5TH INTERNATIONAL EURASIAN CONFERENCE ON MATHEMATICAL SCIENCES AND APPLICATIONS
August 16-19, 2016 - Belgrade, Serbia
FOREWORD
The “5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)’’ jointly organized by Sakarya University, Kocaeli University, Bilecik Seyh Edebali University, Namik Kemal University, International Novi Pazar University, Baku State University, Institute of Applied Mathematics, Turkic World Mathematical Society, and International Balkan University, will be held on August 16-19, 2016 in Belgrade, Serbia.
The series of the International Eurasian Conference on Mathematical Sciences and Applications provide communication between the members of the mathematics community, interdisciplinary researchers, educators, statisticians and engineers. These conferences are held every year in different countries with distinguished participants from all over the world and they build agelong Cultural Bridges.
After the following four very successful international conferences the IECMSA-2012, Prishtine, Kosovo, IECMSA-2013, Sarajevo, Bosnia and Herzegovina, IECMSA-2014, Vienna, Austria, and IECMSA-2015, Athens, Greece now IECMSA-2016, Belgrade, Serbia, hosts esteemed participants from different countries.
While the preparations of IECMSA-2016 were going on, An unexpected and worrisome event happened in Turkey (the homeland of some organizers of IECMSA). Turkey has experienced a coup attempt to overthrow its democratically-elected and legitimate government on July 15th, 2016. After this tragic incident now July 15th becomes 'the “Day to Commemorate Martyrs” in memory of civilians and police officers who gave their lives fighting tanks, helicopters, and heavily armed soldiers who attempted to overthrow the government. We remember with respect our martyrs. We believe that the international public will support the democracy in Turkey as all Turkish citizens with different worldviews have done.
Although these sad events have affected the participations, IECMSA-2016 has taken a lot of applications from all over the world. Moreover, ten worldwide distinguished speakers have been invited to the conference and the abstracts of the plenary talks have been substituted in this book. Also, the electronic version of the abstracts of all presentations can be found in the Conference Abstracts Book at www.iecmsa.org
I wish to thank all members of scientific committee and sponsors for their continued support to the IECMSA-2016. And finally, I would like to sincerely thank all the participants of IECMSA-2016 for contributing to this great meeting in many different ways. I believe and hope that each of them will get the maximum benefit from the conference.
Welcome to Belgrade!
Prof. Dr. Murat TOSUN Chairman On behalf of the Organizing Committee
HONORARY COMMITTEE
Prof. Dr. Muzaffer ELMAS (Rector of Sakarya University)
Prof. Dr. Sadettin HULAGU (Rector of Kocaeli University)
Prof. Dr. Osman SIMSEK (Rector of Namık Kemal University)
Prof. Dr. Azmi OZCAN (Rector of Bilecik Seyh Edebali University at 2007-2016)
Prof. Dr. Ismail KOCAYUSUFOGLU (Rector of Inter. Balkan Univeristy)
Prof. Dr. Suad BEĆIROVIĆ (Rector of International Novi Pazar University)
SCIENTIFIC COMMITTEE
Prof. Dr. Abdel SALHI University of Essex
Prof. Dr. Alfonso CARRIAZO University of Sevilla
Prof. Dr. Altay BORUBAEV Moscow State University
Prof. Dr. Cristina FLAUT Ovidius University
Prof. Dr. Debasis GIRI Haldia Institute of Technology
Prof. Dr. Dragan DJORDJEVIC University of Nis
Prof. Dr. Fikret ALIYEV Baku State University
Prof. Dr. Hans Peter KUNZI University of Cape Town
Prof. Dr. Harry MILLER Int. University of Sarajevo
Prof. Dr. Hellmuth STACHEL Vienna Technical University
Prof Dr. Josef ŠLAPAL Brno University of Technology
Prof. Dr. Kailash MADAN Ahlia University
Prof. Dr. Kudratillo FAYAZOV Nat. University of Uzbekistan
Prof. Dr. Ljubisa KOCINAC Nis University
Prof. Dr. Mukhtarbay OTELBAEV L.N. Gumilev Eur. Nat. University
Prof. Dr. Pavle BLAGOJEVIC Free University
Prof. Dr. Ravi P. AGARWAL Texas A&M University
Prof. Dr. Slavica Ivelić BRADANOVIĆ University of Split
Prof. Dr. Sidney A. MORRIS Federation University Australia
Prof. Dr. Taras BANAKH Ivan Franko Lviv Nat. University
Prof. Dr. Tynysbek KALMENOV Al-Farabi Kazakh Nat.University
Prof. Dr. Uday Chang DE Calcutta University
Prof. Dr. Vasile BERINDE Universitatea de Nord Baia Mare
Prof. Dr. Wolfgang SPROESSING Freiberg Uni. of Mining and Tech.
Prof. Dr. Yaudat SULTANAEV Bashkir State Pedagogical Uni.
Prof. Dr. Zuhair NASHED University of Central Florida
ORGANIZING COMMITTEE
GENERAL COORDINATOR: Murat TOSUN (Sakarya University)
VICE -GENERAL COORDINATOR: Ljubisa KOCINAC (Nis University)
Almatbek KYDYRBEKULY Al-Farabi Kazakh National University
Antonio FERNANDEZ University of Sevilla
Bianca SATCO Stefan cel Mare University of Suceava
Bo Wun HUANG Cheng Shiu University
Byoung Soo KIM Seoul National Uni. of Sci. and Tech.
Chekeev ASYLBEK Kyrgyz National University
Chiun Chieh HSU National Taiwan Uni. of Sci.and Tech.
Edgar Martinez MORO Valladolid University
Gamar MAMMADOVA Baku State University
Hector Luna GARCIA Universidad Autonoma Metropolitana
Jumageldy ALIMOV Turkmenistan
Laura VENTURA University of Padova
Lubica HOLA Math. Inst. Slovak Academy of Sciences
Miroslava ANATIC University of Belgrade
Nargiz SAFAROVA Baku State University
Nihan ALIEV Baku State University
Nirmal C. SACHETI Sultan Qaboos University
Pallath CHANDRAN Sultan Qaboos University
Pranesh KUMAR University of Northern British Columbia
Ramil BAKHTIZIN Ufa State Petroleum Tech. University
Robert NIGMATULIN Russian Academy of Sciences
Victor Jimenez LOPEZ Universidad de Murcia
Walter RACUGNO University of Cagliari
Yue Kuen KWOK Hong Kong University of Sci. and Tech.
Yusif GASIMOV Baku State University
Ziyaviddin YULDASHEV National University of Uzbekistan
CHAPTERS
INVITED TALKS………………………………...…………………………... 1
ALGEBRA………………………………………………………………………… 12
ANALYSIS……………………………………………………………………….. 31
APPLIED MATHEMATICS……………………………………………. 76
DISCRETE MATHEMATICS…………………………………………... 130
GEOMETRY……………………………………………………………………… 136
MATHEMATICS EDUCATION……………………………………… 170
STATISTICS……………………………………………………………………… 184
TOPOLOGY……………………………………………………………………….. 196
THE OTHER AREAS………………………………………………………. 207
POSTERS………………………………………………………………………….. 213
PARTICIPIANTS…………………………………………………………….. 240
5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
CONTENTS
INVITED TALKS
The Plant Propagation Algorithm: Presentation, Implementation and Convergence Analysis
Abdel Salhi ...... 2
Some Applications of Algebras Obtained by the Cayley Dickson Process
Cristina Flaut ...... 3
Some Inverse Problems of the Spectral Theory for the Differential Operators and Their Applications
Etibar S. Panakhov and Ahu Ercan ...... 4
Partially Ordered Metric Spaces Produced with the Help of T0-Quasimetrics
Hans-Peter A. Kunzi...... 5
Jordan Curves in the Digital Plane
Josef Slapal ...... 6
Submanifolds in Euclidean Spaces
Kadri Arslan ...... 7
Hyperplane Mass Partitions via Relative Equivariant Obstruction Theory
Pavle V. M. Blagojevic ...... 8
The Topology of Compact Groups and Pro-Lie Groups
Sidney A. Morris ...... 10
-Bases in Topological and Uniform Spaces
Taras Banakh ...... 11
ALGEBRA
On Some Combinatorial Identities and Harmonic Sums
Necdet Batir ...... 12
Operational Matrix to Solve Black-Scholes Equation for European Option by Using Block Pulse
Functions
A. Jafari Shaerlar and M. Hoseini ...... 13
Some Remarks on the Third Order Linear Recurrent Sequences and Pell Equation
Arzu Ozkoc...... 14 1
5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Results on 2-(0-) Primary Fuzzy Ideals of Commutative Semirings Deniz Sonmez and Gursel Yesilot ...... 15
One Sided (σ,τ)-Lie Ideals and Generalized Derivations in Prime Rings Evrim Guven ...... 16
-Primary Hyperideals on Commutative Hyperrings Elif Ozel Ay, Gursel Yesilot and Deniz Sonmez ...... 18
Semicommutativity of Amalgamated Rings H. Kose, Y. Kurtulmaz, B. Ungor and A. Harmanci ...... 19
The Padovan Sequences in Centro-Polyhedral Groups Hasan Ozturk and Engin Ozkan ...... 20
Codes Defined via Especial Matrices over the Ring and Hadamard Codes Mustafa Ozkan and Figen Oke ...... 21
Cyclic Codes over the New Ring Mustafa Ozkan and Figen Oke ...... 22
The Group Inverses of Some Combinations of Three Tripotent Matrices Murat Sarduvan and Nurgul Kalayci ...... 23
A Note on 2-Absorbing Primary Hyperideals of Multiplicative Hyperrings Neslihan Suzen and Gursel Yesilot ...... 24
Matrix Representation of Octonions and Their Geometry Serpil Halici and Adnan Karatas ...... 26
Generalized Fibonacci Octonions and Their Vector Matrix Representation Serpil Halici and Adnan Karatas ...... 27
On k- Conjugate of Quaternions Serpil Halici and Sule Curuk ...... 28
Some New Classes of Permutation Polynomials over a Finite Field Suphawan Janphaisaeng ...... 29
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
A Note on Neutrosophic Subring of a Ring
Vildan Cetkin and Halis Aygun ...... 30
ANALYSIS
A Certain Subclass of Meromorphic Functions with Positive and Fixed Second Coefficients Associated with the Rafid-Operator
Arzu Akgul...... 31
Iterative Roots of a Linear Function
Boonrod Yuttanan ...... 32
Gegenbauer Polynomials and Positive Definiteness
Christian Berg and Emilio Porcu ...... 33
Homogeneous B-Potential Type Integrals on Hardy Spaces
Cansu Keskin and Ismail Ekincioglu ...... 34
Ideal Convergent Sequence Spaces via Orlicz Function
Emrah Evren Kara and Merve Ilkhan ...... 36
On the Characterizations of Variable Exponent Hardy Spaces According to Riesz Bessel Transform
Esra Kaya, Ismail Ekincioglu and Cansu Keskin ...... 37
On the Lipschitz Stability of Inverse Nodal Problem for p-Laplacian Bessel Equation
Etibar S. Panakhov, Emrah Yilmaz and Tuba Gulsen ...... 39
Factorization through Lorentz Spaces Associated to a Vector Measure
Fernando Mayoral A. Fernández, F. Naranjo, R. Del Campo and E.A. Sánchez-Pérez.... 41
On New Inequalities of Hermite-Hadamard Type for B-1-convex Functions
Gabil Adilov and Ilknur Yesilce...... 42
On Generalized Lupas Operators
H. Gul Ince Ilarslan ...... 43
A New Characterization of Bessel Potential Spaces
Ilham A. Aliev ...... 44
Necessary and Sufficient Conditions for the Boundedness for Singular Integral Operators with in the Lorentz Spaces
Ismail Ekincioglu, Cansu Keskin and Sedat Pak ...... 45
On the Convergence of Implicit Iteration Processes in Convex Metric Spaces
Isa Yildirim ...... 47
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Approximating Common Fixed Points of I-Asymptotically Quasi-Nonexpansive Mappings Isa Yildirim ...... 48
Products of Weighted Composition Operators and Differentiation Operators between Weighted Banach Spaces and Weighted Zygmund Spaces of Analytic Functions Jasbir S. Manhas ...... 49
New Type of Lacunary Ideal Convergent Sequence Spaces Mahmut Dastan, Emrah Evren Kara and Merve Ilkhan ...... 50
The Dual Multiple Knot B-spline Frames Maryam Esmaeili ...... 51
On Multiple Knot B-spline Frames Maryam Esmaeili ...... 52
New Type Open Sets in Bitopological Spaces Merve Ilkhan and Emrah Evren Kara ...... 53
Cofinally Quasi Cauchy Continuity Merve Ilkhan, Pınar Zengin Alp and Emrah Evren Kara ...... 54
A Version of Popovici’s Inequality Mirea Mihaela Mioara ...... 55
Maximal Hyponormal Differential Operators of First-Order with Smooth Coefficients Meltem Sertbas ...... 56
Constructions of the Determinantal Representations of Hyperbolic Forms Mao-Ting Chien ...... 57
The Refinements of Hilbert-Type Inequalities Predrag Vuković ...... 58
Real Interpolation of p -th Power Factorable Operators R. Del Campo, A. Fernández, F. Naranjo, E.A. Sánchez and A. Fernández ...... 59
Reflexivity of Function Spaces Associated to A Σ-Finite Vector Measure R. Del Campo , Antonio Fernández , F. Mayoral and F. Naranjo ...... 60
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On a New Class of k-Uniformly Starlike Functions with Negative Coefficients Based on q-Derivative Operator Sahsene Altinkaya and Sibel Yalcin Tokgoz ...... 62
On the Chebyshev Polinomial Coefficient Estimates for a Subclass of Analytic and Bi-Univalent Functions Sahsene Altinkaya and Sibel Yalcin Tokgoz ...... 63
On New Hermite-Hadamard-Fejér Type Inequalities for Harmonically Quasi-Convex Functions Selahattin Maden, Imdat Iscan and Sercan Turhan ...... 65
On Hermite-Hadamard-Fejér Type Inequalities with Applications for Quasi Geometrically Convex Functions Sercan Turhan, Selahattin Maden and Imdat Iscan ...... 67
The Hankel Determinant for a Certain Subclass of Univalent Functions Sibel Yalcin Tokgoz and Sahsene Altinkaya ...... 69
A New Subclass of Analytic Functions Defined by Using Symmetric Q-Derivative Operator Sibel Yalcin Tokgoz and Sahsene Altinkaya ...... 70
A New Class of Salagean-type Univalent Functions as Related to Sigmoid Function Sibel Yalcin Tokgoz and Sahsene Altinkaya ...... 71
A New Study on Summability of Factored Fourier Series Sebnem Yildiz ...... 72
On the Degenerate Poly-Genocchi Numbers and Polynomials Veli Kurt ...... 74
Arithmetic Properties about Quotients and Powers of Exponential Polynomials Vichian Laohakosol and Pinthira Tangsupphathawat ...... 75
APPLIED MATHEMATICS 14-Point Averaging Operator for the Aproximation of the First Derivatives of a Solution of Laplace’s Equation in a Rectangular Parallelepiped Adiguzel A. Dosiyev and Hediye Sarikaya ...... 77
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On a Solution of Radial Schrödinger Equation for Special Potentials Arzu Guleroglu, Cengiz Dane and Hasan Akbas ...... 78
On the Feedback Linearization for 2D Dynamical System Associated to the Mixing Flow Model Adela Ionescu ...... 79
Principle of Mass Conservation for the Boltzmann’s Moment System of Equations in Fourth Approximation Aizhan Issagali and Auzhan Sakabekov ...... 80
Attainable Set of a Sir Epidemiological Model with Constraintson Vaccination and Treatment Stocks Ali Serdar Nazlipinar, Anar Huseyin and N. Huseyin ...... 81
Solving Container Terminal Scheduling Problems with the Plant Propagation Algorithm Birsen Irem Selamoglu, Abdellah Salhi and Ghazwan Alsoufi ...... 82
A Study on the Dynamics of the Solution of a Riemann Type Problem in a Class of Discontinuous Functions Bahaddin Sinsoysal, Hasan Carfi and Mahir Rasulov ...... 84
Estimation of the Eigenvalues for Neumann Boundary Value Problems Bulent Yilmaz ...... 86
11yxnn Solutions of the Maximum of Difference Equations xynn11max , ; max , xn1 x n 3 yn1 y n 3 Dagistan Simsek and Burak Ogul ...... 88
xnk(2 1) Solutions of the Rational Difference Equations xn1 1 xnk Dagistan Simsek and Burak Ogul ...... 89
Upper and Lower Solution Method for Fourth-order Three point BVPs on an Infinite Interval Erbil Cetin ...... 90
Asymptotic Behavior and Global Nonexistence of a Solution for a System of Nonlinear Higher-Order Wave Equations with Weak Damping Erhan Piskin ...... 91
Decay and Blow up of Solutions for Nonlinear Hyperbolic Equations with Nonlinear Damping Terms Erhan Piskin ...... 92
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Frictional Contact Problem for an Orthotropic Elastic Strip and Layer Elcin Yusufoglu and Huseyin Oguz ...... 93
Sweep Method for Solving the Roesser Type Equation Describing the Motion in the Pipeline Fikret A. Aliev, N. A. Aliev , N. A. Safarova , R. M. Tagiev and G. H. Mammadova ...... 94
Free Vibration of Timoshenko Beam with 3D Tip Mass Subject to Bending in Orthogonal Planes and Torsional Deformation Hilal Doganay Kati and Hakan Gökdag ...... 95
Solving Benjamin-Bona-Mahony Equation by Using the sn-ns Method and the Tanh-Coth Method Hami Gundogdu and Omer Faruk Gozukizil ...... 97
Order Conditions of Symplectic Partitioned Runge-Kutta Method for Stochastic Optimal Control Problems Hacer Oz, Gerhard-Wilhelm Weber and Fikriye Yilmaz ...... 98
Influence of the Impedance Coated Groove on the TEM Wave Radiation from Coaxial Waveguide Hulya Ozturk ...... 99
Determination of an Unknown Heat Source from Integral Overdetermination Condition for Quasilinear Parabolic Equation Irem Baglan and Fatma Kanca ...... 100
The Solutions with Generator and Schrödinger Equation which have Sphere Symmetry for a potential Kismet Kasapoglu, Cengiz Dane and Hasan Akbas ...... 101
Buoyancy Effects on Unsteady Reactive Variable Properties Fluid Flow in a Channel Filled with a Porous Medium Lazarus Rundora and Oluwole Daniel Makinde ...... 102
Qualitative Properties of Solutions of a Partial Differential Equation with a Piecewise Constant Argument Mehtap Lafci and Huseyin Bereketoglu ...... 103
The Modeling of Electric Power and Electric Contact Systems M. N. Kalimoldayev and M. T. Jenaliyev ...... 104
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Numerical Investigation of the Interaction of Waves for 2D Conservation Laws in a Class of Discontinuous Functions with Three-Piece Constant Condition Mahir Rasulov, Hakan Bal and Bahaddin Sinsoysal ...... 105
Experimental and Computational Studies of (2Z, 3E)-3-(((E)-3-Ethoxy-2-Hydroxybenzylidene) Hydrazono)Butan-2-One Oxime Nezihe Caliskan, Cigdem Yuksektepe Ataol, Humeyra Bati, ...... 107
New Identities on the Generalized Exponential and Mellin Integral Transforms and Their Applications Nese Dernek, Fatih Aylikci and Osman Yurekli ...... 108
On Some Applications of the Generalized Laplace Transform Ln Nese Dernek and Sevil Kivrak ...... 110
Perturbation Solution for Systems with Strong Quadratic and Cubic Nonlinearities Nedret Elmas ...... 112
An Asymptotical Method for Determining Hydraulic Resistance’s Coefficient of Gas-Lift Process by the Method of Lines N. S. Hajieva, N. A. Safarova and M. F. Rajabov ...... 114
Numerical Solutions to Initial and Boundary Value Problems for Fractional Telegraph Equations Ozan Ozkan ...... 116
Fractional Calculus Models for Some Bioengineering Problems Ozlem Ozturk Mizrak and Nuri Ozalp ...... 117
Statistical Physics and Thermodynamics Approach to EEG Time Series Sergey Borisenok ...... 118
Quasi-Periodic Wave Solutions of (2+1) Dimentional Breaking Wave Solutions Using with Riemann Theta Functions Secil Demiray and Filiz Tascan ...... 119
Existence of Positive Solutions for Four Point Fractional Boundary Value Problems Serife Muge Ege and Fatma Serap Topal ...... 121
Modelling Problems of Dynamics Using Differential Games S. N. Amirgaliyeva ...... 122
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
New Hybrid Conjugate Gradient Method as a Convex Combination of LS and FR Methods Snezana S. Djordjevic ...... 123
Explicit Finite-Difference Method For Solution of Heat Diffusion-Wave Equation with Mix-Fractional Derivatives Vildan Gulkac ...... 124
Qualitative Behavior of ODE Systems Solutions Corresponding to First Order Chemical Kinetics Mechanisms Victor Martinez-Luaces ...... 125
About One Approach for the Group Synthesis of Recognition and Classification Tasks Yedilkhan Amirgaliyev and Salim Mustafin ...... 126
Stochastic and Random Behavior Model for Immune System Response and Bacterial Resistance with Antibiotic Therapy Zafer Bekiryazici, Mehmet Merdan, Tahir Khaniyev and Tulay Kesemen ...... 128
DISCRETE MATHEMATICS Mathematical Programming for Computing Padmakar-Ivan Index of Graphs Amir Bahrami ...... 131
On Roman {2} Domination in Graphs Nader Jafari Rad and Elahe Shabani ...... 132
Incidence Coloring of Sierpinski Graphs Ummahan Akcan, Emrah Akyar and Handan Akyar ...... 133
The Positivity Problem for Linear Recurrence Sequences of Order Six Vichian Laohakosol and Pinthira Tangsupphathawat ...... 134
Trigonometric Factorizations of the Horadam Sequence and Its Companion Sequence Zafer Siar ...... 135
GEOMETRY Similarity Relation on Bidegenarate Quaternions, Bipseudodegenerate Quaternions and Bidoubly Degenarate Quaternions Abdullah Inalcik ...... 137
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Characterizations of Slant Helices as Rectifying Curves Bulent Altunkaya ...... 138
The Representation of Some Special Homothetic Motions in Lorentz Space Dogan Unal, Mehmet Ali Gungor and Murat Tosun ...... 139
Ruled Surfaces with Type-2 Bishop Frame in E3 Esra Damar and Nural Yuksel ...... 141
On k-Type Pseudo Null Darboux Helices According to Bishop Frame in Minkowski 3-Space Emilija Nešović ...... 142
A Smarandache Curves an Application to Spherical Images Erdal Ozusaglam ...... 143
A Characterization of Involutes and Evolutes of a Given Curve in En Gunay Ozturk, Kadri Arslan and Betul Bulca ...... 144
A Note on Hypersurfaces of a Riemannian Manifold with a Ricci-Quarter Symmetric Metric Connection Hulya Bagdatli Yilmaz ...... 145
Geodesics on the Tangent Sphere Bundle of a Pseudo Hyperbolic 3-Space Ismet Ayhan ...... 147
A New Characterization of General Helix in Minkowski 3-Space Kazim Ilarslan ...... 148
On Surfaces That Intersection Curves are Special Curves Mesut Altinok, Benen Akinci and Levent Kula ...... 149
A Study on Bertrand Curves in E3 Melek Masal and Ayse Zeynep Azak ...... 150
Conharmonic Curvature Tensor of Almost Contact, K-Contact and Sasakian Finsler Manifolds Nesrin Caliskan ...... 151
On the Kinematics for the Closed Planar Homothetic Inverse Motions in Complex Plane Onder Sener, Ayhan Tutar and Serdar Soylu ...... 153
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On the Curves of AW(k) in Minkowski 3- Space Pelin Tekin and Erdal Ozusaglam ...... 154
Pointwise Slant Submersions from Sasakian Manifolds Sezin Aykurt Sepet and Mahmut Ergut ...... 156
Decomposition of Out-of-Control Signals in Multivariate Process Semra Boran, Halil Ibrahim Cebeci, Deniz D. Diren and Dogan Unal ...... 158
On the nth order Bertrand Mate Curves in E3 Seyda Kilicoglu and Suleyman Senyurt ...... 160
On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E³ Seyda Kilicoglu and Suleyman Senyurt ...... 161
An Examination on Mannheim Frenet Ruled Surface Based on Normal Vector Fields in E³ Seyda Kilicoglu ...... 162
Smarandache Curves of Bertrand Curve Pair According to Frenet Frame Suleyman Senyurt and Abdussamet Caliskan ...... 164
A New Approach on the Striction Curves along Bertrandian Frenet Ruled Surfaces Suleyman Senyurt and Abdussamet Caliskan ...... 165
Some Properties of the Inverse Homothetic Motion: Characteristic Point and Minimal Action Serdar Soylu, Ayhan Tutar and Onder Sener ...... 166
The Applications of the Taxicab Metric in Real Life Suleyman Yuksel and Kadir Kanat and A. Murat Aksoy ...... 168
Applications on the Magnetic Fields Zehra Ozdemir, Ismail Gok, F. Nejat Ekmekci and Yusuf Yayli ...... 169
MATHEMATICS EDUCATION Teachers’ Competences for a Successful Implementation of Technology in Mathematics Instruction Ana Donevska-Todorova and Katja Eilerts...... 170
A Scale to Determine Parents' Expectation from Mathematics Education (PEME): Development,
Reliability and Validity
Cahit Aytekin, Bulent Altunkaya, Serdal Baltaci, Yasemin Kiymaz and Avni Yildiz ...... 171
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Teaching of Permutation and Probability with Exchange of Knowledge Method Didem Nimet Berkun and Tuba da ...... 172
Pattern Generalization Strategies of Elementary Students Deniz Ozen and Nilufer Yavuzsoy Kose ...... 173
Development of Algebraic Thinking from the Perspective of Algebraic Habits of Mind Dilek Tanisli ...... 174
Teacher Candidates’ Applications of the Topic of Similarity: A Perspective from the Point of View of Realistic Mathematics Education Isil Bozkurt, Tugce Kozakli and Murat Altun ...... 175
Student Presentations on the History of Mathematics and Science in Math Classes and Its Impact on the Learning Process Irina Peterburgsky ...... 177
The Examination of Mathematical Literacy Skill Levels of 8th Grade Students Murat Altun, Nalan Aydin Gumus, Recai Akkaya and Isil Bozkurt ...... 178
Thinking Further: Mathematics as a Second Language Marjorie Chaves and Victor Martinez-Luaces ...... 180
Perceptions of Students about Functioning of Mathematics Sakir Isleyen ...... 181
Analysis of the Process of the Empty Number Line Usage on Mental Operations Tugce Kozakli, Isil Bozkurt and Murat Altun ...... 182
STATISTICS Limit Distribution for a Semi-Markovian Inventory Model of Type (s,S) Under Heavy Tailed Demand with Infinite Variance Aslı Bektas Kamislik, Tahir Khaniyev and Tulay Kesemen ...... 185
New Methods of Selection Biasing Parameter for Modified Ridge Estimator Hasan Ertas ...... 187
McDonald Extended Weibull Distribution Mustafa Cagatay Korkmaz ...... 188
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Cluster Analysis Using Gower’s Distance for Panel Data Ozlem Akay and Guzin Yuksel ...... 190
A Novel Approximation for Computation Bivariate Distribution Functions in Polygonal Area Orhan Kesemen, Buğra Kaan Tiryaki and Tuncay Uluyurt ...... 191
Linear Regression Model Predictions Using 푳풑-norm: An Application to Explain Plant-available Phosphorus of Corn using Chemical Determination of Inorganic Phosphorus in the Soil Pranesh Kumar ...... 192
On the Asymptotic Expansion for the Moments of an Inventory Model of Type (s,S) with Subexponential Demand and Uniform Distributed Interference of Chance Tulay Kesemen, Asli Bektas Kamislik and Zafer Kucuk3 ...... 194
TOPOLOGY On Completely δ-b-Irresolute Functions Aynur Keskin Kaymakci ...... 197
New Approaches About I Continuous in Ideal Topological Spaces Ayse Cobankaya ...... 199
Investigation of Some Properties of the Cohomology of SO(N) and Its Classifying Space Ayse Cobankaya and Dogan Donmez ...... 200
Recurrence Relations for Unoriented Knot Polynomials of 2, n -Torus Links Kemal Taskopru and Ismet Altintas ...... 201
Unoriented Knot Polynomials of -Torus Links as Vieta Polynomials
Kemal Taskopru and Ismet Altintas ...... 202
Semi-Hurewicz Spaces Ljubiša D.R. Kočinac, Amani Sabah and Moiz ud Din Khan ...... 204
Linear Operator Equations in Hilbert Space Mohammad Saeed Khan ...... 205
Suzuki Type Fixed Point Theorems in Uniform Spaces Vildan Ozturk ...... 206
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
THE OTHER AREAS Numerical Simulation of Flow Boiling Heat Transfer in a Single Horizontal Microchannel Arunabha Chanda ...... 208
Applications of White Noise Calculus to the Computation of Greeks Farai Julius Mhlanga ...... 209
A Literature Review of the Decision Making Problems Studied in the Theories of Intuitionistic Fuzzy, Neutrosophic and Soft Sets Murat Ibrahim Yazar ...... 210
Divisor Problem in Special Sets of Gaussian Integers Olga Savastru ...... 211
Numerical Algorithms for R&D Stochastic Control Models Yue Kuen Kwok ...... 212
POSTERS Direct and Inverse Problems for Fourth-Order Mixed Type Equation with Fractional Derivative Abdumauvlen Berdyshev ...... 214
Numerical Study of the Effect of Wettability Alteration on the Imbibition and Recovery Processes Abdumauvlen Berdyshev, Bakhbergen Bekbauov and Zharasbek Baishemirov ...... 216
Smarandache Curves in Three Dimensional Lie Groups Caner Degirmen, Osman Zeki Okuyucu and Onder Gokmen Yildiz ...... 217
Microspheres Based on Chitosan as Drug Delivery Systems Gabriela Ciobanu, Octavian Ciobanu and Selman Hizal ...... 218
The Test of the Finite-Size Scaling Relations for the Linear Lattice Size 24≤ L≤ 28 of the Four- Dimensional Ising Model on the Creutz Cellular Automaton Ganimet Mulazimoglu Kizilirmak and Fatih Yalcin ...... 219
Numerical Results on a Refinement of Hölder Inequality Gultekin Tinaztepe ...... 220
Equation of Equilibrium and Dilation Field Hulya Ozturk and Nezihe Caliskan ...... 221
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Separation Axioms in Čech Closure Ordered Spaces Irem Eroglu and Erdal Guner ...... 222
Sequentially Topological Groups Ibrahim Ince and Soley Ersoy ...... 223
The Dependence of the Process of Crystal Growth on the Type of the Initial Distribution Ilya Starodumov and Nikolai Kropotin ...... 224
New Learning Dynamic with Rational and Naive Forecasting Strategies in Cobweb Model Katarina Kukić and Jelena Stanojević ...... 225
Vocational High Scholl Students and Mathematics Kamile Sanli Kula and Elif Gunden ...... 226
The Natural Lift Curve of the Spherical Indicatrix of a Spacelike Curve with Null Binormal in Minkowski 3-Space Mustafa Caliskan, Evren Ergun and Mustafa Bilici ...... 227
On the Semisimilarity and Consemisimilarity of Split Quaternions Onder Gokmen Yildiz, Hidayet Huda Kosal and Murat Tosun ...... 228
Evolution of Generalized Space Curve in Minkowski Space Onder Gokmen Yildiz and Murat Tosun ...... 229
A Subclass of Convex Univalent Functions as Related to Sigmoid Function Sahsene Altinkaya and Sibel Yalcin Tokgoz ...... 230
On an Equilibrium State in Two-Sector Model of Economic Dynamics Sabir Isa Hamidov ...... 231
On the Striction Curves of Involutive Frenet Ruled Surfaces in E3 Seyda Kilicoglu, Suleyman Senyurt and Abdussamet Caliskan ...... 232
The Cauchy-Length Formula and the Holditch Theorem for the Homothetic Motion in p Tulay Erisir and Mehmet Ali Gungor ...... 233
On the Study of the Holditch-Type Theorem for the Non-Linear Three Points in
Tulay Erisir and Mehmet Ali Gungor ...... 234
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Generalized Holditch-Type Theorem for the Homothetic Motion in p Tulay Erisir, Mehmet Ali Gungor and Mahmut Akyigit ...... 235
Split Dual Fibonacci and Lucas Octonions Umit Tokeser and Zafer Unal ...... 236
Exact Solutions of Nonlinear Schrödinger Equation by New Type of Generalized F-Expansion Method Yusuf Ali Tandogan and Yusuf Pandir ...... 237
Differential Invariants of Two Affine Curve Families in Plane Yasemin Sagiroglu, Demet Aydemir and Ugur Gozutok ...... 238
Dual Pell and Pell-Lucas Quaternions Zafer Unal and Umit Tokeser ...... 239
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
INVITED TALKS
INVITED TALKS
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Plant Propagation Algorithm: Presentation, Implementation and Convergence Analysis Abdel Salhi 1
Abstract. The Plant Propagation Algorithm (PPA) also known as the Strawberry Algorithm is a search/optimisation heuristic based on the way plants and in particular the strawberry plant propagate. It is characterised by its simplicity, a low number of arbitrary parameters, robustness and ease of implementation. Moreover it is competitive with well- established heuristics and metaheuristics. Indeed, it has now been tested on a variety of optimisation problems both, continuous and discrete, constrained and unconstrained and has been compared to the Genetic Algorithm, Ant Colony Algorithm, Particle Swam Optimisation, the Firefly Algorithm and others; in most cases it has shown superior performance. However, its convergence has not yet been formally established. In my talk, I will present PPA and explain how it works. I will illustrate it on a number of well-known test problems and report some experimental and comparative results. Moreover, I will sketch a proof of convergence for a variant of PPA for continuous global optimisation.
Fig. 1: Strawberry Plant in flower Fig. 2 Ackley’s Function References
[1] Abdellah Salhi and Eric Fraga, Nature-Inspired Optimisation Approaches and the New Plant Propagation Algorithm, Proceedinds of the ICeMATH2011, pp. K2-1:K2-8, Jogjakarta, Indonesia, 2011.
[2] Mohammad Sulaiman, Abdellah Salhi, Irem B. Selamoglu, and Omar B. Kirikchi, A Plant Propagation Algorithm for Constrained Engineering Optimisation Problems, Journal of Mathematical Problems in Engineering, Vol. 2014, pp.1-10, 2014.
1 Department of Mathematical Sciences, University of Essex, CO4 3SQ, UK, [email protected], http://privatewww.essex.ac.uk/~as/
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Applications of Algebras Obtained by the Cayley Dickson Process Cristina Flaut1
Abstract. This talk summarizes original results obtained, in several papers, by the author in the study of algebras obtained by the Cayley-Dickson process. An important problem in the theory of quadratic forms is to determine which anisotropic forms become (or not) isotropic when such forms are extended to the function field of a given form. The study of isotropy of some special quadratic forms over some function field produced a good result: it was proved that for any positive integer n there is an algebra A , obtained by the Cayley-Dickson process with the norm form anisotropic over a suitable field, which has level n -0 . Some computational mathematical methods in the study of these algebras allow us to develop algorithms used to obtain new examples, supplementary relations, new identities or properties which can be very useful for the study of the algebras obtained by the Cayley-Dickson process. These algorithms provide us some new applications, as for example in the Coding Theory.
1 Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania, http://cristinaflaut.wikispaces.com/; http://www.univ-ovidius.ro/math/, cristina\[email protected]; [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Inverse Problems of the Spectral Theory for the Differential Operators and Their Applications Etibar S. Panakhov1 and Ahu Ercan2
Abstract. In this study, we examine certain stability of Sturm-Liouville problem with ll1 A having singularity of type at x 0 . We consider two such problems with xx2 potentials q1 and q2 and discuss proximity of their spectral functions given that the first N 1 eigenvalues of the two spectral problems coincide. Similar stability questions were discussed for regular Sturm-Liouville operators in [2]. Marchenko and Maslow deal with similar issue for regular Sturm-Liouville problem in the case of the spectral functions pj coincide on given interval in [1]. As far as we know, the stability problems for singular operators have not been studied. Keywords. Singular problem, stability, norming constant. AMS 2010. 34B09, 34D20.
References
[1] Marchenko, V. A., Maslov, K. V., Stability of the problem of reconstruction of the SturmLiouville operator in terms of the spectral function, Mathematics of the USSR Sbornik, 81, 525-51, 1970. (in Russian)
[2] Ryabushko, T. I., Stability of the reconstruction of a Sturm-Liouville operator from two spectra, II. Teor. Funksts. Anal., Prilozhen., 18, 176-85, 1973. (in Russian)
[3] Amirov, R. Kh., Gülyaz, S., Inverse spectral problem for the differential equation of the second order with singularity, Proceedings of the Eighth International Colloquium on Differential Equations, 17-241998, 1997.
[4] Panakhov, E. S., Yilmazer, R., On inverse problem for singular Sturm-Liouville operator from two spectra, Ukrainian Mathematical Journal, 58 (1), 147-154, 2006.
1 Baku State University, Baku, Azerbaijan, [email protected] 2 Firat University, Elazig, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Partially Ordered Metric Spaces Produced with the Help of T0-Quasimetrics Hans-Peter A. Kunzi1 (joint work with Yae Ulrich Gaba)
Abstract. We survey investigations on those (partially) ordered metric spaces
1 Xm,, for which there exists a T0 -quasi-metric d on X such that maxd , d m and such that for any x, y X we have that xy if and only if d x,0 y . In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a -quasi-uniformity with a countable base. Among other things we observe that the compatibility conditions between partial order and metric on the investigated spaces become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on are assumed to hold.
1 Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Jordan Curves in the Digital Plane Josef Slapal1
Abstract. Digital images may be considered to be approximations of the real ones and, therefore, their study needs the digital plane Z2 to be equipped with a structure satisfying digital analogues of the basic properties of the Euclidean plane. In particular, we need that a digital analogue of the Jordan curve theorem be valid. Providing the digital plane with a convenient structure is one of the basic problems of digital topology [1], a theory that has arisen for the study of geometric and topological properties of digital images. Despite its name, digital topology is based on using graph theory (adjacency graphs) rather than topology. The most often employed adjacencies for structuring the digital plane are the 4- and 8-adjacencies [2]. Since neither of these two adjacencies itself allows for a Jordan curve theorem, none of them may alone be used for structuring the digital plane but a combination of these two must be employed instead. It was shown in [3] that this disadvantage may be eliminated by using some special adjacencies that it selves provide convenient structures on Z2. In the present note, we study the adjacencies that are coarser than the 8-adjacency and have the property that certain natural cycles in the corresponding adjacency graphs are Jordan curves, i.e., separate Z2 into two connected components. Of these adjacencies, we determine the minimal ones and study their quotients. The results obtained are then used to show that some of the quotient adjacencies satisfy a digital Jordan curve theorem. Thus, these adjacencies may be used as background structures on the digital plane Z2 for the study of digital images. Keywords. Adjacency graph, quotient adjacency, digital plane, Jordan curve theorem. AMS 2010. 05C40, 68R10.
References
[1] Rosenfeld, A., Digital topology, Amer. Math. Monthly 86, 621-630, 1979.
[2] Rosenfeld, A., Picture Languages, Academic Press, New York, 1979.
[3] Slapal, J., Adjacencies for structuring the digital plane, Lect. Notes Comput. Sci. 7655, 123-137, 2012.
1 Brno University of Technology, Brno, Czech Republic, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Submanifolds in Euclidean Spaces Kadri Arslan1
Abstract. In the present section we consider n-dimensional submanifolds Mⁿ in m- dimensional Euclidean space Eⁿ. We present four type of represantations of the submanifolds, which are parametric, explicit, implicit and mixed representations. The important geometric charecteristics of a submanifold are its first and second fundamental forms. So, we consider Gauss and Weingarten decompositions of submanifolds in Euclidean spaces which are related with first and second fundamental forms. Keywords. Submanifold, Second Fundamental form, Mean curvature vector. AMS 2010. 53C40, 53C42.
References
[1] Aminov, Yu. A. The Geometry of Submanifolds, Gordon and Breach Science Publ. 2001.
[2] Aminov, Yu. A. and Rabelo M. L., On toroidal submanifolds of constant negative curvature, Mat. Fiz. Anal. Geom., 2(1995), 275--283.
[3] Chen, B.Y., Geometry of Submanifolds, Dekker, New York, 1973.
[4] Gor'kavyi V.A. and Nevmerzhitskaya E.N., Two-dimensional Pseudospherical surfaces with degenerate bianchi transformation, Ukrainian Mathematical Journal, Vol. 63(2012), No. 11, Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 63(2011), No. 11, pp. 1460-- 1468.
1 Uludag University, Bursa, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Hyperplane Mass Partitions via Relative Equivariant Obstruction Theory Pavle V. M. Blagojevic1
Abstract. In 1960 Branko Grünbaum suggested the following innocent-looking problem: The Grünbaum hyperplane mass partition problem. Can any convex body in d be cut into 2d pieces of equal volume by d suitably-chosen affine hyperplanes? As Grünbaum noted, this is quite easy to prove for d 2 . In 1966 Hadwiger answered Grünbaum’s question (positively) for d 3, while solving a problem raised by J. W. Jaworowski (Oberwolfach, 1963). Grünbaum’s question was independently raised in Computational Geometry, motivated by the construction of data structures for range queries. In 1984 Avis answered Grünbaum’s problem negatively for d 5 . Indeed, one cannot expect a positive answer there, since d hyperplanes in can be described by d 2 parameters, while the hyperplanes one is looking for need to satisfy 21d independent conditions, and 21d d 2 for d 4 . The case d 4 was and still is an open problem. In 1996 Ramos formulated the following general version of the hyperplane mass partition problem for several masses: The Grünbaum–Hadwiger–Ramos problem. For each j 1 and k 1, determine the smallest dimension d j, k such that for every collection M of j masses on there are k affine hyperplanes that cut each of the masses into 2k equal pieces.
The special case jj,1 of the Grünbaum–Hadwiger–Ramos problem, for a single hyperplane k 1 , is settled by the ham-sandwich theorem, which was conjectured by Steinhaus and proved by Banach in 1938. The following lower bound for the function jk, was derived by Avis forj 1 and Ramos, while the upper bound was obtained by Mani-Levitska, Vrecica & ˇZivaljevic:
k 21 k1 log j j j, k j 2 1 2 2 . k
j 1 log j Here 2log2 is rounded down to the nearest power of 2, so jj22 . 2
1 Freie Universität Berlin, Berlin, Germany, Mathematical Institute SASA, Belgrade, Serbia, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
In addition to the general lower and upper bounds, a number of papers have treated special cases, reductions, and relatives of the problem. It was recently documented that, however, quite a number of published proofs do not hold up upon critical inspection, and indeed some of the approaches employed cannot work. In this talk, using the relative equivariant obstruction theory in combination with the “join configuration space / test map scheme” and the study of Gray codes we prove that: Theorem. For t 1 the following instances of the Ramos conjecture hold: (1) 2tt 1,2 3 21 1,
(2) 2tt ,2 3 21 ,
(3) 2tt 1,2 3 21 2,
(4) 2,3 5,
(5) 4,3 10.
Consequently, 4 1,4 5 and 8 2,4 10. (This is a joint work with Florian Frick, Albert Haase, and Günter M. Ziegler)
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Topology of Compact Groups and Pro-Lie Groups Sidney A. Morris1
Abstract. The speaker is well known for his gentle introduction to topology known as “Topology Without Tears”, available on the web: www.topologywithouttears.net and used in over 100 countries and which has been translated (at least partially) into Arabic, Chinese, Greek, Persian, Russian, Spanish and Turkish and also for his very large books: "The Structure of Compact Groups" and "The Lie Theory of Connected Pro-Lie Groups." The presentation should have something for both the specialist and the non-specialist. It is an expository talk describing the topology of compact groups and of pro-Lie groups. Beginning with the definition of topological group, the speaker surveys – but does not prove – some quite deep results.
1 Federation University Australia and La Trobe University, Australia,
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
-Bases in Topological and Uniform Spaces Taras Banakh1
Abstract. In the talk we shall discuss properties of topological spaces with a local - base and of uniform spaces whose uniformity has an -base. Given a partially ordered set P we shall say that a topological space X has a local P-base if each point xX has a neighborhood base (U)P such that UU for any elements of P. A uniform space X is defined to have a P-base if its uniformity has a base (U)P such that UU for any in P. Observe that a topological space is first-countable if and only if it has a local -base; a uniform space is metrizable if and only if its uniformity has an -base. By we denote the poset of all functions from to endowed with the partial order fg iff f(n)g(n) for all n. Topological spaces X with a local -base can be considered as generalized metric spaces. In partiacular, at each point x X they have a countable Pytkeev* network. This is a contable family F of subsets of X such that for any neighborhood U of x and any sequence (xn)n accumulating at x there exists a set F F such that xF U and F contains infinitely many points of the sequence. Uniform spaces with an -base also have many features of generalized metric spaces. In particular, for a uniform space X with an -base the following conditions are equivalent: (i)
X is separable, (ii) X is cosmic, (iii) X is an 0-space, (iv) X is a 0-space, (v) X is a -space with countable extent, (vi) X contains a dense -space with countable extent. If the universal uniformity of a Tychonoff space has a -base, then the conditions (i)-(vi) are equivalent to each of the following: (vii) the function space Cp(X) is cosmic, (vi) Cp(X) is analytic, (ix)
Cp(X) is K-analytic, (x) Cp(X) has a compact resolution, (xi) X is -compact. The universal uniformity of a metrizable space X has an -base if and only is X is -compact. The study of uniform spaces with an -base was motivated by a recent result of Leiderman, Pestov and Tomita, who proved that a free abelian topological group A(X) of a Tychonoff space X has a local -base iff the universal uniformity of X has an -base. Keywords. Uniform space, an -base, a local -base. AMS 2010. 54E15, 54E18, 54E20.
