.1974. On strongly continuous mappings, Kyung pook Math. J.14, 131-143

[2] Balachandran, K., Sundram, P., and Maki, H. 1991. On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. 12, 5-13.

[3] Dontchev, J., Ganster, M. and Noiri, T.1999. Unıfied operation approach of generalized closed sets via topological ideals, Math Japonica 49(3), 395-401.

[4] Dontchev, J. and Noiri, T.1999. Contra-semicontinuous functions, Math.Pannonica 10,159-168

[5] Ekici, E.2007. On almost πgp -continuous functions, Chaos,Solitons and Fractals 32(5),1935-1944

[6] Ekici, E.2008. On contra πg -continuous functions, Chaos,Solitons and Fractals 35(1), 71- 81

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[7] Hayashi, E. 1964. Topologies defined by local properties, Math. Ann. 156, 205-215.

[8] Janković, D.S. 1983. On locally irreducible spaces, Ann. Soc. Sci. Bruxelles Ser. I, 97, 59- 72.

[9] Janković, D. and Hamlet, T.R.1990. New topologies from old via ideals, Amer. Math. Monthly, 97, 295-310.

[10] Kasara, S. 1979.Operation-compact spaces, Math.Japon.,24,97-105

[11] Keskin, A. 2003. New decompositions of contiunity in ideal topological spaces, PhD. Thesis, Konya.

[12] Keskin, A., Noiri, T. and Yuksel S. 2004. f I -sets and decomposition of I CR -continuity, Acta Math. Hungar. 104(4), 307-313.

[13] Kuratowski, K, 1933, Topologie I, Warszawa.

[14] Levine, N.1970. Generalized closed sets in topology, Rend. Circ. Math. Palermo (2), 19, 89-96.

[15] Levine, N. 1960. Strong continuity in topological space, Amer. Math. Monthly 67, 269.

[16] Noiri, T.1996. Mildly normal spaces and some functions, Kyungpook Math. J.36, 183- 190.

[17] Noiri, T. 1984.Supercontinuity and some strong forms of continuity, Indian J. Pure Appl. Math.15(3), 241-250.

[18] Ogata, H. 1983. Operations on topological spaces and associated topology, Math.Japon. 38, 981-985.

[19] Palaniappan, N. and Rao, K.C. 1993. Regular generalized closed sets, Kyungpook Math. J. 33, 211-219

[20] Rani, A. and Balachandran, K.1997. On regular Generalized Continuous Maps in Topological Spaces, Kyunpook Math. J. 37, 305-314.

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[21] Samuels, P.1975. A topology formed from a given topological space, J. London Math. Soc. (2), Studies 10,409-416.

[22] Tong, T. 1989. On decomposition of continuity in topological spaces, Acta Math. Hungar. 54(1-2), 51-55.

[23] Vaidyanathaswamy, R.1945. The localization theory in set-topology, Proc. Indian Acad. Sci. Studies 20,51-61.

[24] Vaidyanathaswamy, R.1960. Set Topology, Chelsea Publishing Company, New York.

[25] Yuksel, S., Keskin, A. and Noiri, T. 2004. Idealization of decomposition theorem, Acta Math. Hungar. 102 (4), 269-277

244 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Extension of Shi’s Quasi-Uniformity to the Fuzzy Soft Sets Vildan Çetkin (1) and Halis Aygün (2) (2) Kocaeli University, Kocaeli, Turkey, [email protected] (2) Kocaeli University, Kocaeli, Turkey, [email protected]

Abstract. The theory of uniform structures is an important area of topology which in a certain sense can be viewed as a bridge linking metrics as well as topological groups with general topological structures. Therefore, it is not surprising that the attention of mathematicians interested in fuzzy topology constantly addressed the problem to give an appropriate definition of a uniformity in fuzzy context. (see [3,4,7,8,10]). The purpose of this talk is to investigate the extension of Shi’s (quasi-)uniformity[10] in the context of fuzzy soft set. We introduce the concept of fuzzy soft remote neighborhood system and give the relations between the fuzzy soft quasi-uniformity and fuzzy soft remote neighborhood system. Keywords. Fuzzy soft set, fuzzy soft topology, quasi-uniformity. AMS 2010. 54A40, 54E15.

References

[1] Aktaş, H., Çağman, N., Soft sets and soft groups, Inform. Sciences, 177(13) 2726-2735, 2007

[2] Aygünoğlu, A., Çetkin, V., Aygün, H., An introduction to fuzzy soft topological spaces, Hacettepe J. of Maths. and Statistics, to be published.

[3] Höhle, U., Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math. 38, no.3 289-323, 1982.

[4] Hutton, B., Uniformities on fuzzy topological spaces, J. Math. Anal. Appl., 58, 559-571, 1977.

[5] Maji, P.K., Biswas, R., Roy, A. R., Fuzzy soft sets, J. of Fuzzy Maths., 9(3), 589-602, 2001.

[6] Molodtsov, D., Soft set theory-First results, Comput. and Maths. with Appl., 37, 19-31, 1999.

[7] Lowen, R., Fuzzy uniform spaces, J. Math. Anal. Appl., 82(2), 370-385, 1981.

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[8] Shi, F.-G., Pointwise uniformities in fuzzy set theory, Fuzzy Sets and Syst., 98(1), 141- 146, 1998.

[9] Sostak, A.P., On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser. II 11, 89-103, 1985.

[10] Yue, Y., Shi, F.-G., L-fuzzy uniform spaces, J. Korean Math. Soc., 44(6), 1383-1396, 2007.

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STATISTICS

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Weight Function Efficiency Against Outliers in Nonlinear Regression Ahmet Pekgör (1) and Aşır Genç (2) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. Detecting outliers in a multivariate point cloud is not trivial, particularly when there are several outliers. The classical identification method deoes not always find them, because it is based on the sample mean and covariance matrix, which are themselves affected by the outliers. In this work, It is compared the efficiency of confirmetion outliers of weight functions, using M estimator’s different weight functions which arent affected by outliers in nonlinear regression. It is used Monte Carlo simulation for determining efficiency of these weight functions in confirming outliers. Keywords. Nonlinear Regression, Weight Function, M-Estimator, Outlier, Robust AMS 2010. 62J02, 62J15.

References

[1] Huber, P.J, “Robust Statistics”, John Willey&Sons, Inc., USA, 1981.

[2] Motulsky, H.J ve Brown, R.E. (2006), Detecting Outliers when Fitting Data with Nonlinear regression – A new Method Based on Robust Nonlinear Regression and the False Discovery Rate, BMC, Bioinformatics, 1471-2105/7/123.

[3] Pekgör, A. (2010). Computation of Breakdown Points in Nonlinear Regregression and an Application. Philosophy of Doctora (Ph.D.) Thesis, Selcuk University, Graduate School of Natural and Applied Sciences, Department of Mathematics, Konya, Turkey. Turkish Government Higher Education Board National Thesis Center Thesis Number: 247007. http://tez2.yok.gov.tr/.

[4] Rousseeuw, P. J. ve Zomeren, B. C. (1990), Unmasking Multivariate Outliers and Leverage Points, Journal of the American Statistical Association, vol. 85, no. 411, 633-639.

[5] Seber, G. A. F. ve Wild, C. J. (1989), Nonlinear Regression, John Willey&Sons, Inc., USA.

[6] Serbert, D.M., Montgomery, D.C. and Rollier, D. (1998), “A Clustering Algorithm for Identifying Multiple Outliers in Linear Regression”, Computational Statistics & Data Analysis, vol. 27, pp. 461-484.

247 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Transmuted Exponentiated Exponential Distribution Faton Merovci University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. In this article, the exponentiated exponential probability distribution is embedded in a larger family obtained by introducing an additional parameter. We generalize the exponentiated exponential distribution using the quadratic rank transmutation map studied by Shaw et al. [7] to develop a transmuted exponentiated exponential distribution. We provide a comprehensive description of the mathematical properties of the subject distribution along with its reliability behavior. The usefulness of the exponentiated exponential distribution distribution for modeling reliability data is illustrated using real data. Keywords. Exponentiated exponential distribution, hazard rate function, reliability function, parameter estimation. AMS 2010. 62N05; 90B25.

References

[1] Gokarna R. Aryal and Chris P. Tsokos. On the transmuted extreme value distribution with application. Nonlinear Analysis: Theory, Methods and Applications, 71:1401–1407, 2009.

[2] R. Ihaka and R. Gentleman. R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5:299–314, 1996. [3] Saralees Nadarajah, The exponentiated exponential distribution: a survey, AStA Advances in Statistical Analysis, Volume 95, Number 3 (2011), 219-251, DOI: 10.1007/s10182-011-0154-5

[4] Zakaria Y. AL-Jammal*, Exponentiated Exponential Distribution as a Failure Time Distribution, Iraqi Journal of Statistical Science (14) 2008 p.p. [63-75]

[5] Gupta, R. D. and Kundu, D., 2001, "Exponentiated Exponential Family; An Alternative to Gamma and Weibull ", Biometrical Journal, vol. 33, no. 1, pp.117-130.

[6] Gupta, R. D. and Kundu, D., 2007, " Generalized exponential distribution: existing methods and recent developments " Journal of the Statistical Planning and Inference, vol. 137, no. 11, pp.3537 – 3547.

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[7] C. P. Quesenberry and Jacqueline Kent. Selecting Among Probability Distributions Used in Reliability. Technometrics, 24(1):59–65, 1982.

[8] W. Shaw and I. Buckley. The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map..Research report, 2007.

249 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Comparisons of Imputation Methods for Missing Data in Mixture Discriminant Analysis Murat Erişoğlu (1) and İsmet Kürşat Sarı (2) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. Missing data is a problem that permeates much of resarch biring done today. In the literature many methods have been proposed to estimate missing value. In this study, we present an evaluation of the performance of current imputation methods across a variety of types and sizes of data sets in mixture discriminant analysis(MDA). Keywords. Stochastic Regression Imputation, Hot-Deck Imputation, MDA AMS 2010. 62-07

References

[1] Halbe, Z. and Aladjem, M., Model-Based Mixture Discriminant Analysis– An Experimental Study, Pattern Recognition 38, 437–440, 2005.

[2] Hastie, T., Tibshirani, R., Discriminant Analysis by Gaussian Mixtures, Journal of the Royal Statistical Society. Series B (Methodological), 58 (1), 155-176, 1996.

[3] Jackson E.C., Missing Values in Linear Multiple Discriminant Analysis, Biometrics, 24(4), 835-844, 1968.

[4] Lim, E. A., Zarita, Z., A comparative Study of Missing Value Estimation Methods Which Method Performs Beter?, International Conference on Electronic Design, Penang, Malaysia, 1-3 December, 2008.

[5] Nguyen, D.V., Wang, N., Carroll, R.I., Evaluation of missing value estimation for microarray data, Journal ofData Science, 2, 347-370, 2004.

[6] Olga, T., Michael, C., Gavin, S., Pat, B., Trevor, H., Robert, T., David, B. Russ, B. A., Missing value estimation methods for DNA microarrays, Bioinformatics, 17(6), 520-525, 2001.

[7] Schafer, J. L., Analysis of Incomplete Multivariate Data, Chapman&Hall/CRC, 2000.

250 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Determining Number of Clusters with Analytic Hierarchy Process Murat Erişoğlu Selcuk University, Konya, Turkey, [email protected]

Abstract. Cluster analysis is an important statistical tool with a wide field of application in data analysis. In cluster analysis, determining the number of clusters is one of the most important problems. Although there are many methods have been proposed for this manner, unfortunately there is no generally accepted procedure. In this study, we proposed analytic hierarchy process(AHP) for determining number of clusters. AHP, since its invention, has been a tool at the hands of decision makers and researchers; and it is one of the most widely used multiple criteria decision-making tools. The basis of this study, determine the number of clusters using together various estimation methods with AHP according to multiple criteria. Keywords. Number of Clusters, GAP, Silhoutte. AMS 2010. 62-07.

References

[1] Erişoğlu, M., Erişoğlu, Ü., Servi, T., Sakallıoğlu, S., A new aproach for determination number of clusters, Pak. J. Statist., 28(1), 141-158 2012.

[2] Ravi, J., Andy, K., Innovation in the cluster validating techniques, Fuzzy Optim. Decis. Making, 7, 233-242, 2008.

[3] Omkarprasad, S. V., Sushil, K., Analytic hierarchy process: An overview of applications, European Journal of Operational Research, 169, 1-29, 2006.

[4] Thomas, L. S., Decision makinh with the analytic hierarchy process, Int. J. Services Sciences, 1(1), 83-98, 2008.

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Analysis of Divorces Using Survival Models Nihal Ata (1) and Tuğça Poyraz Tacoğlu (2) (1) Hacettepe University, Ankara, Turkey, [email protected] (2) Hacettepe University, Ankara, Turkey, [email protected]

Abstract. Survival analysis is a class of statistical methods for studying the occurrence and timing of events in both social and natural sciences. Although these methods are widely used in medical statistics, its use in actuarial sciences, economics, marketing research, and sociology gain importance in recent years. In survival analysis, survival (failure) time is defined as the time to the occurrence of a given event. This event can be the death, development of a disease, response to a treatment, equipment failures, births, marriages, divorces, promotions, retirements, and so forth. When subjects have not experienced the event of interest at the end of the study, the exact survival times of these subjects are unknown and these are called censored observations or censored times. Survival data have some features that are difficult to handle with traditional statistical methods: censoring and time-dependent covariates. The most common approach to model covariate effects on survival is the proportional hazards model [1] and parametric regression models [2, 3, 4]. The scope of this study focuses on usage of survival models in sociology which is less well known by social scientists. In Turkey, the number of divorces is increased from 94219 in 2007 to 118568 in 2010 and crude divorce rate is increased from 1.34 to 1.62 in the same period [5]. Thus, survival models can be a useful tool to determine the predicting factors (covariates) that might associate to divorce and provide a solution to reduce the divorce rate. In this study, divorce is taken as failure and time between marriage and divorce dates are taken as survival time [6, 7]. The undivorced individuals at the end of the follow-up time are called as censored observations. We analyze the survival data by fitting into the non- parametric method by using Kaplan-Meier estimation followed by log-rank test, and semi- parametric and parametric methods by applying the proportional hazards models, accelerated failure models and frailty models. The models are compared to obtain the best model that could predict the factors of divorce, then significance of the factors are discussed for fitted model. Keywords. Divorce, failure time, frailty models, parametric regression models, proportional hazards model.

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AMS 2010. 62N01, 62N02, 62P25.

References

[1] Cox, D.R., Regression models and life-tables, Journal of the Royal Statistical Society, Series B, 34, 187-220, 1972.

[2] Hougaard, P., Frailty models for survival data, Lifetime Data Analysis, 1, 255–273, 1995.

[3] Lee, E.T., Statistical methods for survival data analysis, Wiley&Sons, New York, 2003.

[4] Nardi, A. and Schemper, M., Comparing Cox and parametric models in clinical studies, Statistics in Medicine, 22, 3597-3610, 2003.

[5] Turkish Statistical Institute, Marriage and Divorce Statistics, 2010.

[6] Mortelmans, D., Snoeckx, L, Dronkers, J., Cross-Regional Divorce Risks in Belgium: Culture or Legislative System?, Journal of Divorce & Remarriage, 50 (8), 541-563, 2009.

[7] Rootalu, K., The Effect Of Education On Divorce Risk in Estonia, Trames, 14(1), 21–33, 2010.

253 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Study on an Additive Decomposition of the Blues in a Partitioned Linear Model and Its Two Small Models Nesrin Güler Sakarya University, Sakarya, Turkey, [email protected]

2 Abstract. The general partitioned linear model denoted by  ={,yX11β + X 2 βσ 2 , V }

2 2 known as full model with its two small models  = {,yX11βσ , V } and  = {,yX22βσ , V } is considered. The necessary and sufficient conditions for the additive decomposition of the best linear unbiased estimators (BLUEs) under the full and small models are given. Furthermore, some results are obtained for the equalities between the BLUEs under the full and small models. Some of the obtained results recently examined by [13]; however, distinctly the results obtained in this study are based on a generalized inverse of a symmetric matrix which is obtained from the Pandora's Box called by [9]. Keywords. BLUE, orthogonal projector, partitioned linear model, small model, additive decomposition of estimator. AMS 2010. 62J05, 62H12, 62F30.

References

[1] Alalouf, I. S., Styan, G. P. H., Characterizations of estimability in the general linear model, Ann. Statist. 7, 194-200, 1979.

[2] Baksalary, J. K., Mathew, T., Rank invariance criterion and its application to the unified theory of least squares, Lin. Algeb. Applic. 127, 393-401, 1990.

[3] Bhimasankaram, P., Saharay, R., On a partitioned linear model and some associated reduced models, Lin. Algeb. Applic. 264, 329-339, 1997.

[4] Chu, K. L., Isotalo, J., Puntanen, S., Styan, G. P. H., On decomposing the Watson efficiency of ordinary least squares in a partitioned weakly singular linear model, Sankhyā 66, 634-651, 2004.

[5] Gross, J., Puntanen, S., Estimation under a general partitioned linear model, Lin. Algeb. Applic. 321, 131-144, 2000.

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[6] Nurhonen, M., Puntanen, S., A property of partitioned generalized regression, Commun. Statist. Theor. Meth. 21, 1579-1583, 1992.

[7] Puntanen, S., Some matrix results related to a partitioned singular linear model, Commun. Statist. Theor. Meth. 25(2), 269-279, 1996.

[8] Puntanen, S., Some further results related to reduced singular linear models, Commun. Statist. Theor. Meth. 26(2), 375-385, 1997.

[9] Rao, C. R., Unified theory of linear estimation, Sankhyā Ser. A 33, 371-394, 1971. [Corrigendum: 34, 194, 477, 1972.]

[10] Rao, C. R., A note on the IPM method in the unified theory of linear estimation, Sankhyā Ser. A 34, 285-288, 1972.

[11] Rao, C. R., Representations of best linear unbiased estimators in the Gauss-Markov model with a singular dispersion matrix, J. Multivariate Anal. 3, 276-292, 1973.

[12] Rao, C., R., Mitra, S. K., Generalized Inverse of Matrices and its Application, Wiley, New York, 1971.

[13] Tian, Y., Some decompositions of OLSEs and BLUEs under a partitioned linear model, International Statist. Rev. 75, 224-248, 2007.

[14] Tian, Y., On an additive decomposition of the BLUE in a multiple partitioned linear model, J. Multivariate Anal. 100, 767-776, 2009.

[15] Tian, Y., Takane, Y., On sum decompositions of weighted least-squares estimators under the partitioned linear model, Commun. Statist. Theor. Meth. 37, 55-69, 2008.

[16] Zhang, B. X., Liu, B. S., Lu, C. Y., A study of the equivalence of the BLUEs between a partitioned singular linear model and its reduced singular linear models, Acta Mathematica Sinica, English Ser. Vol. 20(3), 557-568, 2004.

255 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Properties of Systems with Nonidentical Components Nuria Torrado University Public of Navarra, Pamplona, Spain, [email protected]

Abstract. Today, systems with nonidentical components permeate our modern society. They are embedded in air traffic control, nuclear reactors, aircraft, real-time sensor networks, industrial process control, automotive mechanical and safety control, and hospital health care, among others. As the functionality of components becomes more essential, there is a greater need for a high reliability of the systems. Based on Torrado et al. [3, 4, 5], in this talk we will show properties of systems with nonidentical components. Keywords. Heterogeneous distributions, systems with components, reliability theory. AMS 2010. 60E15, 60K10, 62G30.

References

[1] Kochar, S.C. and Kowar, R., Stochastic orders for spacings of heterogeneous exponential random variables, Journal of Multivariate Analysis, 57, 69-83, 1996.

[2] Shaked, M. and Shanthikumar, J.G., Stochastic orders, Springer, New York, 2007.

[3] Torrado, N., Lillo, R.E. and Wiper, M.P., On the conjecture of Kochar and Kowar, Journal of Multivariate Analysis 101, 1274-1283, 2010.

[4] Torrado, N., Lillo, R.E. and Wiper, M.P., A Sequential order statistics: ageing and stochastic orderings, Methodology and Computing in Applied Probability, 2011, DOI: 10.1007/s11009-011-9248-5.

[5] Torrado, N. and Veerman, J.J.P., Asymptotic reliability theory of k-out-of-n systems, Journal of Statistical Planning and Inference 142, 2646-2655, 2012.

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Bayesian Approach of Structural Equation Models with Conjugate Priors for Discrete Distributions Sanem Şehribanoğlu (1) and Hayrettin Okut (2) (1) Yuzuncu Yil University, Van, Turkey, [email protected] (2) Yuzuncu Yil University, Van, Turkey, [email protected]

Abstract. Structural equation modelling (SEM) is the multivariate statistic modelling technique which presents the cause and effect relation between measurable and latent variables. SEM has the power to include numerous conventional different models in it. Besides the classical estimation methods, estimation methods, which use the Bayesian approach, are also included in SEM. In Bayesian approaches, posterior knowledge is attempted to obtain using prior knowledge. Knowledge obtained from previous experiences about unknown parameters is named as prior knowledge. Dispersions obtained using prior knowledge and the probability function relating the observation data together are named posterior dispersions. If prior dispersion and posterior dispersion are coming from the same dispersion family, in Bayes method this is called conjugate prior and conjugate priors facilitates the procedures for the user in deductions. In statistics we classify the parameters used in models in the groups as intermittent and continuous variables. In this study, conjugate priors, which are used in situations that the data are intermittent and how they are obtained is explained. Keywords. Bayesian structural equation model, conjugate prior

References

[1] Congdon,P., Applied bayesian modelling. John Wiley & Sons, LTD, London, 2001.

[2] Gilks W.R., Richardson S., Spiegelhalter D.J., Markov chain monte carlo in practice. Chapman and Hall, London, 1996.

[3] Gill,J., Bayesian methods (a social and behavioral sciences approach). Chapman & Hall/CRC Statistics in the Social and Behavioral Sciences, USA., 2002.

[4] Hayashi, K.,Bentler,P.,M., Yuan, K., Structural equation modeling. handbook of statistics, 27,2008.

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[5] Hoff,D.,P., A first course in bayesian statistical methods. 2. Springer, VIII, USA, 2009

[6] Jöreskog,K.,G., Structural equation modeling with ordinal variables, IMS Lecture Notes, Monograph Series ,24 ,1994.

[7] Lee, S., Structural equation modeling: a bayesian approach. John Wiley & Sons, London, 2007.

[8] Rupp,A.,A., Dey,D.,K., Zumbo.B.,D., To bayes or not to bayes, from whether to when:applications of bayesian methodology to modeling. Structural Equation Modeling, 11,3, 424-451, 2004.

258 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

L-Moments Estimations for Mixture of Two Weibull Distributions Ülkü Erişoğlu (1) and Murat Erişoğlu (2) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. Mixture of two Weibull distributions have wide application in modelling of heterogeneous data. In this paper, we first propose an L-moment estimator for the mixture of two Weibull distributions. Then, a comprehensive comparison is made of the metod of L- moments and the method of maximum likelihood estimation (MLE). Keywords. MLE, L-moment, EM Algorithm. AMS 2010. 62-07, 62F10.

References

[1] Abdul-Moniem, I.B., L-moments and TL-moments estimation for the exponential distribution, Far East J. Theoret. Statist. 23, 51–61, 2007.

[2] Erişoğlu, Ü., Erol, H., Modeling Heterogeneous Survival Data Using Mixture of Extended Exponential-Geometric Distributions, Communications in Statistics - Simulation and Computation, 39, 1939-1952, 2010.

[3] Hosking, J., L-moments: Analysis and estimation of distributions using linear combinations of order statistics, J. R. Statist. Soc. B 52, 105–124, 1990.

[4] Hosking, J.R.M., On the characterization of distributions by their L-moments, J. Statist. Plann. Inference 136, 193-198, 2006.

[5] Jiang, R., Murthy, D. N. P.,. Modeling failure-data by mixture of 2 Weibull distributions: a graphical approach, IEEE Transactions on Reliability 44:477–488, 1995.

[6] Mahdi, T., Seyed, M. H., Saralees, N., Comparison of estimation methods fort he Weibull distribution, Statistics: A Journal of Theoretical and Applied Statistics, iFirst, 1-17, 2011.

[7] Murthy, D. N. P., Xie, M., Jiang, R., Weibull Models,Wiley, NewYork, 2004.

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MATHEMATICS EDUCATION

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Analysis of the Effect of Using Interactive Learning Objects on Critical Thinking Skills of Math Teacher Candidates Ahmet Ş. Özdemir (1) and Ahmet Bilal Yaprakdal (2) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected]

Abstract. The general purpose of this study is to reveal the effects of using interactive learning objects on the critical thinking skills of elementary school math teacher candidates within the scope of web aided course of Teaching Mathematics. In the related course, the teacher candidates that are elected to be in the test group, have been intended to design interactive learning tools, to discuss the learning tools designed by their other colleagues over Web 2.0 platform and as such to create opportunities for more effective learning tools to be developed and to take active part in the course to be given. Within the framework of the objectives stated herein, the subject matter study is essentially very important. In the study an experimental design with pre- test and post- test control groups was used. In the pre-test/post-test control group model, there were two groups that are objectively appointed. In both groups measurements were taken pre and post application. In the study, the effect of the independent variable – “Using Interactive Learning Tools Approach” – of the experimental design utilized on the teacher candidates making up the test group have been analyzed. On the other hand with the control group, an approach based on conventional learning approach has been followed. In both groups the critical thinking skills of the teacher candidates were taken into account as the dependant variable and comparisons between the groups were made by using both the pre and post test scores. In the study, for purposes of determining the level of critical thinking levels of the students in control and test groups, the California critical thinking disposition inventory (CCTDI) has been used. The subject matter measuring tool has been established as a result of the Delphi project that was arranged by the American Philosophy Association in the year 1990. This measuring tool has 7 sub-scales and 75 articles that have been theoretically established and psychometrically tested. The adoption of this tool to Turkish has been accomplished by Kökdemir (2003) in a study that included 913 students. In this present study, two different groups of students receiving web based mathematics education have been compared in terms of their critical thinking skills. Among these two groups, significant differences have been observed in favour of the group designing

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interactive learning tools regarding critical thinking skills. In conclusion, it has been revealed that the design and utilization of interactive learning tools in the education of candidate math teachers would affect their critical thinking skills favorably. Keywords. Interactive learning tools/objects, Education in Mathematics, Critical Thinking

References

[1] Bratina, T., Hayes, D., Blumsack, S., Preparing Teachers to Use Learning Objects, Technology Source, 2002.

[2] Karaman, S. (2005), Preparation of a Content Development System Based on Learning Tools and Determination of Content Development Profiles of Teacher Candidates by means of Tools Approach, Doctorate Thesis, Atatürk University, Social Sciences Institute, Erzurum, 2005.

[3] Kökdemir, D., Belirsizlik Durumlarında Karar Verme ve Problem Çözme (Decision making and Problem Solving in Situations of Uncertainty), Doctorate Thesis, Ankara University, Social Sciences Institute, Ankara, 2003.

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The Relationship between 7th Grade Students Visual Literacy Competencies and Their Academic Achievement Ahmet Şükrü Özdemir (1) and Sevda Göktepe (2) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected]

Abstract. Literacy can be expressed in general terms as a must-have knowledge and skills about a discipline. One of the areas of literacy is visual literacy. Visual elements are included in almost all of the mathematics topics. Graphics, diagrams, concept maps and tables are frequently included as a part of teaching-learning activities in mathematics lessons. Therefore developing students’ visual literacy competencies is an important issue. The aim of the study is to examine the relationship between 7th grade students’ visual literacy skills, their mathematic year-end achievement scores and their gender. Descriptive research model was used in this study. A visual literacy scale which was developed by Kiper and his friends (2012) and students’ mathematic year-end achievement scores are used as data collection tools. 76 students from two different public primary schools in 2011-2012 academic year participated to this study. A statistic program was used to analyze the data and formed tables to show the results. Recommendations were made accordance with these results.

Keywords. Literacy, visual literacy skills, academic achievement.

