4 th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY 11-15 MAY 2017

Abstract Book

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4th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY 11-15 MAY 2017

Abstract Book

List of Major Sponsors

 Istanbul Commerce University  The Turkish Academy of Sciences  Republic of Turkey Prime Ministry Turkish Cooperation and Coordination Agency  Albaraka Turk  Simurg

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4th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY, 11-15 MAY 2017

Honorary Chairs of Scientific Committee

Prof. Dr. B. E. Rhoades, USA Prof. Dr. H. M. Srivastava, Canada Prof. Dr. Ljubisa Kocinac, Serbia Prof. Dr. Mubariz Tapdigoglu Garayev, Saudi Arabia Prof. Dr. Sadek Bouroubi, Algeria Prof. Dr. Ali M. Akhmedov, Azerbaijan Prof. Dr. Werner Varnhorn, Germany Prof. Dr. Emine Mısırlı, Turkey Prof. Dr. Huseyin Cakalli, Turkey Prof. Dr. G. Das, India Prof. Dr. M. Perestyuk, Ukraine Prof. Dr. O. Boichuk, Ukraine Prof. Dr. I. Shevchuk, Ukraine Prof. Dr. Anatoliy M. Samoilenko, Ukraine Prof. Dr. V. Guliyev, Turkey Prof. Dr. M. Abbas, S.Africa Prof. Dr. M. Mursaleen, India Prof. Dr. W. Sintunavarat, Thailand Prof. Dr. V. Kalantarov, Turkey Prof. Dr. Cihan Orhan, Turkey Prof. Dr. Metin Basarir, Turkey

Organizing Committee

Prof. Dr. Ekrem Savas, Istanbul Commerce University Prof. Dr. Richard Patterson, North Florida University Prof. Dr. Mehmet Gürdal, Suleyman Demirel University Prof. Dr. Martin Bohner, Missouri S&T Prof. Dr. Ram Mohapatra, Uni. of Central Florida Prof. Dr. Fairouz Tchier, King Saud University Prof. Dr. Mehmet Dik, Rockford University Prof. Dr. Lubomira Softova, Second University of Naples Prof. Dr. Agron Tato, Polytechnic Uni. of Tirana Prof. Dr. Debasis Giri, Haldia Institute of Technology Prof. Dr. Naim Braha, Uni. of Prishtina Assoc. Prof. Dr. Yusuf Zeren, Yildiz Technical University Assoc. Prof. Dr. Rahmet Savas, Istanbul Medeniyet University Assoc. Prof. Dr. Erhan Deniz, Kafkas University Assoc. Prof. Dr. Mahpeyker Ozturk, Sakarya University Assoc. Prof. Dr. Esra Duman, Gazi University Assist. Prof. Dr. Sukran Konca, Bitlis Eren University Assist. Prof. Dr. Gokhan Cuvalcioglu, Mersin University Assist. Prof. Dr. Emel Asici, Karadeniz Technical University Assist. Prof. Dr. Arzu Akgul, Kocaeli University Assist. Prof. Dr. Hafize Gumus, Necmettin Erbakan University Dr. Veli Capali, Süleyman Demirel University Dr. Lakhdar Ragoub, Alyammah University Dr. Nora Mahloul, University of Scien. and Tech. Houari Boumedien

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4th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY, 11-15 MAY 2017

Local Organizing Committee

Rabia Savas, Sakarya University Sefa Anil Sezer, Istanbul Medeniyet University Dr. Ulas Yamanci, Suleyman Demirel University Sevim Ertug, Ankara University A. Buyukkaya, Sakarya University Oya Mert, Kemerburgaz University Gulsemay Yigit, Kemerburgaz University

Scientific Committee

Prof. Dr. Huseyin Cakalli, Turkey Prof. Dr. Reza Saadati, Iran Prof. Dr. Jeff Connor, USA Prof. Dr. Salih Aytar, Turkey Prof. Dr. Lubomira Softova, Italy Prof. Dr. Charles Swartz, USA Prof. Dr. Reza Langari, USA Prof. Dr. Yagub A. Sharifov, Azerbaijan Prof. Dr. Mikail Et, Turkey Prof. Dr. Niyazi A. Ilyasov, Azerbaijan Prof. Dr. S. A. Mohiuddine, S. Arabia Prof. Dr. Aref Jeribi, Tunisia Prof. Dr. Narendra Kumar Govil, USA Prof. Dr. Husamettin Coskun, Turkey Prof. Dr. T. A. Chishti, India Prof. Dr. Maria Zeltser, Estonia Prof. Dr. Ayhan Serbetci, Turkey Prof. Dr. Kamalmani Baral, Nepal Prof. Dr. Bilal Altay, Turkey Prof. Dr. Ants Aasma, Estonia Prof. Dr. Ismail Ekincioglu, Turkey Prof. Dr. Ismail N. Cangul, Turkey Prof. Dr. A. Sinan Cevik, Turkey Prof. Dr. Murat Tosun, Turkey Prof. Dr. Leiki Loone, Estonia Prof. Dr. Yilmaz Simsek, Turkey Prof. Dr. Akbar B. Aliyev, Azerbaijan Prof. Dr. Harry Miller, Bosnia Prof. Dr. Vali M. Gurbanov, Azerbaijan Prof. Dr. Ali Fares, France Prof. Dr. Faqir M. Bhatti, Pakistan Prof. Dr. Ibrahim Canak, Turkey Prof. Dr. Said Melliani, Morocco Prof. Dr. Naim Braha, Kosova Prof. Dr. Abdalah Rababah, Jordan Prof. Dr. Mustapha Cheggag, Algeria Prof. Dr. Radouane Yafia, Morocco Prof. Dr. Fahrettin Abdullayev, Kırgizistan Prof. Dr. Sudarsan Nanda, India Prof. Dr. Praveen Agarwal, India Prof. Dr. Seyit Temir, Turkey Prof. Dr. P. D. Srivastava, India Prof. Dr. Halit Orhan, Turkey Prof. Dr. Naila Rozi, Pakistan Prof. Dr. Vatan Karakaya, Turkey Prof. Dr. Emine Can, Turkey Prof. Dr. Amir Khosravi, Iran Prof. Dr. Hemen Dutta, India Prof. Dr. Seifedine Kadry, Kuwait Assoc. Prof. Dr. Bayram Ali Ersoy, Turkey Prof. Dr. Ali M. Akhmedov, Azerbaijan Assoc. Prof. Dr. Ayhan Aydın, Turkey Prof. Dr. Ziyatkan Aliyev, Azerbaijan Asst. Prof. Vishnu Narayan Mishra, India Prof. Dr. Poom Kumam, Thailand Dr. Lejla Miller Van-Wieren, Bosnia Prof. Dr. Agacik Zafer, Kuwait Dr. Mayssa Alqurashi, S. Arabia Prof. Dr. Tunay Bilgin, Turkey Dr. Fardous Taoufic, S. Arabia Prof. Dr. Gangaram S. Ladde, USA Dr. Ammar Edress Mohamed, Iraq Prof. Dr. Claudio Cuevas, Brazil

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Dear Collogues; First of all I wish to offer you a warm welcome to the fourth International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2017). The last conference of this series was organized in Bodrum, Turkey, during 19-23 May 2016 and it was attended by 350 scientists from 48 different countries, contributing 300 oral presentations and 50 posters. As the past conference, the aim of this conference is to provide a platform for mathematicians to present their recent Works, exchange ideas and new methods in several important areas of Mathematics and to provide an opportunity to improve collaboration between local and international participants in the wonderful historic city of Istanbul. Further we believe that, the development in various fields of Mathematics lead to new research areas in Mathematics and the richness of the new results can also provide basis for interdisciplinary collaborations. That is why; we have planned to provide a common forum for scientists to communicate their original results in various fields of analysis and applied mathematics. We would like to thank all the invited speakers who have kindly accepted our invitation and have come to spend their precious time by sharing their ideas during the conference. Finally, we would also like to thank all of the members of the Scientific Advisory Committee and the Organizing Committee of this conference. Again we would like to convey our heartiest welcome to each of you who have come to attend this conference and we wish for an enjoyable high scientific level conference and hope to meet you again in the future. With our best wishes and warm regards,

Prof. Dr. Ekrem SAVAS Chair of the Conference

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4th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY, 11-15 MAY 2017

INVITED TALKS

The Sturm-Liouville Theory and Fourier1 Analysis 1 Prof. Dr. Mohammed Al-Gwaiz

Topological Spaces with an -base 2 Prof. Dr. Taras Banakh

Schwarz Problem for Higher-order Equations in a Polydisc 3 Prof. Dr. A. Okay Celebi

Gelfand Theory Unplugged 4 Prof. Dr. Robin Harte

Decision Models for Autonomous Vehicles 5 Prof. Dr. Reza Langari

Summation and Quadrature Processes for Slowly Convergent Series 6 Prof. Dr. Gradimir V. Milovanović

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4th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED MATHEMATICS, KUSADASI, TURKEY, 11-15 MAY 2017

LIST OF TALKS

New Iterative Methods for Solving Non-Linear Equations 7 Osama Y. Ababneh

Solution of Animplicit Complementarity Problem on Isotone Projection Cones 8 Mujahid Abbas

Analysis and Modeling the Drought Hydrologic by the Copulas in North Algeria 9 Rassoul Abdelaziz

Rayleigh-Marangoni Convection in a Layer of Nanofluid 10 Abdullah A. Abdullah

On the Properties of the Orthogonal Polynomials along a Contour 11 Fahreddin.G.Abdullayev and Gülnara.A.Abdullayev

Modeling and Classifying by using Binary Logistic Regression Analysis Application on Hepatitis Disease 13 Data Qais Mustafa Abdulqader

Residual Power Series Approach to Handle a Class of Fractional Differential Equations 14 Ayed H. Adamat

On the Existence and Uniqeness of Positive Solution for a Fractional Boundary Value Problem with New 15 Fractional Derivative Asghar Ahmadkhanlu

On Derivation of a Subclasses of Filiform Leibniz Algebras 16 AL-Nashri AL-hossain Ahmed

Global Existence and Uniqueness of Weak Solution to a Chemotaxis Model 17 N. Aïssa and A. Balehouane

A Generalization of Hereditary Noetherian Prime Rings 18 Evrim Akalan

On Some Inequalities of Analytic and Biunivalent Functions Given by Subordination 19 Arzu Akgul

Compact Finite Differences Method for Burgers-Huxley Equation 20 Canan Akkoyunlu

Approximation Solution of System of Volterra Integro Differential Using Finite Element Method 21 Adel Alamarashi

Fixed Point Theorem in FM-Spaces 22 Rateb AlBtoush

vii Bounds for the Zeros of Polynomials by Using Similar Matrices 23 Mohammad. Al-Hawari

An Application of Decision Tree for Evaluating a Classroom Teaching Practice 24 Sadri Alija and Halil Snopce

Existence of Positive Solution of Boundary Value Fractional Quadratic Differential Equations 25 Youssef Allaoui, Khalid Hilai and Guida Karimi

Some Applications of Cozero Sets in Topological Spaces 26 Ahmad Al-Omari

An Efficient Analytical-Numerical Technique for Handling Model of Fuzzy Differential Equations of 27 Fractional-Order Mohammad Aloroud, Mohammed Al-Smadi, Rokiah Rozita Ahmad, Ummul Khair Salma Din

Adaptation of Fractional Power Series Method for Solving Fuzzy BVPs 29 Mohammad Alaroud, Rokiah Rozita Ahmad, Mohammed Al-Smadi and Ummul Khair Salma Din

Solving Fuzzy Mixed Integral Equations of Second Kind in Hilbert Spaces 31 Mohammed Al-Smadi

Numerical Algorithm for Solving Time-Fractional Bvps in a Simplified Reproducing Kernel Space 33 Mohammed Al-Smadi

Semiregularization of Almost Countably Compact Spaces 35 Zuhier Altawallbeh

Generalization on Countably Compact Spaces via Hereditary Classes 36 Zuhier Altawallbeh

Hybrid Master Equation of the Jump Diffusion Approximation 37 Derya Altintan and Heinz Koeppl

Coefficient Bounds for a Subclass of Analytic Functions with Respect to Symmetric Points 39 Osman Altintas and Oznur Ozkan Kilic

On The Relationship Between A Family of Fibonacci And Lucas Numbers 40 Ipek Altun, Ali Aydogdu and Engin Ozkan

Convergence, Consistency and Stability in Intuitionistic Fuzzy Differential Equations 41 Bouchra Ben Amma, Said Melliani and Lalla Saadia Chadli

SQCQP Descent Scheme for Multi-objective Optimization Problem 42 Md Abu Talhamainuddin Ansary and Geetanjali Panda

On Gauss Balancing and Gauss Cobalancing Numbers 43 Mustafa Asci and Mustafa Yilmaz

An Overview of Ordering Based on Nullnorms 44 Emel Asici

Analytical-Numerical Solutions for a class of Systems of Differential Equations Using Reproducing Kernel 45 Method Ali Mahmud Ateiwi

A Fuzzy Project Scheduling with Constrained Resources 47 Lyazzat Atymtayeva, Ardakbek Kungaliyev and Daniyar Artykov

viii A Convergent Two-Level Linear Scheme for the Generalized Rosenau-Kdv-RLW Eqution 48 Ayhan Aydin

Fuzzy Soft Metric and Fuzzifying Soft Topology Induced by Fuzzy Soft Metric 49 Ebru Aydogdu, Abdulkadir Aygunoglu, Halis Aygun

Some Identities Associated With Hecke Operators 50 Aykut Ahmet Aygunes

Some Rough Convergence Criteria for the Sequences of Intervals of Fuzzy Numbers 51 Salih Aytar

Modified Simple Equation Method and its Applications to Some Nonlinear Physical Equations 52 Gizel Bakicierler and Emine Misirli

Jacobi Elliptic Function Solutions of the Space-Time Fractional Symmetric Regularized Long Wave 53 Equation Dilek Varol Bayram, Sevil Çulha and Ayşegül Daşcıoğlu

1 + Functional Quadratic Integral Equations in the L loc (R ) space 54 Latifa Benhamouche and Smail Djebali

Suborbital Graphs for a Non-Transitive Action of the Normalizer 55 Murat Besenk, Bahadır Ozgur Guler and Abdurrahman Buyukkaya

On Spherically Symmetric Solutions of the Einstein-Maxwell Field Equations 56 Rashida Bibi and Azad A. Siddiqui

A Numerical Approximation Based on Collocation Method for the Solutions of Telegraph Equation 57 Kübra Erdem Bicer

Rotating Disk Cryptosystem: RDC 58 Sadek Bouroubi and Louiza Rezkallah

Free Convection inside a Porous Enclosure 59 Canan Bozkaya

Tauberian Theorems for the Cesáro Second Order Operators for Sequences of Fuzzy Numbers 60 Naim L. Braha

Estimating the Distortion Parameter of the Proportional Hazards Premium for Heavy-Tailed Losses 61 Brahimi Brahim

On Some Fixed Point Results Related to Almost Generalized (α,β)-(ψ,ϕ)-Weakly Contractive Mappings in 62 S Metric Spaces

Abdurrahman Buyukkaya and Mahpeyker Ozturk

Beyond statistical quasi Cauchy sequences 63 Huseyin Cakalli

A Study on Strongly Lacunary Ward Continuity 64 Huseyin Cakalli and Huseyin Kaplan

A Study on Abel Statistical Quasi Cauchy Sequences 65 Huseyin Cakalli and Iffet Taylan

Graph Theoretical Applications of Molecular Graphs 66 Ismail Naci Cangul

ix A Study on Public Transit Users’ Route Choice and Assignment Function 67 Buket Capali and Halim Ceylan

A New Developed Semi–Empirical Formula for Nuclear Reaction Cross–Section Calculations 68 Veli Capali, Mert Sekerci, Hasan Ozdogan and Abdullah Kaplan

On Uninorms on Bounded Lattices 69 Gül Deniz Cayli and Funda Karacal

On Statistical Dunford and Pettis Integration 71 Anita Caushi

On The Cardinality of Category Spaces 72 Bahaettin Cengiz and Banu Gunturk

Remarks and Observations on Some Special Arithmetical Sums 73 Elif Cetin and Yilmaz Simsek

S-Generalized Mittag-Leffler Function 74 Aysegul Cetinkaya, I. Onur Kiymaz and M. Baki Yagbasan

Jacobi elliptic function solutions of time-fractional KdV-Zakharov-Kuznetsov equation 75 Sevil Culha, Dilek Varol Bayram and Aysegul Dascioglu

A Fix-And-Optimize Heuristic for the Integrated Fleet Sizing and Replenishment Planning Problem with 76 Predetermined Delivery Frequencies Niousha Karimi Dastjerd and Kadir Ertogral

Catalogue of Degree Sequences of Molecular Graphs 77 Sadik Delen and Ismail Naci Cangul

Streamline Topology of Vortex Breakdown Bubbles near the Re-Entrant Corner 78 Ali Deliceoglu and Ebutalib Celik

Optimality Conditions for a Linear Differential System with Two Delays 79 Hanna Demchenko

α-Convexity of Some Struve and Lommel Functions 80 Erhan Deniz, Halit Orhan and Murat Çağlar

Fuzzy Soft Topolical Spaces and the Related Category FST 81 Tugbahan Simsekler Dizman and Naime Tozlu

Fuzzy Soft Ditopological Spaces 82 Tugbahan Simsekler Dizman, Naime Tozlu and Şaziye Yüksel

On The Difference Method for Approximating of Second Order Derivatives of a Solution of Laplace's 83 Equation in a Rectangular Parallelepiped Adiguzel A. Dosiyev and Hediye Sarıkaya

Kolmogorov Inequality on Variable Exponent Lebesgue Spaces 84 Ismail Ekincioglu, Esra Kaya

An Application of Functional Variable Method For Semi-Analytical Solutions of Nonlinear Evolution 85 Equations Berfin Elma and Emine Misirli

New Types of Soft Separation Axioms and Soft Compactness in Soft Topological Spaces 86 M. E. El-Shafei, M. Abo-Elhamayel and T. M. Al-shami

x On Quaternion n-Spaces 87 Fatma Ozen Erdogan and Atilla Akpinar

Decomposition of Soft Continuity via Soft Locally b-Closed Set 88 Zehra Guzel Ergul, Naime Tozlu and Saziye Yuksel

Periodic Solutions for a Third-Order Delay Differential Equation 89 Nouioua Farid and A. Ardjuoni

Copula Conditional Tail Expectation for Multivariate Financial Risks 90 Benatia Fatah and Brahim Brahimi

New Properties of Fractional Derivatives Defined Using Mittag Leffler Kernel 91 Arran Fernandez and Dumitru Baleanu

Nodal Solutions for Indefinite Robin Problems 92 Michael Filippakis

On Slowly Oscillating Double Sequences 93 Goksen Findik, Ibrahim Canak and Umit Totur

A Bayes Minimax Result for a Large Class of Distributions 94 Dominique Fourdrinier, Fatiha Mezoued and William E. Strawderman

A Search for Designs with the Same Parameters as 2-(256,64,21) Design with 2-Rank 25 95 Mustafa Gezek

θ Compact and Matrix Mappings on the Space |Af |k 96 Fadime Gokce and G.Canan Hazar Gulec

On Some Classes of Fractional Differential Equations of Parabolic Type 97 Dilovar Guljonov

Some Results about ΔI-Statistically Pre-Cauchy Sequences with an Orlicz Function 98 Hafize Gumus, Omer Kisi and Ekrem Savas

A Numerical Analysis of FLMM for Semilinear Time Fractional Schrödinger Equations 99 Betul Hicdurmaz

On Modified FLMM Methods for Fractional Population Equations 100 Betül Hicdurmaz and Emine Can

Coincidence Best Proximity Points for Geraghty Type Proximal Cyclic Contractions 101 Azhar Hussain

Estimation of a Loss Function for Spherically Symmetric Distribution with Constraints on the Norm 102 Ouassou Idir

An overview on Fuzzy AHP and Its Priority Derivation 103 Iftikhar and Musheer Ahmad

104 Solutions of odod(n)(n1)When n1 Has Three Distinct Odd Primes Nazli Yildiz Ikikardes, Daeyeoul Kim and Lianrong Ma

Hypersurfaces of a Kenmotsu Space Form 105 Mohammad Ilmakchi

On Some Classes of Fractional Integrodifferential Equations in Hilbert Space 106 Mamadsho Ilolov

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The Generators of Regular, Quasi-regular Representations and Casimir Operator 108 Yasemin Isik and Mehmet Sezgin

Some Properties of Cartan Null Curves in Semi-Euclidean 4-space with index 2 109 Esen Iyigün

Proximal Point Algorithms Involving Cesaro Type Mean of Total Asymptotically nonexpansive Mappings 110 in CAT(0) Spaces Amna Kalsoom and Hafiz Futhar ud Din 111 Semi–Empirical Systematic Development for Photon Induced Nuclear Reaction Cross–Section Calculations Abdullah Kaplan, Hasan Ozdogan, Mert Sekerci and Veli Capali

Positive Solutions for Fractional-Order Boundary Value Problems 112 Ilkay Yaslan Karaca

The Existence of Positive Solutions of Boundary Value Problems with P-Laplacian on the Half-Line 113 Ilkay Yaslan Karaca and Aycan Sinanoglu

The Dimension of Digital Khalimsky Manifolds 114 Ismet Karaca and Gokhan Temizel

Some Properties of Persistent Homology Groups 115 Ismet Karaca and Hatice Sevde Denizalti

On Digital Cohomology Groups 116 Ismet Karaca and Ozgur Ege

Some Common Fixed Point Theorems on Complex Valued G -Metric Spaces b 118 Ismet Karaca and Ozgur Ege

On Some Deddens Subspaces of Banach Algebras 119 Mubariz Karaev, Mehmet Gurdal and Havva Tilki

Duhamel Operator and Existence of Invariant Subspace 120 Mubariz Karaev, Mehmet Gurdal and Mualla Birgul Huban

On Extended Eigenvalues and Extended Eigenvectors of Toeplitz Operators 121 Mubariz Karaev, Mehmet Gurdal and Mualla Birgul Huban

A generalization on the Incidence Energy and Laplacian-Energy-Like Invariant 122 Ezgi Kaya and A. Dilek Maden

A Data Mining Approach: Application to the Extraction of the Characteristics of IARD Products in the 123 Insurance Sector

Sadi Khadidja and Lounici Mosbah Nora

The Le Corbusier Approach in the Relationship between Architecture and Mathematics 124 Murat Kilic and Melih Kurnali

The Absolute Möbius Divisor Function and Euler   function 126 Daeyeoul Kim, Umit Sarp and Sebahattin Ikikardes

S-Generalized Lauricella’s Hypergeometric Functions 127 I. Onur Kiymaz, M. Baki Yagbasan and Aysegul Cetinkaya

Some Remarks on Fuzzy Anti-Normed Spaces 128 Ljubiša D.R. Kočinac

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On Asymptotically f-Statistical Equivalent Sequences 129 Sukran Konca and Mehmet Kucukaslan

Some Permanents of Hessenberg Matrices 130 Sibel Koparal, Nese Omur and Cemile Duygu Sener

Best Proximity Points for Generalized Geraghty Proximal Contraction Mapping in Elliptic Valued 131 Metric Space Isil Arda Kosal, Hidayet Huda Kosal and Mahpeyker Ozturk