1 Jan Kochanowski University in Kielce (Poland) and Ivan Franko National University of Lviv (Ukraine), e-mail: [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
ALGEBRA
ALGE BRA
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On Some Combinatorial Identities and Harmonic Sums
Necdet BATIR1
Abstract. For any naturel numbers m and n we first give new proofs for the following well known combinatorial identities
n n 1 1 mS )1()( k n k m k1 k 21 rrrn m 1... 21 ...rrr m and
n n nk )1( nk !, k 1 k and then we produce the generating function and an integral representation for n mS )( . Using them we evaluate many interesting finite and infinite harmonic sums in closed form. Keywords. Harmonic sums, Riemann zeta function, combinatorial identities. AMS 2010. 05A10, 05A19.
References
[1] Adamchik, V., On Stirling numbers and Euler sums, J. Comput. Appl. Math., 79(1997), 119-130.
[2] Bang, S-J., Amer. Math. Monthly, 102(10), 1995, 930.
[3] Boyadzhiev, K. N., Close encounters with the Stirling numbers of the second kind, Math. Magazine, v. 85, no. 4, 2012, 252-266.
[4] Choi, J. and Srivastava, H. M., Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005), 51-70.
[5] Coppo, M-A. and Candelpergher, B., The Arakawa-Kaneko zeta function, Ramanujan J. 22 (2010), 153-162.
n [6] Guo B-N. and Qi, F., An inductive proof for an identity involving and the partial k sums of some series, Int. J. Educ. Sci. Technol., 33(2), 2002, 249-253.
[7] Hofman, M. E., Harmonic number summation identities, symmetric functions, and multiple zeta values, Ramanujan J., to appear; preprint arXiv 1602.03198.
1 Nevşehir HBV University, Nevşehir, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Operational Matrix to Solve Black-Scholes Equation for European Option by Using Block Pulse Functions A. Jafari Shaerlar1 and M. Hoseini 2
Abstract. In financial mathematics,the fair price of option can be achieved by solution of parabolic differential equation. The main porpose of this work is presented new direct method to solve Black-Scholes (B-S) equation. By using Block Pulse Functions (BPFs) and its operational matrix of integration, (B-S) equation can be transformed to linear system of Algebra. We demonstrate the accuracy of our method through numerical examples. Keywords. Black-Scholes equation, Block Pulse Functions, Operational Matrix, Option Pricing.
1 Department of Mathematics, Parsabad Moghan Branch, Islamic Azad University – Parsabad Moghan , Iran. [email protected] 2 Department of Mathematics, Parsabad Moghan Branch, Islamic Azad University – Parsabad Moghan , Iran. [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Remarks on the Third Order Linear Recurrent Sequences and Pell Equation Arzu Ozkoc1
Abstract. We give a recurrence relation by the third order recurrent sequences
2 2 n n1 )14()14( n2 UUtUtU n3 for n 3 with two different initial value. Also we defined the circulant matrices for these sequences and we obtained applications for them. Finally we compute all integer solutions of the Pell equation 22 ytx 2 1)1( for every t 1. Keywords. Third order recurrent sequences, linear recurrence, circulant matrices, Pell equations. AMS 2010. 05A19, 11B37, 11B39, 11D09, 11D41, 11D45
References
[1] Barbeau E.J. Pell's Equation. Springer-Verlag New York, Inc, 2003.
[2] Jacobson M. and Williams H. Solving the Pell Equation, CMS Books in Mathematics. Springer, 2010.
[3] Mollin R.A. Fundamental Number Theory with Applications. Second Edition (Discrete Mathematics and Its Applications) Chapman & Hall/ CRC, Boca Raton, London, New York, 2008.
[4] Ribenboim P. My Numbers, My Friends, Popular Lectures on Number Theory, Springer- Verlag, New York, Inc. 2000.
[5] Shabani A. S. The Proof of Two Conjectures Related to Pell’s Equation x 2 Dy 2 4. Int. Comput. Math. Sci. 2, 1, 24-27, 2008.
[6] Yazlık Y., Taşkara N. Spectral Norm, Eigenvalues and Determinant of Circulant Matrix Involving the Generalized k-Horadam Numbers, Ars Combinatoria 104, 505-512, 2012.
1 Duzce University, Duzce, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some Results on 2-(0-) Primary Fuzzy Ideals of Commutative Semirings Deniz Sonmez1 and Gursel Yesilot2
Abstract. Let R be a commutative semiring with a nonzero identity and L be a complete lattice. The aim of this paper is to introduce the notion of 2-primary (0-primary) fuzzy ideals of R and investigate several properties of these concepts. Keywords. 2-primary fuzzy ideals, and 0-primary fuzzy ideals. AMS 2010. 16Y60,03E72.
References
[1] Baik S. I., and Kim H. S., On fuzzy k-ideals in semirings, Kangweon-Kyungki Math. Jour., 8, 147-154, 2000.
[2] Darani A. Y. and Hashempoor A., L-Fuzzy 0-(1- or 2- or 3-) 2-absorbing ideals in semirings, Annals of Fuzzy Math. and Informatics, 7, 303-311, 2014.
[3] Dheena P. and Coumaressane S., Fuzzy 2-(0- or 1-) prime ideals in semirings, Bull. Korean Math. Soc., 43, 559-573, 2006.
[4] Dixit V. N., Kumar R., and Ajmal N., Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Sets and Systems, 44, no. 1, 127-138, 1991.
[5] Dutta T. K. and Biswas B. K., Fuzzy k-ideals of semirings, Bull. Calcutta Math., Soc. 87, 1, 91-96, 1995.
[6] Ghosh S. , Fuzzy k-ideals of semirings, Fuzzy Sets and Systems, 95, 103-108, 1998.
[7] Liu W.J., Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8, 133- 139, 1982.
[8] Malik D.S. and Mordeson J.N., Fuzzy maximal, radical, and primary ideals of a ring, Information Sciences, 53, 237-250, 1991.
[9] Mukherjee T.K. and Sen M.K., Primary fuzzy ideals and radical of fuzzy ideals, Fuzzy Sets and Systems, 56, 97-101, 1993.
[10] Mukherjee T. K. and Sen M. K., On fuzzy ideals of a ring (1), Fuzzy Sets and Systems, 21, 1, 99-104, 1987.
[11] Zadeh L. A. , Fuzzy sets, Information and Control, 8, 338-353, 1965.
1 Yildiz Technical University, Department of Mathematics, Davutpasa-Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Department of Mathematics, Davutpasa-Istanbul, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
One Sided (σ,τ)-Lie Ideals and Generalized Derivations in Prime Rings Evrim Guven 1
Abstract. Let R be a prime ring with characteristic not 2 and σ,τ,λ,μ, automorphisms of R. Let V be a nonzero left (σ,τ)-Lie ideal ,U a nonzero right (σ,τ)-Lie ideal of R and a,b∈R. In this paper we have given some results on one sided (σ,τ)-Lie ideals and left (or right)- generalized (σ,τ)-derivation in prime rings: (1) aVb⊂C (2) (i) bV⊂ C (or Vb⊂ C), (ii) bU⊂ C (or Ub⊂C C), (iii) (V,b ) =0 (or (b,V) =0), (iv) (U,b) =0, (3) (i) (h(I),a) =0,
(ii) ah(I)⊂C(J), (iii) ah(I)b⊂C ,(iv) h[I,a] =0, (v) h(V)=0 where I,J are ideals and h is a left (or right)-generalized (σ,τ)- derivation of R. Keywords. (σ,τ)-Lie ideal, Prime ring, Commutativity AMS 2010. 16N60, 16W25
References
[1] M. Ashraf, N.-ur Rehman, S. Ali and M. R. Mozumder, On (σ,τ)-Lie Ideals with Generalized Derivations in Rings, International Journal of Algebra, Vol. 3, 2009.
[2] N. Aydın, On One Sided (σ,τ)-Lie Ideals in Prime Rings, Tr. J. of Math., 21, (1997).
[3] N. Aydın and Kandamar. H, (σ,τ)-Lie Ideals in Prime Rings, Turkish J. Math., 18, (1994), No: 2.
[4] N. Aydın and K. Kaya, Some Generalizations in Prime Rings with (σ,τ)-Derivation, Doğa- Tr. J. of Math. 16, (1992).
[5] J. Chang, On the Identity h(x)=af(x)+g(x)b, Taiwanese Journal of Mathematics, 7 ,(2003), No.1.
[6] Ö. Gölbası and E. Koc¸ On (σ,τ)-Lie ideals with Generalized Derivation, Bull. Korean Math. Soc. 47 (2010).
[7] E. Güven, Some Results on Generalized (σ,τ)-derivations in Prime Rings, Beitrage zur Algebra and Geometry, 54, (2013)
[8] E. Güven, K. Kaya and Soytürk. M, Some Results on (σ,τ)-Lie Ideals, Math. J. Okayama Univ. 49, (2007).
1 Kocaeli University, Kocaeli- Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
[9] K. Kaya, (σ,τ)-Lie ideals in Prime Rings, An.Univ. Timisoara, Stiinte Mat. ,30, (1992).
[10] K. Kaya, On Prime Rings With (σ,τ)-Derivations, Doga T.U. Mat. D. C., 12, (1988).
[11] K. Kaya and N. Aydın, Some Results on Generalized Lie Ideals, A Scientific Journal Issued by Jordan University for Women, Vol. 3, No. 1 (1999).
[12] J. H. Mayne, Centralizing Mappings of Prime Rings, Canad. Math. Bull, 27, (1984), 122- 126.
[13] K. H. Park and Y. S. Jung, J., Some Results Concerning (σ,τ)-Derivations on Prime Rings, J. Korea Soc. Math. Educ. Ser.B, Pure and App. Math., 10, (2003).
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
-Primary Hyperideals on Commutative Hyperrings Elif Ozel Ay1, Gursel Yesilot2 and Deniz Sonmez3
Abstract. The purpose of this paper is to define the hyperideal expansion. Hyperideal expansion is associated with prime hyperideals and primary hyperideals. Then, we define some of their properties. Prime and primary hyperideals' numerous results can be extended into expansions. Keywords. Krasner Hyperring, hyperideal, ideal expansion AMS 2010. 13A99, 13-XX
References
[1] Dongsheng, Z., -primary ideals of commutative rings, Kyunkpook Math. J. 41, 17-22, 2001
[2] Davvaz B., Leoreanu V., Hyperring Theory and Applications, Int. Academic Press, U.S.A., 2007
[3] Ameri R., Norouzi M., On Commutative Hyperrings, Int. Journal of Algebraic Hyperstructures and its Applications, 1, 1, 45-58, 2014
[4] Davvaz B., Salasi A., A Realization of Hyperrings, Communications in Algebra, 34, 4389- 4400, 2006
1 Yildiz Technical University, Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected] 3 Yildiz Technical University, Istanbul, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Semicommutativity of Amalgamated Rings H. Kose1, Y. Kurtulmaz, B. Ungor and A. Harmanci
Abstract. In this work, we study some cases when an amalgamated duplication of a ring A along a proper ideal I of a ring B , AI f , is prime, semiprime, semicommutative, nil-semicommutative and weakly semicommutative. Keywords. Semicommutative ring, nil-semicommutative ring, weakly semicommutative ring, amalgamated ring. AMS 2010. 16S70, 16D80, 16S99.
This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF. A3. 16. 008
References
[1] Chen, W. and Cui, S., On weakly semicommutative rings, Commun. Math. Res., 27(2)179- 192, 2011.
[2] Chhiti, M., Mahdou, N., and Tamekhante, M., Clean property in amalgamated algebras along an ideal, Hacet. J. Math. Stat., 44(1), 41-49, 2015.
[3] D’Anna, M., and Fontana, M, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6(3), 443-459, 2007.
[4] Kim, N. K., Kwak, T. K., and Lee, Y., Semicommutative property on nilpotent products, J. Korean Math. Soc., 51(6), 1251-1267, 2014.
1 Ahi Evran University, Kirsehir, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Padovan Sequences in Centro-Polyhedral Groups Hasan Ozturk 1 and Engin Ozkan 2
Abstract. In [1], the authors defined the Padovan orbit of a 2 generator group G for generatoring pair x, y G . In this study, we obtain the generalized order-k Padovan orbit in the centro-polyhedral groups. Keywords. Padovan sequences, Centro-Polyhedral Groups, Length.
References
[1] Taş S. and Karaduman E., The Padovan Sequences İn Finite Groups, Chiang Mai J. Sci. 2014; 41(2): 456 - 462 .
1 Erzincan University, Erzincan, Turkey, [email protected] 2 Erzincan University, Erzincan, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Codes Defined via Especial Matrices over the Ring and Hadamard Codes Mustafa Ozkan1 and Figen Oke2
Abstract. In this study, certain matrices are obtained using the elements of a finite chain ring. Then using these matrices as generator matrices; certain codes and their duals are obtained. Moreover relations between these codes, binary codes and Hadamard codes are explained. Keywords. Hadamard Codes, Gray Map, Codes Over Rings. AMS 2010. 94B05, 94B15.
References
[1] Özkan, M., Öke, F., A relation between Hadamard codes and some special codes over
22 u App. Mathematics and Inf Sci.10 No 2,701-704, 2016.
[2] Jian-Fa, Q. , Zhang L.N. and Zhu S.X., u _)1( cyclic and cyclic codes over the ring ,Applied Mathematics Letters,19,820-823,2006.
[3] S. Zhu, Y. Wang, M. Shi, Some Result On Cyclic Codes Over 22 v , IEEE Trans. Inf. Theory, 56,no 4 ,1680-1684,2010.
[4] Krotov, D. S. Z4-linear perfect codes, Diskretn. Anal. Issled. Oper. Ser.1.Vol. 7, 4. P. 78– 90, 2000.
[5] Krotov, D. S., Z4-linear Hadamard and extended perfect codes, Procs. of the International Workshop on Coding and Cryptography, Paris, pp. 329-334,2001.
[6] Vermani, L. R., Elements of Algebraic Coding Theory, Chapman Hall , India., 1996.
[7] J. Wolfmann, Negacyclic and cyclic codes over , IEEE Trans. Inf. Theory, 45, 2527- 2532, 1999.
[8] A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes , IEEE Trans. Inf. Theory, 45, 1250-1255, 1999.
1 Trakya University, Edirne, Turkey, [email protected], [email protected] 2 Trakya University, Edirne, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Cyclic Codes over the New Ring Mustafa Ozkan1 and Figen Oke2
Abstract. In this study, a non-Frobenious ring over the Galois field p is defined.
Consequences obtained by Özkan M. and Öke F. for the case p 3 are generalized for the prime numbers p 3 . The ring wR is given, two Gray map are defined and the images of cyclic codes under these maps are investigated. Keywords. Hadamard Codes, Gray Map, Codes Over Rings. AMS 2010. 94B05, 94B15.
References
2 [1] Özkan M. and Öke F. Some Special Codes Over 3v 3 u 3 u 3 , Mathematical Sciences and Applications E-Notes .Volume 4 No 1, pp 40-44, 2016.
[2] B. Yıldız, S. Karadeniz, Linear codes over 2u 2 v 2 uv 2 , Des. Codes Cryptgr., 54, 61-71,2010.
[3] S. Karadeniz, B. Yıldız, On v)1( -constacyclic codes over , Journal of the Franklin Institude , 348, 2625-2632, 2011.
[4] Xu Xioafang, -constacyclic codes over IF2 vIFuIF 22 , Computer Engineering and Applications,49,12,77-79,2013.
[5] Udomkavanich P. and Jitman S., On the Gray Image of (1um )-Cyclic Codes uu m pk p k ... p k , Int. J. Contemp. Math. Sciences, Vol.4, No.26, 1265-1272, 2009.
[6] A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes 22 u , IEEE Trans. Inf. Theory, 45, 1250-1255, 1999.
[7] S. Roman, Coding and Information Theory, Graduate Texts in Mathematics, Springer Verlag, 1992.
1 Trakya University, Edirne, Turkey, [email protected], [email protected] 2 Trakya University, Edirne, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Group Inverses of Some Combinations of Three Tripotent Matrices Murat Sarduvan1 and Nurgul Kalayci2
Abstract. We establish an explicit expression of the group inverse of
cX11 cX 22 cX 33 cXX 412 cXX 513 cXX 623 , where X1 , X 2 , X 3 are tripotent matrices and c j , j 1,2, ,6 . Moreover, we study some subcombinations of this combination.
Keywords. Group inverse, tripotent matrix. AMS 2010. 15A09, 15A57.
References
[1] Liu, X., Wu, L., Yu, Y., The group inverse of the combinations of two idempotent matrices, Linear and Multilinear Algebra, 59(1), 101-115, 2011.
[2] Yinlan, C., Kezheng, Z., Tao, X., On nonsingularity and group inverse of linearcombinations of generalized and hypergeeralized projectors, Wuhan University Journal of Natural Sciences, 19(6), 469-476, 2014.
[3] Deng, C.Y., Characterizations and representation of the group inverse involving idempotents, Linear Algebra and its Applications, 434, 1067-1079, 2011.
[4] Liu, X., Wu, L., Benítez, J., On the group inverse of linear combinations of two group invertible matrices, Electronic Journal of Linear Algebra, 22, 490-503, 2011.
[5] Benítez, J., Sarduvan, M., Ülker, S., Özdemir, H., On nonsingularity of combinations of three group invertible matrices and three tripotent matrices, Linear and Multilinear Algebra, 61,463-481, 2013.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
A Note on 2-Absorbing Primary Hyperideals of Multiplicative Hyperrings Neslihan Suzen1 and Gursel Yesilot2
Abstract. The hyperstructure theory was first introduced by Marty in 1934 [7]. Since then, algebraic hyperstructures have been investigated by many researchers with applications in both pure and applied sciences [1], [4], [6], [8] .The multiplication hyperring which is one of the three types of hyperrings was introduced by Rota in 1982 [9], [10]. The concept of primary hyperideal of a multiplicative hyperring have been introduced and studied by Dasgupta[5]. In this study, we will introduce 2-absorbing hyperideal and 2- absorbing primary hyperideals in commutative multiplicative hyperrings, which were studied by Badawi, Tekir and Yetkin in ordinary algebra [2], [3]. We show that every 2-absorbing hyperideal is a 2- absorbing primary hyperideal, but the converse is not true. Also, we prove that every primary hyperideal is 2- absorbing primary hyperideal but a 2- absorbing primary hyperideal need not to be a primary hyperideal. We also obtain some results and examples that show the importance of being 2-absorbing primary hyperideal in a multiplicative hyperring. Keywords. Multiplicative hyperring, primary hyperideal, 2- absorbing hyperideal, 2-absorbing primary hyperideal AMS 2010. 13A99, 13A15
References
[1] Ameri, R., Norouzi, M., On commutative hyperrings, Int. Journal of Algebraic Hyperstructures and Its Applications, 1, no.1, 45-58, 2014.
[2] Badawi, A., On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75, no.3, 417-429, 2007.
[3] Badawi, A., Tekir, U., Yetkin, E., On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51, no.4, 1163-1173, 2014.
[4] Davvaz, B., Leoreanu,V., Hyperring Theory and Applications, International Academic Press, 2007.
[5] Dasgupta, U., On prime and primary hyperideals of a multiplicative hyperring, Analele Stiintifice aleUniversitatii Al I Cuza din Iasi-Matematica, 58, no.1, 19–37, 2012.
1 Yildiz Technical University, Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected]
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[6] Krasner, M., A class of hyperrings and hyperfields, Internat. J. Math. Math Sci.,6, 307- 312, 1983.
[7] Marty, F., Sur une generalization de la notion de groupe, 8iem congress des Mathematiciens Scandivanes, Stockholm, 45-49, 1934.
[8] Procesi, R., Rota, R., On some classes of hypertructures, Discrete Math., 208/209, 485- 497, 1999.
[9] Rota, R., Sugli iperanelli moltiplicativi, Rend. Di Mat., Series VII, 2(4), 711-724, 1982.
[10] Rota, R., Strongly distributive multiplicative hyperrings, J.Geom., 39, 130-138, 1990.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Matrix Representation of Octonions and Their Geometry Serpil Halici1 and Adnan Karatas2
Abstract. In this study, firstly we consider vector matrix representation of octonion algebra. Zorn defined, a special matrix multiplication contains dot product and vector product in order to represent alternative octonion algebra. We investigate this multiplication and then we consider geometry of octonions. And for this purpose we use vector matrix representation. Keywords. Octonions, Split Octonions, Matrix Representations. AMS 2010. 17A20, 17A75, 15A15
References
[1] Baez, J. C. The Octonions. Bull. Amer. Math. Soc., 39 (2002), pp 145-205.
[2] Tian, Y. Matrix Representation of Octonions and Their Applications. Advances in Applied Clifford Algebras, 10 (2000), pp 61-90.
[3] Ward, J. P. Quaternions and Cayley Numbers, Mathematics and Its Applications, Kluwer Academic Publishers, 1997.
[4] Daboul, J., Delbourgo, R. Matrix representation of octonions and generalizations. Journal of Mathematical Physics, 40.8 (1999): 4134-4150.
[5] Zorn, M. Alternativkörper und quadratische Systeme. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Springer Berlin/Heidelberg, pp. 395-402, 1933.
[6] Chanyal, B. C. Split octonion reformulation of generalized linear gravitational field equations. Journal of Mathematical Physics, 56.5 (2015): 051702.
[7] Halıcı, S., Karataş, A. Some Matrix Representations of Fibonacci Quaternions and Octonions. Advances in Applied Clifford Algebras, 1-10. DOI 10.1007/s00006-016-0661-2.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Pamukkale University, Denizli, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Generalized Fibonacci Octonions and Their Vector Matrix Representation Serpil Halici1 and Adnan Karatas2
Abstract. Octonion algebra is non-commutative, non-associative, eight dimensional algebra so octonion algebra cannot be isomorphic to an associative matrix algebra. In [5], author defined a new vector matrix representation for octonion algebra. In this study, we give the matrix representation of generalized Fibonacci octonions by using this vector matrix representation. To explain the issue, we give some examples. Keywords. Octonions, Split Octonions, Matrix Representation, Fibonacci Sequences. AMS 2010. 17A20, 17A75, 11B39.
References
[1] Baez, J. C. The Octonions. Bull. Amer. Math. Soc., 39 (2002), pp 145-205.
[2] Flaut, C., and Vitalii Shpakivskyi., Real matrix representations for the complex quaternions. Advances in Applied Clifford Algebras 23.3 (2013): 657-671.
[3] Keçilioğlu, O., and Akkus, İ., The Fibonacci Octonions. Advances in Applied Clifford Algebras. 25.1 (2015): 151-158
[4] Koshy, T., Fibonacci and Lucas numbers with applications. Vol. 51. John Wiley and Sons, 2011.
[5] Zorn, M. Alternativkörper und quadratische Systeme. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Springer Berlin/Heidelberg, 1933. pp. 395-402.
[6] Smith, J. DH., An introduction to quasigroups and their representations CRC Press, 2006.
[7] Halici, S., Karataş, A. Some Matrix Representations of Fibonacci Quaternions and Octonions. Advances in Applied Clifford Algebras, 1-10, DOI 10.1007/s00006-016-0661-2.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Pamukkale University, Denizli, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On k- Conjugate of Quaternions Serpil Halici1 and Sule Curuk2
Abstract. k-conjugate is defined as a mapping its own inverse. Since k-conjugate is an involution and involutions have the additional properties of distribution over addition and multiplication operations, we review formal axioms of k-conjugate. We discuss the concept of quaternion involutions and then consider some special conjugates of quaternions. Moreover, we give various identities by using the complex matrix representation of quaternions. Keywords. Quaternion, Quaternion matrix, Involution. AMS 2010. 16H05, 15A15, 11M55
References
[1] Todd A. Ell, Stephen J. Sangwine, Quaternion Involutions and Anti- Involutions, Computers and Math. With Appl. 53(2007),137-143.
[2] Ling, Si-Tao, Xue-Han Cheng, and Tong-Song Jiang. Consimilarity of quaternions and coneigenvalues of quaternion matrices. Applied Mathematics and Computation 270 (2015): 984-992.
[3] Kosal, Hidayet Huda, Mahmut Akyigit, and Murat Tosun. Consimilarity of Split Quaternion Matrices and a Solution of the Split Quaternion Matrix Equation X-AX_B= C. arXiv preprint arXiv:1406.7241 (2014).
[4] Ward, J.P. Quaternions and Cayley Numbers, Algebra and Appl. vol. 403, Kluwer, Dortrecht, 1997.
[5] Schulz, Dominik, and Reiner S. Thomä. Using Quaternion-Valued Linear Algebra. arXiv preprint arXiv:1311.7488 (2013).
[6] Zhang, Fuzhen. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Pamukkale University, Denizli, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Some New Classes of Permutation Polynomials over a Finite Field Suphawan Janphaisaeng1
Abstract. A permutation polynomial (over a finite field Fq ) is a polynomial
f()F[] x q x which induces a bijection map from to itself. One important research problems is to find criteria for a polynomial to be a permutation polynomial. A class of
k s permutation polynomials of the form ()x2 x x was derived by Zeng-Zhu-Hu [2] in
psk ptk 2010, and this was generalized to the forms ()()x x L x and ()()x x L x , where Lx() is a linearized polynomial, by Zha-Hu [3] in 2012 and by Sun [1] in 2014, respectively. Using techniques inspired by the work of Zeng-Zhu-Hu, new classes of
pts pjk p t permutation polynomials of the forms ()ax bx c x and ()ax bx cx x are derived. Keywords. Permutation polynomial, finite field. AMS 2010. 05A05, 11T06
References
[1] Sun, G. H., Several classes of permutation polynomials over finite fields, Journal of Computer and Communications, 2, 18-24, 2014.
[2] Zeng, X., Zhu, X. and Hu, L., Two new permutation polynomials with the form 2k s ()()x x L x over F n , Appl. Algebra Engrg. Comm. Comput. 21(2), 145-150, 2010. 2
[3] Zha, Z., Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields and Their Applications, 18, 781-790, 2012.
1 Naresuan University, Phitsanulok, Thailand, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
A Note on Neutrosophic Subring of a Ring Vildan Cetkin1 and Halis Aygun2
Abstract. In this talk, we propose the definition of a neutrosophic subring of a classical ring and investigate its characterizations and fundamental properties. Then, we describe the notion of a neutrosophic ideal and study its basic characteristics. Keywords. Neutrosophic set, ring, ideal, homomorphism of rings. AMS 2010. 05C25, 06E20.
References
[1] Arockiarani, I., Sumathi, I.R.Jency, J.M., Fuzzy neutrosophic soft topological space, Int. J. of Mathematical Archive, 4(10), 225-238, 2013.
[2] Çetkin, V., Aygün, H., An approach to neutrosophic subgroup and its fundamental properties, J. of Intell. & Fuzzy Systs., 29, 1941-1947, 2015.
[3] Dixit, V.N., Kumar, R., Ajmal, R., On fuzzy rings, Fuzzy Sets and Systs., 49, 205-213, 1992.
[4] Majumdar, P., Samanta, S.K., On similarity and entropy of neutrosophic sets, J. of Intell. & Fuzzy Syts., 26(3), 1245-1252, 2014.
[5] Smarandache, F., A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic, rehoboth:American Research Press, 1999.
[6] Wang, H. et al. Single valued neutrosophic sets, Proc. Of 10th Int. Conf. On Fuzzy Theory and Technology, Salt Lake City, Utah, July 21-26, 2005.
[7] Zadeh, L., Fuzzy sets, Information and Control, 8, 87-96, 1965.
1Kocaeli University, Kocaeli, Turkey, [email protected] 2Kocaeli University, Kocaeli, Turkey, [email protected]
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ANALYSIS
ANALYSIS
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
A Certain Subclass of Meromorphic Functions with Positive and Fixed Second Coefficients Associated with the Rafid-Operator
Arzu AKGUL 1
Abstract. In this study it is introduced that a new subclass of meromorphic funtions with positive coefficients defined by Rafid-operator and a necessary and sufficient condition for a function f to be in this class. We obtain coefficient inequality, distortion properties, meromorphically radii of close-to-convexity, starlikeness and convexity, convex linear combinations and integral transformation for the functions f in this class. Keywords. Meromorphic functions; positive coefficients; coefficient inequality; convex linear combination; extreme points; meromorphically starlikeness, convexity and close-to-convexity AMS 2010. Primary 30C45.
References
[1] W. G. Athsan and R. H. Buti, Fractional Calculus of a class of univalent functions with negative coefficients defined by Hadamard product with Rafid-Operator, Eur. J. Pure Appl. Math.,4(2), (2011), 162-173.
[2] M. K. Aouf and H.E.Darwish, Certain meromorphically starlike functions with positive and fixed second coefficients, Turkish J. Math., 21(3), (1997), 311-316
[3] M. K. Aouf and S. B. Joshi, On certain subclass of meromorphically starlike functions with positive coefficients, Soochow journal of Mathematics, 24(2), (1998),79-90.
[4] E. Aqlan, J. M. Jhangiri and S. R. Kulkarni, Class of K-uniformly convex and starlike functions, Tamkang J. Math. 35(2004), 1-7.
[5] W.G. Athsan and S.R. Kulkarni, Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivate I, J. Rajasthan Acad. Phys. Sci., 6(2) (2007), 129-140.
[6] S. K. Bajpai, A note on a class of meromorphic univalent functions, §Rev. Roumaine Math. Pures Appl., 22(1977), 295-297.
[7] J. Clunie, On meromorphic schilict functions, J. London Math.Soc.,34(1959), 115-216
1Kocaeli University, Kocaeli- Turkey, [email protected]
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[8] P. L. Duren, Univalent Functions, Springer, New York, NY, USA, 1983.
[9] F. Ghanim and M. Darus, On class of hypergeometric functions with fixed second positive coefficients, General Mathematics, 17(4),(2009), 13-28
[10] R. M. Goel and N. S. Sohi, On a class of meromorphic functions, Glasnik Mathematicci, 17 (1981).
[11] O. P. Junea and T. R. Reddy, Meromorphic starlike univalent functions with positive coefficients, Ann. Univ. Mariae Curie Sklodowska, Sect.A, 39 (1985), 65-76.
[12] I. B. Jung, Y. C. Kim, H.M. Srivastava, The Hardy spaces of analytic functions associated with certain one parameter families of integral operators, J.Math.Anal.Appl. 176 (1), (1993), 138-147.
[13] A. Y. Lashin, On certain subclasses of meromorphic functions associated with certain integral operators, Comput. Math. Appl. 59(1), (2010), 524-531.
[14] J.E. Miller, Convex meromorphic mapping and related functions, Proc. Amer. Math. Soc. 25(1970), 220-228
[15] S. S: Mller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol.225 of Monographs and Textbooks in Pure and applied Mathematics, Marcel Dekker, Newyork, NY,USA,2000
[16] Ch. Pommerenke, On meromorphic starlike functions, Pac.J.Math.13(1963), 221-235
[17] T. Ram Reddy, R. B. Sharma and K. Saroja, A new subclass of meromorphic functions with positive coefficients, Indian J. Pure Appl. Math., 44(1),(2013),29-26,
[18] W. C. Royster, Meromorphic starlike univalent functions, Trans. amer. Math. Soc., 107(1963), 300-308
[19] Thomas Rosy and Sunil Varma, On a subclass of meromorphic functions defined by Hilbert space operator, Geometry, Vol.( 2013), Article ID 671826, 4 pages,
[20] H. M. Srivastava and S. Owa, Current topics in analytic function theory, World Scientific Publishing, New Jersey,1992.
[21] B. A. Ureagaldi, Meromorphically starlike functions with positive and fixed second coefficients, Kyungpook Math. J.,29(1),(1998), 64-68
5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Iterative Roots of a Linear Function Boonrod Yuttanan1
Abstract. Various forms of polynomial-like functional equations have been studied extensively, see [1]–[7]. In this talk, we determine all continuous functions 푓: ℝ → ℝ of the 푞th order iterative polynomial-like functional equation 푓푞(푥) = 푏푥 + 푐 where 푏 and 푐 are real numbers with 푛 ≠ 0 extending earlier results of Sarkaria in [6]. Keywords. Functional equation, iteration, continuous function. AMS 2010. 39B12, 26A18.
References
[1] Jarczyk , W., On an equation of linear iteration, Aequationes Math. 51(1996), 303–310.
[2] Kuczma, M., Functional Equations in a Single Variables, Monografic Mat.46, PWN, Warszwa, 1966.
[3] Matkowski, J. and Zhang, W., On the polynomial-like iterative functional equation, in: T.M. Rassias (Ed.), Functional Equations and Inequalities, in: Math. Appl., vol. 518, Kluwer Academic, Dordrecht, 2000, pp. 145–170.
[4] Matkowski, J. and Zhang, W., On linear dependence of iterates, J. Appl. Anal. 6 (2000) 149–157.
[5] Nabeya, S., On the functional equation 푓(푝 + 푞푥 + 푟푓(푥)) = 푎 + 푏푥 + 푐푓(푥), Aequationes Math. 11(1974), 199–211.
[6] Sarkaria, K., Roots of translations, Aequationes Math. 75 (2008) 304–307.
[7] Zhang, W., Nikodem, K. and Xu, B., Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl. 315 (2006) 29–40.
1 Department of Mathematics and Statistics, Prince of Songkla University, Songkhla 90110, Thailand, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Gegenbauer Polynomials and Positive Definiteness Christian Berg1 and Emilio Porcu2
Abstract. Let d denote the d-dimensional unit sphere in d1 . The normalized ultraspherical polynomials associated to are defined as
cn d, x Cn x / Cn 1 , for d 1 / 2 , dn2,3,..., 0,
where Cn are the Gegenbauer polynomials given by the generating function
2 n 1 2xr r Cn x r, r 1, x n0 Let G denote an arbitrary locally compact group written multiplicatively and with neutral
d element e. We consider the class P ,G of continuous functions fG: 1,1 n d which are positive definite in the sense that for any and any 11,u ,..., nn , u G
n 1 the nn -matrix fk l, u k u l is hermitian and positive semi-definite. kl,1 These functions are precisely those with a (uniformly convergent) expansion
f x, u n, d u c n d, x , x , u 1,1 G , n0 where nd, is a sequence of continuous positive definite functions on G satisfying
e G e n0 nd, . When is the trivial group { } of one element, this theorem reduces to a famous result of Schoenberg [2]. The special case where G = has applications in geostatistics as covariance functions of time-dependent Gaussian random fields. Keywords. Positive definite function, Space-Time covariance. AMS 2010. 43A35, 33C55.
References
[1] C. Berg and E. Porcu, From Schoenberg coefficients to Schoenberg functions. ArXiv1505.05682. To appear in Constr. Approx.
[2] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108.
1 University of Copenhagen, Denmark, [email protected] 2 Universidad Tecnica Federico Santa Maria, Valparaiso, Chile, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Homogeneous B-Potential Type Integrals on Hardy Spaces Cansu Keskin1 and Ismail Ekincioglu2
Abstract. The authors establish some boundedness of homogeneous fractional 퐻푝 퐻푝 integrals on 훥휈Hardy space. By applying atomic-molecular decomposition of 훥휈Hardy space, they obtain the boundedness of homogeneous fractional type integrals which extends the Stein-Weiss and Taibleson-Weiss's results for the boundedness of the B-Riesz potential 퐻푝 operator on 훥휈Hardy space. Keywords. Laplace-Bessel operator, Bessel generalized shift operator, B-Riesz potential operator, atomic-molecular decomposition, Hardy space. AMS 2010. 34B30, 42B30, 47B06, 47F05.
References
[1] Chanillo, S., Watson, D., Wheeden, R.L., Some integral and maximal operators related to star-like, Studia Math. 107, 223-255, 1993.
[2] Ding, Y., Lu, S., The 퐿푝1푥 … 푥퐿푝푘 boundedness for some rough operators, J. Math. Anal. Appl. 203, 166-186, 1996.
[3] Ding, Y., Lu, S., Weighted norm inequalities for fractional integral operators with rough kernel, Can. J. Math. 50, 29-39, 1998.
[4] Ding, Y., Lu, S., Homogeneous fractional integrals on Hardy spaces, Tohoku Math. J., 52, 153-162, 2000.
[5] Gadzhiev, A. D., Aliev, I. A., Riesz and Bessel potentials generated by the generalized shift operator and their inversions, Theo. of Func. and Appr. (Russian) 1, 47-53, 1990.
[6] Guliyev, V. S., Sobolev theorems for the Riesz B-potentials, Dokl. RAN, (Russian), 358, 4, 450-451, 1998.
[7] Guliyev, V. S., Serbetci, A., Ekincioglu, I., On boundedness of the generalized B-potential integral operators in the Lorentz spaces, Int. Trans. and Spec. Funct., 18, 12, 885-895, 2007.
[8] Kipriyanov, I. A., Fourier-Bessel transformations and imbedding theorems, Trudy Math. Inst. Steklov, 89, 130-213, 1967.
1 Dumlupinar University, Kutahya, Turkey, [email protected] 2 Dumlupinar University, Kutahya, Turkey, [email protected]
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[9] Lyakhov, L. N., Multipliers of the Mixed Fourier-Bessel transform, Proc. Steklov Inst. Math., 214, 234-249, 1997.
[10] Levitan, B. M., Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk., (Russian) , 6, 2, 102-143, 1951.
[11] Muckenhoupt, B., Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 161, 249-258, 1971.
[12] Stein, E. M., Singular Integrals And Differentiability Properties of Functions, Princeton New Jersey, Princeton Uni. Press, 1970.
[13] Taibleson, M. H., Weiss, G., The molecular characterization of certain Hardy spaces, Asterisque 77 Societe Math. de France, 67-149, 1980.
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Ideal Convergent Sequence Spaces via Orlicz Function Emrah Evren Kara1 and Merve Ilkhan2
Abstract. In this presentation, we introduce some new sequence spaces via ideal convergence and an Orlicz function. Also we investigate some properties of the resulting spaces. Keywords. Ideal, ℐ-convergence, orlicz function. AMS 2010. 40A05, 40A35.
References
[1] Esi, A. Some new sequence spaces defined by Orlicz functions, Bull. Inst. Math. Acad. Sin. 27, 71-76, 1999.
[2] Esi, A. Et, M. Some new sequence spaces defined by a sequence of Orlicz functions, Indian J. Pure Appl. Math. 31(8), 967-972, 2000.
[3] Güngör, M. Et, M. Altın, Y. Strongly (푉휎, 휆, 푞)-summable sequences defined by Orlicz functions, Appl. Math. Comput. 157, 561-571 2004.
[4] Tripathy, BC, Hazarika B. Some I-convergent sequence spaces defined by Orlicz functions, Acta Math. Appl. Sin. 27(1), 149-154, 2011.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On the Characterizations of Variable Exponent Hardy Spaces According to Riesz Bessel Transform Esra Kaya, Ismail Ekincioglu1 and Cansu Keskin2
Abstract. Let 푝(∙):ℝ푛 → (0, ∞) be a variable exponent function satisfying that there exists a constant 푝0 ∈ (0, 푝−), where 푝− ∶= ess inf푥∈ℝ푛 푝(푥), such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space 퐿푝(∙)⁄푝0(ℝ푛). In this article, via investigating relations between boundary valued of harmonic functions on upper half space and elements of variable exponent Hardy space 퐻푝(∙)(ℝ푛). We characterize 퐻푝(∙)(ℝ푛) with respect to the first order Riesz Bessel transforms generated by generalized by 푛−푘−|훾| shift operator when 푝 ∈ ( , ∞). − 푛 Keywords. Hardy space, Riesz-Bessel operator, variable exponent. AMS 2010. 42B30, 47B06, 42B25, 47F05.
References
[1] Aliev, I.A., On Riesz transformations generated by a generalized shift operator, Ivestiya Acad. Of Sci. Azerbaydian 1, 7-13, 1987.
[2] Cruz-Uribe, D., Wang, L. A. D., Variable Hardy spaces, Indiana Univ. Math. J. 63, 447- 493, 2014.
[3] Cruz-Uribe, D. V., Fiorenza, A., Variable Lebesgue spaces. Foundations and harmonic analysis, Applied and numerical Harmonic Analysis, Birkhauser, Springer, Heidelberg, 2013.
4] Ekincioğlu, I., Riesz transformations generated by a generalized shift operator (Turkish), Ph. D. Thesis, Ankara University, Institute of Science and Technology, Ankara, 1994.
5 Ekincioğlu, I., The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted 퐿푝,푤,훾-spaces with general weights, Acta Appl. Math., 109, 591-598, 2010.
[6] Fefferman, C., Stein, E. M., 퐻푝 spaces of several variables, Acta Math. 129, 137-195, 1972.
7 García-Cuerva, J., Robio de Francia, J., Weighted norm inequalities and related topics, Amsterdam, North-Holland, 1985.
1 Dumlupinar University, Kutahya, Turkey, [email protected], [email protected] . 2 Dumlupinar University, Kutahya, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
8 Grafakos, L., Modern Fourier analysis, Second edition, Graduate Texs in Mathematics, 250, Springer, New York, 2009.
9 Hou, S., Yang, D., Yang, S., Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math., 15, 6, 1350029, 37, 2013.
10 Hunt, R., Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176, 227-251, 1973.
11 Keskin, C., The singular integral operators related to generalized shift operator in Hardy spaces (Turkish), Ph. D. Thesis, Dumlupınar University Science Institute, Kutahya, 2015.
푃 12 Ky, L. D., A note on 퐻휔-boundedness of Riesz transforms and θ-Calderón-Zygmund operators through molecular characterization, Anal. Theory Appl. 27, 251-264, 2011.
13 Levitan, B.M., Bessel function expansions in series and Fourier integrals, Ushki Math. Nauk (Russian) 6, 2, 102-143, 1951.
14 Lyakhov, L.N., Multipliers of the mixed Foyrier-Bessel transformation, Proc. V.A.Steklov Inst. Math. 214, 234-249, 1997.
15 Nakai, E., Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262, 3665-3748, 2012.
16 Peloso, M., Secco, S., Local Riesz transforms characterication of local Hardy spaces, Collect. Math. 59, 299-320, 2008.
17 Stein, E. M., On the theory of harmonic functions of several variables. II. Behavior near the boundary, Acta Math. 106, 137-174, 1961.
18 Stein, E. M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N. J., 1970.
19 Stein, E. M., Harmonic Analysis: Real-variable methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993.
20 Stein, E. M., G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N. J., 1971.
21 Wheeden, R. L., A boundary value characterization of weighted 퐻1, Enseignement Math. 22, 2, 121-134, 1971.
22 Yang,D., Zhuo, C., Nakai, E., Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Math. Complutense, 29, 2, 245-270, 2016.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On the Lipschitz Stability of Inverse Nodal Problem for p-Laplacian Bessel Equation Etibar S. Panakhov1, Emrah Yilmaz2 and Tuba Gulsen3
Abstract. In this study, reconstruction and stability issues of inverse nodal problem for p Laplacian Bessel equation are considered and Lipschitz stability of inverse nodal problem for this Laplacian operator is studied. Also, it is shown that the space of all potential functions w is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation. Let us consider following Laplacian eigenvalue problem; ll (pp 1) ( 1) ( 1) (u ) ( p 1) w0 ( x ) u, 1 x a , (1) x2 with the Dirichlet conditions u(1) u ( a ) 0, (2) where l0,1,2,..., a , a , p 1 are constants, is a spectral parameter; w02( x ) L (1, a ) is real valued function and
(p 1) u(p 1) usgn u ,
(see [3]). Normally the equation (1) is considered by a condition at the origin. In this case, the problem becomes singular and it is not easy to solve inverse nodal problem in Laplacian case. Therefore, we will study the Lipschitz stability of inverse nodal problem for Laplacian Bessel operator on an interval that problem is not singular. Keywords. p-Laplacian Bessel Operator, Inverse Nodal Problem, Lipschitz Stability. AMS 2010. 34A55, 34L05, 34L20.
References
[1] Gulsen, T., Yilmaz, E., Inverse nodal problem for p_Laplacian diffusion equation with polynomially dependent spectral parameter, Communications, Series A1; Mathematics and Statistics, 65(2), 23-36, 2016.
[2] Koyunbakan, H. and Yilmaz, E., Reconstruction of the potential function and its derivatives for the diffusion operator, Z. Naturforch, A 63, 127-130, 2008.
1 Firat University, Elazig, Turkey, [email protected] 2 Baku State University, Baku, Azerbaijan, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
[3] Koyunbakan, H., Inverse nodal problem for p-Laplacian energy-dependent Sturm- Liouville equation, Bound. Value Probl., 2013:272, 2013 (Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation, Bound. Value Probl., 2014:222, 2014.
[4] Law, C. K. and Tsay, J., On the well-posedness of the inverse nodal problem, Inverse Probl., 17, 1493-1512, 2001.
[5] Marchenko, V. A. and Maslov, K. V., Stability of the problem of recovering the Sturm- Liouville operator from the spectral function, Sb. Math., 81 (123), 475-502, 1970.
[6] McLaughlin, J. R., Stability Theorems for two inverse spectral problems, Inverse Probl., 4, 529-540, 1988.
[7] Yantir, A., Oscillation theory for second order differential equations and dynamic equations on time scales, Master of Science, Izmir institue of Technology, Izmir, 2004.
[8] Yilmaz, E. and Koyunbakan, H., On the high order Lipschitz stability of inverse nodal problem for string equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal., 21, 79- 88, 2014.
[9] Wang, W. C., Cheng, Y. H. and Lian, W. C., Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions, Math. Comput. Modelling, 54 (11-12), 2718-2724, 2011.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Factorization through Lorentz Spaces Associated to a Vector Measure Fernando Mayoral1 A. Fernández2, F. Naranjo3, R. Del Campo4 and E.A. Sánchez-Pérez5
Abstract. In the context of operators defined on Banach function spaces, we analyze the factorization through Lorentz type spaces defined on the vector measure associated to the operator. The talk is based in the joint paper [1] of the cited authors. The main properties of the considered Lorentz type spaces can be found in [2]. Keywords. Banach function spaces, vector measures, real interpolation, factorization. AMS 2010. Primary 46E30; Secondary 47B38, 46B42.