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The Effects of History of Mathematics on Mathematics Education Derya Demiroğlu (1), Emin Aydın (2) and Ali Delice (3) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected] (3) Marmara University, Istanbul, Turkey, [email protected]

Abstract. Knowledge of the history of a dicipline is important not only for prescribing the rules of a dicipline, but also for clear understanding of a subject, This paper which is about the relationship between history of mathematics and mathematics education is classified into four areas; nature of mathematics knowledge and nature of mathematics dicipline, advantages on learning mathematics, advantages on teaching mathematics, using history mathematics in mathematics lessons; history as a tool and history as a goal. Firstly, history of mathematics gives idea about the nature of mathematical knowledge and the nature of the mathematics dicipline. By means of learning history of mathematics, it’s easier to understand creation, growth and nature of mathematical knowledge and comprehend the dynamic, continuing and incomplete progress of mathematics as a human artifact (Siu & Siu, 1979; Liu, 2009). In talking about the advantages of learning of mathematics, it is essential, we believe, that there is paralellisim between history of mathematics and improvement of mathematical thinking in mind. The genetical principle states that the conceptual progress of a learner matches with how mathematical knowledge develops historically (Bagni, Furinghetti ve Spagnolo, 2004), so history of mathematics describes the improvement of mathematical thinking. Besides, the history of mathematics has some motivational, attitudational and cultural effects on learning mathematics. On the advantages on teaching of mathematics, history of mathematics helps teachers with knowing about interconnections between whys and hows of a concept. It makes easier also to organise presentation of a lesson, use teaching materials effectively and reduce students’ confusion about the concepts taught. Studies report that distinction between the use of history of mathematics in a mathematics lesson as a tool or as a goal is extremely important. If the aim is to show students the evolution of mathematical knowledge with the cultural effects of different civilizations during centuries, history of mathematics is a goal. If, on the other hand, the aim

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is to provide students with better conceptual understanding, and historical aproaches are used in solving mathematics problems, history of mathematics is a tool (Jankvist, 2009). In this paper, we surveyed the literature on the the effects of history of mathematics on mathematics education. There is also a surwey part about problems and suggestions to problems about using and implementing history of mathematics in mathematics lessons and mathematics education.

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The Correlation Between Achievement Test Scores and Creative Thinking Skills of Non- gifted Students Esra Altıntaş (1), Ahmet Şükrü Özdemir (2) and Abdulkadir Kerpiç (3) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected] (3) Marmara University, Istanbul, Turkey, [email protected]

Abstract. The purpose of the study is to determine the correlation between achievement test scores and creative thinking skills of non-gifted students. We prepared an activity concerned with Conscious Consumption Unit. This activity was prepared based on Purdue model for 7th grade gifted students in our country. We applied this activity to non- gifted students. We investigated effects of this activity on mathematics achievement and creative thinking skills of non-gifted students. Then we determined the correlation between achievement test scores and creative thinking skills of non-gifted students. Problem sentence can be expressed as: What is the correlation between achievement test scores and creative thinking skills of non-gifted students? Sub-problems are like this: Is there any significant difference between scores of achievement test and creative thinking skills of students in control group and experimental group before and after the application? Is there correlation between achievement test scores and creative thinking skills of students in control group and experimental group? Do achievement test scores of students in control group and experimental group differ according to their creative thinking skills? Do creative thinking skills of students in control group and experimental group differ according to their achievement test scores? In this study, we used a prepost test model with a control group. While the unit was taught with the activities developed based on Purdue model to experimental group, the same unit was taught by using the activities included in National Education Curriculum to control group. For this purpose, we applied mathematics achievement test and a creativity scale called “How creative are you?” to two groups. After collecting data, we analyzed the results. Keywords. Purdue model, creative thinking skills, mathematics education.

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The Changes in the Cognitive and Affective Properties of the Pre-Service Elementary Mathematics Teachers from the Beginning of the Teacher Training Program to Graduation

Ercan Masal (1), Melek Masal, Erkan Akgün, Mithat Takunyacı (1) Sakarya University, Sakarya, Turkey, [email protected]

Absract. Trained manpower is important for the future of the country. That’s why, teacher training becomes important. The aim of this study is to investigate the effects of the instruction on cognitive and affective abilities of pre-service teachers who enrolled the elementary mathematics education and graduated from this program by comparing their levels before the instruction and after the instruction. It is longitudinal research. Also, correlative investigation model was used. The independent variable is the effect of instruction which was measured before the instruction and after the graduation and obtained by repeated measures design. The dependent variables are the strategies of motivation (arrangement of inner target, arrangement of external target, the value of task, self-efficacy, control toward learning, exam anxiety), the strategies of learning (iteration, paraphrase, regulation, critical thinking, meta- cognition, seeking for help, peer cooperation, management of effort, time and working condition), multiple intelligence (verbal, logic, spatial, kinesthetic, music, social, inner and naturalistic), attitudes toward mathematics and abilities of scientific process. The data was collected in the second week of the term and after the graduation. The data was analyzed by Wilcoxon test which was non-parametric test. It was found that the value which was given to the task was decreased. Also, according to the results of this study, there was a significant increase in the use of learning strategies which were regulation, critical thinking, seeking for help and cooperation of the peers and decrease in the use of paraphrase, management of effort and the strategies related to the meta-cognition. Furthermore, it was found that there was no change in the multiple intelligence type of them, but small increase in social intelligence was found. It was found that the attitudes toward mathematics decreased throughout their education. This means that the positive attitudes toward the mathematics were declined. These results showed that there was no change in the abilities about cognitive process of the participants. According to these results, it can be said that the teacher training program is not sufficient to develop cognitive and affective abilities of the students. However, in order to obtain accurate results, it is necessary that new studies should be conducted.

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References

[1] Acat MB, Demiral S. Türkiye’de yabancı dil öğreniminde motivasyon kaynakları ve sorunları. Kuramdan Uygulamaya Eğitim Yönetimi 2002; 8: 312-329. [2] Akar, K. (2006). İlköğretim 6.,7.,8. sınıf öğrencilerinin çoklu zeka kuramına göre sahip oldukları zeka alanları ve akademik başarılarının karşılaştırılması).Yayımlanmamış Yüksek Lisans Tezi. [3] Armstrong,M.A.(1991). Handbook Personel Management Practice.London:Kogan Page Limited. [4] Aşkar, P.(1986). Matematik Dersine Yönelik Tutum Ölçen Likert Tipi Ölçeğin Geliştirilmesi.Eğitim ve Bilim dergisi, (11), 31-34. [5] Ayas, A., Çepni, S., Jhonson, D., & Turgut, M.F. (1997). Kimya öğretimi. Ankara: YÖK/Dünya Bankası, Milli Eğitimi Geliştirme Projesi Hizmet Öncesi Öğretmen Eğitimi. [6] Aydın, F. (2010). Ortaöğretim Öğrencilerinin Coğrafya Derslerindeki Güdülenmelerinin İncelenmesi’, Turkish Studies International Periodical For the Languages, Literature and History of Turkish or Turkic, 5/4. [7] Aydoslu, U. (2005). Öğretmen Adaylarının Yabancı Dil Olarak İngilizce Dersine İlişkin Tutumlarının İncelenmesi (B.E.F. Örneği). Süleyman Demirel Üniversitesi, Sosyal Bilimler Enstitüsü, Yayımlanmamış Yüksek Lisans Tezi, Isparta. [8] Bacanlı, Hasan; Sosyal Beceri Eğitimi, Nobel Yay, Ankara,1999.

[9] Bloom, B.S. (1979). İnsan nitelikleri ve öğrenme. (Çev. Özçelik, D.A.). Ankara, Milli Eğitim Basımevi. [10] Bozanoğlu, İ. (2004). ‘Akademik Güdülenme Ölçeği: Geliştirilmesi, Geçerliği, Güvenirliği’, Ankara Üniversitesi Eğitim Bilimleri Fakültesi Dergisi, 37(2), 83-98. [11] Caine, R.N. ve Caine, G. (1991). Making connections: Teaching and human brain, Alexandria, VA.: Association for Supervision and Currriculum Development. [12] Çanakçı, O., Özdemir, A.Ş.(2011). ‘Matematik Problemi Çözme Tutum Ölçeğinin Geliştirilmesi’, AİBÜ Eğitim Fakültesi Dergisi, 11(1), 119-136. [13] Çoban, A.(1989). ‘Ankara, Merkez Ortaokullarındaki Son Sınıf Öğrencilerinin Matematik Dersine İlişkin tutumları’’Yayınlanmış yüksek lisans tezi, Gazi Üniversitesi Sosyal Bilimler Enstütüsü, Ankara. [14] Dutton, W. (1962). Attitude change of prospective elementary school teachers toward arithmetic teacher. Restob, Virginia: NCTM.

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Open Source Softwares in Mathematics Education Erdal Özüsağlam (1), Gülsüm Biçer (1), Ali Atalay (2) and Pelin Poşpoş (1) (1) Aksaray University, Aksaray, Turkey, [email protected] (2) Eskisehir Osmangazi University, Eskişehir, Turkey, [email protected]

Abstract. Nowadays, the use of computers and communication technologies has become increasingly common, especially in mathematics education. Using computer-aided software offers practical solutions to instructors and students. It helps by reinforcing contrete applications of the subject and supplying an effective course materials. In this study, free and open source alternatives for expensive and licenced softwares which are used in teaching undergraduate, particularly in courses of mathematics and its subbranches are examined. These softwares are totally free, easy to install and use, can run on Linux and Windows platform. Maxima and Xcas which count as one of the most comprehensive softwares are examined, explained and supported by sample applications. Keywords. Open Source Software, computer-aided mathematics education. AMS 2010. 97U70, 97U50.

References

[1] Baki, A. Bilgisayar Destekli Matematik İstanbul: Bitav Yayınları, 2002.

[2] Özüsağlam, E., Mathematica Destekli On-line Matematik Dersi Sunumu Üzerine Bir Çalışma, BTIE ODTU, Ankara, 2001.

[3] Özüsağlam, E. Teknoloji Destekli Matematik Öğretiminin Öğretimi, Matematik Etkinlikleri 2004, Ankara, 2004.

[4] P.N. de Souza and et al., The Maxima Book, 2004.

[5] Özüsağlam, E. , Atalay , A. , Eğitimde Açık Kaynak Kodlu Yazılımlar: Edubuntu, İzmir VII Matematik Etkinlikleri, İzmir, 2008.

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Sequences and Series: What Are These Concepts Through the Eyes of Pre Service Secondary Mathematics Teachers? Gönül Yazgan-Sağ (1) and Ziya Argün (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. Calculus is thought one of the fundamental subject areas of mathematics [1]. Calculus is necessary to conduct research studies and to be successful in university courses which are taken to have careers in science, engineering, medicine, business, and mathematics related fields [2]. Therefore, concepts in calculus and calculus itself have been one of the areas attracted many researchers in mathematics education. If our aim as educators is to construct a well developed understanding of the notion of approximation, the concepts of sequence and series can be considered part of building this notion [3]. It is important to understand the concepts of sequence and series for pre service secondary mathematics teachers who will teach this concept to their students in the future. The purpose of the study was to explore pre service mathematics teachers’ conceptual understanding of the concepts of sequence and series. 55 second year pre service teachers who were enrolled in a Multivariable Calculus course in secondary mathematics teaching programme at a university located in Ankara, Turkey participated in this study. Data was collected through pre service teachers’ responses to open-ended questions about the concept of sequence ans series. Data was analyzed by using constant comparative analysis [4] which is one of the techniques in grounded theory. The results indicated that pre service secondary mathematics teachers had some difficulties such as writing mathematical symbols formally and using logical explanations. In conclusion, most of pre service secondary mathematics teachers confused these concepts with each other. Keywords. Pre service secondary mathematics teachers, the concept of sequence, the concept of series AMS 2010. 97B50, 97D70.

References

[1] Buccino, A. (2000) Politics and professional beliefs in evaluation: The case of calculus renewal. In S. Ganter, (Ed.) Calculus renewal: Issues for undergraduate mathematics

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education in the next decade (pp. 121-146). Kluwer Academic Press/Plenum Publishers: New York.

[2] Frid, S. (1994). Three approaches to undergraduate calculus instruction: Their nature and potential impact on students’ language use and sources of conviction. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education I (pp. 69-100). Providence, RI: American Mathematical Society.

[3] Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp.153-166). Dordrecht: Kluwer Academic Press.

[4] Strauss, A. &, Corbin, J. (1998). Basics of qualitative research: Grounded theory procedures and techniques. London: Sage.

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The Concept of Proof is Examined the Size of Teachers İlyas Yavuz (1) and Mehtap Taştepe (2) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected]

Abstract. Several aims are determined to raise the quality of mathematics educational the every stage of the education system. With the changes of mathematics curriculum in 2005, students' knowing mathematical terms, gaining problem solving ability, having self confidence on math, having positive attitudes to mathematic, having the ability of interpreting the mathematical knowledge gained in lessons in daily life are among this aims (Meb, 2005). In this aspect, mathematic is aimed to be extracted from being a group of formulas, procedures and processes, and its an indispensable tool in daily life, and a science including logical thinking. Through the point of all these desires and aims, the importance of the term 'proof' on mathematic education and teaching is highlighted. Through the point of all these desires and aims, the importance of the term 'proof' on mathematics education and teaching is highlighted. In this aspect, students' –especially department of mathematics and mathematics school teacher education in universities- are expected to have the ability of doing proofs. For this reason, in this study the term of proof in mathematics lessons at universities and attitudes to this term is examined. The most important factor affecting students’ feelings, thoughts, preferences about proof and ability to prove are faculty members of organizing the status of each size class, teaching to prove that students and waiting from them to be able to do to prove. The purpose of this study includes, according to opinions and suggestions of faculty members to examine the concept of proof. Qualitative research carried out in the direction of describing the current situation model is used in this study. Course in abstract algebra, which runs eight different universities and different faculty members conducted semi-structured interview, the obtained data were analyzed using content analysis method. After data analysis; faculty members expressed in general that they give importance to prove in their lessons and a student who is in the field of mathematics should be able to prove definitely. But they expressed that students worry at proof and proving. Even this level of anxiety from section to section (department of mathematics, secondary / primary mathematics section) also have made the determination to be different. They expressed that students are inadequate at proving and students try to memorize proofs in general. They emphasized that especially in the beginning stage, but at every stage of proof, they have difficulty because they more results-oriented work. Keywords. The concept of proof, faculty members

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Analysis of the Elementary 8th Grade Students’ Mathematical Ability Levels from the Perspective of PISA Murat Altun (1), Nalân Aydın (2), Recai Akkaya (3) and Devrim Uzel (4) (1) Uludag University, Bursa, Turkey, [email protected] (2) Uludag University, Bursa, Turkey, [email protected] (3) Abant Izzet Baysal University, Bolu, Turkey, [email protected] (4) Balikesir University, Balikesir, Turkey, [email protected]

Abstract. Our country’s having underachieved one after the other in the international examinations such as PISA, TIMSS etc., has made us consider that there are some problems in our education system. Improvement in the system requires primarily knowing about what kinds of information and skills are lacking. For this purpose, in the present study, 25 questions selected from among those made optional in PISA examinations were asked to 969 8th grade students from three different socio-economic levels and 324 teacher candidates from the department of mathematics and the results were analyzed. In the study carried out by using the descriptive survey model, 7 of the questions were multiple-choice and the others were open-ended ones. In the analysis of the open-ended questions, for each question rubrics which were to form a basis for a scoring of 0, 1, 2 were used: the multiple-choice type of questions were scored as 0, 2. The questions which both students and teacher candidates had difficulty solving showed similarities and it was observed that they had difficulty writing the algebraic statement posed by the problem and explaining the conclusion by solving it, making sense of statistical data and building suggestions by using available data. It is striking that the students at lower and moderate success levels scored very low at the questions involving the interpretation of the scale using strategies, the sense-making of algebraic statements and making suggestions, and the using of them goal-oriented. Considering these results, some suggestions were made for the teaching system. Key Words. Mathematics education, mathematical literacy, problem solving, pisa, mathematics curriculum

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Proof Schemes Used by Mathematics Teacher Candidates Sare Şengül (1) and Pınar Güner (2) (1) Marmara University, Istanbul, Turkey, [email protected] (2) Marmara University, Istanbul, Turkey, [email protected]

Abstract. Proving and problem solving have an important place in the process of understanding mathematics and using it. In mathematics proof related to a theorem, the rule, or a problem in realizing the starting point of knowledge, understanding relationships at steps and making sense of results have an effective role. Constructing concepts as comprehending cause effect relationship also is an important factor in reducing rote learning. Therefore proving is a subject has to be emphasised. According to Hart (1994) in order to put forward properly proving processes of students and reasons of the mistakes made by them in this process, studies by reference to proof intended to research thinking ways of students and cognitive based should be carried out (Weber, 2001). From this point of view the aim of this study is determination of proof schemes used by primary preservice teachers attending first and last grade of Primary Education Mathematics Teaching program of a state university in 2011-2012. The study was carried out on a total of 105 teacher candidates. In order to describe current situation scanning model was used among the descriptive research methods. Teacher candidates were asked to solve five problems within general mathematics lesson and we determinated that their answers which proof schemes were belonged to. So as to set down proof schemes classification which was constituted by Harel and Sowder (1998) was used. As part of investigation the data was collected, percentage and frequency tables belong to schemes used by students were formed. In accordance with the tables, proof schemes predominantly used by candidates were examined. With regard to results some suggestions were offered. Keywords. proof, proof schemes, teacher candidates.

References

[1] Harel, G., Sowder, L., Students’ Proof Schemes: Results From Exploratory Studies, In A. Schoenfeld, J. Karput, and E. Dubinsky (Eds.), Research in Collegiate Matehematics Education, III, 234-283, Providence, RI, American Mathematical Society.

[2] Weber, K. (2001) Student difficulty in Constructing proof: The need for strategic knowledg. Educational Studies in Mathematics, 48(1), 101-119.

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Preservice Elementary Mathematics Teachers’ Proving Skills Yasemin Kıymaz (1), Bekir Kürşat Doruk (2) and Tuğba Horzum (3) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected] (3) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. Proof is an indispensable part of mathematics. Therefore, mathematics education can not be considered without proof. Students at every stage of mathematics education need to deal with different levels of proof. Preservice elementary mathematics teachers meet formal proof which is the highest level of proof in their first year of undergraduate education. It is important to determine preservice teachers’ skill levels about proving in terms of correcting their deficiencies and reconsidering existent education. This study aims to identify preservice teachers’ understanding skills related some given proofs and their ability on proving. The present study is an ongoing research in which we conduct content analysis using the answers given by preservice teachers for their exams. We are willing to present overlapping themes emerged and discuss our results with respective literature. Keywords. proof, understanding proof, mathematics education AMS 2010. 97E50

References

[1] Güler, G., Özdemir, E., Dikici, R., Öğretmen Adaylarının Matematiksel Tümevarım Yoluyla İspat Becerileri ve Matematiksel İspat Hakkındaki Görüşleri, Kastamonu Eğitim Dergisi, 20 (1), 219-236, 2012.

[2] Hanna, G., Mathematical Proof. In: D. Tall (Ed.) Advanced Mathematical Thinking, Kluwer, Dordrecht., 1991.

[3] Hanna, G., The Ongoing Value of Proof. Proceedings of International Group for the Psychology of Mathematics Education, Valencia, Spain, Vol I., 1996.

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[4] Raman, M., Beliefs about proof in collegiate calculus. In Robert Speiser (Ed.), Proceedings of the Twenty Second Annual Meeting, North American Chapter for the International Group for the Psychology of Mathematics Education, Snowbird, Utah, 2001.

[5] Tall, D., Cognitive Development, Representations & Proof. In Justifying and Proving in School Mathematics, Institute of Education, London, 27-38, 1995.

[6] Tall, D., The Cognitive Development of Proof: Is Mathematical Proof For All or For Some? In Z. Usiskin (Ed.), Developments in School Mathematics Education Around the World, Vol, 4, 117-136. Reston, Virginia: NCTM, 1999.

[7] Türker, B., Alkaş, Ç., Aylar, E., Gürel, R., Akkuş İspir, O., The Views of Elementary Mathematics Education Preservice Teachers on Proving. International Journal of Human and Social Sciences (423-427) 5-7, 2010.

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The Conceptual Methodical Approach in Training as a Mean of Increase of Degree of Social Protection of Disabled Children Yuldashev Ziyavidin Khabibovich (1) and Yuldasheva Mina Anishevna (2) National University of Uzbekistan, Specialised labour school №25, Tashkent city

Abstract.In the previous work of authors [1] put forward the following thesis “the attitude to invalids at all times was the basic criterion of degree of humanisation of public arrangement”. Developing this thesis, in the given work authors prove the necessity of presence in each country, where the principles of humanism are proclaimed, of materialised multicomponent and multilevel system of protection of disabled children. Further, for brevity, the specified system of measures of protection of invalid children is called the System. Characterising the components of such System, conclusion that it can be transformed into institute of social protection of invalid children which creates the foundation for integration of invalid children into a modern society. The following items are specified as the components of the System:

 Presence of legal norms - articles in the law protecting rights of invalid children;

 Packages of social protection for children of preschool age (realisation of the regulated payments, free provision of invalid wheelchairs, medicines or other items facilitating process of opposition to illness, free provision of clothes, food, possibility of having free or preferential sanatorium treatment, etc.);***

 Packages of social protection intended to pupils of general educational specialised schools, including pupils of general schools at absence at invalid children of mental development lag (all measures of social protection of the above-stated preschool period expanded with free provision of school accessories, including textbooks, books, office equipment, possibility of having free or preferential recreation in a period of vacation);

 Effective subsystem of measures on employment, adaptation of invalid children in labour team and monitoring of protection of their work;

 Free provision of consultations on problems of marriage, integration of family into society. It is obvious that the presented list can be expanded and detailed, and its actuality is defined by level of investment and presence of the experts really empathizing to the problems and needs of invalid children.

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System levels are variants of filling of components which content is defined by the social status of a family where the child lives, the level of disease degree or affectedness of body parts and psychic. In the System the substructure of specialised training which links should have scientific well-founded metodik and didactic filling and guaranteed sufficient investment should take the central place. It is necessary to mention that in a number of coutries as an alternative to separate education of invalid children the co-education of invalid children with usual peers is practiced or planned. Both of these two approaches have as supporters, so opponents as well. The authors of the given work are sceptical about universal applicability of co-education. If there are 30 or more pupils in a class it is very difficult of a without a damage to knowledge and skills of other pupils to give the same time and methodically directed attention for 2-3 problem children having the corresponding diagnosis of experts. In the cases when problem children are trained indirectly, namely when they solve problems and answer questions of teacher, imitating to other children it is hardly possible to talk about steady knowledge and skills since the action deprived of deep understanding is not capable to generate steady knowledge or skills. Besides that in the mixed group an invalid child is more vulnerable, than in the group of "similar cildren" since in the mixed group a problem child becomes an object of the constant natural curiosity sometimes generating unpleasant remarks or even sneers. One more antithesis of the combined training of invalid children together with usual children, is the complexity of composite organization of educational methodical work since in this case the right to teach in the mixed classes will be provided to teacher of a specific subject and specialists of general profile who do not necessarily have higher or average specialized education in speech pathology. Additional education in speech pathology of specialists of general profile, especially teachers of specific subjects, requires breaking or reorganization of existing system of speech pathology education. The new theoretical base and long approbation of new techniques is necessary for these purposes. Even the richest states can not afford itselves such experiments and expenses for the sake of is conditional human, and most likely, populist ideas and even the most sincere love or care are not able to solve at once objective problems in education of invalid children, for example in mathematics [2]. Certainly, each society has its particular mentality and in the conditions of low birth rate and incomplete satisfaction of parental instinct, the corresponding part of adult population will have arguments in favour of such mixed training, but to transform the mixed training in omnipresent and obligatory, without the scientifically-proved researches and corresponding

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economic base, is not only irrational, but also inhuman, as correct and effective training of invalid children is one of the basic form of their social protection. Further in work the questions of choice of concepts are considered at organization and realization of specialized education of invalid children, in the context of new option of a paradigm of education: «education thoughout the whole life» [3], and also questions of didactic filling of process of specialized training of invalid children. In the final part of work as a technology the new “methodology of basic problems” is offered and its efficiency is proved on the example of education of invalid children to native language, mathematics and computer science.

References [1] Yuldashev Z. Kh., Yuldasheva M.A. Mathematics and information technologies a tool for rehabilitation of disabled people//Book of Abstracts IV CONGRESS OF THE TURCKIC WORLD MATHEMATICAL SOCIETY, Baky, - 2011., p.506. [2] Yuldashev Z. Kh., Yuldasheva M.A. Mathematics and information technologies as a tool of integration of invalids in modern society.//The Collection of materials of III international scientific-theoretical conference «THE ROLE of PHYSICAL AND MATHEMATICAL SCIENCES IN MODERN EDUCATIONAL FIELD», Atirau, Kazakhstan, 2011, pp. 261- 263. [3] Yuldashev Z. Kh., Ashurova D.N. Innovative-Didactic Program Complex and New Formalized Model of Education//Malaysian Journal of Mathematical Sciences 6 (1: 97-103) (2012)

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THE OTHER AREAS

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IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Quartic B-spline Collocation Method for Numerical Solutions of the Klein-Gordon Equation Ahmet Boz (1) and İdris Dağ (2) (1) Dumlupinar University, Kutahya, Turkey, [email protected] (2) Eskisehir Osmangazi University, Eskişehir, Turkey, [email protected]

Abstract. The collocation method using quartic B-splines for solving the nonlinear Klein- Gordon equation has been implemented. The effect of the quartic B-splines in the collocation methods is sought. To do so, the nonlinear KG equation is solved numerically by a method of collocation using quartic B-splines as an approximate function over the finite intervals. Numerical comparison of results of algorithms and some other published numerical results are done by studying test problems. Keywords. Klein-Gordon Equation, Collocation Method ,B-Spline Function AMS 2010. 65L60, 41A15.

References

[1] Ablowitz M.J.,.Kruskal M.D,.Ladik J.F, Solitary wave collisions, SIAM J. Appl. Math,36,428 (1979)

[2] Prenter P.M., Splines and Variational Methods, Wiley, New York, 1975.

[3] Zaki S. I ., Gardner L. R. T. and Gardner G. A., Numerical simulations of Klein-Gordon solitary wave interactions, Il Nuovo Cimento,112B,N.7, 1997.

[4] Dağ˙I., Doğan A., Saka B., B-spline collocation methods for numerical solution of RLW equation, ˙Int. J. Comput. Math, 80,743-757(2003)

[5] Khalife M.E, Elgamal M., A numerical solution to Klein-Gordon equation with Dirichlet boundary condition, Appl. Math. and Comput.160,451-475(2004)

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Vehicle Crashworthiness Optimization Using Differential Evolution Approach Ali R. Yıldız Bursa Technical University, Bursa, Turkey, [email protected]

Abstract. Vehicle crashworthiness is a vital issue to ensure passengers safety and reduce vehicle costs in the early design stage of vehicle design. The aim of the crashworthiness design is to provide an optimized structure that can absorb the crash energy by controlled vehicle deformations while maintaining enough space of the passenger compartment. Optimization techniques have been used to reduce the vehicle design cycle. In this paper, the DE algorithm is applied to crashworthiness optimization of a full-vehicle model to demonstrate its effectiveness. Keywords. Vehicle crashworthiness, differential evolution algorithm, optimization

References

[1] Osuga, R., Fuel economy and weight reduction of motor vehicles, Japan Society of Automotive Engineers. 55(4): 4–8, 2001 [2] Yildiz, A.R., Saitou, K., Topology Synthesis of Multi-Component Structural Assemblies in Continuum Domains, Transactions of ASME, Journal of Mechanical Design 133 (1): 011008-1– 011008-9, 2011. [3] Yildiz, A.R., A novel particle swarm optimization approach for product design and manufacturing, International Journal of Advanced Manufacturing Technology, 40(5-6): 617- 628, 2009. [4] Yildiz, A.R., A novel hybrid immune algorithm for global optimization in design and manufacturing, Robotics and Computer-Integrated Manufacturing, 25(2): 261-270, 2009. [5] Yildiz, A.R., A new design optimization framework based on immune algorithm and Taguchi method, Computers in Industry, 60(8): 613-620, 2009. [6] Yildiz, A.R., An effective hybrid immune-hill climbing optimization approach for solving design and manufacturing optimization problems in industry, Journal of Materials Processing Technology, 50(4): 224-228, 2009. [7] Yildiz, A.R., Hybrid immune-simulated annealing algorithm for optimal design and manufacturing, International Journal of Materials and Product Technology, 34(3): 217-226, 2009.