Characterizations of  pq,,  - Convex Sequences 132 Xhevat Z. Krasniqi

A Note on the Numbers Yn(Λ) and the Polynomials Yn(X;Λ) and Their Generating Functions 133 Irem Kucukoglu and Yilmaz Simsek

Modelling Worldwide CO2 Emissions and Oil Consumption based on the L1, L2 and L∞-norm Regressions 134 Pranesh Kumar and Mohamadtaghi Rahimi

Some Symmerty Identities for Modified Degenerate Apostol-Bernoulli and Modified Degenerate Apostol- 135 Euler Polynomials Related to Multiplier Sums Burak Kurt

Univalency Conditions of a General Nonlinear Integral Operator of Analytic Functions with Different 136 Domains Shuhai Li and Huo Tang

Certain Subclasses of Harmonic Univalent Functions Defined By Convolution and Subordination 137 Shuhai Li and Huo Tang

Refinement of Some Inequalities Concerning to Bn-Operator of Polynomials with Restricted Zeros 138 A. Liman

Gaussian Approximation to the Estimator of the Mean of a Heavy-Tailed Distribution under Random 139 Censoring Djamel Mearghni

Industrial Application of Fuzzy Logic Control for Torque-ripple Minimization in Electricals Machines 140 Zineb Mekrini and Seddik Bri

On Optimal Control of Stochastic Mean Field Systems 141 Brahim Mezerdi

Ring Theory Approaches to Solve Cauchy-Euler Differential Equations of Several Variables 142 Assal Miloud

Subgradient Method of Solving the Problem of Linear Stochastic Programming with Bifurcation Effect 143 Fakhriddin Mirzoahmedov

Modelling Asymmetric Magnetic Recording Heads with an Underlayer Using Superposition 144 Ammar Edress Mohamed

G-compactness for Topological Groups with Operations 145 Osman Mucuk and Huseyin Cakalli

Topological Aspects of Monodromy Groupoids for Group-Groupoids 147 Osman Mucuk and Serap Demir

xiii A Characterization of the Two-Weight Inequality for Riesz Potentials on Cones of Radially Decreasing 149 Functions Ghulam Murtaza

On The Fourth Geometric-Arithmetic Index of Graphs 150 Y. Nacaroglu and A. Dilek Maden

Laplace Transform of Fractional Differential Equations 151 Khaled I. Nawafleh

Gaussian Approximation of a New Tail Index Estimator for Right-Censored Pareto-Type Distributions 152 Abdelhakim Necir

On Residual Algebraic Free Extensions of Valuations 153 Figen Oke

Statistically (C,1,1) Summable Double Sequences of Fuzzy Numbers and a Tauberian Theorem 154 Zerrin Onder, Ibrahim Canak and Umit Totur

The Sheffer Stroke Basic Algebras on the Intervals 155 Tahsin Oner and Tugce Katican

A Reduction of Basic Algebras: Sheffer Stroke Basic Algebras 156 Tahsin Oner and Ibrahım Senturk

Applications on Weak and Strong Forms of Fuzzy α-Open (Closed) Sets 157 Hakeem A. Othman

Code Verification Using Method of Manufactured Solutions for CFD Problems 158 Hatice Ozcan

On the Existence of Einstein Weyl Manifold with a Special Metric Connection 159 F. Ozdemir and M. D. Turkoglu

Repeat Codes, Even Codes, Odd Codes and Their Equivalence 160 Mustafa Ozkan and Figen Oke

Tauberian Theorems for the Weighted Mean Summability Methods of Integrals 161 Firat Ozsarac and Ibrahim Canak

On Lifting Polynomials and Distinguished Pairs 162 Burcu Ozturk, Figen Oke

On Some Fixed Point and Common Fixed Theorems in b-Metric-Like Spaces 163 Mahpeyker Ozturk

Experimental Evidence of Landau Damping in a Fluid at a Macroscopic Scale 164 Eric Padilla, William Cody Wilson and Andrei Ludu

A Boundary Value Problem for an Irrational Order Partial Equation 165 A.A. Pashavand, N.A. Aliyev and A.Y. Delshad Gharegheshlaghi

Generalized Close to Convex Functions with q-Properties 166 Yasar Polatoglu, Oya Mert and Asena Cetinkaya

Steady-State Modeling of the Biological Network via Long-tailed Symetric Distribution 167 Vilda Purutcuoglu and Melih Agraz

xiv Quadrature Formula with High Degree of Exactness 169 Abedallah Rababah

Best Cubic Spline Interpolation Based on Minimizing the Error 170 Abedallah Rababah and Mohammed Bani Khalid

On Chebyshev Collocation Method and Applications to Nonlinear Integral Equations 171 Abdalah Rababah, Benferhat Leila, Hichem Ramoul and Nora Mahloul

The Treatment of Fractional Singular Lagrangian 172 Eqab Rabei

MHD Convective Flow due to a Curved Surface with Thermal Radiation and Chemical Reaction 173 Madiha Rashid

On The Isolated Points of the Spectrum of M-Paranormal Operators 174 Mohammad Rashid

Existence of Homoclinic Orbit in Generalized Planar System of Lienard Type 175 Vahid Roomi

Some Results of the Picard-Krasnoselskii Hybrid Iterative Process 176 Aynur Sahin and Metin Basarir

Optimal Coincidence Best Proximity Point Results in Fuzzy Metric Spaces 177 Naeem Saleem

New Concept of Determinants with Three Indexes (3D Determinants) and Possibilities of Use 178 Armend Salihu

d Boundedness Properties of Some Operators on M (P, Q) ( ) 179 Ayse Sandikci

On Approximate Biprojectivity of Banach Algebras 180 M. H. Sattari

On Some New Sequence Spaces Defined By Almost Lacunary Bounded Variation 181 Ekrem Savas

On Filter Convergence of Nets in Uniform Spaces 182 Ekrem Savas and Ulas Yamanci

On E-J Hausdorff Transformations for Double Sequences 183 Rabia Savas and Hamdullah Sevli

Double Lacunary Statistical Boundedness of Order α 184 Rabia Savas and Mahpeyker Ozturk

Statistical Convergent Functions Via Ideals With Respect To The Intuitionistic Fuzzy 2-Normed Spaces 185 Rahmet Savas

Submanifolds in m1 with Finite Type Pseudo-Hyperbolic Gauss Map 186 H(1)2  Ruya Yegin Sen and Ugur Dursun

On the Determination of Validity of Categorical Syllogisms by Using a Mathematical Model 187 Ibrahım Senturk and Tahsin Oner

xv Converse Theorems for Statistical Convergence 188 Sefa Anil Sezer, Rahmet Savas and Ibrahim Canak

Cesàro Summability of Sequences in 2-Normed Spaces 189 Sefa Anil Sezer and Rahmet Savas

Applications of the Schwarz Lemma to Inequalities for Polynomials with Restricted Zeros 1 9 0 Lubna Wali Shah

Growth of Maximum Modulus of Polynomials and Rational Functions in the Complex Domain 191 Wali Mohammad Shah

Trace Formula for Witt Vector Rings 192 Mokhfi Siham

On Parametrization of the q-Bernstein Basis Functions 193 Yilmaz Simsek

Control of the Performance of the Panel of Judge in Sensory Analysis by a Functinal Principal Component 194 Analysis of Probability Densities Function Yousfi Smail

Optimization of an Execution Time for Parallel Matrix Multiplication by adding a New Set of Processors 195 on the Array Halil Snopce, Sadri Alija, Azir Aliu and Artan Luma

From Dido to Morrey: Variational Problems and Regularity Theory! 196 Lubomira G. Softova

A Publicly Verifiable Authenticated Encryption Scheme Based on Chaotic Maps and Factoring Problems 197 Nedal Tahat

Third-Order Differential Sandwich-type Results Involving the Liu-Owa Operator 198 Huo Tang, M. K. Aouf, Shigeyoshi Owa and Shu-Hai Li

Second-Order Differential Superordination for Analytic Functions in the Upper Half-Plane 199 Huo Tang, H. M. Srivastava, Guan-Tie Deng and Shu-Hai Li

Capacity Sizing and Pricing with Heterogeneous Products and Flexible Resources 201 Salih Tekin

A New Class of the r-Stirling Numbers and the Generalized Bernoulli Polynomials 202 Meriem Tiachachat

Degree Sequences and Inverse Problem on Graphs 203 Muge Togan, Aysun Yurttas and Ismail Naci Cangul

Notes on Permuting tri-derivations on Prime and Semi-prime Rings 204 Seda Oguz Unal, Hasret Durna

Weakly Invariant Subspaces for Multivalued Linear Operators on Banach Spaces 205 Gerald Wanjala

Speech Quality Analysis with Respect to Noise Corruption by a Kalman Filter to Estimation the Parameters 206 of the SWLP Method Ervenila Xhaferraj (Musta)

S-Generalized Srivastava’s Triple Hypergeometric Functions 207 M. Baki Yagbasan, Aysegul Cetinkaya and I. Onur Kiymaz

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Berezin Number Inequality for Convex Function in Reproducing Kernel Hilbert Space 208 Ulas Yamanci, Mehmet Gurdal and Mubariz T. Garayev

On Power Inequalities for Berezin Number of Operators and Convex Functions 209 Ulas Yamanci, Mehmet Gurdal and Ceren Celik

Spectral Properties of Discrete Klein-Gordon Equations 210 Nihal Yokus and Nimet Coskun

On the Inverse Problem on Graphs 211 Aysun Yurttas, Muge Togan and Ismail Naci Cangul

Summability of Subsequences of Divergent Sequences 212 Maria Zeltser and Johann Boss

Many to one Embedding Crossed Cube into Pancake 213 Mohamed Faouzi Zerarka

xvii INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

The Sturm-Liouville Theory and Fourier Analysis

Mohammed Al-Gwaiz

Department of Mathematics, King Saud University, Riyadh [email protected]

Abstract: According to the Sturm-Liouville Theory, the eigenfunctions of a self adjoint linear differential operator of second order form an infinite sequence which is orthogonal and complete in L^2. Thus, depending on the choice of the differential operator and the boundary conditions, we obtain an assortment of bases for L^2. This provides a convenient approach for expanding any function in L^2 in terms of these eigenfunctions. It turns out that the classical Fourier series expansion on (-pi,pi) in terms of sin nx and cos nx is the result of choosing the differential operator to be d^2 /dx^2, with appropriate boundary conditions. For other choices we arrive at a more generalized theory of Fourier series based on other orthogonal bases

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Topological Spaces with an -base

Taras Banakh

Jan Kochanowski University in Kielce (Poland) and Ivan Franko National University of Lviv (Ukraine), Kielce-Lviv, Poland-Ukraine [email protected]

Abstract: Given a partially ordered set P we shall discuss properties of topological spaces X admitting a P-base, i.e., an indexed family (U)P of subsets of XX such that U  U for all  in P and for every x X the family (U[x])P of balls U[x]={yX:(x,y) U} is a neighborhood base at x.

A P-base (U[x])P for X is called locally uniform if the family of entourages -1 (UU U)P remains a P-base for X. A topological space is first-countable if and only if it has an -base. By Moore's Metrization Theorem, a T0-space is metrizable if and only if it has a locally uniform -base. In the talk we shall discuss topological spaces possessing a (locally uniform) - base. Our results show that spaces with an -base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight - based topological spaces. On the other hand, topological spaces with a locally uniform -base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an -base and show that such spaces are close to being -compact. More information can be found in the paper-book [1].

Keywords: generalized metric space, partially ordered set, neighborhood base. References: [1] T. Banakh, “Topological spaces with an -base”, 105 pp. preprint (https://arxiv.org/abs/1607.07978).

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Schwarz Problem for Higher-order Equations in a Polydisc

A. Okay Celebi

Department of Mathematics, Yeditepe University Istanbul, Turkey [email protected]

Abstract: In this presentation, we discuss the Schwarz boundary value problem for higher order linear complex differential equations in the unit polydisc. Firstly we state the results obtained in ℂ (see for example [1]). Secondly the integral representation for functions in ℂ푛[2,3,4] is improved. Then we give the solution of the model equation with homogeneous Schwarz conditions posed in a polydisc, which enables us to define an integral operator. Thus we can convert the linear complex differential equations into an integral equation. The solution is obtained via Fredholm theory.

Keywords: Schwarz problem, Polydisc, P.S: This is a joint work with Umit Aksoy; Atilim University, Department of Mathematics, Ankara, Turkey References: [1] U. Aksoy, A. O. Celebi; A survey on the boundary value problems for complex partial differential equations, Adv. Dyn. Syst. Appl., 5(2010), 133-158. [2] Begehr, H. and Dzhuraev, A., An Introduction to several complex variables and partial differential equations, Pitman Monographs and Surveys1 in Pure and Applied Mathematics, Addison Wesley Longman Limited (1997). [3] Begehr, H., Boundary value problems in C and C^n, Acta Mathematica Vietnamica, 22(1997), 407-425. [4] Begehr, H., Dai, D.-Q. and Li, X., Integral representation formulas in polydomains, Comp. Var., 47(2002), 463-484

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Gelfand Theory Unplugged

Robin Harte

Trinity College, Dublin, Ireland [email protected]

Abstract: It was Norbert Wiener who observed that whenever a periodic continuous function which never vanishes has an absolutely convergent Fourier series, then so does its reciprocal. Pointwise multiplication generates “convolution” of their coefficient sequences, with a homomorphism from sequences to functions; according to Wiener, if the function is pointwise invertible then also the sequence is “convolution invertible”. When Israel Gelfand looked at these sequences he saw for the first time what would come to be known as a commutative “Banach algebra”. He went on to extend Wiener’s observation from absolutely summable sequences to these Banach algebras, with a completely different and abstract proof. The electricity that powers this “Gelfand theory” is Zorn’s lemma and “maximal ideals”, together with the Gelfand-Mazur lemma, which says that maximal ideals are always generated by bounded multiplicative linear functionals. The “unplugged” version bypasses maximal ideals, and proceeds via the superficially more concrete spectral mapping theorem for finite and infinite systems of Banach algebra elements.

Keywords: Fourier series, Banach algebra, maximals ideals, Gelfand characters, several variable spectral mapping theorem. References: [1] Robin Harte, Invertibility and singularity, Dekker (New York) 1988. [2] Robin Harte, Spectral mapping theorems - a bluffer’s guide, Springer Briefs in Mathematics, 2014. [3] R,E. Harte, Non-commutative Taylor invertibility, Operators and Matrices (to appear). [4] Vladimir Mu¨ller, Spectral theory of linear operators, Birkh¨auser Basel, 2007.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Decision Models for Autonomous Vehicles

Reza Langari

Engineering Technology and Industrial Distribution Texas A&M University [email protected]

Abstract: The market for autonomous vehicles (Level 4+ autonomy, or L4+) is expected to reach $42B by 2025 (Bloomberg) and upwards of $85B by 2030. The key issues in this context are i) localization, ii) perception, ii) decision logic, iv) control execution as well as v) validation/verification. These are central to effective functioning of autonomous vehicles and remain both research topics as well as subjects of significant development effort by industry. The presentation provides an overview of the issues listed above and the outlook for future development in the relevant areas. In particular, we focus on decision and control for autonomous vehicles where matters of expediency and safety have to be balanced in a sensible manner in view of uncertainty in the behavior of other vehicles. We present approaches based on classical optimization as well as game theory, which offers a unique means of dealing with multi-player decision processes. The benefits and drawbacks of this approach and future outlook for its use in autonomous driving will also be discussed.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Summation and Quadrature Processes for Slowly Convergent Series

Gradimir V. Milovanović

Serbian Academy of Sciences and Arts, Belgrade, Serbia [email protected]

Abstract: An account on summation/integration methods for computation of slowly convergent series and finite sums, as well as some new results on this subject and new applications, are presented. Methods are based on Gaussian quadrature formulas with respect to some non-classical weight functions over the real line or the halfline. For constructing such quadrature rules we use recent progress in symbolic compuation and variableprecision arithmetic, implemented through our Mathematica package “OrthogonalPolynomials” [1], [2]. Some details on these methods can be found in [3], [4], [5].

Keywords: Summation, Gaussian quadrature rules, weight function, convergence, orthogonal polynomials. References: [1] A. S. Cvetković and G. V. Milovanović, “The Mathematica Package OrthogonalPolynomials”, Facta Univ. Ser. Math. Inform. 19 (2004), 17-36. [2] G. V. Milovanović and A. S. Cvetković, “Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type”, Math. Balkanica 26 (2012), 169-184. [3] G. V. Milovanović, “Summation of series and Gaussian quadratures”, In: Approximation and Computation (R.V.M. Zahar, ed.), ISNM Vol. 119, pp. 459- 475, Birkhäuser Verlag, Basel-Boston-Berlin, 1994. [4] G. Mastroianni and G. V. Milovanović, Interpolation Processes - Basic Theory and Applications, Springer Monographs in Mathematics, Springer Verlag, Berlin - Heidelberg - New York, 2008. [5] G. V. Milovanović, “Summation formulas of Euler-Maclaurin and Abel- Plana: old and new results and applications”, In: Progress in Approximation Theory and Applicable Complex Analysis – In the Memory of Q,I. Rahman (N.K. Govil, R.N. Mohapatra, M.A. Qazi, G. Schmeisser, eds.), Springer, 2017 (to appear).

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

New Iterative Methods for Solving Non-Linear Equations

Osama Y. Ababneh

Department of Mathematics, Zarqa University, Zarqa, Jordan [email protected]

Abstract: Solving the non-linear equation f(x) = 0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non-linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose new modified Newton’s method for solving non-linear equations and free from second derivative. Convergence results show that the order of convergence of the proposed iterative methods is four. Finally, several numerical examples are given to illustrate that the new iterative algorithms are effective.

Keywords: non-linear equations, Newton’s method.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Solution of Animplicit Complementarity Problem on Isotone Projection Cones

Mujahid Abbas

Department of Mathematics, University of Management and Technology, C-II Johar Town, Lahore, Pakistan, and Department of mathematics and applied mathematics, University of Pretoria, South Africa [email protected]

Abstract: In this talk an iterative algorithm is presented in connection with an implicit complementarity problem. It is shown that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are discussed for this sequence to be convergent for implicit complementarity problems defined by isotone projection coneses.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Analysis and Modeling the Drought Hydrologic by the Copulas in North Algeria

Rassoul Abdelaziz

Department of Irrigation and Drainage, National High School of Hydraulics, Blida, Algeria [email protected]

Abstract: In this work, we use the three dimensional copula for modeling the dependence of the drought variables, severity–duration–frequency (S–D–F). Drought is a natural event, which has huge impact on both the society and the natural environment. Drought events are mainly characterized by their severity, duration and intensity. The study adopts standardized precipitation index (SPI) for drought characterization, and copula method for multivariate risk analysis of droughts. The Beni-Behdel River basin was selected as an example to illustrate the copulas. Results indicates that the Student copula was more appropriate for drought analysis in the selected area. Drought probabilities and return periods were calculated and analyzed based on the three dimensional.

Keywords: Copula, Drought, SPI index, return period. References: [1] C. Genest, K. Ghoudi, L.P. Rivest (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3):543–552 [2] H. Joe (1997) Multivariate models and dependence concepts, vol 73. Monographs on statistics and applied probability. Chapman and Hall, London, p 399 [3] W.C. Palmer WC (1965) Meteorological drought. Research paper no. 45, US Weather Bureau, Washington, DC. [4].A. Tawn (1988) Bivariate extreme value theory: models and estimation. Biometrika 75(3):397–415 [5] A. Sklar (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université´ de Paris 8:229–231

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Rayleigh-Marangoni Convection in a Layer of Nanofluid

Abdullah A. Abdullah

Department of Mathematical Sciences, Umm Al-Qura University, Makkah, Saudi Arabia [email protected]

Abstract: A linear stability analysis for the onset of Rayleigh-Marangoni convection in a horizontal layer of a nanofluid heated from below is investigated. The model employed for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The lower boundary of the layer is assumed to be a rigid surface at fixed temperature while the top boundary is assumed to be a non- deformable free surface cooled by convection to an exterior region at a fixed temperature. The boundaries of the layer are assumed to be impenetrable to nanoparticles with their distribution being determined from a conservation condition. The linear analysis uses spectral methods based on the expansion of eigenfunctions as Chebyshev series. Stability boundaries for Rayleigh number and temperature and nanofraction Marangoni numbers are obtained for several nanofluids.

Keywords: Rayleigh-Marangoni convection, Nanofluid, Linear stability, Brownian motion, Thermophoresis.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

On the Properties of the Orthogonal Polynomials along a Contour

Fahreddin.G.Abdullayev1, 2, Gülnara.A.Abdullayev²

¹Kyrgyz-Turkish Manas University, Bishkek, KYRGYZSTAN, [email protected]; [email protected] ²Mersin University, Mersin, TURKEY [email protected]

Abstract: Let be a complex plane; L be a closed rectifiable Jordan curve: Let hz() be a non-negative, summable on L and non-zero except possible on a set of measure zero function. The systems of polynomials n Kzn (); Kzznn()... ; deg Knn  ; n ; satisfying the orthonormality condition: h()()() zKzKzdz   ,  nmn m , L are called orthonormal polynomials for the pair ( ,Lh ) . These polynomials are determined uniquely if the major coeficient n  0 . These polynomials were first studied in [6]. Some properties of the polynomials Kzn ()under the various conditions on the weight function hz() and contour were investigated in [1]- [5], [7]-[9] and others (also, references in therein).

In this work, we investigated the order of growth of the modulus of polynomials in the weighted space, where the contour and the weight functions have some singularities on the finite points on the contour. Exact estimations for the growth of the modulus of orthogonal polynomials were obtained.

Keywords: Orthogonal Polynomial, Quasiconformal Curve, Coformal Mapping.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

References: [1] F.G. Abdullayev.,G.A. Abdullayev, On the Sharp Inequalities for Orthonormal Polynomials Along a Contour, Complex Analysis and Operator Theory, 1, 2017, DOI: 10.1007/s11785-017-0640-1. [2] G. Fauth, Über die Approximation analytischer Funktionen durch Teilsummenihrer Szegö-Entwicklung, Mitt. Mathem. Semin. Giessen, No:67, (1966), pp.1-83. [3] Ya.L. Geronimus, Polynomials Orthogonal on a Circle and Interval. IX + 210 S. m. 9 Tafeln. Oxford/London/New York/Paris 1960. [4] P.P. Korovkin, Sur les polynomes orthogonaux le long d.un contour recti.able dans le cas de la présence d.unpoids, Rec. Math. [Mat. Sbornik] N.S., (1941) , Vol. 9(51), No: 3, pp. 469.485. [5] A.L Kuz.mina, Asymptotic representation of polynomials orthogonal on a piecewise-analytic curves, Proc. "Functional Analysis and theory of Functions", I,.Kazan., (1963), pp. 42-50. [6] G. Szegö, Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören, Mathem. Zeitschr. 9 (1921), pp.218-270. [7] G. Szegö, Orthogonal Polynomials, Fizmatgis,1962, (in Russian). [8] P.K. Suetin, Main properties of the orthogonal polynomials along a circle. Uspekhi Math. Nauk, Vol.21, No:2 (128), (1966), pp.41-88.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Modeling and Classifying by using Binary Logistic Regression Analysis Application on Hepatitis Disease Data

Qais Mustafa Abdulqader

Department of Information Technology, Duhok Polytechnic University, Dohuk, Zakho, Iraq [email protected]

Abstract: Logistic Regression Analysis analyze the relationship between multiple explanatory variables and a single binary response variable, a categorical variable with two categories [1]. Many medical applications have been done in this area such as [2,3,4 and 5]. In this paper, the binary logistic regression analysis technique has been used and applied for building a suitable model for hepatitis disease data using stepwise procedure and depending on some laboratory tests which represents explanatory variables. Also, the technique has used for classifying persons into two groups which are infected and uninfected with viral hepatitis disease. The evaluation was depending on some statistical criteria. The results of the analysis have been discussed and mentioned.