References
[1] del Camp R.; Fernández, A.; Mayoral F.; Naranjo F.; Sánchez-Pérez, E.A.; Lorentz spaces of vector measures and real interpolation of operators, Preprint (2016).
[2] Fernández, A.; Mayoral F.; Naranjo F.; Real interpolation method on spaces of scalar integrable functions with respect to vector measures, J. Math. Anal. Appl., 376, 203-211, (2011).
1 Universidad de Sevilla, Spain, [email protected] 2 Universidad de Sevilla, Spain, [email protected] 3 Universidad de Sevilla, Spain, [email protected] 4 Universidad de Sevilla, Spain, [email protected] 5 Universidad Politécnica de Valencia, Spain, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On New Inequalities of Hermite-Hadamard Type for -1-convex Functions Gabil Adilov 1 and Ilknur Yesilce 2
Abstract. Let f : a,b be a convex function, then the inequality
ab 1 b f a f b f f x dx 22ba a is known as the Hermite-Hadamard inequality that was proven by Hermite in [1] and then, ten years later, Hadamard rediscovered in [2] (see also [3]). Then, various types of Hermite- Hadamard Inequalities were studied ([3,4,5]). -1-convex sets and functions were introduced in [6,7]. In this work, we examine a new inequality of Hermite-Hadamard type for -1-convex functions. Keywords. -1-convexity, -1-convex functions, Hermite-Hadamard Inequality. AMS 2010. 52A40, 52A41.
References
[1] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d'une fonction consideree par Riemann, Journal des Mathematiques Pures et Appliquees, 58, 171- 215, 1893.
[2] Ch. Hermite, Sur deux limites d'une integrale dene, Mathesis, 3, 82, 1883.
[3] S.S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
[4] I. Yesilce, G. Adilov, Hermite-Hadamard Inequalities for L(j)-convex Functions and S(j)- convex Functions, Malaya Journal of Matematik, 3, 3, 346-359, 2015.
[5] H. Kavurmaci, M. Avci, M.E. Ozdemir, New Inequalities of Hermite-Hadamard Type for Convex Functions with Applications, Journal of Inequalities and Applications, 2011:86 2011, doi: 10.1186/1029-2011-86.
[6] G. Adilov and I. Yesilce, -1-convex sets and -1-measurable maps, Numerical Functional Analysis and Optimization, 33, 2, 131-141, 2012.
[7] G. Adilov and I. Yesilce, -1-convex functions, Journal of Convex Analysis, 24, 2, 2017.
1 Akdeniz University, Antalya, Turkey, [email protected] 2 Mersin University, Mersin, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On Generalized Lupas Operators H. Gul Ince Ilarslan1
Abstract. In this work, by taking a continuously differentiable, increasing and unbounded function𝜌, we consider an extension of the Lupas operator 퐿푛 in the form −1 퐿푛(푓표𝜌 )표𝜌 for convenient functions푓 on [0, ∞). We give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation. Keywords. Generalized Lupaş operator; Weighted approximation; Voronovskaya type theorem. AMS 2010. 41A36, 41A25.
References
[1] Agratini, O., On a sequence of linear and positive operators, Facta Univ. Ser. Math. Inform. No. 14 (1999), 41--48.
[2] Aral, A., Inoan, D. and Raşa, I., On the generalized Szász-Mirakyan operators, Results Math. 65 (2014), no. 3-4, 441- 452.
[3] Cárdenas-Morales, D., Garrancho, P. and Raşa, I., Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62 (2011), no. 1, 158-163.
[4] Holhoş, A., Quantitative estimates for positive linear operators in weighted spaces, Gen. Math. 16 (2008), no. 4, 99--110.
[5] Lupas¸ A., The approximation by some positive linear operators, In: Proceedings of the International Dortmund Meeting on Approximation Theory (M.W. Müller et al.,eds.),Ak ademie Verlag, Berlin, (1995), 201-229.
1 Gazi University, Ankara, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
A New Characterization of Bessel Potential Spaces Ilham A. Aliev1
∝ 푛 Abstract. The Bessel potentials 퐽 휑, (휑 ∈ 퐿푝 (푅 )) and the corresponding spaces ∝ 푛 ∝ 푛 퐻푝 (푅 ) = {푓|푓 = 퐽 휑, 휑 ∈ 퐿푝 (푅 )} of these potentials were introduced by N. Aronszajn, K. Smith, and A. Calderon. These potentials play important role in analysis and applications ∝ ([1,p.121-141]; see, also, [2,3]). In this work we define new bi-parametric potentials 퐽훽 휑 =
훽 −∝/훽 푛 ∝ 푛 (퐸 + (−∆)2 ) 휑, (휑 ∈ 퐿푝 (푅 )), and introduce corresponding function spaces 퐻훽,푝(푅 ). ∝ 푛 Although these spaces actually coincide with the classical Bessel potential spaces 퐻푝 (푅 ) up to equivalence of norms (see [4]) and various characterizations of the latter spaces is known (see, e.g. [2],[3]) we give here a new characterization via special wavelet-like transform. ∝ 푛 Note that most of known characterizations of the spaces 퐻푝 (푅 ) are given in terms of finite differences, the order of which increases with parameter α. In the "wavelet language" finite differences are replaced by wavelet measures, the number of vanishing moments of which serves as a substitute for the order of a finite difference. The additional parameter β enables us to minimize the number of vanishing moments as much as we please, no matter how big a parameter α is. Keywords. Bessel potentials, semigroup, wavelet transforms, Sobolev spaces. AMS 2010. 26A33, 46E35, 42C40.
References
[1] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.
[2] S.G.Samko, Hypersingular integrals and their applications; ser.:Analytical Methods and Special Functions, Taylor&Francis, London, 2002.
[3] B. Rubin, Fractional Integrals and Potentials, Pitman Monographs and Surveys in Pure and Appl. Math., 82, Longman, 1996.
[4] I.A. Aliev, Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms, Integral Equations and Operator Theory, 65(2009), 151-167.
1Akdeniz University, Antalya, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Necessary and Sufficient Conditions for the Boundedness for Singular Integral Operators with in the Lorentz Spaces Ismail Ekincioglu, Cansu Keskin1 and Sedat Pak2
Abstract. In this paper, we characterize when a variety of singular integral operators including the Hardy-Littlewood maximal function and Riesz-Bessel transforms are bounded from 퐿푝,ν to 퐿푞,푤 , for arbitrary weghts w, 휈 and indices 1 < 푝 ≤ 푞 < ∞. Keywords. Bessel differential operator, Bessel generalized shift operator, Riesz- Bessel transformations AMS 2010. 47G10, 45E10, 47B37.
References
[1] Aliev, I. A., Gadjiev, A. D., On classes of operators of potential types, generated by a generalized shift. (Russian) Reports of enlarged Sessionof the Seminar sofI.N.Vekua Inst.of Applied Mathematics, Tbilisi, 3, 2, 21-24, 1988.
[2] Bennet, C., Sharpley, R., Interpolation of Operators, Academic Press, 1988.
[3] Carro, M., Pick, L., Soria, J., Stepanov, V. D., On embeddings between classical Lorentz spaces, Math. Ineq. and Appl., 4, 3, 397-428, 2001.
[4] Edmunds, D. E., Evans, W. D., Hardy Operators, Function Spaces And Embeddings, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.
[5] Ekincioglu, I., The Boundedness of High Order Riesz-Bessel Transformations Generated by The Generalized Shift Operator in Weighted 퐿푝,훾,휔 spaces with General Weights, Acta Appl. Math., 109, 2, 591-598, 2010.
[6] Ekincioglu, I., Keskin, C., Er, S., Hölder weight estimates of Riesz Bessel Singular Integrals Generated By a Generalized Shift Operator, Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 19, 1, 75-92, 2011.
[7] Ekincioglu, I., Sayan, H. H., Keskin, C., High Order Riesz Transforms and Mean Value Formula for Generalized Translate Operator, Journal of Ineq. And Appl., 148, 1, 1-18, 2014.
[8] Guliyev, V. S., Sobolev theorems for the Riesz B-potentials, Dokl. RAN, (Russian), 358, 4, 450-451, 1998.
1 Dumlupınar University, Kutahya, Turkey, [email protected], [email protected] 2 Necmettin Erbakan University, Konya, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
[9] Guliyev, V. S., On maximal function and fractional integral, associated with the Bessel differential operator, Math.Inequal. Appl., 6, 2, 317-330, 2003.
[10] Kipriyanov, I. A., Fourier-Bessel transformations and imbedding theorems for weight classes, Trudy Math. Inst. Steklov, 89, 130-213, 1967.
[11] Kipriyanov, I. A., Ivanov, L. A., The obtaining of fundamental solutions for homogeneous equations with singularities with respect to several variables. (Russian), Trudy Sem. S. L. , Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk. Sobolev, 1, 55-77, 1983.
[12] Kolyada, V. I., Rearrangments of functions and embedding of anisotropic spaces of Sobolev type, East J. Approx., 4, 2, 111-199, 1998.
[13] Levitan, B.M., Bessel function expansions in series and Fourier integrals, Uspekhi Mat. Nauk., (Russian) , 6, 2, 102-143, 1951.
[14] Löfström, L., Peetre, J., Approximation theorems connected with generalized translations, Mathematische Ann., 181, 255-268, 1969.
[15] Sawyer, E., Boundedness of classical opertors on classical Lorentz spaces, Studia Math., T. XCVI, 1990.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On the Convergence of Implicit Iteration Processes in Convex Metric Spaces Isa Yildirim 1
Abstract. In this work, we consider a new implicit iteration process and prove that it is faster than the other implicit iteration processes. We prove some convergence theorems for generalized contraction mappings in convex metric spaces. Finally, we present a numerical example to support the result proved herein. Keywords. Fixed point, convergence rate, implicit iteration processes. AMS 2010. 47H10, 54H25.
References
[1] Berinde, V., Picard iteration converges faster than Mann iteration for a class of quasi contractive operators, Fixed Point Theory and Applications, 2004 (2004), 97-105.
[2] Ciri´c, Lj. B., Rafiq, A., Caki´c, N., Ume, J. S., Implicit Mann fixed point iterations for pseudo-contractive mappings, Applied Mathematics Letters, 22, 581.584.
[3] Khan, S.H., Yildirim, I., Ozdemir, M., Convergence of an implicit algorithm for two families of nonexpansive mappings, Comput. Math. Appl. 59 (2010) 3084-3091.
[4] Zamfirescu, Z., Fix point theorems in metric spaces. Arch. Math. 23, 292-298 (1972).
1 Ataturk University, Erzurum, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Approximating Common Fixed Points of I-Asymptotically Quasi-Nonexpansive Mappings Isa Yildirim1
Abstract. In this presentation, we construct an iteration scheme for approximating common fixed points of two finite families of non-self I-asymptotically quasi-nonexpansive mappings. Furthermore, we prove some strong convergence theorems for such mappings in uniformly convex Banach spaces. Keywords. Non-self I-asymptotically quasi-nonexpansive mappings, common fixed point, uniformly convex Banach spaces. AMS 2010. 47H09, 47H10.
References
[1] Goebel, K., Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972) 171-174.
[2] Kiziltunc, H., Ozdemir, M., Akbulut, S., Common Fixed Points for Two Nonexpansive Nonself-Mappings With Errors in Banach Spaces, The Arabian Journal for Science and Engineering, 35 (2010) 215-224.
[3] Rhoades, B.E., Temir, S., Convergence theorems for I-nonexpansive mappings, Int. J. Math. and Mathematical Sciences, 2006, doi:10.1155/IJMMS/2006/63435:1-4.
[4] Tan, K.K., Xu, H.K., Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993) 301-308.
[5] Yildirim, I., On the Convergence Theorems of an Implicit Iteration Process for Asymptotically Quasi I-Nonexpansive Mappings, Hacettepe Journal of Math. and Statistics, 42 (6) (2013) 617-626.
1 Ataturk University, Erzurum, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Products of Weighted Composition Operators and Differentiation Operators between Weighted Banach Spaces and Weighted Zygmund Spaces of Analytic Functions Jasbir S. Manhas1
Abstract. Let v and w be weights on the unit disc D. Let Hv(D) be the weighted Banach space of analytic functions and Hw(D) be the weighted Zygmund space of analytic functions. In this paper, we investigate the analytic mappings φ : D →D and ψ :D→₵ which characterize the boundedness and compactness of products of weighted composition operators and differentiation operators 퐷푊휓,φ and 푊휓,φ퐷 between the weighted spaces Hv(D) and Hw(D). Keywords. Weighted Banach Spaces and Weighted Zygmund Spaces, Weighted Composition and Differentiation Operators, Bounded and Compact Operators. AMS 2010. 47B38, 47B33.
1 Sultan Qaboos University, Muscat, Oman, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
New Type of Lacunary Ideal Convergent Sequence Spaces Mahmut Dastan1, Emrah Evren Kara2 and Merve Ilkhan3
Abstract. The main purpose of this presentation is to define and study on some new spaces of lacunary ideal convergent and lacunary ideal bounded sequences. We discuss some inclusion relations of the resulting spaces and give some properties of these spaces. Keywords. Ideal, ideal convergence, Orlicz function, Lacunary sequence. AMS 2010. 40A05, 40A35.
References
[1] P. Kostyrko, W. Wilczynski, T. Salat: I-convergence, Real Anal. Exchange 26(2), 669- 686, 2000.
[2] B. C. Tripathy, B. Hazarika, B. Choudhary: Lacunary I-convergent sequences, Kyungpook Math. J. 52, 473-482, 2012.
[3] E. E. Kara, M. İlkhan: On some paranormed A-ideal convergent sequence spaces defined by Orlicz function, Asian J. Math. Comput. Research 4(4), 183-194, 2015.
[4] E. E. Kara, M. İlkhan: Lacunary I-convergent and lacunary I-bounded sequence spaces defined by an Orlicz function, Electron. J. Math. Anal. Appl. 4(2), 150-159, 2016.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected] 3 Duzce University, Duzce, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
The Dual Multiple Knot B-spline Frames Maryam Esmaeili1
Abstract. We provide a method for constructing dual Multiple Knot B-spline equiangular frames. We further apply the algorithm to several numerical examples. Finally, we discuss the derived results. Keywords. B-spline, dual frame, Multiple Knot. AMS 2010. 53A40, 20M15.
References
[1] Chui, C.K., Quak, E., Wavelets on a Bounded Interval, In: Braess D, Schumaker LL,editors.Numerical methods of approximation theory. Basel: Birkhauser Verlag, pp. 57-67, 1992.
[2] Christensen, O., Frames And Bases, An Introductory Course, Birkhӓuser, Boston, 2008.
[3] Esmaeili, M., Tavakoli, A., Construction of multiple knot B-spline wavelets on the interval, The Rocky Mountain Journal of Mathematics, to appear.
[4] Tavakoli, A., Esmaeili, M., Construction of dual multiple knot B-spline wavelets on the interval, (Submitted).
[5] Esmaeili, M., Construction of multiple knot B-spline equiangular frames, (Submitted).
1Hormozgan University, Bandar Abbas, Iran, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On Multiple Knot B-spline Frames Maryam Esmaeili1
Abstract. This paper deals with construction of non-uniform multiple knot B-spline frames. These frames are equiangular on a bounded interval. noserore ror serer rore re tru trst e s. Finally, some examples of multiple knot B-spline frames are also presented. Keywords. B-spline, frame, Multiple Knot. AMS 2010. 53A40, 20M15.
References
[1] Chui, C.K., Quak, E., Wavelets on a Bounded Interval, In: Braess D, Schumaker LL,editors.Numerical methods of approximation theory. Basel: Birkhauser Verlag, pp. 57-67, 1992.
[2] Christensen, O., Frames And Bases, An Introductory Course, Birkhӓuser, Boston, 2008.
[3] Esmaeili, M., Tavakoli, A., Construction of multiple knot B-spline wavelets on the interval, The Rocky Mountain Journal of Mathematics, to appear.
[4] Esmaeili, M., Construction of multiple knot B-spline equiangular frames, (Submitted).
1Hormozgan University, Bandar Abbas, Iran, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
New Type Open Sets in Bitopological Spaces Merve Ilkhan1 and Emrah Evren Kara2
Abstract. In this presentation, we define (푖, 푗)-훿-preopen, (푖, 푗)-훿-semiopen, (푖, 푗)-푎- open, (푖, 푗)-푒-open and (푖, 푗)-푒 ∗-open sets in bitopological spaces and study the relation between them. We give analogues results as [4] for the interiors and closures of these sets. Also, we introduce weaker forms of continuous functions between bitopological spaces by using these new open sets. Keywords. Bitopological space, 훿-open sets, preopen sets. AMS 2010. 54A05, 54E55.
References
[1] Kelly, J. C. Bitopological spaces, J. Proc. London Math. Soc. 13, 71-89 1963.
[2] Ekici, E. On e-open sets, 풟풫*-sets and 풟풫휀 ∗-sets and decompositions of continuity, Arab. J. Sci. Eng. 33(2A), 269-282, 2008.
[3] Ekici, E. On 푒 ∗-open sets, (풟, 풮) ∗-sets, Math. Morav. 13(1), 29-36 2009.
[4] Thamizharasi, G. Thangavelu, P. Remarks on closure and interior operators in bitopological spaces, J. Math. Sci. Comput. Appl. 1(1) , 1-8, 2010.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Cofinally Quasi Cauchy Continuity Merve Ilkhan1, Pınar Zengin Alp2 and Emrah Evren Kara3
Abstract. In this presentation, we define cofinally quasi Cauchy sequences and give the relations between these sequences with cofinally Cauchy and quasi Cauchy sequences by examples. Using these sequences, we define cofinally quasi Cauchy continuity as a generalization of forward continuity suggested in [1]. Further, we obtain some other results related to these concepts. Keywords. Cofinally Cauchy sequences, forward continuity, forward compactness AMS 2010. 40A05, 54C08
References
[1] H. Çakallı, Forward continuity, J. Comput. Anal. Appl. 13(2) (2011) 225-230.
[2] D. Burton, J. Coleman, Quasi Cauchy sequences, Amer. Math. Monthly, 117(4) (2010) 328-333.
[3] N. R. Howes, Modern Analysis and topology, Springer Verlag, New York, 1995.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected] 3 Duzce University, Duzce, Turkey, [email protected]
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A Version of Popovici’s Inequality Mirea Mihaela Mioara1
Abstract. In this paper, we present new inequalities based on convexity. T. Popovici proved in 1965 an interesting characterization of convex functions on integrals, relating the values at the barycenters of different subset of given finite set of points. In this case his result reads as follows: Theorem 1 Let 푓: 퐼 → 푅 be a continuous functions. Then f is convex if and only if, 푓(푥)+푓(푦)+푓(푧) 푥+푦+푧 2 푥+푦 푥+푧 푧+푦 + 푓 ( ) ≥ [푓 ( ) + 푓 ( ) + 푓 ( )], for all 푥, 푦, 푧 ∈ 퐼. 3 3 3 2 2 2 A Riemann integral analogue of Theorem 1 concerning convex functions defined on compact intervals is present of C. Niculescu. An important step in deriving the analogue in the following remark: 1 푏 Lemma Let 푓: [푎, 푏] → 푅 be a convex functions. The inequality ∫ 푓(푥)푤(푥)푑푥 + 푏−푎 푎 푎+푏 4 푏 푥 푥+푡 푓 ( ) ≥ ∫ ∫ 푓 ( ) 푑푡푑푥 holds for all functions f whose restrictions to the interval 2 (푏−푎)2 푎 푎 2 5푎+3푏 3푎+5푏 [ , ] all affine functions. 8 8
1 University of Craiova, Romania, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Maximal Hyponormal Differential Operators of First-Order with Smooth Coefficients Meltem Sertbas1
Abstract. In this paper, in terms of boundary values all maximal hyponormal extensions of a class of a minimal operator, generated by a linear differential-operator expression of first-order with smooth operator coefficients are described in Hilbert space of vector-functions in a finite interval. The structure of the spectrum of the maximal hyponormal extensions is examined. Keywords. Differential operator, hyponormal operator, Minimal and maximal operators, Extension, Spectrum of an operator. AMS 2010. 47A10, 47A20.
References
[1] Berezansky, Yu. M., Expansions in eigenfunctions of self-adjoint operators, Amer. Math. Soc. Providence, RI, 1968.
[2] Daleckii, J.U., Krein, M.G., Stability of solutions of differential equations in Banach space, Amer. Math. Soc., Providence, RI, 1974.
[3] Gorbachuk, V. I., Gorbachuk, M. L., Boundary value problems for operator differential equations, Kluwer Academic, Dordrecht, 1991.
[4] Ismailov, Z., Unluyol, E., Hyponormal differential operators with discrete spectrum, Opuscula Mathematica, 30, 79-94, 2010.
[5] Stochel, J., Szafraniec, F. H., The normal part of an unbounded operator, Nederl. Akad. Wetensch. Proc. Ser. A, 92, 495–503, 1989.
1 Karadeniz Technical University, Trabzon, Turkey, [email protected] tr
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Constructions of the Determinantal Representations of Hyperbolic Forms Mao-Ting Chien1
Abstract. Let A be an n-by-n matrix. A ternary form associated to A, defined by F(t, x, y; A) = det(t I + x H + y K), is hyperbolic with respect to (1, 0, 0), where H = (A + A*)/2 and K = (A – A*)/(2i). The Fiedler-Lax conjecture is recently proved by Helton and Vinnikov, namely, for any real ternary hyperbolic form G(t, x, y), there exist real symmetric matrices H and K such that G(t, x, y) = F(t, x, y; H+iK). We construct real symmetric matrices for the determinantal representations of some ternary forms and algebraic curves. Keywords. Determinantal representation, symmetric matrices, Fiedler-Lax conjecture. AMS 2010. 14J17, 15A60.
References
[1] Chien, M.T., Nakazato, H., Determinantal representations of hyperbolic forms via weighted shift matrices, Applied Math. Comput., 258, 172-181, 2015.
[2] Helton, J.W., Vinnikov V., Linear matrix inequality representations of sets, Comm. Pure Appl. Math., 60, 654–674, 2007.
1 Soochow University, Taipei, Taiwan, [email protected]
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The Refinements of Hilbert-Type Inequalities Predrag Vuković1
Abstract. In this article we establish a class of more accurate Hilbert-type inequalities based on an improved form of the Young inequality, known from the literature. We obtain refined and reversed relations in a general multidimensional case. As an application, we give improved versions of the classical Hilbert and Hardy inequalities. Keywords. Hilbert inequality, Hardy inequality, Young inequality AMS 2010. 26D10, 26D15.
References
[1] Krnić, M., Lovričević, N., Pečarić, J., Jessen's functional, its properties and applications, An. St. Univ. Ovidius Constanta, 20, 225-248, 2012.
[2] Krnić, M., Pečarić, J., Perić, I., Vuković, P., Recent Advances in Hilbert-type Inequalities, Element, Zagreb, 2012.
[3] Kufner, A., Maligranda, L., Persson, L. E., The Hardy inequality --About its history and some related results, Vydavatelsky servis, Pilsen, 2007.
1 University of Zagreb, Zagreb, Croatia, [email protected]
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Real Interpolation of p -th Power Factorable Operators
R. Del Campo1, A. Fernández2, F. Naranjo3, E.A. Sánchez4 and A. Fernández 5
Abstract. Let X be an order continuous Banach function space on a finite measure space (,,) and let E be a Banach space. Given a continuous linear operator TXE: we
p consider the vector measure mTA():() A T and the spaces Lm()T , with p 1, of p - scalar integrable functions with respect to the vector measure mT . The operator T is said to be -th power factorable if it factorizes through the space .
In this talk we study if the property being -th power factorable is inherited by interpolated operator in the real interpolation method. Keywords. Real interpolation, vector measure, -th power factorable operator. AMS 2010. 46B70, 46G10, 46E30.
References
[1] del Campo R., Fernández A.. Galdames O., Mayoral F., Naranjo F., Complex interpolation of operators and optimal domains, Intgr. Equ. Oper. Theory, 80, 229--238, 2014.
[2] del Campo R., Fernández A., Mayoral F., Naranjo F., Sánchez E.A., Lorentz spaces of vector measures and real interpolation of operators, preprint.
1Universidad de Sevilla, Sevilla, Spain, [email protected] 2Universidad de Sevilla, Sevilla, Spain, [email protected] 3Universidad de Sevilla, Sevilla, Spain, [email protected] 4Universidad de Sevilla, Sevilla, Spain, [email protected] 5Universidad de Sevilla, Sevilla, Spain, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
Reflexivity of Function Spaces Associated to A Σ-Finite Vector Measure R. Del Campo 1, Antonio Fernández 2, F. Mayoral 3 and F. Naranjo 4
Abstract. From the point of view of functional analysis the second most desired property of infinite spaces is reflexivity (the first one is completeness) and probably it is the most used in applications due to the weak compactness of its unit ball. Typical undergraduate examples of reflexive Banach spaces are Lebesgue Lp-spaces (1 < p < ∞) of a positive σ- finite measure. The corresponding scalar function spaces associated to a vector measure ν with values into a Banach space have been long studied. In this new context the things are really different. There appear several Lp-spaces associated to the vector measure: in the weak sense Lpw(ν), in the strong sense Lp(ν), and finally, integrability in the Choquet sense Lp(∥ν∥), of course for 1 ≤ p < ∞. These kind of spaces are, in general, different from each other and non reflexive, even for 1 < p < ∞. When the vector measure ν is defined on a σ- algebra the reflexivity of Lpw(ν) and Lp(ν) has been studied in [1]. Roughly speaking, for 1 < p < ∞, the space Lpw(ν), or equivalently Lp(ν), is reflexive if and only if they coincide. Also in the same context of a vector measure defined on a σ-algebra, the reflexivity of Lp(∥ν∥) is obtained as a byproduct of a general result about interpolation from [2], namely, Lp(∥ν∥) is always reflexive for all 1 < p < ∞. In the present paper we study the reflexivity of these spaces when the measure is defined on a δ-ring, a more general (but natural) structure than a σ- algebra. In this new context we can say that a similar result characterizing reflexivity of Lpw(ν) and Lp(ν) holds. Nevertheless Lp(∥ν∥) is not always reflexive. We characterize those vector measures for which Lp(∥ν∥) is reflexive as the locally strongly additive vector measures. Much of this work deals with this kind of measures. Keywords. reflexivity, integrable function, vector measure, δ-ring, locally strongly additive measure. AMS 2010. 46A25, 46G10, 46E30, 28B05.
References
[1] A. Fernández, F. Mayoral, F. Naranjo, C. Sáez, E.A. Sánchez-Pérez, Spaces of p- integrable functions with respect to a vector measure, Positivity, 10, 1, 1–16, 2006.
1Universidad de Sevilla, Sevilla, Spain, [email protected] 2Universidad de Sevilla, Sevilla, Spain, [email protected] 3Universidad de Sevilla, Sevilla, Spain, [email protected] 4Universidad de Sevilla, Sevilla, Spain, [email protected]
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[2] A. Fernández, F. Mayoral, F. Naranjo, Real interpolation method on spaces of scalar integrable functions with respect to vector measures, J. Math. Anal. Appl., 376, 1, 203–211, 2011.
[3] R. del Campo, A. Fernández, F. Mayoral, F. Naranjo, Reflexivity of function spaces associated to a σ-finite vector measure, J. Math. Anal. Appl., 438, 1, 339–350, 2016.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On a New Class of k-Uniformly Starlike Functions with Negative Coefficients Based on q-Derivative Operator Sahsene Altinkaya1 and Sibel Yalcin Tokgoz2
Abstract. In this work, we introduce a class of k-uniformly starlike functions of analytic univalent functions with negative coefficients. This class is based on a derivative ~ operator q zfD )( . Also we study coefficient bounds and extreme points.
Keywords. Analytic functions, Univalent functions, q-derivative. AMS 2010. 30C45, 33D15.
References
[1] Altıntaş, O., On a subclass of certain starlike functions with negative coefficient, Math. Japon. 36, 489-495, 1991.
[2] Aydoğan, M., Kahramaner, Y., Polatoğlu, Y., Close-to-Convex Functions Defined by Fractional Operator, Appl. Math. Sci., 7, 56, 2769-2775, 2013.
[3] Bharati, R., Parvatham, R., Swaminathan, A., On subclasses of uniformaly convex functions and correspondding class of starlike functions, Tamkang J. Math., 28, 17-32, 1997.
[4] Brahim , K. L., Sidomou, Y., On some symmetric q-special functions, Le Matematiche, LXVIII, 107-122, 2013.
[4] Biedenharn, L. C., The Quantum Group SU q )2( and a q-Analogue of the Boson Operators, J. Phys., A, 22, L873-L878, 1984.
[5] Goodman, A. W., On uniformly starlike functions, J. Math. Anal. Appl. 155, 364-370, 1991.
[6] Jackson, F. H., On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 46, 253-281, 1908.
[7] Kanas, S., Srivastava, H.M., Linear operators associated with k-uniformly convex functions, Integral Transform. Spec. Funct., 9, 121-132, 2000.
[8] Ronning, F., A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Sklodowska, 47, 13, 123-134, 1993.
[9] Srivastava, H.M., Mishra, A.K., Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl., 39, 3-4, 57-69, 2000.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On the Chebyshev Polinomial Coefficient Estimates for a Subclass of Analytic and Bi- Univalent Functions Sahsene Altinkaya1 and Sibel Yalcin Tokgoz2
Abstract. In this work, we establish a new subclass of analytic and bi-univalent functions. Using the Chebyshev polynomials, we obtain coefficient expansions for functions in this subclass. Keywords. Analytic functions, Bi-univalent functions, Chebyshev polynomials. AMS 2010. 30C45, 30C50.
References
[1] Altınkaya, Ş., Yalçın, S., Initial coefficient bounds for a general class of bi-univalent functions, International Journal of Analysis, Article ID 867871, 4 pp, 2014.
[2] Altınkaya, Ş., Yalçın, S., Coefficient bounds for a subclass of bi-univalent functions, TWMS Journal of Pure and Applied Mathematics, 6, 2, 180-185, 2015.
[3] Brannan, D. A., Taha, T. S., On some classes of bi-univalent functions, Studia Universitatis Babeş-Bolyai, Mathematica, 31, 2, 70-77, 1986.
[4] Brannan, D. A., Clunie, J. G., Aspects of comtemporary complex analysis, (Proceedings of the NATO Advanced Study Instute Held at University of Durham:July 1-20, 1979). New York: Academic Press, 1980.
[4] Dziok, J., Raina, R. K., Sokol, J., Application of Chebyshev polynomials to classes of analytic functions, C. R. Acad. Sci. Paris, Ser. I, 353, 433-438, 2015.
[5] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, USA, 259, 1983. [6] Doha, E. H., The first and second kind Chebyshev coefficients of the moments of the general-order derivative of an infinitely differentiable function, Intern. J. Comput. Math. 51, 21-35, 1994.
[7] Hamidi, S. G., Jahangiri, J. M., Faber polynomial coefficient estimates for analytic bi- close-to-convex functions, C. R. Acad. Sci. Paris, Ser.I, 352, 1, 17-20, 2014.
[8] Lewin, M., On a coefficient problem for bi-univalent functions, Proceeding of the American Mathematical Society, 18, 63-68, 1967.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
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[9] Magesh, N. Yamini, J., Coefficient bounds for a certain subclass of bi-univalent functions, International Mathematical Forum, 8, 27, 1337-1344, 2013.
[10] Mason, J. C., Chebyshev polynomials approximations for the L-membrane eigenvalue problem, SIAM J. Appl. Math., 15, 172-186, 1967.
[11] Netanyahu, E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1, Archive for Rational Mechanics and Analysis, 32, 100-112, 1969.
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On New Hermite-Hadamard-Fejér Type Inequalities for Harmonically Quasi-Convex Functions Selahattin Maden1, Imdat Iscan2 and Sercan Turhan3
Abstract. In this paper, we give the theorems and results of the both left and right side of new Hermite-Hadamard-Fejér type inequalities for harmonically-quasi convex functions via fractional integrals. Keywords. Harmonically-Quasi-convex function, Hermite-Hadamard-Fejér type inequalities, Fractional Integral. AMS 2010. 26D15, 26A51, 26D10, 26A15.
References
[1] Chen F., Wu S., Fejér and Hermite-Hadamard- Fejér type inequalities for harmonically convex functions, Journal of Applied Mathematics, Volume 2014, Article ID 386806.
[2] Chen, F., Extensions of the Hermite Hadamard inequality for harmonically convex functions via fractional integrals, Applied Mathematics and Computation., 268, 121-128, 2015.
[3] Iscan, I., Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Journal of Mathematics and Statistics 43 (6), 935-942, 2014.
[4] Iscan, I., Numan, S., Ostrowski typw inequalities for harmonically quasi-convex functions, Electronic Journal of Mathematical Analysis and Applications, Vol. 2(2), 189-198, 2014.
[5] Iscan, I., Ostrowski type inequalities for harmonically s-convex functions, Konuralp Jurnal of Mathematics, Volume 3, No 1, 63-74, 2015.
[6] Iscan, I., Kunt, M., Hermite-Hadamard- Fejér type inequalities for harmonically convex functions via fractional integrals, RGMIA Research Report Collection, 18, 1-16, 2015.
[7] Iscan, I., Fejér and Hermite-Hadamard-Fejer type inequalities for harmonically s- convex functions via Fractional Integrals, The Australian Journal of Mathematical Analysis and Applications, Vol: 12, 1, 1-6, 2015.
1 Ordu University, Ordu, Turkey, [email protected] 2 Giresun University, Giresun, Turkey, [email protected] 3 Giresun University, Giresun, Turkey, [email protected]
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[8] Latif, M.A., Dragomir, S.S., Momoniant, E., Some Fejér type inequalities for harmonically cnvex function special means, http://rgmia.org/papers/v18/v18a24.pdf.
[9] Latif, M.A., Dragomir, S.S., Momoniant, E., Fejér type inequalities for harmonically convex functions with applications, http://rgmia.org/papers/v18/v18a67.pdf.
[10] Niculescu, C. P., Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2), 155167, 2000.
[11] Niculescu, C. P., Convexity according to means, Math. Inequal. Appl. 6 (4), 571579, 2003. Available online at http://dx.doi.org/10.7153/mia-06-53.
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5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016)
On Hermite-Hadamard-Fejér Type Inequalities with Applications for Quasi Geometrically Convex Functions Sercan Turhan1, Selahattin Maden2 and Imdat Iscan3
Abstract. In this paper, the new identities are given for differentiable functions. As a result of this identities, both right and left sides of Hermite-Hadamard-Fejér type inequalities for differentiable quasi geometrically convex functions via fractional integrals and hypergeometric functions are obtained. Some applications to special means of real numbers are also given. Keywords. Quasi-Geometrically-convex, Hermite-Hadamard-Fejér type inequalities, Fractional Integral. AMS 2010. 26A51, 26D15, 26D10, 26A15.
References
[1] Iscan, I., Hermite-Hadamard type inequalities for GA-s-convex functions, Le Matematiche,, Vol. LXIX, Fasc. II, 129-146, 2014.
[2] Iscan, I., New general integral inequalities for quasi-geometrically convex functions via fractional integrals, Journal of Inequalities and Applications, 2013:491, 15 pages, 2013.
[3] Iscan, I., Some Generalized Hermte-Hadamard Type Inequalities for Quasi-Geometrically Convex Functions, American Journal of Mathematical Analysis, Vol. 1, No. 3,48-52, 2013.
[4] Iscan, I., Kunt, M., Hermite-Hadamard-Fejér Type Inequalities for Quasi-Geometrically Convex Functions via Fractional Integrals, Hindawi Publishing Corporation Journal of Mathematics, Article ID 6523041, 7 pages, 2016.
[5] Iscan, I., Turhan, S., Generalized Hermite-Hadamard-Fejér Type Inequalities for GA- Convex Functions Via Fractional Integral, (Submitted). http://arxiv.org/pdf/1511.03308v2.pdf.
[6] Ji, A. P., Zhang, T. Y., Qi, F., Integral Inequalities of Hermite Hadamard Type (α;m)-GA convex Functions, Journal of Function Space and Applications, Article ID 823856, 2013.
[7] Kilbas, A. A., Srivastava, H. M., Trujillo ,J. J., Theory and application of fractional differential equations. Elsevier, Amsterdam, 2006.
1 Giresun University, Giresun, Turkey, [email protected] 2 Ordu University, Ordu, Turkey, [email protected] 3 Giresun University, Giresun, Turkey, [email protected]
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[8] Latif, M.A., New Hermite Hadamard type integral inequalities for GA-convex functions with applications. Volume 34, Issue 4, 2014.
[9] Latif, M.A., Dragomir, S.S., Momoniant, E., Some Fejér type integral inequalities related with geometrically-arithmetically-convex functions with applications, (Submitted).
[10] Maden, S., Turhan, S., Iscan, I., New Hermite-Hadamard-Fejer Type Inequalities for GA-Convex Functions, AIP 1726, 020043-1–020043-4; doi: 10.1063/1.4945869, 2016.
[11] Niculescu, C. P., Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2), 155-167, 2000. Available online at http://dx.doi.org/10.7153/mia-03-19.
[12] Niculescu, C. P., Convexity according to means, Math. Inequal. Appl. 6 (4), 571-579, 2003. Available online at http://dx.doi.org/10.7153/mia-06-53.
[13] Shuang, Y., Yin, H.P., Qi, F., Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions, Analysis (Munich) 33 (2), 197-208, 2013. Available online at http://dx.doi.org/10.1524/anly.2013.1192.
[14] Zhang, T.Y., Ji, A.P., Qi, F., Some inequalities of Hermite-Hadamard type for GA- convex functions with applications to means, Le Matematiche, Vol. LXVIII, Fasc. I, 229-239, 2013.
[15] Zhang, X.M., Chu, Y.M., Zhang, X.H., The Hermite-Hadamard Type Inequality of GA- Convex Functions and Its Application, Journal of Inequalities and Applications, Volume 2010, Article ID 507560, 11 pages. doi:10.1155/2010/507560.
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The Hankel Determinant for a Certain Subclass of Univalent Functions Sibel Yalcin Tokgoz1 and Sahsene Altinkaya2
Abstract. In this work, making use of the generalized Hankel determinant, we investigate a certain subclass of univalent functions. Moreover, we obtain the Fekete-Szegö inequalities for this function class. Keywords. Univalent functions, Fekete-Szegö inequality, Hankel determinant. AMS 2010. 30C45, 30C50.
References
[1] Ding, S.S., Ling, Y., Bao, G.J., Some properties of a class of analytic functions, J. Math. Anal. Appl., 195 (1), 71-81, 1995.
[2] El-Ashwah, R., Kanas, S., Fekete-Szegö Inequalities for Quasi-Subordination Functions Classes of Complex Order, Kyungpook Math. J., 55, 679-688, 2015.
[3] Fekete, M., Szegö, G., Eine Bemerkung Über Ungerade Schlichte Funktionen, J. Lond. Math. Soc., 2, 85-89, 1933.
[4] Hayami, T., Owa, S., Hankel determinant for p-valently starlike and convex functions of order α, General Math., 17(4), 29-44, 2009.
[5] Hayami, T., Owa, S., Generalized Hankel determinant for certain classes, Int. Journal of Math. Analysis, 4 (52), 2473-2585, 2010.
[6] Kowalczyk, B., Lecko, A., Fekete-Szegö Problem for a Certain Subclass of Close-to- convex Functions, Bull. Malays. Math. Sci. Soc., 38, 1393-1410, 2015.
[7] Noonan, J. W., Thomas, D. K., On the second Hankel determinant of areally mean p- valent functions, Trans. Amer. Math. Soc., 223 (2), 337-346, 1976.
[8] Noor, K. I., Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., 28(c), 731-739, 1983.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
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A New Subclass of Analytic Functions Defined by Using Symmetric Q-Derivative Operator Sibel Yalcin Tokgoz1 and Sahsene Altinkaya 2
Abstract. In this work, we introduce a new subclass of analytic functions defined by using symmetric q-derivative operator. Moreover, for functions belonging to this function class, we investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points. Keywords. Analytic functions, coefficient estimates, distortion bounds, extreme points. AMS 2010. 30C45, 33D15.
References
[1] Aydoğan, M., Kahramaner, Y., Polatoğlu, Y., Close-to-convex functions defined by fractional operator, Appl. Math. Sci., 7, 56, 2769 – 2775, 2013.
[2] Biedenharn, L.C., The Quantum Group SU(2)q and a q-Analogue of the Boson Operators, J. Phys., A 22, L873-L878, 1984.
[3] Brahim, K. L., Sidomou, Y., On some symmetric q-special functions, Le Matematiche, LXVIII, 107-122, 2013.
[4] Gasper, G., Rahman, M., Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, MA, 1990.
[5] Jackson, F. H., On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 46, 253-281, 1908.
[6] Mohammed, A., Darus, M., A generalized operator involving the q-hypergeometric function, Mat. Vesnik, 65, 4, 454-465, 2013.
[7] Purohit, S. D., Raina, R. K., Fractional q-calculus and certain subclass of univalent analytic functions, Mathematica, 55, 1, 62-74, 2013.
[8] Özkan Uçar, H. E., Coefficient inequalties for q-starlike functions, Appl. Math. Comp., 276, 122-126, 2016.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
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A New Class of Salagean-type Univalent Functions as Related to Sigmoid Function Sibel Yalcin Tokgoz1 and Sahsene Altinkaya2
Abstract. In this paper, we define and investigate a new class of Salagean-type univalent functions in the open unit disk U {z∈ℂ:|z|<1}. Making use of Sigmoid function, we find estimates on the coefficients ,, aaa 432 . Moreover, upper bounds are obtained for
2 aa 23 , where μ∈ℂ.
Keywords. Sigmoid function, Salagean operator, Fekete-Szegö inequality. AMS 2010. 30C45.
References
[1] Bucur, R., Andrei, L., Breaz, D., Coefficient Bounds and Fekete-Szegö Problem for a Class of Analytic Functions Defined by Using a New Differential Operator, Appl. Math. Sci., 9, 1355 – 1368, 2015.
[2] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York , USA, 259, 1983.
[3] Fekete, M., Sezegö, G., Eine Bemerkung Über Ungerade Schlichte Funktionen, J. of the London Math. Soc., 2, 85-89, 1933.
[4] Fadipe-Joseph, O. A., Oladipo, A.T., Uzoamaka Ezeafulukwe, A., Modified sigmoid function in univalent function theory, International J. of Math. Sci. & Engg. Appls., 7, V, 313- 317, 2013.
[5] Kanas, S., Darwish, H. E., Fekete-Szegö problem for starlike and convex functions of complex order, Appl. Math. Let., 23, 777-782, 2010.
[6] Kowalczyk, B., Lecko, A., Fekete-Szegö Problem for a Certain Subclass of Close-to- convex Functions, Bull. Malays. Math. Sci. Soc., 38, 1393-1410, 2015.
[7] Salagean, G.S., Subclasses of univalent functions, in: Complex Analysis- Fifth Romanian Finish Seminar, Bucharest, 1, 362-372, 1983.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
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A New Study on Summability of Factored Fourier Series Sebnem Yildiz1
Abstract. In the present paper, the author presents a matrix generalization of two main results on the Np, n summability factors for the Ap, summability method by using k n k Fourier series. Some new results have also been obtained dealing with some other summability methods. Keywords. Summability factors, absolute matrix summability, Fourier series, infinite series. AMS 2010. 26D15, 40F05, 40G99, 42A24.
References
[1] Bor, H., On two summability methods, Math. Proc. Cambridge Philos. Soc. 97, 147-149, 1985.
[2] Bor, H., A note on two summability methods, Proc. Amer. Math. Soc. 98, 81-84, 1986.
[3] Bor, H., Multipliers for summability of Fourier series., Bull. Inst. Math. Acad. Sinica 17, 285-290, 1989.
[4] Bor, H., On the relative strength of two absolute summability methods. Proc. Amer. Math. Soc. 113, 1009-1012, 1991.
[5] Bor, H., On the absolute summability factors of Fourier Series, J. Comp. Anal. Appl. 8, 223-227, 2006.
[6] Bor, H., Factors for Np,,nn summability of Fourier series, Bull. Inst. Math. Acad. k Sin. (N.S.) 3, 399-406, 2008.
[7] Bor, H., Some new results on infinite series and Fourier series, Positivity 19, 467-473, 2015.
[8] Bor, H., Some new results on absolute Riesz summability of infinite series and Fourier series, Positivity 20, (in press), 2016.
1Ahi Evran University, Kirsehir, Turkey, [email protected]
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[9] Hardy, G. H., Divergent Series, Oxford University Press Oxford, 1949.
[10] Özarslan, H. S., A note on Np, n summability factors. Int. J. Pure Appl. Math. 13, 485- k 490, 2004.
[11] Özarslan, H. S., Öğdük, H. N., Generalizations of two theorems on absolute summability methods, Aust. J. Math. Anal. Appl., 1, 7 pp, 2004.
[12] Özarslan, H. S., Yıldız, Ş., A new study on the absolute summability factors of Fourier series. J. Math. Anal. 7, 31-36, 2016.
[13] Sulaiman, W. T., Inclusion theorems for absolute matrix summability methods of an infinite series, IV Indian J. Pure Appl. Math., 34 (11), 1547-1557, 2003.
[14] Tanoviç-Miller, N., On strong summability, Glas. Mat., 34 (14), 87-97, 1979.
Acknowledges. This work supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.E2.16.010.
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On the Degenerate Poly-Genocchi Numbers and Polynomials Veli Kurt1
Abstract. Last years, many mathematicians worked on the degenerate Bernoulli numbers and the degenerate Euler numbers. In this article, we define and investigate the degenerate Poly-Genocchi numbers and polynomials. We prove some theorems for these numbers and polynomials. We also give a relation between these numbers and Stirling numbers of the second kind. Keywords. Bernoulli polynomials and numbers, Genocchi polynomials and numbers, generating function, Stirling numbers of the second kind, degenerate Bernoulli numbers, degenerate Euler numbers. AMS 2010. 11B68, 11B83.