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[8] Yildiz, A.R., Hybrid Taguchi-harmony search algorithm for solving engineering optimization problems, International Journal of Industrial Engineering Theory-Applications and Practice, 15(3): 286-293, 2008. [9] Hamza, K., Saitou, K., Automated vehicle structural crashworthiness design via a crash mode matching algorithm, Transactions of ASME, Journal of Mechanical Design, 133 (1): 011003-1– 011003-9, 2011. [10] Wang, H., Li, G.Y., Li, E.Y., Time based metamodeling techniques for vehicle crashworthiness optimization, Computer Methods in Applied Mechanics and Engineering 199(37-40):2497-2509, 2010. [11] Liao, X.T., Li, Q., Yang, X.J., Li, W., Zhang, W.G., A two-stage multi-objective optimisation of vehicle crashworthiness under frontal impact, International Journal of Crashworthiness, 13(3): 279-288, 2008. [12] Acar, E., Solanki, K.N., Improving accuracy of vehicle crashworthiness response predictions using ensemble of metamodels, International Journal of Crashworthiness, 14(1): 49 – 61, 2009a. [13] Acar, E., Solanki, K.N., System reliability based vehicle design for crashworthiness and effects of various uncertainty reduction measures, Structural and Multidisciplinary Optimization, 39(3): 311-325, 2009b. [14] Fang, H., Solanki, K.N., Horstemeyer, M.F., 2004a, Numerical simulations of multiple vehicle crashes and multidisciplinary crashworthiness optimization, International Journal of Crashworthiness, 10(2), 161-171, 2004a [15] Fang, H., Solanki, K.N., Horstemeyer, M.F., Rais-Rohani, M., Multi-impact crashworthiness optimization with full-scale finite element simulations, Computational Mechanics, The 6th World Congress on Computational Mechanics in Conjunction APCOM’04, Beijing, China, September 5-10, 2004b. [16] Solanki, K.N., Oglesby, D.L., Burton, C.L., Fang, H., Horstemeyer, M.F., Crashworthiness simulations Comparing PAM-CRASH and LS-DYNA, The 2004 SAE World Congress, Paper No. 148, Detroit, Michigan, March 8-11, 2004. [17] Fang, H., Solanki, K.N., Horstemeyer, M.F., Energy-based crashworthiness optimization for multiple vehicle impacts, IMECE2004-59123, Proceeding of IMECE 2004: International Mechanical Engineering Congress & Exposition, Anaheim, California, 2004. [18] Yildiz, A.R., Optimal structural design of vehicle components using topology design and optimization, Materials Testing 50(4):224-228, 2008.

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[19] Hamza, K., Saitou, K., Design for Structural Crashworthiness using Equivalent Mechanism Approximations, Transactions of ASME, Journal of Mechanical Design 127(3):485-492, 2005. [20] Storn R., Price, K., Differential Evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces Technical Report TR-95-12, International Computer Science, Berkeley, California, 1995. [21] Yildiz, A.R., Hybrid Taguchi-differential evolution algorithm for optimization of multi- pass turning operations, Applied Soft Computing, DOI: http://dx.doi.org/10.1016/j.asoc.2012.01.012, 2012. [22] Yildiz, A.R., A new hybrid particle swarm optimization approach for structural design optimization in the automotive industry, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, DOI: 10.1177/0954407012443636, 2012. [23] Yildiz, A.R., A new hybrid differential evolution algorithm for the selection of optimal machining parameters in milling operations, Applied Soft Computing, DOI: http://dx.doi.org/10.1016/j.asoc.2011.12.016, 2012. [24] Yildiz, A.R., A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing, Applied Soft Computing, http://dx.doi.org/10.1016/j.asoc.2012.04.013, 2012 [25] Lee, K.S., Geem, Z.W., A new structural optimization method based on the harmony search algorithm, Computers and Structures 82: 781–798, 2004. [26] Fourie, P.C., Groenwold, A.A., The particle swarm optimization algorithm in size shape optimization, Struct Multidiscipl Optim 23:259–267, 2002. [27] Eberhart, R., Kennedy, J., A new optimizer using particle swarm theory, In: Proc IEEE Sixth International Symposium on Micro Machine Human Science. Nagoya, Japan, pp 39-43, 1995. [28] Saka, M.P., Optimum geometry design of geodesic domes using harmony search algorithm, Advances In Structural Engineering 10: 595-606, 2008. [29] Yildiz, A.R., A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing, Applied Soft Computing, http://dx.doi.org/10.1016/j.asoc.2012.04.013, 2012 [30] Yildiz, A.R., K. Solanki, Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach, International Journal of Advanced Manufacturing Technology, 59(1-4): 367-376, 2012.

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Optimization of Vehicle Components Using Artificial Bee Colony Algorithm Ali R. Yıldız Bursa Technical University, Bursa, Turkey, [email protected]

Abstract. There is an increasing interest to design light-weight and cost-effective vehicle components. In this research, artificial bee colony algorithm (ABC) is used to solve structural design optimization problems. The ABC algorithm is applied to the structural design optimization of a vehicle component to illustrate how the present approach can be applied for solving structural design problems. The results show the effectiveness of the ABC to find better optimal structural design. Keywords. Structural design, Artificial bee colony algorithm, Optimization, Vehicle Design

References

1. A. R. Yildiz, K. Solanki, Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach, International Journal of Advanced Manufacturing Technology, 59(1-4): 367-376. 2012. 2. A.R. Yildiz, K. Saitou, Topology Synthesis of Multi-Component Structural Assemblies in Continuum Domains, Transactions of ASME, Journal of Mechanical Design 133(1) (2011) 011008-1– 011008-9. 3. S Bureerat, J Limtragool (2006), "Performance enhancement of evolutionary search for structural topology optimisation", Finite Elements in Analysis and Design, 42(6), pp.547- 566. 4. S Bureerat, S Srisomporn (2010), "Optimum plate-fin heat sinks by using a multi-objective evolutionary algorithm", Engineering Optimization, 42(4), pp.305-323. 5. G Rennera, A Eka´r (2003), "Genetic algorithms in computer aided design", Computer- Aided Design 35, pp.709–726. 6. A.R. Yildiz, A new hybrid particle swarm optimization approach for structural design optimization in the automotive industry, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, DOI: 10.1177/0954407012443636, 2012 7. E Ferhat, D Erkan, MP Saka (2011), "Optimum design of cellular beams using harmony search and particle swarm optimizers", Journal of Constructional Steel Research 67(2) pp.237-247. 8. SN Omkar, J Senthilnath, R Khandelwal, G Narayana Naik, S Gopalakrishnan (2011), "Artificial Bee Colony (ABC) for multi-objective design optimization of composite structures", Applied Soft Computing 11, pp.489–499.

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9. CV Camp, BJ Bichon, SP Stovall (2005), "Design of steel frames using ant colony optimization", Asce-Journal of Structural Engineering 131, pp.367-525. 10. MP Saka (1998), "Optimum design of steel grillage systems using genetic algorithms", Computer Aided Civil Infrastruct Eng 13, pp.233-238. 11. MP Saka (2008), "Optimum geometry design of geodesic domes using harmony search algorithm", Advances in Structural Engineering 10, pp.595-606. 12. SN Omkar, J Senthilnath, R Khandelwal, G Narayana Naik, S Gopalakrishnan (2008), "Artificial immune system for multi-objective design optimization of composite structures", Engineering Applications of Artificial Intelligence 21, pp.1416–1429. 13. A.R. Yildiz, Hybrid immune-simulated annealing algorithm for optimal design and manufacturing, International Journal of Materials and Product Technology, 34(3) (2009) 217- 226. 14. GC Luh, CH Chueh (2004), "Multi-modal topological optimization of structure using immune algoritm", Computer Methods in Applied MechaniDE and Engineering 193, pp. 4035-4055. 15. A.R. Yildiz, A new design optimization framework based on immune algorithm and Taguchi method, Computers in Industry, 60(8) (2009) 613-620. 16. AR Yildiz (2009), An effective hybrid immune-hill climbing optimization approach for solving design and manufacturing optimization problems in industry, Journal of Materials Processing Technology 209, pp.2773-2780. 17. A.R. Yildiz, Hybrid Taguchi-Harmony Search Algorithm for Solving Engineering Optimization Problems, International Journal of Industrial Engineering Theory, Applications and Practice, 15(3) (2008) 286-293. 18. AR Yildiz (2009), "A novel particle swarm optimization approach for product design and manufacturing", International Journal of Advance Manufacturing Technology 40, pp.617-628. 19. AR Yildiz (2009), "A novel hybrid immune algorithm for global optimization in design and manufacturing", Robotic and Computer Integrated Manufacturing 25, pp.261-270. 20. D. Karaboga, An idea based on honey bee swarm for numerical optimization. Technical report TR06. Computer Engineering Department. Erciyes University, Turkey, 2005. 21.D. Karaboga, C. Ozturk, A novel clustering approach: Artificial Bee Colony (ABC) algorithm, Applied Soft Computing, 2011; 11(1):652-657. 22. D. Karaboga, B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm, Journal of Global Optimization 2003; 39:459-471. 23. S.N. Omkar, J. Senthilnath, R. Khandelwal, N.G. Narayana, S. Gopalakrishnan, Artificial Bee Colony (ABC) for multi-objective design optimization of composite structures, Applied Soft Computing 2011;11:489-499.

24. Ali R. Yildiz, A new hybrid artificial bee colony algorithm for robust optimal design and manufacturing, Applied Soft Computing, http://dx.doi.org/10.1016/j.asoc.2012.04.013, 2012

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Modeling of Flux Decline at Nanofiltration Membrane by Artificial Neural Networks Beytullah Eren Sakarya University, Sakarya, Turkey, [email protected]

Abstract. Accumulation of substances on membrane surfaces leads to clogging of pores and decrease of flux across. This phenomenon adversely affects separation performances of the membrane [1]. In order to avoid the clogging of the membrane, operating conditions (cross flow velocity, pH, feed pressure, etc.) must be optimized in accordance with the characteristics of the feed solution [2]. A number of theoretical models are developed for process optimization and for prediction of flux decline in different membrane processes. The theoretical models are very complex and they require a very detailed knowledge of the membrane, and the properties of solutes and solvent in the membrane [3]. Therefore, simple models also are needed to evaluate flux decline in membrane processes. In this study, flux decline is predicted by a neural network model using only experimental data. For this purpose, a total of 150 experimental data are grouped to three parts as the training set (90 data), validation set (20 data), and test set (20 data). The optimal network architecture is developed through trial and error approach. Optimal network architecture was determined as one hidden layer with 18 neurons using levenberg-marquardt training algorithm. Results with the optimum network showed that the developed ANN model predictions and experimental data matched well and the model can be employed successfully for prediction of flux decline. Keywords. Flux decline, Nanofiltration Membrane, Neural Network, Modeling.

References

[1] Van der Bruggen, B., L. Braeken, and C. Vandecasteele, Flux decline in nanofiltration due to adsorption of organic compounds, Sep. Purif. Technol., 29(1), 23-31, 2002.

[2] Koyuncu, I., D. Topacik, and M.R. Wiesner, Factors influencing flux decline during nanofiltration of solutions containing dyes and salts. Water Res., 38(2), 432-440, 2004.

[3] Bowen, W.R., M.G. Jones, and H.N.S. Yousef, Prediction of the rate of crossflow membrane ultrafiltration of colloids: A neural network approach, Chem. Eng. Sci., 53(22), 3793-3802, 1998.

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Reduction of Dimensionality in the Problem of Diffraction From a Half-Plane Burim Kamishi (1) and Rasim Bejtullahu (2) (1) University of Prishtina, Prishtina, Kosovo, [email protected] (2) University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. In this paper we consider the reduction of dimensionality in the problem of diffraction from a half-plane. The integral equation formulation is of particular advantage in the case of boundary-value problems associated with partial differential equations. In this paper, a second-order partial differential equation in two dimensions will be restated as an integral equation in only one dimension. The integral equation obtained is one-dimensional and includes all the boundary conditons. Keywords. Diffraction, partial differential equations, boundary conditions, singular integral equations. AMS 2010. 78A45, 45E05, 35A18.

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Estimation of Lake Sapanca Daily Water Level Using Artificial Intelligent Techniques Emrah Doğan (1) and Sümeyra Demir (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya Municipality, Water and Sewerage Authority, Sakarya, Turkey

Abstract. Lake Sapanca, which is a water resource for Sakarya City and its surrounding, is also an important water resource for industrial water supply. Lake Sapanca is one of the lake, that is rarely used as drinking water resource of our country [1]. Because of the natural beauty and closeness to the Istanbul, it subjects to intense structuring. Global warming causes to the climate changes. As a result of there are serious problems in the water resources of the world. In this study, the effects of the global climate change in Turkey and in our world, was analyzed. It was emphasized where the Lake Sapanca of this change is. In this study scripts were built between factors of climate and changing lake level of Lake Sapanca. Factors of climate are data like daily average rainfall, daily average evaporation, daily average relative humidity, daily average temperature, daily average wind speed, which are taken from Sakarya management of meteorological station [2]. The value of the daily lake level was taken from the data of the state hydraulic works (DSİ, 32. Branch Office, SAKARYA). These data were compared with the result that was obtained by subjecting to artificial neural network (ANN) under the various scripts and the best model was determined. Keywords. Lake Sapanca, Artificial Neural Network, Global Warming, Lake Level.

References

[1] DOĞAN,E., IŞIK, S., Sapanca Gölü Günlük Buharlaşma Miktarının Radyal Temellli Yapay Sinir Ağı Modeli Kullanılarak Tahmin Edilmesi, Bilimde Moders Yöntemler Sempozyumu, BMYS 2005, Kocaeli, 2005

[2] DEMİR, S., Uzun Vadeli Su Temininde Meteorolojik Faktörlerin Sapanca Gölü Su Kalitesine Etkisi,Yüksek Lisans Tezi, Sakarya Üniversitesi Fen Bilimleri Enstitüsü, 2011.

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Seismic Q Estimation Using Artifical Neural Network Eray Yıldırım (1), Ruhi Saatçılar (1), Gündüz Horasan (1) and Semih Ergintav (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) TÜBİTAK-MAM, Kocaeli, Turkey, [email protected]

Abstract. The purpose of this study is to estimate the attenuation characteristics of seismic waves propagation in the shallow layers of the ground using Artificial Neural Network (ANN), to investigate traditional attenuation estimation methods and to determine the relation between the estimated values and soil types. A number of different applications for attenuation are available in the literature. Seismic quality factor is defined as the inverse of absorption. Various methods are used for the estimation of seismic Q in time and frequency domains [1], [2], [3]. In this study, ANN is used for estimation of seismic Q parameter as an alternative of traditional methods [4]. Performance of Q parameter estimation using ANN firstly investigated in theoretical models created by synthetic seismograms. A number of ANN models that have different training algorithm and neuron number were created for estimation of seismic Q. Levenberg- Marquardt training algorithm gave the best performance for Q estimation. The performance of ANN results was compared with amplitude decay, spectral ratio and Wiener filter methods. The ANN model that has the best performance for Q estimation was also applied to the field seismic reflection data. Estimated Q values were statistically analyzed using standard deviation and confidence interval. The more confident seismic Q values are obtained as a result of these analyzes. The Q values obtained from the ANN method were used for determining of soil types. Keywords. Attenuation, Seismic Quality Factor, Artificial Neural Networks.

References

[1] Johnston, D.H. and Toksöz, M.N., Seismic Wave Attenuation, SEG, Geophysics Reprint Series No:2, 1981.

[2] Saatcılar, R. and Coruh, C., 1995. Seismic Q Estimations for Lithologic Interpretation, 65 th. SEG Meeting, Houston, October 8-13, 1995.

[3] Tonn R., The determination of the seismic quality factor from VSP data: A comparison of different computational methods, Geophysical Prospecting, 39, 1-27, 1991.

[4] Leung Hui, C., Artificial Neural Networks – Application, Publisher: Intech, ISBN 978- 953-307-188-6, 586 Pages, 2011.

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A Real-Time Virtual Sculpting Application with a Haptic Device Gülüzar Cit (1), Kayhan Ayar (2), Soydan Serttaş (3),Cemil Öz (4) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected] (4) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this paper, a 3D virtual sculpting application is developed for 3D virtual models with removing or adding materials using Boolean operations. Virtual sculpting simulation reads 3D virtual models in a variety file formats such as raw and stl consisting of a triangle poligon mesh and voxelizes its outer surface and interiror to generate its volumetric dataset. We used octree and hashing teqhniques to reduce the memory requirement needed for volumetric dataset. The surface is locally reconstructed using Marching Cubes algorithm known as the most popular isosurface extraction algorithm after removing or adding material to the 3D virtual model. The user interacts with the model using a haptic device to give the force- feedback like real-life sculpting. . Keywords. Virtual sculpting, voxelization, octree, haptic

References

[1] Akenine-Möller, T., Fast 3d triangle-box overlap testing, Journal of Graphics Tools, vol. 6, no. 1, pp. 29-33, 2001

[2] Galyean, A., Hughes, J. F., Sculpting: An Interactive Volumetric Modeling Technique, Computer Graphics, Volume 25, Number 4, pp. 268-274, July 1991.

[3] Akenine-Möller, T. and Haines, E., Real-Time Rendering, AK Peters Ltd, 2nd edition, 2002.

[4] O'Neill, G.T., Lee, W.S., William, J., Haptic-Based 3D Carving Simulator, Advances in Haptics, p.299-314, April 2010

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[5] Williams, J, O’Neill, G.T., Lee, W.S., Interactive 3D Haptic Carving using Combined Voxels and Mesh, HAVE 2008 – IEEE International Workshop on Haptic Audio Visual Environments and their Applications, 2008

[6] Thon, S., Gesquière, G., Raffin, R., A low Cost Antialiased Space Filled Voxelization of Polygonal Objects, GraphiCon 2004, pp. 71—78, 2004

[7] Jones, M. W., The production of volume data from triangular meshes using voxelisation, Computer Graphics Forum, 15(5):311–318, 1996.

[8] Niu, Q., Chi, X., Leu, M.C., Ochoa, J., Image processing, geometric modeling and data management for development of a virtual bone surgery system, Computer Aided Surgery, vol. 13(1), p.30–40, 2008

[9] Foley, J.D., Van Dam, A., Feiner, S.K, Hughes, J.F. , Computer Graphics : Principles and Practice, 2nd Edition, Addison-Wesley, pp.92-99, 1990

[9] Perng, K.L., Wang, W.T., Flanagan, M., and Ouhyoung, M., A real-time 3d virtual sculpting tool based on modified marching cubes. In Proceedings of International Conference on Artificial Reality and Teleexistence, p.64–72, 2001

[10] Ho, C.C., Tu, C.H. and Ouhyoung, M., Detail Sculpting using Cubical Marching Squares, ICAT 05 - Proceedings of the 2005 international conference on Augmented tele-existence, p.10-15, 2005

[11] Lorensen, W. E. and Cline, H. E., Marching Cubes: A High Resolution 3D Surface Construction Algorithm, Computer Graphics, vol. 21, no. 4, p. 163-169, 1987

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Pre-service Student Teachers’ Point, Line and Plane Concept Knowledges in Geometry and Used Multiply Representation Güler Tuluk [email protected]

Abstract. The purpose of the study is to determine pre-service clasroom teachers’ views of geometry pre-kowledge of point, line, plane and objects. Participants were first year students in education faculty. Ten open-ended questions were given the students and asked to explain this concepts, the method of descriptive analysis was used for the open-ended question. The quantitative data were tabulated by using frequency and percentage The results indicated that the students used the criteria of plane position in general and also wide range of criteria of one and two dimension. The study suggests that mathematics lessons attempt to reinforce definition and symbolic approaches in order to develop pre-service students’ teachers thinking in geometry instruction. Keywords. Pre-service student teacher, plane geometry, objects of geometry knowledge.

References

[1] Aksu, H.H., (2005), İlköğretimde Aktif Öğrenme Modeli ile Geometri Öğretiminin, Başarıya, Kalıcılığa, Tutuma ve Geometrik Düşünme Düzeyine Etkisi, Dokuz Eylül Üni. Eğitim Bilimleri Enstitüsü, Yayınlanmış Doktora Tezi, İzmir.

[2] Allendoerfer, C. B. 1969, “The Dilemma in Geometry”, Mathematics Teacher, vol. 62, pp. 165-169.

[3] Aslan , S., Yıldız, C., (2010), “11. Sınıf Öğrencilerinin Matematiksel Düşünmenin Aşamalarındaki Yaşantılarından Yansımalar”, Eğitim ve Bilim, Cilt 35, Sayı 156 H.Ü, Ankara

[4] Baki, A., (2008). “Kuramdan uygulamaya Matematik Eğitimi”, Harf Eğitim Yayıncılık, Ankara

[5] Ball, D. L., Hill, H. C. and Bass, H. 2005, “Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide?” American Educator, Winter 2005/06, 14-22 & 43-46.

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[6] Ball, D. L. (2003). What mathematics knowledge is needed for teaching mathematics? Paper presented at the US Department of Education, Secretary’s Mathematics Summit, Washington, DC.

[7] Battista, M. (1999). Geometry results from the Third International Mathematics and Science Study. Teaching Children Mathematics, 5(6), 367-373.

[8] Blitzer, B. (2005). “Thinking Mathematically”, Third Edition. Pearson Education, Inc. USA p:491- 557

[9] Burns, M., About Teaching Mathematics. Second Edition. Math Solutions Publication, California, 2000.

[10] Cankoy, O. (2002). “Matematik ve Günlük Yaşam Dersi ile ilgili Görüşler”. Paper presented at the 5.Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi.

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Self-Dual Codes from Self-Dual Codes of Smaller Lengths and Recursive Algorithm Hatice Topcu (1) and Hacı Aktas (2) (1) Nevsehir University, Nevsehir, Turkey, [email protected] (2) Nevsehir University, Nevsehir, Turkey, [email protected]

Abstract. Self-dual codes have been received great attention by researchers since the beginning of the coding theory [1,3,4,6]. In this work, some construction methods for this kind of codes are given which produces new self-dual codes from self-dual codes of smaller lengths [2,5,7,8,9,10]. A special one of these methods that is called recursive algorithm is also given [9]. For the binary case, it has shown that recursive algorithm is actually same with another so-called building-up construction method [11,12]. This comparison is mentioned here. Keywords. Self-dual codes, building-up construction, recursive algorithm. AMS 2010. 94B05.

References

[1] Shannon, C., A mathematical theory of communication, Bell System Tech. J., 27, 379-473, 1948.

[2] Kim, J.-L., New extremal self-dual codes of length 36,38 and 58, IEEE Trans. Inform Theory, 47, 386-393, 2001.

[3] Pless, V., On the classification and enumeration of self-dual codes, J. Combin Theory Ser. A, 18 no.3 , 313-335, 1975.

[4] Rains E, Sloane N.J.A., Self-dual codes, in: Pless V., Huffman W.C. (Eds.), “Handbook of Coding Theory”, Elsevier, Amsterdam. Netherlands, 1998.

[5] Harada M., The existence of a self-dual [70,35,12] code and formally self-dual codes, Finite Fields Appl., 3 , 131-139, 1997.

[6] Huffman, W.C., On the classification and enumeration of self-dual codes, Finite Fields Appl., 11, 451-490, 2005.

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[7] Kim, J.-L., Lee, Y., Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combin Theory Ser. A, 105 , 79-95, 2004.

[8] Gulliver, T.A., Kim, J.-L., Lee, Y., New MDS or Near MDS self-dual codes, IEEE Trans. Inform. Theory, 54 no.9 , 4354-4360, 2008.

[9] Aguilar-Melchor, C., Gaborit, P., On the classification of extremal [36,18,8] binary self- dual codes, J. Combin Theory Ser. A, 18 no.3 , 313-335, 1975.

[10] Kim, J.-L., Lee, Y., An efficent construction of self-dual codes, preprint, 2012.

[11] Aguilar-Melchor, C., Gaborit, P., Kim, J.-L., Sok, L., Solé, P., Classification of extremal snd s-extremal binary self-dual codes of length 38, IEEE Trans. Inform. Theory, 58 , 2253- 2262, 2012.

[12] Aktaş, H., Topcu H., Self-dual Kodlar ve İnşa Yöntemleri, Nevşehir Üniversitesi Fen Bilimleri Enstitüsü, 2012.

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Theoretical Calculation of the Ground-State Magnetic Moments in the Odd-Mass Nuclei Hakan Yakut(1), Emre Tabar(1), Ekber Guliyev(2) and Osman Örnek(1) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Institute of Physics, National Academy of Sciences, Azerbaijan

Abstract. In this study the spin polarization effects on magnetic moments of the deformed odd-A nuclei were investigated by using a microscopic method called as Quasiparticle-phonon nuclear model (QPNM). The model operates with the strength function method, calculating the fragmentation of single quasiparticle, one phonon states and the quasiparticle-phonon states over a large number of nuclear levels. Therefore, in this model the

j wave functions Φ K of odd-mass nuclei consist of the sum of one-quasiparticle and one- quasiparticle ⊗ phonon [1].

j Firstly, we find the mean value using Φ K function, employ the variational principle

(the method of Lagrange multipliers), and deduce the secular equation for the energies ηK for an odd nucleus, P η K = .0)( Secondly, we also obtain the set of equations for the coefficients, including the normalizations of the wave functions. Then, the intrinsic magnetic moment

µ = KK Kg is derived from the expectation value of z component of the magnetic dipole operator for a K>1/2 state of an odd-mass nucleus. Finally, we present theoretical calculations of intrinsic magnetic moments for the few Gadolinium and Dysprosium nuclei, as an application. Calculations show that the spin-spin interaction in this isotopes leads to polarization effect influencing the magnetic moments. The detailed information about the explanations represented above is in ref. [1]. Keywords. Magnetic moment, QPNM, Spin interaction AMS 2010. 81S05, 00A79.

References

[1] H. Yakut, E. Guliyev, M. Güner, E. Tabar and Z. Zenginerler, QPNM Calculation fort he Ground State Magnetic Moments of Odd-mass Deformed Nuclei: 157-167Er Isotopes, Nuclear Phys. A, 888, 23-33, 2012.

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Performans Ranking of Turkish Insurance Companies : The Anp Application İlyas Akhisar Marmara University, Istanbul, Turkey, [email protected]

Abstract. Decision making concept can be defined as selection process choosing one of the options, if the outcomes are not certain. In practice it is not sufficient to evaluate the options according to the simplest form of decision making only based on a criterion. In such cases, we must consider all the variety of different information to decide to best option. It arise the requirement of multi-criteria decision making methods. AHP(Analytical Hierarch Process) is a technique that models the relations which are different stage. On the other hand that technique is insufficient to evaluate interdependency relations. The ANP (Analytic Network Process) has the ability to add all the criteria related with the issue. The method of the ANP can be described as follows. The first phase of the ANP is to compare the criteria in whole system to form the supermatrix. This is done through pairwise comparisons by asking experts. The first step of the ANP is to compare the importance between each criterion. The next step is to calculate the influence of the elements (criteria) in each component (matrix) using the eigenvalue method. In this research, financial performans ranking of Turkish Insurance companies, which large- scaled business in non-life branches, are obtained for the period 2000 -2010 using financial ratios of companies and applying the ANP model which was developed with the superdecision software. Keywords: ANP, performance ranking, insurance sektor, financial ratios

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Relations on FP-Soft Sets Applied to Decision Making Problems İrfan Deli (1), Naim Çağman (2) and İrfan Şimşek (3) (1) Kilis 7 Aralik University, Kilis, Turkey, [email protected] (2) Gaziosmanpasa University, Tokat, Turkey, [email protected] (3) Gaziosmanpasa University, Tokat, Turkey, [email protected]

Abstract. Soft set theory, initiated by Molodtsov, is a tool for modeling various types of uncertainty. In this work, we first define relations on fuzzy parametrized soft sets (FP-soft sets) and study some of their properties. We also give a decision making method based on the relations. Finally, we give an example which shows that it can be successfully applied to problems that contain uncertainties. Keywords. Soft sets, fuzzy sets, FP-soft sets, relations on FP-soft sets, decision making. AMS 2010. : 03E15, 03E75, 91B06, 20F10.