Keywords: Binary logistic regression, Hepatitis, Stepwise procedure, statistical criteria. References: [1] S. Sweet, and K. Martin, "Data Analysis with SPSS: A First Course in Applied Statistics," 4th ed., Pearson publisher, 2011. [2] S. Javali, and P. Pandit, " Multiple logistic regression model to predict risk factors of oral health diseases," Romanian statistical review journal, 5(2012), 73- 86. [3] P. Reeda, and Y. Wub, " Logistic regression for risk factor modelling in stuttering research," Journal of Fluency Disorders, 38 (2013), 88-101. [4] M. T., Dev Mukherji, N. Padalia, and A. Naidu, " A heart disease prediction model using SVM-decision trees-logistic regression (SDL)," International Journal of Computer Applications, 68 (2013), 11-14. [5] W. M. Amir et al., " Association of Hypertension with Risk Factors Using Logistic Regression, " Applied Mathematical Sciences, 8(2014), 2563 – 2572. ______13

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Residual Power Series Approach to Handle a Class of Fractional Differential Equations

Ayed H. Adamat

Department of Mathematics, Faculty of Science, Al-Hussein Bin Talal University, P.O. Box 20, Ma'an, Jordan [email protected]

Abstract: In this paper, we present a computational algorithm to find the coefficients of the fractional power series solutions for linear and nonlinear differential equations of fractional order. This approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software [1-5]. The proposed method based on the generalized Taylor's formula that involving Caputo fractional derivative which constructs an analytical solution in the series form and reproduces the exact solution when the solution is a finite series. This technique is applied to a few test examples to illustrate the accuracy, efficiency and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.

Keywords: Fractional differential equations; Fractional power series method; Initial value problems; Generalized Taylor series approximation. References: [1] A. El-Ajou, O. Abu Arqub, M. Al-Smadi, A general form of the generalized Taylor’s formula with some applications, Applied Mathematics and Computation, 256 (2015), 851-859. [2] Z. Odibat, N. Shawagfeh, Generalized Taylor’s formula. Appl. Math. Comput., 186 (2007), 286-293. [3] K. Moaddy, M. Al-Smadi, I. Hashim, A Novel representation of the exact solution for differential algebraic equations system using residual power-series method, Discrete Dynamics in Nature and Society, 2015 (2015), Article ID 205207, pp.1-12. [4] I. Komashynska, M. Al-Smadi, A. Ateiwi, S. Al-Obaidy, Approximate analytical solution by residual power series method for system of fredholm integral equations. Applied Math. Inform. Sci., 10 (2016) 975-985. [5] I. Komashynska, M. Al-Smadi, O. Abu Arqub, S. Momani, An efficient analytical method for solving singular initial value problems of nonlinear systems. Applied Math. Inform. Sci., 10 (2016), 647-656. DOI: 10.18576/amis/100224 ______14

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

On the Existence and Uniqeness of Positive Solution for a Fractional Boundary Value Problem with New Fractional Derivative

Asghar Ahmadkhanlu

Department of Mathematics, Azarbaijan Shahid Madani University, Km 35 Tabriz-Maragheh rod, Tabriz, Iran [email protected]

Abstract: We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem:

퐴퐵푅 훼 푡퐷0 푢(푡) + 푓(푡, 푢(푡)) = 0 0 < 푡 < 1, 1 < 훼 < 2 퐴퐵푅 훼−1 푡퐷0 푢(0) = 푢(1) = 0

퐴퐵푅 훼 where 푡퐷0 denotes the Atangana-Baleano fractional derivative in sense of Reimman-liouville. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results

Keywords: Boundary value problem, Positive solution, fixed point theorems References: [1] A. Atangana, “On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation”, App. Math. Comp., 273(2016) 948-956. [2] N. Al-Salti, E. Karimov and S. Kerbal, “Boundary-value problems for fractional heat equation involving Caputo-Fabrizio derivative”, NTMSCI, 4.4 (2016), 79-89. [3] T. Salat, “Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer”, J. Nonlinear Sci. Appl., 9 (2016), 3647-3654. [4] D. Baleanu1, B. Agheli, M. M. Al Qurashi, “Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives”, Advances in Mechanical Engineering 8.12(2016), 1–8. ______15

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

On Derivation of a Subclasses of Filiform Leibniz Algebras

AL-Nashri AL-hossain Ahmed

Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, KSA. [email protected]

Abstract: In this paper, dimension the derivations of a Third class of Leibniz algebras are studied and discussed. For 퐿 ∈ 푇퐿푏푛 , we have 푛 + 1 ≤ dim 퐷푒푟(퐿) ≤ 2푛 + 1, are deduced.

Keywords: Filiform Leibniz algebra, numerical validation, gradation, derivation.. References [1] Albeverio, S.; Ayupov, Sh. A.; Omirov, B. A., On nilpotent and simple Leibniz algebras, Comm. in Algebra Vol. 33(2005), 159-172. [2] Albeverio, S., Omirov, B. A., Rakhimov, I. S., (2006), Classi_cation of 4- dimensional nilpotent complex Leibniz algebras, Extracta Math., 3(2006), 197- 210. [3] AL-hossain, A. A.; Khiyar, A. A.,Derivations of some Filiform Leibniz algebras. Pure and Applied mathematics Journal.Vol. 3,No.6,(2014), 121-125. [4] Alnashri. A. A., Derivations of one type of algebra of First class Filiform Leibniz algebras of Dimension Derivation (n+1), International Journal of Advanced Scienti_c and Technical Research,Vol. 1,No.5,(2015), 41-55. [5] Alnashri. A. A., Derivations of Second type of algebra of _rst class Filiform Leibniz algebras of Dimension Derivation (n+1), International Journal of Advanced Scientific and Technical Research,Vol. 3,No.5,(2015), 29-43. [6] Ayupov, Sh. A.; Omirov, B. A., On Leibniz algebra, Algebra and Operator Theory. Proceeding of the Colloquium in Tashkent (1997), Kluwer (1998), Doi 10.1007/978-94-011-5072-9-1 Springer, p 1-13. [7] Ayupov, Sh. A.; Omirov, B. A., On 3-dimensional Leibniz algebra, Uzbek Math. J. (1999), 9-14. [8] Dixmier. J. and Lister. W. G. , Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8(1957), 155-158. [9] Jacobson. N. , A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc. 6(1955), 281{283.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Global Existence and Uniqueness of Weak Solution to a Chemotaxis Model

N. Aïssa and A. Balehouane

Laboratoire AMNEDP, Faculté de Mathematiques USTHB

BP 32, El Alia, Bab Ezzouar, 16111 Alger, Algeria.

Abstract: We prove global existence and uniqueness of weak solutions to a chemotaxis model with nonlinear diffusion.

First, we use a fixed point theorem to prove local in time existence of a weak solution.

In order to prove that the solution is global in time, we provide a priori estimates of the solution in an appropriate Lebesgue space by adapting the method of [3].

Then, we obtain uniform bounds of the solution by using the result [1] based on Moser's iterative method.

KeyWords: Quasilinear Parabolic Systems, Reaction-Diffusion Systems, Chemotaxis.

References: [1] N. D. Alikakos, Lp bounds of solutions of reaction-diffusion equations. Comm. Partial Differential Equations 4, (1979), 827-868. [2]R. Kowalczyk, Z. Szymanska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl. 343 (2008), 379-398. [3] L. Wang, C. Mu, P. Zheng, Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real. World. Appl 14, 1634-1642 (2013).

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

A Generalization of Hereditary Noetherian Prime Rings

Evrim Akalan

Department of Mathematics, Hacettepe University, Ankara, Turkey [email protected]

Abstract: In this talk, we will introduce generalized hereditary noetherian prime rings (G-HNP rings for short) which generalizes the class of hereditary noetherian prime (HNP rings for short) rings. We will describe the structure of projective ideals of G-HNP rings and some over rings of G-HNP rings. Examples will be given to illustrate and delimit the theory.

Keywords: HNP Rings, projective ideals, invertible ideals. References: [1] D. Eisenbud, J. C. Robson, “Hereditary Noetherian Prime Rings”, J. Algebra 16 (1), (1970), 86-104. [2] H. Marubaysahi, “A Krull type generalization of HNP rings with enough invertible ideals”, Comm. in Algebra 11 (5) (1983), 469-499.

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On Some Inequalities of Analytic and Biunivalent Functions Given by Subordination

Arzu Akgul

Department of Mathematics, Kocaeli University, Kocaeli, Turkey [email protected]

휑 Abstract: In the present investigation, the subclass MΣ (훾, 휆, 훿)_ of analytic biunivalent functions is defined and established bounds for the coefficients for this subclass. Also several related classes are considered and connections to earlier known results are made. Keywords: Analytic and bi-univalent functions, subordination, coefficient estimate References: [1] R. M. Ali, S. K. Lee, V. Ravichandran and S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25, (2012), 344-351. [2] D.A. Brannan, T.S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Math. Anal. and Appl., Kuwait; February 18.21, 1985, in: KFAS Proceedings Series, vol. 3, Perg- amon Press, Elsevier Science Limited, Oxford, 1988, pp. 53.60. see also Studia Univ. Babe¸s-Bolyai Math. 31 (2) (1986) 70.77. [3] D. A. Brannan and J. G. Clunie, 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] S. Bulut, Faber polynomial coe¢ cient estimates for a subclass of analytic bi- univalent functions, Filomat, Vol. 30, No. 6, (2016),1567-1575 . [5] S. S. Ding, Y. Ling, and G. J. Bao, Some properties of a class of analytic functions, Journal of Mathematical Analysis and Applications, vol. 195, no. 1, pp. 71.81, 1995 [6] B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Applied Mathematics Letters, 24, (2011), 1569-1573. [7] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, Journal of Classical analysis, 2, 1, (2013), 40-60. [8] J. M. Jahangiri and S. G. Hamidi, Coe¢ cient estimates for certain classes of bi-univalent functions, Int. J. Math. Sci., ArticleID 190560, (2013), 4 pp. ______19

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Compact Finite Differences Method for Burgers-Huxley Equation

Canan Akkoyunlu

Department of Mathematics and Computer Sciences, Istanbul Kultur University, Bakırkoy, Istanbul, Turkey [email protected]

Abstract: In this paper, a numerical solution for the Burgers-Huxley equation is presented by using compact finite differences method. In the solution of the problem, finite differences discretization along the time, and fifth-order compact finite differences scheme along the spatial coodinate are applied. Dispersive properties for the compact finite difference method are investigated for the linearized equations to examine the nonlinear dynamics after discretization. The result shows that the applied method in this study is an applicable tecnique and approximates the exact solution very well.

Keywords: Burger-Huxley equation, compact finite differences method, dispersion analysis. References: [1] AG. Bratsos, “A fourth order improved numerical scheme for the generalized Burgers-Huxley equation’’, J. Comput Math, 1(2011), 152-158. [2] B. Batiha, MSM. Noorani and I. Hashim, “Application of variational iteration method to the generalized Burgers-Huxley equation’’, Chaos Solitons Fractals, 36(2008), 660-663. [3] HNA, Ismail, K. Raslan, AAA. Rabboh, “Adomain decomposition method for Burgers-Huxley and Burgers-Fisher equation’’, Appl Math Comput, 159(2004), 291-301. [4] I. Hashim, MSM. Noorani and MRS. Al-Hadidi, “Solving the generalized Burgers-Huxley equation using the adomain decomposition method’’, Math Comput Model, 43(2006), 1404-1411. [5] M. Javidi, “A numerical solution of the generalized Burgers-Huxley equation by pseudospectral method and darvishi’s preconditioning’’, Appl Math Comput, 175(2006), 1619-1628. [6] M. Javidi, “A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method’’, Appl Math Comput, 178(2006), 338-344. ______20

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Approximation Solution of System of Volterra Integro Differential Using Finite Element Method

Adel Alamarashi*

*Department of Mathematics, College of Science Jazan University, KSA. *Department of Mathematics, College of Education, Thamar University, Yemen. [email protected] ; [email protected]

Abstract: In the present paper, we first obtain variational form of the problem, and then, finite element method and basis functions will be used. Also, the error analysis of the method is considered. Furthermore, we give numerical computational example to test and validate the proposed method.

Keywords: System Of Volterra Integro-Differential; Finite element method- error analysis. References: [1] Adel. Al-Marashi and Al-Faour.O.M. "Approximate solution for system of multi- term initial value problem of Fractional Differential Equations by Spline functions". Sana'a University Journal of Science & Technology, (2008), 1, pp.251-264. [2] Adel A. Al-Marashi ,” Approximate Solution of the System of Linear Fractional Integro-Differential Equations of Volterra Using B- Spline Method”, American Review of Mathematics and Statistics December 2015, Vol. 3, No. 2, pp. 39-47 [3] Adel Al-Marashi. “Numerical Solution For Multi-term Fractional Orders Differential Equations By Splin Functions”, Journal Focuses on Human Knowledge and Applied Sciences, Issue no 9,June (2008),pp.170 – 187 [4] Z. Mahmoodi , J. Rashidinia & E. Babolian, B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations, pp. 1787-1802, Journal Applicable Analysis An International Journal Volume 92, 2013 - Issue 9. [5] Hermann Brunner, The numerical treatment of Volterra integro-differential equations with unbounded delay, Journal of Computational and Applied Mathematics, Volume 28, December 1989, Pages 5-23. ______21

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Fixed Point Theorem in FM-Spaces

Rateb AlBtoush

Department of Mathematics& Statistics Faculty of Science P.O. Box (7) Mu'tah University Al-Karak-Jordan [email protected]

Abstract: The goal of this section is to study the existence problems of fixed points or common fixed points for some varieties of single-valued mappings in non-Archimedean probabilistic fuzzy metric space.

Keywords: Fixed Point, Compatible mappings, Non-Archimedean Menger probabilistic normed spaces. References: [1] Y. J. Cho, Fixed points in fuzzy metric spaces, J. Fuzzy Math. 5(1997), 949- 962. [2] S. S. Chang, On the theory of probabilistic metric spaces with applications, Acta Math. Sinica, New Series, 1(4) (1985), 366-377. [3] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Set Syst. 27(1998), 385-390. [4] V. Gregori, S. Morillas and A. Sapena, Examples of fuzzy metrics and applications, Fuzzy Sets Syst. 170(2011), 95{111. [5] P.J. He, The variational principle in fuzzy metric spaces and its applications, Fuzzy Sets and Systems, 45 (1992), 389{394. [6] M. Hegedus and T. Szilagyi, Equivalence conditions and a new _xed point theorem in the theory of contraction mappings, Math. Japonica, 25 (1) (1980), 147-157.

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Bounds for the Zeros of Polynomials by Using Similar Matrices

Mohammad. Al-Hawari

Department of Mathematics, Irbid National University, Irbid, Jordan [email protected]

Abstract: We apply several matrix inequalities to the similar Frobenius companion matrices of monic polynomials to derive new bounds for the zeros of these polynomials which are better than other bounds.

Keywords: Bounds for the zeros of polynomials; Companion matrix; Spectral norm; Spectral radius

References: [1] Horn, R.A. and Johnson, C.R., 1985, Matrix Analysis (Cambridge: Cambridge University Press). [2] Hou, J.C. and Du, H.K., 1995, Norm inequalities of positive operator matrices. Integral Equations Operator Theory, 22, 281–294. [3] Kittaneh, F. and Shebrawi, K .,2007,Bounds for the zeros of polynomials from matrix inequalities – II, Linear and Multilinear Algebra., 55:2, 147-158. [4] Kittaneh, F., 2003, Bounds for the zeros of polynomials from matrix inequalities. Archiv des Mathematik, 81, 601–608. [5] Kittaneh, F., 2003, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Mathematica, 158, 11–17. [6] Linden, H., 2000, Bounds for zeros of polynomials using traces and determinants. Seminarberichte Fachbereich Mathematik FeU Hagen., 69, 127– 146. [7] Mohammad H. Al-Hawari.,2016, ZEROS OF POLYNOMIALS BY USING SOME INEQUALITIES. Far East Journal of Mathematical Sciences (FJMS)., Volume 100, Number 10, 2016, Pages 1545-1550.

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An Application of Decision Tree for Evaluating a Classroom Teaching Practice

Sadri Alija and Halil Snopce

Faculty of Business and Economics, South East European University, Tetovo, Macedonia [email protected] Faculty of Contemporary Sciences and Technologies, South East European University, Tetovo, Macedonia [email protected]

Abstract: The aim of this research is to use the decision-making tree in order to give the meaning for the classification of some activities done during the realization of the math subject lesson. Actually, the aim is to identify which activity is more important in achieving the objective of the lesson and at the same time, which one has bigger impact in achieving the final results on the math course. In total we have used 22 variables divided into three different groups: The first one is about some activities concerning the beginning of the lesson, the second groups are the variables about the continuation of the lesson and the third groups are the variables concerning the exercises and assessment. For all of these three activities we have chosen the averages of the answers for every group separately. In this research are shown the results gathered from 48 randomly chosen students of the Faculty of Business Economics at the SEE-University in Macedonia, who have attended the math subject on the winter semester of the year 2016. For the construction of the decision making tree, we have used the ID3 algorithm by using the Weka software. We have found that the root of the decision-making tree is the attribute concerning the activities taken at the beginning of the lesson. This attribute at the same time is the most important attribute from all 4 attributes.

Key words: Teaching practice, decision tree, ID3 algorithm, Weka software ______24

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Existence of Positive Solution of Boundary Value Fractional Quadratic Differential Equations

Youssef Allaoui1, Khalid Hilai2 and Guida Karimi3

Laboratory of Applied Mathematics and Scientific Computing 1,2,3 Faculty Of Sciences and Technologies1,2,3 Sultan Moulay Slimane University, PO Box 523, 23000 Beni Mellal, Morocco1,2,3 [email protected],[email protected], [email protected]

Abstract: In this work, we prove the existence as well as approximate of the positive solutions for boundary value problem of nonlinear fractional quadratic differenrtial equations. We use some porprieties of the Mittag-Leffler functions and its relationship with fractional calculus. Also we obain some results regarding the existence of positive solutions using the Dhage iterative method enbodied in a recent hybrid fixed point theorem of Dhage in partially ordered normed lineair spaces.

Keywords: Fractional quadratic differential equation, Mittag-leffler equation, Dhage iterative method, Approximate positive solution. References: [1] Bapurao C.Dhage, Lakshmikantham.V, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid syst. 4, 414-424 (2010). [2] Bapurao C.Dhage, Lakshmikantham, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations. Differ. Equ. Appl. 2, 465-486 (2010). [3] Bapurao C.Dhage and Shyam B. Dhage: Approximating positive solutions of nonlinear first order ordinary quadratic differential equations: Applied and Interdisciplinary Mathematics, Cogent Mathematics 2,1023671(2015). [4]Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergi V. Rogostin: Mittag-Leffler functions, Related Topics and Applications. Springer-Verlag Berlin Heidelberg 2014. [5]K.Hilal, A. Kajouni: Boundary value problems for hybrid differential equations. Mathematical Theory and modeling. 2224-5804 (2015).

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Some Applications of Cozero Sets in Topological Spaces

Ahmad Al-Omari

Al al-Bayt University, Faculty of Sciences, Department of Mathematics P.O. Box 130095, Mafraq 25113, Jordan [email protected]

Abstract: An ideal on a set X is a nonempty collection of subsets of X withz heredity property which is also closed finite unions. The concept of ideal topological spaces via cozero sets was introduced by Al-Omari [10]. In this paper, we introduce and study some an operator via cozero sets and we construct a topology τ∗ for X by using the cozero sets and an ideal I on X. Moreover, we obtain characterizations and preserving theorems of quasi compact spaces.

Keywords: cozero set, zero set, quazi compact space, ideal topological space. References: [1] S. Bayhan and I. L. Reilly, On some variants of compactness, Hacettepe Journal of Mathematics and Statistics 43 (6) (2014), 891–898. [2] S. Bayhan, A. Kanibir, A. McCluskey, and I. L. Reilly, On almost z- supercontinuity, Filomat 27(6) (2013), 965–969. [3] Z. Frolik, Generalizations of compact and Lindelof sppaces (Russian), Czechoslovak Math. J. 9(84) (1959), 172–217. [4] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nos- trand Co., Inc., Princeton, N. J. 1960. [5] J. K. Kohli and R. Kumar, Z-supercontinuous functions, Indian J. Pure Appl. Math. 33 (7) (2002), 1097 [6] J. K. Kohli, D. Singh and R. Kumar, Generalizations of z- supercontinuous functions and Dδ -supercontinuous functions, Appl. Gen. Topology 9 (2008), 239-251. [7] J. K. Kohli, D. Singh and J. Aggarwal, F-supercontinuous functions, Appl. Gen. Topol. 10 (1) (2009) [8] M. K. Singal and S. B. Niemse, z-continuous mappings, Math. Student 66 (1997), 193-210. [9] W. T. Van Est and H. Freudenthal, Trennung durch stetige Funktionen in topologischen Rau¨men, Indagationes Math. 15 (1951), 359-368. [10] A. Al-Omari, On ideal topological spaces via cozero sets, Questions and Answers in General Topology 34 (2) (2016), 83–91. [11] D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97 (4) (1990), 295-310. ______26

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An Efficient Analytical-Numerical Technique for Handling Model of Fuzzy Differential Equations of Fractional-Order

Mohammad Aloroud1, Mohammed Al-Smadi2,*, Rokiah Rozita Ahmad1, Ummul Khair Salma Din1

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 2Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan [email protected]

Abstract: This paper proposes an efficient analytical numerical technique for finding approximate solutions for a class of fuzzy fractional initial value problems. The algorithm is based on the generalized Taylor series residual power series (RPS), which is extended to handle such problems. The analytical solution is calculated in the form of multiple fractional power series expansion of a rabidly convergent series with easily computable components. In addition, description of the RPS method is discussed [1-6]. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient for exploring fuzzy fractional models.

Keywords: Generalized Taylor series, Residual power series method, Initial value problem, Fuzzy fractional differential equations. References: [1] K. Moaddy, M. Al-Smadi and I. Hashim, A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-series Method, Discrete Dynamics in Nature and Society, Vol. 2015 (2015), Article ID 205207, 1-12. http://dx.doi.org/10.1155/2015/205207 [2] M. Al-Smadi, Solving initial value problems by residual power series method, Theoretical Mathematics & Applications 3(1), (2013) 199-210. [3] I. Komashynska, M. Al-Smadi, O. Abu Arqub and S. Momani, An efficient analytical method for solving singular initial value problems of nonlinear systems, Applied Mathematics & Information Sciences, 10(2), (2016) 647-656.