References
[1] Carlitz L, A degenerate Staudt-Clausen theorem, Arch. Math. (Besel), 7, 28-33, 1956.
[2] Carlitz L, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math., 15, 51-88, 1979.
[3] Kim D. S., Kim T., A note on degenerate poly-Bernoulli numbers and polynomials, arXiv: 1503.08418, 2015.
[4] Zhang Z., Yang J., On sums of products of the degenerate Bernoulli numbers, Integral Transforms Spe. Func., 20, 751-755, 2009.
1 Akdeniz University, Antalya, Turkey, [email protected]
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Arithmetic Properties about Quotients and Powers of Exponential Polynomials Vichian Laohakosol1 and Pinthira Tangsupphathawat2
Abstract. The following two arithmetic properties are well-known for polynomials. 1. ([1]) If Ax() and Bx() are two real polynomials taking integer values for each positive integer s and if B()() s A s , then A()/() x B x is a real polynomial.
2. ([5]) If is a real polynomial and m( 2) is an integer such that As()1/m is integral for each positive integer , then Ax()1/m is a real exponential polynomial. These two arithmetic properties have also been considered in [1], [2], [3] and [4] for exponential polynomials whose exponent bases are integers and whose coefficients are polynomial with integer coefficients. We investigate here the case of general real exponential polynomials, i.e., exponential polynomials with real exponent bases and real polynomial coefficients. This research is supported by the Faculty of Science and Kasetsart University through the grant No. RFG-13. Keywords. Exponential polynomial, quotient, power. AMS 2010. 33B10, 11B37.
References
[1] Lewis, D. J., Morton, P., Quotients of polynomials and a theorem of Pisot and Cantor, J. Fac. Sci. Univ. Tokyo 28, 813-822, 1982.
[2] Lovász, L., On finite Dirichlet series, Acta Math. Acad. Sci. Hungarica 22(1-2), 227-231, 1971.
[3] Perelli, A., Zannier, U., A note on arithmetic properties of certain recurring sequences, Colloquia Mathematica Societatis János Bolyai, 34. Topics in Classical Number Theory, Budapest (Hungary), 1209-1215, 1968.
[4] Perelli, A., Zannier, U., Arithmetic properties of certain recurrence sequences, J. Austral. Math. Soc. 37(Ser. A) , 4-16, 1984.
[5] Pólya, G., Szegö, G., Problems and Theorems in Analysis, Vol. II, Springer, 1976.
1 Kasetsart University, Bangkok, Thailand, [email protected] 2 Phranakhon Rajabhat University, Bangkok, Thailand, [email protected]
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APPLIED MATHEMATICS
APPLIED MATHEMATICS
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14-Point Averaging Operator for the Aproximation of the First Derivatives of a Solution of Laplace’s Equation in a Rectangular Parallelepiped Adiguzel A. Dosiyev1 and Hediye Sarikaya2
Abstract. A 14-point averaging operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace's equations in a rectangular parallelepiped. The boundary functions 휑푗 on the faces 훤푗 , j=1,2,...,6 of the parallelepiped are supposed to have fourth derivatives satisfying 4,휆 the Hölder condition, i.e. , 휑푗휖퐶 (훤푗), 0 < 휆 < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error 푢ℎ − 푢 of the approximate solution 푢ℎ at each grid point (x₁,x₂,x₃), |푢ℎ − 푢| 1 Near East University, Nicosia, North Cyprus, [email protected] 2 Near East University, Nicosia, North Cyprus, [email protected] 77 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On a Solution of Radial Schrödinger Equation for Special Potentials Arzu Guleroglu1, Cengiz Dane2 and Hasan Akbas3 Abstract. The solutions of eigenvalue problems which are given by radial Schrödinger 푐 equation with the potential 푉(푟) = 푎푟2 + 푏푟 − are determined using variational method. 푟 For this, we propose a trial function as the solution to eigenvalue problem. The eigenvalues are determined so that trial function become the solution of the corresponding variational problem. Keywords. Variational method, eigenvalue. AMS 2010. 34B09, 35A24, 76M30. References [1] Rektory, K., Variational Methods in Mathematics, Science and Engineering, D. Reidel Publishing Company, London, 1980. [2] Squires, G.L., Problems in Quantum Mechanics With Solutions, Cambridge University Press, 1995. [3] Paredes-Gutiérrez, H., Cuero-Yépez, J.C., Porras-Montenegro, N., Effect of spatially dependent screening on the binding energy of shallow impurities in spherical GaAs- (Ga,AI)As quantum dots, J. Appl. Phys., 75, 10, 5150-5153, 1994. [4] Ikhdair, S.M., Hamzavi, M., Spectral properties of quantum dots influenced by a confining potential model, Physica B, 407, 24, 4797–4803, 2012. [5] Chaudhuri, R.N., Mondal, M., Eigenvalues of anharmonic oscillators and the perturbed Coulomb problem in N-dimensional space, Phys. Rev. A, 52, 3, 1850-1856, 1995. 1 Trakya University, Edirne, Turkey, [email protected] 2 Trakya University, Edirne, Turkey, [email protected] 3 Trakya University, Edirne, Turkey, [email protected] 78 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Feedback Linearization for 2D Dynamical System Associated to the Mixing Flow Model Adela Ionescu1 Abstract. The mixing flow theory appears in an area with far from complete solving problems: the flow kinematics [4]. Its methods and techniques developed the significant relation between turbulence and chaos. The turbulence is an important feature of dynamic systems with few freedom degrees, the so-called “far from equilibrium systems”. These are widespread between the models of excitable media. The challenge is great, as the simulation parameters involve strong nonlinearities for the models, both in 2D and 3D case. [3] In the present paper a special standpoint is taken into account, namely the feedback linearisation of the dynamical system associated to the mixing flow model. The theory contains two fundamental nonlinear controller design techniques: input-output linearization and state-space linearization ([1], [2]). The approach is usually referred as input-output linearization or feedback linearization and is based on concepts from nonlinear systems theory. The resulting controller includes the inverse of the dynamic model of the process, providing that such an inverse exists. The results will be used for further analysis of 3D mixing flow dynamical system. Keywords. Mixing flow model, feedback linearization, control law. AMS 2010. 74F20; 93A30; 93B52; 93C15 References [1] Henson M, Seborg D. Nonlinear Process Control, Prentice Hall, Englewood Cliffs, New Jersey, 2005. [2] Isidori A. Nonlinear Control Systems, Springer-Verlag, New York, 1989 [3] Ionescu, A, Recent trends in computational modeling of excitable media dynamics. New computational challenges in fluid dynamics analysis, Lambert Academic Press, Saarbrucken, Germany, 2010 [4] Ottino, J.M: The kinematics of mixing: stretching, chaos and transport, Cambridge University Press. U.K., 1989. 1University of Craiova, Romania, [email protected] 79 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Principle of Mass Conservation for the Boltzmann’s Moment System of Equations in Fourth Approximation Aizhan Issagali1 and Auzhan Sakabekov2 Abstract. We study the one-dimensional non-linear non-stationary Boltzmann’s moment system of equations in fourth approximation with the tools developed by Sakabekov in [1-2]. For the third approximation Sakabekov proves the mass conservation law (cf. Theorem in [2]). We consider the initial value problem for the system of hyperbolic partial differential equations in the conservative form. We extend the proof of the principle of mass conservation to the fourth approximation. The principle of mass conservation allows to prove the global existence in time of the solution of the system of nine hyperbolic partial differential equations representing the fourth approximation of the one-dimensional non-linear non- stationary Boltzmann’s moment system of equations. Keywords. Boltzmann equation, moment system, mass conservation, initial value problem, hyperbolic partial differential equations AMS 2010. 35Q20, 35L65, 35L45 References [1] Sakabekov, A. (2002). Initial and boundary value problems for the Boltzmann’s moment system of equations. Almaty: Gylym. [2] Sakabekov, A., Auzhani, E. (2013). Analogue of the mass conservation law for third approximation of one-dimensional nonlinear Boltzmann’s moment system equations. J. Math. Phys., 54(5), 053512. doi:10.1063/1.4805363 1 Nazarbayev University, Astana, Kazakhstan, [email protected] 2 Kazakh National Research Technical University after K.I. Satpayev, Almaty, Kazakhstan, [email protected] 80 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Attainable Set of a Sir Epidemiological Model with Constraintson Vaccination and Treatment Stocks Ali Serdar Nazlipinar1, Anar Huseyin2 and N. Huseyin3 Abstract. In this study the controllable spread of some infectious disease is considered. The evolution model of the disease is described by the 3-dimensional nonlinear ordinary differential equations system. Vaccination and treatment are accepted as control parameters of the system. It is assumed that the stocks of vaccination and treatment is limited. Attainable sets of the system are approximately calculated for different control stocks. Graphical results are presented and possible biological applications are discussed. Keywords. SIR model, attainable set, control system, integral constraint. AMS 2010. 93B03, 93C15, 92D25 References [1] Kh.G. Guseinov, A. S. Nazlipinar, On the continuity property of Lp balls and an application, J.Math. Anal. Appl. 335, 1347-1359 (2007). [2] Kh.G. Guseinov, A.S. Nazlipinar, On the continuity properties of attainable sets of nonlinear control systems with integral constraint on controls, Abstr. Appl. Anal. (2008) doi:10.1155/2008/295817. Article ID 295817, 14 pages [3] Kh.G. Guseinov, Approximation of the attainable sets of the nonlinear control systems with integral constraints on control, Nonlinear Analysis, TMA 2009. 71. 622-645. [4] Kh.G. Guseinov, A.S. Nazlipinar, An Algorithm for Approximate Calculation of the Attainable Sets of the Nonlinear Control Systems with Integral Constraint on Controls, Comp. Math. Appl., Vol.62, No.4, P.1887-1895, 2011. doi:10.1016/j.camwa.2011.06.032 1 Dumlupinar University, Kutahya, Turkey, [email protected] 2 Cumhuriyet University, Sivas, Turkey, [email protected] 3 Cumhuriyet University, Sivas, Turkey, [email protected] 81 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Solving Container Terminal Scheduling Problems with the Plant Propagation Algorithm Birsen Irem Selamoglu1, Abdellah Salhi2 and Ghazwan Alsoufi3 Abstract. The Plant Propagation Algorithm (PPA) epitomised by the way the strawberry plant propagates, has been demonstrated to work well on both unconstrained and constrained continuous optimization problems [1], [2]. PPA emulates the strategy that plants deploy to survive by colonising new places which have good conditions for growth. Plants, like animals, survive by overcoming adverse conditions. The strawberry plant, for instance, has a survival and expansion strategy which is to send short runners to exploit the local area if the latter offers good conditions, and to send long runners to explore new and more remote areas otherwise. Recently, PPA has been extended to solve discrete optimization problems such as the well-known Travelling Salesman Problem (TSP), [3]. In this paper, we investigate its use to solve another class of discrete optimization problems, namely scheduling, [4], [5]. Here, the concern is with maritime operations arising at container ports, in particular Quay Crane Assignment and Quay Crane Scheduling problems, when combined. The main objective is to minimise vessel working time, which is the time it takes to discharge containers from a vessel and load it up when necessary [6], [7]. The performance of the algorithm on a representative list of test problems is compared to that of an exact algorithm, Branch-and-Cut in particular, as implemented in CPLEX, [8], and an adaptation of the Genetic Algorithm, [9]. Computational results are included. Keywords. Combinatorial optimisation, Quay crane assignment, Quay crane scheduling, Nature-inspired algorithms, CPLEX References [1] Salhi, A., Fraga, E., Nature-inspired optimisation approaches and the new plant propagation algorithm. In Proceedings of the ICeMATH2011 pp. K2-1 to K2-8, 2011. [2] Sulaiman, M., Salhi, A., Selamoğlu, B. İ., and Kirikchi, O. B., A Plant Propagation Algorithm for Constrained Engineering Optimisation Problems, Mathematical Problems in Engineering, vol. 2014, Article ID 627416, 10 pages, 2014. doi:10.1155/2014/627416 1 University of Essex, Colchester, UK, [email protected] 2 University of Essex, Colchester, UK, [email protected] 3 University of Essex, Colchester, UK, [email protected] 82 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [3] Selamoğlu, B. İ., Salhi, A., The Plant Propagation Algorithm for Discrete Optimisation: The Case of the Travelling Salesman Problem, In: Nature-Inspired Computation in Engineering, pp. 43-61. Springer, 2016. [4] Pinedo, M., Scheduling: Theory, Algorithms, and Systems, Prentice Hall, Englewood Cliffs, NJ, 1995. [5] Rodriguez, J.A.V., Salhi, A., Hybrid Evolutionary Methods for the Solution of Complex Scheduling Problems, In: Advances in Artificial Intelligence, pp. 17-28. Institu to Politecnico Nacional, 2006. [6] Diabat, A., Theodorou, E., An Integrated Quay Crane Assignment and Scheduling Problem, Computers & Industrial Engineering, Volume 73, July 2014, pp. 115-123, ISSN 0360-8352, http://dx.doi.org/10.1016/j.cie.2013.12.012. [7] Fu, Y.M., Diabat, A., Tsai, I. T., A multi-vessel quay crane assignment and scheduling problem: Formulation and heuristic solution approach, Expert Systems with Applications, Volume 41, Issue 15, 1 November 2014, pp. 6959-6965, ISSN 0957-4174, http://dx.doi.org/10.1016/j.eswa.2014.05.002. [8] IBM ILOG CPLEX Optimization Studio CPLEX User’s Manual, 2011. [9] Bierwirth, C., Frank, M., A survey of berth allocation and quay crane scheduling problems in container terminals, European Journal of Operational Research, 202(3):615-627, 2010. 83 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Study on the Dynamics of the Solution of a Riemann Type Problem in a Class of Discontinuous Functions Bahaddin Sinsoysal1, Hasan Carfi2 and Mahir Rasulov3 Abstract. In this study a new original method for founding a numerical solution Riemann problem for 2D Burgers equation with two piecewise constant initial condition in a class of discontinuous functions is proposed, that is, the following problem is considered. u u u u u ,0 (1) t x y u 0, , 1 rruyxu )cos,sin()0,,( 2 (2) u2 ,02,0, 2 It is known that the classical solution of the problem (1), (2) may not exist. In order to find the weak solution, the following auxiliary problem which has some advantages over the initial problem is introduced. y tyxw 1),,( 1 x 2 2 ),,(),,( 2 2 dtutyudtutxu 0),,(),,( (3) t 2 2 0 0 yxwyxw ),()0,,( (4) Here 0 yxw ),( is any continuously differentiable solution of the auxiliary problem (3), (4), then 2 tyxw ),,( tyxu ),,( yx tyxu ),,( , defined above, is a solution of the main problem. Using the advantages of the auxiliary problem, an efficient numerical algorithm to obtain the numerical solution is suggested. Some computer tests have been carried out. Keywords. Riemann problem, Shock and rarefaction waves, finite differences scheme in a class of discontinuous functions. AMS 2010. 35L65, 35L67, 65M06. 1 Beykent University, Istanbul, Turkey, bsinsoysal @beykent.edu.tr 2 Nisantasi University, Istanbul, Turkey, [email protected] 3 Beykent University, Istanbul, Turkey, [email protected] 84 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Guckenheimer, J., Shocks and rarefactions in two space dimensions, Arch. Rational Mech. Anal., 59, 3, 281-291, 1975. [2] Kruzkov, S. N., First order quasilinear equation in several independent variables, Math. USSR Sb., 10, 217-243, 1970. [3] Lindquist, W. B., Construction of solutions for two-dimensional Riemann problems, Comput. Math. Appl., 12A, 615-630, 1986. [4] Rasulov, M. A., Coskun, E., Sinsoysal, B., Finite differences method for a two- dimensional nonlinear hyperbolic equations in a class of discontinuous functions, App. Math. and Comp., 140, 1, 279-295, 2003. [5] Zhang, P., Zhang, T., Generalized characteristic analysis and Guckenheimer structure, Journal of Differential Equations, 152, 409-430, 1999. [6] Zhang, T., Zheng, Y., Two-dimensional Riemann problem for a single conservation law, Transactions of the American Mathematical Society, 312, 2, 589-619, 1989. [7] Sheng, W., Two-dimensional Riemann problem for scalar conservation laws, Journal of Differential Equations, 183, 239-261, 2002. 85 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Estimation of the Eigenvalues for Neumann Boundary Value Problems Bulent Yilmaz1 Abstract. In this article we obtain the asymptotic formulas and the numerical estimations of arbitrary order for eigenvalues of the non-selfadjoint Sturm-Liouville operators with Neumann boundary conditions, when the potential has singularity. Then using these we compute the main part of the eigenvalues in special cases. We obtain the comparison of the approximate eigenvalues. Keywords. Eigenvalues, Sturm-Liouville problems, Asymptotic method, Numerical method AMS 2010. 34L16, 34L20 References [1] G. D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Transactions of the American Math. Society, vol. 9, no. 4, pp. 373- 395, 1908. [2] N. Dunford, J.T. Schwartz Linear Operators, Part 3, Spectral Operators, New York, 1988. [3] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser Verlag, Basel, 1986. [4] M. A. Naimark, Linear Differential Operators, George G. Harrap and Company, 4th edition, 1967. [5] Ya. D. Tamarkin, Some General Preblem of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions, Math. Zeit. 27 (1927), 1-54. [6] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 76-90, 2002. [7] B. Yilmaz and O. A. Veliev, Asymptotic formulas for Dirichlet boundary value problems, Studia Scientiarum Mathematicarum Hungarica, vol.42, no.2, pp.153-171, 2005. 1 Marmara University, Istanbul, Turkey, [email protected] 86 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [8] Guldem Yildiz, Bulent Yilmaz, and O. A. Veliev., Asymptotic and Numerical Methods in Estimating Eigenvalues, Mathematical Problems in Engineering. Volume 2013, Article ID 415479. [9] Pryce, J.D., Numerical Solution of Sturm-Liouville Problems, Clarendon, Oxford, 1993 87 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Solutions of the Maximum of Difference Equations 11yxnn xymax , ; max , nn11x x y y n1 n 3 n1 n 3 Dagistan Simsek1 and Burak Ogul2 Abstract. The behaviour of the solutions of the following system of difference equations is examined. (1) where the initial conditions are positive real numbers. Keywords. Difference Equation, Maximum Operations, Semicycle AMS 2010. 39A12, 39A20. References [1] Cinar C., Stevic S. and Yalcinkaya I., On The Positive Solutions Of A Reciprocal Difference Equation With Minumum, J. Appl. Math. Computing, 17, 307-314, 2005. [2] Janowski E. J., Kocic V. L., Ladas G. and Tzanetopoulos G., Global behaviour of k maxxAn , solutions of xn1 , Journal of Difference Equations and Applications, 3, 297- xn1 310, 1998. [3] Papaschinopoulos G. and Hatzifilippidis V., On a max difference equation, Journal of Mathematical Analysis and Applications, 258, 258-268, 2001. [4] Patula W. T. and Voulov H. D., On a max type recursive relation with periodic coefficients, Journal of Difference Equations and Applications, 10, 3, 329-338, 2004. [5] Simsek D., Demir B. and Cinar C., On the Solutions of the System of Difference Equations x{n+1}=max{A/x{n},y{n}/x{n}}, y{n+1}=max{A/y{n},x{n}/y{n}}, Discrete Dynamics in Nature and Society, Volume 2009, Article ID 325296 (2009), 11 pages. 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan, [email protected]; Selcuk University, Konya, Turkey, [email protected] 2 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan, [email protected] 88 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) xnk(2 1) Solutions of the Rational Difference Equations xn1 1 xnk Dagistan Simsek1 and Burak Ogul2 Abstract. In this paper the solutions of the following difference equation is examined, xnk(2 1) xn1 , n=0,1,2,... (1) 1 xnk where the initial conditions are positive real numbers. Keywords. Difference Equation, Period 2k+2 Solution AMS 2010. 39A10, 39A12. References [1] Amleh A. M., Grove E. A., Ladas G. and Georgiou D. A., On the recursive sequence xn1 xn1 , J. Math. Anal. Appl., 233, no. 2, 790-798, 1999. xn xn1 [2] Cinar C., On the positive solutions of the difference equation xn1 , Appl. 1 axnn x 1 Math. Comp., 158 (3), 809–812, 2004. [3] Elabbasy E. M., El-Metwally H. and Elsayed E. M., Qualitative behavior of higher order difference equation, Soochow Journal of Mathematics, 33(4), 861-873, 2007. [4] Elsayed E. M., On the Dynamics of a higher order rational recursive sequence, Communications in Mathematical Analysis, 12 (1), 117–133, 2012. xn1 [5] Stevic S., On the recursive sequence xn1 , Taiwanese J. Math., Vol.6, No. 3, 405- gx()n 414, 2002. xn3 [6] Şimşek D., Çınar C. and Yalçınkaya İ., On the recursive sequence xn1 , Int. J. 1 xn1 Contemp. Math. Sci., 1, no. 9-12, 475-480, 2006. 1 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan, [email protected]; Selcuk University, Konya, Turkey, [email protected] 2 Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan, [email protected] 89 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Upper and Lower Solution Method for Fourth-order Three point BVPs on an Infinite Interval Erbil Cetin 1 Abstract. In this presentation, we deal with existence of solutions for fourth-order three-point boundary value problems on a half line. The existence result of at least one solution between a pair of lower and upper solutions is shown by using upper and lower solution method, the Schauder fixed point theorem Also by topological degree theory existence of at least three between two pairs of lower and upper solutions is proved. An example is given to illustrate the main results. Keywords. Fourth order boundary value problem, Schauder fixed point theorem, topological degree theory, upper and lower solution. AMS 2010. 35A15, 35B38 References [1] Graef, J. R., Qian, C., Yang, B., A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl, 287, 217-233, 2003. [2] Lian, H., Zha, J., Agarwal, R. P., Upper and lower solution method for n-th order BVPs on an infinite interval, Boundary Value Problems, 100, 17 pages, 2014, Doi:10.1186/1687- 2770-2014-100. [3] Ma, D., Yang, X., Upper and lower solution method for fourth-order four-point boundary value problems, J. Comput. Appl. Math., 223, 543—551, 2009 [4] Biriki, M., Moussaoui, T., O’Regan D., Existence of solutions for a fourth-order boundary value problem on the half-line via critical point theory, Electron. J. Qual. Theory Differ. Equ., 24, 1–11, 2016. [5] Agarwal, R. P., On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 91-110, 1989 1Ege University, Izmir, Turkey, [email protected] 90 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Asymptotic Behavior and Global Nonexistence of a Solution for a System of Nonlinear Higher-Order Wave Equations with Weak Damping Erhan Piskin1 Abstract. This work studies a initial-boundary value problem of the weak damped nonlinear higher-order wave equations. We prove that the solution decays exponentially and blows up with negative initial energy. Keywords. Higher-order wave equation, asymptotic behavior, global nonexistence. AMS 2010. 35B44. References [1] Adams, R. A., Fournier, J. J. F., Sobolev Spaces, Academic Press, 2003. [2] Georgiev, V., Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (2), 295-308, 1994. [3] Zhou Y., Global existence and nonexistence for a nonliear wave equation with damping and source terms, Math. Nacht, 278, 1341-1358, 2005. 1 Dicle University, Diyarbakir, Turkey, [email protected] 91 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Decay and Blow up of Solutions for Nonlinear Hyperbolic Equations with Nonlinear Damping Terms Erhan Piskin1 Abstract. In this talk, we consider the global existence, energy decay, and blow up of solutions for a nonlinear hyperbolic equation. We prove the energy decay estimates of the energy function by using Nakao’s inequality. Also, we study the blow up of solutions for the equation. Keywords. Hyperbolic equation, decay, blows up. AMS 2010. 35B44. References [1] Messaoudi, S.A., On the decay of solutions for a class of quasilinear hyperbolic equations with nonlinear damping and source terms, Math. Methods Appl. Sci. 28, 1819-1828, 2005. [2] Nakao, M., Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl. 58(2), 336-343, 1977. [3] Pişkin, E., Polat, N., Global existence, decay and blow up solutions for coupled nonlinear wave equations with damping and source terms, Turk. J. Math. 37, 633-651, 2013. [4] Wu, Y., Xue, X., Uniform decay rate estimates for a class of quasilinear hyperbolic equations with nonlinear damping and source terms, Appl. Anal. 92(6), 1169-1178, 2013. 1 Dicle University, Diyarbakir, Turkey, [email protected] 92 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Frictional Contact Problem for an Orthotropic Elastic Strip and Layer Elcin Yusufoglu 1 and Huseyin Oguz 2 Abstract. The frictional contact problem for an elastic strip under a rigid punch system is considered. The elastic strip is fixed to the bottom of surface. The rigid punch system is applied to top surface of orthotropic and homogenous strip with thickness h . The contact problem is reduced to a system of singular integral equations by using the theory of elasticity and the Fourier integral transforms. The singular integral equation is solved with the help of Series Method. The contact problem is numerically investigated. Elastic strip system under the pressure distribution due to the mechanical properties are examined and the results are shown in the table and tabular form Keywords. Contact Problem, Singular İntegral Equations, Fourier İntegral Transfrom, Series Method References [1] Muskhelishvili, N.I, 1958, Singular İntegral Equations, Wolters-Noordhoff Publishing, Groningen [2] Vorovich I.I., Aleksandrov V.M., Babeshko V.A.,1974, Nonclassical Mixed Problems of the theory of Elasticity, Nauka,Moscow [3] Erdoğan, F., Gupta, G.D., Contact and Crack Problems for an Elastic Wedge, International Journal of Applied Merhanics 60(1993) 663-639 1Usak University, Usak, Turkey, [email protected] 2Dumlupınar University, Kütahya, Turkey, [email protected] 93 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Sweep Method for Solving the Roesser Type Equation Describing the Motion in the Pipeline Fikret A. Aliev1, N. A. Aliev 1, N. A. Safarova 1, R. M. Tagiev 1 and G. H. Mammadova 1 Abstract. The initial problem for the system of hyperbolic equations describing the motion in oil production with gas lift method is considered [1]. Introducing a new variable which is the difference of gas pressure and volume (or gas-liquid mixture (GLM)) multiplied by a constant number (balancing unit of measurements), the original system of equations is reduced to the such form of equations which after appropriate discretization becomes a Roesser type discrete equation [2]. Searching the new variable as a linear function of the volume of gas (or GLM), it is shown that the coefficients satisfy the two difference equations of the first order, one of which corresponds to a quadratic equation and the second is a linear difference equation of the first order whose coefficients depend on the solution of the first h one. For the special case when the subintervals at time and at height of the well leading to zero the analytical expression for the sweep coefficients at each point of the domain of definition of the gas volume (or GLM) and pressure is obtained. In the case when the volume of the assessment gas and the motion (initial conditions) are constant at the mouth, it is shown that the results obtained by the Roesser model [3] coincide with the known results, where the concrete analytical expression for the parameter of the balancing unit of measurements is provided. Keywords. Gas lift, system of hyperbolic equations, balancing parameter, discretization, Roesser model. AMS 2010. 35L20, 65N22. References [1] Aliev, F.A., Ismailov, N.A., Namazov, A.A. Asymptotic method for finding the coefficient of hydraulic resistance in lifting of fluid on tubing, J. Inv. Ill-Posed Probl., 23, 511-518. 2015. [2] Aliev, F.A., Aliev, N.A., Mutallimov, M.M., Tagiev, R.M., An algorithm for constructing models of Roesser for gas lift process in oil production, Proceedings of IAM., 3, 2, 173-184, 2014. [3] Roesser, R.P., A discrete state space model for linear image processing, IEEE Trans. Autom. Contr., 20, 1975, 1-10. 1Institute of Applied Mathematics Baku State University, Baku, Azerbaijan, e-mail [email protected], [email protected], [email protected], [email protected] 94 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Free Vibration of Timoshenko Beam with 3D Tip Mass Subject to Bending in Orthogonal Planes and Torsional Deformation Hilal Doganay Kati1 and Hakan Gökdag2 Abstract. Vibration analysis of beam with tip mass has received great attention in the relevant literature. In this study mathematical modeling of a Timoshenko beam carrying a 3D rigid tip mass whose center of gravity is not coincident with beam end and undergoes bending deformation in orthogonal planes plus torsional deformation is derived. To this end, Hamilton’s Principle is applied to define the governing equations and all possible boundary conditions. Then, analytically solution is obtained, and results are compared with those belonging to Euler-Bernoulli beam theory. Moreover, well-known finite element software, ANSYS, is employed to check accuracy of the results. It is observed that analytical results are in good agreement with reference values. Figure1: Timoshenko beam with 3D tip mass Keywords. Timoshenko Beam, Bending and Torsional deformation, Tip mass. AMS 2010. 53A40, 20M15. References [1] Oguamanam D.C.D., Free Vibration of beams with Finite Mass Rigid Tip Load and flexural-torsional coupling, International Journal of Mechanical Sciences, 45, 963-979, 2003. 1 Bursa Technical Universisty, Bursa, Turkey, [email protected] 2 Bursa Technical Universisty, Bursa, Turkey, [email protected] 95 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [2] Oguamanam D.C.D., Arshad M., On the natural frequencies of a flexible manipulator with a tip load, Proceedings of the Institution of Mechanical Engineers, 219, 1199-1205, 2005. [3] Salarieh H., Ghorashi M., Free Vibration of Timoshenko Beam with Finite Mass Rigid Tip Load and Flexural Torsional Coupling, International Journal of Mechanical Science, 48, 763- 779, 2006. [4] Gökdağ H., Kopmaz O., Coupled Bending and Torsional Vibration of a Beam with in- span and tip attachments, Journal of Sound and Vibration, 287, 591-610, 2005. 96 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Solving Benjamin-Bona-Mahony Equation by Using the sn-ns Method and the Tanh- Coth Method Hami Gundogdu1 and Omer Faruk Gozukizil2 Abstract. In this study, we consider the Benjamin Bona Mahony equation (BBM) which is in the form of 푢푡 + 푢푥 + 푢푢푥 − 푢푥푥푡 = 0. We use the sn-ns method and the tanh- coth method to solve this equation. And, exact solutions are gained. Then, we compare the solutions which are found by using these methods. In addition to hyperbolic solutions that obtained by the tanh method, we obtain trigonometric solutions and elliptic function solutions by the sn-ns method. Keywords. Benjamin Bona Mahony equation (BBM), the sn-ns method, tanh-coth method, elliptic function solution, trigonometric solution. 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 97 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Order Conditions of Symplectic Partitioned Runge-Kutta Method for Stochastic Optimal Control Problems Hacer Oz1, Gerhard-Wilhelm Weber2 and Fikriye Yilmaz3 Abstract. In this work, we obtain the symplectic partitioned Runge-Kutta (SPRK) scheme for the optimal control problem of stochastic differential equations (SDEs). In order to discretize the optimal control problem, there are two basic approaches: discretize-then- optimize and optimize-then-discretize. We mainly focus on SPRK scheme for the optimal control problem of SDEs by following the discretize-then-optimize approach. After we present Hamiltonian formulations for the stochastic optimal control problem, we discretize the cost functional and the state equation with the help of Runge-Kutta schemes. To obtain the optimality system, we state the discrete Lagrangian. Then, we get the stochastic adjoint pair (푝푡, 푞푡). Our main contribution is to obtain an implicit Runge- Kutta scheme for the adjoint pair. After we get the SPRK scheme [3], we derive the order conditions of the SPRK scheme for the stochastic optimal control problems. We compare Stratonovich-Taylor expansion of the exact solution and Stratonovich-Taylor expansion of the approximation method defined by the SPRK scheme successively to get the order conditions. We have found additional order conditions to the classical Runge-Kutta schemes for SDEs [1,2] for both order-1 and order- 1.5. As applications, we choose some problems from finance. We compare the numerical results with the exact solutions and numerical examples confirm our results. Keywords. stochastic optimal control, Runge-Kutta discretization, symplectic partitioned Runge-Kutta scheme, Lagrange multiplier, Stratonovich-Taylor expansion. References [1] Burrage, K. and Burrage, P.M., High strong order explicit RungeKutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22(13), pp. 81-101, 1996. [2] Burrage, P.M., Runge-Kutta Methods for Stochastic Differential Equations, PhD Thesis, Department of Mathematics, University of Queensland, Australia, 1999. [3] Yılmaz, F., Öz, H., Weber, G.-W., Simulation of Stochastic Optimal Control Problems with Symplectic Partitioned Runge-Kutta Scheme, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms 22, pp. 425-440, 2015. 1 Middle East Technical University and Atilim University, Ankara, Turkey, [email protected] 2 Middle East Technical University, Ankara, Turkey, [email protected] 3 Gazi University, Ankara, Turkey, Ankara, Turkey, [email protected] 98 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Influence of the Impedance Coated Groove on the TEM Wave Radiation from Coaxial Waveguide Hulya Ozturk1 Abstract. In this work, Wiener-Hopf technique is used to reveal the effect of the coating grooves on the radiation of TEM waves in a coaxial waveguides. By applying Fourier transformation to the fields and the boundary conditions, problem is reduced to the solution of simultaneous modified Wiener-Hopf equations. In the numerical results, some simulations that illustrate the effect of impedance loading on the reflection coefficient are presented. Keywords. Diffraction, Wiener-Hopf Technique, Coaxial Waveguide. AMS 2010. 53A40, 20M15. References [1] Mittra, R., Lee, S. W., Analytical Techniques in the Theory of Guided Waves, McMillan, 1971. [2] Hacıvelioğlu, F, Büyükaksoy, A., Uzgören, G., Radiation characteristics of a coaxial waveguide with opposing dielectric-filled grooves, IET Science, Measurement and Technology, 4, 28-39, 2008. 1Gebze Technical University, Kocaeli, Turkey, [email protected] 99 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Determination of an Unknown Heat Source from Integral Overdetermination Condition for Quasilinear Parabolic Equation Irem Baglan1 and Fatma Kanca2 Abstract. In this research, we consider a coefficient problem of an inverse problem of a quasilinear parabolic equation with periodic boundary and integral over determination conditions. We prove the existence, uniqueness and continuously dependence upon the data of the solution by iteration method. Also we consider numerical solution for this inverse problem by using linearization and finite difference method and convergence analysis of this numerical method is proved. Keywords. Quasilinear Parabolic Equation, Inverse Problem, Periodic Boundary Condition, Finite Difference Method, Integral Overdetermination Condition. AMS 2010. 35K05, 35K29, 65M06, 65M12. References [1] Sakınc I., Numerical Solution of a Quasilinear Parabolic Problem with Periodic Boundary Condition, Hacettepe Journal of Mathematics and Statistics, 39(2):183-189, 2010. [2] Cannon J,R., Lin Y., Determination of parameter p(t) in Hölder classes for some semilinear parabolic equations. Inverse Problems, 4, 595-606, 1988. [3] Pourgholia R, Rostamiana M and Emamjome M., A numerical method for solving a nonlinear inverse parabolic problem. Inverse Problems in Science and Engineering, 18(8), 1151-1164, 2010. [4] Hill G.W., On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Mathematica, 8, 1-36, 1886. [5] Ozbilge E., Demir A., Inverse problem for a time-fractional parabolic equation, Journal of Inequalities and Applications, 81, (Mar 2015), 2015. [6] Dehghan, M., Implicit locally one-dimensional methods for two-dimensional diffusion with a nonlocal boundary condition, Math. And Computers in simulation, 49, 331-349, 1999. 1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kadir Has University, Istanbul, Turkey, [email protected] 100 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Solutions with Generator and Schrödinger Equation which have Sphere Symmetry for a potential Kismet Kasapoglu1, Cengiz Dane2 and Hasan Akbas3 Abstract. In this study, it is found transformation special form of a Radial Schrödinger equation which has got certain potential, and it is found certain symmetry of equation which is written special form and its solutions by its symmetry. These solutions are compared with solutions which is made by order methods. Keywords. Symmetry, Schrödinger Equations. AMS 2010. 76M60, 34B60, 34A05. References [1] Flügge S., Practical Quantum Mechanics, Springer- Verlag, Berlin Heidelbergs, 1994. [2] Hans Stephani, Differential Equations Their Solution Using Symmetries, Cambridge University Press, New York, 2003. [3] A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC. , New York, 2003. [4] John Starrett, Solving Differential Equations by Symmetry Groups, Mathematical Association of America, 114, 9, 778-792, 2007. [5] Francesco Oliveri, Lie Symmetries of Differential Equations: Classical Results and Recent Contributions, Symmetry, 2, 8, 658-706, 2010. [6] Gilmore Robert, Lie Groups, Lie Algebras, and Some of Their Applications, John Wileyond. Sons, Inc., New York, 1974. [7] Peter E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Prees, New York, 2000. 1Trakya University, Edirne, Turkey, [email protected] 2 Edirne, Turkey, [email protected] 3 Edirne, Turkey, [email protected] 101 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Buoyancy Effects on Unsteady Reactive Variable Properties Fluid Flow in a Channel Filled with a Porous Medium Lazarus Rundora 1 and Oluwole Daniel Makinde 2 Abstract. This article studies the flow and heat transfer properties in a vertical channel between two uniformly porous plates with suction/injection under the influence of constant pressure gradient, convective cooling and buoyancy force. The channel is filled with a porous medium and the flow is assumed to be unsteady and incompressible with variable viscosity and variable thermal conductivity. The coupled nonlinear partial differential equations for momentum and energy balance are numerically solved to obtain the flow velocity and fluid temperature profiles. The effects of the important thermophysical parameters on the flow velocity, fluid temperature, skin friction and nusselt number are simulated and explained. The buoyancy force was observed to increase the flow velocity, the skin friction, the wall heat transfer rate and the fluid temperature. Injection and suction as well as the increase in thermal conductivity parameter were found to have a significant cooling effect on the flow system. Keywords. buoyancy force, suction/injection, porous medium, variable viscosity, variable thermal conductivity. AMS 2010. 53A40, 20M15. References [1] Al-Hadhrami, A.K., Elliott, L. and Ingham, D.B., A new model for viscous dissipation in porous media across a range of permeability values, Transport in Porous Media, 53, 117 – 122, 2003. [2] Bernabe, Y. and Maineult, A., Physics of Porous Media: Fluid Flow Through Porous Media, Elsevier B.V. 2015. [3] Turner, J.S., Bouyancy effects in fluids, Cambridge University Press, 1973. [4] Xuan, Y. and Li, Q., Heat transfer enhancement of nanofluids, Int. J. Heat and Fluid Flow., 21-1, 58 – 64, 2000. 1 University of Limpopo, South Africa, [email protected] 2 University of Limpopo, South Africa, [email protected] 102 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Qualitative Properties of Solutions of a Partial Differential Equation with a Piecewise Constant Argument Mehtap Lafci1 and Huseyin Bereketoglu2 Abstract. In this study, we consider a partial differential equation with a piecewise constant argument. We research existence and uniqueness of the solutions of this equation. Moreover, we investigate oscillation, unstability and stability of the solutions. Keywords Partial differential equation, Piecewise constant arguments, Oscillation, Stability AMS 2010 35B05; 35B35; 35K05; 39A11 References [1] J. Wiener and L. Debnath. A parabolic differential equation with unbounded piecewise constant delay. Internat. J. Math Math, Sci. 15, 2, 339-346, 1992. [2] J. Wiener. Generalized Solutions of Functional Differential Equations. World Scientific, Singapore, 1994. [3] T. Veloz and M. Pinto. Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument. J. Math. Anal. Appl. 426, 1, 330-339. 2015. 1 Ankara University, Ankara, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected] 103 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Modeling of Electric Power and Electric Contact Systems M. N. Kalimoldayev1 and M. T. Jenaliyev2 Abstract. This work describes the issues of mathematical modeling of electric power and electric contact systems. As known, electric power and electric contact systems are among the most important elements of energy supply of consumers. Ensuring reliable and trouble-free functioning of these systems is an important problem in applied and theoretical aspects. The consistently improving algorithms optimal stabilization with an illustration of the obtained results by numerical examples are applied for models of electric power systems [1-3]. The peculiarities of emerging thermal conditions in the contact space were studied based on the theory of parabolic equations in non-cylindrical domains for the electric contact systems [4]. Keywords. modeling, optimal stabilization, electric power system, electric contact system, parabolic equation. AMS 2010. 35K05, 35K20, 49J15, 93B05, 93C15. References [1] Kalimoldayev, M.N., Aisagaliev, S.A., Certain problems of synchronization theory, Journal of Inverse and III-posed Problems, 21, Issue 1, 159-175, 2013. [2] Akhmetzhanov, M.A., Jenaliyev, M.T., and Kalimoldayev, M.N., On the controllability of the multidimensional phase system, AIP Proceedings, 1611, 190-193, 2014. [3] Kalimoldayev, M.N., Jenaliyev, M.T., Abdildayeva, A.A., and Kopbosyn, L.S., On the optimality one power system, AIP Proceedings, 1611, 194-198, 2014. [4] Amangaliyeva, M.M., Jenaliyev, M.T., Kosmakova, M.T., and Ramazanov, M.I., About one of the homogeneous problem for the heat equation in an infinite angular region, Sib.Math.J., 56, 6, 1234-1248, 2015. 1 Institute of Information and Computational Technologies, Almaty, Kazakhstan, [email protected] 2 Institute of Information and Computational Technologies, Almaty, Kazakhstan, [email protected] 104 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Investigation of the Interaction of Waves for 2D Conservation Laws in a Class of Discontinuous Functions with Three-Piece Constant Condition Mahir Rasulov1, Hakan Bal2 and Bahaddin Sinsoysal3 Abstract. In this paper the following problem u uf ug )()( ,0 (1) t x y u 0, , 1 2 3 yxu )0,,( u2 , , 2 2 3 u , 2 3 2 (2) u 2 u 3 is considered. Here uf )( , ug )( , 0 . 2 3 2 Since the classical solution of (1), (2) does not exist, in order to find a weak solution, an auxiliary problem has been introduced as follows. tyx 2 ),,(),,( tyxtyxu 3 ),,(),,( tyxtyxu ),,( tyxu ),,( dxdydt 2RR t 2 x 3 y dxdyyxuyx 0)0,,()0,,( (3) R2 y y x 1 1 x ),,( ddtu 2 ),,( 3 ),,( dtyudtxu t 2 3 y 1 1 x 2 ),,( 3 ),,( dtudtu . (4) 2 3 As seen from (4), the derivatives of tyxu ),,( with respect to x, y and t are not involved, which is one of the advantages of the auxiliary problem. On the basis of the suggested auxiliary problem, an efficient algorithm for finding the numerical solution has been developed, and some computer tests have been carried out. Keywords. Shock and rarefaction waves, Riemann problem. AMS 2010. 35L65, 35L67, 65M06. 1 Beykent University, Istanbul, Turkey, [email protected] 2 Beykent University, Istanbul, Turkey, [email protected] 3 Beykent University, Istanbul, Turkey, [email protected] 105 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Conway, E., Smoller, J., Global solutions of the Cauchy problem for quasilinear first- order equations in several variables, Comm. Pure Appl. Math., 19, 95-105, 1966. [2] Lindquist, W. B., The scalar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal., 17, 1178-1197, 1986. [3] Ben-Artzi, M., Falcovitz, J., Li, J., Wave interactions and numerical approximation for two-dimensional scalar conservation laws, Comp. Fluid Dynamics J., 14, 4, 401–418, 2006. [4] Rasulov, M. A., Identification of the saturation jump in the process of oil displacement by water in a 2D domain, Dokl RAN, 319, 4, 943-947, 1991. [5] Wagner, D., The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 38, 534-559, 1983. [6] Zheng, Y., Systems of conservation laws, two-dimensional Riemann problem, Birkhauser, in the Series of Progress in Nonlinear Differential equations, 2001. 106 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Experimental and Computational Studies of (2Z, 3E)-3-(((E)-3-Ethoxy-2- Hydroxybenzylidene) Hydrazono)Butan-2-One Oxime Nezihe Caliskan1, Cigdem Yuksektepe Ataol2, Humeyra Bati3, Numan Kurban1 and Pelin Kurnaz3 Abstract. In this work, the structure of (2Z, 3E)-3-(((E)-3-Ethoxy-2- Hydroxybenzylidene)Hydrazono)Butan-2-One Oxime (1) has been synthesized and characterized by IR, UV/vis, NMR and X-ray diffraction. Single crystal X-ray diffraction results show that 1 crystallizes in the monoclinic system, space group P21/c. The monomer and dimer molecular structures of the title compound in the ground state (in vacuo) were optimized by Density Functional Theory (DFT) to include correlation corrections with the 6– 311G(d, p) and B3LYP/6-31G basis sets. In DFT calculations, hybrid functionals are also used, including the Becke’s three-parameter functional (B3) [1], which defines the exchange functional as the linear combination of Hartree-Fock, local, and gradient-corrected exchange terms. The B3 hybrid functional was used in combination with the correlation functionals of Lee et al. [2]. In addition to the experimental studies, the optimized structures, vibrational parameters, chemical shifts, molecular orbital energies, thermodynamic properties, ionization energy, electron affinity, electronegativity, global chemical hardness and chemical softness, fukui functions, nonlinear optical properties and natural bond orbital analysis of the molecule have been investigated by using DFT. The HOMO and LUMO energies were calculated by time-dependent TD-DFT approach. To estimate the chemical reactivity of the molecule, the molecular electrostatic potential (MEP) surface map of the title molecule and PES scan were investigated with theoretical calculations at the B3LYP/6-31G and B3LYP6–311G(d, p) levels. The experimental results of the compound have been compared with theoretical results and it is found to show good agreement with calculated values. Keywords. Oxime, hydrazone, DFT. References [1] Becke, A. D., Density-functional thermochemistry. III. The role of exact Exchange, J. Chem. Phys. 98, 5648-5652, 1993. [2] Lee, C., Yang, W., Parr, R. G., Development of the Colle-Salvetti correlation energy formula into a functional of the electron density. Phys. Rev. B37, 785-789, 1988. 