References

[1] Ali, M.I., Feng, F., Liu, X., Min, W.K. and Shabir, M., On some new operations in soft set theory, Computers and Mathematics with Applications, 57, 1547-1553, 2009.

[2] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96, 1986.

[3] Çağman, N., Erdoğan F. and Enginoğlu, S., FP-soft set theory and its applications, Annals of Fuzzy Mathematics and informatics, 2/2, 219-226, 2011.

[4] Çağman, N. and Enginoğlu, S., Soft matrices and its decision makings, Computers and Mathematics with Applications 59, 3308-3314, 2010.

[5] Çağman, N. and Enginoğlu, S., Soft set theory and uni-int decision making, European Journal of Operational Research 207, 848-855, 2010.

[7] Jiang, Y., Tang Y., Chen, Q., An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling, 35, 824-836, 2011.

[8] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999) 19-31.

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Assessing the Relationship between Educational Performance and Attitudes of Turkish Students İbrahim Demir (1) and Serpil Kılıç (2) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. In this study, the main purpose was to determine the relationship between general education performance and attitudes towards math, science and reading on the Turkish sample in Programme for International Student Assesment (PISA) using Structural Equation Modeling (SEM). This approach is used to propose direct and indirect effects to predict students’ achievement. PISA survey takes place in every three years since 2000. In 2009, Turkish PISA dataset contains 4996 students in 170 schools in 6 regions in Turkey. In the study general education performance was modeled as a common latent variable of the three observed achievement scales which were math, science and reading. In conclusion, although direct and indirect effect sizes of different attitudes using in the analysis on general education performance could be different, these factors were the important factors for the success of students. Keywords. PISA, Students’ achievement, Structural equation modeling AMS 2010. 62-07, 62H99.

References

[1] Ceylan, E., Berberoğlu, G., Öğrencilerin fen başarısını açıklayan etmenler: Bir Modelleme Çalışması, Eğitim ve Bilim, 32.144, 36-48, 2007.

[2] Hoon, K. C., Fah, L. Y., An investigation of factors that contribute to rural students’ mathematics achievement: A Structural Equation Modeling Approach, Third International Conference on Science and Mathematics Education, 10-12 November 2009.

[3] Uzun, B., Öğretmen, T., Fen başarısı ile ilgili bazı değişkenlerin TIMSS-R Türkiye örnekleminde cinsiyete gore ölçme değişmezliğinin değerlendirilmesi, Eğitim ve Bilim, 35.155, 26-35, 2010.

[4] Hox, J. J., Bechger, T. M., An introduction to Structural Equation Modeling, Family Science Review, 11, 354-373, 1998.

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Comparing Turkey’s Domestic Debt Stock Increment with Linear Regression, Ridge Regression and Principle Component Regression Ibrahim Demir (1), Gizem Altunel (2) and Erhan Cene (3) (1) Yildiz Technical University, Istanbul, Turkey, [email protected] (2) Yildiz Technical University, Istanbul, Turkey, [email protected] (3) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. There are many variables that affect domestic debt. But when variables are put into the model together, a strong relationship between variables that refers to a multicollinearity problem occurs. In a multiple linear regression model, in order to decrease the standard error of relationship coefficient between dependent and independent variables and in order to have more sensitive predictions, multicollinearity problem had to be solved. Some data gathering methods and variable elimination methods are being used in order to overcome multicollinearity. Also some of the methods use biased predicting methods since they correct multicollinearity problem without eliminating variables but those methods give biased results. Two of the methods that deals with the multicollinearity are Ridge Regression and Principal Component Regression. In this study, an application to domestic debt increment by using both Ridge Regression and Principal Component Regression has been conducted. In analysis, a multi linear regression model with 8 variables constructed between years 1985-2010 intuitively thought to be affecting domestic debt increment: the public sector borrowing requirement, the exchange rate, the commercial deposit interest rate, the internal debt interest rate, the annual inflation rate, the government budget deficit, the internal debt service. Analysis showed that, while Ridge Regression and Principal Component Regression coefficients matched with the theoretical expectations, least square coefficients gave inconsistent results. Keywords. Domestic Debt, Ridge Regression, Multicollinearity, Principle Component Regression AMS 2010. 62J07, 62J05

References

[1] Abdul-Wahaba S.A, Bakheitb, C.S., Al-Alawia, S.M., “Principal component and multiple regression analysis in modelling of ground-level ozone and factors affecting its

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concentrations” Environmental Modelling & Software, Volume 20, Issue 10, October 2005, Pages 1263–1271, 2005.

[2] Albayrak, A., “Çoklu Doğrusal Bağlantı Halinde En Küçük Kareler Tekniğinin Alternatifi Yanlı Tahmin Teknikleri Ve Bir Uygulama”, ZKÜ Sosyal Bilimler Dergisi, 1- 1., 2005.

Alpar, R., “Uygulamalı Çok Değişkenli İstatistiksel Yöntemlere Giriş 1”, Nobel Yayın, No: 452, Ankara, 2003.

[3] Büyükuysal, M., “Ridge Regresyon Analizi ve Bir Uygulama”, Yüksek Lisans Tezi, Uludağ Üniversitesi, Sağlık Bilimleri Enstitüsü, 2010.

[4] Canküyer, E. and Sönmez H., “Regresyon Analizinde Çoklu Doğrusal Bağlantı Sorununun İncelenmesi Ve Uygulanması”, Anadolu Üniversitesi Fen Fakültesi Dergisi, 173- 187., 1996.

[5] El-Deren, M and Rashwan, N. I., “Solving Multicollinearity Problem Using Ridge Regression Models”, Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 12, 585 – 600, 2011.

[6] Liu, R.X, Kuang J, Gong Q, Hou XL. “Principal component regression analysis with SPSS”, Comput Methods Programs Biomed. 2003 Jun;71(2):141-7., 2003.

[7] Sonuvar, E., “Türkiye ithalatının Ridge Regresyonla analizi”, Yüksek Lisans Tezi, YTÜ, Fen Bilimleri Enstitüsü., 2007.

[8] Vigneau E., Devaux F., Qannari, E. M. and Robert, P. “Principal Component Regression, Ridge Regression And Ridge Principal Component Regression In Spectroscopy Calibration”, Journal of Chemometrics, vol. 11, 239–249, 1997.

[9] Yavuz, A., “Türkiye’de İç Borç Stoğundaki Değişimin Analizine Yönelik Bir Regresyon Analizi Çalışması”, Süleyman Demirel Üniversitesi, İktisadi Ve İdari Bilimler Fakültesi, C.8, S.1 s.339-356, 2003.

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The Adomian Decomposition Method for Solving Nonlocal Boundary Value Problems for First-Order Linear Hyperbolic Equation Lazhar Bougoffa Al-Imam University, Riyadh, Saudi Arabia, [email protected]

Abstract. In this paper, we present a new approach to resolve nonlocal boundary value problems of linear hyperbolic equation of first-order with nonlocal conditions of integral type by first transforming the given nonlocal boundary value problem into an equivalent boundary value problem such a manner that the Adomian's decomposition method with a modification can be applied. Keywords. Non-local boundary value problem, Hyperbolic equation of first-order, Adomian decomposition method.. AMS 2010. 35L02, 35L03.

References

[1] von Mises, R., Festschrift Heinrich Weber, pp. 252-282, Leipzig, 1912.

[2] G. Adomian, Nonlinear Stochastic Operator Equations, Academic, Orlando, FL, 1986.

[3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994. [4] L. Bougoffa, A weak solution for hyperbolic equations with a nonlocal condition, The Australian Journal of Mathematical Analysis and Applications (AJMAA), Vol. 2, Issue 1, pp. 1-7 (2005).

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Genetic Algorithm and Financial Optimisation Mehmet Altay Ünal Ankara University, Ankara, Turkey, [email protected]

Abstract. Genetic algorithm is an innovative approach to artifical intelligence, capable of solving complex problems with almost no human supervision. The algorithm was first introduced by John Holland in 1975. This method utilizes the survival of the fittest to select the best solution for the problem under consideration. In this work we present the main features of genetic algorithm and give an example how to use genetic algorithm for financial optimisiation. Keywords. Genetic algorithm, Financial optimisation AMS 2010. 91G10, 91G60.

References

[1] Holland, J. H. (1975). Adaptation in natural and artificial systems. Ann Arbor: The University of Michigan Press

[2] Davis, L. (1991). Handbook of genetic algorithms. New York: Van Nostrand Reinhold.

[3] Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading, MA: Addison-Wesley.

[4] Richard J. Bauer, Jr. (1950) Genetic Algorithms and Investment Strategies, Willey Finance

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Uniform Time Controllability of Affine Control Systems on Semisimple Lie Groups Memet Kule University of 7 Aralik, Kilis, Turkey, [email protected]

Abstract. In this study, we deal with an affine control system Σ defined on a simply- connected semisimple Lie group G with Lie algebra L(G). We provide uniform time controllability properties of such systems on the Semisimple Lie group established by relating to their associated invariant part. Keywords. Semisimple Lie Groups, Affine Control Systems, Uniform Time Controllability AMS 2010. 22E46, 93B05, 93C10.

References

[1] Ayala, V., San Martin, L.A.B., Controllability properties of a class of control systems on Lie groups, In: Procedings of second Nonlinear Control Network (NCN) Workshop, France, 2000.

[2] Ayala, V., Ayala-Hoffmann, J., and Tribuzy, I., Controllability of invariant control systems at uniform time, Kybernetika, 45, (3), pp. 405-416, 2009.

[3] Jurdjevic, V., and Sussmann, H.J., Control Systems on Lie Groups, Journal of Differential Equations, 12, pp. 313-329, 1972.

[4] Jurdjevic, V., Geometric Control Theory, Cambridge University Press, Cambridge, 1997.

[5] Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups, Scott Foresman and Company, London, 1971.

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The Effect of Instruction States Designed According to Van Hiele Geometrical Thinking Levels on the Geometrical Success Mustafa Terzi (1) and Şeref Mirasyedioğlu (2) (1) Gazi University, Ankara, Turkey, [email protected]

(2) Baskent University, Ankara, Turkey, [email protected]

Abstract. The aim of this research is to determine the effect of instruction states designed in accordance with the Van Hiele geometrical thinking levels on the geometrical success. “First test – last test control group model” was used among the trial models in the research. The subjects of this research were 18 students from Ankara eighth grade students during 2008 – 2009 education year as the experiment group and 20 students from the same school as the control group. In the research, “Geometrical Success test” was used as data collection tool. In the analysis of the data, double sided Kikare test, for unrelated measurements Mann Whitney U- test, for related measurements Wilcoxon Marked Orders test and for unrelated measurements t test were applied. The results of the research are as follows: A meaningful pre-education difference between the geometrical success level of the students instructed with traditional education and the geometrical success level of students instructed with education designed according to Van Hiele geometrical thinking levels could not be found. On the other hand the education designed according to Van Hiele geometrical thinking levels was effective in increasing the geometrical success levels of students. Key Words: Van Hiele, Geometry, Geometrical Success

References

[1] Altun, M. (2005). Eğitim fakülteleri ve ilköğretim öğrencileri için matematik öğretimi. Bursa: Erkam Matbaası.

[2] Baki, A. (2001). Bilişim Teknolojisi Eşiği Altında Matematik Eğitiminin Değerlendrilmesi, Milli Eğitim Dergisi, 149,26-31.

[3] Bennie, K. (2005), MALATI “SHAPE and SPACE”, An Approach to the Study of Geometry in the Intermediate Phase.

[4] Burger, W., ve Shaughnessy, J.M. (1986). Characterizing the van Hiele levels of development in geometry. Journal For Research in Mathematics Education, 17, 31- 48.

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[5] Burns, M. (2000). About teaching mathematics. (Second edition). California: Math Solutions Publication.

[6] Çelebi Akkaya, S. (2006). Van Hiele düzeylerine göre hazırlanan etkinliklerin ilköğretim 6. sınıf öğrencilerinin tutumuna ve başarısına etkisi. Yayımlanmamış Yüksek Lisans Tezi, Abant İzzet Baysal Üniversitesi Sosyal Bilimler Enstitüsü, Bolu.

[7] Erkoç, N. (2008). Çocuklarda düşünme becerileri nasıl geliştirilir? http://www.gulucek. 01.2009 tarihinde indirilmiştir.

[8] Frerking, B. Giddens. (1994). Conjecturing and Proof-Writing in Dynamic Geometry. Dissertation Abstracts International. 55:12.

[9] Genz, R. (2006). Determining high school geometry students’ geometric understanding using Van Hiele levels: Is there a difference between standarts-based curriculum students and non standarts-based curriculum students. Unpublished Master Thesis, Brigham Young University, Department of Mathematics Educations.

[10] Hoffer, A. (1981). Geometry is more than proff. Mathematics Teacher, 74,11-18.

[11] Idris, N. (2007). The effect of geometers’ sketchpad on the performance in geometry of Malaysian students’ achievement and Van Hiele geometric thinking. Malaysian Journal of Mathematical Sciences, 1(2), 169 – 180.

[12] Karakuş,F.(2001).Ortaöğretim düzeyi için tasarlanan fraktal Geometri öğretim programının değerlendirilmesi.Doktora Tezi.Karadeniz Teknik Üniversitesi Eğitim Bilimleri Enstitüsü,Trabzon.

[13] Kılıç, Ç. (2003). İlköğretim 5. sınıf matematik dersinde Van Hiele düzeylerine göre yapılan geometri öğretiminin öğrencilerin akademik başarıları, tutumları ve hatırda tutma düzeyleri üzerindeki etkisi. Yayımlanmamış Yüksek Lisans Tezi, Anadolu Üniversitesi Eğitim Bilimleri Enstitüsü, Eskişehir.

[14] Larew, L., W. (1999). The effects of learning geometry using a computer-generated automatic draw tool in the levels of reasoning college developmental students. Unpublished doctoral dissertation. College of Human Resources and Education. Morgantown, West Virginia.

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[15] Lowry, J. A. (1988). An Investigation of Nine-Yaer Olds' Geometric Concepts of Area and Perimeter. Dissertation Abstracts International. 48: 8.

[16] Mayberry, J. (1983). The van Hiele levels of geometric thought in undergraduate preservice teachers. Journal for Research in Mathematics Education, 14(1), 58- 69.

[17] M.E.B. (2000). İlköğretim okulu matematik dersi programı 5. sınıflar. İstanbul: Milli Eğitim Basımevi.

[18] NCTM, (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

[19] Sheard, W. H. (1981). “Why is Geometry a Basic Skill?”, Mathematics Teacher. 74, 1: 19-21,

[20] Toluk, Z. & Olkun, S. (2004). Sınıf Öğretmeni Adaylarının Geometrik Düşünme Düzeyleri. Eğitim ve Bilim, 134, 55-60.

[21] Usiskin, Z. (1982). Van Hiele Levels and Achievement in Secondary School Geometry. (Final Report of the Cognitive Development and Achievement in Secondary School Geometry Project). Chicago: University of Chicago. (ERIC Document Reproduction Service No. ED 220288.

[22] Van Hiele, P. M. (1986). Structure and Insight: A Theory of Mathematics Education

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On a Problem of Thermal Convection with Unset Flow Rate Danaev N.T., Darybaev.B.S., Urmashev B.A. Institute of Mathematics and Mechanics, Kazakh National University, Al-Farabi, Kazakhstan, [email protected]

Abstract. In the two-dimensional region Ω, as shown in the figure, we consider a system of equations of thermal convection in the following dimensionless form /1/:   ∂u 1 Grg   −∆=∇+∇+ )( upuu 2  T, ∂t Re Re g  ,0 udiv = ,0 ∂T  1 )( Tu ∆=∇+ T, ∂t Pr Re

3  ∆θβ Lg ρ∆pL ν where −= gg ),,0( Gr = , Re = , Pr = ν 2 μ λ dimensionless parameters of Grashof, Reynolds and Prandtl, ∆θ - a characteristic temperature difference, v - kinematic viscosity, λ - coefficient of thermal diffusivity. The boundary conditions are as follows: on top of the solid wall (ВС, СD): Tvu === ,0,0 on the bottom wall (АА′ ): Tvu === ,1,0 entrance boundary (АВ): ∂T ;0 ,1 ,0 v ,0 p == ,1 = ;0 ∂x on the outflow boundary (DO′): ∂T 0 ,0 ,0 pu == ,0 = 0 . ∂y On the basis of the proposed iterative algorithm /2/, carried out numerical calculations and obtained the flow pattern for different Grashof and Reynolds numbers. It was established that at sufficiently high Reynolds number (Re(δp)=500-700), that is, for sufficiently strong flow, caused by the pressure drop, increasing the temperature difference between the walls (eg, numbers Gr=5∗105) does not lead to a marked increase in flow rate.

1. Alibiev D.B., Danaev N.T., Smagulov Sh.S. Numerical solution of a heat problem with the consumption of unset flow rate // Computational technologies. – Novosibirsk, 1995. -B.4. - №12. –P.10-28. 2. Danaev N.T., Urmashev B.A. Iterative schemes for solving the auxiliary grid of Navier- Stokes equations // Journal of KSU, a series of mathematics, mechanics, computer science. – 2000. - №4. - P.74-78.

The work was supported by the Scientific Committee of MES RK (contract № 967 dated 02.03.2012)

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Solution of Spherical Triangles in Geodesy with the Equations Used in Spherical Trigonometry Nihat Ersoy (1), Erol Yavuz (2) and R. Gürsel Hoşbaş (3) (1) Yildiz Technical University, Istanbul, [email protected] (2) Yildiz Technical University, Istanbul,[email protected] (3) Yildiz Technical University, Istanbul, [email protected]

Abstract Spherical trigonometry is the trigonometry in which the computations are performed over a sphere instead of a plane. The science disciplines of Geography, Cartography, Geodesy and Astronomy which deal with celestial sphere and the globe (terrestrial sphere) utilize the spherical trigonometry. Spherical trigonometry, which is a part of trigonometry, is the science in which spherical triangle solutions are performed on a sphere using the mathematical relations between the elements of a triangle (angles and lengths) and using spherical trigonometric equations. The aim of this presentation is to describe the spherical triangle solutions by an application using trigonometric theorems and formulas of spherical trigonometry which are employed through computations on a sphere. Keywords. Küresel Trigonometri, Jeodezi, Astronomi, Ekses, Coğrafi Koordinatlar

References

[1] Ulusoy, E., Düzlem ve Küresel Trigonometri, İDMMA Basımevi, İstanbul, 1969.

[2] Ayres, Jr. F., Theory and Problems Plane and Spherical Trigonometry, Newyork, 1954.

[3] Ferval, H., Elements de Trigonometrie Hachette Etc, Paris, 1905.

[4] Yaşayan, A; Hekimoğlu Ş., Küresel Trigonometri, KTÜ Basımevi, Trabzon, 1982.

[5] Ersoy, N., Trigonometri, MEB Basımevi, Ankara, 2001.

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Application of Incomplete Cylindrical Functions in the Diffraction of Gaussian Beam from a Half-Plane Rasim Bejtullahu (1), Burim Kamishi (2), Zeqë Tolaj (3) and Fisnik Aliaj (4) (1) University of Prishtina, Prishtina, Kosovo, [email protected] (2) University of Prishtina, Prishtina, Kosovo, [email protected] (3) University of Prishtina, Prishtina, Kosovo, [email protected] (4) University of Prishtina, Prishtina, Kosovo, [email protected]

Abstract. Diffraction of a Gaussian beam on the edge of an half-plane obstacle which absorbs completely the incident beam, is investigated. The diffraction pattern is registered on a parallel plane screen separated at a distance from the plane of the obstacle. The problem is treated in polar coordinates (r, θ) on the plane of the diffraction obstacle, and (ρ, θ) on the plane where the diffraction pattern is observed. The solution is seeked using incomplete cylindrical functions. Mathematical analysis shows that the diffractional intensity distribution represents a system of ellipses with the principal semi-axis parallel to the edge of the obstacle. This result is confirmed by the numerical analysis and the experimental verification. Keywords. Diffraction, incomplete cylindrical functions, Gaussian beam. AMS 2010. 78A45, 33B20, 34B30.

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Asynchronous Motor with Finite Element Method Nonlinear Analysis Sevcan Aytaç Korkmaz (1) and Hasan Kürüm (2) (1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this study, finite element method (FEM) using mathematical formulas that the induction motor magnetic vector potential and magnetic flux density changes in the solution are examined [1],[2],[9],[11]. For this, a program developed using Matlab programming language. When the shape obtained in the screening literatör obtained from this study data to other programming languages is made shorter than the command lines, and the degree of accuracy was higher than. Keywords. Induction Motor, Finite element Method, Matlab

References

[1] Chari, M.V.K., Silvister, P., 1970, “Finite Element Solution of Saturable Magentic Field Problems”, IEEE Transactions on Power Apparatus and Systems, Vol pas-89, No:7 (1642- 1650)

[2] Selçuk, A. H., 2003, “Lineer Asenkron Motorlarda Uç Etkilerinin Sonlu Elemanlar Yöntemiyle İncelenmesi”, Doktora Tezi, Fırat Üniversitesi, Elazığ.

[3] Silvester, P., Cabayan, H.S., Browne, B.T., 1973, “Efficient Techniques For Finite Element Analysis Of Electric Machines”, IEEE PES Winter Meeting, New York.

[4] Freeman, E. M., Lawther, D. A., 1973, “ Normal Force In Single Sided Linear Induction Motors”, Proc. Iee, Vol.120, No.12, Dec

[5] Chari, M.V.K., 1973, “Finite Element Solution Of The Eddy Current Problem In Magnetic Structures” IEEE PES Summer Meeting And EHV/UHV Conference, Vancouver, B.C. Canada.

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[6] Demirchian, K. S., Chechurin, V., Sarma, M. S., 1976, “Scalar Potential Concept For Calculating The Steady Magnetic Fields And Eddy Currents”, IEEE Trans. On Mag., Vol. MAG-12, No.6 Nov.

[7] I.., S.., Boldea, A.., Nasar, The Induction Machine Handbook, CRC Pres LLC, Washington D.C., 133 159, 2002.

[8]J.., J.., Cathey, Electric machines analysis and design applying matlab, Mc Graw Hill, Singapore, 317-420, 2001.

[9]. Hasan KÜRÜM, (1990) “Çift Yanlı Lineer Asenkron Motorların Sonlu Elemanlar Yöntemi İle Analizi”, Doktora Tezi, Fırat Üniversitesi, Elazığ.

[10]. KÜRÜM H., (2002) “ Bir Lineer Asenkron Motorun Çelik Sekonderinin Manyetik Özelliklerinin Matematiksel Olarak Modellenmesi”, F.Ü. Fen ve Müh. Bilimleri Dergisi, Elazığ

[11]. AYTAC KORKMAZ S. (2009) “ Sonlu elemanlar yöntemi ile asenkron motorun 3 boyutlu analizi ”

313 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Relationship Between Classical and Quantum Physic

Skender Ahmetaj (1), Skender Kabashi (2) and Sadik Bekteshi (3) (1) University of Prishtina, Kosovo, [email protected] (2) University of Prishtina, Kosovo, [email protected] (3) University of Prishtina, Kosovo, [email protected]

Abstract. The subject of this work is the relationship between classical relativistic physics and the quantum physics. Keywords. Dirac’s equation, Quantum Physics, Schrödinger’s equation. AMS 2010. 70S99, 81T99.

Modeling of gas flow through a rectangular channel (variable leak valve)

Sefer Avdiaj(1), Naim Syla(2) and Fisnik Aliaj(3) (1)University of Prishtina, Prishtina, Kosova, [email protected] (2)University of Prishtina, Prishtina, Kosova, [email protected] (3)University Prishtina, Prishtina, Kosova, [email protected]

Abstract. Gas flow through a bounded space varies obviously with the density of the gas. Knudsen was among the first to investigate, both theoretically and experimentally, the case wherein the gas is rarefied to the extent that the average distance traveled by each molecule between collisions with other molecules is of the same order as, or greater than, the lateral dimensions of the flow channel. If the gas is flowing through a pipe of characteristic diameter D and the mean free path of the molecules is λ , the ratio λ / D will determine what is called flow regime. The examples of applications of continuum and kinetic approaches which are available for the simulation especially in transition regime together with conditions of their applications will be given. Keywords. Gas flow, Knudsen number, flow regimes. AMS 2010. 76P05, 82C40.

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Modelling Kosovo’s Power System and Scenarios for Sustainable Development Sadik Bekteshi (1), Skender Kabashi(2) and Skender Ahmetaj(3) (1) University of Prishtina, Kosovo, [email protected] (2) University of Prishtina, Kosovo, [email protected] (3) University of Prishtina, Kosovo, [email protected]

Kosovo’s energy system depends mainly on the electric energy sector, where 95-97% is produced by the lignite power plants which are also one of the largest sources of air pollution. So, for studying of possible baseline energy developments and available options to mitigate emissions, we developed an integrated energy supply–demand and emission model (ESDE model) for time period 2010-2025 with user defined variables as: population, electricity demand, electricity generation, penetration of renewable energy etc. This model was constructed in the STELLA program, which makes use of Systems Dynamics Modelling as a methodology. It consists of four sectors: residential sector, industrial sector, service sector and electric power generation sector. These sectors are interrelated with a number of equations and assumptions. In various scenarios the potential for growth of the renewable energy sector and its integration into the energy system of Kosovo is investigated. The analysis of scenarios shows that there exists a large potential to reduce emissions compared to the business-as-usual (BAU) scenario by 10%. We find that orientation on environmentally friendly energy sources would pose near-term costs that are relatively modest compared to their contribution to sustainable energy development of Kosovo. Keywords. Power system, modelling, scenarios, sustainable development. AMS 2010. 00A69, 91B76

References

[1] The energy strategy of Republic of Kosovo 2009– 2018, MEM (Ministry of Energy and Mining) https://www.ks-gov.net/mem

[2] MEM (Ministry of Energy and Mining) Forecast of energy demand in Kosovo for the period 2007 –2016 https://www.ks-gov.net/mem

[3] STELLA. Copyright©1985-2009 by isee systems, inc. www.iseesystems.com

[4] Forrester, J. W. Dynamic models of economic systems and industrial organizations (archive paper from 1956). System Dynamics Review, 19(4):331–345, 2003.

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[5] Sterman, J. D. Business Dynamics: Systems Thinking and Modeling for a Complex World. McGraw-Hill, Boston, MA, 2000.

[6] Maani, K. E and R. Y. Cavana. Systems Thinking and Modeling: Understanding Change and Complexity.Prentice Hall, Auckland, N.Z., 2000.

[7] Gjonca A: “Demography of Kosovo Before the War,” 1999

[8] World Bank: “2001 World Development Indicators (WDI) database : Population dynamics”

[9] ESTAP (Energy Sector Technical Assistance Project) Kosovo, World Bank Grant No.TF- 027791

[10] Transmission, System and Market Operator-KOSTT ‘Long Term Energy Balance (2009 – 2018)

[11] IPCC, Climate Change 2001 The Scientific Basis, Cambridge University Press, Cambridge. 2001. http://www.ipcc.ch/

[12] Bekteshi, S: Kabashi, S; Slaus, I; Zidansek, A; Najdovski, D, Modelling rapid climate changes and analyzing their impacts, Management of Environmental Quality. An International Journal, 19, 422-432, 2008.