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[4] I. Komashynska, M. Al-Smadi, A. Ateiwi and S. Al-Obaidy, Approximate Analytical Solution by Residual Power Series Method for System of Fredholm Integral Equations, Applied Mathematics & Information Sciences 10 (3), (2016) 975-985. [5] I. Komashynska, M. Al-Smadi, A. Al-Habahbeh, A. Ateiwi, Analytical approximate Solutions of Systems of Multi-pantograph Delay Differential Equations Using Residual Power-series Method, Australian Journal of Basic and Applied Sciences 8 (10), (2014) 664-675. [6] O. Abu Arqub, Series solution of Fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, (2013) 31–52.

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Adaptation of Fractional Power Series Method for Solving Fuzzy BVPs

Mohammad Alaroud1, Rokiah Rozita Ahmad1, Mohammed Al-Smadi2,*, Ummul Khair Salma Din1

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 2Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan [email protected]

Abstract: In this paper, we propose a computational iterative technique, the fractional power seies method (FPR) for finding numeric-analytic solutions for a class of fuzzy BVPs. The approach constructs to express the solutions in form of a series expansion in terms of elementary α-level representation [1-6]. By linguistic of fuzzy terms, the fuzzy fractional equation is converted to system of fractional equations in crisp case, whereas the crisp results are mapped to fuzzy output using the membership functions. Further, numerical examples are provided and discussed quantitatively and graphically to show the performance features, generality and superiority of the FRP algorithm. The results reveal that the method is quite accurate, simple, straightforward, and convenient for exploring fuzzy fractional models.

Keywords: Generalized Taylor series, Residual power series method, Boundary value problem, Fuzzy fractional differential equations. References: [1] K. Moaddy, M. Al-Smadi and I. Hashim, A Novel Representation of the Exact Solution for Differential Algebraic Equations System Using Residual Power-series Method, Discrete Dynamics in Nature and Society, Vol. 2015 (2015), Article ID 205207, 1-12. http://dx.doi.org/10.1155/2015/205207 [2] M. Al-Smadi, Solving initial value problems by residual power series method, Theoretical Mathematics & Applications 3(1), (2013) 199-210. [3] I. Komashynska, M. Al-Smadi, O. Abu Arqub and S. Momani, An efficient analytical method for solving singular initial value problems of nonlinear systems, Applied Mathematics & Information Sciences, 10(2), (2016) 647-656. doi:10.18576/amis/100224. [4] I. Komashynska, M. Al-Smadi, A. Ateiwi and S. Al-Obaidy, Approximate Analytical Solution by Residual Power Series Method for System of Fredholm

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Integral Equations, Applied Mathematics & Information Sciences 10 (3), (2016) 975-985. doi:10.18576/amis/100315 [5] I. Komashynska, M. Al-Smadi, A. Al-Habahbeh, A. Ateiwi, Analytical approximate Solutions of Systems of Multi-pantograph Delay Differential Equations Using Residual Power-series Method, Australian Journal of Basic and Applied Sciences 8 (10), (2014) 664-675. [6] O. Abu Arqub, Series solution of Fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, (2013) 31–52.

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Solving Fuzzy Mixed Integral Equations of Second Kind in Hilbert Spaces

Mohammed Al-Smadi

Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan [email protected]

Abstract: In this paper, we propose a computational iterative technique for finding numeric-analytic solutions for a class of fuzzy mixed integral equations of the second kind based on orthonormal basis sets derived from Gram-Schmidt process in reproducing-kernel Hilbert spaces [1-6]. The approach constructs to express the solutions in form of a series expansion in terms of elementary 훼-level representation in the Sobolev space 휔2[푎, 푏]. By linguistic of fuzzy terms, the fuzzy integral equation is converted to system of integral equations in crisp case, whereas the crisp results are mapped to fuzzy output using the membership functions. Further, numerical examples are provided and discussed quantitatively and graphically to show the performance features, generality and superiority of the reproducing-kernel algorithm.

Keywords: Fuzzy calculus, Reproducing-kernel theory, Fredholm-Volterra integral equations, Fourier series expansion. References: [1] O. Abu Arqub, M. Al-Smadi and N. Shawagfeh, “Solving Fredholm integro- differential equations using reproducing kernel Hilbert space method”, Applied Mathematics and Computation, 219 (2013), 8938-8948. [2] M. Al-Smadi, O. Abu Arqub and S. Momani, “A Computational Method for Two-Point Boundary Value Problems of Fourth-Order Mixed Integrodifferential Equations”, Mathematical Problems in Engineering, 2013 (2013), Article ID 832074, 1-10. http://dx.doi.org/10.1155/2013/832074 [3] O. Abu Arqub and M. Al-Smadi, “Numerical algorithm for solving two- point, second-order periodic boundary value problems for mixed integro differential equations,Applied Mathematics and Computation,243(2014)911-922

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[4] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Numerical Solutions of Fuzzy Differential Equations using Reproducing Kernel Hilbert Space Method”, Soft Computing, (2015), 1-20. http://dx.doi.org/10.1007/s00500-015- 1707-4 [5] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh and S. Momani, “Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method”, Applied Mathematics and Computation, 291 (2016) 137-148. [6] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems”, Soft Computing, 2016 (2016), 1-16. doi:10.1007/s00500-016-2262-3

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Numerical Algorithm for Solving Time-Fractional Bvps in a Simplified Reproducing Kernel Space

Mohammed Al-Smadi

Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan [email protected]

Abstract: In this paper, we propose a computational iterative technique for finding numeric-analytic solutions for a class of of time-fractional boundary value problem within favorable aspects of the reproducing kernel Hilbert space in Caputo sense. The algorithm methodology is based on generating an orthonormal basis sets derived from Gram-Schmidt process [1-6]. The approach constructs to express the solutions in form of a series expansion in terms of elementary representation in the Sobolev space. Error estimates are obtained as well as numerical examples are provided and discussed quantitatively and graphically to show the performance features, generality and superiority of the reproducing-kernel algorithm. The numerical results indicate that the IRKA is a significant development tool for handling such issues arising in computer, physics and engineering fields.

Keywords: Fractional differential equations; Reproducing kernel theory; Inner product spaces; Error estimation and error bound. References: [1] O. Abu Arqub, M. Al-Smadi and N. Shawagfeh, “Solving Fredholm integro- differential equations using reproducing kernel Hilbert space method”, Applied Mathematics and Computation, 219 (2013), 8938-8948. [2] M. Al-Smadi, O. Abu Arqub and S. Momani, “A Computational Method for Two-Point Boundary Value Problems of Fourth-Order Mixed Integrodifferential Equations”, Mathematical Problems in Engineering, 2013 (2013), Article ID 832074, 1-10. http://dx.doi.org/10.1155/2013/832074 [3] O. Abu Arqub and M. Al-Smadi, “Numerical algorithm for solving two- point, second-order periodic boundary value problems for mixed integro- differential equations”, Applied Mathematics and Computation, 243 (2014), 911- 922. ______33

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

[4] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Numerical Solutions of Fuzzy Differential Equations using Reproducing Kernel Hilbert Space Method”, Soft Computing, (2015), 1-20. http://dx.doi.org/10.1007/s00500-015- 1707-4 [5] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh and S. Momani, “Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method”, Applied Mathematics and Computation, 291 (2016) 137-148. [6] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems”, Soft Computing, 2016 (2016), 1-16. doi:10.1007/s00500-016-2262-3

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Semiregularization of Almost Countably Compact Spaces

Zuhier Altawallbeh

Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan [email protected]

Abstract: Among various covering properties of topological spaces a lot of attention has been made to those covers. Weakly Lindelöf spaces were introduced by Frolik [1] and nearly compact spaces were defined by Singal and Mathur [2]. Other generalizations of Lindelöfness and compactness are given by many authors as Balasubramanian [3], and Altawallbeh and Al-Momany in [4]. Bonanzinga, Matveev and Pareek [5] defined almost countably compact spaces as a generalization of countably compact spaces. In this paper, we investigate this new class of spaces, almost countably compact spaces, also we study some other properties in the view of regular cover notion and semiregularization topology relating to this class of spaces.

Keywords: regularly open sets, regularly closed sets, countably compact, nearly countably compact, semiregularization topology. References: [1] Z. Frolik, “Generalizations of compact and Lindelöf spaces”, Czechoslovak Math. J., 9.84(1959), 172-217. [2] M.K Singal and A. Mathur, “On nearly compact spaces”, Boll. Un. Mat Ital., 2.4(1969), 702-710. [3] G. Balasubramanian, “On some generalizations of compact spaces”, Glas. Mat. Ser. III, 17.37(1982), 357-380. [4] Z. Altawallbeh and A. Al-Momany, “Nearly countably compact spaces”, International Electronic Journal of Pure and Applied Mathematics, 8.4(2014), 59-56. [5] M. Bonanzinga, M.V Matveev and M. Pareek, “Some remarks on generalizations of countably compact spaces and Lindelöf spaces”, Rend.Circ.Mat.palermo, 2.51.1(2002), 163-174.

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Generalization on Countably Compact Spaces via Hereditary Classes

Zuhier Altawallbeh

Department of Mathematics, Tafila Technical University, Tafila 66110, Jordan [email protected]

Abstract: A lot of work has been taken in account to generalize different covering properties of spaces as compact and countably compact spaces in different ways either by taking topologies with respect to an ideal as in [1] which was presented by Hamlett and Jankovic or by giving weaker condition in the definition as nearly countably compact spaces which is done by Altawallbeh and Al-Momany in [2]. In this article, we use the notions of generalized topologies and hereditary classes that was presented by Csa ́sza ́r in [3] and [4]. Here, we define and characterize the countably compact spaces taking in account generalized topologies in terms of hereditary classes. In particular, by setting a generalized topology μ on a nonempty set X, we define new concept of countably compactness, μԨ-countably compact space in generalized topology μ in the sense of a hereditary class Ԩ, called ԨGTS. The space ԨGTS is μԨ- countably compact if for every countable μ- covering of X there exists a finite subset such that the complement, in X, of the union of sets of that subset belongs to Ԩ.

Keywords: hereditary class, generalized topology. References: [1] T.R. Hamlett and D. Jankovic, “Ideals in general topology”, General Topology and Applications, 1988, 115-125. [2] Z. Altawallbeh and A. Al-Momany, “Nearly countably compact spaces”, International Electronic Journal of Pure and Applied Mathematics, 8.4(2014), 59-56. [3] Ả. Csẚszẚr, “Generalized topology, generalized continuity”, Acta Math. Hungar, 96 (2002), 351-357. [4] Ả. Csẚszẚr, “Modification of generalized topologies via hereditary classes”, Acta Math. Hungar, 1-2. 115 (2007), 29-36.

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Hybrid Master Equation of the Jump Diffusion Approximation

Derya Altintan* and Heinz Koeppl**

*Department of Mathematics, Selcuk University, Konya, Turkey **Department of Electrical Engineering and Information Technology, Technische Universitat Darmstadt, Darmstadt, Germany [email protected]

Abstract: Most often biochemical reactions are multi-scale processes because of the differences in the abundance of species and the reaction rates. To exploit this multi-scale nature, hybrid models which combine the deterministic and stochastic approaches are needed. In [1], we propose a jump diffusion approximation to model and simulate these types of systems. The idea of the method is to partition the reactions into fast and slow subgroups and model the fast group by Langevin equation while slow group is modelled by continuous time Markov chains. In this study, we define a new vector whose components represents the number of occurrences of reactions in the hybrid model given in [1]. Similar to the idea of splitting the reactions into two different groups, we consider this vector as a combination of random vectors represent the continuous and discrete random processes which count the occurrence of reactions in the fast group and the slow group, respectively. Based on the studies in [4], we prove that the time derivative of the probability distribution of this new vector which is called hybrid master equation is a summation of the Fokker-Planck equation (FPE) [2] which represents the time evolution of the conditional probability distribution of the continuous random variable given discrete random variable and chemical master equation (CME) [3] which is the time derivative of the marginal distribution of the discrete variable.

Keywords: Multi-Scale Processes, Jump-Diffussion Approximation, Fokker- Planck Equation (FPE), Chemical Master Equation (CME).

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References: [1] A. Ganguly, D. Altintan, H. Koeppl, “Jump-diffusion approximation of stochastic reaction dynamics: Error bounds and algorithms”, SIAM journal of Multiscale Modeling and Simulations, 13. 4 (2015), 1390-1419. [2] D. T. Gillespie, “The Chemical Langevin and Fokker-Planck Equations for the Reversible Isomerization Reaction”, The Journal of Physical Chemistry A, 106.20 (2002), 5063-5071. [3] D. T. Gillespie, “A rigorous derivation of the chemical master equation,” Physica A, 188 (1992), 404–425. [4] R. F. Pawula. “Generalizations and extensions of the fokker-planck- kolmogorov equations”, IEEE Transactions on Information Theory , 13.1, 1967, 33-41. Acknowledgments This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) Program no:3501 Grant, no. 115E252.

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Coefficient Bounds for a Subclass of Analytic Functions with Respect to Symmetric Points

Osman Altintas1, Oznur Ozkan Kilic2

1 Department of Mathematics Education, Baskent University,, Ankara, Turkey [email protected] 2 Department of Technology and Knowledge Management, Baskent University,, Ankara, Turkey [email protected]

Abstract: In this paper, we determine the coefficient bounds for a subclass of analytic functions with respect to symmetric points which is introduced here several corollaries are also considered.

Keywords: Analytic function, Close-to-convex function, Close-to-star function, Symmetric points. References: [1] O. Altıntaş, “On a subclass of certain starlike functions with negative coefficients”, Math. Japonica 36, No. 3, (1991), 489-495. [2] O. Altıntaş, “Certain applications of subordination associated with neighborhoods”, Hacettepe J.Math. Statist. 39, No. 4, (2010), 527-534. [3] R. Bucur, D. Breaz and L. Georgescu, “Third hankel determinant for a class of analytic functions with respect to symmetric points”, Acta Univer. Apulensis, No.42, (2015), 79-86. [4] R.N. Das, P. Sing, “On subclasses of schlicht mappings”, Indian. J. Pure Appl. Math. 8 (1997), 864-872. [5] K. Sakaguchi, “On certain univalent mappings”, J. Math. Soc. 11 (1959), Japan, 72-75. [6] C. Selveraj, N. Vasanthi, “Subclasses of analytic functions with respect to symmetric and conjugate Points”, Tamkang J. Math. 42 (1) (2011), 87-94 [7] H. M. Srivastava, O. Altinta_s and S. K. Serenbay, “Coefficient bounds for certain subclasses of starlike functions of complex order”, Appl. Math. Lett. 24 (2011), 1359-1363.

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On The Relationship Between A Family of Fibonacci And Lucas Numbers

Ipek Altun2, Ali Aydogdu1, Engin Ozkan2

1Department of Mathematics and Computing, Beykent University 2Department of Mathematics. Erzincan University [email protected], [email protected], [email protected]

Abstract: In this work, we prove some properties of a family of Fibonacci numbers. Also some relationship between the family of Fibonacci and Lucas numbers are given.

Keywords: Fibonacci Numbers, Generalized Fibonacci Numbers, Lucas Numbers. References: [1] V.E. Hoggatt, “Fibonacci and Lucas numbers”, Houghton Mifflin, 1969 [2]. J. lvie , “A General Q-Matrix”, Fibonacci Quarterly, Vol. 10, No. 3, April, 1972, 255-261 [3] T. Koshy, “Fibonacci and Lucas numbers with applications”, John Wiley & Sons, Inc.; 2001.

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Convergence, Consistency and Stability in Intuitionistic Fuzzy Differential Equations

Bouchra Ben Amma1, Said Melliani2 and Lalla Saadia Chadli3

Laboratory of Applied Mathematics and Scientific Computing 1,2,3 Faculty Of Sciences and Technologies1,2,3 Sultan Moulay Slimane University, PO Box 523, 23000 Beni Mellal, Morocco1,2,3 [email protected] , [email protected], [email protected]

Abstract: In this work, we consider first-order intuitionistic fuzzy differential equations with initial value conditions. The convergence, consistency and stability of difference method for approximating the solution of intuitionistic fuzzy differential equations are studied. Then the local truncation error is defined and sufficient conditions for convergence, consistency and stability of difference method are provided and some examples are presented to illustrate the accuracy of our proposed concepts.

Keywords: Intuitionistic fuzzy differential equations, Convergence, Consistence, Stability, Local truncation error References: [1] K. Atanassov, Intuitionistic fuzzy sets. VII ITKR’s session, Sofia (deposited in Central Science and Technical Library of the Bulgarian Academy of Sciences 1697/84) (1983) [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), pp. 87-96. [3] K. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets Syst. 64(2) (1994), pp. 159-174. [4] S. Abbasbandy, T. Allahviranloo, Numerical Solution of fuzzy differential equations by Taylor method, J.of Comp.Methods in Appl. Math. 2 (2002), 113- 124. [5] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, J.J. Nieto, Numerical methods for fuzzy differential inclusions, Journal of Computer and Mathematics With Applications 48 (2004), pp. 1633–1641.

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SQCQP Descent Scheme for Multi-objective Optimization Problem

Md Abu Talhamainuddin Ansary 1, Geetanjali Panda2

Department of Mathematics, Indian Institute of Technology Kharagpur Kharagpur, India 1 [email protected], 2 [email protected]

Abstract: Developing numerical approximation techniques for multi-objective programming problem (MOP): (f1, f2,…fm) , f: is a growing research area in recent time. Some of these techniques for unconstrained MOP include steepest descent scheme by Fliege and Svaiter [1] in 2000, Newton scheme by Fliege, Drummond, and Svaiter [2] in 2009, Quasi-Newton scheme by Qu, Goh and Chan [3] in 2011 and for constrained MOP include SQP scheme by Fliege and Vaz [4] in 2015. The SQP scheme due to Fliege and Vaz considers the linear approximation of the constraint functions and quadratic approximation of all objective functions. This paper has focused on nonlinear MOP with inequality constraints and developed a descent converging sequence considering quadratic approximation of both constraints and all objective functions at every iterating point. This method is follows the idea of sequential quadratically constrained quadratic program (SQCQP) technique. A non- differentiable penalty function l∞ is used to restrict the constraint violations. It is proved that the descent sequence generated in this process converges to a critical point under Slater constraint qualifications. Global convergence of this scheme is justified with some mild assumptions. This process is free from any kind of priori chosen parameters and ordering information of objective functions as in scalarization processes. The proposed scheme is verified and compared with existing methods using a set of test problems. Keywords: Sequential quadratically constrained quadratic programing; Slater constraint qualification; Penalty function, Critical point. References: [1] J .Fliege and B.F. Svaiter, “Steepest descent scheme for multicriteria optimization”, Math. Methods Oper. Res., 51.3(2000), 479-494. [2] J. Fliege, L. M. G. Drummond, and B.F. Svaiter, “Newton's method for multiobjective optimization”, SIAM J. Optim, 20.2(2009), 602-626. [3] S. Qu and M. Goh and F. T. S. Chan , “Quasi-Newton methods for solving multiobjective optimization”, Oper. Res. Lett., 39(2011), 139-150. [4] J. Fliege and A. I. F. Vaz, “A method for constrained multiobjective optimization based on SQP techniques”, SIAM J.Optim., 24.4(2015), 2091-2119. ______42

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On Gauss Balancing and Gauss Cobalancing Numbers

Mustafa Asci and Mustafa Yilmaz

Department of Mathematics, Pamukkale University, Kınıklı, Denizli, Turkey [email protected]

Abstract: A. Behera and G.K. Panda [1] defined new number sequence Balancing numbers as following: They call a natural number 푛 a balancing number if 1 + 2 + ⋯ + (푛 − 1) = (푛 + 1) + (푛 + 2) + ⋯ + (푛 + 푟) for some natural number 푟, while they call 푟 the balancer corresponding to the balancing number 푛. According to Panda and Ray [2] the values of 푛 satisfying the Diophantine equation 1 + 2 + ⋯ + 푛 = (푛 + 1) + (푛 + 2) + ⋯ + (푛 + 푟) for some natural number 푟 are known as cobalancing numbers while 푟 is the cobalancer corresponding to the cobalancing number 푛. Many authors study the interesting properties of Balancing and Cobalancing numbers. In this paper we define and study the Gaussian Balancing and Gaussian Cobalancing numbers. We give many properties of these numbers and we prove these properties by matrix methods. Keywords: Balancing Numbers, Cobalancing Numbers, Gauss Fibonacci Numbers. References: [1] A. Behera and G.K. Panda. On the square roots of triangular numbers. The Fib. Quart, 49 (1)28-33, [2] G.K. Panda and P.K. Ray Cobalancing Numbers and Cobalancers. Int. J. Math. Sci., 2005 (8): 1189-1200. [3] Liptai, K.; Panda, G. K.; Szalay, L. A balancing problem on a binary recurrence and its associate. Fibonacci Quart. 54 (2016), no. 3, 235–241. [4] Rout, S. S.; Panda, G. K. k-gap balancing numbers. Period. Math. Hungar. 70 (2015), no. 1, 109–121. [5] Davala, R. K.; Panda, G. K. On sum and ratio formulas for balancing numbers. J. Indian Math. Soc. (N.S.) 82 (2015), no. 1-2, 23–32. [6]Panda, G. K.; Panda, A. K. Balancing-like sequences associated with integral standard deviations of consecutive natural numbers. Fibonacci Quart. 52 (2014), no. 5, 187–192. [7] Panda, G.K.; Rout, S. S. Gap balancing numbers. Fibonacci Quart. 51 (2013), no. 3, 239–248. ______43

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An Overview of Ordering Based on Nullnorms

Emel Asici

Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, Trabzon, Turkey [email protected]

Abstract: Nullnorms and t-operators were introduced by Calvo and et all [1] and Mas and et all [2], respectively, which are also generalizations of the notions of t-norms and t-conorms. In [3] Mas and et all it is pointed out that nullnorms and t-operators are equivalent since they have the same block structures in [0,1]2. In [4] Asici was define and discuss an order induced by nullnorms on bounded lattices. In this paper, we investigate some properties an order induced by nullnorms on bounded lattices. We determine with the examples the relationship between the order induced by a nullnorm and the order on the lattice. So, 푆- partial order and 푇-partial order are extended to a more general form.

Keywords: Nullnorm, Bounded Lattice, Partial order. References: [1] T. Calvo, B. De Baets and J. Fodor, “The functional equations of Frank and Alsina for uninorms and nullnorms”, Fuzzy Sets Syst., 120(2001), 385-394. [2] M. Mas, G. Mayor and J. Torrens, “t-operators”, Int. J. Uncertain. Fuzz. Knowl.-Based Syst., 7(1999), 31-50. [3] M. Mas, G. Mayor and J. Torrens, “The distributivity condition for uninorms and t-operators”, Fuzzy Sets Syst., 7(1999), 31-50. [4] E. Aşıcı, “An order induced by nullnorms and its properties”, Inf. Sci., 267 (2014), 323-333. [5] E. Aşıcı and F. Karaçal, “On the T-partial order and properties”, Fuzzy Sets Syst., [6] E. P. Klement, R. Mesiar and E. Pap, “Triangular Norms ”, Kluwer Academic Publishers, Dordrecht 2000. [7] J. Drewniak, P. Drygas and E. Rak, “Distributivity between uninorms and nullnorms”, Fuzzy Sets Syst., 159(2008), 1646-1657. [8] J. Casasnovas and G. Mayor, “Discrete t-norms and operations on extended multisets”, Fuzzy Sets Syst., 159(2008), 1165-1177.