1Gazi University, Ankara, Turkey, [email protected] 2Cankiri Karatekin University, Cankiri, Turkey, [email protected] 3Ondokuz Mayis University, Samsun, Turkey, [email protected] 107 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) New Identities on the Generalized Exponential and Mellin Integral Transforms and Their Applications Nese Dernek1, Fatih Aylikci2 and Osman Yurekli3 Abstract. In the present paper, it is shown that the third iterate of the L2n - transform is the generalized exponential integral transform E2n,1. A Parseval-Goldstein type theorem involving the L2n - transform, P2n - transform and the E2n,1 - transform is given. Using this theorem and its corollaries, a number of interesting infinite integrals and generalized Mellin transforms of elemantary and special functions are presented. Some illustrative examples are also given. Keywords. Laplace transforms, L2n-transforms, G2n-transforms, Fs,n-transforms, Fc,n-transforms, Hv,n-transforms, Kv,n-transforms, E2n,1-transforms, Parseval-Goldstein type theorems. AMS 2010. Primary 44A10, 44A15, secondary 33B15, 44B25. References [1] Brown, D., Dernek, N., Yürekli, O., Identities for the exponential integral and the complementary error transforms, Applied Mathematics and Computation, 182 (2), 1377- 1384, 2006. [2] Brown, D., Dernek, N., Yürekli, O., Identities for the E2,1-transform and their applications, Applied Mathematics and Computation, 187 (2), 1557-1566, 2007. [3] Dernek, N., Aylıkçı, F., Balaban, G., New identities for the generalized Glasser transform, the generalized Laplace transform and the E2n,1-transform, International Eurasian Conference on Mathematical Sciences and Applications, Book of Abstracts, 135-137, 2015. [4] Dernek, N., Aylıkçı, F., Identities for the Ln-transform, the L2n-transform and the P2n- transform and their applications, Journal of Inequalities and Special Functions, Vol.5 Issue 4, 1-17, 2014. [5] Dernek, N., Aylıkçı, F., Some results on the Pv,2n, Kv,n and Hv,n integral transforms, Turkish Journal of Mathematics, (to appear), 2016. [6] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms Vol. 1.,Mac Graw-Hill Book Company Inc., New York-Toronto-London, 1954. 1 Marmara University, Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected] 3 Department of Mathematics, Ithaca College, Ithaca, NY-14850, USA 108 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [7] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms Vol. 2.,Mac Graw-Hill Book Company Inc., New York-Toronto-London, 1954. [8] Gradshteyn, I.S., Rhyzik, I.M., Table of integrals, series and products, Academic Press, New York, 1980. [9] Magnus, W., Oberhettinger, F., Soni, R.P., Formulas and theorems for the special functions of mathematical physics, Springer-Verlag New York Inc., 1966. [10] Yürekli, O., A Parseval-type theorem applied to certain integral transforms, IMA Journal of Applied Mathematics, 42, 241-249, 1989. [11] Yürekli, O., Identities, inequalities, Parseval-type relations for integral transforms and fractional integrals, Ph.D.thesis, University of California, Santa Barbara, 1988. [12] Yürekli, O., New identities involving the Laplace and L2-transforms and their applications, Applied Mathematics and Computation, 99 (2-3), 141-151, 1999. 109 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On Some Applications of the Generalized Laplace Transform Ln Nese Dernek1 and Sevil Kivrak2 Abstract. In the present paper, the writers aim to expand the applicability of the generalized Laplace transform to partial and ordinary differential equations with non-constant coefficients. In the first section, Efros Theorem, Generalized Product Theorem, for the Ln and L2n accompanies a number of preliminary definitions, properties and theorems. The second section contains new theorems which help the reader to distinguish the equations that can be easily and effectively solved by using Ln transform by generalizing them. Following these lemmas some descriptive examples are supplied, at the end of this section. Keywords. Laplace Transforms, Generalized Laplace Transforms, Partial Differential Equations, Ordinary Differential Equations, Convolution Theorem, Efros Theorem AMS 2010. Primary 44A10, 44A35, secondary 35C10, 34B05. References [1] Aghili, A., Ansari, A., Sedghi A., An inversion technique for the Ln-transform with applications, Int. J. Contemp. Math. Sciences, 2.28, 1387-1394, 2007. [2] Aghili, A., Ansari, A., A new approach to solving SIEs and system of PFDEs using the L2- transform, Differential Equations and Control Processes, N3, 1817-2172, 2010. [3] Dernek, N., Aylıkçı, F., Identities for the Ln-transform, The L2n-transform and the P2n- transform and their applications, Journal of Inequality and Special Functions, 5.4, 1-16, 2014. [4] Dernek, N., Aylıkçı, F., Laplace ve L2-dönüşümleriyle kısmi türevli denklemlerin çözümleri, Marmara University, Master Thesis, 2014. [5] Dernek, N., Aylıkçı, F., Kıvrak, S., An alternative technique for solving ordinary differential equations, Konuralp Journal of Mathematics, 4.1, 68-79, 2016. [6] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms Vol. 1, NewYork,NY,USA, McGraw-Hill, 1954. [7] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms Vol. 2, New York,NY,USA, McGraw-Hill, 1954. 1 Marmara University, Istanbul, Turkey, [email protected] 2 Marmara University, Istanbul, Turkey, [email protected] 110 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [8] Gaugler, T., Applications of the Ln-transform to partial differential equations, arşiv: 1202.2402v2, 2012. [9] Yürekli, O.,Wilson, S., A new method of solving Bessel’s differential equation using the L2-transform, Applied Mathematics and Computation, 130.2,587-591,2002. 111 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Perturbation Solution for Systems with Strong Quadratic and Cubic Nonlinearities Nedret Elmas1 Abstract. A perturbation algorithm using any time transformation is introduced. To account for the nonlinear dependence of the function, we exhibit the function f of the system in the differential equation. To this end, we introduce the transformation e , ttwfT , where f is a function that depends on t or w. The new perturbation algorithm with any time transformation is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical. Keywords: Perturbation Methods, Time transformation, Multiple Scales method, Numerical Solutions, systems with quadratic and cubic nonlinearities References [1] H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981. [2] Nayfeh, A.H., Nayfeh S.A. and Mook, D.T., On Methods for continuous systems with quad-ratic and cubic nonlinearities, Nonlinear Dynamics, 3, 145-162, 1992. [3] M. Pakdemirli and H. Boyaci, Comparison of direct-perturbation methods with discretization-perturbation methods for nonlinear vibrations, Journal of Sound and Vibration 186, 837-845, 1995. [4] Nayfeh, A.H., Reduced order models of weakly nonlinear spatially continuous systems, Nonlinear Dynamics 16,105–125, 1998. [5] C F Chan Man Fong and D D Kee, Perturbation Methods, Instability, Catastrophe and Chaos, World Scientific Publishing, 1999. 1Celal Bayar University, Manisa, Turkey, [email protected] 112 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [6] J.I. Ramos, On Linstedt–Poincaré techniques for the quintic Duffing equation, Appl. Math. Comput., 193 , pp. 303–310, 2007. [7] M. Pakdemirli, M. M. F. Karahan and H. Boyacı, A new perturbation algorithm with better convergence properties: Multiple Scales Lindstedt Poincare Method, Mathematical and Computational Applications 14, 31-44, 2009. [8] M. Pakdemirli, M. M. F. Karahan, A new perturbation solution for systems with strong quadratic and cubic nonlinearities, Mathematical Methods in the Applied Sciences,33, 704- 712, 2010. [9] M. Pakdemirli and N. Elmas, Perturbation theorems for estimating magnitudes of roots of polynomials, Applied Mathematics and Computation 216, 1645-1651, 2010. 113 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) An asymptotical method for determining hydraulic resistance’s coefficient of gas-lift process by the method of lines Hajieva N. S.1, Safarova N. A.1 and Rajabov M.F. 2 Abstract. A mathematical model in oil production is formulated to determine the coefficient of hydraulic resistance [1] during the motion of the gas-liquid mixture in the lift. As it is known, the motions of the gas and gas liquid mixture in the tubes are described by the system of partial differential equations of hyperbolic type in the assymptotic case [2]: P c 2 Q , t F z (1) Q P F aQ,2 t z where the parameters in (1) defined as [3]. Applying the straight line method, we obtain from (1) x А VuBxA 0 c 1 ))(( , with initial condition х 0 0 0 QPQP 0 .,,, 10 1 2 2 It is required to minimize the functional N ~ ii 2 f c )( QQ 22 . i1 On a concrete example the comparison of the values of the obtained hydraulic resistance coefficient with the statistical value of hydraulic resistance is given. It is shown that they differ from each other to the order 10-3. Keywords. Gas-lift, the coefficient of hydraulic resistance, the method of lines AMS 2010. 49J15, 49J35. References [1] Aliev, F. A., Ismailov, N. A., Inverse problem to determine the hydraulic resistance coefficient in the gaslift process, Appl. Comput. Math., 12, 3, 306-31, 2013. 1 Baku State University, Baku, Azerbaijan, e-mail: [email protected], [email protected] 2 Institute of Control Systems of ANAS, Baku, Azerbaijan, e-mail: [email protected] 114 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [2] Mutallimov, M. M, Askerov, I. M., Ismailov, N. A., Rajabov, M. F., An asympotical method to construction a digital optimal regime for the gaslift process, Appl. Comput. Math., 9, 1, 77-84, 2010. [3] Mukhtarova, N. S., Algorithm to solution the identification problem for finding the coefficient of hydraulic resistance in gas-lift process, Proc. of IAM, 4, 2, 206-213, 2015. 115 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Solutions to Initial and Boundary Value Problems for Fractional Telegraph Equations Ozan Ozkan1 Abstract. In this article, the Fractional Laplace Differential Transform Method (FLDTM) has been employed to obtain approximate analytical solutions of boundary value problems for fractional telegraph equations. The FLDTM is a combined form of the Laplace transform and Differential Transform Method (DTM). The numerical solutions constructed by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is a promising toll for solving fractional telegraph equations. Keywords. Fractional telegraph equations, Fractional Laplace differential transform method, fractional derivatives, series solutions. AMS 2010. 35R11, 34K28, 35C10 1 Selcuk University, Konya, Turkey, [email protected] 116 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Fractional Calculus Models for Some Bioengineering Problems Ozlem Ozturk Mizrak1 and Nuri Ozalp2 Abstract. The main idea in this work is to answer the question: Why one should prefer to use fractional calculus tools in some bioengineering problems (the electrical impedance of the electrode-tissue interface (a key problem in pacemaker design), the stress-strain behavior of arterial viscoelasticity and hysteresis (important predictors of heart disease), and the bulk elastic properties of normal and cancerous breast tissue (malignant and benign)) rather than conventional calculus’? In this manner we give the model structures and numerical simulations on these problems and some conclusions are drawn in the final section. Keywords. Bioengineering, electrode-tissue interface, stress-strain behavior, viscoelasticity, tissue, fractional calculus. References [1] Magin, R. L., Fractional calculus models of complex dynamics in biological tissues, Computers and Mathematics with Applications, 59, 1586-1593, 2010. [2] Magin, R. L., R.L. Magin, Crit. Rev. Biomed. Eng., 32, 1-104, 2004. [3] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications ofFractional Differential Equations, Elsevier, Amsterdam, 2006. [4] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (4), 367-386, 2002. 1 Ankara University, Ankara, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected] 117 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Statistical Physics and Thermodynamics Approach to EEG Time Series Sergey Borisenok1 Abstract. A set of statistical physics and thermodynamics characteristics, like entropy [1], specific heat [1,2], temperature [3] and others, can be defined for time series and, as we demonstrate here, applied for the data registered from electroencephalogram (EEG) channels after the elimination of artifacts. Then one can introduce an analog of condensed matter ‘phase state’ for EEG channel series in the specific time interval. We show that, like in the case of informational code [4], there is a correspondence between the functional state of human brain and this statistically and thermodynamically defined ‘phase’; and the evolution of one brain functional state into another can be described as a phase transition. Thus, a statistical ‘phase’ of EEG series represents the corresponding brain functional state. Keywords. Time series, nonlinear statistical analysis, electroencephalogram (EEG). AMS 2010. 62M10, 62P10, 82B30. References [1] Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L., Stanley, H. E., Econophysics: Financial time series from a statistical physics point of view, Physica A 279, 443-456, 2000. [2] Ausloos, M., Financial time series and statistical mechanics, 153-168, in ‘Computational Statistical Physics. From Billiards to Monte Carlo’, Springer-Verlag, New York, 2002. [3] Hasegawa, H., Washio, T., Ishimiya, Y., Inductive thermodynamics from time series data analysis, Lecture Notes in Computer Science 2281, 384-394, 2002. [4] Mekler, A., Borisenok, S., EEG informational code dependence on the functional state: General trends and characteristic period, International Journal of Psychophysiology 94, 190, 2014. 1 Abdullah Gul University, Kayseri, Turkey, [email protected] 118 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Quasi-Periodic Wave Solutions of (2+1) Dimentional Breaking Wave Solutions Using with Riemann Theta Functions Secil Demiray1 and Filiz Tascan2 Abstract. The exact solutions of PDE’s can help us to understand complicated physical models and unreval the general structure of the complex nonlinear phenomena. So, there are some successful methods to obtain exact solutions [1], [2], [5], [8], [11]. In 1980's Nakamura proposed a method to construct periodic or quasi periodic wave solutions of nonlinear equations by combining Riemann theta function and Hirota's bilinear method in [6,7]. Just using this method both periodic wave solutions and soliton wave solutions can be obtained directly. Recently, these methods have extended to investigate periodic wave solutions of some nonlinear differential, discrete and supersymmetric equations. For example, Tian and his colaborators obtained periodic wave solutions by Riemann theta functions of some nonlinear differential equations [10]. Wu obtained one and two periodic solutions for the (2 + 1) dimensional Toda lattice equation [12] and the periodic solutions of some supersymmetric equations were investigated too [9]. In literature, other studies are available. Finally Demiray and Tascan obtained one, two and three periodic solutions of (3+1) Generalized BKP equation [3]. Finding N=3 soliton solutions is important for PDE's. If an equation is known to have a soliton solution including at least three solitons, then this has been observed to co-inside with integrability. Because of some difficulties in the calculation and the lack of unknown parameters, it is not possible to find three-periodic solutions always. But in Demiray's Phd Thesis [4], she showed a way how to find N=3 periodic solutions easily. In this study one-periodic, two-periodic and three periodic solutions of (2+1) Dimensional Breaking Soliton Equation using with Hirota-Riemann Method are obtained. The asymptotic behavior of the periodic wave solutions are analysed and it is shown that periodic solutions tend to the known soliton solutions under a small amplitude limit. Finally, to show the propogation of the wave and to understand the structure of the wave, graphics of the wave solutions are plotted. 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 119 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Keywords. Quasi-periodic wave solutions, Riemann Theta Function, Hirota’s Bilinear Form, (2+1) Dimensional Breaking Soliton Equation. AMS 2010. 35G20, 35B10, 14K25. References [1] Belokolos, E.D., Bobenko, A.I., Enol'skii, V.Z., Its, A.R., Matveev, V.B., Algebrogeometric approach to non-linear integrable equations, Springer, 1994. [2] Bluman, G.W., Kumei, S., Symmetries and Differential Equations, New York-Springer Verlag, 1989. [3] Demiray, S., Tascan, F., Quasi-periodic solutions of (3+ 1) generalized BKP equation by using Riemann theta functions, Applied Mathematics and Computation, 273, 131-141, 2016. [4] Demiray, S., Kısmi Diferensiyel Denklemlerin Riemann Theta Fonksiyonları İle Periyodik Çözümleri, PhD, Eskisehir Osmangazi University, Eskisehir, Turkey, 2016. [5] Miura, M.R., Bäcklund Transformation, Berlin-Springer Verlag, 1978. [6] Nakamura, A., A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evolution Equations. I. Exact Two-Periodic Wave Solution, J. Phys. Soc. Jpn. 47, 1701, 1979. [7] Nakamura, A., A Direct Method of Calculating Periodic Wave Solutions to Nonlinear Evolution Equations. II. Exact One- and Two-Periodic Wave Solution of the Coupled Bilinear Equations, J. Phys. Soc. Jpn., 48, 1365, 1980. [8] Ryogo, H., Exact solution of the Korteweg De Vries equation for multiple collisions of solitons, Physical Review Letters, 27, 1192-1194, 1971. [9] Tian, S., Zhang, H., Super Riemann Theta Function Periodic Wave Solutions and Rational Characteristics For a Supersymmetric KdV-Burgers Equation, Theoretical and Mathematical Physics, 170, 287-314, 2012. [10] Tian, S., Zhang, H., Riemann theta functions periodic wave solutions and rational characteristics for the (1+1)-dimensional and (2+1)- dimensional Ito equation, Chaos, Solitons and Fractals, 47, 27-41, 2013. [11] Wang, M., Li, X. and Zhang, J., The G′/G expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A., 372, 417-423, 2008. [12] Wu, Y., Quasi-periodic wave solution and asymptotic behavior for the (2+1)- dimensional Toda lattice equation, Applied Mathematics and Computation, 219, 3065-3072, 2012. 120 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Existence of Positive Solutions for Four Point Fractional Boundary Value Problems Serife Muge Ege1 and Fatma Serap Topal2 Abstract. In this presentation, we study the existence and multiplicity of positive solutions to the four point boundary value problems of nonlinear fractional differential equations. Our results extend some recent works in the literature Keywords. Caputo fractional derivative, Fractional boundary value problem, Positive solutions, Fixed-point theorems. AMS 2010. 34B10, 26A33, 34B15. References [1] Benchohra M, Hamania S, Ntouyas S. K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71, 2391-2396, 2009. [2] Krasnoselskii M. A., Topological Methods in the Theory of Nonlinear Integral Equations (A. H. Armstrong, Trans.) Pergamon: Elmsford, 1964. [3] Leggett R.W., Williams L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces., Indiana Univ. Math. J., 28, 673-688, 1979. [4] Zhang S. Q., Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Diff. Eqns. , 36, 1-12, 2006. [5] Zhao X, Chai C, Ge W., Positive solutions for fractional four-point boundary value problems., Commun Nonlinear Sci. Numer. Simulat., 16, 3665-3672, 2011. 1 Ege University, Izmir, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected] 121 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Modelling Problems of Dynamics Using Differential Games S. N. Amirgaliyeva1 Abstract. The research investigates differential games, the dynamics of which is described by ordinary differential equations. These game models consider problems of pursuit-evasion, which are defined by terminal set and the set of phase constraints. Strategies of different players are described: -strategy [1], its modifications, and connection between are established. Structure of differential games is described using one-parameter semigroups of operators [2, 3] on the basis of which - strategies can be built [1] and operators describe the set of initial positions, favourable for a particular player in the game models with terminal set. Keywords. Differential games, pursuit-evasion, terminal set, set of phase constraints, -strategy, one-parameter semigroups of operators. AMS 2010. 34H, 49B, 90D. References [1] Pshenichnyi B. N., Ostapenko V. V., Differential Games [in Russian], Naukova Dumka, Kiev,1992. [2] Ostapenko V.V., Amirgaliyeva S.N., Ostapenko E.V. Convex analysis and differential games [in Russian], Science, Almaty, 2005. [3] Ostapenko V. V. Convexity in differential games. Springer Book. “Pareto-Optimality. Game Theory and Equilibria”, ed. P. Pardalos. 2008. 1 Institute of Information and Computational Technologies, Almaty, Kazakhstan, [email protected] 122 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) New Hybrid Conjugate Gradient Method as a Convex Combination of LS and FR Methods Snezana S. Djordjevic 1 Abstract. In this paper we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. An interesting fact is that the search direction of this method satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition and this property doesn't depend on any line search. Furthermore, we are going to prove that this fact holds for any hybrid conjugate gradient method presented as a convex combination of two conjugate gradient methods. The strong Wolfe line search conditions are used. The global convergence of this new method is proved. Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one. Keywords. Hybrid conjugate gradient method; Convex combination; Dai-Liao Conjugacy condition. AMS 2010. 90C30. References [1] N. Andrei, 40 Conjugate Gradient Algorithms for unconstrained optimization, A survey on their de nition, ICI Technical Report, 13/08, 2008. [2] N. Andrei, New hybrid conjugate gradient algorithms for uncon- strained optimization, Encyclopedia of Optimization, 2560-2571, 2009. [3] Y. H. Dai,, L. Z. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim. 43 (2001), 87-101. [4] J.K. Liu, S.J. Li, New hybrid conjugate gradient method for un-constrained optimization, Applied Mathematics and Computa- tion, 245 (2014), 36-43. 1 College for technology and art 16000 Leskovac, Serbia, [email protected] 123 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Explicit Finite-Difference Method For Solution of Heat Diffusion-Wave Equation with Mix-Fractional Derivatives Vildan Gulkac1 Abstract. Explicit finite-difference method to solve heat diffusion-wave equation with mix-fractional derivatives has been prepared. Time-fractional derivative is shown Jumarie’s modified Riemann-Liouville derivative and fractional derivative for space-coordinate is shown Riesz derivative [1, 2, 3]. Stability analysis of method is also performed and efficiently exemplified by two problems. Keywords: Fractional Derivative, Finite Difference Methods, Heat Diffusion-Wave Equation, Stability Analysis. AMS 2010. 65R10, 26A33, 35B35. References [1] Jumaric G., On the Fractional Solution of the Equation f(x+y)=f(x)f(y) and Its Application to Fractional Laplace I. Applied Mathematics and Computation, 219 (2012) 1625-1643. [2] Jumaric G., Table of Some Basic Fractional Calculus Formulae Derived from a Modified Rieamann-Liouville derivative for non-Differentiable Functions. Applied Mathematic Letters, 22 (2009) 378. [3] Jumaric G., Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non- Differentiable Functions Further Results. Comput. Math. Appl., 51 (2006) 1367-1376. 1 Kocaeli University, Kocaeli, Turkey, [email protected] 124 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Qualitative Behavior of ODE Systems Solutions Corresponding to First Order Chemical Kinetics Mechanisms Victor Martinez-Luaces1 Abstract. First Order Chemical Kinetics Mechanisms (FOCKM) and their corresponding ODE systems were studied in a book chapter published four years ago [1] and after that, the associated FOCKM matrices were deeply analyzed in a later book chapter released last year [2]. Recently, several journal papers examined other aspects of particular cases, like FOCKM involving reversible and irreversible reactions [3], with and without final products [4]. This new article has two different characteristics, since it can be considered as a survey paper – where the main results about FOCKM systems matrices are analyzed – and at the same time they are interpreted in terms of the qualitative behavior of the solutions corresponding to the ODE linear systems involved. Several qualitative aspects like increase, decrease, inflection points, asymptotic behavior and stability are studied from the mathematical viewpoint and the chemical consequences of these results are remarked. Keywords. Chemical reactions, ODE systems, Solutions qualitative behavior. AMS 2010. 34A30, 34D05, 34D20. References [1] Martinez-Luaces, V., First Order Chemical Kinetics Matrices and Stability of O.D.E. Systems, in Advances in Linear Algebra Research, Nova Science Publishers, New York, U.S.A., 2015. [2] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical Kinetics, In Tech Open Science, Rijeka, Croatia, 2012. [3] Martinez-Luaces, V., Stability of O.D.E. solutions corresponding to chemical mechanisms based-on unimolecular first order reactions, Mathematical Sciences and Applications E- notes, Vol. 3, No. 2, pp. 58-63, 2014. [4] Martinez-Luaces, V., Stability of O.D.E. systems associated to first order chemical kinetics mechanisms with and without final products, Konuralp Journal of Mathematics, in press. 1UdelaR, Montevideo, Uruguay, [email protected] 125 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) About One Approach for the Group Synthesis of Recognition and Classification Tasks Yedilkhan Amirgaliyev1 and Salim Mustafin2 Abstract. The main task of the group synthesis of algorithms of classification is to construct the optimal partitioning of the result of the test object in a plurality of sets of partitions generated by each algorithm, a basic set of components of the group. The notion of optimality is specified for the specific tasks of recognition and classification, using a variety of criteria (functional) quality [1-2]. We formulate the problem of group synthesis. Given a set of object classification 21 ...,,, SSSS m . Each object i SS t is uniquely described by a set of n numerical parameters S ii 1,..., in , called attributes of characterizing objects. Thus, S - a subset of n-dimensional space. There are many 21 ,...,, AAAA t - a set of classification algorithms, forming a group. With regard to the set of each algorithm u AA builds a partition u uu 21 KKKRRSA l ;,...,,; ji tuljijiKK .,...,2,1,,...,2,1,,,0 The task of constructing the resulting decomposition, combining the results of each algorithm. As a result of this partitioning we can consider. There is actual issue of construction of optimal partitions resulting in multiple partitions in a basic set of algorithms Committee. Consider the problem of the group synthesis, which consists in the construction of the partition resulting in a set of algorithms that form the ccc c centers of classes. Let 21 ,...,, АААА t be the set of central algorithms. Each algorithm cc i АА is applied to the object a plurality of subsets of building 1,...,SSS q - a set of central objects. In other words, according to the principle of operation of the algorithms from Ac , each subset c c c Si consists of objects defined as central objects using the algorithms A1 ,...,Аt . The algorithm is implemented and tested in solving real-world problems [3]. Keywords. Pattern recognition, functional quality, group synthesis. AMS 2010. 68W40, 68Q25. 1 Institute of Information and Computational Technologies, Almaty, Kazakhstan, [email protected] 2 Institute of Information and Computational Technologies, Almaty, Kazakhstan, [email protected] 126 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Zhuravlev Y.I. On the algebraic approach to solving the problems of recognition and classification. Problems of Cybernetics, 33, 93-103, 1978. [2] Amirgaliyev Y. N., Mukhamedgaliyev A. F. Optimization model of classification algorithms (taxonomy), Jour. Calculated. Math. and Math. Physics, No.11, 1733-1737, 1985. [3] Amirgaliyev Y., Yunussov R. Pattern recognition systems in the of automatic person identification using the passport data, Computer Modeling and New Technologies, 19, No.2, 27-32, 2015. 127 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Stochastic and Random Behavior Model for Immune System Response and Bacterial Resistance with Antibiotic Therapy Zafer Bekiryazici1, Mehmet Merdan2, Tahir Khaniyev3 and Tulay Kesemen4 Abstract. In this paper, we consider a stochastic model and stochastic effect addition model of immune system response and bacterial resistance with antibiotic therapy. Random noise is added to the model to investigate the random behavior of the model. The numerical characteristics such as the expectation, variance and confidence interval are calculated for random effects with two different distributions from the results of numerical simulations. In addition, the compliance of the random behavior of the model is examined. Furthermore we apply Euler-Maruyama, Milstein’s method to simulate the solution of stochastic model for immune system response and bacterial resistance with antibiotic therapy. Finally we compare stochastic model and stochastic effect addition model of immune system. Keywords. Stochastic differential equation systems, Bacterial resistance, Euler- Maruyama, Random effect. AMS 2010. 60H10, 65C30. References [1] Ternent, L., Dyson, R.J., Krachler, A-M. et al, Bacterial fitness shapes the population dynamics of antibiotic resistant and susceptible bacteria in a model,Antibiotic, J Theor Biol, 372, 1-11, 2014. [2] Daşbaşı, B., Öztürk, İ., Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response, Springer Plus 5, 408, 1-17, 2016. [3] Higham, D.J., An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Rev. 43, 3, 525-546, 2001. [4] Clatworthy, A., Pierson, E., Hung, D., Targeting virulence: a new paradigm for antimicrobial therapy, Nat. Chem. Biol. 3, 541-548, 2007. [5] Merdan, M., Khaniyev, T., On the Behavior of Solutions Under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model, International Journal of Biotechnology and Biochemistry 4, 1, 75-100, 2008. 1 Department of Mathematics, Recep Tayyip Erdogan University, Rize, Turkey, [email protected] 2 Department of Mathematical Engineering, Gümüşhane University, Gümüşhane, Turkey, [email protected] 3 Department of Industrial Engineering, TOBB University of Economics and Technology, Ankara, Turkey, [email protected] 4 Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey, [email protected] 128 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [6] Bekiryazici, Z., Merdan, M., Kesemen, T., Khaniyev, T., Random Modeling of Biochemical Reactions under Gaussian Random Effects, Abstracts Book: International Conference on Mathematics and Mathematics Education, 192-193, 2016. [7] Liu, P., Müller, M., Derendorf, H., Rational dosing of antibiotics: the use of plasma concentrations versus tissue concentrations, Int. J. Antimicrob. Agents 19, 285-290, 2002. [8] Butler, M., Buss, A., Natural products—the future scaffolds for novel antibiotics?, Biochem. Pharmacol. 71, 919-929, 2006. [9] Øksendal, B., Stochastic Differential Equations, 5th ed., Springer-Verlag, Berlin, 1998. 129 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) DISCRETE MATHEMATICS DISCRETE MATHEMATICS 130 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Mathematical Programming for Computing Padmakar-Ivan Index of Graphs Amir Bahrami1 Abstract. The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = Σ[neu (e|G) + nev (e|G)], where neu (e|G) is the number of edges of G lying closer to u than to v, nev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. This topological Index is developed very recently. In this paper, an exact mathematical program for PI index of the some nanotubes is given. Keywords. Padmakar-Ivan index, Molecular graph, Chemical structures. AMS 2010. 05C12, 92E10, 05C75. References [1] Khadikar, P. V.; Deshpande, N. V.; Kale, P. P.; Gutman, I. Spectral Moments of Polyacenes, J. Chem. Inf. Comput. Sci. 1994, 34, 1181-1183. [2] Gutman, I.; Gaurilovic, N.; Nankovic, D.; Khadikar, P. V.; Deshpande, N. V.; Kale, P. P. Dependence of Spectral Moments of Benzenoid ydrocarbons on Molecular Structure. The Case of Linear Polyacenes, J. Serb. Chem. Soc. 1994,59 , 519-524. [3] Gutman, I. Formula for the Wiener Number of Trees and Its Extension to Graphs Containing Cycles, Graph Theory Notes New York 1994, 27, 9-15. [4] Balaban A. T. Chemical Applications of Graph Theory, Academic Press, London (UK), 1976. 1 Department of Mathematics, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran., [email protected] 131 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On Roman {2}Domination in Graphs Nader Jafari Rad1 and Elahe Shabani2 Abstract. A Roman dominating function (or just RDF) on a graph GVE (,) is a function fV: {0,1,2}satisfying the condition that every vertex u for which fu( ) 0 is adjacent to some vertex v with fv( ) 2. The weight of an RDF f is the value w()() f f v . The Roman domination number of a graph G, denoted by R ()G , is the vV minimum weight of a Roman dominating function on G. A Roman {2} dominating function on a graph G is a function satisfying the condition that every vertex for which is adjacent to some vertex with or at least two vertices vv12, for which f( v12 ) f ( v ) 1. The Roman domination number {R 2} ()G of G is defined as expected. In this talk we present various bounds and characterizations for this variant of a graph. Keywords. Roman domination number, Roman domination number. AMS Subject Classification. 05C69 References [1] E. W. Chambers, W. B. Kinnersley, N. Prince and D.B. West, Extremal Problems for Roman Domination, Siam J. Discrete Math. 23(3)(2009), 1575{1586. [2] M. Chellali, T. W. Haynes, S. T. Hedetniemi, A.A. McRae, Roman f2g-domination, Discrete Applied Mathematics (2016), In press. [3] M. Chellali and N. Jafari Rad, Trees with unique Roman dominating functions of minimum weight, Discrete Math. Alg. Appl. 6(3)(2014) 1450038 (10 pages) [4] E. J. Cockayne, P. A. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi, Roman domination in graphs, Discrete Mathematics 278 (2004), 11{22. [5] A. Hansberg, N. Jafari Rad and L. Volkmann, Vertex and edge critical Roman domination in graphs, Util. Math. 92 (2013), 73{88. [6] T. W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, 1998. 1 Department of Mathematics, Shahrood University of Technolog, Shahrood, Iran, [email protected] 2 Department of Mathematics, Shahrood University of Technolog, Shahrood, Iran, 132 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Incidence Coloring of Sierpinski Graphs Ummahan Akcan1, Emrah Akyar2 and Handan Akyar3 Abstract. Incidence coloring of a graph 퐺 was introduced by Brualdi and Massey in 1993. 퐼(퐺) = {(푢, 푒) | 푢 ∈ 푉, 푒 ∈ 퐸, 푢 is incident with 푒 } be the set of incidences of 퐺 and two incidences (푢, 푒) and (푣, 푓) are adjacent if one of the following holds: (i) 푢 = 푣 (ii) 푒 = 푓 (iii) 푢푣 ∈ {푒, 푓}. An incidence coloring of 퐺 is a coloring of its incidences in which adjacent incidences are assigned different colors. The incidence chromatic number of 퐺, denoted by 푋푖(퐺), is the smallest number of colors in an incidence coloring. Incidence coloring conjecture says that every graph can be incidence colored with ∆ + 2 colors. In this presentation, we show that 푆(푛, 3) Sierpinski graph holds incidence coloring conjecture. Keywords. Sierpinski Graph, Incidence Coloring, Incidence Chromatic Number. AMS 2010. 05C15, 05C62, 05C75. References [1] Brualdi, R. A., and Massey, J. J. Q. Incidence and strong edge colorings of graphs. Discrete Math., 122, 51-58, 1993. [2] Guiduli, B., On incidence coloring and star arboricity of graphs, Discrete Math., 163, 275-278, 1997. [3] Hinz A. M., and Parisse D., Coloring Hanoi and Sierpinski graphs, Discrete Mathematics, 312, 1521-1535, 2012. [4] Klavzar S., and Milutinovic U., Graphs 푆(푛, 푘) and a variant of the Tower of Hanoi problem, Czechoslovak Mathematical Journal, 47, 95-10, 1997. 1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected] 3 Anadolu University, Eskisehir, Turkey, [email protected] 133 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Positivity Problem for Linear Recurrence Sequences of Order Six Vichian Laohakosol1 and Pinthira Tangsupphathawat2 Abstract. By a linear recurrence sequence ()un of order k ( 2), we mean a real sequence satisfying a linear recurrence relation of the form un a1 u n 1 a 2 u n 2 ak u n k ( n k ), where a12, a , , ak ( 0) and the initial values u0,,, u 1 uk 1 are given integers. The “positivity problem” asks whether all the terms of such a sequence are positive? The decidability of this problem still remains open. Yet, certain partial results have already appeared, namely, the positivity problem for sequences satisfying a second order linear recurrence was shown to be decidable by Halava-Harju-Harvensalo, [1], in 2006. The positivity problem for sequences satisfying a third or a fourth order linear recurrence has been shown to be decidable in [2], [3], and [4]. Our objective here is to discuss the decidability of this problem for sequences of order six. Keywords. Positivity problem, recurrence sequence, decidability. AMS 2010. 11B37, 03D20. References [1] Halava, V., Harju, T., and Hirvensalo, M., Positivity of second order linear recurrent sequences, Discrete Applied Mathematics 154 , 447-451, 2006. [2] Laohakosol V., Tangsupphathawat, P., Positivity of third order linear recurrence sequences, Discrete Applied Mathematics 157 , 3239-3248, 2009. [3] Laohakosol, V., Tangsupphathawat, P., A remark about the positivity problem of fourth order linear recurrence sequences, Acta et Commentations Universitatis Tartuensis de Mathematica 18 , 1-6, 2014. [4] Tangsupphathawat, P., Punnim, N., and Laohakosol, V., The positivity problem for fourth order linear recurrence sequences is decidable, Colloq. Math. 128 , 133-142, 2012. 1 Kasetsart University, Bangkok, Thailand, [email protected] 2 Phranakhon Rajabhat University, Bangkok, Thailand, [email protected] 134 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Trigonometric Factorizations of the Horadam Sequence and Its Companion Sequence Zafer Siar1 Abstract. In this study, we consider the Horadam sequence {푊푛} and its companion sequence {푋푛} satisfying the recurrence of the second order. As generalizations of the earlier results, we give new results about factorizations of these sequences. In order to obtain these results, we use the connections between determinants of tridiagonal matrices and the terms of these sequences, and also will benefit from Chebyshev polynomial of the second kind. Keywords. Horadam sequence, determinant, tridiagonal matrix, eigenvalue. AMS 2010. 11B39, 11C20, 15B05, 15A18. References [1] Ş. B. Bozkurt, F. Yımaz, and D. Bozkurt, On the Complex Factorization of the Lucas Sequence, Appl. Math. Lett., 24 (2011), 1317-1321. [2] N. D. Cahill, J. R. D'Errico, and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41(1) (2003), 13-19. [3] N. Cahill and D. Narayan, Fibonacci and Lucas numbers as Tridiagonal matrix determinant, Fibonacci Quart., 42(3) (2004), 216-221. [4] A. F. Horadam, Basic Properties of Certain Generalized Sequence of Numbers, Fibonacci Quart., 3 (3) (1965), 161--176. [5] E. Kılıç and P. Stănică, Factorizations and Representations of Second order Recurrences with indices in arithmetic progressions, Bulletin of the Mexican Mathematical Society, 15 (1) (2009), 23-26. [6] A. Nallı and H. Civciv, A Generalization of Tridiagonal Matrix Determinants, Fibonacci and Lucas Numbers, Chaos, Solitons and Fractals, 40 (2009), 355-361. [7] Z. Şiar and R. Keskin, Some new identities concerning the Horadam sequence and its companion sequence. (Submitted to Journal) 1 Bingol University, Bingol, Turkey, [email protected] 135 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) GEOMETRY GEOMETRY 136 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Similarity Relation on Bidegenarate Quaternions, Bipseudodegenerate Quaternions and Bidoubly Degenarate Quaternions Abdullah Inalcik 1 Abstract. In this paper, the concept of similarity for elements of bidegenarate quaternions, bipseudodegenerate quaternions and bidoubly degenarate quaternions is given by solving ax xb . Keywords. quaternions, biquaternions, similarity, bidegenarate quaternions, bipseudodegenerate quaternions, bidoubly degenarate quaternionsis, generalized inverse. AMS 2010. 53A40, 20M15. References [1] Hamilton, W.R., Lectures on Quaternions, Hodges and Smith, Dublin (1853). [2] Yaglom, I.M., Comlex Numbers in Geometry, Academic Press, New York (1968). [3] Agrawal, O.P., Hamilton operators and dual-number-quaternions in spatial kine- matics, Mech. Mach. Theory, 22 (6), 569-575 (1987). [4] Flaut, C., Some equation in algebras obtained by Cayley-Dickson process, An. St. Univ. Ovidius Constanta, 9 (2), 45-68 (2001). [5] Tian, Y., Universal factorization equalities for quaternionic matrices and their Applications, Math. J. Okayama Univ., 42, 45-62 (1999). [6] Tian, Y., Biquaternions and their complex matrix represations, Beitr. Algebra Geom, 54, 575-592 (2013). [7] Tian, Y., Solving Two Pairs of Quaternionic Equations in Quaternions, Adv. Appl. Clifford Algebras, 20, 185-193 (2010). 1 Artvin Coruh University, Artvin, Turkey, [email protected] 137 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Some Characterizations of Slant Helices as Rectifying Curves Bulent Altunkaya1 Abstract. In this paper, we study some characterizations of rectifying slant helices in 퐸3. First we determine the curvature and the torsion of rectifying slant helices. By the help of them we construct linear differential equations and by their solutions, we determine families of slant helices that lie on cones. Keywords. Rectifying Curve, Curvature, Torsion, Slant Helix. AMS 2010. 53A04, 53A05. References [1] Ali T. Ahmad, Position vector of a slant helix in Euclidean space 퐸3, Journal of the Egyptian Mathematical Society, 20, 1, 1-6, 2012. [2] Chen, B. Y., When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly 110, 147-152, 2003. [3] Chen, B. Y., Dillen, F. Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Academia Sinica 33, No. 2, 77-90, 2005. [4] Izumiya S., Takeuchi N., New special curves and developable surfaces, Turk. J.Math. 28, 153-163, 2004. [5] Kula, L. and Yayli, Y., On slant helix and its spherical indicatrix, Applied Mathematics and Computation, 169, 600-607, 2005. This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: PYO-EGF.4001.15.001 1 Ahi Evran University, Kirsehir, Turkey, [email protected] 138 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Representation of Some Special Homothetic Motions in Lorentz Space Dogan Unal1, Mehmet Ali Gungor2 and Murat Tosun3 Abstract. In this study, Rodrigues parameters will be defined for spacelike and 3 timelike axes separately and they will be represented with figures in 1 , 3 dimensional Lorentz space. On the other hand, the motions which corresponding to homothetic rotation matrices in Lorentz space will be represented with the figures for spacelike and timelike axes again. Keywords. Homothetic Rodrigues equation, homothetic motions, Lorentz space. AMS 2010. 53A17, 15A30. References [1] Hacısalihoğlu, H. H. and Arslan, İ., The Sets of Homothetic Motions. Commun. Fac. Sci. Univ. Ank. Series A1, 39, 9-14 (11990), 1990. [2] Birman, G. S. and Nomizu, K., Trigonometry in Lorentzian Geometry. Am. Math. Mon., 91, 9, 543-549, 1984. [3] Bottema, O. and Roth, B., Theoretical Kinematics. North-Holland Press, New York, 1979. [4] Ergin, A. A., The Kinematic Geometry On The Lorentzian Plane. Ankara University Graduate School of Naturel and Applied Sciences Deparment of Mathematics, Ph.D. Thesis, (1989). [5] Bükcü, B., Cayley Formula and its Applications in Lorentz Spaces. Ankara University Graduate School of Naturel and Applied Sciences Deparment of Mathematics, Ph.D. Thesis, (2003). [6] Bükcü, B., General Cayley Mapping at Euclidean Space and Rotation Matrices. Erciyes University Graduate School of Naturel and Applied Sciences Press, 22, 1-2, 194-202, 2006. [7] Bükcü, B., On the Rotation Matrices in the Semi-Euclidean Space. Commun. Fac. Sci. Univ. Ank. Series A1, 55, 1, 7-13, 2006. [8] Keçilioğlu, O., Özkaldı, S. and Gündoğan, H., Rotations and Screw Motion with Timelike Vector in 3-Dimensional Lorentzian Space. Adv. App. Clifford Alg., 22, 1081-1091, 2012. 1 Sakarya University, Distance Education Research and Training Center, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected] 139 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [9] Özkaldı, S. and Gündoğan, H., Cayley Formula, Euler Parameters and Rotations in 3- Dimensional Lorentzian Space. Adv. App. Clifford Alg., 20, 367-377, 2010. [10] Güngör, M. A., and Tosun, M., One Parameter Lorentzian Motions In Lorentz 3-Space, Kragujevac Jour. of Math. , 31, 95 - 109, 2008. [11] Tosun, M., Küçük, A. and Güngör, M. A., The Homothetic Motions In The Lorentz 3- Space. Acta Math. Sci., 26, 711-719, 2006. 140 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Ruled Surfaces with Type-2 Bishop Frame in E3 Esra Damar 1 and Nural Yuksel 2 Abstract. In this paper, we research the theory of the ruled surfaces according to type- 2 Bishop frame. Firstly, we calculated Darboux vector of type-2 Bishop motion for fixed and moving spaces in E3 and then we find the distribution parameter of a ruled surfaces generated by a this vector . Also we show that the ruled surfaces whose generated by a darboux vector in type-2 Bishop Frame is developable but there is no developable ruled surfaces generated by the straight line in type-2 Bishop trihedron moving along a curve. Keywords. Ruled surfaces, type-2 Bishop frame, darboux vector, distribution parameter, developable ruled surfaces. AMS 2010. 53A40, 20M15. References [1] Bishop, R. L., There is More than one way to Frame a Curve, The American Mathematical Monthly, 82, 246-251, 1975. [2] Yılmaz, S., Turgut, M., A new version of Bishop Frame and application to spherical images, J.Math. Anal. Appl., 371, 764-776, 2010. [3] Bükcü¸ B., Karacan, M. K., Special Bishop motion and Bishop Darboux rotation axis of the space curve, J. Dyn. Syst. Geom. Theor., 6 , 27-34, 2008. [4] Karacan, M. K., Bükcü¸ B., Yuksel, N., On the dual Bishop Darboux rotation axis of the dual space curve, Appl. Sci., 10 , 115-120, 2008. [5] Yuksel, N., The Ruled Surfaces according to Bishop Frame in Minkowski 3-Space, Hindawi Publishing Corporation Abstract and Applied Analysis, 2013. 1 Hitit University, Corum, Turkey, [email protected] 2 Erciyes University, Kayseri, Turkey, [email protected] 141 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On k-Type Pseudo Null Darboux Helices According to Bishop Frame in Minkowski 3-Space Emilija Nešović1 Abstract. A k-type pseudo null Darboux helices, according to Frenet frame and for k=1,2,3 in Minkowski 3-space are introduced in [1]. In this paper, we derive Bishop frame (relatively parallel adapted frame) for pseudo null curve in Minkowski 3-space and characterize a k-type pseudo null Darboux helices according to such frame. We obtain the relationships between 1-type, 2-type and 3-type pseudo null Darboux helices and give some examples. Keywords. Bishop frame, pseudo null curve, Darboux vector, Darboux helix, Minkowski 3-space AMS 2010. 53A04, 53C40. References [1] Nešović E., Ozturk U., Betul Koc Ozturk E., On k-type pseudo null Darboux helices in Minkowski 3-space, J. Math. Anal. Appl. 439 (2016), 690-700. [2] Ozturk U., Nešović E., On pseudo null and null Cartan Darboux helices in Minkowski 3- space, Kuwait J. Sci. 43(2) (2016), 161-179. [3] Walrave J., Curves and surfaces in Minkowski 3-space, Ph. D. Thesis, University of Leuven, 1995. [4] Ziplar E., Senol A., Yayli Y., On Darboux helices in Euclidean 3-space, Global J. Sci. Front. Res. 12 (2012), 72-80. 1 University of Kragujevac, Faculty of Science, Serbia, [email protected] 142 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Smarandache Curves an Application to Spherical Images Erdal Ozusaglam1 Abstract. In this work, we introduce some Smarandache curves in Euclidean space according to new version of Bishop frame. Also we give some applications to spherical images. Keywords. spherical images, Bishop frame, Smarandache curves. AMS 2010. 