[13] Zidanšek, A.; Blinc,R.; Jeglič,A.; Kabashi,S; Bekteshi,S.; Šlaus,I. Climate changes, biofuels and the sustainable future. International Journal of Hydrogen Energy, doi: 10.1016/j.ijhydene.2008

[14] Ministry of Economic Development - Appendices to ‘Study on the Potential for Climate Change Combating in Power Generation in the Energy Community– South East Europe Consultants, Ltd. Prishtina, 2011, http://www.energycommunity.org/pls/portal/docs/1006177.PDF

[15] Wei, M., Patadia, S. and Kammen, D. M. "Putting renewables and energy efficiency to work: How many jobs can the clean energy industry generate in the U. S.?" Energy Policy, 38, 919 - 931, 2010

[16] Kammen M. D., Mozafari M. and Prull D., Sustainable Energy Options for Kosovo An analysis of resource availability and cost, University of California, Berkeley. 2012

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The Specific Exponential Stability of Solutions of Linear Homogeneous Volterra Integro-differential Equation of Sixth Order S. Iskandarov Institute of Theoretical and Applied Mathematics of National Academy of Sciences of Kyrgyz Republic, Bishkek, Kyrgyzstan, [email protected]

Abstract. All presented functions are continuous and relations is true ttt≥00,; ≥≥τ t IDE – integro-differential equation; DE – differential equation; under the exponential stability of solutions of linear homogeneous Volterra IDE of sixth-order mean tends to zero as t →∞ all its solutions and their derivatives up to fifth order inclusive, i.e. for each of its solutions xt() estimates: ()k −λ x( te )=t O (1) (λ >= 0; k 0,1,2,3, 4,5) . In this paper we study PROBLEM. To establish sufficient conditions for exponential stability of solutions of the IDE sixth order: x(6) () t− a () tx(5) () t + a () tx(4) () t + a () tx′′′ () t + a () tx ′′ () t +++ a () txt ′ () a () txt () 5 4 3 2 10 t 5 +=τ()k ττ ≥ ∫ ∑Qtk ( , ) x ( ) d 0, t t0 (1) k =0 t0 in the case of the condition:

at5 ( )≥ 0, ()a5 i.e. when any non-trivial solution of the corresponding linear homogeneous differential equation of sixth order: 4 (6) (5) ()k x() t−+ a50 () tx () t∑ ak () tx () t =≥ 0, t t (10) k =0 is not exponentially stable, as evidenced by Ostrogradskii-Liouville formula. These conditions are called specific.

Let 0,< pqkk- some auxiliary parameters; 0< Wtk ()- some of the weight functions (k = 1, 2) ; yt(), ut () - the new unknown functions. To solve this problem IDE sixth-order (1) by replacing:

x′′() t+ px11 ′ () t += qxt () W 1 () t yt (), (3)

y′′() t+ py22 ′ () t += qyt () W 2 () tut () (4) reduced to a system of two second-order DE (3), (4) and a second order IDE for the ut(), and to develop a method to this system of equations in the construction of a square (for the DE (3), (4)) and the method of cutting functions by transformations on scheme of ABC)))→→ the author. At the end we apply the lemma of Lusternik-Sobolev. Keywords. integro-ordinary differential equation, specific sufficient conditions, exponential stability. AMS 2010. 45 J05.

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Rendering Virtual Welding Seam Form Soydan Serttaş (1), Kayhan Ayar (2), Cemil Öz (3) and Gülüzar Cit (4) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected] (3) Sakarya University, Sakarya, Turkey, [email protected] (4) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. In this study, a tree-dimensional welding seam form designed for virtual welding simulator with using basic geometrical operations. Training of a welding operator is a long period work with high cost and exertion. To decrease the cost, welding simulators are being used. The most important part of welding simulators is the construction of realistic weld seam in real time. In this paper, a welding seam form that we improved for our welding simulator is explained with bead geometry. Welding seam that we developed, is seen fit by welding authorities, and they pointed out that this form reflects the real welding seam. Keywords. Virtual welding simulator, welding seam form, virtual reality.

References

[1] Ellis, S.R., Nature and Origin of Virtual Environments: A Bibliographic Essay, Computing Systems in Engineering, Cilt: 2, Volume: 4, 321-347, 1991.

[2] Astheir, P., Dai, Göbel, M., Kruse, R., Müller, S. ve Zachmann, G., Realism in Virtual Reality, in: (Magnenat Thalmann N and Thalmann D), Artificial Life and Virtual reality, John Wiley & Sons, New York, 189-209, 1994.

[3] Slater, M. ve Usoh, M., Body Centred Interaction in Immersive Virtual Environments, Artificial Life and Virtual Reality, N. Magnenat Thalmann and D. Thalmann, John Wiley& Sons, UK, 125-147, 1994.

[4] Wu, C., Microcomputer-based Welder Training Simulator, Computers in Industry, Cilt: 20, 321-325, 1992.

[5] Wu, C., Wen, C. ve Wu, L., A Microcomputer-Controlled Welder Training System, Computers Education, Cilt: 20, No: 3, 271-274, 1993.

[6] Heston, T., Virtually Welding, The Fabricator, FMA Puplication, March 2008.

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[7] Hamide, M., Masoni, E. ve Bellet, M.; Adaptive Mesh Technique for Thermal Metallurgical Numerical Simulation of Arc Welding Processes, Int. J. for Numerical Methods in Eng., Cilt: 73, 624-641, 2008.

[8] Palani, P. K. ve Murugan, N., Modeling and Simulation of Wire Feed Rate for Steady Current and Pulsed Current Gas Metal Arc Welding Using 317l Flux Cored Wire, Int. J. Adv. Manuf. Technol., Cilt: 34, 1111-1119, 2007.

[9] Top, Y., Simülasyon ve Temrinle Ark Kaynakçısı Yetiştirme Programı, Yüksek Lisans Tezi, Sakarya Üniversitesi, Haziran 1997.

[10] Top, Y. ve Findik, F., Ark Kaynakçısının Eğitiminde Simülatör Kullanımı, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Sayı:1, Cilt:2, sayfa:93-97, Sakarya, 1998.

[11] Denison, T. G., Arc Welding Simulator, US Patent No: 4.452.589, Jun. 5, 1984.

[12] Blair, B. A., Device for Teaching and Evaluating Person’s Skill as a Welder, US Patent No: 4.124.944, Nov.14, 1978.

[13] Paton, B. E., Vasiliev, V. V., Bogdanovsky, V. A., Danilyak, S. N., Gavva, V. M., Roiko, J.P., Nushko, V.A., Electric-Arc Trainer For Welders, US Patent No : 4.716.273, Dec. 29, 1987.

[14] Vasiliev, V. V., Sergei, N. D., Levina, A.I., Nushko, V.A., Roiko, J.P., Spark Trainer for Welders, US Patent No: 4.689.021, Aug.25, 1987.

[15] Schow, H. B., Welding Simulator Spot Designator System, US Patent No: 4.132.014, Jan 2, 1979.

[16] Mavrikios D, Karabatsou V, Fragos ve Chryssolouris, G., A Prototype Virtual Reality- Based Demonstrator for Immersive and Interactive Simulation of Welding Process, Int J Comput Integr Manuf, Cilt: 19, 294–300, 2006.

[17] Porter, N C., Cote, J. A., Gifford, T. D. ve Lam, Wim., Virtual Reality Welder Training, J Ship Prod, Cilt: 22, 126-138. 2005

[18] Donglin, L. ve Qiang, W., Several Key Technologies of the Computer Simulation of the Welding Process, Second International Symposium on Computational Intelligence and Design, ISCID, Cilt 1, 393-396, 2009.

[19] Fast K., Gifford T. ve Yancey R., Virtual Training for Welding, International Symposium on Mixed and Augmented Reality, 0-7695-2191-6/04. 2004.

[20] Oz, C., Findik, F., Iyibilgin, O., Soy, U., Kiyan, Y., Serttas, S., Ayar, K., Uslu, S., Yasar, Y., Geçmişten Günümüze Kaynak Simülatörleri, Metal Dünyası 201(2011), ISSN: 1305-3701, s 108-111.

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[21] Terrence L. Chambers, Amit Aglawe, Dirk Reiners, Steven White, Christoph W. Borst, Mores Prachyabrued, Abhishek Bajpayee, Real-time simulation for a virtual reality- based MIG welding training system, Virtual Reality, DOI 10.1007/s10055-010-0170-x

[22] Zeng Z, Wang L, Wang Y, Zhang H (2009) Numerical and experimental investigation on temperature distribution of the discontinuous welding. Comput Mater Sci 44(4):1153– 1162

[23] Wu CS, Zhang MX, Li KH, Zhang YM (2007) Numerical analysis of double-electrode gas metal arc welding process. Comput Mater Sci 39:416–423

[24] Lee CK, Candy J, Tan CPH (2004) Measurement and finite element analysis of temperature distribution in arc welding process. Int J Comput Appl Technol 21(4):171–177

[25] Chan B, Pacey J, Bibby M (1999) Modelling gas metal arc weld geometry using artificial neural network technology. Can Metall Q 38:43–51

[26] I.S. Kim, K.J. Son, Y.S. Yang, P.K.D.V. Yaragada, Sensitivity analysis for process parameters in GMA welding processes using a factorial design method, International Journal of Machine Tools & Manufacture 43 (2003) 763–769

[27] Harris, K. R., The Absolute Beginner's Guide to Direct3D, http://www.codesampler.com/d3dbook/chapter05/chapter_05.htm,2003, 09.09.2009

[28] Aleksas R., Approximation Of A Cubic Bezier Curve By Circular Arcs And Vice Versa, Information Technology and Control, Cilt: 35, Sayı:4, 2006.

[29] Yang U.,. Lee G. A., Kim Y., Jo D., Choi J., Kim K-H., Virtual Reality based Welding Training Simulator with 3D Multimodal Interaction, 2010 International Conference on Cyberworlds, p 150-154.

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The Relationships Among Interest Rate, Exchange Rate and Stock Price: A BEKK - MGARCH Approach Serpil Türkyılmaz (1) and Bengi Yıldız (2) (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]

Abstract. This paper employs a BEKK-MGARCH model approach to generate the conditional variances of monthly stock exchange prices, exchange rates and interest rates for Turkey. For the sample period 2002:M1-2009:M1, in which the changes is followed before global economic crisis effects, the results indicate significant transmission of shocks and volatility among three financial sectors. This finding points to presence of cross-market hedging and sharing of common information by investors. Keywords. BEKK-MGARCH Model, Volatility Transmission, Conditional Variance. AMS 2010. 91B84, 62M10.

References

[1] Bollerslev, T., 1986, “Generalized Autoregressive Conditional Heteroscedasticity”, Journal of Econometrics, 31, 307-327.

[2] Engle, R.F., 1982, “Autoregressive Conditional Heteroskedasticity with estimates of the variance of United Kingdom Inflation”, Econometrica, 50, 987-1007.

[3] Chen, H., Lobo, B.J. and Wong, W.K., 2005, “Links Between the Indian, U.S. and Chinese Stock Markets”, National University of Singapore, Department of Economics Working Paper, No.0602, 1-27.

[4] Glosten, L.R., Jagannathan, R. and Runkle, D., 1993, “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks”, Journal of Finance, 48, 1779-1801.

[5] McAleer, M., 2005, “Automated Inference and Learning in Modeling Financial Volatility”, Econometric Theory, 21, 232-261.

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[6] Nelson, D.B., 1991, “Conditional Heteroscedasticity in assets returns: A new approach, Econometrica, 55, 703-708.

[7] Zakoian, J.M., 1994, “Threshold Heteroscedastic Models”, Journal of Economic Dynamic and Control, 18, 931-955.

[8] Wei, C.C., 2008, “Multivariate GARCH Modeling Analysis of Unexpected U.S.D, Yen and Euro-dollar to Reminibi Volatility Spillover to Stock Markets, “Economics Bulletin”, vol.3, No.64, pp.1-15.

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The Causality Relationship Among Inflation, Output Growth and Their Uncertainties: Evidence for Turkey Serpil Türkyılmaz (1) and Mesut Balıbey (2) (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Tunceli University,Tunceli, Turkey, [email protected]

Abstract. This paper investigates the causality relationships among inflation, growth, inflation uncertainty and growth uncertainty in Turkey for the period of 2003:M01-2011:M12 by using monthly data. For this purpose, firstly, we employ EGARCH model to estimate time varying conditional variance of inflation and output growth, as a measures of inflation and growth uncertainties. Secondly, the causality relationships among these variables are analyzed by VEC Granger Causality Tests. Consequently, findings indicate that inflation leads to inflation uncertainty. This result supports Friedman(1977) and Ball(1992) hypothesis. In addition, inflation uncertainty affects growth in line with Friedman(1977) hypothesis. Furthermore, our estimation results show that growth uncertainy causes growth. This finding supports Pindyck(1991) hypothesis. However, our estimation results contradict the Cukierman-Meltzer(1986) and Holland(1995) hypothesis. Keywords. Inflation, Inflation Uncertainty, Output Growth, Output Uncertainty, EGARCH Modelling, VEC Granger Causality Test. AMS2010. 91B84, 62M10.

References

[1] Ball, L., 1992, Why Does Higher Inflation Raise Inflation Uncertainty, Journal of Monetary Economics, 29, 371-378.

[2] Berument, H. and Nergiz Dimcer,N.2005, Inflation and Inflation uncertainty inflation Uncertainty in the G-7 countries, Physica A, 348, 371-370.

[3] Black, F., 1987, Business cycles and equilibrium,New York:Basil Blackwell.

[4] Caporale, B. and Caporale, T., 2002, Asymmetric effects of inflation shocks on inflation uncertainty, AEJ,Vol.30,No.4,385-388.

[5] Caporale, T.and Mckierman, B., 1996, The Relationship between Output variability and growth: evidence from past war UK data, Scottish Journal of Political Economy, 43, 229-36.

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[6] Cukierman, A and A. Meltzer, 1986, A theory of ambiguity, credibility, and inflation under discretion and asymetric information, Econometica, 54,1099-1128.

[7] Deveraux, M., 1989, A positive theory of inflation and inflation variance, Economic Inquiry, 27:105-116.

[8] Friedman, M., 1977, Nobel Lecture:Inflation and Unemployment, Journal of Political Economy, 85/3, 451-72.

[9] Fountas,S., 2001, The relationship between inflation and inflation uncertainty in the UK :1885-1998, Economics Letters, 74,77-83.

[10] Fountas, S., and M. Karanasos, 2006, The relationship between economic growth and real uncertainty in the G3., Economic Modelling, Vol.23, 638-647.

[11] Fountas, S., and Karanasos, M. and Kim, J. 2002, Inflation and Output Growth Uncertainty and their relationship with inflation and Output Growth, Economic Letters, 75, 293-301.

[12] Heidari, H., and Bashiri, S., 2011, Revisiting the effects of Growth Uncertainty on İnflation in Iran:An Application of GARCH-in-mean models, International Journal of Business and Development Studies, Vol.3, No.1,123-140.

[13] Holland, A.S., 1995, Inflation and Uncertainty:Tests for Temporal Ordering, Journal of Money, Credit and Banking, 27/3, 827-37.

[14] Jiranyakul, K., 2011, The Link between Output Growth and Output Volatility in five Crisis-Affected Asian Countries, Middle Eastern Finance and Economics, ISSN:1450-2889 Issue 12, 101-108.

[15] Karanasos, M., and Kim, J., 2005, The Inflation output variability relationship in the G3: A Bivariate GARCH (BEKK) Approach, Risk Letters, Vol.1(2), 17-22.

[16] Korap, L., 2010, Threshold GARCH Modeling of the Inflation and Inflation Uncertainty relationship: Historical evidence from the Turkish Economy, MPRA (Munich Personal RePeC Archive, Paper No.31765, 1-22.

[17] Pindyck, R. (1991), Irreversibility, uncertainty and investment, Journal of Economic Literature, 29, pp. 1110-1148.

[18] Türkyılmaz, S. And Özer, M., 2010, MGARCH Modelling of the Relationships among Inflation, Output, Nominal and Real Uncertainty in Turkey, “MIBES Transactions International Journal”, Vol.4, Issue 1, 125-138.

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Bounds on the Largest Eigenvalue of the Distance Signless Laplacian of Connected Graphs Ş. Burcu Bozkurt (1) and Durmuş Bozkurt (2) (1) Selcuk University, Konya, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. The signless Laplacian for the distance matrix of a connected graph, called the distance signless Laplacian, was introduced in [1]. In this paper, we present some upper and lower bounds on the largest eigenvalue of the distance signless Laplacian of connected graphs. Keywords. Distance matrix, signless Laplacian, largest eigenvalue. AMS 2010. 05C12, 05C50.

References

[1] Aouchiche M., Hansen P., A signless Laplacian for the distance matrix of a graph, Les Cahiers du GERAD, G-2011-78, V+1-10, 2011.

[2] Buckley F., Harary F., Distance in graphs, Addison-Wesley, Red-wood, 1990.

[3] Shu J., Wu Y., Sharp upper bounds on the spectral radius of graphs, Linear Algebra Appl., 377, 241-248, 2004.

[4] Zhou B., Trinajstić N., Further results on the largest eigenvalues of the distance matrix and some distance-based matrices of connected (molecular) graphs, Internet Electron. J. Mol. Des., 6, 375-384, 2007.

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An Historical Overview of Visual Mathematical Arts Zeki TEZ Marmara University, Istanbul, Turkey, [email protected]

Abstract. Visual mathematical arts cover a wide range of subjects, including polyhedra, tiling/tesselation, impossible figures, Möbius-strips, art of anamorphosis and fractals. There are five types of convex polyhedron (“Platonic polyhedra”), each facet of which is made up of the same regular polygons, and thirteen types of convex semi-regular polyhedron (“Archimedean polyhedra”), comprised of two to three different regular polygons. The covering of a surface can be achieved by periodic or aperiodic tiling without leaving any space on it. One can come across the examples of periodic tiling in Alhambra Palace of Granada, the Alcazar of Seville and the shrine of Darb-ı Imam in Iran. The ornaments in Dar al-Shifa of Divriği and “Roll of Topkapı” are created by craftsman, known to have brought some patterned samples with them as they travelled from Iran and Horasan to Asia Minor. Various crystallographers, through examination of mathematical features of examples of ornamentation in Medieval Islamic Art, observed similarities between geometric patterns and crystal patterns. [1] Among the designers who distinguish themselves from their contemporaries of the 19th century are Owen Jones, Prisse d’Avennes, Christopher Dresser and William Morris. In tiling of a plane without any space, only an equilateral triangle, a square, and a regular hexagon can be used. While Roger Penrose studied aperiodic and periodic sets of tile, it was his father, L. S. Penrose, who created “Penrose’s Staircase” [2] Maurits Cornelis Escher took a lively interest in various mathematical subjects, such as tile patterns, non-Euclidean geometry and visual games, and inspired by the studies of graphic artist S. J. de Mesquita, geometrician H. S. Coxeter and mathematician G. Pólya, he majored in symmetry. While René Magritte, the surrealist artist, worked on illusional impossible objects, Victor Vasarely, having reached a poly-chrome abstraction by means of various geometric shapes, pioneered the “Optical Art” (“Op-Art”) that typically favors illusion.

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“Möbius strips”, a one-sided, non-orientable abstract surface, is named after August Ferdinand Möbius. A robust sample of “Möbius strips” can be obtained when one end of a flat strip is twisted 180o (half turn) and adhered to the other end. There may exist two or three “vanishing points” at twisted and unusual perspective systems. Such a system is known as “art of anamorphosis” or “twisted/distorted perspective”. With mirror-anamorphosis, a cylindrical or a conical mirror is placed on the object to be drawn to transform a flat distorted image. An outstanding example of perspective- anamorphosis is the skull image in the “The Ambassadors” (1533), a painting by Hans Holbein the Younger. If a piece were cut off a brocoli and magnified, its every cutting would seem to be a copy of a whole. The concept of “fractal” was first suggested by Benôit B. Mandelbrot in 1975, in his search for an answer to the question “How long is the coast of Britain?” [3]. Key Words. Visual mathematical arts, polyhedra, tiling AMS 2010. 97A30 Acknowledge. This study was supported by Marmara University-BABKO-Turkey with Project No. FEN-D-130612-0231

References

[1] U. Ilktürk, “An Attempt to Combine Mathematics and Visual Arts – A Research on Islamic Geometric Patterns”, Master Thesis, IT University of Göteborg / Chalmers University of Technology and University of Gothenburg, Göteburg, Sweden (2008).

[2] M. Arık, M. Sancak, Pentapleks Kaplamalar, TÜBİTAK, Ankara (2007).

[3] http://im-possible.info/english/articles/vis_math_art/

327 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Processing of Non-determined Results of Observations with Interval Interpolation Polynoms Ziyavidin YuldasheV (1), Alimzhan Ibragimov (2) and Shukhrat Tadjibaev (3) (2) National University of Uzbekistan, Tashkent, Uzbekistan, [email protected]

Abstract. In the given work the third method of research of measures of indeterminate form of parametres of installations - a method of an interval sizing up at which use it is considered is observed that not determined parametres have the restricted amplitude of oscillation. The problem of an interpolating within the limits of the interval analysis [1] in the assumption is particularly observed that base interval arithmetics can be any. In-process [2] problem of interpolation is observed in classical interval arithmetics. Thus the method of an inclusion of operation of subtraction of intervals is applied to warranting of a condition of 2 interpolation from  in space  . Research of the formulated problems is conducted, numerical experiments are made. The built machine alternatives of interpolational interval polynomials are issued in the form of a complete set of modules with the matching interface of expansive system [3]. Scaling of values of interval interpolation formulae in points or an interpolating x ∈ I([ ab , ]) is made for any point by system [4] by an overloading of matching interval operations of suitable alternative of interval arithmetics. Keywords. interval interpolation, interval extension. AMS 2010. 53A40, 20M15.

References

[1] Moore, R.E., Interval Analysis, Englewood Cliffs. N.J.: Prentice-Hall, 1966. [2] Kalmykov, S.А., Shokin, Yu.I., Yuldashev, Z.Kh., Methods of interval analysis, Nauka, Novasibirsk, 1986. [3] Yuldashev, Z.Kh., Ibragimov, A.A., Kalhanov, P.Zh. A package of interval algorithms for the wide user. It is registered in the state catalogue of Republic Uzbekistan, №DGU 02201, Tashkent, 5/19/2011. [4] Yuldashev, Z.Kh., Ibragimov, A.A., Kalhanov, P.Zh. A complex of programs for scaling of values of interval algebraicly admissible expressions within the limits of various interval arithmetics. It is registered in the state catalogue of Republic Uzbekistan, №DGU 02202, Tashkent, 5/19/2011.

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POSTER

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Generalized Incomplete Pochhammer Symbols and Their Applications to Exponential Functions Ayşegül Çetinkaya (1) and Onur Kıymaz (2) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In [9], the incomplete Pochhammer symbols and their applications to a class of hypergeometric and other related functions are given. In this paper, we introduce the generalized incomplete Pochhammer symbols by means of generalized incomplete Gamma functions. With the help of these, a generalization of incomplete exponential functions and some properties such as integral representations are given. Keywords. Incomplete Gamma Functions, Incomplete Pochhammer Symbols, Generalized Incomplete Gamma Functions, Generalized Incomplete Pochhammer Symbols. AMS 2010. 33B15, 33B20, 33C20.

References

[1] Chaudhry, M. A., Zubair, S. M., Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55, 99–124, 1994.

[2] Chaudhry, M. A., Zubair, S. M., On a Class of incomplete gamma functions with applications, Chapman & Hall/CRC, Boca Raton, 2002.

[3] Chaudhry, M. A., Zubair, S. M., Extended incomplete gamma functions with applications, J. Math. Anal. Appl., 274, 725–745, 2002.

[4] Chaudhry, M. A., Qadir, A., Srivastava, H. M., Paris, R.B., Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159, 589–602, 2004.

[5] Chaudhry, M. A., Qadir, A., Incomplete exponential and hypergeometric functions with applications to non-central χ2-distribution, Comm. Statist. Theory Methods, 34, 525–535, 2005.

[6] Miller, A. R., Moskowitz, I. S., On certain generalized incomplete gamma functions, J. Comput. Appl. Math., 91, 179–190, 1998.

329 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

[7] Miller, A. R., Reductions of a generalized incomplete gamma function Lamé de Fériet functions and incomplete integrals, Rocky Mountain J. Math., 30, 703–714, 2000.

[8] Nadarajah, S., A note on incomplete exponential functions, Arab. J. Sci. Eng., 32, 223– 224, 2007.

[9] Srivastava, H. M., Chaudhry, M. A., Agarwal, R. P., The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms and Special Functions, 1–25, 2011.

[10] Zubair, S.M., Chaudhry, M. A., Heat conduction in semi-infinite solid subject to time dependent surface heat fluxes: An analytic study, Wärme-und-Stoffübertragung, 28, 357–364, 1993.

[11] Zubair, S.M., Chaudhry, M. A., Temperature solutions due to steady and non-steady, movingpoint-heat sources in an infinite medium, Internat. Comm. Heat Mass Transfer, 21, 207–215, 1994.

[12] Zubair, S.M., Chaudhry, M. A., Temperature solutions due to time-dependent moving- heat-line sources, Heat Mass Transfer, 31, 185–189, 1996.

[13] Zubair, S.M., Chaudhry, M. A., Non-quasi-steady analysis of heat conduction from a moving heat source, Discussion note, ASME, J. Heat Transfer, 118, 511–512, 1996.

330 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Solution of the Radial Schrödinger Equation for the Potential Family A B ()rV ++−= rDrC 2 r 2 r A. Güleroğlu (1), C. Dane (2) and H. Akbaş (3) (1) Trakya University, Edirne, Turkey, [email protected] (2) Trakya University, Edirne, Turkey, [email protected] (3) Trakya University, Edirne, Turkey, [email protected]

Abstract. Using the exact analytical solutions of the radial Schrödinger equation with A B ()rV +−= rC κ potential family for the values of κ −= 1,0 and − 2 by asymptotic r 2 r iteration method, we obtained an analytical solution of the radial Schrödinger equation for the A B potential family ()rV ++−= rDrC 2 by using series expansion [1], [2], [3], [4] ,[5]. r 2 r Keywords. Asymptotic iteration method, eigenfunctions, analytical solution.

References

[1] Ciftci, H., Hall, R. L., Saad, N., Asymptotic iteration method for eigenvalue problems, J. Phys. A: Math. Gen., 36, 11807-11816, 2003.

[2] Barakat, T., The asymptotic iteration method for the eigenenergies of the Schrödinger equation with the potential V(r) = −Z/r + gr + λr2, J. Phys. A Math. Gen., 39, 823-831, 2006.

[3] Bayrak, O., Boztosun, I., Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method , J. Phys. A Math. Gen., 39, 6955-6963, 2006.

[4] Aygun, M., Bayrak, O., Boztosun, I., Solution of the radial Schrödinger equation for the A B potential family ()rV +−= rC κ using the asymptotic iteration method, J. Phys. B At. r 2 r Mol. Opt. Phys., 40, 537-544, 2007.

[5] Besis, D., Vrscay, E. R., Handy, C.R., Hydrogenic atoms in the external potential V(r)=gr+λr2: exact solutions and ground-state eigenvalue bounds using moment methods, J. Phys. A: Math. Gen., 20, 419-428, 1987.

331 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Modeling of Concentrators Influence on Stress Condition of Elastic-plastic Structures A.M. Polatov National University of Uzbekistan, Tashkent, Uzbekistan, [email protected]

Abstract. Using of mathematical modeling for solving of practical problems and conducting numerical experiments [1] allows to estimate strength of structural elements characteristics with various parameters for reliable structural elements resources assessment. Majority of technical and engineering problems in construction, aerodynamics, nuclear energy and space technologies are connected with problems of strength and reliability determination of structures’ and machines’ carrying elements [2]. This is connected with realization computing experiments and with research of stress - deformation elements condition, for saving of used materials structures strength retention. Elastic - plastic conditions of constructions with one, two and three vertically located circular cavities is considered [3]. Influence on stress - deformation structure structure’s condition and concentrators’ mutual influence studied [4]. Also influences of straight-line cracks and cracks with rounded tops are analyzed. The results presented in tables and figures. Keywords. Modeling, stress-deformation, influence of cavities. AMS 2010. 53A40, 20M15.

References

[1] Popov, Yu.P., Samarskiy, A.A.. Computing experiments, Computers, models and computing experiments Journal, Moscow, Nauka, 1988, 18-78 p.