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Analytical-Numerical Solutions for a class of Systems of Differential Equations Using Reproducing Kernel Method

Ali Mahmud Ateiwi

Department of Mathematics, Faculty of Science, Al-Hussein Bin Talal University, P.O. Box 20, Ma'an, Jordan [email protected]

Abstract: This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method [1-6]. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

Keywords: System of differential equations, Reproducing kernel method, Boundary value problem, Gram-Schmidt process. References: [1] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems”, Soft Computing, 2016 (2016), 1-16. doi:10.1007/s00500-016-2262-3 [2] M. Al-Smadi, O. Abu Arqub and S. Momani, “A Computational Method for Two-Point Boundary Value Problems of Fourth-Order Mixed Integrodifferential Equations”, Mathematical Problems in Engineering, 2013 (2013), Article ID 832074, 1-10. http://dx.doi.org/10.1155/2013/832074 [3] M. Al-Smadi, O. Abu Arqub, N. Shawagfeh and S. Momani, “Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method”, Applied Mathematics and Computation, 291 (2016) 137-148. [4] O. Abu Arqub, M. Al-Smadi, S. Momani and T. Hayat, “Numerical Solutions of Fuzzy Differential Equations using Reproducing Kernel Hilbert Space ______45

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Method”, Soft Computing, (2015), 1-20. http://dx.doi.org/10.1007/s00500-015- 1707-4 [5] O. Abu Arqub and M. Al-Smadi, “Numerical algorithm for solving two- point, second-order periodic boundary value problems for mixed integro- differential equations”, Applied Mathematics and Computation, 243 (2014), 911- 922. [6] O. Abu Arqub, M. Al-Smadi and N. Shawagfeh, “Solving Fredholm integro- differential equations using reproducing kernel Hilbert space method”, Applied Mathematics and Computation, 219 (2013), 8938-8948.

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A Fuzzy Project Scheduling with Constrained Resources

Lyazzat Atymtayeva, Ardakbek Kungaliyev, Daniyar Artykov

Department of Computer Engineering, Satbayev Kazakh National Research Technical University, Almaty, Kazakhstan [email protected]

Abstract: The problems of project scheduling optimization in the conditions of constrained resources have been researched by many authors including Arvind Sathi et al. [1], Xiaoqing (Frank) Liu et al. [2], Wang [3], Herroelen, W. et al. [4], Javad Nematian et al. [5] and others. They have been focused on the using of fuzzy logic and systems as an intelligent component in project management tools for scheduling. The main purpose for their researches was the development of an algorithm for solving Fuzzy Random Resource-Constrained Project Scheduling problem. This problem concerned the using of linear programming approach that was proposed by Alvarez-Valdes and Tamarit in 1993 [6]. In this paper we use the mentioned concepts for development of intelligent tools and algorithms in fuzzy project scheduling and discuss the ways how to make it enabled the converting of original complex scheduling model to a mixed integer programming model. Keywords: fuzzy project random resource-constrained scheduling, linear programming, mixed interger programming. References: [1] Arvind Sathi, Thomas E. Morton, and Steven F. Roth.Callisto: An Intelligent Project Management System, Journal AI Magazine. 7(5), 34-52 (1986) [2] Xiaoqing (Frank) Liu, Gautam Kane, Monu Bambroo. An intelligent early warning system for software quality improvement and project management, Journal of Systems and Software. 79(11), 15521564 (2006) [3] Wang, J. A fuzzy project scheduling approach to minimize schedule risk for product development. Fuzzy Sets andSystems, 127, 99116 (2002). [4] Herroelen W., and R. Leus. Project Scheduling under Uncertainty: Survey and Research Potentials. European Journal of Operational Research 165, 289306 (2005). [5] Javad Nematian, Kourosh Eshghi, Abdolhamid Eshragh- Jahromi. A Resource-Constrained Project Scheduling Problem with Fuzzy Random Duration. Journal of Uncertain Systems Vol.4, No.2, pp.123-132, 2010 [6] Alvarez-Valdes, R. and J. Tamarit. The project Scheduling Polyhedron: Dimension, Facets and Lifting Theorems. European Journal of Operational

Research 67, 1993, pp. 204-220 ______47

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A Convergent Two-Level Linear Scheme for the Generalized Rosenau-Kdv-RLW Eqution

Ayhan Aydin

Department of Mathematics, Atilim University, Incek, Istanbul, Turkey [email protected]

Abstract: A new convergent two-level finite difference scheme is proposed for the numerical solution of initial value problem of generalized Rosenau-KdV- RLW equation. The new scheme is linear and conservative. It contains one free parameter. The impact of the parameter to error of the numerical solution is studied. The prior estimate of the finite difference solution is obtained. The existence, uniqueness and convergence of the scheme are proved. Accuracy and reliability of the scheme is tested by simulating the solitary wave graph of the equation. Numerical experiments indicate the efficiency of the method..

Keywords: conservative scheme, convergence, solitary wave. References: [1] Wongsaijai, B., Poochinapan, K., A tree-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and the Rosenau-RLW equation, Applied Mathematics and Computation, 245, (2014), 289-304. [2] Pan, X., Wang, Y., Zhang, L., Numerical analysis of a pseudo-compact C-N conservative scheme for the Rosenau-KdV equation coupling with the Rosenau- RLW equation, Boundary Value Problem, (2015):65. [3] Y.L. Zhou, Application of Discrete Functional Analysis to the Finite Difference Methods, International Academic Publishers, Beijing, 1990

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Fuzzy Soft Metric and Fuzzifying Soft Topology Induced by Fuzzy Soft Metric

Ebru Aydogdu, Abdulkadir Aygunoglu, Halis Aygun

Department of Mathematics, Kocaeli University, Kocaeli, Turkey [email protected] , [email protected] , [email protected]

Abstract: The aim of this study is to define fuzzy soft metric compatible to soft theory and investigate fuzzifying soft topology induced by fuzzy soft metric. For this, firstly we introduce a fuzzy soft metric on soft set and by using this we construct a fuzzy metric on soft set. Then we investigate fuzzifying soft topology characterized by this fuzzy metric and studied their properties.

Keywords: Soft metric, Fuzzy metric, Fuzzifying soft topology. References: [1] A. Aygünoğlu and H. Aygün, “Some notes on soft topological spaces”, Neural Computing and Applications, 21.1(2012),113-119. [2] A. Aygünoğlu, V. Cetkin and H. Aygün, “An introduction to fuzzy soft topological spaces”, Hacettepe Journal of Mathematics and Statistics, 43.2(2014), 197-208. [3] A. George and P. Veeramani, “On some results in fuzzy metric spaces”, Fuzzy sets and systems, 64.3(1994), 395-399. [4] A. Shostak, “Two decades of fuzzy topology: basic ideas, notions and results”, Russ.Math.Surv., 44(1989), 125-186. [5] B.P. Varol and H.Aygün, “Fuzzy soft topology”, Hacettepe Journal of Mathematics and Statistics, 41.3(2012), 407-419. [6] J.J. Minana and A. Šostak, "Fuzzifying topology induced by a strong fuzzy metric." Fuzzy Sets and Systems 300 (2016), 24-39. [7] M. Shabir and M. Naz, “On soft topological spaces” Comput. Math. Appl.”, 61(2011), 1786-1799. [8] P.K. Maji, R. Biswas and A.R. Roy, “Soft set theory” Computers & Mathematics with Applications, 45(2003), 555-562. [9] S. Das and S.K. Samanta, “Soft real sets, soft real numbers and their properties”, J. Fuzzy Math., 20.3(2012), 551-576. [10] S. Das and S.K. Samanta, “Soft metric”, Annals of Fuzzy Mathematics and Informatics, 6.1(2013), 77-94. [11] S. Das and S.K. Samanta, “On soft metric spaces”, J. Fuzzy Math., 21.3(2013), 707-734. ______49

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Some Identities Associated With Hecke Operators

Aykut Ahmet Aygunes

Department of Software Engineering, Faculty of Engineering and Architecture, Antalya Akev University, Antalya, Turkey [email protected]

Abstract: In this paper, we introduce the Hecke operators and we give some properties of these operators. Then we deal with a new approach, based on Hecke operators, to study a large class of special polynomials including Bernoulli and Euler polynomials. From this new approach we obtain some identities associated with Hecke operators.

Keywords: Hecke operators, partial Hecke operators, Bernoulli and Euler polynomials. References: [1] L. Euler, Methodus generalis summandi progressiones, Comment. acad. sci. Petrop., v.6 (1738) 68-97. [2] Y. Hellegouarch, Invitation aux mathématiques de Fermat-Wiles, Dunod. (2001). [3] T. Kim, Symmetry identities for the twisted generalized Euler polynomials , Adv. Stud. Contemp. Math. Vol 19 ( 2009 ), 111-118. [4] T. Kim, Some identities of symmetry for the generalized Bernoulli numbers and polynomials , ( 2009) Arxiv, http://arxiv.org/pdf/0903.2955. [5] J. L. Raabe, Zurückführung einiger Summen and bestimmten Integrale auf die Jacob Bernoullische Function, Journal für die reine and angrew. math., 42 (1851) 348-376. [6] J-P. Serre, Cours d'arithmétique, PUF, 1970.

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Some Rough Convergence Criteria for the Sequences of Intervals of Fuzzy Numbers

Salih Aytar

Department of Mathematics, Süleyman Demirel University, lsparta, Turkey [email protected]

Abstract: An interval of fuzzy number is a set of fuzzy numbers with the property that any fuzzy number that lies between two fuzzy numbers in the set is also included in the set. In this talk, we examine the rough convergence relations between a sequence of intervals of fuzzy numbers and a sequence of fuzzy numbers included these intervals.

Keywords: Rough convergence; Fuzzy interval numbers References: [1] F.G. Akçay, S. Aytar (2015), Rough convergence of a sequence of fuzzy numbers, Bulletin of Mathematical Analysis and Applications, 7(4): 17-23. [2] S. Aytar (2008). Rough statistical convergence, Numer. Funct. Anal. and Optimiz. 29(3-4):291-303. [3] H.X. Phu (2001), Rough convergence in normed linear spaces, Numer. Funct. Anal. and Optimiz., 22:201-224.

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Modified Simple Equation Method and its Applications to Some Nonlinear Physical Equations

Gizel Bakicierler and Emine Misirli

Department of Mathematics, Ege University, Bornova, İzmir, Turkey [email protected] , [email protected]

Abstract: The model of many problems encountered in various fields of engineering and science such as plasma physics, mathematical physics and fluid mechanics is expressed by nonlinear partial differential equations. Therefore, it is very important to investigate the solutions of these equations. In this paper, wave solutions of some nonlinear physical equations are obtained by Modified Simple Equation Method and wave types are determined. This method have been applied (2+1) dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and (3+1) dimensional Jimbo-Miwa equation as a result we obtained exact solution functions. The graphics of solution functions have been drawn using the Mathematica program and these solutions of equations are interpreted.

Key Words: nonlinear partial differential equations, evolution equations, modified simple equation method

References: [1] Ayati, Zainab, "Exact Solutions of Nonlinear (2+ 1)-Dimension Nonlinear Dispersive Long Wave and Coupled Boiti–Leon–Pempinelli Equations by using the Modified Simple Equation Method." World Applied Programming, WAP journal, Vol (3), Issue (12), December 2013. 565-571. [2] Jawad, Anwar Ja’afar Mohamad, Marko D. Petković and Anjan Biswas, "Modified simple equation method for nonlinear evolution equations." Applied Mathematics and Computation 217 (2010): 869-877. [3] Khan, M. Ashrafuzzaman and M. Ali Akbar, "Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method." Applied and Computational Mathematics 4,3 (2015) [4] Zayed, Elsayed ME, and Hoda Ibrahim SA, "Modified simple equation method and its applications for some nonlinear evolution equations in mathematical physics." International Journal of Computer Applications 67, 6 (2013).

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Jacobi Elliptic Function Solutions of the Space-Time Fractional Symmetric Regularized Long Wave Equation

Dilek Varol Bayrama, Sevil Çulhab, Ayşegül Daşcıoğlua

aDepartment of Mathematics, Faculty of Science and Arts, Pamukkale University, Denizli, 20070, Turkey bInstitute of Science, Pamukkale University, Denizli, 20070, Turkey [email protected], [email protected], [email protected]

Abstract: In this work, new families of analytical exact solutions of the fractional nonlinear symmetric regularized long wave (SRLW) equation are presented by using the Jacobi elliptic function expansion method. By this method, the solutions are found in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions and soliton solutions are obtained.

Keywords: Jacobi elliptic function, fractional differential equation, SRLW equation References: [1] S. Ahmadian, M.T. Darvishi, New exact traveling wave solutions for space- time fractional (1+1)-dimensional SRLW equation, Optik 127 (2016) 10697–10704. [2] R. Abazari, Application of 퐺′⁄퐺-expansion method to traveling wave solutions of three nonlinear evolution equations, Comput. Fluids 39 (2010)1957–1963. [3] F. Xu, Application of exp-function method to symmetric regularized long wave (SRLW) equation, Phys. Lett. A 372 (2008) 252–257. [4] H. Jafari, N. Kadkhoda, C.M. Khalique, Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Computers and Mathematics with Applications 64 (2012) 2084–2088. [5] J.F.Alzaidy, The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs, American Journal of Mathematical Analysis, 1 (1) 2013 14-19. [6] O. Guner, D. Eser, Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods Advances in Mathematical Physics, 2014 Article ID 456804.

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1 + Functional Quadratic Integral Equations in the L loc (R ) space

Latifa Benhamouche and Smail Djebali

Department of Mathematics, Faculty of Sciences, Saad Dahlab University Blida, Algeria [email protected]

Abstract : In this work we study the existence of solutions of functional 1 + quadratic integral equations of Volterra type in the space L loc (R ) consisting of all real functions locally integrable on the positive real half axis. The main result of this paper is obtained by using the new concept of family of measures of weak noncompactness recenlty introduced by Olszowy in [1] a 2014 paper, which is applied here in conjection with the Schawder- Tychonov fixed point theorem. In fact many authors studied the solvability of different types of integral equations on the Banach space BC (R+) consisting of all real functions defined, bounded and continuous on the positive real half axis, while in some practical situations integral equations are well understood in 퐿1 settings see for example Taoudi [3]. Recently in a 2015 paper [2], Wang and Zhou developed some fixed point theorems in locally convex spaces with Krein-Smulian property. As an 1 + application authors gave existence result in the space L loc (R ) to a general Volterra integral equation. Unfortunately the quadratic case is not covered by their result. The quadratic case was followed with interest in [4] but only on bounded intervals. In the present paper we focus on the question of existence of solutions of functional quadratic integral equations on unbounded intervals.

References: 1 + [1] L. Olszowy, A family of measures of noncompactness in the space L loc (R ) and its application to some nonlinear Volterra integral equation,}Mediterr. J. Math. 11, 2014, no. 2, 687--701. [2] F. Wang, H. Zhou,Fixed point theorems and the Krein-Smulian property in locally convex spaces, Fixed Point Theory Appl. 2015, 2015:154. [3] N. Salhi and M.A Taoudi, Existence of integrable solutions of an integral equation of Hammerstein type on unbounded interval, M. J of mathemathiques, [4] A. Bellour, D. O'Regan, M.A. Taoudi, On the existence of integrable solutions for a nonlinear quadratic integral equation, J. Appl. Math. Comput. 2014, 46:67-77.

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Suborbital Graphs for a Non-Transitive Action of the Normalizer

Murat Besenk, Bahadır Ozgur Guler, Abdurrahman Buyukkaya

Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey

[email protected], [email protected], [email protected]

Abstract: The notion of suborbital graphs was introduced by Sims[6]. We can summarize Sims’theory as follows: a permutation group G act on a set H, suborbital graphs formed by this action with vertex-set H on which G induces automorphisms. After it was shown that this idea is also useful in the study of the modular group whichis a finitely generated Fuchsian group[3], some other finitely generated groups have been studied by suborbital graphs[1,2,4]. In most of them, it has been emphasized the connection between elliptic elements in group and circuits of the same order in graph closely related with the signature problem. It is known that the graph of a group provides a method by which a group can be visualized; in many cases it suggests an economical algebraic proof for a result and it gives same information but in a much more efficient way [5]. In this view, the idea of suborbital graph has been used mainly by finite group theorists. The aim of this paper is to examine the action of the normalizer which produce some congruence equations with solutions. Actually, the suborbital graphs of the normalizer were studied for some special cases[2]. In here, taking a different case, we obtained new results.

Keywords: Normalizer, Imprimitive action, Suborbital graphs. References: [1] M. Beşenk, “Circuit lengths of graphs for the Picard group”. J Inequal Appl (2013), 106:8. [2] B.O. Güler et al., “Elliptic elements and circuits in suborbital graphs”, Hacet J Math Stat 40(2) (2011), 203–210. [3] G.A. Jones, D. Singerman, K. Wicks, “The modular group and generalized Farey graphs”, Lond Math Soc Lect Note Ser 160(1991), 316–338. [4] S. Kader, B.O. Güler (2013) “On suborbital graphs for the extended modular group”, Gr Comb, 29(2013), 1813–1825. [5] M. Magnus, A. Karrass, D. Solitar, “Combinatorial group theory”. Wiley, New York, (1966). [6] C.C. Sims, “Graphs and finite permutation groups”, Math. Z. 95(1967), 76– 86. ______55

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On Spherically Symmetric Solutions of the Einstein-Maxwell Field Equations

Rashida Bibi and Azad A. Siddiqui

Department of Mathematics, School of Natual Sciences, National University of Science and Technology (NUST), Islamabad, Pakistan [email protected]

Abstract: In literature many solution of the Einstein-Maxwell equations have been found [1-8]. We consider the spherically symmetric geometry and classify the solutions of the Einstein-Maxwell equations by considering the null/non-null electromagnetic field and isotropic/anisotropic mater with the help of Segre type of the spacetime.

Keywords: Einstein-Maxwell Equations, null/non-null Electromagnetic Field, Segre Type, and Isotropic/Anisotropic geometry. References: [1] H. Stephani, D. Kramer, M. MacCallum and C. Hoenselaers, Exact Solutions to Einstein Field Equations, (2009). [2] C. B. G. Mclntosh, J. M. Foyster and A. W. C. Lun, The classi_cation of the Ricci and Plebanski Tensors in General Relativity using Newman-Penrose Formalism, J. Math. Phys., 22 (1980). [3] Tooba Feroze. Exact solutions of the Einstein-Maxwell equations with linear equation of state, Can.J.Phys (2012). 90 1179-1183. [4] G. S. Hall, and D. A. Negm, Physical structure of the energy-momentum tensor in General Relativity, Int. J. Theo Phys. Volume 25, (1986). [5] Rashida Bibi, Tooba Feroze, and Azad A. Siddiqui, Solution of the Einstein– Maxwell equations with anisotropic negative pressure as a potential model of a dark energy star, Canadian Journal of Physics 2016. [6] S. Thirukkanesh and S. D. Maharaj, Charged Anisotropic Matter with Linear Equation of State, Class. Quant. Grav., 25 (2008). [7] M. K. Mak and T. Harko, Quark Stars Admitting a One Parameter Group of Conformal Motions, Int. J. Mod. Phys. D, 13 (2004). [8] J. M. Sunzu, S. D. Maharaj and S. Ray, Quark Star Model with Charged Anisotropic Matter, Astrophysics and Space Science, 354 (2014). ______56

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A Numerical Approximation Based on Collocation Method for the Solutions of Telegraph Equation

Kübra Erdem Bicer

Department of Mathematics, Manisa Celal Bayar University, Manisa, Turkey [email protected]

Abstract: In this paper, a numerical method based on Bernoulli polynomials is developed to solve telegraph equations. By using Bernoulli polynomials and collocation points, the main problem is reduced to the system of algebric equations, after evaluating some matrix operations. By solving this system, we obtain the coefficients of approximate solutions of the main problem. And also to demonstrate the validity and applicability of this method, an error analysis based on residual function is developed and this method is applied to some examples.

Keywords: Telegraph Equations, Partial Differential Equations, Numerical Methods, Bernoulli Polynomials. References: [1] H. M Lieberstein, “Theory of Partial Differential Equations”, Academic Press, New York (1972). [2] G . Doetsch, “Introduction to the Theory and Application of the Laplace Transformation”, Springer, Berlin (1974). [3] D. R. Bland, “Wave Theory and Applications”, Oxford Clarendon press, London (1988). [4] V. H. Weston and S. He, “Wave splitting of the telegraph equation in R3 and its application to inverse scattering”, InverseProblems, 9 (1993), 789–812. [5] P. M. Jordan and A . Puri, “Digital signal propagation in dispersive media”, Journal of Applied Physics, 85.3 (1999), 1273–1282. [6] J. Banasiak and J. R. Mika, “Singular perturbed telegraph equations with applications in the random walk theory” Journal of Applied Mathematics and Stochastic Analysis, 11.1(1998), 9–28. [7] P. R. Wallace, “Mathematical Analysis of Physical Problems”, Dover, New York (1984). [8] L. Debnath and P. Mikusinski, “Introduction to Hilbert Spaces with Applications”, Elsevier Academic Press, London (2005).

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Rotating Disk Cryptosystem: RDC

Sadek Bouroubi and Louiza Rezkallah

Department of Mathematics, USTHB University, Bab Ezzoua, Algier, Algeria [email protected]

Abstract: Cryptography is a hot and active topic, it plays a crucial role in many aspects nowadays, from internet banking and e-commerce to email and webbased business processes. An important area of research today is to test the security of cryptosystems, a cryptographic system is safe as long as no one could not break it. Modern cryptographic algorithms are based on mathematical problems, known to be difficult, so breaking the security of a cryptosystem requires the solution of a such problem, as in the case of RSA [1] which is based on the factorization problem in number theory. This paper proposes a symmetric- key cryptosystem named Rotating Disk Cryptosystem and is denoted as RDC used for encryption and decryption of text as well as images which is based on chaotic function and the factorization problem.