53A40, 20M15. References [1] A.T. Ali, Special Smarandache curves in the Euclidean space, International Journal of Mathematical Combinatorics, 2(2010), 30-36. [2] B. Bükçü and M.K. Karacan, On the slant Helices according to Bishop frame, International Journal of Computational and Mathematical Sciences, 3(2) (Spring) (2009), 1039-1042. [3] B. Bükçü and M.K. Karacan, Special Bishop motion and Bishop Darboux rotation axis of the space curve, Journal of Dynamical Systems and Geometric Theories, 6(2008), 27-34. [4] M. Çetin, Y. Tunçer, M.K. Karacan, Smarandache Curves According to Bishop Frame in Euclidean 3-Space, Gen. Math. Notes, Vol. 20, No.2, (Feb. 2014) 50-66. [4] M. Turgut and S. Yilmaz, Smarandache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, 3(2008), 51-55. [5] S. Yilmaz and M. Turgut, On the diferential geometry of the curves in Minkowski space- time I, Int. J. Contemp. Math. Sciences, 3(27) (2008), 1343-1349. 1 Aksaray University, Aksaray, Turkey, [email protected] 143 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Characterization of Involutes and Evolutes of a Given Curve in En Gunay Ozturk1, Kadri Arslan2 and Betul Bulca3 Abstract. The orthogonal trajectories of the first tangents of the curve are called the involutes of x. The hyperspheres which have higher order contact with a curve x are known osculating hyperspheres of x. The centers of osculating hyperspheres form a curve which is called generalized evolute of the given curve x in n-dimensional Euclidean space En. In the present study, we give a characterization of involute curves of order k (resp. Evolute curves) of the given curve x in n-dimensional Euclidean space En. Further, we obtain some results on these type of curves in E3 and E4, respectively. Keywords. Frenet curve, involutes, evolutes. AMS 2010. 53A04, 53A05 References [1] D. Blaženka and MŠ. Željka , Involutes and evolutes in n-dimensional simply isotropic space, Journal of information and organizational sciences 2(3) (1999), 71-79. [2] H. Gluck, Higher curvatures of curves in Euclidean space, Am. Math. Monthly 73 (1966), 699-704. [3] O. A. Goncharova, Ruled surfaces in E⁴ with constant ratio of the Gaussian curvature and Gaussian torsion, Journal of Mathematical Physics, Analysis, Geometry 4(3) (2008), 371- 379. [4] G. Öztürk, K. Arslan and H. H. Hacisalihoglu, A characterization of ccr-curves in R^{m}, Proc. Estonian Acad. Sci. 57(4) (2008), 217-224. [5] M. C. Romero-Fuster and E. Sanabria-Codesal, Generalized evolutes, vertices and conformal invariants of curves in Rⁿ⁺¹, Indag. Mathem., N.S. 10 (1999), 297-305. [6] M. Turgut and T. A. Ali, Some characterizations of special curves in the Euclidean space E⁴, Acta Univ. Sapientiae, Mathematica 2(1) (2010), 111-122. 1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected] 3 Uludag University, Bursa, Turkey, [email protected] 144 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Note on Hypersurfaces of a Riemannian Manifold with a Ricci-Quarter Symmetric Metric Connection Hulya Bagdatli Yilmaz1 Abstract. The object of this paper is to study some properties of hypersurfaces of a Riemannian manifold endowed with a Ricci-quarter symmetric metric connection. Among others, Gauss, Weingarten and Codazzi equations for such a connection have been derived. Keywords. Ricci-quarter symmetric metric connection, Totally geodesic, Totally umbilical, Quasi-umbilical, Gauss equation, Weingarten equation, Codazzi equation, Sectional curvature. AMS 2010. 53C05, 53C07, 53C40, 53B05. References [1] Ahmad, M., Jun, J., Haseeb, A., Hypersurfaces of almost r-paracontact Riemannian manifold endowed with a quarter symmetric metric connection, Bull. Korean Math. Soc., 46, 477-487, 2009. [2] Chen, B., Geometry of submanifolds, Marcel Dekker. INC., New York, 1973. [3] Chen, B., Yano, K., Hypersurfaces of a conformally flat space, Tensor N.S., 26, 318-322, 1972. [4] De, U. C, Mondal, A. K., Hypersurfaces of Kenmotsu manifolds endowed with a quarter symmetric non-metric connection, Kuwait J. Sci. Eng., 39 (1A), 43-56, 2012. [5] Demirbağ, S. A., Yılmaz, H. B., Uysal, S. A., Zengin, F. Ö., On quasi Einstein manifolds admitting a Ricci quarter symmetric metric connection, Bull. of Math. Anal. and App., (3) 4, 84-91, 2011. [6] Golab, S., On semi-symmetric and quarter symmetric linear connections, Tensor N. S., 29, no:3, 249-254, 1975. [7] Kamilya, D., De, U. C., Some properties of a Ricci-quarter symmetric metric connection, Indian J. pure appl. Math., 26 (1), 29-34, 1995. [8] Misra, R. S., Pandey, S. N., On quarter symmetric metric F- connections, Tensor, 34, 1-7, 1934. 1 Marmara University, Istanbul, Turkey, [email protected] 145 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [9] Yano, K., On semi-symmetric metric connection, Rev. Roumanie, Math. Pure Appl., 15, 1579-1586, 1970. [10] Yano, K., Imai, T., Quarter symmetric metric connections and their curvature tensors, Tensor N. S., 38, 13-18, 1982. 146 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Geodesics on the Tangent Sphere Bundle of a Pseudo Hyperbolic 3-Space Ismet Ayhan1 3 Abstract. In this paper, geodesics on a pseudo hyperbolic 3-space H1 have been considered. Then, the Sasaki semi-Riemann metric on the tangent sphere bundle with radius , 3 3 HT 1 of H1 has been obtained and non-null geodesics on are classified into horizontal, vertical and oblique type. Moreover, the geodesics of oblique type have been classified with 3 respect to the principle curvatures of projected curve on of the geodesics on HT 1 Keywords. Tangent Sphere Bundle, Sasaki Semi Riemann Metric, Geodesics. AMS 2010. 55R25, 53C07, 53C22. References [1] Ayhan, I., Geodesics On The Tangent Sphere Bundle of 3-Sphere, International Electronic Journal of Geometry, 6(2), 100-109, 2013. [2] Ayhan, I, On The Sphere Bundle with The Sasaki semi Riemann Metric of a Space Form, GJARCMG, 3(1), 25-35, 2014. [3] Ayhan, I., On The Tangent Sphere Bundle of The pseudo Hyperbolic two Space, GJARCMG, 3(2), 76-90, 2014. [4] Free, P., Introduction to General Relativity, http://personalpages.to.infn.it/~fre/PPT/ virgolect.ppt.3, 2003. [5] Kilingenberg, W., and Sasaki, S., On the tangent sphere bundle of a 2-sphere, Tohuku Math. Journ., 27, 49-56, 1975. [6] Nagy, P.T., Geodesics on the tangent sphere bundle of a Riemann manifold, Geometriae Dedicata, 7, 233-243, 1978. [7] Sasaki,S., Geodesic on the tangent sphere bundles over space forms, Journ. Für die reine und angewandte math., 288, 106-120, 1976. [8] Sasaki, S., On the Differential Geometry of Tangent Bundle of Riemann Manifolds II, Tohuku Math. Journ., 14, 146-155, 1962. 1 Pamukkale University, Denizli, Turkey, [email protected] 147 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A New Characterization of General Helix in Minkowski 3-Space Kazim Ilarslan1 Abstract. In this talk, we discuss the answer of the question whether there exist any general helix whose curvatures 푘1 and 푘2 satisfying the condition |푘1| = |푘2| in Minkowski 3-space. Then we show that the answer of the question is related to the casual character of slope axis of given curve. Furthermore, we give a nice relation between the general helix whose curvatures satisfying the condition |푘1| = |푘2| and the biharmonic curves in Minkowski 3-space. This talk based on the following joint papers with Professors Ç. CAMCI and A. UÇUM. Keywords. General helix, Minkowski 3-space, slope axis, biharmonic curve. AMS 2010. 53C30, 53C52. References [1] Uçum A., Camcı Ç. and İlarslan K., On general helices with spacelike slope axis in Minkowski 3-space, submitted (2015). [2] Uçum A., Camcı Ç. and İlarslan K., On general helices with timelike slope axis in Minkowski 3-space, Advances in Applied Clifford Algebras, June 2016, Volume 26, Issue 2, pp 793-807. [3] Camcı Ç. and İlarslan K. and Uçum A., On general helices with lightlike slope axis in Minkowski 3-space, to appear (2016) 1 Kirikkale University, Kirikkale, Turkey, [email protected] 148 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On Surfaces That Intersection Curves are Special Curves Mesut Altinok1, Benen Akinci2 and Levent Kula3 Abstract. We investigate surfaces that intersection curve are special curves and we obtain characterization for these surfaces. Also related examples and their illustrations are drawn with Mathematica. Keywords. Surfaces, intersection curve, curvatures. AMS 2010. 53A04, 14H52. References [1] Düdül, B. U. and Çalışkan, M., The Geodesic Curvature and Geodesic Torsion of The Intersection Curve of Two Surfaces, Acta Universitatis Apulensis 24, 161-172, 2010. [2] Do Carmo, P.M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ., 1976. [3] Sabuncuoğlu, A., Diferensiyel Geometri, Nobel Yayınevi, 2004. [4] Hacısalihoğlu, H. H. and Ekmekçi, N., Tensör Geometri, Fen Fakültesi, Beşevler-Ankara, 2003. [5] Ye, X. and Maekawa, T., Differential geometry of intersection curves of two surfaces, Computer-Aided Geometric Desing 16, 767-788, 1999. This work is supported by Ahi Evran University Scientific Research Project Coordination Unit. Project number: PYO-FEN.4001.15.011. 1 Ahi Evran University, Kirsehir, Turkey, [email protected] 2 Ahi Evran University, Kirsehir, Turkey, [email protected] 3 Ahi Evran University, Kirsehir, Turkey, [email protected] 149 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Study on Bertrand Curves in E3 Melek Masal1 and Ayse Zeynep Azak 2 Abstract. In this paper, Bertrand curves in the three dimensional Euclidean space are defined according to Bishop frame which has been introduced by Richard L. Bishop in 1975 and it is proven that the distance between the corresponding points of these curves is constant. Moreover, the relationships between Bishop vectors, Frenet vectors and Bishop curvatures of these curves are established. Keywords. Bertrand curves, Bertrand B-pair, Bishop frame. AMS2010. 53A04. References [1] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125, 1503- 1509, 1997. [2] Bertrand, J., La theories de courbes a double courbure, J. Math. Pures et Appl., 15, 332- 350, 1850. [3] Bishop, L. R., There is more than one way to frame a curve, Amer. Math. Monthly 82, 3, 246–251, 1975. [4] Clauvelin, N., Olson, W. K., Tobias I., Characterization of the geometry and topology of DNA pictured as a discrete collection of atoms, J. Chem. Theory Comput. 8, 3, 1092-1107, 2012. [5] Han, C. Y., Nonexistence of rational rotation-minimizing frames on cubic curves, Comput. Aided Geom. Design 25, 4-5, 298-304, 2008. [6] Hanson, A. J., Ma, H., Parallel transport approach to curve framing, Indiana University II, Techreport 425, 1995. [7] Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, J. Geom., 74, 97-109, 2002. [8] Shoeemake K., Animating rotation with quaternion curves, in Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques, 245-254, 1985. 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 150 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Conharmonic Curvature Tensor of Almost Contact, K-Contact and Sasakian Finsler Manifolds Nesrin Caliskan1 Abstract. Conharmonic curvature tensor of almost contact, K-contact and Sasakian Finsler manifolds is studied. In this manner, the notions of quasi-conharmonically flatness, ξ- conharmonically flatness and ϕ-conharmonically flatness of almost contact, K-contact and Sasakian Finsler manifolds are discussed. Some structure theorems that satisfy these conditions are clarified. In this regard it is shown that conharmonically flatness and ϕ- conharmonically flatness of Sasakian Finsler manifolds are equivalent. Keywords. Conharmonic curvature tensor; quasi-conharmonically flatness; ξ- conharmonically flatness; ϕ-conharmonically flatness; Sasakian Finsler manifold. AMS 2010. Mathematics Subject Classification (2010): 53D15, 53C05, 53C15, 53C60 References [1] De, U. C., Singh, R. N., Pandey, S. K., On conharmonic curvature tensor of generalized Sasakian-space-forms, International Scholarly Research Network, 1-14, 2012. [2] Doric, M., Petrovic-Turgasev, M., Versraelen, L., Conditions on the conharmonic curvature tensor of Kahler hypersurfaces in complex space forms, Publications De L’institut Mathematique, 44, 97-108, 1988. [3] Dwivedi, M. K., Kim, J. S., On conharmonic curvature tensor in K-contact and Sasakian manifolds. Bull Malays Math Sci Soc, 34, 171-180, 2011. [4] Ghosh, S., De, U. C., Taleshian, A., Conharmonic curvature tensor on N(K)-contact metric manifolds, International Scholarly Research Network, 1-11, 2011. [5] Khan, Q., On conharmonically and special weakly Ricci symmetric Sasakian manifolds, Navi Sad J Math, 34, 71-77, 2004. [6] Kirichenko, V. F., Rustanov, A. R., Shihab, A. A., Geometry of conharmonic curvature tensor of almost Hermitian manifolds, Journal of Mathematical Sciences, 90 79-93, 2011. 1 Usak University, Faculty of Education, Department of Elementary Mathematics Education, 64200, Usak-TURKEY. e-mail: [email protected] 151 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [7] Kirichenko, V. F., Shihab, A. A., On geometry of conharmonic curvature tensor for nearly Kahler manifolds, Journal of Mathematical Science, 177, 675-683, 2011. [8] Mishra, R. S., Conharmonic curvature tensor in Riemannian, almost Hermite and Kahler manifolds, http://www.dli.gov.in/rawdataupload/upload/insa/INSA_2/20005a8a_330.pdf , 1, 330-335, 1969. [9] Shihab, A. A., On geometry of conharmonic curvature tensor of nearly Kahler manifold, Journal of Researches(Sciences), 37, 39-48, 2011. [10] Yaliniz, A. F., Caliskan, N., Sasakian Finsler manifolds, Turk J Math, 37, 319-339, 2013. 152 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Kinematics for the Closed Planar Homothetic Inverse Motions in Complex Plane Onder Sener1, Ayhan Tutar2 and Serdar Soylu3 Abstract. In this study, the Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic inverse motions in complex plane. The Steiner point was defined when the rotation number was different zero and the Steiner normal was defined when the rotation number was equal to zero. The moving pole point was given with its components and its relation between Steiner point or Steiner normal was explained. The sagittal motion of a telescopic crane was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of telescopic crane and the moving arm of telescopic crane. The results obtained in the first section of this study were applied for this motion. Keywords. Steiner formula, polar moment of inertia, planar kinematics, homothetic inverse motions, complex plane. AMS 2010. 53A17, 70B10. References [1] Blaschke, W., Müller, H. R., Ebene Kinematik, R. Oldenbourg, München, 1956. [2] Dathe, H., Gezzi, R., Characteristic directions of closed planar motions, Zeitschrift für Angewandte Mathematik und Mechanik, 92(9), 731-748, 2012. [3] Kuruoğlu, N., Düldül, M., Tutar, A., Generalization of Steiner formula for the homothetic motions on the planar kinematics, Applied Mathematics and Mechanics (English Edition), 24(8), 945-949, 2003. [4] Müller, H.R., Verall gemeinerung einer Formel von Steiner, Abh. Braunschweig. Wiss. Ges., 29, 107-113, 1978. [5] Müller, H.R., Uber Tragheitsmomente bei Steinerscher Massenbelegung, Abh. Braunschweig. Wiss. Ges., 29, 115-119, 1978. [6] Steiner, J., Von dem Krümmungs-Schwerpuncte ebener Curven, Journal für die reine und angewandte Mathematik, 21, 33-63, 1840. [7] Tutar, A., Kuruoğlu, N., The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics, Mechanism and Machine Theory, 34, 1-6, 1999. 1 Ondokuz Mayis University, Samsun, Turkey, [email protected] 2 Present address: Kyrgyz-Turk Manas University, Faculty of Science, Mathematics Department, Bishkek, Kyrgyzstan Permanent address: Ondokuz Mayis University, Faculty of Art and Science, Samsun, Turkey, [email protected] 3 Giresun University, Giresun, Turkey, [email protected] 153 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Curves of AW(k) in Minkowski 3- Space Pelin Tekin 1 and Erdal Ozusaglam 2 Abstract. In this study, we give the curves of AW(k)-type by using type-2 Bishop frame in Minkowski 3-Space. Also, the characterizations of curves of AW(k)-type were given. Finally, we discuss curvature conditions of kind of curves to the type-2 Bishop frame. Key Words. AW(k)-type curves, Type-2 Bishop Frame AMS 2010. 53A04, 53B30 References [1] K. Arslan and A. West, Product submanifolds with pointwise 3-planar normal sections, Glasgow Math. J., 37, (1995), 73-81. [2] K. Arslan and C. Özgür, Curves and surfaces of AW(k)-type, Geometry and Topology of Submanifolds IX, World Scientific, (1997), 21-26. [4] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, N. J., 1976. [5] M. K. Karacan and B. Bükçü, On natural curvatures of Bishop frame, Journal of Vectorial Relativity, 5, (2010), 34-41. [6] B. Kılıç and K. Arslan, On curves and surfaces of AW(k)-type, BAU Fen Bil. Enst. Dergisi, 6(1), (2004), 52-61. [7] İ. Kişi and G. Öztürk, AW(k)-type curves according to the Bishop frame, arXiv: 1305.3381, (2013). [8] R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int Elec Journ Geom, 3 (2), 67-101, 2010. [9] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983. [10] C. Özgür and F. Gezgin, On some curves of AW(k)-type, Differential Geometry- Dynamical Systems, 7, (2005), 74-80. 1 Aksaray University, Aksaray, Turkey, [email protected] 2 Aksaray University, Aksaray, Turkey, [email protected] 154 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [11] Y. Ünlütürk, and M. Çimdiker, Some characterizations of curves of AW(k)-type according to the Bishop frame, New Trends in Math. Scie. 2(3), (2014) 206-215. [12] S. Yılmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371, (2010) 764-776. 155 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Pointwise Slant Submersions from Sasakian Manifolds Sezin Aykurt Sepet1 and Mahmut Ergut2 Abstract. In this paper, we study the pointwise slant submersions from Sasakian manifolds onto Riemannian manifolds. We investigate the harmonicity of the pointwise slant submersions and obtain necessary and sufficient conditions for such maps to be totally geodesic. We also find curvature relations between the total manifold and the base manifold. Keywords. Riemannian submersion, Sasakian manifold, pointwise slant submersion. AMS 2010. 53C15, 53D15, 53C43. Acknowledgements. This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.A3.16.007 References [1] Baird, P., Wood, J.C., Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs, No.29, Oxford University Press, The Clarendon Press, Oxford, 2003. [2] Blair, D.E., Riemannian geometry of contact and symplectic manifolds, Springer, 2010. [3] Chinea, D., Almost contact metric submersions, Rend. Circ. Mat. Palrmo, 34(1), 89-104, 1985. [4] Falcitelli, M., Ianus, S., Pastore, A.M., Riemannian Submersions and Related Topics, World Scientific, River Edge, NJ, 2004. [5] Gunduzalp, Y., Slant submersions from almost product Riemannian manifolds, Turk. J. Math., 37, 863-873, 2013. [6] Kupeli, I., Murathan, C., Slant Riemannian submersions from Sasakian manifolds, arXiv:1309.2487v1 [Math.DG], 2013. [7] Lee, J.W., Sahin, B., Pointwise slant submersions, Bull. Korean Math. Soc., 51(4), 1115- 1126, 2014. [8] O'Neill, B., The fundamental equations of a submersions, Mich. Math. J., 13, 459-469, 1966. 1 Ahi Evran University, Kirsehir, Turkey, [email protected] 2 Namik Kemal University, Tekirdag, Turkey, [email protected] 156 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [9] Park, K.S., H-Semi-Slant Submersions from almost quaternionic Hermitian manifolds, Taiwan. J. Math., 18(6), 1909-1926, 2014. [10] Park, K.S., H-Slant submersions, Bull. Korean Math. Soc. 49(2), 329-338, 2012. [11] Park, K.S., Pointwise slant and pointwise semi-slant submanifolds in almost contact metric manifolds, arXiv:1410.5587v2 [math.DG], 2014. [12] Sahin, B., Anti-invariant Riemannian submerisons from almost Hermitian manifolds, Cent. Eur. J. Math., 8(3), 437-447, 2010. [13] Sahin, B., Riemannian Submersions from almost Hermitian manifolds, Taiwanese J. Math., 17(2), 629-659, 2013. [14] Sahin, B., Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie, 54(102), no.1, 93-105, 2011. [15] Watson, B., Almost Hermitian submersions, J. Differential Geom., 11(1), 147-165, 1976. [16] Tastan, H.M., Anti-holomorfic semi-invariant submersion from Kaehlerian manifolds, arXiv: 1404.2385. 157 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Decomposition of Out-of-Control Signals in Multivariate Process Semra Boran, Halil Ibrahim Cebeci, Deniz D. Diren and Dogan Unal1 Abstract. Quality control charts are the most effective techniques of statistical process control which is a tool of total quality management. Quality control charts are used in order to monitor the processes and audit the quality of processes if they are under control or not. Multivariate control charts are utilized when there are more than one simultaneously parameter which affects the process. Although these control diagrams are capable of discovering the process being under control or not, they are not capable of finding out which parameter causes this situation. This study uses Mason-Young –Tracy decomposition method that provides the decomposition and estimate of the out-of control signal so as to identify where out-of control situations issue from. Results of the methodology are evaluated through an implementation. Keywords: Statistical process control, Multivariate control chart, Decomposition method. AMS 2010. 62H86, 65C60. References [1] Mason, R. L., Tracy, N. D., Young, J. C., Decomposition of T2 for Multivariate control Chart Interpretation, Journal of Quality Technology, Vol.27, No.2, April 1995 [2] Yu, J., Xi, L., Zhou, X., Identifaying source(s) of out-of-control signals in multivariate manufacturing processes using selective neural network ensemble, Engineering Applications of Artificial Intelligence 22 (2009) 141-152 [3] Li, T., Hu, S., Wei, Z., Liao, Z., A Framework for Diagnosing the Out-of-Control Signals in Multivariate Process Using Optimized Support Vector Machines, Mathematical Problems in Engineering, Volume 2013, Article ID 494626, 9 pages [4] Montgomery, Introduction to Statistical Quality Control, 6th Edition, [5] Oktay E., A survey of multivaiate control chart pattern-Regognition literatüre, 07-12 May 2015, Edirne/Turkey [6] Çetin, S., Birgören, B., Çok değişkenli Kalite Kontrol Çizelgelerinin Döküm Sanayiinde Uygulanması, Gazi Üniv. Müh. Mim. Fak. Der.,Volume 22, No 4, 809-818,2007 1 Sakarya University, Sakarya, Turkey, [email protected] 158 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [7] Aparisi.F., Sans.,J., Interpreting the Out-of-Control Signal of Multivariate Control Charts Employing Neural Networks, International Journal of Computer, Electrical, Automation, control and Information Engineering, Vol:4, No:1, 2010 [8] PrabhuS., Runger.G, Designing a Multivariate EWMA Control Chart, Journal of Quality Technology 29 8-15 Ja’97 [9] ALEB. H., Control Charts Applications For Multivariate Attribute Processes, Computers & Industrial Engineering 56 (2009) 399–410 159 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the nth order Bertrand Mate Curves in E3 Seyda Kilicoglu1 and Suleyman Senyurt2 Abstract. In this study, firstly we worked on the Bertrand mate α2 of Bertrand mate α1 and α3 Bertrand mate α2. We called them second and third order Bertrand mate α3 and α2 of a Bertrand curve α. Secondly we try to give an equation of nth order Bertrand of curve α. Further the orthogonality conditions of Frenet vector fields are given with simple matrix product. Finally we give an equation of nth modified Darboux vector order Bertrand curve of α. Keywords. Bertrand curve, Frenet apparatus, modified Darboux vector, second Bertrand curve. AMS 2010. 53A04 - 53A05. References [1] Hacısalihoğlu, H. H., Differential Geometry(in Turkish), Cilt 1, İnönü University Publications, Malatya, 1994. [2] Izumiya, S., Takeuchi, N., Special curves and Ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44, 1, 203-212, 2003. [3] McCleary J., Geometry from a Differentiable Viewpoint, Vassar College, Cambridge University Press, 1994. [4] Senyurt, Ş., Kılıçoğlu Ş., On the differential geometric elements of the involute ˜D scroll, Adv. Appl. Clifford Algebras Springer Basel, doi:10.1007/s00006-015-0535-z,25(4), 977- 988, 2015. 1Baskent University, Ankara, Turkey, [email protected] 2Ordu University, Ordu, Turkey, [email protected] 160 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Differential Geometric Elements of Mannheim Darboux Ruled Surface in E³ Seyda Kilicoglu 1 and Suleyman Senyurt2 Abstract. In this paper we consider two special ruled surfaces associated to Mannheim curve pair {α,α^{∗}} . First, Mannheim Darboux Ruled surface of the curve α be defined and examined in terms of the Frenet-Serret apparatus of the Mannheim curve α, in E³. Further we will examine the differential geometric elements such as, Weingarten map S, Gauss and mean curvatures K and H, of Darboux ruled surface and Mannheim Darboux ruled surface relative to each other. Also first, second and third fundamental forms of Mannheim Darboux ruled surface will be examined in terms of the Mannheim curve α too. Keywords. Ruled surface, Darboux vector, Mannheim curve. AMS 2010. 53A04, 53A05 References [1] Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 205, 1997. [2] Izumiya S., Takeuchi N., Special curves and Ruled surfaces. Beitr¨age zur Algebra und Geometrie Contributions to Algebra and Geometry, Volume 44, No. 1, 203-212, 2003. [3] Kılıcoglu Ş., and Şenyurt S., On the Differential Geometric Elements of Bertrandian Darboux Ruled surface in E³. (Submited) [4] Lipschutz M.M., Differential Geometry. Schaum's Outlines. [5] Liu H. and Wang F., Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-126(7) , 2008. [6] Orbay K. and Kasap E., On mannheim partner curves, International Journal of Physical Sciences, Vol. 4 (5), 261-264, 2009. [7] Orbay K., Kasap E., and Aydemir İ., Mannheim Offsets of Ruled Surfaces. Mathematical Problems in Engineering. Article Number: 160917, 2009. [8] Senyurt S., and Kılıçoglu Ş.¸ On the differential geometric elements of the involute D scroll, Adv. Appl. Clifford Algebras , Springer Basel, doi:10.1007/s00006-015-0535-z. 25(4), pp. 977-988, (2015). [9] Springerlink, Encyclopaedia of Mathematics. Springer-Verlag, Berlin Heidelberg New York 2002. 1Baskent University, Ankara, Turkey, [email protected] 2Ordu University, Ordu, Turkey, [email protected] 161 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) An Examination on Mannheim Frenet Ruled Surface Based on Normal Vector Fields in E³ Seyda Kilicoglu1 Abstract. In this paper we consider six special Frenet ruled surfaces along to the Mannheim pairs { α, α∗}. First we define and find the parametric equations of Frenet ruled surfaces which are called Mannheim Frenet ruled surface, along Mannheim curve α, in terms of the Frenet apparatus of Mannheim curve α. Later using only one matrix, we give all nine positions of normal vector fields of these six Frenet ruled surfaces and Mannheim Frenet ruled surface in terms of Frenet apparatus of Mannheim curve α too. Further using that matrix we have some results such as; normal ruled surface and Mannheim normal ruled surface of Mannheim curve α have perpendicular normal vector fields along the curve ϕ₂(s)=α+((tanθ)/(k₁tanθ-k₂))V₂, under the condition tanθ≠((k₂)/(k₁)). Keywords. Mannheim curve, Frenet Ruled surface. AMS 2010. 53A04, 53A05 References [1] Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, isbn 0-13- 212589-7, 1976. [2] Ergüt M., Körpınar T. and Turhan E., On Normal Ruled Surfaces of General Helices In The Sol Space Sol³, TWMS J. Pure Appl. Math., 4(2), 125-130, 2013. [3] Graves L.K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc., 252, 367--392, 1979. [4] Kılıçoğlu Ş., Some Results on Frenet Ruled Surfaces Along the Evolute-Involute Curves, Based on Normal Vector Fields in E3. Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, 296-308, Avangard Prima, Sofia, Bulgaria,2016.doi:10.7546/giq-17-2016-296-308. http://projecteuclid.org/euclid.pgiq/1450194164. [5] Kılıçoğlu Ş., On the Involute B-scrolls in the Euclidean Three-space E^{ 3}. XIII^{th}, Geometry Integrability and Quantization, Varna, Bulgaria: Sofia, 205-214, 2012. 1Baskent University, Ankara, Turkey, [email protected] 162 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [6] Kilicoglu S., Senyurt S., Hacisalihoglu H. H., An examination on the positions of Frenet ruled surfaces along Bertrand pairs {α, α∗}} according to their normal vector fields in E³ Applied Mathematical Sciences, Vol. 9, 2015, no. 142, 7095-7103 http://dx.doi.org/10.12988/ams.2015.59605 [7] Liu H. and Wang F., Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No 1-2, 120-126(7) , 2008. [8] Orbay K. and Kasap E., On mannheim partner curves, International Journal of Physical Sciences, Vol. 4 (5), 261-264, 2009. [9] Orbay K., Kasap E., and Aydemir İ., Mannheim Offsets of Ruled Surfaces. Mathematical Problems in Engineering. Article Number: 160917, 2009. [10] Senyurt S., and Kılıçoglu Ş.¸ On the differential geometric elements of the involute D scroll, Adv. Appl. Clifford Algebras , Springer Basel, doi:10.1007/s00006-015-0535-z. 25(4), pp. 977-988, (2015). 163 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Smarandache Curves of Bertrand Curve Pair According to Frenet Frame Suleyman Senyurt1 and Abdussamet Caliskan2 Abstract. In this paper, let be (α, 훼∗) Bertrand curve pair, when the Frenet vectors of 훼∗ curve are taken as the position vectors, the curvature and the torsion of Smarandache curves are calculated. These values are expressed depending upon the α curve. Besides, we illustrate example of our main results. Keywords. Bertrand curve pair, Smarandache Curves, Frenet invariants. AMS 2010. 53A04. References [1] Ali, A. T.,. Special Smarandache Curves in the Euclidean Space, Intenational Journal of Mathematical Combinatorics, 2, 30-36, 2010. [2] Çalışkan, A., Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve, Boletim da Sociedade parananse de Mathemtica, 34, 2, 53—62, 2016. [3] Hacısalihoğlu, H. H., Diferensiyel Geometri, Cilt 1, İnönü Üniversitesi Yayınları, Malatya, 1994. [4] Turgut, M., Yılmaz, S., Smarandache Curves in Minkowski space-time, International Journal of Mathematical Combinatorics, 3, 51-55, 2008. 1Ordu University, Ordu, Turkey, e-mail: [email protected] 2Ordu University, Ordu, Turkey, e-mail: [email protected] 164 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A New Approach on the Striction Curves along Bertrandian Frenet Ruled Surfaces Suleyman Senyurt1 and Abdussamet Caliskan2 Abstract. In this paper, we consider six special ruled surfaces associated to the Bertrand curves pair {훼, 훼∗}. We describe Bertrandian Frenet ruled surfaces and striction curves of these surfaces depending on the angle between the tangent vectors of the Bertrand curves pair {훼, 훼∗}. Also, we examined the situation of the tangent vectors belonging to Striction curves of Frenet and Bertrandian Frenet ruled surfaces. Keywords. Bertrand curves pair, Striction curves, Ruled surfaces, Frenet ruled surface, Bertrandian Frenet ruled surface. AMS 2010. 53A04 - 53A05. References [1] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0- 13-212589-7, 1976. [2] Hacısalihoğlu, H. H., Diferensiyel Geometri, Cilt 1, İnönü Üniversitesi Yayınları, Malatya, 1994. [3] Izumiya, S., Takeuchi, N., Special curves and Ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44, 1, 203-212, 2003. [4] Kılıçoğlu, Ş, Şenyurt, S. Hacısalihoğlu H.H., On the striction curves of Involute and Bertrandian Frenet ruled surfaces in E^3, Applied Mathematical Sciences, 9, 142, 7081 - 7094, 2015. [5] Sabuncuoğlu, A., Differential Geometry, Nobel Publications, Ankara, 2006. 1Ordu University, Ordu, Turkey, [email protected] 2Ordu University, Ordu, Turkey, [email protected] 165 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Some Properties of the Inverse Homothetic Motion: Characteristic Point and Minimal Action Serdar Soylu1, Ayhan Tutar2 and Onder Sener3 Abstract. In this study, the formula of the kinetic energy is expressed during one parameter closed inverse homothetic motion. Then we mentioned about The Principle of least action of the closed planar motion. Finally we found characteristic points of this motion. References [1] Blaschke W., Müller H. R., 1956. Ebene Kinematik, R. Oldenbourg, München. [2] Dathe, H., Gezzi, R. Characteristic directions of closed planar motions. Zeitschrift für Angewandte Mathematik und Mechanik, 92(9), 731-748 (2012) [3] Blaschke, W., Müller, H. R. Ebene Kinematik, R. Oldenbourg, München (1956) [4] Düldül M., Kuruoğlu N., 2008. Computation of polar moments of inertia with Holditch type theorem, Applied Mathematics E-Notes, 8, 271-278. [5] Müller H.R., 1978. Verall gemeinerung einer Formel von Steiner. Abh. Braunschweig. Wiss. Ges., 29, 107-113. [6] Inan, E. and Tutar. A., (2014) Characteristic Directions of Closed Planar Motions. [7] Nagerl H., Kubein-Meesenburg D., Fanghanel J., Thieme K.M., Klamt B., Schwestka- Polly R., 1991. Elements of general theory of joints. 6. General kinematical structure of mandibular movements. Anat. Anz., 173, 249-264. [8] Steiner J., 1840. Von dem Krümmungs-Schwerpuncte ebener Curven, Journal für die reine und angewandte Mathematik, 21, 33-63. [9] Thieme K. M., Kubein-Meesenburg D., Ihlow D., Nagel H., 2006. Is a “movable hinge axis” used by the human stomatognathic system? Acta of Bioengineering and Biomechanics, 8, 13-25. [10] Tölke J., 1978. Steiner-Formein für die Bahnflachen geschlossener Aquiaffinbewegungen, Sitzungsber. Österr. Akad. Wiss, Math.-Nat. Klasse, 187, 325-337. 1 Giresun University, Giresun, Turkey, [email protected] 2 Present address: Kyrgyz-Türk Manas University, Faculty of Science, Mathematics Department, Bishkek, Kyrgyzstan Permanent address:Ondokuz Mayis University, Faculty of Art and Science, Samsun, Turkey, [email protected] 3 Ondokuz Mayıs University, Samsun, Turkey, [email protected] 166 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [11] Tutar A., Kuruoğlu N., 1996. The Steiner formula and the Holditch theorem for the homothetic motions on the planar kinematics, Mechanism and Machine Theory, 34, 1-6 [12] Dathe, H., Gezzi, R, 2014. Characteristi points and cycle in planar kinematics with application to the human gait. Acta of Bioengineering and Biomechanicss. [13] Tutar, A Sener O., 2015, The Steiner Formula and Polar Moment of Inertia for the Closed Planar Homothetic Motions In Complex Plane. 167 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Applications of the Taxicab Metric in Real Life Suleyman Yuksel1 and Kadir Kanat2 and A. Murat Aksoy3 Abstract. In this presentations, some applications of the Taxicab metric which are modelling electrical installations for the buindings, robotic rotation through a line in real life are given. Keywords. Taxicab geometry, Taxicab rotation, Euclidean rotation, Elektrik tesisatı. AMS 2010. 97G50, 51K05, 51K99, 51F99. References [1] H. Minkowski, Gesammelte Abhandlungen, Chelsa Publishing Co., New York, 1967. [2] K. Menger, You Will Like Geometry, Guildbook of the Illinois Institute of Technology Geometry Exhibit, Museum of Science and Industry, Chicago, IL, 1952. [3] E.F. Krause, Taxicab Geometry; An Adventure in Non-Euclidean Geometry, Dover Publications, Inc., New York, 1986. [4] Bayar, A., and R. Kaya. On a Taxicab Distance on a Sphere. Missouri J. Math. Sci 17 (2005): 41-51. [5] Yüksel, S. Spherical Taxicab Geometry. it is submitted. [6] Akça, Ziya, and Rüstem Kaya. On the distance formulae in three dimensional taxicab space. Hadronic Journal 27.5 (2004): 521-532. [7] Odası, Ø. TMMOB Elektrik Mühendisleri. Elektrik İç Tesisleri Yönetmeliği. Ankara, Haziran (2005). [8] Akça, Ziya, and Rüstem Kaya. On the taxicab trigonometry. Jour. of Inst. of Math. Comp. Sci.(Math. Ser.) 10 (1997): 151-159. [9] Sabuncuoğlu, Arif. Analitik Geometri. Nobel Yayın Dağıtım, 2002. Support: This presentation is supported by a project named “63/2016-03 - Taksikab metriği yardımıyla elektrik tesisatı modelleme” from Gazi University BAP. 1 Gazi University, Ankara, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected] 3 Gazi University, Ankara, Turkey, [email protected] 168 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Applications on the Magnetic Fields Zehra Ozdemir1, Ismail Gok2, F. Nejat Ekmekci3 and Yusuf Yayli4 Abstract. Magnetic fields are produced by electric currents and magnetic materials such as moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. Magnetic fields are widely used throughout modern technology, particularly in electrical engineering and electromechanics. Also, a magnetic vector field has many useful applications in physics and geometry. In this study we give some examples and applications related to magnetic fields. Keywords. Special curves, Killing vector field, magnetic flows, differential equations. AMS 2010. 37C10, 53A04. References [1] Bird RB, Stewart WE, Lightfoot EN (1960) Transport Phenomena. John Wiley&Sons. ISBN 0-471-07392-X. [2] Hazeltine RD, Meiss J D (2003) Plasma Confinement. Dover publications, inc. Mineola, New York. [3] Boozer AH (2004) Physics of magnetically confined plasmas. Rev Mod Phys 76:1071- 1141. [4] Barros M, Romeo A (2007) Magnetic vortices. EPL 77: 1-5. [5] Barros M, Cabrerizo JL, Fernández M, Romero A (2007) Magnetic vortex filament flows, J Math Phys, 48: 082904. [6] Hasimoto HA (1972) Soliton on a vortex filament, J Fluid Mech 51: 477-485. [7] Drut-Romanius SL, Munteanu MI (2011) Magnetic curves corresponding to Killing magnetic fields in E3 , J Math Phys 52: 113506. 1 Ankara University, Ankara, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected] 3 Ankara University, Ankara, Turkey, [email protected] 4 Ankara University, Ankara, Turkey, [email protected] 169 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) MATHEMATICS EDUCATION MATHEMATICS EDUCATION 170 Teachers’ Competences for a Successful Implementation of Technology in Mathematics Instruction Ana Donevska-Todorova1and Katja Eilerts2 Abstract. As technologies increasingly influence students’ everyday occupation, research in mathematics education faces new challenges in responding whether and how educational media impact learning outcomes in mathematics. Which teachers’ competences are requred for a successful implementation of technology in mathematics education [3]. In particular, we look at teachers’ content knowledge according to the theory of pedagogical content knowledge (PCK) and the technological-pedagogical content knowledge framework specific for mathematics education (M-TPACK). This relates the content knowledge of linear algebra considering epystemological, historical and didactical issues of concepts as vector spaces, bilinear and multilinear forms [1], [2]. We offer a qualitative analysis on the question: in which way could dynamic geometry environments (DGE) improve students’ learning outcomes and how does it depend on the role of instructors. Keywords. Mathematics education, linear algebra, technology, teaching. AMS 2010. 97U60, 15A03, 15A63, 15A15, 01A05, 15-03. References [1] Donevska-Todorova, A. (2015). Conceptual Understanding of Dot Product of Vectors in a Dynamic Geometry Environment. The Electronic Journal of Mathematics & Technology 9(3). [2] Donevska-Todorova, A. (2014). Three Modes of Description and Thinking of Linear Algebra Concepts at Upper Secondary Education. Proceedings of the 48. Jahrestagung der Gesellschaft für Didaktik der Mathematik (GDM) 48(1), 305-308. [3] Eilerts, K.; Israel, G. & Seifert, A. (2011): Entwicklung eines niveaustufenbezogenen, phasenübergreifenden Berufsfähigkeitsprofil für angehende Lehrkräfte im Bereich allgemeiner pädagogischer Kompetenz. In: Eilerts, K.; Hilligus, A. H.; Kaiser, G. & Bender, P. (Hrsg.), Kompetenzorientierung in Schule und Lehrerbildung - Perspektiven der bildungspolitischen Diskussion, der empirischen Bildungsforschung und der Mathematik- Didaktik (S. 215-230). Münster: LIT Verlag. 1Humboldt-University of Berlin, Berlin, Germany, [email protected] 2Humboldt-University of Berlin, Berlin, Germany, [email protected] 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Scale to Determine Parents' Expectation from Mathematics Education (PEME): Development, Reliability and Validity Cahit Aytekin1, Bulent Altunkaya2, Serdal Baltaci3, Yasemin Kiymaz4 and Avni Yildiz5 Abstract. Main purpose of scale development is to generate a valid measure of an underlying construct [1]. In this study, it was aimed to develop a scale to measure parent expectation from mathematics education. Participant were 321 parents whose children from 4 different public middle school. The total variance explained was 62%. For item pool, 29 items were prepared based on the parents’ expectation literature and views of the experts. Exploratory (EFA) and confirmatory factor analysis (CFA)were used to ensure the construct validity of the scale. In order to determine whether the scale and its factors are valid or not, the Cronbach alpha consistency coefficients were calculated. EFA showed that the PEME scale consist of three factors named as "Conceptual Understanding and Student Active Engaging", "Expectation of positive behavior and attitude", "Teaching Mathematics as a rule and teacher centered instruction". Scree plot have confirmed the number of factor of PEME. After CFA analysis, it was seen that this three factor structure has high or acceptable level of fitting indexes. Cronbach alpha of PEME was calculated as 0,843. This means that PEME is a valid scale to determine parents' expectation from mathematics education. It will be suggested to examine by using the PEME to investigate parents’ expectation from mathematics education. Keywords. Parent Expectation, Mathematics Education, Scale Development. References [1] Clark, L. A., Warson, D., Constructing validity: Basic issues in objective scale development. Psychological, Assessment., 7 , 309-319, 1995. *This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: EGT.A3.16.013” 1Ahi Evran University, Kirsehir, Turkey, [email protected] 2Ahi Evran University, Kirsehir, Turkey, [email protected] 3 Ahi Evran University, Kirsehir, Turkey, [email protected] 4 Ahi Evran University, Kirsehir, Turkey, [email protected] 5 Bulent Ecevit University, Eregli, Zonguldak, Turkey, [email protected] 171 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Teaching of Permutation and Probability with Exchange of Knowledge Method Didem Nimet Berkun1 and Tuba Ada2 Abstract. The aim of this study was to compare The Exchange of Knowledge Method (EKM) to teacher centered teaching methods on learning success and the recall level in teaching Permutation and Probability. The research has been designed in test model with pre- test and post-test control groups. The participants of the study were 36 students that were divided into one experimental (18) and one control group (18). For the purpose of this study, the experimental groups were instructed by using “Exchange of Knowledge Method” technique of cooperative learning whereas the control group was instructured by using teacher centered teaching methods. “Achievement Test” prepared from Permutation and Probability unit in a seventh grade math class were given both groups, at the beginning of the study as a pre-test, at the end of the studyas a post-test and then also given 3 week later as a recall test. t- test and 3x2 Mixed-Design ANOVA was applied for comparing control and experimental groups. At result of study is not founded out a relevant difference as of statistics between post-tests and recall tests of groups. Also It was found that most of the students in the experimental group held positive views about The Exchange of Knowledge Method. However, according to the opinion of some students and the observations have emerged as some of the negative situations such as the students in the low group don’t do all the tasks, some of the students match with friends they do not want to work, to occur more noise in the classroom, to create chaos exchange of peer and group. Keywords. Cooperative Learning, Exchange of Knowledge Method, Teaching Permutation and Probability. AMS 2010. 05A05, 03B48, 91C. References [1] Johnson, D. W. and Johnson, R. T. (1999). Making Cooperative Learning Work. Theory into Practice, 38(2), 67-73. [2] Leikin, R. & Zaslavsky O. (1999). Cooperative Learning in Mathematics. Mathematics Teacher, 92(3), 240-247. [3] Tanışlı, D. ve Sağlam, M. (2006). Matematik öğretiminde işbirlikli öğrenmede bilgi değişme tekniğinin etkililiği. Eğitimde Kuram ve Uygulama, 2(2), 47-67. 1 Middle school teacher (MEB), Afyonkarahisar, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected] 172 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Pattern Generalization Strategies of Elementary Students Deniz Ozen1 and Nilufer Yavuzsoy Kose2 Abstract. Although algebraic thinking is a way of thinking began to develop at an early age, it has seen in the mathematics curriculum that introduction to early algebra with the pattern concept starts from 1st grade of primary school, following this; variable concept starts from 6th grade of elementary school. Generalizing the functional relationship of recognized pattern by using mathematical language is one of the ways to think algebraically described as generalizing mathematical ideas, making statements on this and explaining this process in formal and appropriate ways [1]. This study was aimed to investigate the students' pattern generalization process through the given geometric pattern tasks. To that end, the focus of this study was to determine students' strategies of the recognition and generalization of linear and nonlinear geometric patterns. Participants of this qualitative research were 8 students from grades 5 to 8. In data collecting process, 5 geometric pattern tasks [2] were used and clinical interviews were conducted. In these tasks, students were asked to build first 3 stages of the patterns with pattern blocks, to describe the unknown stages (for example 4th, 10th and 37th stages) of the patterns in words and then to find a general rule for these functions. As a result, the statements and strategies used by students in the recognition and generalization process of patterns in each tasks will be presented with drawings from students' interviews. Keywords. Patterns, Geometric Patterns, Pattern Generalization, Algebraic Reasoning References [1] Blanton, M. L., Kaput, J. J,. Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 412-446, 2005. [2] Markworth, K. A., Growing and growing: Promoting functional thinking with geometric growing patterns, Unpublished doctoral dissertation, University of North Carolina at Chapel Hill, 2010. 1 Adnan Menderes University, Faculty of Education, Aydin, Turkey, [email protected] 2 Anadolu University, Faculty of Education, Eskisehir, Turkey, [email protected] 173 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Development of Algebraic Thinking from the Perspective of Algebraic Habits of Mind Dilek Tanisli1 Abstract. Creative, critical and flexible thinking and before anything else thinking about thinking are important and necessary skills in the complex world of technology. For this reason, it is an important problem confronting educators how students’ thinking skills could be developed, how these skills could be transformed into habits, and how they could be integrated into the learning-teaching environment are confronted as an important problem for educators. Moreover, giving students the mental habits specific to each discipline is mentioned as an anchor that will lead the way in the problem-solving process. Starting from this premise, what the habits of mind are and how they could be used and improved in educational environment seems to worth thinking on. The main purpose of this study is to introduce the conceptual framework of algebraic habits of mind that is revised from the main components of Driscoll’s algebraic habits of mind and the components of algebraic thinking arose from the various research studies. Furthermore, sample activities will be presented to show what could be done for the purpose to acquire algebraic habits of mind and to provide algebraic thinking in the context of habit of mind. Keywords. mathematical habits of mind, algebraic habits of mind, teaching practice. AMS 2010. 