[2] Zenkevich, O., Final elements methods in technics, Moscow, Mir, 1975, 541 p.

[3] Ilyushin, A.A., Flexibility, Moscow, Gostekhizdat, 1948, 376 p.

[4] Neyber, G., Stress concentration, OGIZ, Gosttekhizdat, 1947, 204 p.

332 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Special Solution of Schrödinger Equation with by Symmetries C. Dane (1), K. Kasapoğlu (2) and H. Akbaş (3) (1) Trakya University, Edirne, Turkey, [email protected] (2) Trakya University, Edirne, Turkey. [email protected] (3) Trakya University, Edirne, Turkey, [email protected]

Abstract. The one dimensional time-independent Schrödinger equation

2  2 ψ +∇− = ψψ xExxVx )()()()( was transformed into equation ′′ yxcy =+ 0)( . 2m Symmetries of ′′ yxcy =+ 0)( were found for 2 nxnxc =−+= ,...2,1,0,)12()( and

− 41 α 2 nxc α 224)( x 2 −−++= n α == ,...2,1,0,....2,1, , respectively. 4x 2 By using these symmetries, the special analytic solution of the equation ′′ yxcy =+ 0)( was obtained under the some conditions [1], [2], [3], [4], [5]. Keywords. Schrödinger equation, symmetry, analytical solution.

References

[1] Stephani, H., Differential equations Their Solution using Symmetries, Cambridge University Pres, New York, 1999.

[2] Polyanin, A.D., Zaitge, V.F., Handbook of Exact solution for ordinary differential equations, Chapman and Hall/CRC, Washington D.C.,2003.

[3] Zwillinger, D., Handbook of Differential Equations, Academic Pres, San Diego,1989.

[4] Abramowitz, M.,Stegun, I.A, Handbook of Mathematical Functions With formulas Graphs, and Mathematical Tables,1964.

[5] Gradshteyn, I.S., Ryzhik, I.M., Table of integrals, series, and product, Academic Pres, INC., New York, 1980.

333 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

An Analysis of Blood flow in the Human Descending Aorta with Different Reynolds Number Dilek Pandır (1) and Yusuf Pandır (2) (1) Bozok University, Yozgat, Turkey, [email protected] (2) Bozok University, Yozgat, Turkey, [email protected]

Abstract. Blood flow through the aorta is one of the most complex flow situations found in the cardiovascular system. A property of flow transition in the mammalian aorta is generally characterized by Reynolds number Re [1-4]. From the point of fluid mechanics, the effect of Reynolds numbers are observed where Reynolds number change, the reverse flow in the primary flow, symmetrical and asymmetrical vortices in the secondary flow and streamline of flow increase in the descending aorta. Keywords. Reynolds number, Navier-Stokes Equation, aorta, stream lines. AMS 2010. 35C30, 76B47, 76B10.

References

[1] Li, J. K., Laminar and turbulent flow in the mammalian aorta: Reynolds number, Theor Biol., 7; 135, 409-414, 1988.

[2] Chen, J., Lu, X. Y., Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch. J. Biomech.39, 818-832, 2006.

[3] Johnston, B. M., Johnston, P.R., Corney, S., Kilpatrick, D., Non-Newtonian blood flow in human right coronary arteries: steady state simulations. J Biomech. 37, 709-720, 2004

[4] Taylor, C. A., Hughes, T. J. R., Zarins, C. K., Finite element modeling of blood flow in arteries. Comput. Methods Appl. Mech. Engrg. 158, 155-196, 1998.

334 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Ideals and Their Characterizatios in Factor Γ - Near – Rings Eduard Domi (1) and Islam Braja (2) (1) University”Aleksander Xhuvani", Elbasan, Albania, [email protected] (2) University”Aleksander Xhuvani", Elbasan, Albania, braja [email protected]

Abstract. The purpose of this paper is to provide a link between prime ideals and maximal ideals in gamma-near-rings through factor gamma-near -rings. Initially it is presented the concept of homomorphism of Γ-near-rings [3] and the factor Γ-near-ring is defined. Furthermore, we describe the theorem of epiomorphism and the theorem of izomorphism of the Γ-near-rings. These theorems will be used after we introduce in a natural way the concept of the prime ideal and maximal ideal [4], providing their characterizations as well as the link between these two types of ideals. Once the definition of factor near-ring M / I is given, where for its ideal I is given the link of gamma factor near-rings with maximal ideals by the assertion: The ideal of the Γ-near-rings M is the maximal ideal then and only then when M / I is simple Γ-near-rings. We make the characterization of these two ideals through factor gamma-near -rings by theorem: If (M, +, (⋅)Γ) is a Γ-near-ring such that for a γ ∈ Γ exists an element which is γ-unit, then every maximal ideal I of M is prime. Key words. Gamma-near-rings, gamma factor near-rings, prime and maximal ideals. Mathematics 2010. 16Y30

References

[1] Nobusawa, N., On a generalization of the ring theory, Osaka, J. Math., 18 (1964), 411 – 422.

[2] Pilz, G., Near-Rings. The Theory and Applications, New York, 1977.

[3] Satyanarayana, Bh., A Note on -near-rings, Indian J. Math. (B. M. Prasud Birth Centenary Commemoration volume) 41 (1999) 427 – 433.

[4] Satyanarayana, Bh., Contributions to Near-ring Theory, Bh. D. Thesis, Nagarjuna University, 1985.

335 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Numerical Solution of Some Dynamic Problems in Nanomechanics Elman Hazar (1) and Mustafa Kemal Cerrahoğlu (2) (1) Sakarya University, Adapazarı, Turkey, [email protected] (2) Sakarya University, Turkey, [email protected]

Abstract. In this study, a new method has been developed for one dimensional quasistatic probelem of Thomlinson monoatomic chain using the Newton-Raphson method. In this process, numerical solutions has been obtained and an algorithm has been set to determine the friction forces between the atoms in the chain. Various models has been taken into account in manomechanical level regarding to the number of atoms included and the approximated solutions have been attained by employing the equilibrium conditions for each models. Taking the Morse potential into account and considering the mutual interactions of atoms we have reached the following generalized equations:

   where, F R F L are the right and left part of the mutual interaction forces, F exp is called the i , i i reaction force produced by the moving atomic chain and affecting the fixed monoatomic chain, u u are the displacements of the atoms, β is the angle between the radius vector r i , p il il and the axis OX, D and α are the Morse constants to be determined. These equation system has been aproximately solved by using Mathematica 6 software and the solutions obtained with these two methods have been compared. Keywords: Nanomexanika, potentials, reaction force

References

[1] Tomlinson G.A. A molecular theory of friction .Mag..Series-1929.-P.935-939.

[2] Robbins M.O, Muser H. Computer simulations of Friction,Lubrication and Wear.Handbook of Modern Trybology (Bhushan B.,ed.)-CRC Press,2001.-43p.

[3] Rieth M. Nano-Engineering in Science and Technology.-New Jersey:WSP,2003.- 146p.Verlag, New-York, 1979.

336 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Inclined Curves in the Lightlike Cone E.Selcen Yakıcı, İsmail Gök, F.Nejat Ekmekci and Yusuf Yaylı Ankara University, Ankara, Turkey

Abstract. In [ 4 ], Liu characterized curves whose tangent vector field makes a constant

3 angle with a fixed vector in E1 . Also, he characterize some curves in the 2 − dimensional and 3 − dimensional lightlike cone. In this study, we consider inclined curves in the 2 − dimensional and 3 − dimensional lightlike cone. Thus, we confirm the characterization of helices in the 2 − dimensional lightlike cone and we generalize such curves 3 − dimensional lightlike cone by using their harmonic curvature functions and Darboux vector. Keywords. lightlike cone, inclined curves, harmonic curvature functions, darboux vector. AMS 2010. 53A40, 20M15.

References

[1] Ali, A., and Lopez, R., Slant helices in Euclidean 4 − space E 4 , arXiv: 0901.3324 [math.DG] 9 Jan 2009.

[ 2 ] Camcı, Ç., Ilarslan, K., Kula, L., Hacsalihoğlu, H., H., Harmonic curvatures and generalized helices in E n . Chaos, Solitons & Fractals (2007). doi:10.1016/j.chaos.2007.11.001.

[3] Gök, İ., Camc, Ç., Hacsalihoğlu, H., H., Vn − slant helices in Euclidean n − space , Math. Commun., Vol. 14, No. 2, pp. 317-329 (2009).

[ 4 ] Izumuya S., and Takeuchi, N., New special curves and developable surfaces, Turk. J. Math. Vol. 28, (2004), 153-163.

[5] Liu, H., Curves in the lightlike cone, Contributions to Algebra and Geometry, 45 (2004), No. 1, pp. 291-303.

[ 6 ] Liu, H., and Meng, Q., Representation Formulas of Curves in a Two- and Three-Dimensional Lightlike Cone, 59 (2011), pp. 437-451.

[ 7 ] O'Neill, B., Semi-Riemannian Geometry, Academic Press, NewYork 1983. Zbl 0531.53051.

337 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A Study on The Spherical Curves and Bertrand Curves in Minkowski 4 Spacetime Gül Güner(1), Nejat Ekmekci(2), Yasemin Sağıroğlu (3) (1) Karadeniz Technical University, Trabzon, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected] (3) Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. In [1], we gave a method for constructing Bertrand curves from the spherical curves in 3 dimensional Minkowski space. In this work, we generalized the method of constructing Bertrand curves to the curves in Minkowski 4 spacetime. Keywords. Bertrand curve, Minkowski space. AMS 2010. 53A35.

References

[1] Güner, G., Ekmekci, N., On the Spherical curves and Bertrand curves in Minkowski-3 space, J. Math. Comput. Sci. 2, 4, 898-906, 2012. [2] Çöken, A. C., Çiftçi Ü., On The Cartan Curvatures of a Null Curve in Minkowski Spacetime, Geometriae Dedicata, 114, 71-78, 2005. [3] Ferrandez, A., Gimenez, A., Lucas, P., Characterization of null curves in Lorentz- Minkowski spaces, Publicaciones de la RSME, 3, 221-226, 2001. [4] Liu, H., Curves in the Lightlike Cone, Contributions to Algebra and Geometry, 1, 291-303, 2004. [5] İlarslan, K., Nesovic, E., Some Characterizations of Null, Pseudo Null and Partially Null Rectifying Curves in Minkowski Space-Time, Taiwanese Journal of Mathematics, 5, 1035- 1044, 2008. [6] Fusho, T., Izumiya, S., Lightlike surfaces of spacelike curves in de Sitter 3-space, 2006. [7] Matsuda, H., Yorozu, S., Notes on Bertrand curves, Yokohama Mathematical Journal, 50, 41-58, 2003. [8] Yılmaz, S., Turgut, M., On the Differential Geometry of the Curves in Minkowski Space- time I, Int. J. Contemp. Math. Sciences, Vol. 3, no. 27, 1343 – 1349, 2008.

338 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Reconstruction in Education Hülya Bozyokuş Uludag University, Bursa, Turkey, [email protected]

Abstract. In Turkish society, the opinion which everybody accepts is that our education system should be changed. From the past to the now, the changes have been made by examining the European models. What do the youngs at High-school want? Or What do they motive with? The 120 students studying at the Vocational Scholl of Technical Sciences of Uludağ University have been applied questionnaires. Based on the solutions and propasals, it has been decided that the restructing is important and necessary. As a result, the purpose of the Vocational Schools is to educate the indivuduals who is creative, informed, specialized in his or her subject, developed. Keywords. reconstruction, reform in education, quality of education.

References

[1] 2000’li Yıllarda Lise Eğitimine Çağdaş Yaklaşımlar Sempozyumu, 8-9 Haziran (2002).

[2] Mesleki ve Teknik Eğitimde Öğretmen Eğitimi Uluslararası Konferansı, 22 Ocak (2004).

[3] Eğitimde Stratejik Planlama, “Makaleler”, Ankara, (2010).

[4] Doğan, H., A.Ü Eğitim Bilimleri Fakültesi, Eğitim Programları ve Öğretimi Bölümü.

[5] Mesleki ve Teknik Eğitimin Yeniden Yapılandırılması, Fırat Üniversitesi Haber Dergisi Ocak (2005).

339 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

State Estimation in Induction Motors via Block-Pulse Functions Hakan Kızmaz (1) and Saadettin Aksoy (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. Field-oriented control techniques can be classified as indirect or feedforward method. In the first case, the coordinate transformation is obtained via the slip speed which is indirectly determined using some parameters. In the second case, the flux components, which are determined by an observer, are used to obtained the mentioned transformation [1,2]. Consequently, estimation of the rotor flux components from terminal variables such as stator voltage current and rotor angular speed has been a major task in the theory and practice of the field-oriented control of the induction motors [1,2,3,4]. However, the rotor flux can not be directly measured in squirrel-cage induction motors. Therefore, it is required an estimator or state observer to provide the rotor flux. Block pulse functions (BPFs) are defined on the interval tT∈[0, ] and have the orthogonality property like the Walsh, Chebyshev and Legendre series [5,6,7]. BPFs, like well known Walsh functions, constitutes a complete set of orthogonal basis functions and appeared to have application various of fields such as design and analysis linear and nonlinear systems, control theory and estimation problems [8,9]. In this paper, a simple algorithm that uses input-output measurements such as stator voltage, current and rotor angular speed for the simultaneous estimation of the rotor flux components of induction motors is presented. The proposed algorithm, which is simple in form and convenient for computer uses is based on the Block Pulse Functions (BPFs) and consist of three steps. The algorithm consists of three steps. In the first step, the feedback gain matrix G is a function of speed and it should be determined for each different speed measurement values. In the second step, stator voltage and current measurements and unknown state vector are expressed in a block-pulse series. Thus, we approximate the continuous-time functions with a set of piecewise constant functions. In this approximation, the total numbers of steps in the interval [0,T ], m can be chosen as any positive integer number. Finally, in the last step, the observer state equations are converted into integral equations by integrating the terms on either side of the equations. Then, unknown state estimation vector together with the block-pulse function approximation of known stator

340 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

voltage and current measurement vectors are substituted in the integral equation. After some algebraic manipulations, state observer equation is transformed into a computationally convenient algebraic form whose solution can be obtained easily by a computer program. Consequently, resulting this recursive solution unknown state vector is easily calculated. The proposed algorithm has been implemented in MATLAB and it has been applied to a squirrel-cage induction motor which was fed from various supply sources such as sinusoidal, six-steps and PWM waveforms at different operation conditions. Results obtained by proposed algorithm are in harmony with the real results. Keywords. Block-Pulse Functions, Observers, State Estimation, Induction Machine.

References

[1] B. K. BOSE, Modern Power Electronics And AC Drives. USA: Prentice-Hall, 2002. [2] G. C. VERGHESE and S. R. SANDERS, "Observer For Flux Estimation In Induction Machines," IEEE Transactions On Industry Electronics, vol. 35, no. 1, February 1988.

[3] Jingchuan LI, Longya XU, and Zheng ZHANG, "An Adaptive Sliding-Mode Observer For Induction Motor Sensorless Speed Control," IEEE Transactions On Industry Applications, vol. 41, no. 4, July/August 2005.

[4] M. G. SIMOES and B. K. BOSE, "Neural Network Based Estimation Of Feedback Signals For A Vector Controlled Induction Motor Drive," IEEE Transactions On Industry Applications, vol. 31, no. 3, pp. 620-629, May/June 1995.

[5] K. R. PALASINAMY and D. K. BHATTACHARYA, "System identifications via block- pulse functions," International Journal of System Science, vol. 12, no. 5, pp. 643-647, 1981, DOI: 10.1080/00207728108963772.

[6] G. SANSONE, Orthogonal Functions. New York: Interscience Publishers, 1991.

[7] P. STAVROULAKIS and S. TZAFESTAS, "Walsh Series Approach To Observer an Filter Design In Optimal Control Systems," International Journal of Control, vol. 26, no. 5, pp. 721-736, 1977.

[8] Murlan S. CORRINGTON, "Solution of Differential and Integral Equations with Walsh Functions," IEEE Transactions On Circuit Theory, vol. CT-20, no. 5, pp. 470-476, September 1973, DOI: 10.1109/TCT.1973.1083748.

[9] N. GOPALSAMI and B. DEEKSHATULU, "Comments On Design of piecewise constant gains for optimal control via Walsh functions," IEEE Transactions On Automatic Control, vol. 21, no. 4, pp. 634-636, August 1976, DOI: 10.1109/TAC.1976.1101253.

341 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Geodesics on the Tangent Sphere Bundle of the 3-sphere Ismet Ayhan Pamukkale University, Denizli, Turkey, [email protected]

Abstract. In this study, the Sasaki Riemann metric g S on the unit tangent sphere

3 3 bundle 1ST of the unit 3-sphere S is obtained with respect to the geodesic polar coordinate of the 3-sphere S 3 . The coefficients of the Levi-Civita connection of the Sasaki Riemann

3 S manifold on tangent sphere bundle 1 gST ),( are found. Moreover, the differential equations

3 S of geodesics of 1 gST ),( is calculated. Keywords. The tangent sphere bundle of the 3-sphere, Geodesics of the Sasaki Riemann metric. AMS 2010. 55R25, 53C25.

References

[1] A.L.Yampolski, The curvature of the sasaki metric of tangent sphere bundles, Geometric Sbornic, 28, 132-145, 1985.

[2] Klingenberg, W., and Sasaki, S., On the tangent sphere bundle of a 2-sphere. Tohuku Math. Journ. 27, 49-56, 1975.

[3] Nagy, P.T., On the tangent sphere bundle of a Riemannian 2-manifold. Tohuku Math. Journ. 29, 203-208, 1977.

[4] Nagy, P.T., Geodesics on the tangent sphere bundle of a Riemann manifold, Geometriae Dedicata 7, 233-243, 1978.

[5] O’Neill, B. Semi-Riemannian Geometry with applications to relativity. Academic Press, New York, 1997 [6] Sasaki, S., Geodesic on the tangent sphere bundles over spaces forms. Journ. Für die reine und angewandte math.288, 106-120, 1976.

342 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

One-Sided Similar Ideals in Γ-Semigroups and Some Properties of the Principal Quasi- Ideals Islam Braja (1) and Eduard Domi (2) (1) University”Aleksander Xhuvani", Elbasan, Albania, [email protected] (2) University”Aleksander Xhuvani", Elbasan, Albania, [email protected]

Abstract. In this paper are provide proofs of some propositions related to the properties of H-classes in a Γ –semigroup [1] and the properties of the generator classes of principal quasi-ideals [2]

Secondly, we give the definitions of the (left and right) similar ideals, where the left ideals L1 and L2 (also right ideals R1 and R2) of a Γ-semigroup are similar from the left (similar from the right) if exists a bijection λ from L1 to L2 (from R1 to R2) for which:

(1.1) for ∀ l ∈ L1 and m ∈ M, λ(mαl) = mαλ(l),

(1.2) for ∀ r ∈ R1 and m ∈ M, λ(rαm) = λ(r)αm] Finally, we state the condition for which to idempotent elements e = eαe, f = fαf (e, f ∈ M, α, β ∈ Γ) of a Γ-semigroup M are D-equivalent. Keywords. Γ-semigroup, left ideal (right ideal, double-sided ideal), principal quasi- ideal, similar ideals (from the left or from the right), idempotent elements, H-class, D-class, Γ– group.

References

[1] Sen, M.K.Saha,N.K:” On Γ-semigroup” I. Bull. Col. Math. Soc. 80, Saha, N. K., (1988), pp. 180-

[2] Steinfeld, O.:”Quasi-ideals in rings and semigroups” Akadenia Kido Budapest,

[3] Braja, I. “Characterizations of regular Γ-semigroups using quasi-ideals” Internacional Journal of Mathematical Analisis, vol.3, no. 33-36 ,2009.(Bord Editorial)

[4] Xhillari, Th.: “Quasi-idealet në -semigroupe”, BSHN, no. 5, (2008), 16 – 24.

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A Different Characterization of U1(ZC8) Kamil Arı (1) and Merve Görgülü (2) (1) Karamanoglu Mehmetbey University, Karaman, Turkey, [email protected] (2) Karamanoglu Mehmetbey University, Karaman, Turkey, [email protected]

Abstract. In this paper, torsion-free part of U1(ZC8) is characterized by using a different description. That is, unlike the previous works, a classical ring and number theory is used in this characterization. Keywords. Characterization, torsion-free unit AMS 2010. 16U60.

References

[1] Aleev, R.Zh., Panina, G.A., The units of cyclic groups of order 7 and 9, Russian Math., No:11, (2000), 80-83.

[2] Dickson, E.L., History of the Theory of Numbers, vol. II, Chelsea Publishing Company, New York, 1992.

[3] Karpilovsky, G., Unit group of group rings, Longman Scientific & Technical, Essex, England. 1989.

[4] Sehgal, S.K., Units in integral group rings, Longman Scientific & Technical, Essex, England, 1.

[5] Arı, Kamil, A Different Characterization of U1(ZC8) , International Journal of Applied Mathematics, Vol.12, No:2 109–113, 2003

344 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

The Dependence Described by Copulas in the Reinsurance Treaties Kleida Haxhi (1) and Oriana Zacaj (2) (1) Drejtore e Menaxhimit te Riskut, Intersig, Vienna Insurance Group, Tirane, Albania, [email protected] (2) Universiteti Politeknik i Tiranes, Tirane, Albania, [email protected]

Abstract. This paper analyzes the reinsurance of portfolios of dependent risks and in particular risk sharing between cedent and reinsurer expressed in terms both of the expected value and of the variance of the retained aggregate claim and the ceded aggregate claim. Usually the premiums principles used for sharing the gross premiums between the two companies are based on both such indicators. Such issue is developed for classical reinsurance treaties and for dependence structures of the risks that consider that consider scenarios from the independence to the comonotonicity. The dependence between random variables is described in terms of Copulas, which are multivariate distributions with uniformly distributed on the interval [0,1], under not restrictive conditions , for whatever multivariate distribution it can be obtained the associated Copula. The issue is analyzed in order to determine optimal combinations of various reinsurance treaties, according to a “mean –variance” optimization criterion. Keywords. Copulas, Reinsurance treaty, Dependent risks

References

[1] R. B. Nelsen, An introduction to Copulas, Springer, New York 1999 [2] J. Dhaene, M.J. Goovaerts, Dependency of risk and stop-loss order, Astin Bulletin, 1996 [3] H. Joe, Multivariate Models and Dependence Model Concepts, Chapman & Hall, London 1997 [4] M.J. Goovaerts, R. Kaas, A.E. Van Heerwaarden, Optimal reinsurance in relation to ordering of risks, Insurance: Mathematicsand Economics, 1989 [5] P. Embrechts, F. Klüppelberg, A. McNeil, Modelling Dependance with Copulas and Applications to Risk Management, ETHZ, Zürich [6] J. Dhaene, M. Denuit, M.J. Goovaerts, R. Kaas, D. Vyncke, The concept of comonotonicity in actuarial science and finance, Insurance: Mathematics and Economics, 2002

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Algebraic Hyperstructure of Soft Sets Associated to Ternary Semihypergroups

Kostaq Hila (1), Sabri Sadiku (2) and Krisanthi Naka (3) (1) University of Gjirokastra, Gjirokastra, Albania, [email protected] (2) University of Prishtina, Prishtina, Kosovo, [email protected], [email protected] (3) University of Gjirokastra, Gjirokastra, Albania, [email protected]

Abstract. Hyperstructure theory was introduced in 1934, at the eighth congress of Scandinavian Mathematicians, when F. Marty defined hypergroups based on the notion of hyperoperation. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians such as in fuzzy sets and rough set theory, cryptography, codes, automata, combinatorics, artificial intelligence, probability, graphs and hypergraphs, geometry, lattices and binary relations. Many complicated problems in economics, engineering, environment, social science, medical science and many other fields involve uncertain data. These problems which come face to face with in life cannot be solved using classical mathematic methods. There are several well- known theories to describe uncertainty. For instance fuzzy sets theory, rough sets theory and other mathematical tools. But all these theories have their inherited difficulties as pointed out by Molodtsov. In 1999, Molodtsov initiated the novel concept of soft sets as a new mathematical tool for dealing with uncertainties that is free from the difficulties affecting existing methods and which theory has rich potential for applications in several directions. Applications of Soft Set Theory in other disciplines and real life problems are now catching momentum. The algebraic structure of soft sets has been studied by several authors. In this paper we introduce and initate the study of the concept of soft ternary semihypergroups, soft ternary subsemihypergroups and soft left (right, lateral, quasi, bi-) hyperideals using soft sets and several related properties are investigated.

Keywords. soft ternary semihypergroup, soft hyperideal.

AMS 2010. 08A72, 20N20, 16Y99.

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Solution of Time-Independent One Dimensional Schrödinger Equation Using Symmetries K. Kasapoğlu (1), C. Dane (2) and H. Akbaş (3) (1) Trakya University, Edirne, Turkey, [email protected] (2) Trakya University, Edirne, Turkey, [email protected] (3) Trakya University, Edirne, Turkey, [email protected]

Abstract. Using the symmetry condition for the second order linear homogeneous Schrödinger equation, the symmetries of the one dimensional Schrödinger equation with potentials )( = nxV 2 and )( −= xxV have been determined. The symmetries were used to find analytical solutions [1], [2], [3], [4], [5]. Keywords. Analytical solution, symmetry, Schrödinger equation.

References

[1] Stephani, H., Differential equations Their Solution using Symmetries, Cambridge University Pres, New York, 1999.

[2] Polyanin, A.D., Zaitge, V.F., Handbook of Exact solution for ordinary differential equations, Chapman and Hall/CRC, Washington D.C.,2003.

[3] Zwillinger, D., Handbook of Differential Equations, Academic Pres, San Diego,1989.

[4] Abramowitz, M.,Stegun, I.A, Handbook of Mathematical Functions With formulas Graphs and Mathematical Tables,1964.

[5] Gradshteyn, I.S., Ryzhik, I.M., Table of integrals, series, and product, Academic Pres, INC., New York, 1980.

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Parameter Estimation Based on Type-II Fuzzy Logic Kamile Şanlı Kula (1) and Türkan Erbay Dalkılıç (2) (1) Ahi Evran University, Kırşehir, Turkey, [email protected] (2) Karadeniz Technical University, Trabzon, Turkey, [email protected]

Abstract. Regression analysis is an area of statistics that deals with the investigation of the dependence of a variable upon one or more variables. Recently, much research has studied fuzzy estimation. The fuzzy regression method can be used to obtain unknown parameters of regression models based fuzzy data. In this study we will use the ANFIS for parameter estimation and propose an algorithm in case where the independent variables are fuzzy sets. These sets are type-II fuzzy sets because of characterized by a Gaussian membership function whit fuzzy mean or fuzzy standard deviation. Keywords. Type-II fuzzy logic, membership function, parameter estimation. AMS 2010. 62A86

References

[1] Castillo, O., Melin, P. Type-2 fuzzy logic: theory and applications, Sipringer., 2008.

[2] Aisbett, J., Rickart, J. T., Morgenthaler, D., Multivariate modeling and type-2 fuzzy sets, Fuzzy Sets and Systems 163 (2011) 78-95.

[3] Mendel, J. M., Type-2 Fuzzy Sets and Systems: An Overview IEEE Computational Intelligence Magazine Fabuary, 2007.

[4] Mendel, J. M., Advances in type-2 fuzzy sets and systems, Information Sciences 177 (2007) 84-110.

This work is supported by Ahi Evran University BAP Project with Project number FBA-11- 20.

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Intrinsic Geometry of the NLS Equation According to Bishop Frame in Euclidean 3-Space Mahmut Ergüt (1), Handan Öztekin (1) and Sezin Aykurt (2) (1) Firat University, Elazig, Turkey, [email protected], [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In this study, we investigate a general intrinsic geometry according to Bishop frame in Euclidean 3-space. Furthermore, we obtain the NLS Equation by using intrinsic derivatives of orthonormal triad.

Keywords. Bishop frame, intrinsic geometry, the NLS Equation. AMS 2010. 83A05, 53B99.

References

[1] Rogers, C. and Schief, W. K., Intrinsic Geometry of the NLS Equation and Its Auto- Bäcklund Transformation, Studies in Applied Mathematics 101: 267--287, 1998.