Keywords: Rotating Disk, Symmetric-key Cryptosystem, Chaotic function, Linear Congruential Generator Modified, Factorization Problem. References: [1] R. Rivest, A. Shamir and L. Adleman, « A Method for Obtaining Digital Signatures and Public-Key Cryptosystems », Communications of the ACM, vol. 21, no 2, 1978, p. 120-126. [2] Arjen K. Lenstra and H. W. Lenstra, Jr. (eds.). "The development of the number field sieve". Lecture Notes in Math. (1993) 1554. Springer-Verlag. [3] Laurence T. Yang, Li Xu, Sang-Soo Yeo, Sajid Hussain, An integrated parallel GNFS algorithm for integer factorization based on Linbox Montgomery block Lanczos method over GF(2). Computers Mathematics with Applications, Volume 60, Issue 2, July 2010, Pages 338–346. [4] Delfs, Hans Knebl, Helmut (2007). "Symmetric-key encryption". Introduction to cryptography: principles and applications. Springer. ISBN 9783540492436 ______58

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Free Convection inside a Porous Enclosure

Canan Bozkaya

Department of Mathematics, Middle East Technical University, Ankara, Turkey [email protected]

Abstract: The steady free convection in a rectangular enclosure filled with a fluid-saturated porous medium is numerically investigated in the present study. The left vertical wall of the enclosure is maintained at a temperature greater than the temperature of the right vertical wall. The top and bottom horizontal walls of the enclosure are adiabatic. The incompressible, laminar flow inside the homogeneous and isotropic porous medium is assumed to obey Darcy law. The fluid physical properties are constant except the density in the body force term which is treated according to Boussinesq approximation. The fluid and porous medium are in thermal equilibrium, and the thermal radiation flux in y-direction is considered negligible compared to that in x-direction. Thus, the temperature gradient in x-direction is higher than that in y-direction. For this reason, the thermo-diffusion velocity only in the x-direction is considered. The governing equations, obtained under these assumptions, in non-dimensional stream function and temperature formulation are solved using the dual reciprocity boundary element method (DRBEM). Dual reciprocity BEM aims to transform the differential equations into equivalent integral equations only on the boundary of computational domain by treating the nonhomogeneity through a radial basis function approximation. The stream function equation is solved by DRBEM in the usual way, that is by using the fundamental solution of the Laplace equation. On the other hand, the fundamental solution to the modified diffusion (Laplacian) term in the energy equation (that is, the coefficients of the second order derivatives in x- and y-directions are not equal due to higher thermal radiation in x-direction) is used, and moreover the corresponding radial basis functions for the modified diffusion equation are derived. The results of the present numerical model is compared with previously published works. The validated model is employed to investigate the effect of the physical parameters, namely Rayleigh number and radiation parameter, on the flow and heat transfer characteristics.

Keywords: Free convection, porous medium, thermal radiation, DRBEM.

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Tauberian Theorems for the Cesáro Second Order Operators for Sequences of Fuzzy Numbers

Naim L. Braha

Department of Mathematics and Computer Sciences University of Prishtina, Avenue Mother Teresa, No-4, Prishtine, 10000, Kosova [email protected]

Abstract: In this paper we define the Cesáro second order summability method for fuzzy numbers and prove some theorems dealing with weighted statistical converge of fuzzy numbers and their behaviors. In second section we prove Tauberian theorems for this kind of summability method and in the end of the paper we prove Tauberian theorems, related to the Cesáro second order mean- level convergence.

Keywords: Cesáro second order summability method, Tauberian theorems. References: [1] Y. Altin; M. Mursaleen, H. Altinok, Statistical summability $(C,1)$ for sequences of fuzzy real numbers and a Tauberian theorem, Journal of Intelligent and Fuzzy Systems 21 (2010) 379-384. [2]S. Aytar; M. A. Mammadov and S. Pehlivan, Statistical limit inferior and limit superior for sequences of fuzzy numbers, Fuzzy Sets and Systems 157(7) (2006), 976-985. [3] B. Bede and S. G. Gal, Almost periodic fuzzy number valued functions, Fuzzy Sets and Systems 147 (2004), 385-403. [4] Braha, Naim L. Geometric properties of the second-order Ces\'aro spaces. Banach J. Math. Anal. 10 (2016), no. 1, 1-14. [5] N. L. Braha, Tauberian conditions under which $\lambda -$statistical convergence follows from statistical summability $(V,\lambda ),$ Miskolc Math. Notes 16(2) (2015), 695-703. [6] N.L. Braha and Mikail Et, Tauberian theorems for the Euler-N\"orlund mean- convergent sequences of fuzzy numbers, to appear in Iranian Journal of Fuzzy Systems. [7N.L. Braha, Tauberian Theorems under N\"orlund-Ces\'aro summability methods (357-411), Current Topics in Summability Theory and Applications, editors, Hemen Dutta and Billy E. Rhoades, Springer, 2016. ______60

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Estimating the Distortion Parameter of the Proportional Hazards Premium for Heavy-Tailed Losses

Brahimi Brahim

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria. [email protected]

Abstract: Estimating the distorted parameter in the case of non negative heavy- tailed losses has been treated in [1]. In this paper, we extend this work to the case of the real heavy-tailed losses. We derive an asymptotic distribution of the estimator. We construct a practically implemented confidence interval for the distortion parameter and illustrate the performance of the interval in a simulation study with application to real data.

Keywords: Proportional-hazard premium; Distortion risk measure; Distortion parameter; Extreme value; Heavy tail; Risk aversion index; Lévy-stable distribution. References: [1] Brahimi, B., Meraghni, D., Necir, A. and Zitikis, R., (2011). Estimating the distortion parameter of the proportional-hazard premium for heavy-tailed losses. Insurance Math. Econom. 49, 325-334. [2] Brahimi, B; Abdelli, J., (2016). Estimating the distortion parameter of the proportional hazards premium for heavy-tailed losses under Lévy-stable regime.Insurance Math. Econom. 70 (2016), 135--143. [3] Hill, B.M., (1975). A simple approach to inference about the tail of a distribution. Ann. Statist. 3, 1136-1174. [4] Necir, A. and Meraghni, D., (2010). Estimating L-functionals for Heavy- tailed Distributions and Applications. Journal of Probability and Statistics. Volume 2010, ID 707146. [5] Necir, A., Rassoul, A., Zitikis, R., (2010). Estimating the conditional tail expectation in the case of heavy-tailed losses. Journal of Probability and Statistics 2010, ID 596839.

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On Some Fixed Point Results Related to Almost Generalized (휶, 휷) − (흍, 흓) −Weakly Contractive Mappings in 푺 Metric Spaces

Abdurrahman Buyukkaya1, Mahpeyker Ozturk2

1Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey 2Department of Mathematics, Sakarya University, Sakarya, Turkey [email protected], [email protected]

Abstract: In this paper, we establish some fixed and common fixed point theorems by using almost generalized (훼, 훽) − (휓, 휙) −weakly contractive mappings in the settings of 푆 metric spaces, which given results are generalizations of some existing literature.

Keywords: Fixed Point, Almost Contraction, Admissible Mappings, S Metric Spaces.

References: [1] L. Ciric, M. Abbas, R. Saadati and N. Hussain, Common fixed points of almost generalized contractive mappings in ordered matric spaces , Appı. Math. Comput. 217(2011), 5784-5789. [2] M. Abbas, G. V. R. Babu, and G. N. Alemayehu, On common fixed of weakly compatible mappings satisfying generalized condition (B), Filomat, 25(2) (2011), 9-19. [3] S. Alizadeh, F. Moradlou, P. Salimi, Some fixed point results for (Alpha,Beta)-(Psi,Phi)-conrtactive mappings, Filomat, 28(3) (2014) 635-647. [4] S. Sedghi, N. Shobe, S. Aliouche, A generalization of fixed point theorems S metric spaces, Matematicki Vesnik, 64(3) (2012), 258-266. [5]O. Yamaod, W. Sintunavarat, Some Fixed Point Results for Generalized Contraction with Cyclic (Alpha,Beta)-admissible Mapping in Multiplicative Metric Spaces, Journal of Inequalities and Applications, 488 (2014), 1-15. [6]S. Sedghi, N. V. Dung, Fixed point theorems on S metric spaces, Matematicki Vesnik, 66(1) (2014), 113-124

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Beyond Statistical Quasi Cauchy Sequences

Huseyin Cakalli

Maltepe University, Graduate School of Science and Engineering, Marmara Egitim Koyu, TR 34857,Maltepe, Istanbul, Turkey [email protected]

Abstract: A sequence (αk) is called -statistically downward quasi-Cauchy if limn (1/n)|{k≤n: k+1-k}|=0 for every >0, where (n) is a non-decreasing sequence of positive real numbers tending to  such that limsupn (n) /n )<,  n =O(1), and  αk =αk+1 – αk for each positive integer k. A real valued function defined on a subset of the set of real numbers is -statistically downward continuous if it preserves -statistically downward quasi-Cauchy sequences, i.e. (f(k)) is -statistically downward quasi-Cauchy whenever (k) is. It turns out that the set of -statistical downward continuous functions defined on an above bounded set is a proper subset of the set of uniformly continuous fonctions.

Keywords: statistical convergence, -statistical downward quasi Cauchy sequence,. References: [1] D. Burton, J. Coleman, “Quasi-Cauchy sequences”, Amer. Math. Monthly, 117 (2010), 328-333. [2] H.Cakalli, “Forward continuity”,J. Comput. Anal. Appl.,13 2(2011),225-230. [3] H. Cakalli, “Statistical ward continuity”, Appl. Math. Lett., 24 10 (2011), 1724-1728. [4] H. Cakalli, “Statistical quasi-Cauchy sequences”, Math. Comput. Modelling, 54 (2011), 1620-1624. [5] H. Cakalli, “A Variation on Statistical Ward Continuity”, Bull. Malays. Math. Sci. Soc., DOI 10.1007/s40840-015-0195-0 [6] H.Cakalli,”Upward and downward statistical continuities”, Filomat, 29 10 (2015), 2265-2273. [7] H.Cakalli, and H. Kaplan, “A study on N-theta-quasi-Cauchy sequences”, Abstr. Appl. Anal, 2013 (2013), Article ID 836970, 4 pages. [8] H. Cakalli, and H. Kaplan, “A variation on lacunary statistical quasi cauchy sequences”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66 2 (2017),1-9 [9] H. Cakalli, and M.K. Khan, “Summability in topological spaces”, Appl. Math. Lett., 24 (2011), 348-352.

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A Study on Strongly Lacunary Ward Continuity

Huseyin Cakalli*, Huseyin Kaplan**

* Maltepe University, GraduateSchool of Science and Engineering, Marmara Egitim Koyu, Maltepe, Istanbul, Turkey [email protected] ** Niğde University, Faculty of Science and Letters, Niğde, Turkey [email protected]

Abstract: In this paper, the concept of a strongly lacunary 2-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to strongly lacunary 2-ward continuity, and some other kinds of continuities, introducing strongly lacunary 2 ward continuity in the sense that a real valued function f defined on a subset A of IR, the set of real numbers, is strongly lacunary 2 ward continuous on A if it preserves strongly lacunary 2 2 quasi-Cauchy sequences of points in A, i.e. (f(αk)) is a strongly lacunary  2 quasi-Cauchy sequence whenever (αk) is a strongly lacunary  quasi-Cauchy 2 sequences of points in A, where a sequence (αk) is called strongly lacunary  2 2 quasi-Cauchy if ( αk) is a strongly lacunary quasi-Cauchy sequence where 

αk= αk+2 -2 αk+1 + αk for each positive integer k.

Keywords: Summability; series and sequences; continuity and related questions. References: [1] D. Burton, and J. Coleman, “Quasi-Cauchy Sequences”, Amer. Math.

Monthly, 117 4 (2010), 328-333. [2] Naim L. Braha, H. Cakalli, “A new type continuity for real functions”, J Math. Analysis.7 6 (2016), 54-62. [3] H. Cakalli, “On G-continuity”, Comput. Math. Appl., 61 (2011), 313-318. [4] H. Cakalli, “N-theta-Ward continuity”, Abstr. Appl. Anal,. 2012 Article ID 680456 (2012), 8 pp. [5] H. Cakalli, “A variation on arithmetic continuity”, Bol. Soc. Paran. Mat., 35 3 (2017), 195-202. [6] H. Cakalli, C. Gunduz Aras and A. Sonmez, “Lacunary statistical ward continuity”, AIP Conf. Proc. 1676, 020042 (2015)

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A Study on Abel Statistical Quasi Cauchy Sequences

Huseyin Cakalli*, Iffet Taylan**

* Maltepe University, Graduate School of Science and Engineering, Marmara Egitim Koyu, Maltepe, Istanbul, Turkey [email protected] ** Maltepe University, Education Faculty, Maltepe, Istanbul, Turkey [email protected]

Abstract: In this paper, we investigate the concepts of Abel statistical convergence and Abel statistical quasi Cauchy sequences. We also study Abel statistical continuity and Abel statistical ward continuity. A real valued function f is Abel statistically continuous on a subset E of IR, the set of real numbers, if it preserves Abel statistical convergent sequences, i.e. (f (pk)) is Abel statistically convergent whenever (pk) is an Abel statistical convergent sequence of points in E. Some other types of continuities are also studied and interesting results are obtained.

Keywords: Statistical Convergence; Abel series method; continuity. References: [1] D. Burton, J. Coleman, “Quasi-Cauchy sequences”, Amer. Math. Monthly, 117 (2010), 328-333. [2] H. Cakalli, “A study on statistical convergence”, Funct. Anal. Approx. Comput.1 2 (2009) , 19–24. [3] H. Cakalli, “Forward continuity”, J. Comput. Anal. Appl., 13 2 (2011), 225- 230. [4] H. Cakalli, “Statistical ward continuity”, Appl. Math. Lett., 24 10 (2011), 1724-1728. [5] H. Cakalli, “Statistical quasi-Cauchy sequences”, Math. Comput. Modelling, 54 (2011), 1620-1624. [6] H. Cakalli, “On G-continuity”, Comput. Math. Appl. 61 (2011), 313-318. [7] H. Cakalli, “A variation on arithmetic continuity”, Bol. Soc. Paran. Mat. 35 3 (2017), 195-202. [8]H. Çakallı and M. Albayrak, “New Type Continuities via Abel Convergence”, Scientific World Journal, 2014 (2014), Article ID 398379, 6 pages.

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Graph Theoretical Applications of Molecular Graphs

Ismail Naci Cangul

Department of Mathematics, Uludag University, Bursa, Turkey

Joint work with Aysun YURTTAS, Muge TOGAN, Ahmet Sinan CEVIK

Abstract: Most of the papers on graph theory published recently deal with molecular graphs due to their intensive applications in chemistry and pharmacology. In this paper we give some applications of topological graph indices in these two areas.

Keywords: Molecular graph, topological graph index. References: [1] K. Ch. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, A. S. Cevik, On the first Zagreb index and multiplicative Zagreb coindices of graphs, Analele Stiintifice ale Universitatii Ovidius Constanta 24 (1) (2016) [2] K. C. Das, K., Xu, I. N., Cangul, A. S., Cevik, A., Graovac, On the Harary Index of Graph Operations, Journal of Inequalities and Applications, SI: Recent Advances in General Inequalities, DOI: 10.1186/1029-242X-2013-339, 2013, [3] K. Ch. Das, A. Yurttas, M. Togan, I. N. Cangul, A. S. Cevik, The multiplicative Zagreb indices of graph operations, Journal of Inequalities and Applications, (2013), 1-14. [4] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals III, Total $\pi$- electron energy of alternant hydrocarbons, \textit{Chem. Phys. Lett.}, \textbf{17} (1972), 535-538. [5] H. Hosoya, K. Kawasaki, K. Mizutani, 1972, Topological index and thermodynamic properties: Empirical rules on the boiling point of saturated hydrocarbons, Bull. of the Chem. Soc. of Japan, 45, 3415. [6] H. Wiener, 1947, Correlation of heats of isomerization and differences in heats of paraffin hydrocarbons, J. of the American Chemical Society, 69, 2636 [7] H. Wiener, 1947, Structural determination of parafin boiling points, J. of the American Chemical Society, 69, 17 [8] N. Trinajsti?, Chemical graph theory, vols. I and II, CRC, Boca Raton, FL, 1983

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A Study on Public Transit Users’ Route Choice and Assignment Function

Buket Capali1, Halim Ceylan2

1Department of Civil Engineering, Faculty of Engineering, Süleyman Demirel University, Isparta, Turkey 2Department of Civil Engineering, Faculty of Engineering, Pamukkale University, Denizli, Turkey [email protected]

Abstract: Many passengers wants to travel between two points from point A to point B, the next step is to determine the routes to reach their destination. This determination relies on passenger behavior. When there are more than two routes to travel, a rate representing the amount of interest in the travel assignment process for the different transit routes is determined. In this study; a new formula for transit users’ route choice and assignment, which includes travel demand and waiting time, has been developed.

Keywords: Public transportation, route choice, transit assignment.

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A New Developed Semi–Empirical Formula for Nuclear Reaction Cross–Section Calculations

Veli Capali1, Mert Sekerci1, Hasan Ozdogan2, Abdullah Kaplan1

1Department of Physics, Süleyman Demirel University, Isparta, Turkey 2Department of Biophysics, Akdeniz University, Antalya, Turkey [email protected]

Abstract: In this study; a new semi–empirical formula for reaction cross–section calculations, which includes inelastic scattering and Coulomb effects, has been developed. The cross–sections and asymmetry parameters are dependent on target nuclei and the selected energies which are 10-11, 18-19 and 22-23 MeV. The reaction cross–section calculations have been performed for (p,n) reaction for sample materials. The cross–section values calculated from developed semi– empirical formula have been compared with the experimental values exist in the literature and nuclear reaction models’ computation results.

Keywords: Semi–Empirical Formula, Nuclear Cross–Section, Asymmetry Parameter.

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On Uninorms on Bounded Lattices

Gül Deniz Cayli and Funda Karacal

Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey [email protected] [email protected]

Abstract: The triangular norms (t-norms) with 1 as neutral element and triangular conorms (t-conorms) with 0 as neutral element were introduced by Schweizer and Sklar in [14]. Uninorms that were introduced by Yager and Rybalov in [15] are generalization t-norms and t-conorms. These operators allow the freedom for the neutral element e (sometimes called identity) to be an arbitrary element from unit interval [0,1], which is 1 for t-norms and 0 for t- conorms. In [11] the existence of uninorms on an arbitrary bounded lattice 퐿 for the neutral element e to be an arbitrary element from 퐿 ∖ {0,1} has been shown. As a by-product, it has been demonstrated that the existence of the smallest uninorm and of the greatest uninorm on L with a fixed neutral element 푒 ∈ 퐿 ∖ {0,1}. In [6] the existence of idempotent uninorms on given bounded lattice L for any element 푒 ∈ 퐿 ∖ {0,1} playing the role of a neutral element has been proved. By this construction method, the smallest idempotent uninorm and the greatest idempotent uninorm with the neutral element 푒 ∈ 퐿 ∖ {0,1} is obtained. In this paper, we study uninorms on bounded lattices and investigate some properties of these operators. By considering the existence of t-norms and t-conorms on an arbitrary bounded lattice L, we give methods of constructing uninorms with given neutral element 푒 ∈ 퐿 ∖ {0,1}. And we consider some special classes of t- norms and t-conorms by using the presence of uninorms on arbitrary bounded lattice L with a fixed neutral element 푒 ∈ 퐿 ∖ {0,1}.

Keywords: Uninorm, Bounded lattice, T-norm, T-conorm. References: [1] E. Aşıcı, F. Karaçal, “On the T -partial order and properties”, Inf. Sci. 267(2014), 323-333.

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[2] E. Aşıcı, F. Karaçal, “Incomparability with respect to the triangular order”, Kybernetika 52(2016), 15-27. [3] G. Birkhoff, “Lattice Theory”, American Mathematical Society Colloquium Publishers, Providence, RI, 1967. [4] B. De Baets, “Uninorms: the known classes”, Third Internat FLINS Workshop in Fuzzy Logic and Intelligent Technologic, 1998. [5] S. Bodjanova, M. Kalina, “Construction of uninorms on bounded lattices”, IEEE 12th International Symposium on Intelligent Systems and Informatics, SISY 2014, September 1113, 2014, Subotica, Serbia. [6] G.D. Çaylı, F. Karaçal, R. Mesiar, “On a new classes of uninorms on bounded lattices”, Inf. Sci. 367–368(2016), 221-231. [7] G. Deschrijver, “Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory”, Inf. Sci. 244(2013), 48-59. [8] P. Drygaś, “On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums”, Fuzzy Sets Syst. 161(2010), 149-157. [9] P. Drygaś, D. Ruiz-Aguilera, J. Torrens, “A characterization of uninorms locally internal in A(e) with continuous underlying operators”, Fuzzy Sets Syst. 287 (2016), 137-153. [10] J.Fodor, R.R. Yager, A. Rybalov, “Structure of uninorms,” Internat J. Uncertain Fuzziness Knowledge-Based Systems, 5(1997), .411-427. [11] F. Karaçal, R. Mesiar, “Uninorms on bounded lattices”, Fuzzy Sets Syst. 261(2015), 33-43. [12] J. Martin, G. Mayor, J. Torrens, “On locally internal monotonic operations”, Fuzzy Sets Syst. 137(2003), 27-42 [13] R. Mesiar, E. Pap, “Different interpretations of triangular norms and related operations”, Fuzzy Sets Syst. 96(1998), 183-189. [14] B.Schweizer, A.Sklar, “Statistical metric spaces”, Pacific J. Math, 10(1960), 313-334. [15] R.R. Yager, A. Rybalov, “Uninorms aggregation operators”, Fuzzy Sets Syst. 80(1996), 111-120.

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On Statistical Dunford and Pettis Integration

Anita Caushi

Department of Mathematical Engineering, Polytechnic University of Tirana Mother Teresa Square nr.4, Tirana, Albania [email protected]

Abstract: Pettis and Dunford integrals are more important concepts with respect to modern theory of probabilities. We can find them in the definitions of mathematical expectation as well as dispersion. Some random probailitary funktions take the value on vectorial spaces.In this paper we extend the usual concept of Dunford and Pettis integration to a statistical form. We give some essential properties of them and give an example where we find one function that is statistical Dunford integrable but not Pettis integrable. We obtain some special properties of statistical Pettis integration which are well known for usual the Pettis integration.

References: [1] Bhardwaj V., Balai., On Ëeakly Statistical Convergence, international Journal of Mathematics and Mathematical Sciences, 2007, Article iD 38530, 9p [2] Çakalli H., A study on statistical convergence. Functional analysis, approximation and computation 1:2(2009), 19-24 [3] Caushi, A., Tato, A., A statistical integral of Bohner type on Banach space, Hikari Ltd Appl. Math. Sci., Vol. 6, 2012, no. 137-140, 6857-6870. [4] Connor J., Ganichev M., and Kadets V., “A characterization of Banach spaces ëith separable duals via weakly statistical convergence,” Journal of Mathematical Analysis and Applications, vol. 244, no. 1, pp. 251–261, 1989. [5] Fast H., “Sur la convergence statistique,” ColloquiumMathematicum, 2,1951. [6] Fridy J. A., “On statistical convergence,”Analysis, 5-4, 301-313, 1985. [7] Fridy J. A., “Statistical limit points,” Proceedings of the American Mathematical Society, vol. 118, no. 4, pp. 1187–1192, 1993. [8] Schoenberg i. J., “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, 66, 5, 1959. [9] Schëabik S., Guoju Y., Topics in Banach space integration, Series in Analysis vol. 10. Ëorld Scientific Publishing Co. Singapore 2005. [10] Steinhaus H., “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951. [11] Zygmund A., Trigonometric Series, Cambridge University Press, Cambridge, UK, 1979 ______71

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On The Cardinality of Category Spaces

Bahaettin Cengiz1, Banu Gunturk2

1Faculty of Engineering, Baskent University, Ankara, Turkey [email protected] 2Faculty of Engineering, Baskent University, Ankara, Turkey [email protected]

Abstract: We call a compact Hausdorff space Ω a category space if C(Ω), the space of all scalar-valued continuous functions, is a dual space. Category spaces are necessarily extremally disconnected, and on such a space there always exists an essentially unique category (or perfect) Borel measure. If one of the category measures on a category space is σ-finite, then so are the rest. These measures play a crucial role in determining the greatest lower bound for the cardinalities of infinite category spaces. In this paper, among other things, it is proved that for any infinite category space Ω, card(Ω) ≥ 2c if the category measure is σ-finite, 2ℵ1 and 2 otherwise, where c is the cardinality of the continuum, ℵ1 is the first uncountable cardinal number, and for any infinite cardinal number ℵ, 2ℵ denotes the cardinality of the power set of a set of cardinality ℵ. Both of the above mentioned cardinal numbers are actually attained.