53A40, 20M15. References [1] Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, Grades 6-10. Portsmouth, NH: Heinemann. [2] Transition to algebra: A habits of mind approach (2014). Algebraic Habits of Mind. It is retrieved May 26, 2014, from http://ttalgebra.edc.org/AHOM. [3] Moyer, J., Huinker, D., & Cai, J. (2004). Developing algebraic thinking in the earlier grades: A case study of the U.S. Investigations Curriculum. The Mathematics Educator, 8(1), 6-38. 1 Anadolu University, Turkey, [email protected] 174 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Teacher Candidates’ Applications of the Topic of Similarity: A Perspective from the Point of View of Realistic Mathematics Education Isil Bozkurt1, Tugce Kozakli2 and Murat Altun3 Abstract. According to Freudenthal, historically mathematics has started with the real world problems [1] and all the mathematical concepts have emerged by the mathematicization of the real life by people [2]. The concept of “real-world problem” that has a significant place in the RME amounts to the problem situations that can be real or premeditated by the students [3]. The RME that is based on getting the student to confront with these problems has three basic principles. These are the rediscovery of the process, guided discovery and allowing for self-evolving models. Within the scope of the present study, the education was given to teacher candidates in compliance with the RME. In line with this, the topic of similarity was taught to the 8th grade students by the teacher candidates. The education given to the students by the teachers was analyzed in reference to the RME. The study group of this research designed in line with qualitative patterns was composed of 20 (five groups each consisting of four candidates) mathematics teacher candidates. The groups conducted an application for two lesson hours on 8th grade students in five different elementary schools. Within the context of special teaching methods course, a problem was presented through the integration of already-learned knowledge with skills. In the application, first of all, two figures used to teach the topic of similarity theoretically were drawn on the board and the students were asked to tell which previously-learnt mathematics topic those two figures reminded them of. Subsequently, students were asked to debate and express where this particular topic was/could be used in daily life. Following the responses of the students, there were debates about the use of the topic of similarity in calculating the distances which cannot be measured and it was suggested that it could, for instance, be used the width of a river over which a bridge was to be built. In an attempt to carry this real situation into the classroom, considering the gap between the desks in the classroom as a river, it was debated how the width of a river would be measured by using the topic of similarity. A group in the class was picked up and asked to measure the width in question. Meanwhile, all the materials likely to be needed were already made available in the classroom. It was eventually debated in the classroom what purpose the outcome obtained as a result of the application in the classroom 1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected] 3 Uludag University, Bursa, Turkey, [email protected] 175 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) and the topic of similarity would serve in the real world. The teacher candidates were asked to carry out this application with the 8th grade students and present the lesson by video-taping it. Furthermore, the teachers were asked to write an assessment report. Video recordings and assessment reports constituted the data of the study. Applications and the assessment reports were analyzed within the framework of RME principles and presented in the present study. As a result of the analysis, it was found that the teacher candidates fell short of attracting students’ attention to the problem and emphasizing the mathematics side of the application carried out especially at the end of the lesson. As far as guided discovery was concerned, it was observed that when they experienced difficulty, the teachers carried out the application themselves instead of guiding the students. Regarding rediscovery of the process, students’ responses concerning applying the topic of similarity in the real world were noticeable. During these two stages, the teacher candidates stated that since they were given the problem ready-made and the students already knew about the subject, teachers’ application was not effective. Although it was noticeable that the students improved the topic of similarity and managed some solutions reading the concept of equality in process of allowing for the self- evolving models, it was observed that the teacher candidates still failed to deliver the sufficient performance. The students stated that they participated in the lessons with interest and assigned once again a more powerful meaning to an already-known mathematics subject (similarity). Keywords. Real-world problem, realistic mathematics education, similarity. References [1] Altun, M. (2015). Liselerde Matematik Öğretimi. Bursa, Aktüel Alfa Akademi Yayıncılık [2] Gravemeijer, K. (1990) Context problems and realistic mathematic instruction, Gravemeijer, K., Hauvel M. V. & Streefland, L. (Ed.) Contexts Free Productions Tests and Geometry in Realistic Mathematics Education, the State University of Utrecht, Netherlands. [3] Van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In Encyclopedia of mathematics education (pp. 521-525). Springer Netherlands. 176 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Student Presentations on the History of Mathematics and Science in Math Classes and Its Impact on the Learning Process Irina Peterburgsky1 Abstract. Areas of study such as art, language, literature and music are typically taught in conjunction with historical background. For example, poetry is usually presented in tandem with aspects of the prevailing culture. The same should be the norm in mathematics and natural sciences. Understanding of mathematics and natural sciences is improved when students are taught about the time period in which these thinkers lived, how they became interested in their fields of expertise, and how their ideas are related to each other. When students study the history of mathematics and natural sciences, they gain a strong appreciation for the evolution of these fields. At many colleges and universities, the history of mathematics is formally instructed. In smaller ones where it is difficult to find the means to offer courses in the history of mathematics and sciences, there are other opportunities for students to study in an informal setting. Students from both upper and lower courses I teach give presentations on the history of mathematics and sciences in classes and beyond classes each semester. These are structured, guided, and independent research projects. Students learn how mathematics and sciences were developed throughout history, how difficult and non-linear the process of discovery was. They broaden their knowledge of mathematical and scientific concepts as they study about areas which are specific to their presentations. This work brings joy and insight to the students. Each semester a number of them (including those who were majoring in other disciplines) make the decision to choose mathematics or science as their major concentration. I hope that our experiences will be helpful and instructive. I greatly appreciate the willingness of my former students, Philip Oreto and Louis Einhorn, to contribute to this paper. 1 Suffolk University, Boston, USA, [email protected] 177 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Examination of Mathematical Literacy Skill Levels of 8th Grade Students Murat Altun1, Nalan Aydin Gumus2, Recai Akkaya3 and Isil Bozkurt4 Abstract. The concept called mathematical literacy in PISA is defined as the individual’s capacity to understand the role of mathematics in daily life and to employ mathematics in solving his/her problems[1]. The poor performance of Turkish students in PISA[2] indicates the existence of certain problems in the Turkish education system and creates the need of scrutinizing the existing problems. In this study, answers were sought to the following research questions: "(i) What kind of skills do the PISA mathematical literacy questions that Turkish eighth graders have difficulty in solving require? (ii) Is there any similarity between the PISA mathematical literacy performance and the SBS placement exam performance of Turkish students? Mathematical literacy questions were asked to the students at different levels of academic performance without any prior teaching. In doing so, the skills required by the questions and the deficiencies in mathematical abilities were determined. Therefore, this study is different from previous studies, most of which focus on the demographic characteristics of participants. The assessment instrument of the study is the PISA Mathematical Test (PMT) including 12 questions (16 questions in total with sub-questions) chosen among those used in the PISA tests. Prior to the main study, a pilot study was conducted through administering the PMT to 52 eighth grade students, and the assessment instrument was tested in terms of internal and external validity and reliability level. The sample included 726 eighth grade students at low, medium, and high levels of academic performance. Descriptive analysis was performed on the data set. Descriptive survey method was used in this study. Successful results obtained by the students in questions requiring reproduction skills show that they do not have a significant problem in terms of such skills. However, sufficient success was not achieved in the questions requiring connection and reflection skills. Significant differences were observed between the performances of different student groups in the questions requiring the same skill types. Given the fact that incompetence was observed in the problems with different subjects, it is required to prepare contextual problems related to individual, social, professional and scientific situations that will require the use of such skills and to enrich the content of courses. 1 Uludag University, Bursa, Turkey, [email protected] 2 Turkey, [email protected] 3 Abant Izzet Baysal University, Bolu, Turkey, [email protected] 4 Uludag University, Bursa, Turkey, [email protected] 178 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Within the scope of the study, the activities or attempts to increase mathematical literacy success might begin with using contexts for existing questions and establishing relations between mathematical knowledge and real-word situations. Besides, it is also required to incorporate the published mathematical literacy questions in the curriculum Keywords. Mathematical literacy, mathematical literacy skills, problem solving. References [1] McCrone, S.S. & Dossey, J.A. (2007). Mathematical literacy - it’s become fundamental. Principal Leadership, 7 (5), 32-37. [2] Milli Eğitim Bakanlığı (MEB) (2012). PISA Türkiye. Ankara: Yenilik ve Eğitim Teknolojileri Genel Müdürlüğü. 179 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Thinking Further: Mathematics as a Second Language Marjorie Chaves1 and Victor Martinez-Luaces2 Abstract. Mathematics Education researchers [1], [2] remarked the importance of language in the learning, teaching and doing mathematics. The relationship between language and mathematics was already analyzed in a previous paper [3], focused on a particular experience carried out in Germany and Uruguay. Within this research line, new experiences are collected in a recent book chapter [4]. In Uruguay, English is taught as a second language. This paper presents an ex post- facto retrospective research [5] conducted in order to know which were the similarities and differences of the Mathematics and English teaching and learning. Thus, a questionnaire was designed and sent to undergraduate and graduate students of different faculties, mainly to know their point of view concerning with the teaching and learning of both subjects. The responses were coded independently before performing the final coding. These results are analysed and several conclusions and recommendations are proposed. Keywords. English and Mathematics, Second language, Teaching and learning. AMS 2010. 97A40, 97B20, 97C50. References [1] Barton, B., The language of mathematics: Telling mathematical tales. New York, Springer, 2008. [2] Allen, F. B., Language and the Learning of Mathematics. A speech delivered at the NCTM Annual Meeting, Chicago, 1988. [3] Ospitaletche-Borgmann, E. & Martinez-Luaces, V., La Matemática como idioma y su importancia en la enseñanza y aprendizaje del Cálculo. Números. 79, 7 – 16, 2012. [4] Martinez-Luaces, V., Inverse Modeling Problems and their Potential in Mathematics Education, in Teaching and Learning: Principles, Approaches and Impact Assessment. New York: Nova Publishers. 2016. https://www.novapublishers.com/catalog/product_info.php?products_id=58429 [5] Cohen, L., Manion, L., & Morrison, K., Research methods in education, Routledge, 2013. 1IMUC, Montevideo, Uruguay, [email protected] 2UdelaR, Montevideo, Uruguay, [email protected] 180 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Perceptions of Students about Functioning of Mathematics Sakir Isleyen1 Abstract. This study, which is based on the perceptions of the students in economic and administrative sciences, was conducted to determine the idea of the math class; descriptive scanning model was used in the research. Research conducted in Yüzüncü Yıl University, in the Faculty of Economics and Administrative Sciences and in degree programs comprised 329 candidates. In the stage of data collection, the candidates specialized in mathematics were prepared by the researchers to determine their thoughts and perceptions on the content and requirements demanded for "Opinion Survey on Mathematics" and utilized demographic information form. However, to have taken the mathematics course with some questions in the survey which were asked in order to determine the opinions of the candidates for what it gives, the collected data have been analyzed on both a quantitative and qualitative level; descriptive statistics were used. According to the results of the survey candidates in this survey have acquired knowledge of the concepts of space and infinite concepts, especially the mathematics of the properties of space; their interests were centered on the subjects of matrix, differentiation and integration. Also, one of the most striking results of this study was that thirty-five percent of students of this section had interest in mathematics. This has been revealed as a fact that the Education System has still been reconsidered and weighted in terms of employability issues through specialization in mathematics. Keywords. Mathematics Education, the ideal college student, education system. AMS 2010. 97B, 97C. 1 Yuzuncu Yil University, Turkey, [email protected] 181 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Analysis of the Process of the Empty Number Line Usage on Mental Operations Tugce Kozakli1, Isil Bozkurt2 and Murat Altun3 Abstract. In Realistic Mathematics Education, "the empty number line" as an instructional model, was introduced for the first time by Freudenthal [3]. The empty number line is a line or curve which not includes any notation and number on its own and, provides opportunity for marking numbers which students need to calculate [1]. The empty number line which is suggested to use in elementary school (first and second grade) [5], enable a free study environment to students and provide them to produce their own strategy, especially in mental operations, on a linear/nonlinear model that will be drawn randomly. This model is suitable for addition and subtraction operations with the numbers smaller than 100. Besides that, it is a support for students to use informal strategies and bring them to classroom, to make soft computing and to develop problem solving strategies on high levels. The present research was conducted as a "case study" which is a kind of qualitative research design [2]. The study was carried out with 2nd graders during their four hour lesson. In this process, first, it was counted with the bead string and then the real life problems was given to students which were suitable for studying on empty number line. More than one students showed the operations on empty number line on the blackboard at the same time and the strategies they used were discussed in the classroom. In order to determine how this new concept was used, dialogues and discourses of students among themselves and their teacher were focused on. The individual worksheets of students were analyzed used document analysis technique with the purpose of determining strategies students used on the empty number line which it enabled to act freely and provide to use their own strategy. In this study which was researched on mental addition and subtraction operations on the empty number line, it is found that students used different strategies. At first students added or subtracted on by fives and tens and then, during the process, they varied the strategies and found easier paths for themselves. (For ex. Make up 8 to 10 and then add/subtract by 10). We also determined with both researcher observations and classroom teacher opinions that the students participated the lesson actively and willingly. As conclusion in this case study, it is found that the empty line number is an effective tool could be used in elementary grade level for teaching mental operations and problem solving. Keywords. Empty number line, mental operations, operational strategies. 1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected] 3 Uludag University, Bursa, Turkey, [email protected] 182 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Bobis, J., & Bobis, E. (2005). The empty number line: Making children’s thinking visible. Making mathematics vital, 66-72. [2] Cohen, L., Manion, L., & Morrison, K. (2007). Research in mathematics education (6th ed.). New York: Routledge. [3] Freudenthal, H. (1983) Didactical phenomenology of mathematical structures (Dordrecht, D. Reidel). [4] Murphy, C. (2011). Comparing the use of the empty number line in England and the Netherlands. British Educational Research Journal, 37(1), 147-161. [5] Treffers, A. (1991). Didactical background of a mathematics program for primary education, In L. Sttreefland (Ed.) Realistic Mathematics Education İn Primary School, pp.21- 56, Utrecht, THANetherlands. 183 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) STATISTICS STATISTICS 184 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Limit Distribution for a Semi-Markovian Inventory Model of Type (s,S) Under Heavy Tailed Demand with Infinite Variance Aslı Bektas Kamislik1, Tahir Khaniyev22 and Tulay Kesemen3 Abstract. Many problems in stock control, queuing, stochastic finance and reliability theory, can be set down upon investigation of a semi-Markovian inventory model of type (s,S). A semi-Markovian inventory model of type (s,S) has been comprehensively studied in recent years and some of it’s characteristics investigated. Large body of existing literature based on the assumption that the demand distributions are light tailed with finite variance. The main purpose of this study is to investigate the effect of heavy tailed demand quantities on the stochastic process X(t) which represents a semi-Markovian Inventory model of type (s,S). Instead of choosing a single distribution function like Pareto or Cauchy, we worked here with the whole class of regularly varying distributions with infinite variance. Hence we consider the demand random variables are of the form F̅(x) = x−α L(x), 1 < α < 2. Under the assumption of heavy tailed demand and uniform distributed interference of chance, a stochastic process X(t) is constructed. Two term asymptotic expansion for the ergodic X(t)−s S−s distribution of the process Y(t) = , β ≡ → ∞ is obtained as follows when : β 2 2 2 (4υ − υ ) (υ − 2) 1 2 Q (υ) = + [ G (2β) − G (2β − βυ)] + O(β(α−2) −1L (β)), Y 2 4 0 0 2β2 1 here x x t ∞ 1 2 2−α G0(x) = ∫ G(t)dt = ∫ ∫ ∫ F̅(υ)dυds , L1(t) = (L(t)) L(t L(t)). 0 0 0 m1 s Finally by using Karamata theorem, weak convergence theorem is proved for limit distribution. Keywords. Semi-Markovian inventory model, Heavy tailed distributions, Infinite variance, Renewal reward process, Limit distribution, Karamata theorem. AMS 2010. 60K05, 41A60. Acknowledgement. The authors wish to thank to Scientific and Technological Research Council of Turkey TÜBİTAK, for the financial support. Project Number: 115F221. 1Recep Tayyip Erdosan University, Rize, Turkey, [email protected] 2 TOBB University of Economics and Technology, Ankara, Turkey [email protected] 3 Karadeniz Technical University, Trabzon, Turkey, [email protected] 185 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Bekar, N., Aliyev, R., Khaniyev, T., Asymptotic expansions for a renewal-reward process with Weibull distributed interference of chance, Contemporary Analysis and Applied Mathematics, 1(2), 200-211, 2013. [2] Bingham, N.H, Goldie, C.M., Teugels, J.L., Encyclopedia of Mathematics and its Applications-Regular Variation, Cambridge University Press New-York Vol-2, 1987. [3] Borovkov, A.A., Asymptotic Methods in Queuing Theory, John Wiley, New York, 1984. [4] Embrechts, P., Klüppelberg, C., Mikosh, T., Modelling Extremal Events, Springer Verlag, 1997. [5] Feller, W., Introduction to Probability Theory and Its Applications II, John Wiley, New York, 1971. [6] Geluk, J.L., A Renewal Theorem in the finite-mean case., Proceedings of the American Mathematical Society, 125(11), 3407-3413, 1997. [7] Gihman I.I., Skorohod A.V., Theory of Stochastic Processes II, Springer, Berlin, 1975. [8] Seneta, E., Lecture Notes in Mathematics-Regularly Varying Functions, Springer- Verlag New York 1976. 186 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) New Methods of Selection Biasing Parameter for Modified Ridge Estimator Hasan Ertas1 Abstract. In multiple linear regression model, ordinary least square estimator is adversely affected and does not give reliable results in the case where multicollinearity among independent variables are present. To solve these types of problems, biased estimators are used. The most used biased estimator is the ridge estimator proposed by Hoerl and Kennard (1970). Later, Swindel (1976) defined modified ridge estimator (MRE) as an alternative to ridge estimator. Since MRE is a complicated function of biasing parameters k, we are faced with the problem of choice of k. In this study, new methods for the selection of k are proposed. The performances of MRE obtained from using the proposed biasing parameters k, according to mean square error criterion, are examined with a Monte Carlo simulation study. Moreover, a numerical example is given to support simulation results. Keywords. Linear regression, Multicollinearity, Biased estimator. AMS 2010. 62J05, 62J07. References [1] Gultay B., Kaciranlar S., Mean Square Error Comparisons of the Alternative Estimators for the Distributed Lag Models, Hacettepe Journal of Mathematics and Statistics, 44, 1215- 1233, 2015. [2] Hoerl, A.E., Kennard, R.W., Ridge Regression: Biased Estimation for Nonorthogonal Problems, Technometrics, 12, 1, 55-67, 1970. [3] Kibria, B.M.G., Performance of Some New Ridge Regression Estimators, Communications in Statistics, Simulation and Computation, 32(2): 419-435, 2003. [4] Swindel, F. F., Good Ridge Estimators Based on Prior Information, Comm. Statist. Theory Methods, A5 (11), 1065-1075, 1976. 1 Artvin Coruh University, Artvin, Turkey, [email protected] 187 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) McDonald Extended Weibull Distribution Mustafa Cagatay Korkmaz1 Abstract. The extended Weibull distribution (EW) introduced by [4] with the following cumulative distribution function (cdf) / x GEW x; , , 1 exp x e, x 0; , , 0, (1) where and are shape parameters and is scale parameter. From (1), the ordinary Weibull distribution, including exponential and Rayleigh distributions and having monotone hazard rate function (hrf), is obtained when 0 . The EW distribution has increasing or upside-down bathtube hrf. On the other hand, generalized beta-generated or McDonald-G family [1] of the distribution has the following cdf, c Gx,ξ F xabcξξ IGxcb ab 1 ab wa w1 x abc Mc G ;,,, ,,, (,) 1 ,0;,,0, (2) 0 where abc, , 0 are shape parameters, Gx,ξ is cdf of the parent distribution with ξ parameter vector, (,)ab is beta function and I x,, a b is incomplete beta ratio function. The beta-G [3] and Kumaraswamy-G [2] family of the distributions are obtained when c 1 and a 1 respectively. In this paper, we introduce a new extended Weibull model using (1) and (2) with the following cdf, c xe /x 1e c xe /x b1 F x a b c I a b 1 a b wa w x Mc EW ,,,,,, 1e ,, (,) 1 , 0 . 0 (3) Hence, a, b, c>0 are shape parameters in addition to , , 0. Also, we obtain more flexible distribution than ordinary Weibull and extended Weibull distributions for the modelling data. We obtain several properties of this new extended distribution such as special cases, hrf, moments, maximum likelihood estimations. Finally we end the paper with data analysis and conclusions. 1 Artvin Coruh University, Artvin, Turkey, [email protected] 188 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Keywords. Extended Weibull distribution, generalized distribution, hazard rate function. AMS 2010. 60E05, 62H12. References [1] Alexander, C., Cordeiro, G. G., Ortega, E. M. M., Sarabia, J. M., Generalized beta- generated distributions, Computational Statistics and Data Analysis, 56, 1880-1897, 2012. [2] Cordeiro, G.M., de Castro, M., A new family of generalized distributions, Journal of Statistical Computation and Simulation, 81, 883–893, 2011. [3] Eugene, N., Lee, C., Famoye, F., Beta-normal distribution and its applications, Communications in Statistics—Theory and Methods, 31, 497–512, 2002. [4] Peng, X., Yan, Z., Estimation and application for a new extended Weibull distribution, Reliability Engineering and System Safety, 121, 34–42, 2014. 189 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Cluster Analysis Using Gower’s Distance for Panel Data Ozlem Akay1 and Guzin Yuksel2 Abstract. Panel data sets have been increasingly used in economics to analyse complex economic phenomena. The standard panel models are composed of a cross-section character of the data in the time. Through constructing a panel data matrix, the clustering method is applied to panel data analysis. Clustering is a widely used statistical tool to determine subsets in a given data set. Frequently used clustering methods are mostly based on distance measure. So this paper proposes Gower’s distance, which is dedicated to the treatment of mixed data(containing both continuous and binary values). An experimental analysis is illustrated on a real data set by using Stata. Keywords. Panel Data, Cluster Analysis, Gower’s Distance. AMS 2010. 62H30,91C20. References [1] Lu, H., Huang, S., Clustering Panel Data., SIAM International Workshop on Data Mining held in conjunction with the 2011 SIAM International Conference on Data Mining., 1-10., 2011. [2] Mouchart, M., Rombouts, J., Clustered panel data models: An efficient approach for nowcasting from poor data, International Journal of Forecasting., 21, 577-594, 2005. [3] Xu, R., Wunsch, D. C., Clustering, A simple non-Euclidean geometry and its physical basis, John Wiley & Sons, Canada, 2009. [4] Zheng, B., Li, S., Multivariable Panel Data Cluster Analysis and Its Application,, Computer Modeling &New Technologies., 18, 553-557., 2014. [5] Zheng, T., Zhu, D., Wang, X., Yu, B., Panel Data Clustering and Its Application to Discount Rate of B Stock in China., 2009 Second International Conference on Information and Computing Science., 163-166, 2009. 1 Cukurova University, Adana, Turkey, [email protected] 2 Cukurova University, Adana, Turkey, [email protected] 190 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Novel Approximation for Computation Bivariate Distribution Functions in Polygonal Area Orhan Kesemen1, Buğra Kaan Tiryaki2 and Tuncay Uluyurt3 Abstract. Generally bivariate probability density function defined in a rectangular area is used to calculate the cumulative distribution function from the bivariate probability density function [1,2]. However, definition limits of the probability density functions being non-rectangular are in existence in practice [3]. In this paper, primarily arbitrary non- rectangular areas are defined by applying a polygonal approach. The polygonal area obtained as a result of this approach constitutes boundaries of the probability density function. Thus, the bivariate piecewise probability density function can be defined in an arbitrary area. Then the cumulative distribution function is calculated in the obtained area. Two types of approaches are used for these calculations. The first approach is applied to take integral analytically of bivariate continuous probability density function in the polygonal area. The second approach is developed a numerical method since the explicit integral of the selected probability density function cannot be found Keywords. Bivariate distribution functions, cumulative distribution function, bivariate piecewise distribution functions, probability density function based on polygon. AMS 2010. 47N30, 68Q87. References [1] Miller, S. and Childers, D., Probability and random processes: With applications to signal processing and communications, Academic Press, 2012. [2] Roussas, G. G., An introduction to probability and statistical inference, Academic Press, 2003. [3] Kesemen, O. and Doğru, F. Z., Cumulative Distribution Functions of Two Variables In Polygonal Areas, International Statistics Congress, Antalya, Turkey, 2011. 1 Karadeniz Technical University, Trabzon, Turkey, [email protected] 2 Karadeniz Technical University, Trabzon, Turkey, [email protected] 3 Artvin Coruh University, Artvin, Turkey, [email protected] 191 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Linear Regression Model Predictions Using 푳풑-norm: An Application to Explain Plant-available Phosphorus of Corn using Chemical Determination of Inorganic Phosphorus in the Soil Pranesh Kumar1 Abstract. The linear regression models, estimated by using the principle of least squares errors (LSE), are often employed in function fitting and data analysis applications [1, 2, 6, 7 and 8]. These regression models are known to be optimal and perform relatively well under certain assumptions such as when the errors: follow normal distributions, are free of large size outliers and satisfy the Gauss-Markov assumptions. However, in practice, the LSE based linear regression models fail to provide optimal results, for instance, in non- Gaussian situations especially when the errors follow distributions with fat tails and error terms possess a finite variance [9]. There are several applications such as risk analysis which require analyzing tail distributions, where the optimality conditions are not satisfied [10 and 11] and, thus, applications of the LSE based regression models may not be appropriate. Alternatively, we may use 퐿푝-norm to estimate the linear regression model parameters to search for an appropriate regression model [9]. We present the 퐿1, 퐿2 and 퐿∞-norm based regression models for analyzing the experimental data where chemical determination of inorganic phosphorus in the soil is used to explain the plant-available phosphorus of corn grown in the soil. Comparative performance of the 퐿푝-norm based regression models indicate that the LSE based regression model may not be the best choice especially in making tail-end predictions. For making inference from these fitted models, we use resampling methods of Jackknife and Bootstrap [3, 4, 5, 12 and 13]. We discuss the bootstrap confidence intervals, bootstrap percentile intervals and bias-corrected, accelerated (BCa) percentile intervals. Keywords. Least squares estimation, 퐿푝-norm, resampling, confidence interval. AMS 2010. 62J05, 62J20. References [1] Boscovich, R. J., De literaria expeditione per ponticiam ditionem et synopsis amplioris operis, Bononiensi Scientiarum et Artum Instituto atque Academia Commentarii, 4, 353-396, 1757. 1 University of Northern British Columbia, Prince George, Canada, [email protected] 192 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [2] Edgeworth, F. Y., On observations relating to several quantities, Phil. Mag. (5th Series), 24, 222-223, 1887. [3] Efron, B. and Tibshirani, R. J., An Introduction to the Bootstrap, Chapman and Hall, New- York, 1993. [4] Freedman, D. A., Bootstrapping regression models, Annals of Statist., 9, 6, 1218-1228, 1981. [5] Fox, J., Bootstrapping Regression Models, Appendix to An R and S-PLUS Companion to Applied Regression, 2002. http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-bootstrapping.pdf [6] Gauss, C. F., Combinations observationum erroribus minimus obnoxiae pars prior, Printed in Werke, Gottingen IV, 6-7, 1820. [7] Laplace, P. S., Theorie Analytique des Probabilities, Paris, 1812. [8] Legendre, A. M., Nouvelles methodes pour la determination des orbites des cometes Encyclopdia Britannica, 1806. http://www.britannica.com/EBchecked/topic/420949/Nouvelles-methods-pourla determination-des -orbites-des-cometes, 1806 [9] Nyquist, H., Recent Studies on 퐿푝-norm Estimation, Doctoral thesis, University of Umea, 1980. [10] Quenouille, M., Approximate tests of correlation in time series, Journal of the Royal Statist. Soc., B, 11,18-84, 1949. [11] Snappin, S. M. and Small, R. D., Tests of significance using regression models for ordered categorical data, Biometrics, 42, 583-592, 1966. [12] Tukey, J. W., Bias and confidence in not quite large samples, Annals of Math. Statist. 29, 614, 1958. [13] Wu, C. F. J., Jackknife, bootstrap and other resampling methods in regression analysis, Annals of Statist., 14, 4,1261-1295, 1986. 193 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Asymptotic Expansion for the Moments of an Inventory Model of Type (s,S) with Subexponential Demand and Uniform Distributed Interference of Chance Tulay Kesemen1, Asli Bektas Kamislik2 and Zafer Kucuk3 Abstract. Heavy tailed distributions, which tend to produce outlying values attract more and more attention in recent years and the number of publications are systematically growing. The reason of such popularity is, heavy tailed distributions are used to model many physical and economic systems as diverse as medical sciences, civil engineering applications, meteorology, financial risk management and inventory systems. Some studies regarding inventory control models have shown that the random variables which represents the demands tend to heavy-tailed distribution especially when sudden unexpected fluctuations happen in the demand quantities. Hence investigation of some probabilistic characteristics of inventory control systems with heavy tailed demand is of importance. The main purpose of this study is to investigate the asymptotic behaviour of a semi- Markovian inventory model of type (s,S) with heavy tailed demand and uniform distributed interference of chance. Specially we used Weibull distribution with 퐹̅(푥) = exp(−푥훼) , 0 < 훼 < 1 for the demands. It is well known that Weibull distribution belongs to the subexponential subclass of heavy tailed distributions in this case (with shape parameter 0 < 훼 < 1). As a first step we obtained asymptotic expansion for the ergodic distribution function of the process X(t) which represents the considered system here. Then by using the results of the study Geluk and Frenk (2011) we obtained two term asymptotic expansion for the nth order (n=1,2,...) ergodic moments of the process X̃(t) = X(t) − s as follows : n+1 n 2 μ2 2 n E(X̃n) = ( ) βn + ( ) βn−1 + O(βn−2), β ≡ S − s → ∞, (n + 1)(n + 1) μ1 (n + 1)(n + 2) n here μn = E(η ), n = 1,2, … . Finally we tested the accuracy of the approximation formulas by using Monte Carlo simulation methods. Keywords. Semi-Markovian inventory model of type (s,S), Subexponential distri- bution, Renewal reward process, Asymptotic moments, Monte Carlo simulation. AMS 2010. 60K05, 41A60. 1Karadeniz Technical University, Trabzon. Turkey, [email protected] 2Recep Tayyip Erdogan University, Rize, Turkey, [email protected] 3Karadeniz Technical University, Trabzon, Turkey [email protected] 194 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Acknowledgement. The authors wish to thank to Scientific and Technological Research Council of Turkey TÜBİTAK, for the financial support. Project Number: 115F221. References [1] Feller, W., Introduction to Probability Theory and Its Applications II, John Wiley, New York, 1971. [2] Geluk, J.L., Frenk, J.B.G., Renewal Theory for Random Variables with Heavy Tailed Distribution and Finite Varience, Statistics and Probability Letters, 81(1), 77-82, 2011. [3] Khaniyev, T., Kokangül, A., Aliyev, R., An Asymptotic Approach for a Semi-Markovian Inventory Model of Type (s, S), Applied Stochastic Models in Business and Industry, 29(5), 439-453, 2013. [4] Khaniyev, T., Aksop, C., Asymptotic Results for an Inventory Model of Type (s,S) with Generalized Beta Interference of Chance, TWMS J. App. Eng. Math., 1(2), 223-236, 2011. 195 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) TOPOLOGY TOPOLOGY 196 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On Completely δ-b-Irresolute Functions Aynur Keskin Kaymakci1 Abstract. In this talk, we introduce a new class of complete irresoluteness which are called completely δ-b-irresolute functions. Of course, we obtain some characterizations and some properties of them. Also, we give their relationships with other types of functions between topological spaces. Keywords. δ-b-open sets, b-open sets, δ-semi-open sets, semi open sets, completely δ- b-irresolute functions AMS 2010. 54C05, 54C08, 54C10 References [1] G. B. Navalagi, On completely α-irresolute functions, Topology Atlas Preprint#460. [2] E. Ekici and S. Jafari, On a weaker form of complete irresoluteness, Boll. Soc. Paran. Mat., (3s)v. 26 1-2(2008): 81-87. [3] D. Carnahan, Some properties related to compactness in topological spaces, Ph. D. Thesis, Univ. of Arkansas, 1973. [4] S. P. Arya and R.Gupta, On Strongly continuous mappings, Kyungpook Math. J., 14(1974), 131-143. [5] A. Keskin Kaymakci, Weakly δ-b-continuous functions, Gen. Math. Notes, 27(1)(2015), 24-39. [6] P. E. Long and L. L. Herrington, Basic properties of regular closed functions, Rend. Circ. Mat. Palermo, 27(1978), 20-28. [7] J. H. Park, Strongly θ-b-continuous functions, Acta Math. Hungar., 110(4)(2006), 347- 359. [8] A. Keskin Kaymakci, On δ-b-open sets, submitted. [9] G. Navalagi and A. M. Abdul-Jabbar, Some remarks on completely α-irresolute functions, Internat. Jour. Math. Sci.,Vol. 5 No.1 (2006), 1-8. 1 Selcuk University, Konya, Turkey, [email protected] 197 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [10] A. A-El-Atik, A study on some types of mappings on topological spaces, M. Sc. Thesis, Tanta University, Egypt, 1997. [11] D. Andrijević, On b-open sets, Mat. Vesnik, 48(1996), 59-64. [12] L. A. Steen and J. A. Seebach, Jr., Counterexamples in Topology, Holt, Reinhart and Winston, Inc. New York, 1970. [13] M. K. Singal, A. R. Singal and A. Mathur, On nearly compact spaces, Boll. UMI, (4)2(1969), 702-710. [14] E. Ekici, On γ-normal spaces, Bull. Math. Soc. Sci. Math. Roumanie, Tome 50(98) No.3(2007), 259-272. [15] Allam A. A., Zahran A.M., Hasanein I. A., On almost continuous, δ-continuous and set connected mappings, Ind. J. Pure Appl. Math., 18(11)(1987), 991-996. [16] A. S.Mashhour, M. E. Abdel El-Monsef, S. N. El-Deeb, On precontinuous and weak precontinuous functions, Proc. Math. Phys. Soc. Egypt, 53(1982), 47-53. 198 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) New Approaches About I Continuous in Ideal Topological Spaces Ayse Cobankaya1 Abstract. In [8], Özkurt introduced and investigated the new notion continuous. So, we introduce the notions I continuous, I continuous, and I* continuous. In this paper, we investigate the relations among continuous, continuous and these new continuous definitions. Keywords. continuous, continuous, continuous, continuous, continuous. AMS 2010. 54C05, 54C08, 54C10, 54E52. References [1] Abd El-Monsef, M. E.,Lashien E.F. and Nasef A. A., On I− open sets and I− continuous functions, Kyungpook Math. J. 32, 1, 21-30, 1992. [2] Açıkgöz, A., Noiri, T., Yüksel, Ş., A decomposition of continuity in ideal topological spaces, Acta Math. Hungar., 105, 4,285-289, 2004. [3] Fomin, S., Extensions of topological spaces, Ann. of Math., 44, 471- 480, 1943. [4] Hayashi, E., Topologies defined by local properties, Math. Ann., 156, 205-215, 1964. [5] Jankovic, D., Hamlett, T. R., New topologies from old via ideals, Amer. Math. Monthly,97, 295-310,1990. [6] Kuratowski, K., Topology, New- York, Academic Press, 1966. [7] Levine, N., A decomposition of continuity in topological spaces, Amer. Math. Monthly., 68, 44-46, 1961. [8] Özkurt, A., Some generalizations of local continuity im ideal topological spaces, Scientific Studies and Research Series Mathematics and Informatics 24, 1, 75-80, 2014. [9] Vaidyanathaswamy, R., The localisation theory in set topology. Proc. Indian Acad. Sci. 20, 51-61, 1945. 1Cukurova University, Adana, Turkey, [email protected] 199 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Investigation of Some Properties of the Cohomology of SO(N) and Its Classifying Space Ayse Cobankaya1 and Dogan Donmez2 * Abstract. In this paper we determined those generators of H (SO(n), Z2 ) which are * connected by Steenrod operations. We also studied some properties of H(X)G using the Leray - Serre spectral sequence of the fibration XXBGG. Keywords: Steenrod Operations, Spectral Sequence, Borel Cohomology AMS 2010: 18G40, 57T10, 55R10. References [1] McCleary, J., A User's Guide to Spectral Sequences. Cambridge University Press, 2001. [2]Mimura, M., Toda, H., Lie Groups, I And II. American Mathematical Society, 1991. [3] Lucas, E., Théorie des Fonctions Numériques Simplement Périodiques. American Journal of Mathematics 1, 3, 197–240, 1878. 1Cukurova University, Adana, Turkey, [email protected] 2Cukurova University, Adana, Turkey, [email protected] 200 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Recurrence Relations for Unoriented Knot Polynomials of 2, n -Torus Links Kemal Taskopru1 and Ismet Altintas2 Abstract. We prove that the two-variable Kauffman polynomials L and F , and the one-variable BLM/Ho polynomial Q of 2, n -torus link can be determined by a third order recurrence relation, respectively. Also we give the recursive properties of these recurrence relations. Keywords. Kauffman polynomials, BLM/Ho polynomial, Recurrence relation, Generating function, Explicit form. AMS 2010. 57M25, 11B37, 11B83. References [1] Brandt, R. D., Lickorish, W. B. R., Millett, K. C., A polynomial invariant for unoriented knots and links, Invent. Math., 84, 563–573, 1986. [2] Ho, C. F., A new polynomial for knots and links–preliminary report. Abstracts Amer. Math. Soc., 6, 300, 1985. [3] Kauffman, L. H., On knots, Vol. 115 of Annals of Mathematics Study, Princeton University Press, 1987. [4] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 3, 395– 407, 1987. [5] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417–471, 1990. [6] Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, 2011. [7] Shannon, A. G., Horadam, A. F., Some properties of third-order recurrence relations, Fibonacci Quart., 10, 2, 135-145, 1972. 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 201 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Unoriented Knot Polynomials of 2, n -Torus Links as Vieta Polynomials Kemal Taskopru1 and Ismet Altintas2 Abstract. We correlate the two-variable Kauffman polynomials L and F , and the one-variable BLM/Ho polynomials Q of 2, n -torus links with the Vieta polynomials. By using the relations between the BLM/Ho polynomials and Vieta polynomials, we express the Kauffman polynomials in terms of the BLM/Ho polynomials. Keywords. Kauffman polynomials, BLM/Ho polynomial, Vieta polynomials, Recurrence relation, Generating function, Explicit form. AMS 2010. 57M25, 11B37, 11B39, 11B83. References [1] Brandt, R. D., Lickorish, W. B. R., Millett, K. C., A polynomial invariant for unoriented knots and links, Invent. Math., 84, 563–573, 1986. [2] Cereceda, J. L., Determinantal representations for generalized Fibonacci and Tribonacci numbers, Int. J. Contemp. Math. Sci., 9, 6, 269–285, 2014. [3] Ho, C. F., A new polynomial for knots and links–preliminary report, Abstracts Amer. Math. Soc., 6, 300, 1985. [4] Horadam, A. F., Vieta polynomials, Fibonacci Quart., 40, 3, 223-232, 2002. [5] Kauffman, L. H., On knots, Vol. 115 of Annals of Mathematics Study, Princeton University Press, 1987. [6] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 3, 395– 407, 1987. [7] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417–471, 1990. [8] Koshy, T., Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, 2011. [9] Robbins, N., Vieta’s triangular array and a related family of polynomials, Internat. J. Math. Math. Sci., 14, 2, 239-244, 1991. 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 202 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) [10] Shannon, A. G., Horadam, A. F., Some properties of third-order recurrence relations, Fibonacci Quart., 10, 2, 135-145, 1972. [11] Taşçı, D., Yalçın, F., Vieta-Pell and Vieta-Pell-Lucas polynomials, Adv. Difference Equ., 1, 224 , 8pp. 2013. [12] Yalçın, N. F., Taşçı, D., Erkuş-Duman, E., Generalized Vieta-Jacobsthal and Vieta- Jacobsthal-Lucas polynomials, Math. Commun., 20, 2, 241-251, 2015. 203 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Semi-Hurewicz Spaces Ljubiša D.R. Kočinac1, Amani Sabah2 and Moiz ud Din Khan3 Abstract. A topological space 푋 has the Hurewicz property if for every sequence (풰푛 ∶ 푛 ∈ ℕ) of open covers of 푋 there exists a sequence (풱푛 ∶ 푛 ∈ ℕ) such that every 풱푛 is a finite subset of 풰푛 and each 푥 ∈ 푋 belongs to ∪ 풱푛 = ∪ {푉 ∶ 푉 ∈ 풱푛} for all but finitely many 푛. We use semi-open sets to define and study 풔-Hurewicz spaces and their relatives, as well as the star versions of these spaces. Relations with known classes of spaces defined in a similar way are established. Keywords. Semi-open set, s-Hurewicz space, star s-Hurewicz space. AMS 2010. Primary 54D20; Secondary 54C08. 1 University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia, [email protected] 2 COMSATS Institute of Information Technology, Chak Shahzad, Park road, Islamabad, Pakistan, [email protected]. 3 COMSATS Institute of Information Technology, Chak Shahzad, Park road, Islamabad, Pakistan, [email protected] 204 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Linear Operator Equations in Hilbert Space Mohammad Saeed Khan1 Abstract. In this paper we provide existence and uniqueness results for linear operator equations of the form ()I Am x f where A is a self-adjoint operator on a Hilbert space. Some applications to the study of invertible matrices are also presented. Keywords. Linear Operator Equation, Self-Adjoint Operator, Complex Matrix. AMS 2010. 47A05; 47A50. References [1] A. Fonda, J. Mawhin, Iterative and Variational Methods for the Solvability of Some Semilinear Equations in Hilbert Spaces, J. Diff.Equations, 98(1992) 355-375 [2]. J. J. Koliha, Power Convergence and Pseudoinverses of Operators in Banach Spaces, J. Math. Anal. Appl., 48(1974) 446-469 [3]. W. V. Petryshyn, Direct and iterative methods for the solution of linear operator equations in Hilbert space, Trans. Amer. Math. Soc., 105 (1962) 136-175 1 Sultan Qaboos University, Al-Khod, Muscat, Sultanate of Oman, e-mail: [email protected] 205 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Suzuki Type Fixed Point Theorems in Uniform Spaces Vildan Ozturk1 Abstract. In this work, we introduce the notion of Suzuki type contractions in uniform spaces by employing the concept of E-distance functions introduced by Aamri and El- Moutawakil [1]. Also we give existence of a fixed point. Keywords. Fixed point, Suzuki type contraction, E-distance, uniform space. AMS 2010. 47H10, 54H25. References [1] Aamri, M., El Moutawakil, D., Common fixed point theorems for E-contractive or E- expansive maps in uniform spaces, Acta Math. Acad. Paedagogicae Nyiregyhaziensis, 20, 83- 91, 2004. [2] Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 136 (5), 1861–1869, 2008. 