[2] Gürbüz, N., Intrinsic Geometry of the NLS Equation and Heat System in 3-dimensional Minkowski Space, Adv. Studies Theor. Phys., Vol. 4, no. 11, 557-564, 2010.

[3] Bukcu, B., Karacan, K. K., The Slant Helices According to Bishop Frame, International Journal of Computational and Mathematical Sciences, 3:2, 63-66, 2009.

[4] Bukcu, B., Karacan, K. K., On Natural Curvatures of Bishop Frame, Journal of Vectorial Relativity, 5:4, 34-41, 2010.

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On the Mcshane Integral on the Riesz space Mimoza Shkëmbi (1) and Ismet Temaj (2) Elbasan University, Elbasan, Albania, [email protected] Prizren University, Prizren, Kosovo, [email protected]

Abstract. Recently, the intensive research on the integration on Riesz spaces, offer a very important tool in modern mathematics and have many practical applications, for example in economics. It is known that Kantorovich monograph in economy concern with both the theory of of the Kurzweil-Henstock integral and the basic facts on Riesz spaces. It is well-known that integration theory with values in ordered spaces cannot be reduced to the analogous theory for locally convex spaces. This fact justifies the main goal of this book: to investigate and develop a measure and integration theory of the Kurzweil-Henstock type for functions with values in ordered spaces. Afected by the work of Antonio Boccuto, Beloslav Riecan, Marta Vrábelová we studied the same problems for another important type of integration on such space as Mcshane integral. In this paper we present in other way the definition of Mcshane integral on the Riesz space using a very important lemma of famous Fremlin. We reconstruct almost all the propeties of Mchane integral given in [4] and these ones become a little more stronger. We give some new results in relation with Henstock- Kurzweil ones. In the second section we define the strong version of Mcshane integral and give the necessary and sufficent condition of this concept. Keywords. Riesz spaces, compact topological spaces, M- integral. AMS 2010. 28B15, 46G10.

Referenca

[1] Boccuto A., Riecan B.,Vrabelova M. Kurzweil-Henstock Integral in Riesz spaces. Bentham e Books, 2009.

[2] Boccuto, A., Skvortsov, V.A. Hestock-Kurzweil type integration of Riesz-space-valued functions and applications to Walsh series. real, Analysis Exchange, Vol 29(1), 2003/2004, pp.419-439.

[3] M. DUCCHOŇ - B.RIEČ AN,On the Kurzweil-Stieltjes integral in ordered spaces, Tatra Mountains Math. Publ., 8 (1996), 133-141.

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[4] Temaj, I., Tato, A., Mcshane type integration of Riesz-space- valued function and application to sandwich property, Botim i Akademise se Shkencave, 2011.

[5]Schaefer, H.H., Banach Lattices and positive Operators, Springer Verlag 1974

[6] Kurt, D.S., Swartz, C.W, Theories of integration, World Scientific,Vol. 9

351 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Developing Attitude Scale toward the Geometric Objects for the Preservice Teachers Nejla Gürefe (1) and Adnan Kan (2) (1) Gazi University, Ankara, Turkey, [email protected] (2) Gazi University, Ankara, Turkey, [email protected]

Abstract. One of the most important factors that affects on students in education is teacher [1]. The students of teachers having positive attitude toward their lesson will occur positive attitude [2,3], so they will be more successful [4]. For this reason, the attitude of teachers is so important. The attitude is not a phenomenon that may change easily. Therefore, when preservice teachers were on the university, to measure and detect attitudes of teachers’ is more significant mean. In this study, a measuring instrument that aims to measure preservice mathematics teachers’ attitude toward subject of geometric objects that is sub- learning area of geometry was developed. Questionnaire was administered to 306 university students to determine ıts reliability and validity. Exploratory and confirmatory factor analysis was done for construct validity. The findings of analysis showed that this instrument has three factors structure model. This model explains an important part of the variance (%60.016). Cronbach-Alfa and test re-test reliability coefficients were found for reliability. These findings show that this scale is valid and reliable. Keywords. Geometric object, attitude, scale development, validity, reliability.

References

[1] Stipek, D., Motivation to Learn: From Theory to Practice (3rd ed.). Boston: Allyn & Bacon, 1998.

[2] Aiken, L. R., Update on attitudes and other affective variables in learning mathematics, Rev. Educ. Res., 46(2), 293-311, 1976.

[3] Relich, J., Way, J., Martin A., Attitudes to teaching mathematics: further development of a measurement instrument, Math. Educ. Res. J., 6 (1), 56-69, 1994.

[4] Ceungh, K.C., Outcomes of schooling: mathematics achievement and attitudes towards mathematics learning in Hong Kong, Edu. Stud. Math., 19, 209-219, 1988.

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Modeling the RC Electric Circuit with Finite Element Method Naim Syla (1), Fisnik Aliaj (2) and Sefer Avdiaj (3) (1) Universiteti i Prishtinës, Prishtinë, Kosova, [email protected] (2) Universiteti i Prishtinës, Prishtinë, Kosova, [email protected] (3) Universiteti i Prishtinës, Prishtinë, Kosova, [email protected]

Abstract. The electric circuit is composed of the source, condenser and resistance. If the voltage between the condenser plates depends on the time, then the intensity of the electric current will be proportional to condenser capacity and the rate change of the voltage. In principle, this dependence in the case of discharge of a condenser can be represented by a differential equation, where the derivative member dU() t dt appears that expresses the rate change of the voltage between the capacitor plates. Analytic solution of this equation is a function U(t) of the exponential type. In this contribution, we have used the Finite Element Method (FEM) to solve for the differential equation. Software of choice to implement the method was the commercially available finite element program ANSYS. The results obtained from the ANSYS finite element program were compared with the analytical solution. It was shown that the results of the computer model gave similar results to the analytical solution. Keywords. Electric circuit, finite elements, voltage, ANSYS. AMS 2010. 74S05, 78A25.

353 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Extended Caputo Fractional Derivative and Its Applications Onur Kıymaz (1), Ayşegül Çetinkaya (2) and Yusuf Sökmen (3) (1) Ahi Evran University, Kirsehir, Turkey, [email protected] (2) Ahi Evran University, Kirsehir, Turkey, [email protected] (1) Ahi Evran University, Kirsehir, Turkey, [email protected]

Abstract. In [7], the authors give an extension of Riemann-Liouville fractional derivative with including an extra parameter. In this paper, inspired by the same idea, we introduced an extension of Caputo fractional derivative operator. The extended Caputo fractional derivatives of some elementary functions are also calculated. Keywords. Fractional derivative, Hypergeometric functions, Mellin transform. AMS 2010. 26A33, 33C05, 33C20.

Acknowledgement. This work was supported by the Scientific Research Projects Council of Ahi Evran University, Kirsehir, TURKEY under Grant FBA-11-17.

References

[1] Chaudhry, M. A., Zubair, S. M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55, 99-124, 1994.

[2] Chaudhry, M. A., Zubair, S. M., On the decomposition of generalized incomplete gamma functions with applications of Fourier transforms, J. Comput. Appl. Math., 59 (3), 253-284, 1995.

[3] Chaudhry, M. A., Temme, N. M., Veling, E.J.M., Asymptotic and closed form of a generalized incomplete gamma function, J. Comput. Appl. Math., 67, 371-379, 1996.

[4] Chaudhry, M. A., Qadir, A., Rafique, M., Zubair, S. M., Extension of Euler's beta function, J. Comput. Appl. Math., 78, 19-32, 1997.

[5] Chaudhry, M. A., Qadir, A., Srivastava, H. M., Paris, R. B., Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2), 589-602, 2004.

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[6] Miller, A.R., Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, Rocky Mountain J. Math., 30, 703-714, 2000.

[7] Özarslan, M. A., Özergin, E., Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Mathematical and Computer Modelling, 52, 1825-1833, 2010.

[8] Srivastava, H. M., Manocha, H. L., A Treatise on Generating Functions, Halsted, Ellis Horwood, Wiley, New York, Chicester, New York, 1984.

[9] Srivastava, H. M., Saxena, R. K., Operators of fractional integration and their applications, Appl. Math. Comput., 118, 1-52, 2001.

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Non-Symmetric Divisor Problem over the Ring of Gaussian Integers Olga Savastru Odessa I.I.Mechnikov National University, Odessa, Ukraine, [email protected]

Abstract. Let []i be the ring of Gaussian integers. For π xx∈, > 1,αγ , ∈ [ i ], 0 ≤<≤ ϕ ϕ 0 122 consider the summatory function given by

Tx(;,,,)γα0 ϕ 1 ϕ 2 = ∑ τ1,1,2 ( α ), αα≡ 0 (mod γ ) Nx()α ≤ ϕ<≤ αϕ 12arg τα α ∈ α= ααα2 where 1,1,2 ()denote the number of representations of []i as 123, where

ααα123,, are Gaussian integers. Applying the bound for the Kloosterman sum over []i , the method of Vinogradov we get the asymptotic formula in case, when the norm of a difference of progression grows.

Theorem. Let αγ0,∈ [iN ], ( γ ) >≡ 1, α 00/ 0(mod γ ), ( αγ , ) = β. Then for every

1 2 2+ε N ()γ ε > 0, xN≥ ()γ and ϕϕ21−≥ 1 −ε x 4

3 +ε ϕϕ− xx x x4 Tx(;γα , , ϕ , ϕ )=21c (,γα ) log++c (,γα ) , 01 2 0 0 1 0 O1 2πNN() γβ ( ) N () γ N 2 ()γ where cc0(,γα 01 ),(, γα 0 )are computable functions. Analogical problem studied for function τ ()n under the ring of rational integers by Krätzel, 1,1,2 Liu [1]. Keywords. Divisor function, gaussian numbers. AMS 2010. 11D37.

References

[1] Liu H.Q., Divisor problems of 4 and 3 dimensions, Acta Arith., 73, 247-269, 1995.

356 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Harmonic Curvature Function of Special Curves in Lie Groups Osman Zeki Okuyucu (1), İsmail Gök (2), Yusuf Yaylı (3) and F. Nejat Ekmekci (4) (1) Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] (2) Ankara University, Ankara, Turkey, [email protected] (3) Ankara University, Ankara, Turkey, [email protected] (4) Ankara University, Ankara, Turkey, [email protected]

Abstract. In this paper, we define harmonic curvature of a curve in three dimensional Lie groups with a bi-invariant metric. Then, we give some characterizations for special curves in three dimensional Lie groups with a bi-invariant metric with the help of the harmonic curvature. Keywords. Harmonic curvature, special curves, Lie groups. AMS 2010. 14Q05, 22E15.

References

[1] M. A. Lancret, Mémoire sur les courbes à double courbure, Mémoires présentés à l'Institut1 (1806) 416-454.

[2] A. C. Çöken, Ü. Çiftçi, A note on the geometry of Lie groups, Nonlinear Analysis TMA 68 (2008) 2013-2016.

[3] Ü. Çiftçi, A generalization of Lancert's theorem, J. Geom. Phys. 59 (2009) 1597-1603.

[4] S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk. J. Math 28 (2004), 153-163.

[5] L. Kula, N. Ekmekci, Y. Yayl and K. İlarslan, Characterizations of slant helices in Euclidean 3-space, Turk. J. Math. 34 (2) (2010) 261-273.

[6] O. Zeki Okuyucu, İ. Gök, Y. Yaylı and N. Ekmekci, Slant Helices in Three Dimensional Lie Groups, Submitted to publish.

[7] N. Ekmekci and K. İlarslan, On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.

[8] A. Görgülü and E. Özdamar, A Generalization of the Bertrand curves as General Inclined n curves in E , Commun. Fac. Sci. Univ. Ank., Series A1 V.35, pp. 53-60 (1986).

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Applications to Markov Chains: Introducing a BONUS-MALUS Model for the MTPL Portfolio in Albania Oriana Zacaj (1) and Kleida Haxhi (2) (1) Universiteti Politeknik i Tiranes , Tirane, Albania, [email protected] (2) Drejtor i Menaxhimit te Riskut, Intersig, Vienna Insurance group, Tirane, Albania, [email protected]

Abstract. A tariff structure takes account of the various risk attitudes of the policyholders by partitioning them into homogenous classes, namely the tariff classes. The insureds belonging to the same class pay the same premium. The partition into tariff classes is introduced by means of a Bonus-Malus system. In other countries, the Bonus-Malus system is widely used for both compulsory and voluntary insurance. However, in Albania, this system is not totaly implemented in the voluntary insurance and not at all in the compulsory insurance However, the liberalization of the insurance premium tariffs for compulsory motor third party liabilities, soon Bonus Malus system will play an important part in ratemaking for this type of insurance, which holds over 60% of the insurance market Until last year the tariffs of premium fort he MTPL portfolio in Albania were set by the government and all the insurance companies should deal with the same tarifs. But even now that those tariffs are liberalized, still there is not any bonus – malus system offered from any of the insurance companies operating in Albania. With the improvement of the claim database as a support for the application of a Bonus – Malus model, we offer a model which might be appropriate for this case The problem of determining the premium scale when a Bonus-Malus system is superimposed on the premium ratemaking, is dealt with in the actuarial literature according to different approaches. In this paper, we compare two of these approaches with the aim of enlightening that they meet two conceptually different criteria: a financial equilibrium criterion and a minimization criterion. An additional suggestion, still based on a minimization criterion, is proposed. Analytical and numerical results are developed and discussed. Key words. Experience rating, Bonus-Malus systems, Poisson-gamma processes, Stationary distribution, Premium ratemaking, Bonus-Malus scales.

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References

[1] Jean Lemaire, Automobile Insurance, Actuarial Models, 1985.

[2] YIU-KUEN TSE, Nonlife Actuarial Models Theory, Methods and Evaluation, 2009

[3] M. H DeGroot,. M. J. Schervish, Probability and Statistics, 3rd edition, 2002

[4] R.V. Hogg, ,A.T. Craig, Introduction to Mathematical Statistics, 5th edition, 1995

[5] N. L. Johnson, S. Kotz, Distributions in Statistics: Discrete Distributions, 1969

[6] N. L. Johnson, S. Kotz, Distributions in Statistics:Continuous Univariate Distributions-I, 1970

[7] S. Ross, AFirst Course in Probability, 7th edition, 2006

[8] S. A. Klugman, H. H. Panjer, G E. Willmot, Loss Models From Data to Decisions,2nd edition, 2004

[9] O. Zacaj, E. Dhamo Loss Models: Statistical Methods in modeling losses deriving from the insurance contracts, National Conference on advanced studies in the mathematics, chemistry, and physic engineering, 2011

[10] O. Zacaj, E. Dhamo Statistical methods in modelling losses deriving from insurance contracts: Application to Albanian motor insurance data, International Conference “Information Systems and Technology Innovations: their application in Economy, Faculty of Economy, University of Tirana, 2012

[11] J. Lemaire, Bonus - Malus Systems in Automobile Insurance, Kluwer Academoc Publisher, Boston 1995

[12] G. Taylor, Setting a Bonus-Malus scale in the presence of other rating factors, ASTIN Bulletin 27, 1997

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Fractional Calculus Operator for a Singular Sturm-Liouville Equation Reşat Yılmazer (1), Ökkeş Öztürk (2) (1) Firat University, Elazig, Turkey, [email protected] (2) Firat University, Elazig, Turkey, [email protected]

Abstract. In this study, deals with the design of fractional singular Sturm-Liouville equations. We obtain explicit solutions of these equations by means of N- fractional calculus method. We consider the following non-homogeneous singular Sturm-Liouville equation 1 4 [ , , , ] = + + + = , ( ) ( )2 2 훼 � − 훾 퐿 푦 푥 훾 훼 푦2 푦 �훽 2� 푓 푥 ≠ 휋 which has a solution of the form 푥 − 휋 푥 − 휋

= ( ) ( ) 1 훾+ 2 −푖훽 푥−휋 × ( ( ) ( )) 푦 푥 − 휋 ( 푒 ) ( ) 1 1 푖훼 1 푖훼 . 훾− − −훾− + 2−훾 푖훽 푥−휋 1 푖훼 −2푖훽푥 2 2훽 2푖훽푥 2 2훽 �� 푓 푥 − 휋 푒 −훾−2−2훽푒 푥 − 휋 � 푒 푥 − 휋 � 1 푖훼 −1 Keywords. Fractional calculus; Sturm-Liouville equation; Nishimoto's operator.훾−2+ 2훽 AMS 2010. 26A33, 34A08

References [1] de Romero, S.S, Srivastava, H.M., An Application of the N-Fractional Calculus Operator method to a Modified Whittaker Equation, Appl. Math. and Comp. 115, 11-21, 2000.

[2] Nishimoto, K., Fractional Calculus, vols. I, II, III, IV and V, Descartes Press, Koriyama, 1984, 1987, 1989, 1991 and 1996.

[3] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications Mathematics in Science and Enginering, vol. 198, Academic Press, New York, London, Tokyo and Toronto, 1999.

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State Variables Computation Technique for Linear Systems by Using Block-Pulse Functions Saadettin Aksoy (1) and Hakan Kızmaz (2) (1) Sakarya University, Sakarya, Turkey, [email protected] (2) Sakarya University, Sakarya, Turkey, [email protected]

Abstract. State variables that determine a system’s dynamics should be known for analysis and control of dynamical systems [1,2] Specifically, dynamic feedback for pole placement is adaptive control applications [3]. Unfortunately, all of the state variables cannot be measured in practice. As a result, use of a suitable state observer or estimator is unavoidable in order to obtain immeasurable state variables. There exist variety of state observers in the literature [4]. BPFs based the single-input single-output system identification has been studied by Palasinamy and Bhattacharya [5] and Cheng and Hsu [6]. Also parameter estimation by using block-pulse functions are studied by some people such as Kwong and Chen [7], Hsu and Cheng [6]. In this study, a general algorithm that uses only input and output and output measurements is proposed for state variables estimation of linear, time-invariant multi-input multi-output systems. The proposed algorithm, which is simple in form and convenient for computer usage is based on the BPFs and consist of three steps. In the first step, the feedback gain matrix, G which will force the estimation error to go to zero in a short time, can be determined by using a suitable method [4]. In the second step, input-output measurements and unknown state vector are expressed in a block pulse series. Thus, we approximate the continuous-time functions with a set of piecewise-constant functions. In this approximation, the total numbers of steps in the interval is [0,T] is m, and it can be chosen as any positive integer number. Finally, in the last step, the observer state equations are converted into integral equations by integrating the terms on either side of the equations. Then, unknown state estimation vector together with the block-pulse function approximation of known inputs and outputs measurement vectors are substituted in the integral equation. After some algebraic manipulations, state observer equation is transformed into a computationally convenient algebraic form whose solution can be obtained easily by a computer program. Consequently, resulting this recursive solution unknown state vector is easily calculated. Consequently, resulting this recursive solution unknown state vector is easily calculated. Note that this

361 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

algebraic solution form is rather convenient for computer usage and much computing time and storage can be saved, and the recursive algorithms developed also make it possible to real time state estimator for dynamic systems. The proposed estimation algorithm was implemented in MATLAB and it was applied to different cases. Results obtained by the proposed algorithm are in harmony with the real results. Keywords. Orthogonal Functions, Block-Pulse Functions, Observers.

References

[1] Chi-Tsong CHEN, Linear System Theory and Design. USA: CBS Publishing, 1984.

[2] F. M. BRASCH and J. B. PEARSON, "Pole Placement Using Dynamic Compensators," IEEE Transactions on Automatic Control, vol. AC-15, pp. 34-43, 1970.

[3] K. J. ASTRÖM and B. WITTERMARK, Adaptive Control. USA: Addison Wesley Publication Inc., 1989.

[4] O. REILLY, Observers for Linear Systems. London, UK: Academic Press, 1983.

[5] K. R. PALASINAMY and D. K. BHATTACHARYA, "System identifications via block- pulse functions," International Journal of System Science, vol. 12, no. 5, pp. 643-647, 1981, DOI: 10.1080/00207728108963772.

[6] Bing CHENG and Ning-Show HSU, "Single-input-single-output system identification via block-pulse functions," International Journal of Systems Science, vol. 13, no. 6, pp. 697- 702, 1982, DOI: 10.1080/00207728208926380.

[7] C. P. KWONG and C. F. CHEN, "Linear feedback system identification via block-pulse functions," International Journal of System Science, vol. 12, no. 5, pp. 635-642, 1981, DOI: 10.1080/00207728108963771.

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On Fuzzy Soft Γ-ring S. Onar (1) and B.A. Ersoy (1) (1) Yildiz Technical University, Istanbul, Turkey, [email protected]

Abstract. After Molodtsov introduced the concept of soft sets in 1999, soft sets theory has been extensively studied by many authors. Soft set theory has been applied to many different fields, such as function smoothness, Riemann and Perron integration, measurement theory, game theory, decision making. It is well known that the concept of fuzzy sets, introduced by Zadeh, has been extensively applied to many scientific fields. In 1971, Rosenfeld applied the concept to the theory of groupoids and groups. In 1982, Liu defined and studied fuzzy subrings as well as fuzzy ideals. Since then many papers concerning various fuzzy algebraic structures have appeared in the literature. Also, Maji et al. presented the definition of fuzzy soft set, and Roy et al. presented some applications of this notion to the decision- making problems in E. İnan et al. have already introduced the definition of fuzzy soft rings and studied some of their basic properties. In this work, We introduce the concept of fuzzy soft Γ-ring. Then we study some properties of fuzzy soft Γ-rings. Especially the definitions of fuzzy soft Γ-ideals are proposed and some theory of them is considered. we attempt to study fuzzy soft Γ−ring theory by using fuzzy soft sets. We prove that basic theorems in soft fuzzy rings is also valid in fuzzy soft gamma ring. Consequently we study definition of fuzzy soft Γ−ideal and derive some results on them.

363 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

Some Examples of Crossed Modules Tufan Sait Kuzpınarı Aksaray University, Aksaray, Turkey, [email protected]

Abstract. Crossed module concept was defined, by J.H.C. Whitehead (1949) as an algebraic model for homotopy types. The notion of 3-crossed module was introduced by Arvasi, Kuzpınarı and Uslu in [1] as an algebraic model for homotopy connected 4-type. This notion is based on ideas of Conduches' 2-crossed module, [6] and give a way to define n- crossed modules. Keywords. Crossed module, 2-crossed module, simplicial group, Moore complex AMS 2010. 18D35, 18G30, 18G50, 18G55.

References

[1] Arvasi Z., Kuzpınarı T.S. and Uslu E. O. Three Crossed Modules, Homology, Homotopy and Applications, 11:161-187. 2009.

[2] Blakers A.L. Some relations between homology and homotopy groups, Ann. of Math., 49:428-461. 1948.

[3] Brown R. and Loday J-L. Van Kampen Theorems for Diagram of Spaces. Topology 26:311-335. 1987.

[4] Carrasco P. and Cegarra A.M. Group-theoretic algebraic models for homotopy types, Journal of Pure and Applied Algebra,75:195-235. 1991.

[5] Castiglioni J.L. and Ladra M. Peiffer elements in simplicial groups and algebras, Journal of Pure and Applied Algebra, 212:2115-2128. 2008.

[6] Conduche D. Modules croises generalises de longueur 2, J.P.A.A, 34:155-178, 1984.

[7] Curtis E.B. Simplicial homotopy theory, Adv. in Math., 6:107-209, 1971.

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Oscillation Criteria for Second Order Nonlinear Differential Equations Xhevair Beqiri (1) and Elizabeta Koci (2) (1) State University of Tetova, [email protected] (2) University of Tirana, [email protected]

Abstract. We present new oscillation criteria for certain nonlinear differential equations of second order with damping term

++ txftqtxtptxtr = 0))(()()(')())'(')(( ≥ tt 0 (1) that are used oscillatory solutions of differential equatioins (α +β txfttxt = 0))(()())'(')( (2) where they are different from most known ones. Our results extend and improve some previous oscillation criteria and cover the cases which are not covered by known results. In this paper, by using the generalized Riccati technique we get a new oscillation and nonoscillation criteria for (1). The theorems prove to be efficient in many cases and have shown results in the literature. Keywords. nonlinear differential, equations, interval, criteria, damping, second order etc.

References

[1] H. El – Metwally, M. A. El –Moneam, The oscillatory behavior of second order nonlinear functional differential equations, the arabian Jurnal for science and inginering, volume 31. Number 1A, january 2006.

[2]Samir H. Saker , Oscillation theorems for second order nonlinear functional differential equations with damping, dynamic sys. appl. 12 , 2003, 307 - 322

[3]RakJoong Kim, Oscillation and nonoscillation criteria for differential equations of second order, Korean J. Math. 20011, N0.4, pp. 391 – 402 [4] Xh. Beqiri, E. Koci; Interval oscillation criteria for second order nonlinear diferential equations with damping term, Inter. Jurn. of science, innovat. and new technol., Vol.1. N.2, 2011

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[5] Aydin Tiryaki; Oscillation criteria for a certain second order nonlinear differential equations with deviating arguments; 2009, N0 . 61,1-11

[6] Yuri V. Rogovchenko;Interval oscillation of a second order nonlinear differential equation with a damping term, Discrete and continuous dynamical systems supplement , pp. 883 – 891, 2007

[7] M.M.A. El Sheikh, R.A.Sallam, D.I. Elimy; oscillation criteria for nonlinear second order damped differential equations, 2010, vol.10, nr. 3

[8]J. Ohriska, A. Zulova, Oscillation criteria for second order non – linear differential equations, IM preprint series A, N0.10/2004

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Solutions of the Nonlinear Differential Equations by Use of Symmetric Fibonacci Functions Yusuf Ali Tandoğan (1) and Yusuf Pandır (2) (1) Bozok University, Yozgat, Turkey, [email protected] (2) Bozok University, Yozgat, Turkey, [email protected]

Abstract. Based on Kudryashov’s method and the Fibonacci or Lucas Riccati equation method, some new solutions of a non-integrable nonlinear partial differential equation are found. Also, some basic properties of symmetric Fibonacci and Lucas functions are given in this research. For more details, we refer the reader to [1-5]. Keywords. Kudryashov’s method, symmetric Fibonacci functions, exact solutions. AMS 2010. 35C08, 35Q51.

References

[1] Ahmad, T. A., Ezzat, R. H., General Expa-function method for nonlinear evolution equations, Appl. Math. Comput., 217, 451-459, 2010.

[2] Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonl. Sci. Numer. Simulat., 17, 2248-2253, 2012.

[3] Pandir, Y., Gurefe, Y., Misirli, E., A new approach to Kudryashov’s method for solving some nonlinear physical models, Int. J. Phys. Sci., 7 (21), 2860-2866, 2012.

[4] Stakhov, A., Rozin, B., On a new class of hyperbolic functions, Chaos Soliton. Fract., 23, 379-389, 2005.

[5] Abdel-Salam, E. A-B., Al-Muhiameed, Z. I. A., Exotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system, Turk. J. Phys., 35, 241-256, 2011.

367 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

A New Approach to the Trial Equation Method Yusuf Gurefe (1), Yusuf Pandır (2) and Emine Mısırlı (3) (1) Bozok University, Yozgat, Turkey, [email protected] (2) Bozok University, Yozgat, Turkey, [email protected] (3) Ege University, Izmir, Turkey, [email protected]

Abstract. In recent years, some new versions of the trial equation method has been proposed to solve nonlinear partial differential equations [1-6]. In this research, we define a more general trial equation method, and also use this method to obtain a number of exact solutions to the nonlinear differential equations. Keywords. Trial equation method, elliptic function solution, soliton solution. AMS 2010. 35C08, 35Q51.

References

[1] Liu, C. S., Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Commun. Theor. Phys., 45, 219-223, 2006.

[2] Liu, C. S., Applications of complete discrimination system for polynomial for classifications of travelling wave solutions to nonlinear differential equations, Comput. Phys. Commun., 181, 317-324, 2010.

[3] Du, X. H., An irrational trial equation method and its applications, Pramana-J. Phys., 75, 415-422, 2010.

[4] Gurefe, Y., Sonmezoglu, A., Misirli, E., Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana-J. Phys., 77, 1023-1029, 2011.