Keywords: Category space, cardinality, dual space.

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Remarks and Observations on Some Special Arithmetical Sums

1Elif Cetin and 2Yilmaz Simsek

1Department of Mathematics, Faculty of Art and Science, University of Manisa Celal Bayar, Manisa, Turkey, 2Department of Mathematics, Faculty of Science University of Akdeniz TR- 07058, Antalya, Turkey. [email protected] , [email protected]

Abstract: In this talk, we investigate some properties of the Dedekind sums and Hardy sums. These sums are used commonly not only in analytic number theory, but also in other areas of mathematics. The Dedekind sums arise in the behaviour of the Dedekind eta functions under the modular groups. On the other hand, the Hardy sums arise in the behaviour of the Theta functions under the subgroups of the modular groups. Recently, many authors give relation between the Dedekind sums and geometry (lattice point enumeration in polytopes, topology (signature defects of manifolds), algorithmic complexity (pseudo random number generators), character theory, the family of zeta functions, the Bernoulli and Euler functions. Moreover, we give some applications of the special arithmetic sums related to the Hardy sums, the Dedekind sums and the other special arithmetical sums.

Keywords: Dedekind sums, Hardy-Berndt sums, Generating functions, Bernoulli numbers and polynomials. References: [1] T. M. Apostol, “Introduction to Analytic Number Theory”, Springer-Verlag, New York, USA (1976), 340pp. [2] B.C. Berndt, “Analytic Eisenstein series, Theta-functions, and series relations in the spirit of Ramanujan”, J. Reine Angew. Math. 303/304(1978), 332-365. [3] E. Cetin, Y. Simsek and I.N. Cangul, “Some special finite sums related to the three-term polynomial relations and their applications”, Adv. Difference. Equ., 283(2014), 1-18. [4] E. Cetin, “A Note on Hardy type Sums and Dedekind Sums”, FILOMAT, 30.4(2016), 977-983. ______73

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S-Generalized Mittag-Leffler Function

Aysegul Cetinkaya, I. Onur Kiymaz, M. Baki Yagbasan

Department of Mathematics, Ahi Evran University, Kırşehir, Turkey [email protected]

Abstract: In this study, we introduced a new generalization of Mittag-Leffler function by using S-generalized beta function. Furthermore, we investigated some of its properties such as integral representations, recurrence formulas, derivative formulas and Mellin transform.

Keywords: S-Generalized Beta function, Mittag-Leffler function, Mellin transform. Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.A3.16.035 References: [1] Agarwal, P.,Choi, J.,Paris,R.B.“Extended Riemann-Liouville fractional derivative operator and its applications”,J.Nonlinear Sci.Appl,8,(2015): 451-466. [2]Chaudhry M. A., Qadir A., Rafique M., Zubair S. M., “Extension of Euler's beta function”, J. Comput.Appl. Math., 78, (1997): 19-32. [3] Luo, Min-Jie, Milovanovic, G. V., Agarwal, P., “Some results on the extended beta and extended hypergeometric functions”, Applied Mathematics and Computation, 248, (2014): 631-651. [4] Prabhakar, T. R. “A singular integral equation with a generalized Mittag Leffler function in the kernel”, Yokohama Math. J., 19, (1971): 7-15. [5] Srivastava, H. M., Agarwal, P., Jain S., “Generating functions for the generalized Gauss hypergeometric functions”, Applied Mathematics and Computation, 247, (2014): 348-352. [6] Srivastava, H. M., Jain, R., Bansal, M. K. “A Study of the S-Generalized Gauss Hypergeometric Function and Its Associated Integral Transforms”, Turkish Journal of Analysis and Number Theory, 3.5, (2015): 116-119. [7] Srivastava, H. M., Manocha, H.L., “A Treatise on Generating Functions”, Halsted, New York (Ellis Horwood, Chichester), (1984). [8] Özergin, E., Özarslan, M. A., Altın, A., “Extension of gamma, beta and hypergeometric functions”, Journal of computational and applied mathematics, 235.16 (2011): 4601-4610. [9] Özarslan, M. A., Yılmaz, B. “The extended Mittag-Leffler function and its properties”, Journal of Inequalities and Applications, 2014.1, (2014): 85. ______74

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Jacobi Elliptic Function Solutions of Time-Fractional Kdv- Zakharov-Kuznetsov Equation

Sevil Culha, Dilek Varol Bayram, Aysegul Dascioglu

Department of Mathematics, Faculty of Science and Arts, Pamukkale University Denizli, Turkey [email protected]

Abstract: In this study, new analytical exact solutions of the nonlinear evolution equation in mathematical physics, namely the conformable time-fractional KdV– Zakharov–Kuznetsov (KdV–ZK) equation are presented by using the Jacobi elliptic function expansion method.

Keywords: Jacobi elliptic function, fractional differential equation, KdV- ZK equation. References: [1] R.L. Mace, M.A. Hellberg, The Korteweg–de Vries–Zakharov–Kuznetsov equation for electron-acoustic waves, Phys. Plasmas 8 (6) (2001) 2649– 2656. [2] M.H. Islam, K. Khan, M.A. Akbar, M.A. Salam, Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation, Springer Plus 3 (105) (2014) 1–9. [3] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57–66. [4] H. Naher, F.A. Abdullah, M.A. Akbar, Generalized and improved (퐺’/퐺)- expansion method for (3+1)-dimensional modified KdV–Zakharov– Kuznetsov equation, PLoS One 8(5) (2013) 1–7. [5] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the Tzitzeica– Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations using the modified simple equation method, Ain Shams Engr J 4(4) (2013) 903–909. [6] S. T. Mohyud-Din, A. Irshad, On exact solutions of modified KdV-ZK equation, Alexandria Engineering Journal (55) (2016) 3253-3265. [7] E. M. E. Zayed, New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized (퐺′⁄퐺)-expansion method Journal of Physics A: Mathematical and Theoretical (42) (2009). ______75

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A Fix-And-Optimize Heuristic for the Integrated Fleet Sizing and Replenishment Planning Problem with Predetermined Delivery Frequencies

Niousha Karimi Dastjerd, Kadir Ertogral

Department of Industrial Engineering, TOBB University of Economics and Technology, Sogutozu, Ankara, Turkey [email protected], [email protected]

Abstract: We tackled an integrated fleet sizing and replenishment planning problem in a vendor managed inventory system. There is a set of customers which must be replenished based on a given set of predetermined frequencies. The vehicle fleet consists of multiple types of heterogeneous vehicles which differ in carrying capacity, cost per kilometer, and ownership costs. Customer demands are taken as deterministic values. The main decision we make in this problem is the triple asignment of vehicle-frequency-customer. As a result of these assignment decisions, we obtain an annual costs consisting of vehichle ownership cost, routing cost, inventory holding and fixed replenishment costs. A key simplification in the model is the use of linear approxiamation for the routing cost based on the number of customers visited in a tour. The developed model, which is new in the literature, integrates fleet sizing and replenishment planning decisions. Our problem is NP-hard since it can be shown that a special case of our problem is a bin packing problem. In order to solve large problems efficiently, we suggested and applied a fix and optimize heuristic as a solution procedure. This fix and optimize heuristic divides the problem into smaller problems in which some variables are binaries and the others are linearly relaxed, and it fixes the linear decision variable iteratively. We also showed the effectiveness of the suggested heuristic solution procedure on a large set of randomly generated problems.

Keywords: Fleet sizing, Replenishment planning, Predetermined frequencies, Fix and Optimize

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Catalogue of Degree Sequences of Molecular Graphs

Sadik Delen, Ismail Naci Cangul

Department of Mathematics, Uludag University, Bursa, Turkey [email protected], [email protected]

Abstract: Molecular graphs are those graphs which are trees with vertex degrees at most 4. They have applications in chemistry and pharmacology. In this paper we give the catalogue of all molecular graphs and give results on their classification.

Keywords: Molecular graph, degree sequences. References: [1] B. Bollobas, Degree sequences of random graphs, Discrete Mathematics 33 (1981), 1-19. [2] Delen, S., Cangul, I. N., Algebraic Properties of Some Graph Operations in Terms of Degree Sequences, (preprint) [3] Delen, S., Cangul, I. N., Degree Sequences of Join and Corona Products of Graphs, (preprint)

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Streamline Topology of Vortex Breakdown Bubbles near the Re-Entrant Corner

Ali Deliceoglu 1 and Ebutalib Celik 2

1Erciyes University, Kayseri, Turkey, [email protected] 2Erciyes University, Kayseri, Turkey, [email protected]

Abstract: In this paper, topological bahavior of a viscous flow near a re-entrant corner is studied. The method derived in this paper is intended to clarify the understanding flow structures near the re-entrant corner. A numerical-asymptotic matching solution for computing the local singular behavior of a viscous flow around a re-entrant corner is developed. The theory is applied to the vortex brakdown bubbles found numerically in an L-shaped cavity.

Keywords: Vortex Breakdown, Flow structure, Bifurcations. Research supported by the TUBITAK under Grant No: 114F525 References: [1] Deliceoğlu, A. and Aydın, S.H., “ Flow bifurcation and eddy genesis in an L- shaped cavity” . Computers and Fluids, Vol. 73, pp. 24-46, 2013. [2] Deliceoğlu, A. and Aydın, S.H., “Topological flow structures in an L-shaped cavity with the vertical motion of the upper lid” . Journal of Computational and Applied Mathematics, Vol. 259, pp. 937-943, 2014. [3] Hawa T. And Rusak Z., Numerical-Asymptotic expansion matching for computing a viscous flow around a sharp expansion corner, Theoret. and Comp. Fluid Dyn., Vol. 15, pp. 265-281, 2002.

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Optimality Conditions for a Linear Differential System with Two Delays

Hanna Demchenko

Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno,Czech Republic [email protected]

Abstract: In the contribution, for linear differential system with two delays 푑푥(푡) = 퐴 푥(푡) + 퐴 푥(푡 − 휏) + 퐴 푥(푡 − 훿) + 푏푢(푡), (1) 푑푡 0 1 2 푛 where 퐴0, 퐴1 and 퐴2are 푛 × 푛 constant matrices, 푏 ∈ 푅 , 휏 > 0, 훿 > 0, 푢(푡) ∈ 푅 and 푢(푡) is a control function, a problem of minimizing a functional ∞ 푇 푇 푇 푇 퐼 = ∫ (푥 (푡)퐶11푥(푡) + 푥 (푡)퐶12푥(푡 − 휏) + 푥 (푡)퐶13푥(푡 − 훿) + 푥 (푡 − 푡0 푇 푇 푇 휏)퐶21푥(푡) + 푥 (푡 − 휏)퐶22푥(푡 − 휏) + 푥 (푡 − 훿)퐶31푥(푡) + 푥 (푡 − 훿)퐶33푥(푡 − 훿) + 푑푢2(푡))푑푡 (2) where integrand is a positive-definite quadratic form, is considered. To solve the problem, Malkin’s approach and Lyapunov’s second method are utilized.

Theorem. Assume that there exists a positive-definite matrix 퐻 satisfying the matrix equation 1 퐴푇퐻 + 퐻퐴 + 퐶 + 퐶 +퐶 − 퐻푏푏푇퐻 = 휃. 0 0 11 22 33 푑 If, moreover, 퐻퐴1 = −퐶12 and 퐻퐴2 = −퐶13, then for problem (1)-(2) the optimal stabilization control function exists and equals 1 푢 (푡) = − 푏푇퐻푢(푡). 0 푑

Keywords: optimal control, delayed differential system, Lyapunov-Krasovskii functional, integral quality criterion. References: [1] G.A. Dolenko, D.Ya Khusainov, “A partial inverse linear-quadratic optimization problem”, Cybernetics and System Analysis, Vol.41, No.3 (2005), 473-478. [2] F. R. Gantmacher, “The Theory of Matrices”, AMS Chelsea Publishing, Providence, RI, USA (2002). [3] I.G. Malkin, “Theory of Stability of Motion, Second revised edition”, Moscow, Nauka Publisher (1966), 530. ______79

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α-Convexity of Some Struve and Lommel Functions

1 Erhan Deniz, 2 Halit Orhan, Murat Caglar

Department of Mathematics, Kafkas University, Kars, Turkey Department of Mathematics, Atatürk University, Erzurum, Turkey [email protected] (E.Deniz) [email protected] (H.Orhan) [email protected] (M. Çağlar)

Abstract: In this paper our first aim is to determine the radii of α-convexity of some normalized Struve and Lommel different of the first kind. Moreover, necessary and sufficient conditions are also given for the parameters such that the these functions are α-convex in the open unit disk. The key tools in the proof of our main results are the Mittag-Leffler expansion for Struve and Lommel functions, properties of zeros of the Struve and Lommel functions and their derivatives and some inequalities for complex and real numbers.

Keywords: Sruve function, Lommel function, convex functions, starlike functions, α-convex functions, radii of α-convexity, zeros of Stuve and Lommel functions. Acknowledgements: The research of E. Deniz and M. Çağlar was supported by the Commission for the Scientific Research Projects of Kafkas University, project number 2016-FM-67.

References: [1] Á. Baricz, P. A. Kupán, R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind. Proc. Amer. Math. Soc. 142(5) (2014), 2019- 2025. [2] Á. Baricz, R. Szász, The radius of convexity of normalized Bessel functions of the first kind. Anal. Appl. 12(5) (2014), 485-509. [3] A. Baricz, D.K. Dimitrov, H. Orhan, N. Yağmur, Radii of starlikeness of some special functions, Proc. Amer. Math. Soc. (in press) doi:10.1090/proc/13120. [4] A. Baricz, N. Yağmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. (in press) doi:10.1007/s11139-015-9724-6. ______80

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Fuzzy Soft Topolical Spaces and the Related Category FST

Tugbahan Simsekler Dizman, Naime Tozlu

Department of Science and Mathematics Education, Gaziantep University Sehitkamil, Gaziantep, Turkey [email protected]

Abstract: Fuzzy set theory defined by Zadeh in 1968 as a new method for vagueness. Certainly this theory became the most succesful theory for unprecise concepts. Both in mathematics and in engineering a lot of papers based on this theory were published. In 2001, Molodtsov defined soft set theory as a different approach for vague concepts. Molodtsov searched the relations between this theory with the fuzzy set theory and showed that soft set is more general than fuzzy set. In a short time the researchers worked on soft set theory and its applications. Also the hybrid models as fuzzy soft set were defined and studied by several researchers. In this paper we consider fuzzy soft sets with a different approach. We inspire of Sostak’s fuzzy sets and generalize this idea for fuzzy soft sets. This allows us to grade the openness and closedness of a fuzzy soft set in a fuzzy soft topological space. The degree may range from 1 to 0 for each parameter. We define fuzzy soft continuous mappings between two fuzzy soft topological spaces also we define the category FST of fuzzy soft topological spaces and give some properties of this category. Moreover we define the initial and the final fuzzy soft topological spaces. Keywords: fuzzy soft set, fuzzy soft topology, category FST References: [1] A.Aygunoğlu and H.Aygun, Some notes on soft topological spaces, Neural Computing and Applications, doi: 10.1007/s00521-011-0722-3, 2011. [2] A. Kharal and B.Ahmad, Mappings on fuzzy soft classes, Advances in Fuzzy Systems, doi:10.1155/2009/407890, 2009. [3] P. K. Maji, A. R. Roy, R. Biswas, Fuzzy soft sets, J. Fuzzy Math.9.3 (2001), 589-602. [4] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37.45 (1999),19-31. [5] T. Simsekler, S.Yuksel, Fuzzy soft topological spaces, Ann. Fuzzy Math. Inform.,5.1(2013),87-96. [6] A. Sostak, On a fuzzy topological structure, Proceedings of the 13thˇ Winter School on Abstract Analysis, Section of Topology,11 (1985), 89103. ______81

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Fuzzy Soft Ditopological Spaces

Tugbahan Simsekler Dizman, Naime Tozlu, Şaziye Yüksel

Department of Science and Mathematics Education, Gaziantep University, Sehitkamil, Gaziantep, Turkey [email protected]

Abstract: Fuzzy and soft sets are two different approaches for vague concepts. Both of these theories took attention of researchers and were applied several branches of mathematics. Also the hybrid models as fuzzy soft sets were defined and searched by the mathematicians. The concept of a ditopology was introduced by L.M. Brown and studied in a series of papers by L.M. Brown and co-authors. Ditopologies are related to the concept of a bitopology introduced by J.L. Kelly. In this paper we consider the case when two independent fuzzy soft structures on a given set are defined – one of them is realizing the property of openness, and the other is interpreting the property of closedness. Hence we define the concept of a fuzzy soft ditopological space. Some properties of such spaces are studied.

Keywords: Fuzzy set, soft set, ditopology References: [1] L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems 98 (1998), 217–224. [2] L. M. Brown, R. Erturk, ¨ Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy Sets and Systems 110 (2) (2000), 227–236. [3] J.L. Kelly, Bitopological spaces , Proc. Lond. Math.Soc., III Ser. 13 (1963), 71–89 .

[4] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37(4-5) (1999), 19-31. [5] A. Sostak, ˇ On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser II 11 (1985), 89–103. [6] A. Sostak, ˇ Two decades of fuzzy topology: basic ideas, notions and results, Russian Math. Surveys 44:6 (1989), 125–186. [7] S. Roy and T. K. Samanta, A note on fuzzy soft topological spaces, Ann. Fuzzy Math. Inform. 3 (2012) 305–311

[8] L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (1965) 338–353.

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On the Difference Method for Approximating of Second Order Derivatives of a Solution of Laplace's Equation in a Rectangular Parallelepiped

Adiguzel A. Dosiyev and Hediye Sarıkaya

Department of Mathematics, Near East University, Nicosia, KKTC, Mersin 10, Turkey [email protected]

Abstract: A 14-point averaging operator is used to construct finite difference problems for the approximation on a cubic grid with step size h, of the pure and mixed second order derivatives of a solution of the Dirichlet problem of

Laplace's equation on a rectangular parallelepiped. The boundary functions 휑푗 on 푝,휆 the faces 훤푗, 푗 = 1,2, . . . ,6 belong to the Hölder classes 퐶 , 0 < 휆 < 1, 푝 ∈ {4,5}, and on the edges their second and fourth order derivatives satisfy the compatibility conditions.

It is proved that the solutions of the constructed difference problems converge of orders 푂(ℎ푝−2+휆) and O(ℎ푝−2+휆) to the exact value of the second pure and mixed derivatives, respectively.

Numerical experiments are illustrated to support the theoretical results.

Keywords: finite difference method, approximation of the derivatives, error estimations, Laplace's equation on parallelepiped

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Kolmogorov Inequality on Variable Exponent Lebesgue Spaces

Ismail Ekincioglu, Esra Kaya

Dumlupinar University, Department of Mathematics, Kutahya, Turkey, [email protected] Dumlupinar University, Department of Mathematics, Kutahya, Turkey, [email protected]

Abstract: In this study, it is considered the Riesz-Bessel transforms, The Riesz- Bessel transforms are bounded on variable exponent Lebesgue spaces for , but these operators are not bounded on variable exponent Lebesgue spaces for . For this reason, it is necessary to work in variable exponent Hardy spaces to get the boundedness of such operators for . The most important method to show the boundedness of the Riesz- Bessel transforms in variable exponent Hardy spaces is atomic decomposition. The most important inequality used when atomic decomposition is applied is Bessel type Kolmogorov inequality. Therefore, in this paper, Bessel type Kolmogorov inequalities, which is necessary to demonstrate the boundedness of Riesz-Bessel transforms in the variable exponent Hardy spaces, will be proved in variable exponent Lebesgue spaces. We say that is a convolution-type singular integral operator with regularity of order if the distribution coincides with a function on and has the properties that are hold and, for all multi-indices and . Therefore, singular integrals that satisfy above conditions are bounded on , . More importantly, the pointwise smoothness conditions guarantee that they satisfy weighted norm inequalities. In this work, we will obtain weighted Kolmogorov inequality.

Keywords: Generalized Shift Operator, Laplace-Bessel Operator, Kolmogorov Inequality, Variable Exponent Lebesgue Spaces. References: [1] A. Osexkowski, “Sharp Inequalities for Riesz Transforms”, Stud. Math., 222, (2014), 1-18. [2] B. M. Levitan, “Generalized Shift Operators”, Moskow Nauva (1973).

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An Application of Functional Variable Method For Semi- Analytical Solutions of Nonlinear Evolution Equations

Berfin Elma and Emine Misirli

Department of Mathematics, Ege University Bornova, İzmir, Turkey [email protected] , [email protected]

Abstract: In recent years, many powerful methods have been developed to obtain exact solutions of partial differential equations. In this paper, we obtained some semi-analytical solutions of the nonlinear Modified Benjamin-Bona equation and nonlinear Coupled Klein-Gordon system which seems in the various scientific and engineering fields such as fluid mechanics, chemical kinematics, by using Functional Variable method. We specified wave types of these equations. Also the physical behaviors of the obtained solution functions are examined and three dimensional graphics are drawn using the Mathematica program. It is shown that this method is used to solve the equations of evolution in mathematical physics and engineering.

Key Words: evolution equations, nonlinear partial differential equations, functional variable method.

References: [1] Kamruzzaman K.H.A.N., and M. Ali AKBAR," Study of functional variable method for finding exact solutions of nonlinear evolution equations." Walailak Journal of Science and Technology (WJST) 12,11 (2014): 1031-1042. [2] Zayed, Elsayed ME and S.A. Hoda Ibrahim, " The functional variable method and its applications for finding the exact solutions of nonlinear PDEs in mathematical physics." AIP Conference Proceedings. Eds. Theodore E. Simos, et al.Vol.1479.No.1.AIP,2012. [3] Zayed, Elsayed ME, Yaser A. Amer, and Ahmed H. Arnous, " Functional variable method and its applications for finding exact solutions of nonlinear PDEs in mathematical physics." Scientific Research and Essays 8,42 (2013): 2068-2074. [4] Zerarka, A.S.Ouamane, and A. Attaf, " On the functional variable method for finding exact solutions to a class of wave equations." Applied Mathematics and Computation 217,7 (2010): 2897-2904.

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New Types of Soft Separation Axioms and Soft Compactness in Soft Topological Spaces

M. E. El-Shafei1, M. Abo-Elhamayel1 and T. M. Al-shami2

1Department of Mathematics, Mansoura University, Egypt 2Department of Mathematics, Sana'a University [email protected]

Abstract: In the present article, we define partial belong and total non belong relations which are more e_ective to theoretical and application studies in soft topological spaces. Many properties related to these two relations are studied and discussed. We then introduce new soft separation axioms namely p-soft Ti- spaces (i = 0; 1; 3; 4), depending on a totally non belong relation, and study their characterizations in detail. with the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti- spaces are stronger than soft Ti-spaces, for i = 0; 1; 4. Also, we definedne a p- soft regular space, which is weaker than a soft regular space ([1], Def.31), by utilizing a total non belong relation instead of non belong relation (2.2) and verify the equivalent between soft T2-spaces and p-soft T3-spaces when the universe set is _nite. In the last section, we initiate a concept of soft locally compact spaces and study main properties. We investigate under what conditions a soft subset of a soft T2-space is soft compact. Moreover, we derive some important results such that every soft compact T2-space is soft T3-space and a _nite product of soft locally compact spaces is soft locally compact. we illuminate that some _ndings obtained in general topology are not true concerning softitopological spaces such as a _nite soft topological space need not be soft compact.