1 University of Artvin Coruh, Artvin, Turkey, [email protected] 206 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) THE OTHER AREAS THE OTHER AREAS 207 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Simulation of Flow Boiling Heat Transfer in a Single Horizontal Microchannel Arunabha Chanda1 Abstract. Boiling in microchannels is a very efficient mode of heat transfer in which very high heat and mass transfer coefficients can be achieved. Pumping power required for two-phase flows is lesser than single-phase liquid flows to achieve a given heat removal. Applications include heat pumps, automotive air conditioners etc. Although experimental investigations have been carried out on the topic [1] there is a severe scarcity of literature attempting mathematical modelling. A numerical study of two phase flow through the microchannel has been carried out in this study. The objective of the study is to understand the effects of different fluid inlet velocities and fluid inlet temoperatures. The computational fluid dynamics (CFD) model equations are solved using commercial software ANSYS fluent 13.0 to understand the hydrodynamic and thermal behaviour of the two phase flows through microchannels. The numerical model is validated against available literaure [2]. It is found that the incipient heat flux is influenced by both the inlet velocity as well as the fluid inlet temperature independently, to a great deal. Also as the mass flux is increased (by increasing Reynolds number), the region of single phase flow increases. The heat transfer coefficient at the inlet is high and it falls sharply along the flow direction. But once the boiling process starts , the temperature at the heated wall starts fluctuating which causes fluctuation in the heat transfer coefficient as well. Both convective and nucleate boiling components are present in the range of simulation operation ( I.E. Re =200 to 550 and constant temperature and heat flux.) Keywords. Microchannel, Incipient Heat flux, Heat tranfer coefficient References [1] Wang, G.,Cheng, P., Bergles, A.E., Effects of Inlet /Outlet configurations of Flow Boiling Instability in Parallel Microchannels, Iternational Journal of Heat and Mass Transfer 51, 2267-2281 (2008) [2] Liu, D., Lee, P and Garimella, S. V., Prediction of the Onset of Nucleate Boiling in Microchannel Flow, Iternational Journal of Heat and Mass Transfer 48, 5134-5149 (2005) 1 Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India 208 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Applications of White Noise Calculus to the Computation of Greeks Farai Julius Mhlanga2 Abstract. The sensitivities of financial options to changes in price, volatility, interest rates and other relevant parameters requires differentiating functionals of the random price process which are not smooth. We develop a white noise framework for the computation of Greeks. The starting point is to obtain the extension of the Malliavin derivative to the whole L2 . We then give some important results on the S-transform. Based on these together with the Donsker delta function we derive an explicit expression for delta for processes without jumps and processes with jumps. Keywords. Greeks; white noise; Hermite transform; Donsker delta function. AMS 2010. Primary 60H40; Secondary 60H30. References [1] Aase K., Øksendal B., Privault N. and Ubøe J. (2000) White Noise Generalizations of the Clark-Haussmann-Ocone Theorem with Application to Mathematical Finance. Finance and Stochastics. 4, 465−496. [2] Di Nunno G., Øksendal B. and Proske F. (2009) Malliavin Calculus for L´evy Processes with Application to Finance. Springer-Verlag. [3] Elliott R.J. and Van der Hoek J. (2003) A General Fractional White Noise Theory and Applications to Finance. Math. Finance. 13, 301−330. [4] Holden H., Øksendal B., Ubøe J. and Zhang T. (2009) Stochastic Partial Differential Equations - A Modelling, White Noise Functional Approach. Second Edition. Birkh¨auser, Boston. [5] Mhlanga F. J. and Becker R. (2013) Applications of White Noise Calculus to the Computations of Greeks. Communications on Stochastic Analysis. Vol. 7, No. 4, 493-510. 1 University of Limpopo, Sovenga, Africa, [email protected], [email protected] 209 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Literature Review of the Decision Making Problems Studied in the Theories of Intuitionistic Fuzzy, Neutrosophic and Soft Sets Murat Ibrahim Yazar1 Abstract. In this study we survey the latest studies about decision making problems handled with the theories that contain uncertainty and imprecision. The main theory which is the one that most of the studies are handled about this topic is the fuzzy set theory and there are recently literature surveys such as [1] and [2]. Whereas, in this study we only focus to the decision making problems studied in theories of intuitionistic fuzzy, neutrosophic and soft sets which are all almost new theories dealing with uncertainties. Keywords. Decision making, intuitionistic fuzzy sets, neutrosophic sets, soft sets. AMS 2010. 90B50, 03E72. * This study is supported by Karamanoğlu Mehmetbey University BAP, Project no: 26-M-16. References [1] Ervural, B., Kabak, Ö., A taxonomy for multiple attribute group decision making literature, presented at Fuzzy Systems (FUZZ-IEEE), 2015 IEEE International Conference, İstanbul, 2015. [2] Kahraman, C., Çevik, O.S., Oztaysi, B., Fuzzy multicriteria decision-making: A literature Review, International Journal of Computational Intelligence Systems. 8, 4, 637-666, 2015. [3] Molodtsov, D.A., Soft set theory-first results, Computers and Mathematics with Applications, 37, 19-31, 1999. [4] Atanassov ,K.T., Intuitionistic fuzzy sets, Fuzzy Sets Systems. 20, 87–96, 1986. [5] Smarandache, F., Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math. 24, 287-297, 2005. 1Karamanoglu Mehmetbey University, Karaman, Turkey, [email protected] 210 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Divisor Problem in Special Sets of Gaussian Integers Olga Savastru1 Abstract. Let A and A b e f i x e d s e t s o f the Gaussian integers. () is 1 2 AA12, the number of representations of in form , where AA12, . By the S ()we denote the function in case, when A2 []i , AS2 () is a fixed sector of complex plane Si( ) : 0 arg , . 1 2 2 2 1 L e t T( x ; , 0 , S ( )) S () . 0 (mod ), Nx() Applying the method of Vinogradov we get the asymptotic formula in case, when the norm of a difference of progression grows. Theorem. Let 00, [],()1,(,)i N ,()NN () . Then for every 0, 3 4 32 N () xN () and 21 1 the following formula hold: x 2 1 2 221 x x x x T( x ; , , S ( )) c ( , ) log c ( , ) O , 0 0 0 1 0 1 NN()() N() N 4 () where cc0( , 0 ), 1 ( , 0 ) are computable constants. Keywords. Divisor function, Gaussian numbers, asymptotic formula, arithmetic progression. AMS 2010. 11B25, 11N37, 11N60, 11R42. References [1] Huxley M.N., Exponential sums and lattice points, Proc. London Math. Soc., V. 87, 2003. - 591-609. [2] Nowak W.G., Divisor problems in special sets of positive integers, Acta Arith. Univ. Comenianae.- V. 61, 1992. - 101-115. [3] Varbanec P.D., Zarzycki P., Divisors of the Gaussian Integers in an Arithmetic Progression, J.Number Theory. - V.33, 1990.- 152-169. 1 Odessa I.I.Mechnikov National University, Odessa, Ukraine, [email protected] 211 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Algorithms for R&D Stochastic Control Models Yue Kuen Kwok 1 Abstract. We consider the optimal strategy of R&D expenditure adopted by a firm that engages in R&D to develop an innovative product to be launched in the market. The firm faces with technological uncertainty associated with the success of the R&D effort and market uncertainty of the stochastic revenue flow generated by the new product. Our model departs from most R&D models by assuming that the firm’s knowledge accumulation has impact on the R&D progress, so the hazard rate of arrival of R&D success is no longer memoryless. Also, we assume a finite life span of the technologies that the product resides on. In this paper, we propose efficient finite difference schemes that solve the Hamilton-Jacobi-Bellman formulation of the resulting finite time R&D stochastic control models with an optimal control on R&D expenditure and an optimal stopping rule on the abandonment of R&D effort. Special attention has been taken in the choice of discretization of the HJB equation so that convergence of the numerical solution to the viscosity solution of the HJB equation is guaranteed. Theoretical studies on the convergence properties of the numerical schemes are also presented. We performed numerical tests to verify the theoretical results on convergence of the scheme and examine the impact of the far field boundary condition on accuracy of the numerical calculations. The optimal strategies of R&D expenditure with varying sets of model parameters are analyzed. In particular, we observe that R&D expenditure decreases with firm’s knowledge stock and may even drop to zero when the accumulation level is sufficiently high. This is a joint work with Chi Man Leung, Department of Mathematics, Hong Kong Baptist University 1 Department of Mathematics, Hong Kong University of Science and Technology 212 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) POSTER POSTERS 213 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Direct and Inverse Problems for Fourth-Order Mixed Type Equation with Fractional Derivative Abdumauvlen Berdyshev2 Abstract. The paper describes matters the unique solvability of the direct and inverse nonlocal problems for the mixed type equation of the fourth order with fractional derivative in the sense of Caputo. qtpxtx t t Let ,,10:, ,0 ,0 where qp 0, are real numbers. In we consider the equations Lu xftx ,, (1) where 4 u 0tC tuD ,0, x4 Lu,tx 4u 2u t .0, x4 t 2 Here 0tC uD is the operator of fractional differentiation by t orders 1,0 of in the sense of Caputo [1, p.92]. Let us the formulation of the inverse problem. We need to find a couple of functions with the following properties: txu С 1,4 xf С х,t ,, ;1,0 satisfies the equation (2) in ; txu tutu tutu tutu , satisfies the conditions xx ,0,1,0 x x ,,0,1 xxx xxx ,,1,1 xpxu xqxu qtp , ,, ,, x ;10 satisfies the conditions of bonding xu 0, xuD 0, x .10, 0tC t For the stability of the unique solvability of the formulation of problems are used the theory of integral equation methods and spectral analysis. Let us note that analogous problems studied in [2,3]. Keywords. fourth-order mixed type equation, fractional derivative, ınverse problem. AMS 2010. 26A33, 35L35, 35K35. 2 Abay Kazakh National Pedagogical University, Almaty, Kazakhstan, [email protected] 214 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) References [1] Kilbas A. A., Srivastava H. M., Tujillo J.J. Theory and applications of fractional differentional equations. North-Holland Mathematics Studies, 201. Elsevier Science B.V., Amsterdam, 2006. Xvi+523pp. [2] Berdyshev A. S., Cabada A., Kadirkulov B. J. The Samarskii-Ionkin type problem for the fourth order parabolic equation with fractional differential operator. An Inter. J. Computers and Mathematics with Applications. – Elsevier, 2011. v. 62. - №10. –P.3884-3893. [3] Berdyshev A. S., Eshmatov B., Kadirkulov B. J. Bondary value problems for fourth-order mixed type equation with fractional derivative. Electronic journal of differential equations (EJDE). Vol. №36, 2016. –P.1-11. 215 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Study of the Effect of Wettability Alteration on the Imbibition and Recovery Processes Abdumauvlen Berdyshev1, Bakhbergen Bekbauov2 and Zharasbek Baishemirov3 Abstract. This research describes development of the multicomponent multiphase model of technology "alkali-surfactant-polymer". In currently used models due to the explicitness of solving equations for the components, the size of the time steps limited, in order to stabilize the overall procedure. The approach used here is a sequential approach, which solves both pressure and composition implicitly. The flow equations are solved using a block-centered finite-difference scheme. The user has opportunity to select either one-, two- point upstream or third-order spatial discretization. Numerical results show that surfactants with low optimum solubilization ratio have very poor effect on interfacial tension reduction and only play the role for wettability alteration. Keywords. mathematical model, multiphase model, wettability alteration, surfactant. AMS 2010. 76S05, 76T30. 1 Abay Kazakh National Pedagogical University, Almaty, Kazakhstan, [email protected], 2 Al-Farabi Kazakh National University, Almaty, Kazakhstan, [email protected] 3 Abay Kazakh National Pedagogical University, Almaty, Kazakhstan, [email protected] 216 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Smarandache Curves in Three Dimensional Lie Groups Caner Degirmen1, Osman Zeki Okuyucu2 and Onder Gokmen Yildiz3 Abstract. In this paper, we introduce special Smarandache curves in three dimensional Lie groups with a bi-invariant metric. Also, we obtain Frenet apparatus of a Smarandache curve for some special cases of three dimensional Lie groups Keywords. Smarandache curves, Lie groups. AMS 2010. 53A04; 22E15. References [1] Ahmad, T. A., Special Smarandache Curves in the Euclidean Space, International J.Math. Combin., Vol.2, 30-36, 2010. [2] Turgut, M., Yılmaz, S., Smarandache Curves in Minkowski Space-time, International J.Math. Combin., Vol.3, 51-55, 2008. [3] Bektaş Ö., Yüce, S., Special Smarandache Curves According to Darboux Frame in 퐸3, Romanian J. of Math and Comp. Sci., Vol.3 (1), 48-59, 2013. [4] Okuyucu, O. Z., Gök, İ., Yaylı, Y., Ekmekci, N., Slant helices in three dimensional Lie groups, Appl. Math. and Comp., Vol.221, 672-683, 2013. [5] Gök, İ., Okuyucu, O. Z., Ekmekci, N., Yaylı, Y., On Mannheim Partner Curves in three Dimensional Lie Groups, Miskolc Mathematical Notes., Vol.15 (2), 467–479, 2014. [6] Okuyucu, Z., Yıldız, Ö. G., Tosun, M., Spinor Frenet Equations in Three Dimensional Lie Groups, Adv. Appl. Clifford Algebras, DOI 10.1007/s00006-016-0651-4, 2016. 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 3 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 217 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Microspheres Based on Chitosan as Drug Delivery Systems Gabriela Ciobanu1, Octavian Ciobanu2 and Selman Hizal3 Abstract. In this study, the new chitosan microspheres with different hydroxyapatite and nystatin (an antifungal antibiotic) loadings were prepared by coprecipitation method. The hydroxyapatite nanopowder was obtained by the wet chemical precipitation method, showing an average crystallite size of 58.3 nm. The samples have been characterized by XRD and SEM analysis. The microspheres obtained were 15 - 70 μm in diameter. The release of the nystatin into biological media was studied by in vitro method. The results obtained demonstrated that the chitosan - hydroxyapatite matrix can be used as a solid vehicle for nystatin in medical topical application. Keywords. Chitosan, hydroxyapatite, microspheres, nystatin. AMS 2010. 92C40, 92E10, 92E20. References [1] Zargar, V., Asghari, M., Dashti, A., A review on chitin and chitosan polymers: structure, chemistry, solubility, derivatives, and applications, ChemBioEng Reviews 2015;2:204–226. [2] Rinaudo, M., Chitin and chitosan: properties and applications, Prog Polym Sci 2006; 31:603–632. [3] Pillai, CKS., Paul, W., Sharma, CP., Chitin and chitosan polymers: Chemistry, solubility and fiber formation, Prog Polym Sci 2009;34:641–678. [4] Wang, L.Y., Ma, G.H., Su, Z.G., Preparation of uniform sized chitosan microspheres by membrane emulsification technique and application as a carrier of protein drug, J Control Release 2005;106:62–75. 1 Gheorghe Asachi Technical University of Iasi, Iasi, Romania, [email protected] 2 Grigore T. Popa University of Medicine and Pharmacy, Iasi, Romania, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected] 218 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Test of the Finite-Size Scaling Relations for the Linear Lattice Size 24≤ L≤ 28 of the Four-Dimensional Ising Model on the Creutz Cellular Automaton Ganimet Mulazimoglu Kizilirmak1 and Fatih Yalcin2 Abstract. The four dimensional Ising model is simulated on the Creutz Cellular Automaton (CCA) algorithm with the linear lattice size 24≤ L≤ 28. The temperature dependence of Binder parameter (gL) are analyzed for the lattice with the linear lattice size 24 ≤ L ≤ 28. The exponents in the finite-size scaling relations for the order parameter and the magnetic χ susceptibility at the finite lattice critical temperature are computed to be β (Tc (L=24))= χ χ χ χ 0.4983, β (Tc (L=26))= 0.4987, β(Tc (L=28))=0.4990 and γ(Tc (L=24))=1,0263, γ(Tc (L=26))= χ 1,0256, γ (Tc (L=28))= 1,0267, which are consistent with the renormalization group prediction of β = 0.5 and γ = 1. χ Approximate values for the critical temperature of the infinite lattice of Tc (∞) =6,6858 χ (without the logarithmic factor), Tc (∞) =6,6857 (with the logarithmic factor), 6.6819(11) C (with the logarithmic factor), and Tc (∞) =6,6858 (without the logarithmic factor), C Tc (∞) =6,6857 (with the logarithmic factor) are obtained from the finite-size scaling relations. The results are in very good agreement with the results of a series expansion of Tc (∞), 6.6817 and 6.6802, the dynamic Monte Carlo value Tc (∞) = 6.6803, the cluster Monte Carlo value Tc (∞) = 6.6800, and the Monte Carlo value using the Metropolis-Wolff cluster algorithm Tc (∞) = 6.6802 . Keywords. Ising Model, Cellular Automaton, Critical Temperature, Order Parameter, Magnetic Susceptibility, Acknowledgement. This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: PYOFen.4001.14.010. 1Ahi Evran University, Kırşehir, Turkey, [email protected] 2 Bayburt, Turkey, [email protected] 219 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Numerical Results on a Refinement of Hölder Inequality Gultekin Tinaztepe1 Abstract. In this work, the sharp inequality related to Hölder Inequality which is presented [1], [2] and its derivation is summarised. Also, to show the efficiency of given sharpening, some new numerical results carried out are given. Keywords. Hölder inequality, convexity. AMS 2010. 26D07. References [1] Tınaztepe, G., The Sharpening Hölder Inequality via Abstract Convexity, Turk. J. Math., 40, 438-444, 2016. [2] Tınaztepe, G., Tınaztepe, R. and Kemali, S., On Some Inequalities and Their Refinements, 4th International Eurasian Conference on Mathematical Sciences and Applications, Greece, 2015. [3] Beckenbach, E. and Bellman, R., Inequalities, Springer-Verlag, 1961. 1 Akdeniz University, Antalya, Turkey, [email protected] 220 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Equation of Equilibrium and Dilation Field Hulya Ozturk1 and Nezihe Caliskan2 Abstract. When we deform a body we displace the particles of this body. In measuring the amount of deformation we are interested only in displacements which vary with position in the body. Otherwise, the body is just displaced bodily rather than deformed. The displacement gradient 푒푖푗 interrelates two vectors, displacement 푢 and position 푥, which 휕푢푖 we define as 푒푖푗 = . 휕푥푗 Commonly, our stressed body is in static equilibrium, in which case all the components of force and torque must be zero. This places some restrictions on the stress components 𝜎푖푗. The change in volume of the unit volume called the dilatation is approximately the sum of the diagonal components of the strain tensor 휀푖푗 (휀푖푗 = 푒푖푗 + 푒푗푖), and proportional for the isotropic cases with the sum of the diagonal components of the stress tensor 𝜎푖푗. Keywords. static equilibrium, displacement, dilatation Acknowledgement. This study has been supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.E2.16.024. References [1] H. Öztürk, N. Çalışkan, H. Saraçoğlu, M. S. Soylu, Dilatation Fields of Hexagonal Dislocation Network at the Epilayer/Subsrate Interface, Bulletin of Pure and Applied Sciences, 22D, 2, 101-106, 2003. [2] C. N. Reid, Deformation Geometry For Materials Scientists, Pergamon Press., First edition, 1973. [3] E. Flint, Essentials of crystallography, Mir Publishers, second edition, 1971. [4] I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw Hill, second edition, 1956. 1 Ahi Evran University, Kirsehir, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected] 221 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Separation Axioms in Čech Closure Ordered Spaces Irem Eroglu1 and Erdal Guner2 Abstract. Topological spaces can be generalized by many ways. Leopoldo Nachbin [4] developed a way to generalize topological spaces by an order. He defined topological ordered spaces, such that a triple (X,τ,≤) where τ is a topology and ≤ is a partial order on X. In this paper, we generalize closure spaces by a preorder and we give some order separation axioms in Čech closure ordered spaces. Keywords. Closure ordered space, Ti-ordered separation axiom, Čech closure space AMS 2010. 54A05, 54B05. References [1] Čech, E., Topological Spaces, Czechlovak Acad. Of Sciences, Prag, 1966. [2] Davey, B.,A., Priestly, H.A., Introduction to lattices and order, Cambridge University Press, 1999. [3] Mashhour, A. S., Ghanim, M.H., On closure spaces, Indian J. Pure appl.,14, 6, 680-691, 1983. [4] Nachbin, L., Topology and order, Van Nostrand, Princeton, 1965. 1 Ankara University, Ankara, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected] 222 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Sequentially Topological Groups Ibrahim Ince1 and Soley Ersoy2 Abstract. In this paper, we introduce the notion of sequentially topological groups with the sequentially open and closed subsets. Moreover, we characterize the sequentially topological groups and investigate the relationships between sequentially closed, sequentially open sets, sequentially compactness, sequentially continuity and being sequentially topological groups Keywords. Sequentially spaces and topological groups AMS 2010. 54D55; 54H11 References [1] Boone J.R., Siwiec F., Sequentially quotient mappings, Czechoslovak Math. J. 26 (1976), 174–182. [2] Bose, M. K., Lahiri, I., Sequential topological spaces and separation axioms, Bulletin of the Allahabad Math. Soc., 17(2002), 23-37. [3] Caicedo, X., Sequentially closed sets. (Spanish) Rev. Colombiana Mat. 14 (1980), no. 2, 111–130. [4] Franklin, S.P., Spaces in which sequences suffice, Fund. Math., 57 (1965) 107-115. [5] Joshi, K.D., Introduction to general topology, New Age International (P) Limited, 1983. [6] Kelley, J.L., Convergence in topology, Duke Mathematical Journal, 17:277–283, 1950. [7] Malykhin, V. I., Subspaces of sequential spaces, Mathematical Notes, 64(3), 1998, 351- 356. [8] Pontryagin, L.S., Topological groups, Translated from the second Russian edition by Arlen Brown Gordon and Breach Science Publishers, Inc., New York-London-Paris 1966. [9] Wilansky, A., Topology for analysis, R.E. Krieger Pub. Company Inc., Malabar, Florida 1983. 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 223 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Dependence of the Process of Crystal Growth on the Type of the Initial Distribution Ilya Starodumov1 and Nikolai Kropotin2 Abstract. Tasks of crystal growth can be described by a model called the phase field crystal (PFC). The phase field crystal (PFC) model describes a field that is related to the local atomic number density, such that it is spatially periodic in the solid and constant in the liquid. Originally formulated in a parabolic form for the description of purely dissipative dynamics, the PFC model has also been extended to include faster degrees of freedom consistent with inertia due to propagative regimes of phase transformation. In particular, a hyperbolic or modified PFC model was introduced which includes an inertial term, and thus allows for the description of both fast and slow processes in phase transformations [1], [2]. The process of crystal growth can be expressed as a transition of atomic structure to an absolutely stable state or to a metastable state. In the PFC model these states are described by regular distributions of the atomic density. Getting of the system into any metastable condition may be caused by the peculiarities of the computational domain, initial and boundary conditions. However, an important factor in the formation of the crystal structure can be the initial disturbance. In current report we show how different types of initial disturbance can increase or decrease velocity of the setting an absolutely stable state of crystal structure. Keywords. Phase field crystal, crystal grows, simulation AMS 2010. 74E15, 74S05. References [1] P. Stefanovic, M. Haataja, N. Provatas, Phase field crystal study of deformation and plasticity in nanocrystalline materials, Physical Review E 80 (4) (2009) 046107. [2] P. K. Galenko, H. Gomez, N. V. Kropotin, K. R. Elder, Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation, Phys. Rev. E 88 (2013) 013310. 1 Laboratory of Multi-Scale Mathematical Modeling,Ural Federal University, Ekaterinburg, Russia, [email protected] 2 AO NPO MKM, Izhevsk, Russia, [email protected] 224 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) New Learning Dynamic with Rational and Naive Forecasting Strategies in Cobweb Model Katarina Kukić1 and Jelena Stanojević2 Abstract. A simple cobweb model in agriculture in which an income in rural sector is introduced in [1]. In this paper we introduce new learning dynamics with rational and naive forecasting strategies into this simple model and analyse the stability properties of obtained system. In this model we consider the economy consisting of two sectors, urban and rural in which only one agricultural good is produced. In traditional approach in [1] naive expectation is discussed, were expected price in period t + 1 is simply equal to previous period price. Here we introduce some new assumptions into this model by separating agents in rural and in ubran sector and letting agents in urban sector have different strategies of forecasting. Following [3], [2] we suppose that in urban sector agents base decision upon predictions of future values of endogenous variables (prices in ourcase) whose actual values are determined by equilibrium equations. In this paper we will present simple cobweb model with heterogeneous forecasts, by exploiting techniques from [4]. Result of considering two strategies is dynamical system of two variables with 5 parameters and we present stability analysis of steady states for obtained system. We conclude that for the rational forecasting cost C>0, agents choose naive forecasting strategies, while for cost C=0 rational forecasting is dominant choice. References [1] Vahid F. Nowshirvani, The cobweb phenomenon in subsistence agriculture: a theoretical analysis, Center discussion paper no. 14, Economic growth center, Yale university (1966) [2] Brock, W.A., and Hommes, C.H., Hetereogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamic & Control 22 (1998), 1235-1274 [3] Brock, W.A., and Hommes, C.H., A rational route to randomness, Econometrica 65 (1997), 1059-1095 [4] Waters G.A., Chaos in the cobweb model with a new learning dynamic, Journal of Economic Dynamics & Control, 33 (2005), 1201-1216 1 Faculty of Transport and Traffic Engineering, University of Belgrade 2 Faculty of Economics, University of Belgrade 225 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Vocational High Scholl Students and Mathematics Kamile Sanli Kula1 and Elif Gunden2 Abstract. The purpose of this study is to investigate the mathematics anxiety of vocational high scholl students in terms of some sociodemthisographic variants. In research "Personal Information Form" developed by the researcher, "Mathematics Anxiety Rating Scale" developed by Erol (1989) were used. Keywords. Vocational high schools, Mathematics, Anxiety, Students. AMS 2010. 97B30. This work was supported by the Scientific Research Projects Council of Ahi Evran University, Kırşehir, Turkey under Grant PYO-FEN.4003.15.002. References [1] Bekdemir, M.. Meslek Yüksekokulu Öğrencilerinin Matematik Kaygı Düzeylerinin ve Başarılarının Değerlendirilmesi. Erzincan Üniversitesi, Fen Bilimleri Enstitüsü Dergisi, 2(2), 169-189, 2009. [2] Erktin, E., Dönmez, G. ve Özel, S., Matematik Kaygısı Ölçeğinin Psikometrik Özellikleri. Eğitim ve Bilim, 31(140), 26-33, 2006. [3] Erol, E. Prevalance and Correlates of Math Anxiety in Turkish High School Students. Basılmamış Yüksek Lisans Tezi, Boğaziçi Üniversitesi, Sosyal Bilimler Enstitüsü, İstanbul, 1989. [4] Leylek, R. ve Gürlen, E., Comparison of Basic Math Skills of Vocational High Scholl Students Enrolled with and without Examination System, Electronic Journal of Vocational Colleges, December 2015, 40-46, 2015. 1 Ahi Evran University, Kirsehir, Turkey, [email protected] 2 Ahi Evran University, Kirsehir, Turkey, [email protected] 226 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Natural Lift Curve of the Spherical Indicatrix of a Spacelike Curve with Null Binormal in Minkowski 3-Space Mustafa Caliskan1, Evren Ergun2 and Mustafa Bilici3 Abstract. In this study, we dealt with the natural lift curves of the spherical indicatrices of a spacelike curve with null binormal. Furthermore, some interesting results about the original curve were obtained depending on the assumption that the natural lift 2 curves should be the integral curve of the geodesic spray on the tangent bundle ST 1 and T . Keywords. Natural Lift , Geodesic Spray, Spherical Indicatrices . AMS 2010. 53B30, 51B20, 14H50. References [1] B.O'Neill, Semi-Riemannian Geometry, with applications to relativity. Academic Press, New York, 1983. [2] M. Çalışkan, A.İ. Sivridağ, H.H. Hacısalihoğlu, Some Characterizations for the natural lift curves and the geodesic spray, Communications, Fac. Sci.Univ. Ankara Ser. A Math. 28,(33) (1984), 235-242. [3] E. Ergün, M. Çalışkan, The Natural Lift Curve of The Speherical Indicatrixof a Non-Null Curve In Minkowski 3-Space,International Mathematical Forum,7,(15), (2012), 707-717. [4] E. Ergün, M. Çalışkan, On Geodesic Sprays In Minkowski 3-Space,International Journal of Contemp. Math. Sciences, 6,(39), (2011), 1929-1933. [5] J. Walrave, Curves and Surfaces in Minkowski Space K. U. Leuven FaculteitDer Wetenschappen, 1995. [6] J. A. Thorpe, Elementary Topics In Differential Geometry, Springer-Verlag,New York, Heidelberg-Berlin, 1979. [7] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag,New York, Inc., New York, 1994. 1 Gazi University, Ankara, Turkey, [email protected] 2 Ondokuz Mayıs University, Samsun, Turkey, [email protected]. 3 Ondokuz Mayıs University, Kurupelit, Samsun, Turkey, [email protected]. 227 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Semisimilarity and Consemisimilarity of Split Quaternions Onder Gokmen Yildiz1, Hidayet Huda Kosal2 and Murat Tosun3 Abstract. In this study, we introduce the concept of semisimilarity and consemisimilarity of split quaternions. Moreover, we examine the solvability conditions and general solutions of systems xay b, ybx a and xay b, ybx a in split quaternions. If there exist x and y that satisfy first equations system, then a and b are said to be semisimilar, if there exist x and y that satisfy second equations system, then a and b are said to be consemisimilar. Keywords. Split quaternion, semisimilarity, semiconsimilarity. References [1] Alagoz, Y., Oral, K.H., Yuce, S.: Split quaternion matrices. Miskolc Math. Notes 13, 223– 232 (2012) [2] Bevis, J., Hall, F., Harhvig, R.E.: Pseudoconsimilarity and semiconsimilarity of complex matrices. Linear Algebra Appl. 90, 73–80 (1987) [3] Brand, L.: The roots of a quaternion. Am. Math. Mon. 49, 5197520 (1942) [4] Cho, E.: De Moivre’s formula for quaternions. Appl. Math. Lett. 11(6), 33–35 (1998) [5] Cockle, J.: On systems of algebra involving more than one imaginary. Philos. Mag. 35, 434–435 (1849) [6] Erdogdu, M., Ozdemir, M.: Two-sided linear split quaternionic equations with unknowns. Linear Multilinear Algebra 63(1), 97–106 (2015) [7] Flaut, C.: Some equation in algebras obtained by Cayley–Dickson process. An. St. Univ. Ovidius Constanta. 9(2), 45–68 (2001) 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected] 228 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Evolution of Generalized Space Curve in Minkowski Space Onder Gokmen Yildiz1 and Murat Tosun2 Abstract. In this paper we consider the evolution of generalized space curve in n- dimensional Minkowski Space. We expressed evolution equation of the Frenet frame by matrix equation. Finally, we given integrability conditions for the evolution. Keywords. Evolution of curves, curve flows, Minkowski Space. AMS 2010. 53C44, 53A05, 51B20. References [1] All, N. H. A., Mohamed, S. G., Al-Dossary, M. T., Evolution of Generalized Space Curve as a Function of Its Local Geometry, Applied Mathematics, 5, 2381-2392, 2014. [2] Kwon, D. Y., Park, F.C., Chi, D.P., Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18, 1156-1162, 2005. [3] Yıldız, Ö. G., Ersoy, S., Masal, M., A note on inextensible flows of curves on oriented surface, Cubo a Mathematical Journal, 16(03), 11-19, 2014. [4] Yıldız, Ö. G., Tosun, M., Karakuş, S. Ö., A note on inextensible flows of curves in E n , Int. Electron. J. Geom., 6, 118-124, 2013. 1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 229 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) A Subclass of Convex Univalent Functions as Related to Sigmoid Function Sahsene Altinkaya1 and Sibel Yalcin Tokgoz2 Abstract. In this paper, by using convolution between analytic functions, we define a subclass of univalent functions in the open unit disk U {z∈ℂ:|z|<1}. Making use of Sigmoid function, we find estimates on the coefficients ,, aaa 432 . Moreover, upper bounds are 2 obtained for aa 23 , where μ∈ℂ. Keywords. Coefficient bounds, Sigmoid function, convolution. AMS 2010. 30C45, 30C50. References [1] Bucur, R., Andrei, L., Breaz, D., Coefficient Bounds and Fekete-Szegö Problem for a Class of Analytic Functions Defined by Using a New Differential Operator, Appl. Math. Sci., 9, 1355 – 1368, 2015. [2] Duren, P. L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York , USA, 259, 1983. [3] Fekete, M., Sezegö, G., Eine Bemerkung Über Ungerade Schlichte Funktionen, J. of the London Math. Soc., 2, 85-89, 1933. [4] Fadipe-Joseph, O. A., Oladipo, A.T., Uzoamaka Ezeafulukwe, A., Modified sigmoid function in univalent function theory, International J. of Math. Sci. & Engg. Appls., 7, V, 313- 317, 2013. [5] Kanas, S., Darwish, H. E., Fekete-Szegö problem for starlike and convex functions of complex order, Appl. Math. Let., 23, 777-782, 2010. [6] Kumar, S. S., Kumar, V., On Fekete-Szegö Inequality for Certain Class of Analytic Functions, Acta Universitatis Apulensis, 37, 211-222, 2014. [7] Kowalczyk, B., Lecko, A., Fekete-Szegö Problem for a Certain Subclass of Close-to- convex Functions, Bull. Malays. Math. Sci. Soc., 38, 1393-1410, 2015. 1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected] 230 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On an Equilibrium State in Two-Sector Model of Economic Dynamics Sabir Isa Hamidov1 Abstract. The model of economic dynamics, consisting of two units that produce, respectively, means of production and objects of commodities. We investigate the equilibrium state of the model. The technique of super linear mappings is used. The types of equilibrium model are defined. 1 Baku State University, Baku, Azerbajian, [email protected] 231 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Striction Curves of Involutive Frenet Ruled Surfaces in 푬ퟑ Seyda Kilicoglu1, Suleyman Senyurt2 and Abdussamet Caliskan3 Abstract. In this paper, we consider eight special ruled surfaces associated to the evolute curve α and involute α∗. They are called as Frenet ruled surface and involutive Frenet ruled surfaces, cause of their generators are the Frenet vector fields of evolute curve α. First we give the tangent vector fields of striction curves of all Frenet ruled surfaces and the tangent vector fields of the striction curves of involutive Frenet ruled surfaces are given in terms of the Frenet apparatus of evolute curve α. Further we give only one matrix in which we can see sixteen position of these tangent vector fields, such that we can say there is six position the tangent vector fields are perpendicular. Keywords. Involute curve, Striction curves, Ruled surfaces, Frenet ruled surface, Involutive Frenet ruled surface. AMS 2010. 53A04 - 53A05. References [1] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0- 13-212589-7, 1976. [2] Hacısalihoğlu, H. H., Diferensiyel Geometri, Cilt 1, İnönü Üniversitesi Yayınları, Malatya, 1994. [3] Izumiya, S., Takeuchi, N., Special curves and Ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry, 44, 1, 203-212, 2003. [4] Kılıçoğlu, Ş, Şenyurt, S., Hacısalihoğlu H.H., On the striction curves of Involute and Bertrandian Frenet ruled surfaces in E^3, Applied Mathematical Sciences, 9, 142, 7081- 7094, 2015. 1Baskent University, Ankara, Turkey, [email protected] 2Ordu University, Ordu, Turkey, [email protected] 3Ordu University, Ordu, Turkey, [email protected] 232 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Cauchy-Length Formula and the Holditch Theorem for the Homothetic Motion in p Tulay Erisir1 and Mehmet Ali Gungor2 Abstract. Yaglom [1,2] and Harkin [3] studied the generalized complex number system and generalized complex plane. One-parameter homothetic motion in generalized complex plane jp was studied by Gürses et al. [6]. Moreover, the Steiner formula and the Holditch theorem giving the relationship between the areas formed by linear points in the generalized complex plane p were given by Erisir [4,5]. In this paper, we study a new generalization of the Holditch theorem given by [4] in the generalized complex plane . So, we obtain the Holditch theorem for homothetic motion in the generalized complex plane using the Cauchy-length formula giving the length of the enveloping curves of lines formed by points. Keywords. Generalized complex plane, homothetic motion, Holditch theorem. AMS 2010. 53A17, 53B50, 11E88. References [1] Yaglom, I. M., A Simple non-Euclidean Geometry and its Physical Basis, Springer- Verlag, New-York, 1979. [2] Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968. [3] Harkin A. A., Harkin, J. B., Geometry of Generalized Complex Numbers, Math. Mag., 77, 2, 118-129, 2004. [4] Erisir, T., Güngör, M. A., Tosun, M., A New Generalization of the Steiner Formula and the Holditch Theorem, Adv. Appl. Clifford Algebr., 26, 1, 97-113, 2016. [5] Erisir, T., Güngör, M. A., Tosun, M., The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in Generalized Complex Plane, DOI 10.1007/s00006-016-0642- 5, 2016. [6] Gürses, N. (Bayrak), Akbiyik, M., Yüce, S., One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane Cj, Adv. Appl. Clifford Algebr., 26, 1, 115-136, 2016. 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 233 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) On the Study of the Holditch-Type Theorem for the Non-Linear Three Points in p Tulay Erisir1 and Mehmet Ali Gungor 2 Abstract. The generalized complex number system and generalized complex plane were studied by Yaglom [1,2] and Harkin [3]. The Steiner formula and the Holditch-type theorem giving the relationship between the polar moment of inertia formed by linear points for the one-parameter planar motion in the generalized complex plane p were given by Erisir et al. [4]. In this paper, we study a new generalization of the Holditch-type theorem given by [5] for the non-linear points in the generalized complex plane . While we obtain the Holditch-type theorem, we use the Cauchy-length formula giving the length of the enveloping curves of lines formed by these non-linear points. Keywords. Generalized complex plane, Holditch-type theorem, Cauchy-length formula. AMS 2010. 53A17, 53B50, 11E88. References [1] Yaglom, I. M., A Simple non-Euclidean Geometry and its Physical Basis, Springer- Verlag, New-York, 1979. [2] Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968. [3] Harkin A. A., Harkin, J. B., Geometry of Generalized Complex Numbers, Math. Mag., 77, 2, 118-129, 2004. [4] Erisir, T., Güngör, M. A., Tosun, M., The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in Generalized Complex Plane, DOI 10.1007/s00006-016-0642- 5, 2016. [5] Erisir, T., Güngör, M. A., Tosun, M., A New Generalization of the Steiner Formula and the Holditch Theorem, Adv. Appl. Clifford Algebr., 26, 1, 97-113, 2016. 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 234 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) The Generalized Holditch-Type Theorem for the Homothetic Motion in p Tulay Erisir1, Mehmet Ali Gungor 2 and Mahmut Akyigit3 Abstract. In this paper, we study the Holditch-type theorem which gives the relationship between the polar moment of inertia of the orbit curves drawn by three non-linear points for the homothetic motion in the generalized complex plane p . For this, firstly, we calculate the Cauchy-length formula for the homothetic motion in the generalized complex plane . Then, we prove the Holditch-type theorem with the aid of the length of the enveloping curves of lines formed by points. Keywords. Generalized complex plane, homothetic motion, Holditch-type theorem. AMS 2010. 53A17, 53B50, 11E88. References [1] Yaglom, I. M., A Simple non-Euclidean Geometry and its Physical Basis, Springer- Verlag, New-York, 1979. [2] Yaglom, I. M., Complex Numbers in Geometry, Academic Press, New York, 1968. [3] Harkin A. A., Harkin, J. B., Geometry of Generalized Complex Numbers, Math. Mag., 77, 2, 118-129, 2004. [4] Erisir, T., Güngör, M. A., Tosun, M., A New Generalization of the Steiner Formula and the Holditch Theorem, Adv. Appl. Clifford Algebr., 26, 1, 97-113, 2016. [5] Erisir, T., Güngör, M. A., Tosun, M., The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in Generalized Complex Plane, DOI 10.1007/s00006-016-0642- 5, 2016. [6] Gürses, N. (Bayrak), Akbiyik, M., Yüce, S., One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane Cj, Adv. Appl. Clifford Algebr., 26, 1, 115-136, 2016 1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected] 235 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Split Dual Fibonacci and Lucas Octonions Umit Tokeser1 and Zafer Unal2 Abstract. In this work, we present split dual Fibonacci and Lucas octonions and give generating functions, Binet formulas, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity for split dual Fibonacci and Lucas octonions. Keywords. Dual Fibonacci Octoninos, Dual Lucas Octonions, Split Dual Fibonacci Ocronions, Split Dual Lucas Octonions, Binet’s formula, Generating function, Catalan’s identity. AMS 2010. 11B39, 11B37, 15A66. References [1] W.K. Clifford, Preliminary Sketch of Bi-quaternions, Proc. London Math. Soc., 4: 381- 395, 1873. [2] S. Halıcı, On Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 22:321-327, 2012. [3] S. Halıcı, On Dual Fibonacci Octonions, Adv. Appl. Clifford Algebras, 25:905-914, 2015. [4] A.F. Horadam, Complex Fibonacci Numbers and Fibonacci Quaternions, Amer. Math. Monthly, 70:289-291, 1963. [5] O. Keçilioğlu, İ. Akkuş, The Fibonacci Octonions, Adv. Appl. Clifford Algebras, 25:151- 158, 2015. [6] İ. Akkuş, O. Keçilioğlu, Split Fibonacci and Lucas Octonions, Adv. Appl. Clifford Algebras, 25:517-525, 2015. 1 Kastamonu University, Kastamonu, Turkey, [email protected] 2 Kastamonu University, Kastamonu, Turkey, [email protected] 236 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Exact Solutions of Nonlinear Schrödinger Equation by New Type of Generalized F-Expansion Method Yusuf Ali Tandogan1 and Yusuf Pandir2 Abstract. Many research scientists have been used to find Jacobi elliptic function solutions of nonlinear partial differential equations [1-3]. In this study, a new type of generalized F-expansion method is implementing to get exact solutions of nonlinear Schrödinger equation [4]. Consequently, new exact solutions such as single and combined non-degenerate Jacobi elliptic function solutions are acquired by use of this method. The recommended method has contributed to the farther improvement of the family of solutions of the other nonlinear equations. Keywords. A new type generalized of F-expansion method, hyperbolic nonlinear Schrödinger equation, single and combined non-degenerate Jacobi elliptic function solutions. AMS 2010. 35R50, 35C07, 35C08, 33E05. References [1] Abdou, M. A., An improved generalized F-expansion method and its applications, J. Comput. Appl. Math. 214(1), 202-208, 2008. [2] Zhang, S., Xia, T., A generalized F-expansion method with symbolic computation exactly solving Broer–Kaup equations, Appl. Math. Comput. 189, 1, 836-843, 2007. [3] Pandir, Y., Turhan, N., A new version of the generalized F-expansion method and its applications, AIP Conf. Proceed. (In Press). [4] Kaplan, M., Unsal, O., Bekir, A., Exact solutions of nonlinear Schrodinger equation by using symbolic computation, Math. Meth. Appl. Sci. 39, 2093-2099, 2016. 1Bozok University, Yozgat, Turkey, [email protected] 2Bozok University, Yozgat, Turkey, [email protected] 237 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Differential Invariants of Two Affine Curve Families in Plane Yasemin Sagiroglu1, Demet Aydemir2 and Ugur Gozutok3 Abstract. This paper examines the affine differential invariants of the two curves. The equivalence of the two curves is obtained using these invariants in according to Affine group. In addition, differential invariants obtained will be shown to be the minimal system of generators. Keywords. Affine group, differential invariants, equivalence. AMS 2010. 53A15, 53A55. References [1] Liu, H., Curves in Affine and Semi-Euclidean Spaces, Results. Math., 65, 235-249, 2014. [2] Sağıroğlu, Y., Affine Differential Invariants of Curves, Lambert Academic Publishing, Saarbrücken, 2012. [3] Sağıroğlu, Y, Pekşen, Ö., The equivalence of centro-equiaffine curves, Turk. J. Math. 34, 95-104, 2010. [4] Sağıroğlu, Y., The equivalence problem for parametric curves in one-dimensional affine space, Int. Math. Forum, 6, 4, 177-184, 2011. [5] Sağıroğlu, Y., The equivalence of curves in SL(n,R) and its application to ruled surfaces, Appl. Math. Comput., 218, 1019-1024, 2011. [6] Aripov, R.G., Khadziev, D., The Complete System of Global Differential and Integral Invariants of a Curve in Euclidean Geometry, 51, 1-14, 2007. 1 Karadeniz Technical University, Trabzon, Turkey, [email protected] 2 Karadeniz Technical University, Trabzon, Turkey, [email protected] 3 Karadeniz Technical University, Trabzon, Turkey, [email protected] 238 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) Dual Pell and Pell-Lucas Quaternions Zafer Unal1 and Umit Tokeser2 Abstract. The aim of this work is introduce dual Pell and Pell-Lucas quaternions. We give generating functions and Binet formulas for dual Pell and Pell-Lucas quaternions. Moreover, we obtain some identities for dual Pell and Pell-Lucas quaternions including Catalan’s identitiy, Cassini’s identity and d’Ocagne’s identity etc. Keywords. Dual Pell quaternions, Dual Pell-Lucas quaternions, Binet’s formula, Generating function Catalan’s identity. AMS 2010. 05A15, 11B83, 11R52. References [1] A.F. Horadam, Pell identities, Fibonacci Quart. 9, 245-252. 1971. [2] A.F. Horadam, Quaternion recurrence relations, Ulam Quart. 2(2), 23-33, 1993. [3] C.B. Çimen and A. İpek, On Pell Quaternions and Pell-Lucas Quaternions. Adv. Appl. Clifford Algeb. 26, 39-51, 2016. [4] W.K. Clifford, Preliminary Sketch of Bi-quaternions, Proc. London Math. Soc., 4: 381- 395, 1873. 1 Kastamonu University, Kastamonu, Turkey, [email protected] 2 Kastamonu University, Kastamonu, Turkey, [email protected] 239 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016) PARTICIPANTS 240 List of Participants of IECMSA-2016 Title Name- Surname University Prof. Dr. Abdel SALHI University of Essex Prof. Dr. Abdumauvlen SULEIMANOVICH Abay Kazakh National Pedagogical University Prof. Dr. Antonio FERNANDEZ CARRION Universidad de Sevilla Prof. Dr. Arunabha CHANDA Jadavpur University Prof. Dr. Ayhan TUTAR Kyrgyz-Turkish Manas University Prof. Dr. Christian BERG University of Copenhagen Prof. Dr. Cristina FLAUT Ovidius University Prof. Dr. Elcin YUSUFOGLU Usak University Prof. Dr. Emilija NEŠOVIĆ University of Kragujevac Prof. Dr. Etibar PENAKHOV Baku State University Prof. Dr. Faik Nejat EKMEKCI Ankara University Prof. Dr. Fernando MAYORAL MASA Universidad de Sevilla Prof. Dr. Fikret ALIYEV Baku State University Prof. Dr. Francisco NARANJO NARANJO Universidad de Sevilla Prof. Dr. Gabil ADILOV Akdeniz University Prof. Dr. H. 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