[5] Gurefe, Y., Sonmezoglu, A., Misirli, E., Application of an irrational trial equation method to high-dimensional nonlinear evolution equations, J. Adv. Math. Stud., 5, 41-47, 2012.

[6] Pandir, Y., Gurefe, Y., Kadak, U., Misirli, E., Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal., (In press), 2012.

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Elliptic Function Solutions of a Nonlinear Partial Differential Equation Yusuf Pandir (1), Yusuf Gürefe (2) and Emine Mısırlı (3) (1) Bozok University, Yozgat, Turkey, [email protected] (2) Bozok University, Yozgat, Turkey, [email protected] (3) Ege University, Izmir, Turkey, [email protected]

Abstract. Mathematical models in science and engineering can be expressed as nonlinear partial differential equations. Therefore, constructing exact solutions to these equations is very important subject in mathematics. In recent years, a lot of methods has been developed for solving nonlinear differential equations. In this sense, some new versions of the trial equation method are proposed in [1-5] and new class of solutions for nonlinear partial differential equations are obtained. In this study, using the modified trial equation method, we obtain elliptic F-function, elliptic E-function, elliptic Pi-function and soliton solutions. Keywords. Extended trial equation method, elliptic function solution, soliton solution. AMS 2010. 35C08, 35Q51.

References

[1] Liu, C. S., Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Commun. Theor. Phys., 45, 219-223, 2006.

[2] Liu, C. S., Applications of complete discrimination system for polynomial for classifications of travelling wave solutions to nonlinear differential equations, Comput. Phys. Commun., 181, 317-324, 2010.

[3] Gurefe, Y., Sonmezoglu, A., Misirli, E., Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana-J. Phys., 77, 1023-1029, 2011.

[4] Gurefe, Y., Sonmezoglu, A., Misirli, E., Application of an irrational trial equation method to high-dimensional nonlinear evolution equations, J. Adv. Math. Stud., 5, 41-47, 2012.

[5] Pandir, Y., Gurefe, Y., Kadak, U., Misirli, E., Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal., (In press), 2012.

369 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

On Parallel Surfaces in Minkowski 3-Space Yasin Ünlütürk (1) and Erdal Özüsağlam (2) (1) Kirklareli University, Kırklareli, Turkey, [email protected] (2) Aksaray University,Aksaray, Turkey, [email protected]

Abstract. In this paper, we study on some properties of parallel surfaces in Minkowski 3-space. The results given in this paper were given in Euclidean space by Hacisalihoglu, Hicks. By using these two former studies, we try to show these properties in Minkowski 3-space. Also we give the relation among the fundamental forms of parallel surfaces in Minkowski 3-space. Finally we show that how a curve which is geodesic on M become again a geodesic on parallel surface Mr by the normal map in Minkowski 3-space. Keywords. Parallel surface, Coefficients of Fundamental forms, Fundamental forms AMS 2010. 53A05

References

[1] Craig, T., Note on parallel surfaces, Journal für die Reine und Angewandte Mathematik (Crelle's journal), 94, 162--170.

[2] Görgülü, A. and Çöken, C., The dupin indicatrix for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space in semi-Euclidean space R₁ⁿ. Journ. Inst. Math. and Comp. Sci. (Math Series), 7(3), 221-225., 1994.

[3] A. Görgülü and A. C. Çöken, The Euler theorem for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space E ₁ⁿ⁺¹, Journ. Inst. Math. and Comp. Sci. (Math. Series) Vol:6, No.2, 161-165, 1993.

[4] Gray, A., Modern differential geometry of curves and surfaces, CRC Press, Inc., 1993. Hacisalihoglu : Hacısalihoğlu, H. H., Diferensiyel geometri, Cilt I-II, Ankara Üniversitesi, Fen Fakültesi Yayınları, 2000.

[5] Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008.

[6] O'Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.

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Computation of Energies of Atoms over Integer and Non Integer Quantum Numbers Yusuf Yakar (1), Bekir Çakır (2) and Ayhan Özmen (2) (1) Aksaray University, Aksaray, Turkey, [email protected] (2) Selcuk University, Konya, Turkey, [email protected]

Abstract. In this study, we calculated the ground state energies of open- and closed- shell atoms over Slater type orbitals (STOs) with integer and noninteger principal quantum numbers using ab-initio method. For the non integer principal quantum numbers, two-electron coulomb and exchange integrals into total energy expression were defined in terms of incomplete β function. The obtained results are in good agreement with the literature. Keywords. Open- and closed-shell atoms, integer and non integer STOs, incomplete β function. AMS 2010. 26A33, 46F12, 65R13

References

[1] Jones, H. W., Developments in multicenter molecular integrals over STOs using expansions in spherical harmonics, Int. J. Quantum Chem. 51, 417-423, 1994.

[2] Guseinov, I. I., Özmen, A., Atav, Ü., Yüksel, H., Computation of overlap integrals over Slater-type orbitals using auxiliary functions, Int. J. Quantum Chem. 67, 199-204, 1998.

[3] Mamadov, B.A., Kara, M., Orbay, M., A new algorithm for the calculation of two-center overlap integrals over Slater type orbitals, Chinese Journal of Physics, 40, 283-287, 2002.

[4] Guseinov, I.I., Mamedov, B.A., Orbay, M., Özdoğan, T., Calculation of arbitrary overlap integrals over Slater type orbitals using basic overlap integrals, Communication in Theoretical Physics, 33, 161-166, 2000.

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372 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

PARTICIPANTS

No Title Name, Surname University 1 Prof. Dr. Abdugafur Rakhimov National University of Uzbekistan 2 Prof. Dr. Abdullah Altın Ankara University 3 Prof. Dr. Abdullah Avey Sofiyev Suleyman Demirel University 4 Prof. Dr. Abdullah Mağden Ataturk University 5 Prof. Dr. Adnan Mazmanoğlu Istanbul Aydın University 6 Prof. Dr. Ajet Ahmeti University of Prishtina 7 Prof. Dr. Alexander Sostak University of Latvia 8 Prof. Dr. Ali M. Akhmedov Baku State University 9 Prof. Dr. A. Asylkanovich Borubaev Kyrgyz National University 10 Prof. Dr. Amanbek Jainakov National Academy of Sci. of Kyrgyz Rep. 11 Prof. Dr. Arif Salimov Ataturk University 12 Prof. Dr. Askhad Polatov National University of Uzbekistan 13 Prof. Dr. Auzhan Sakabekov Kazakh-British Technical University 14 Prof. Dr. Azmi Özcan Bilecik Seyh Edebali University 15 Prof. Dr. Baki Karlığa Gazi University 16 Prof. Dr. Bayram Şahin Inonu University 17 Prof. Dr. Bilal Sherali MES RK 18 Prof. Dr. Bujar Xh. Fejzullahu University of Prishtina 19 Prof. Dr. B. Asanbekovich Boljiev Kyrgyz Technicall University 20 Prof. Dr. Bülent Karakaş Yuzuncu Yil University 21 Prof. Dr. Cihan Özgür Balıkesir University 22 Prof. Dr. Doost Ali Mojdeh University of Tafresh 23 Prof. Dr. Ekrem Savaş Istanbul Ticaret University 24 Prof. Dr. Emine Mısırlı Ege University 25 Prof. Dr. Etibar Penahlı Firat University 26 Prof. Dr. Fahir Talay Akyildiz Gaziantep University 27 Prof. Dr. Faton M. Berisha University of Prishtina 28 Prof. Dr. Fethi Çallıalp Dogus University 29 Prof. Dr. Fevzi Berisha FNA-Universiteti i Prishtines 30 Prof. Dr. Fikret Aliyev Baku State University 31 Prof. Dr. Gamar Mammadova Baku State University 32 Prof. Dr. H. Hilmi Hacısalihoğlu Bilecik Seyh Edebali University 33 Prof. Dr. H. S. Akhundov Baku State University 34 Prof. Dr. Halis Aygün Kocaeli University 35 Prof. Dr. Harry I.Miller International University of Sarajeevo 36 Prof. Dr. Hasan Akbaş Trakya University 37 Prof. Dr. Hasan Kürüm Firat University 38 Prof. Dr. Iskandarov Samandar Nati. Aca. of Sci. of Kyrgyz Rep. 39 Prof. Dr. İrfan Şiap Yildiz Technical University 40 Prof. Dr. Jamshid Aliyev Baku State University 41 Prof. Dr. K. Ishmakhametov Kyrgyz National University 42 Prof. Dr. Kadri Arslan Uludag Universitesi 43 Prof. Dr. K. Bekbolot Emenovich Kyrgyz National University 44 Prof. Dr. Kazım İlarslan Kırıkkale University

373 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

45 Prof. Dr. Kleida Haxhi Intersig, Vienna Insurance Group 46 Prof. Dr. Kostaq Hila University of Gjirokastra 47 Prof. Dr. Latifa Agamalieva Baku State University 48 Prof. Dr. Lazhar Bougoffa Al-İmam University 49 Prof. Dr. Levent Kula Ahi Evran University 50 Prof. Dr. M. A. Sadygov Baku State University 51 Prof. Dr. Madina Khojimurodova The National University of Uzbekistan 52 Prof. Dr. Mahmut Ergüt Firat University 53 Prof. Dr. Marjan Dema University of Prishtina 54 Prof. Dr. Mehmet Çetin Koçak Ankara University 55 Prof. Dr. Mikail Et Firat University 56 Prof. Dr. Minir Efendija University of Prishtina 57 Prof. Dr. Muhib Lohaj University of Prishtina 58 Prof. Dr. Murat Altun Uludag University 59 Prof. Dr. Murat Tosun Sakarya University 60 Prof. Dr. Muvasharkhan Jenaliyev Inst. of Math., Infor. and Mech. 61 Prof. Dr. Muzaffer Elmas Sakarya University 62 Prof. Dr. N. Ismayilov Baku State University 63 Prof. Dr. Nadir Ibadov Ganja State University 64 Prof. Dr. Nagozy Danaev Al-Farabi Kazakh National University 65 Prof. Dr. Naim Braha Universiy of Prishtina 66 Prof. Dr. Naim Syla University of Prishtina 67 Prof. Dr. Naime Ekici Cukurova University 68 Prof. Dr. Nargiz Safarova Baku State University 69 Prof. Dr. Neşet Aydın Çanakkale Onsekiz Mart University 70 Prof. Dr. Nihal Yılmaz Özgür Balıkesir University 71 Prof. Dr. Olga Savastru Odessa I.I.Mechnikov National University 72 Prof. Dr. Ömer Akın TOBB ETU 73 Prof. Dr. Patrick Sole University of Nice Sophia Antipolis 74 Prof. Dr. Pudji Astuti Institut Teknologi Bandung 75 Prof. Dr. Qamil Haxhibeqiri University of Prishtina 76 Prof. Dr. Qëndrim R. Gashi University of Prishtina 77 Prof. Dr. Rahmangulu Esadullayev Turkmenistan 78 Prof. Dr. Ramadan Zejnullahu University of Prishtina 79 Prof. Dr. Rasim Bejtullahu University of Prishtina 80 Prof. Dr. Ravi P. Agarwal Florida Institute of Technology 81 Prof. Dr. Regjep Gjergji University of Prishtina 82 Prof. Dr. Rifat Güneş Inonu University 83 Prof. Dr. Rugova Muje Pristhine University 84 Prof. Dr. Rüstem Kaya Eskisehir Osmangazi University 85 Prof. Dr. Saadettin Aksoy Sakarya University 86 Prof. Dr. Sabri Sadiku University of Prishtina 87 Prof. Dr. Sadik Bekteshi University of Prishtina 88 Prof. Dr. Sezer Ş. Komsuoğlu Kocaeli University 89 Prof. Dr. Sh.A.Ayupov Tashkent Institute 90 Prof. Dr. Shukri Klinaku University of Prishtina 91 Prof. Dr. Skender Ahmetaj University of Prishtina 92 Prof. Dr. Soltan Aliev NAS of Azerbaijan 93 Prof. Dr. Tynysbek Kalmenov Minis.of Edu. and Sci.of Rep.of Kazak.

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94 Prof. Dr. Xhevat Krasniqi University of Prishtina 95 Prof. Dr. Yusuf Gasimov Baku State University 96 Prof. Dr. Yusuf Yaylı Ankara University 97 Prof. Dr. Zeki Gündüz Sakarya University 98 Prof. Dr. Zeki Tez Marmara University 99 Prof. Dr. Z. Khabibovich Yuldashev National University of Uzbekistan 100 Assoc. Prof. Dr. Ahmet Şükrü Özdemir Marmara University 101 Assoc. Prof. Dr. Ahmet Yıldız Dumlupinar University 102 Assoc. Prof. Dr. Ali Erdoğan Hacettepe University 103 Assoc. Prof. Dr. Ali Karcı Inonu University 104 Assoc. Prof. Dr. Ali Rıza Yıldız Bursa Teknik University 105 Assoc. Prof. Dr. Ayşe Altın Hacettepe University 106 Assoc. Prof. Dr. Dilek Pandır Bozok University 107 Assoc. Prof. Dr. Elçin Yusufoğlu Dumlupinar University 108 Assoc. Prof. Dr. Emin Aydın Marmara University 109 Assoc. Prof. Dr. Emine Can Kocaeli University 110 Assoc. Prof. Dr. Emrah Doğan Sakarya University 111 Assoc. Prof. Dr. F. Nejat Ekmekçi Ankara University 112 Assoc. Prof. Dr. Fatma Özdemir Istanbul Teknik University 113 Assoc. Prof. Dr. Hakan Kasım Akmaz Cankiri Karatekin University 114 Assoc. Prof. Dr. Halit Orhan Ataturk University 115 Assoc. Prof. Dr. İlyas Akhisar Marmara University 116 Assoc. Prof. Dr. İlyas Yavuz Marmara University 117 Assoc. Prof. Dr. İsmail Ekincioğlu Dumlupinar University 118 Assoc. Prof. Dr. Kamile Şanlı Kula Ahi Evran University 119 Assoc. Prof. Dr. Mehmet Ali Güngör Sakarya University 120 Assoc. Prof. Dr. Nejmi Cengiz Ataturk University 121 Assoc. Prof. Dr. Ömer Faruk Gözükızıl Sakarya University 122 Assoc. Prof. Dr. Semra Doğruöz Adnan Menderes University 123 Assoc. Prof. Dr. Soley Ersoy Sakarya University 124 Assoc. Prof. Dr. Şenol Dost Hacettepe University 125 Assoc. Prof. Dr. Ünsal Tekir Marmara University 126 Assoc. Prof. Dr. Yıldıray Keskin Selcuk University 127 Assoc. Prof. Dr. Yusuf Yakar Aksaray University 128 Assoc. Prof. Dr. Zerrin Gül Esmerligil Cukurova University 129 Assist. Prof. Dr. Abdullah Fatih Özcan Inonu University 130 Assist. Prof. Dr. Abdullah Kablan Gaziantep University 131 Assist. Prof. Dr. Ahmet Boz Dumlupinar University 132 Assist. Prof. Dr. Ahmet Pekgör Selcuk University 133 Assist. Prof. Dr. Ali Deniz Usak University 134 Assist. Prof. Dr. Ali Şahin Aksaray University 135 Assist. Prof. Dr. Ali Şenol Cankiri Karatekin University 136 Assist. Prof. Dr. Alimzhan Ibragimov National University of Uzbekistan 137 Assist. Prof. Dr. Ayşe Zeynep Azak Sakarya University 138 Assist. Prof. Dr. Ayşegül Çetinkaya Ahi Evran University 139 Assist. Prof. Dr. Ayten Koç Istanbul Kültür University 140 Assist. Prof. Dr. Bahtiyar Bayraktar Uludag University 141 Assist. Prof. Dr. Cengiz Dane Trakya University 142 Assist. Prof. Dr. Dilek Ersalan Cukurova University

375 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

143 Assist. Prof. Dr. Emrah Evren Kara Bilecik Seyh Edebali University 144 Assist. Prof. Dr. Emre Eroğlu Kirklareli University 145 Assist. Prof. Dr. Ercan Masal Sakarya University 146 Assist. Prof. Dr. Erdal Özüsağlam Aksaray University 147 Assist. Prof. Dr. Ergin Jable Pristhine University 148 Assist. Prof. Dr. Filiz Yıldız Hacettepe University 149 Assist. Prof. Dr. Güler Tuluk Kastamonu University 150 Assist. Prof. Dr. Hakan Gökdağ Bursa Teknik University 151 Assist. Prof. Dr. Hakan Yakut Sakarya University 152 Assist. Prof. Dr. Halis Bilgil Aksaray University 153 Assist. Prof. Dr. İbrahim Demir Yildiz Technical University 154 Assist. Prof. Dr. İsmail Gök Ankara University 155 Assist. Prof. Dr. İsmet Altıntaş Sakarya University 156 Assist. Prof. Dr. İsmet Ayhan Pamukkale University 157 Assist. Prof. Dr. Jeta Alo Beykent University 158 Assist. Prof. Dr. Kevser Köklü Yildiz Technical University 159 Assist. Prof. Dr. M. Fatih Talu Inonu University 160 Assist. Prof. Dr. Mahmut Akyiğit Sakarya University 161 Assist. Prof. Dr. Memet Kule Kilis 7 Aralık University 162 Assist. Prof. Dr. Mevlüt Tunç Kilis 7 Aralık University 163 Assist. Prof. Dr. Muhsin İncesu Muş Alparslan University 164 Assist. Prof. Dr. Murat Candan Inonu University 165 Assist. Prof. Dr. Murat Güzeltepe Sakarya University 166 Assist. Prof. Dr. Murat Sarduvan Sakarya University 167 Assist. Prof. Dr. Mustafa Aşcı Pamukkale University 168 Assist. Prof. Dr. Mustafa Eröz Sakarya University 169 Assist. Prof. Dr. Mustafa Kemal Cerrahoğlu Sakarya University 170 Assist. Prof. Dr. Nazar Şahin Öğüşlü Cukurova University 171 Assist. Prof. Dr. Nesrin Güler Sakarya University 172 Assist. Prof. Dr. Nihat Ersoy Yildiz Technical University 173 Assist. Prof. Dr. Onur Kıymaz Ahi Evran University 174 Assist. Prof. Dr. Özgün Gürmen Alansal Dumlupinar University 175 Assist. Prof. Dr. Reşat Yılmazer Firat University 176 Assist. Prof. Dr. Sare Şengül Marmara University 177 Assist. Prof. Dr. Selcen Yüksel Perktaş Adıyaman University 178 Assist. Prof. Dr. Selda Çalkavur Kocaeli University 179 Assist. Prof. Dr. Semra Ahmetolan Istanbul Teknik University 180 Assist. Prof. Dr. Serkan Demiriz Gaziosmanpasa University 181 Assist. Prof. Dr. Süleyman Şenyurt Ordu University 182 Assist. Prof. Dr. Şahin Ceran Pamukkale University 183 Assist. Prof. Dr. Şeyda Kılıçoğlu Başkent University 184 Assist. Prof. Dr. Tufan Sait Kuzpınarı Aksaray University 185 Assist. Prof. Dr. Tuğça Poyraz Tacoğlu Hacettepe University 186 Assist. Prof. Dr. Ummahan Merdinaz Acar Mugla Sitki Kocman University 187 Assist. Prof. Dr. Vedat Şiap Yildiz Technical University 188 Assist. Prof. Dr. Yasemin Kıymaz Ahi Evran University 189 Assist. Prof. Dr. Yasemin Sağıroğlu Karadeniz Teknik University 190 Assist. Prof. Dr. Yılmaz Gündüzalp Dicle University 191 Assist. Prof. Dr. Yusuf Ali Tandoğan Bozok University

376 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

192 Lecturer Dr. Anita Caushi Polytechnic Univerity of Tirana 193 Lecturer Dr. Eduard Domi University A.Xhuvani 194 Lecturer Dr. Erol Yavuz Yildiz Technical University 195 Lecturer Dr. Esen İyigün Uludag University 196 Lecturer Dr. Hanni Garminia Institut Teknologi Bandung 197 Lecturer Dr. Islam Braja University of Elbasan 198 Lecturer Dr. Mimoza Shkëmbi Elbasan University 199 Lecturer Dr. Murat Canayaz Yuzuncu Yil University 200 Lecturer Dr. Mustafa Terzi Gazi University 201 Lecturer Dr. Nihal Ata Hacettepe University 202 Lecturer Dr. Şerife Yılmaz Karadeniz Teknik University 203 Lecturer Dhurata Valera University of Aleksandër Xhuvani 204 Lecturer Doğan Ünal Sakarya University 205 Lecturer Feyza Esra Erdogan Adıyaman University 206 Lecturer Hülya Bozyokuş Uludag University 207 Lecturer Hürmet Fulya Akız Bozok University 208 Lecturer Hüseyin Oğuz Dumlupinar University 209 Lecturer İqtadar Hussain National Univ. of Comp. and Emer. Sci. 210 Lecturer M. Anishevna Yuldasheva Specialised labour school 211 Lecturer Muhsin Çelik Sakarya University 212 Lecturer Sanem Şehribanoğlu Yuzuncu Yil University 213 Lecturer Sema Servi Selcuk University 214 Lecturer Şamil Akçağıl Bilecik Seyh Edebali University 215 Lecturer Yusuf Gürefe Bozok University 216 Lecturer Yusuf Muştu Bilecik Seyh Edebali University 217 Lecturer Yusuf Pandır Bozok University 218 Rsc. Assist. Dr. Arzu Öğün Ünal Ankara University 219 Rsc. Assist. Dr. Ayşe Betül Koç Selcuk University 220 Rsc. Assist. Dr. Bayram Çekim Gazi University 221 Rsc. Assist. Dr. Beytullah Eren Sakarya University 222 Rsc. Assist. Dr. İrfan Şimşek Gaziosmanpaşa University 223 Rsc. Assist. Dr. Kadir Kanat Gazi University 224 Dr. Bayram Ali Ersoy Yildiz Technical University 225 Dr. Edmundo Huertas University Carlos III of Madrid 226 Dr. Figen Çilingir Cankaya University 227 Dr. Muhammad Aslam Quaid-i-Azam University 228 Dr. Murat Erişoğlu Selcuk University 229 Dr. Nuria Torrado Universidad Pública de Navarra 230 Dr. Rabia Aktaş Ankara University 231 Dr. Saleem Abdullah Quaid-i-Azam University 232 Dr. Ülkü Erişoğlu Selcuk University 233 Rsc. Assist. Abdullah İnalcık Artvin Coruh University 234 Rsc. Assist. Aferdita Lohaj University of Prishtina 235 Rsc. Assist. Ahmet Daşdemir Aksaray University 236 Rsc. Assist. Albert Jonuzaj University of Prishtina 237 Rsc. Assist. Ali Aydoğdu Ataturk University 238 Rsc. Assist. Artan Alidema University of Prishtina 239 Rsc. Assist. Artan Berisha University of Prishtina 240 Rsc. Assist. Arzu Güleroğlu Trakya University

377 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

241 Rsc. Assist. Bashkim Trupaj University of Prishtina 242 Rsc. Assist. Banu Pazar Varol Kocaeli University 243 Rsc. Assist. Behar Baxhaku University of Prishtina 244 Rsc. Assist. Bengi Yıldız Bilecik Seyh Edebali University 245 Rsc. Assist. Beste Madran Yeditepe University 246 Rsc. Assist. Burim Kamishi University of Prishtina 247 Rsc. Assist. Cansu Keskin Dumlupinar University 248 Rsc. Assist. Dilek Bayrak Karadeniz Teknik University 249 Rsc. Assist. Duygu Dönmez Demir Celal Bayar University 250 Rsc. Assist. Ece Yetkin Marmara University 251 Rsc. Assist. Edmond Aliaga University of Prishtina 252 Rsc. Assist. Elçin Gökmen Muğla Sıtkı Koçman University 253 Rsc. Assist. Elver Bajrami University of Prishtina 254 Rsc. Assist. Eray Yıldırım Sakarya University 255 Rsc. Assist. Esra Altıntaş Marmara University 256 Rsc. Assist. Fatih Temiz Yildiz Technical University 257 Rsc. Assist. Faton Merovci University of Prishtina 258 Rsc. Assist. Fisnik Aliaj University of Prishtina 259 Rsc. Assist. Gazmend Nafezi University of Prishtina 260 Rsc. Assist. Gönül Yazgan Sağ Gazi University 261 Rsc. Assist. Gül Güner Karadeniz Teknik University 262 Rsc. Assist. Gülsüm Biçer Aksaray University 263 Rsc. Assist. Hakan Kızmaz Sakarya University 264 Rsc. Assist. Hatice Parlatıcı Sakarya University 265 Rsc. Assist. Hatice Topcu Nevşehir University 266 Rsc. Assist. Hazim Misini University of Prishtina 267 Rsc. Assist. Hidayet Hüda Kösal Sakarya University 268 Rsc. Assist. İlkem Turhan Dumlupinar University 269 Rsc. Assist. İlknur Koca Gaziantep University 270 Rsc. Assist. İrfan Deli Kilis 7 Aralık University 271 Rsc. Assist. Kajtaz Bllaca University of Prishtina 272 Rsc. Assist. Kısmet Kasapoğlu Trakya University 273 Rsc. Assist. Kushtrim Podrimqaku University of Prishtina 274 Rsc. Assist. Mahmut Mak Ahi Evran University 275 Rsc. Assist. Mehmet Altay Ünal Ankara University 276 Rsc. Assist. Mehmet Fatih Karaaslan Yildiz Technical University 277 Rsc. Assist. Merve Görgülü Karamanoğlu Mehmetbey University 278 Rsc. Assist. Mesut Balıbey Tunceli University 279 Rsc. Assist. Nejla Gürefe Gazi University 280 Rsc. Assist. Nihat Yağmur Erzincan University 281 Rsc. Assist. Nurten Bayrak Yildiz Technical University 282 Rsc. Assist. Oğuzhan Bahadır Corum Hitit University 283 Rsc. Assist. Okan Kuzu Ahi Evran University 284 Rsc. Assist. Osman Zeki Okuyucu Bilecik Seyh Edebali University 285 Rsc. Assist. Önder Gökmen Yıldız Bilecik Seyh Edebali University 286 Rsc. Assist. Övgü Gürel Ankara University 287 Rsc. Assist. Pınar Güner Marmara University 288 Rsc. Assist. Seda Göktepe Yildiz Technical University 289 Rsc. Assist. Sefer Avdiaj University of Prishtina

378 IECMSA-2012 1st International Eurasian Conference on Mathematical Sciences and Applications

290 Rsc. Assist. Setenay Akduman Dokuz Eylul University 291 Rsc. Assist. Sevda Göktepe Marmara University 292 Rsc. Assist. Simge Öztunç Celal Bayar University 293 Rsc. Assist. Soydan Serttaş Sakarya University 294 Rsc. Assist. Ş. Burcu Bozkurt Selcuk University 295 Rsc. Assist. Tuğba Mert Cumhuriyet University 296 Rsc. Assist. Tülay Soyfidan Erzincan University 297 Rsc. Assist. Uğur Dağdeviren Dumlupinar University 298 Rsc. Assist. Ulaş Yamancı Suleyman Demirel University 299 Rsc. Assist. Ümit Karabıyık Necmettin Erbakan University 300 Rsc. Assist. Ümit Tokeşer Gazi University 301 Rsc. Assist. Xhevair Beqiri State University of Tetova 302 Rsc. Assist. Vildan Çetkin Kocaeli University 303 Rsc. Assist. Zafer Şiar Bilecik Seyh Edebali University 304 Teacher Güngör Özçeker Turkey 305 Student Alba Vrapi Kosovo 306 Student Arife Aysun Karaaslan Yildiz Technical University 307 Student Armend Sh. Shabani University of Prishtina 308 Student Ezgi Erdoğan Yildiz Technical University 309 Student Hülya Başeğmez Ahi Evran University 310 Student Lütfü Sizer Ahi Evran University 311 Student Mesut Altınok Ahi Evran University 312 Student Mine Öztürk Gazi University 313 Student Oriana Zacaj Universiteti Politeknik i Tiranes 314 Student Tuba Yiğit Suleyman Demirel University 315 Student Zehra Velioğlu Cukurova University 316 Student Valmir Krasniqi University of Prishtina

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