Keywords: Partial belong (Total non belong) relation, Soft regular, p-soft Ti- space (i = 0; 1; 3; 4), Soft T2-space, Soft compactness, Soft locally compactness and soft topological spaces. References: [1] M. Shabir and M. Naz, On soft topological spaces, Computers and Mathematics with Applications 61 (2011) 1786-1799.

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On Quaternion n-Spaces

Fatma Ozen Erdogan, Atilla Akpinar

Department of Mathematics, Uludag University, Bursa, Turkey [email protected] , [email protected]

Abstract: In this presentation, we introduce some combinatorial results related to points and lines of the quaternion n  space P ( J ' ) defined by the special

Jordan matrix algebra J'H (,)Αn J , the set of n by n matrices, with entries in an local ring A :QQ (an quaternion division ring Q ,  Q 2 and   0 ), that are symmetric with respect to the canonical involution J . Keywords: local ring, special Jordan matrix algebra; quaternion; quaternion 3- space; quaternion n-space, projective Klingenberg plane, projective plane.

References: [1] Akpinar, A. and Erdogan, F.O., “On Special Jordan Algebras of Dimension 2n²-n”, 2017 (under review) [2] Akpinar, A. and Erdogan, F.O., “Collineations and Cross-Ratios in Octonion Planes”, 2017 (under review) [3] Baker C.A., Lane N.D., Lorimer J.W., “A coordinatization for Moufang- Klingenberg Planes”, Simon Stevin, 65(1991), 3-22. [4] Bix, R., “Octonion Planes over Local Rings”, Trans. Amer. Math. Soc., 261(2), (1980), 417-438. [5] Faulkner, J.R., “Octonion Planes Defined by Quadratic Jordan Algebras”, Mem. Amer. Math. Soc., 104(1970), 1-71. [6] Faulkner, J.R., “The Role of Nonassociative Algebra in Projective Geometry”, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, R.I., 159(2014). [7] Jacobson, N., “Structure and Representations of Jordan Algebras”, Colloq. Publ., Amer. Math. Soc., Providence, R.I., 39(1968). [8] Jacobson, N., “Basic Algebra I”, W. H. Freeman and Company, New York, 1985. [9]McCrimmon,K., “The Freudenthal-Springer-Tits Constructions of Exceptional Jordan Algebras”, Trans. of the Amer. Math. Soc.,139(1969), 495- 510.

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Decomposition of Soft Continuity via Soft Locally b-Closed Set

Zehra Guzel Ergul, Naime Tozlu, Saziye Yuksel

Department of Mathematics, Ahi Evran University, Kırsehir, Turkey [email protected] Department of Mathematics, Omer Halisdemir University, Nigde, Turkey [email protected] Department of Mathematics, Selcuk University, Konya, Turkey [email protected]

Abstract: In this paper, we introduce soft locally b-closed sets in soft topological spaces, which are defined over an initial universe with a fixed set of parameters, and study some of their properties. We investigate their relationships with different types of subsets of soft topological spaces with the help of counterexamples. Also, the concept of soft locally b-closed continuous functions is presented. Finally, some decompositions of soft continuity are obtained.

Keywords: Soft set, Soft topological space, Soft locally b-closed set, Soft locally b-closed continuous function. Acknowledgements: This work is supported by Ahi Evran University Scientific Research Projects Coordination Unit. (Project Number: FEF. A3.16.020). References: [1] Acikgoz A. and Arabacioglu Tas N., “Some new soft sets and decompositions of some soft continuities”, Annals of Fuzzy Mathematics and Informatics, 9.1(2015), 23-35. [2] Akdag M. and Ozkan A., “Soft α-open sets and soft α-continuous functions”, Abstract and Applied Analysis, http://dx.doi.org/10.1155/2014/891341, (2014). [3] Akdag M. and Ozkan A., “Soft b-open sets and soft b-continuous functions”, Math. Sci., 8.124(2014). [4] Akdag M. and Ozkan A., “On Soft preopen sets and soft pre separation axioms”, Gazi University Journal of Science, 27.4(2014), 1077-1083. [5] Akdag M. and Ozkan A., “On soft β-open sets and soft β-continuous functions”,The Scientific World Journal, http://dx.doi.org/10.1155/2014/843456, (2014). [6] Ali M.I., Feng F., Liu X., Min W.K. and Shabir M., “On some new operations in soft set theory”, Computers and Mathematics with Applications, 57(2009), 1547-1553. ______88

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Periodic Solutions for a Third-Order Delay Differential Equation

Nouioua Farid and A. Ardjuoni

Department of Mathematics, Souk-Ahras University, Souk-Ahras, Algeria [email protected]

Abstract: In this paper, the following third order nonlinear delay differential equation with periodic coefficient

x'''(t)  p(t)x''(t)  q(t)x'(t)  r(t)x(t)  f (t, x(t), x(t (t)))  c(t)x(t (t)) is considered. By employing Green’s function, Krasnoselskii’s …fixed point theorem and the contraction mapping principle, we state and prove the existence and uniqueness of periodic solutions to the third-order delay differential equation. Finally, an example is given to illustrate our results.

A. Ardjouni and A. Djoudi, Existence of periodic soluxions for a second-order non-linear neuxral dixerentixl equation with variable delay, Palesx. J. Math., x(x014),191-197.

Keywords: Fixed point , positive periodic solutions, third-order delay differential equations. References: [1] A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second- order nonlinear neutral differential equation with variable delay,Palesx. J. Math , 3(2014), 191-197 . [2] A. Ardjouni, A. Djoudi, and A. Rezaiguia, Existence of positive periodic solutions for two types of third-order nonlinear neutral differential equation with variable delay, Appl. Math. E-Notes, 14(2014), 86-96 [3] A. Ardjouni and A. Djoudi, Existence of positive periodic solutions for a nonlinear neutral differential equation with variable delay, Appl. Math. E-Notes, 12(2012), 94-101 . [4] A. Ardjouni and A. Djoudi, Existence of periodic solutions for a second- order nonlinear neutral differential equation with functional delay, Electronic. J. Qual. Theory Diff. Eq, 2012, No. 31,9 pp. [5] T. A. Burton, Liapunov functionals, fixed point and stability by Krasnoselskii’s theorem nonlinear stud, 9(2002), 181-190 ______89

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Copula Conditional Tail Expectation for Multivariate Financial Risks

Benatia Fatah and Brahim Brahimi

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria, [email protected]

Abstract: Our goal in this paper is to propose an alternative risk measure which takes into account the fluctuations of losses and possible correlations between random variables. This new notion of risk measures, that we call Copula Conditional Tail Expectation describes the expected amount of risk that can be experienced given that a potential bivariate risk exceeds a bivariate threshold value, and provides an important measure for right-tail risk. An application to real financial data is given.

Keywords: Conditional Tail Expectation; Positive Quadrant Dependence; Copulas; Dependence measure; Risk Management; Market Models. References: [1] Artzner, P. H., Delbaen, F., Eber, J. M., Heath, D., 1997. Thinking Coherently, RISK 10, 68-71. [2] Brahimi, B., Meraghni, D., Necir, A., 2010. Distortion risk measures for sums of dependent losses. J. Afr. Stat. 5, 260-267. [3] Joe, H., 1997. Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman and Hall, London. [4] Nelsen, R. B., 2006. An Introduction to Copulas. second ed. Springer, New York. [5] Sklar, A., 1959. Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8, 229-231.th. Econom. 21(2), 173-183.

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New Properties of Fractional Derivatives Defined Using Mittag Leffler Kernel

Arran Fernandez 1, Dumitru Baleanu 2,3

1 Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, United Kingdom 2 Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey 3 Institute of Space Sciences, Magurele-Bucharest, Romania [email protected]

Abstract: Recent developments in the theory of fractional calculus have included a new definition introduced by Atangana and Baleanu [1] for fractional derivatives in terms of a Mittag-Leffler kernel function; the theory of such derivatives has been further extended in other papers such as [2]. In this presentation we prove a new formula for such fractional derivatives, in terms of an infinite convergent series of standard Riemann-Liouville or Caputo fractional derivatives. This enables us to extend various results from classical calculus, such as the product rule and chain rule, much more easily than otherwise. We also consider the semigroup property for derivatives and integrals and how it applies to these new fractional differintegrals, and we show how to solve certain fractional ODEs using the new definition.

Keywords: Fractional Derivatives, Fractional Integrals, Laplace Transforms, Ordinary Differential Equations References: [1] A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model”, Therm. Sci. 20(2) (2016), 763–769. [2] T. Abdeljawad and D. Baleanu, “Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel”, J. Nonlinear Sci. Appl. 10(3) (2017), 1098-1107.

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Nodal Solutions for Indefinite Robin Problems

Michael Filippakis

Department of Digital Systems Univeristy of Piraeus Piraeus 18536, Greece [email protected]

Abstract: We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite, unbounded potential. The reaction term is a Caratheodory function of arbitrary structure outside an interval [−c, c] (c > 0), odd on [−c, c] and concave near zero. Using a variant of the symmetric mountain pass theorem, together with truncation, perturbation and comparison techniques, we show that the problem has a whole sequence {un}n≥ 1 of distinct nodal solutions converging to zero in C1 (Ω).

Keywords: Indefinite potential, Robin boundary condition, sequence of nodal solution, regularity theory, strong maximum principle

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On Slowly Oscillating Double Sequences

Goksen Findik, Ibrahim Canak, Umit Totur

Department of Mathematics, Ege University, Bornova, Izmir, Turkey Department of Mathematics, Ege University, Bornova, Izmir, Turkey Department of Mathematics, Adnan Menderes University, Aydin, Turkey [email protected]; [email protected] ; [email protected]

Abstract: In this paper, we first examine the relationships between a double sequence and its arithmetic means in different senses (i.e. (C,1,0), (C,0,1) and (C,1,1) means) in terms of slow oscillation in certain senses and investigate some properties of oscillatory behaviors of the difference sequence between the double sequence and its arithmetic means in different senses. Next, we give an alternative proof of the generalized Littlewood Tauberian theorem for Cesaro summability method as an application of the results obtained in the first part.

Keywords: Tauberian conditions and theorems, convergence in Pringsheim's Sense, slow oscillation, double sequences, summability (C, 1, 0), (C, 0, 1) and (C, 1,1) References: [1] R. Schmidt, “Über divergente Folgen und lineare Mittelbildungen”, Math. Z., 22 (1925), 89-152. [2] E. Landau, “Über einen Satz des Herrn Littlewood”, Palermo Rend., 35 (1913), 265-276. [3] T. Vijayaraghavan, “A Tauberian theorem”, J. Lond. Math. Soc., 1 (1926), 113-120. [4] F. Móricz, “Tauberian theorems for Cesaro summable double sequences”, Studia Math., 110 (1) (1994), 83-96.

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A Bayes Minimax Result for a Large Class of Distributions

Dominique Fourdrinier, Fatiha Mezoued and William E. Strawderman

Université de Rouen, France, Ecole Nationale de Statistique et d'Economie Appliquée, Algiers, Algeria Rudgers University, USA [email protected], [email protected], [email protected]

Abstract: We consider Bayesian estimation of the location parameter of a random vector X having a unimodal spherically symmetric density (‖푥−휃‖2) for a spherically symmetric superharmonic prior density ((‖휃‖2)). We consider minimaxity of the Bayes estimator 훿휋(푋) = 푋 + 훻푀(∥푋∥2)/ 푚(∥푋∥2) under quadratic loss, where m is the marginal associated to 푓(∥푥−휃 ∥2) and M is the marginal with respect to 퐹(∥푥−휃 ∥2)=1/2∫푓(푡)∞‖푥−휃‖2푑푡 . In this paper we extend the results given by Fourdrinier, Strawderman in 2008 [3], and Fourdrinier, Mezoued strawderman in 2012 [2].

Keywords: Bayesian estimation, Spherically symmetric density, Minimaxity, Superharmonicity, proper and improper estimators. References: [1]J.O. Berger, “Minimax estimation of location vectors for a wide class of densities”, Annals of Statistics, 3:1318{1328}, (1975). [2]D. Fourdrinier, F. Mezoued, and W. E. Strawderman, “Bayes minimax estimators in the Berger class”, EJS, 6: 783-809, (2012). [3]D. Fourdrinier and W. E. Strawderman, “Generalized bayes minimax estimators of location vector for spherically symmetric distributions”, Journal of Multivariate Analysis, 99(4): 735-750, (2008).

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A Search for Designs with the Same Parameters as 2- (256,64,21) Design with 2-Rank 25

Mustafa Gezek

Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, USA [email protected]

Abstract: Let D be a t − (v, k, λ) design with b blocks. One of the important results in theory of combinatorial designs is that the number of blocks of a design ≥ the number of points, named as the Fisher inequality. This implies that the p- rank of D is not greater than v. The researchers working in this area are interested about the lower bound on the p-ranks of some certain designs and a great amount of research is done on this problem. Let D_1 be a geometric design having as blocks the d -subspaces of P G(n, q) or AG(n, q), and let m be the p- rank of D_1 . Hamada [3] conjectured that if D_2 is a design with the same parameters as D_1 , then the p-rank of D_2 is greater or equal m, and the equality holds if and only if D_2 is isomorphic to D_1 . In [1, 2, 4, 5, 6], there are some proven cases and as well as some counterexamples to the this conjecture. Our work here is using the connection of known 2-(64,16,5) designs with 2-rank 16 to find non-geometric designs with the same parameters as AG 3 (4, 4). References: [1] Clark D., Jungnickel D., Tonchev V.D., Affine geometry designs, polarities, and Hamada’s conjecture, Journal of Combinatorial Theory, Series A 118 231- 239 (2011). [2] Doyen J., Hubaut X., Vandensavel M., Ranks of incidence matrices of Steiner Triple Systems, Math Z., Vol.163 (1978) 251 - 259. [3] Hamada N., On the p-rank of the incidence matrix of a balanced of partially balanced incomplete block design and its applications to error correcting codes, Hiroshima Math J. 3, 153-226 (1973). [4] Hamada N., Ohmori H., On the BIB design having the minimum p-rank, J.Combin Theory Ser. A Vol 18 (1975) 131 - 140. [5] Harada M., Lam C., Tonchev V.D., Symmetric (4,4)-nets and generalized Hadamard matrices over groups of order 4. Des. Codes Cryptogr. 34, 71-87 (2005). [6] Teirlinck T., On the projective and affine hyperplanes, J. Combin. Theory Ser. A, Vol. 28 (1980) 290 - 306.

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훉 Compact and Matrix Mappings on the Space |푨풇|풌

Fadime Gokce, G.Canan Hazar Gulec

Department of Mathematics, Pamukkale University, Denizli, Turkey [email protected], [email protected]

Abstract: In this study, we characterize compact and matrix operator in the θ class (|퐴푓 |푘, |퐵푓| ) for k≥1, and give some topological and algebraic properties θ of the space |퐴푓 |푘, where A and B are factorable matrices, θ = (θ푛) is nonnegative sequence. Also we determine norms and Hausdorff measure of noncompactness of matrix operators in these classes. Thus we extend some recent results of Sarıgöl [8,9] and some well known results.

Keywords: Matrix Transformations; Factorable Matrices; Sequence Spaces; Measure Hausdorff Noncompactness; Norms; Compact Operators. References: [1] Bor, Hüseyin. "On two summability methods." Math. Proc. Cambridge Philos. Soc. 97 (1985), 147-149. [2] Maddox, I.J., Elements of functinal analysis, Cambridge University Press, London, New York, 1970. [3] Malkowsky E., Rakocevic, V., On matrix domains of triangles, Appl. Math. Comp. 189 (2007), 1146-1163. [4] Malkowsky E., Rakocevic, V. An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad.(Beogr) 9 (2000), 143-234. [5] Orhan, C. and Sarıgöl, M.A., On absolute weighted mean summability , Rocky Moun. J. Math. 23 (3) (1993), 1091-1097. [6] Rakocevic, V., Measures of noncompactness and some applications, Filomat 12 (1998), 87-120. [7] Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J. Sci. 42 (2015), 1-8. [8] Sarıgöl, M., A., Characterization of general summability factors and applications, Comp. Math. Appl. 62 (2011), 2665-2670. [9] Sarıgöl, M.A., Matrix transformations on fields of absolute weighted mean summability, Studia Sci. Math. Hungar. 48 (3) (2011), 331-341

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On Some Classes of Fractional Differential Equations of Parabolic Type

Dilovar Guljonov

A.Juraev Insitute of Mathematics, Academy of Sciences of the Republic of Tajikistan Dushanbe, Tajikistan [email protected]

Abstract: The present paper is devoted to studying the equations of parabolic type in which instead of Laplaсe operator considered the differential operator with fractional orders of derivatives by spatial variables. Instead of classical equation of normal diffusion of form 흏풖 = ∆풖 (ퟏ) 흏풕 is consider a fractional equation of isotropic anomaly diffusion 흏풖 = ∆휶/ퟐ풖, ퟏ ≤ 휶 ≤ ퟐ (ퟐ) 흏풕 or an equation of anisotropic anomaly diffusion 흏풖 = 훁 ∙ (퓙ퟏ−휷훁퐮), (ퟑ) 흏풕 푴 ퟏ−휷 where 훁 − gradient operator and operator 퓙푴 [풗] is defined for vector-valued function 풗 and has form

ퟏ−휷 −ퟏ 휷−ퟏ 퓙푴 [풗] = 푭 [ ∫ 풎(풊풎 ∙ 흃) 풎 ∙ 풗̂(흃)푴(풅풎)]. |풎|=ퟏ Here 푴 − probability measure in unit sphere ℝ풏 described an anisotropic −ퟏ 풏 diffusion, 푭 [풗] − an inverse Fourier transform of function 풗 ∈ 푳ퟐ ⊂ ℝ . Equation (1) is correspond to the random walk model and equations (2) and (3) are correspond to the continuous in time random walk and Levi flights models (see, e.g., [1]). Initial-boundary problem for the equations (2) and (3) are invistiges by method of Fourier fractional analysis.

References: [1] V.V. Uchaykin, “Self-similar anomaly diffusion and stable laws”, Uspekhi

Physic.Nauk, 173.8(2003), 847-876.

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Some Results about ΔI-Statistically Pre-Cauchy Sequences with an Orlicz Function

Hafize Gumus, Omer Kisi and Ekrem Savas

Abstract: In this study, we define the concept of I-statistically convergence for difference sequences and we use an Orlicz function to obtain more general results. We also show that an ΔI-statistically convergent sequence with an Orlicz function is ΔI -statistically pre-cauchy.

References: [1] Das, P. and Sava¸s, E., On I-statistically pre-cauchy.sequences, Taiwanese Journal of Mathematics, Vol.18, No.1, 115-126, (2014). [2] Khan, V.A., Ebadullah, K., Ahmad, A., I-Pre-Cauchy sequences and Orlicz functions, Journal of Mathematical Analysis, 3(1), 21-26, (2012). [3] Dutta, A. J. and Tripathy, B.C., Statistically pre-cauchy fuzzy real-valued sequences deÖned by an Orlicz function, Proyecciones Jour. of Mathematics, 33(3), 235-243, (2014).

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A Numerical Analysis of FLMM for Semilinear Time Fractional Schrödinger Equations

Betul Hicdurmaz

Department of Mathematics, Istanbul Medeniyet University, Uskudar, Istanbul, Turkey [email protected]

Abstract: In this study, semi linear time fractional Schrödinger differential equations are solved approximately with three types of difference schemes. Semi-discretization in time variable is provided by an iterated Lubich approximation and two different iterated Fractional linear multi step method (FLMM) methods. Numerical analysis of the obtained theoretical results are presented with a discussion on some particular problems.

Keywords: Semi linear time fractional Schrödinger equation, Fractional linear multi step method, Lubich approximation. References: [1] R. Garrappa, I. Moret, M. Popolizio, “On the time-fractional Schrödinger equation: Theoretical analysis and numerical solution by matrix Mittag-Leffler functions", Computers & Mathematics with Applications, 2016, In press. [2] M. Naber, "Time fractional Schrodinger equation", J. Math. Phys. Vol 45, No. 8 pages 3339-3352, Aug. 2004. [3] A. H. Bhrawy, M. A. Zaky, "An improved collocation method for multi- dimensional space–time variable-order fractional Schrödinger equations", Applied Numerical Mathematics, 111, 2017, 197–218.

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On Modified FLMM Methods for Fractional Population Equations

Betül Hicdurmaz and Emine Can*

Department of Mathematics, Istanbul Medeniyet University *Department of Physics Engineering, Istanbul Medeniyet University Uskudar, Istanbul, Turkey

Abstract: In the present paper, a new modified Fractional Linear Multistep Method (FLMM) is presented for the numerical solution of linear and semi linear differential equations. Constructed methods are implemented on the generalized time fractional-order biological population model (GTFBPM) with one or two dimensions which arise in population dynamics. Selected problems are equations which are applied to population growth in species in biology.

Keywords: Population dynamics, Fractional Linear Multistep Method. References: [1] V. K. Srivastava, S. Kumar, M. K. Awasthi, B. K. Singh, “Two-dimensional time fractional-order biological population model and its analytical solution”, Egyptian Journal of Basic and Applied Sciences, 1 (2014), 71-76. [2] N. V. Mantzaris, P. Daoutidis, F. Srienc, “Numerical solution of multi- variable cell population balance models. II. Spectral methods”, Computers and Chemical Engineering 25 (2001) 1441–1462. [3] S. Allaart-Bruin, J. ter Maten, S. Verduyn Lunel, Modified Extended BDF time-integration methods, applied to circuit equations, RANA Report 02-25, Eindhoven University of Technology, Presented at SCEE-2002, Eindhoven, June 2002.

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Coincidence Best Proximity Points for Geraghty Type Proximal Cyclic Contractions

Azhar Hussain

Department of Mathematics, University of Sargodha Sargodha, Pakistan [email protected]

Abstract: In this talk we introduce the notions of generalized Geraghty proximal cyclic contractions for non-self mapping and obtain coincidence best proximity point theorems in the setting of complete metric space. Some examples are given to show the validity of our results. Our results extended and unify many existing results in the literature.

Keywords: α-Geraghty proximal contraction of first and second kind, α- proximal cyclic contraction, α-proximal admissible maps. References: [1] Abbas, M, Hussain, A, Kummam, P: A Coincidence Best Proximity Point Problem in G-Metric Spaces. Abst. and Appl. Anal. 2015, Article ID 243753, 12 pages (2015). [2] Akbar, A, Gabeleh, M: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 153, 298-305 (2012). [3] Basha, S.S. (2011). Best proximity point theorems generalizing the contraction principle. Nonlinear Anal., 74:5844-5850, (2011). [4] Basha, SS, Shahzad, N, Jeyaraj, R: Best proximity points: approximation and optimization. Optim. Lett. (2011). [5] Shahzad, N, Basha, SS, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148, 69-78 (2011).

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Estimation of a Loss Function for Spherically Symmetric Distribution with Constraints on the Norm

Ouassou Idir

Cadi Ayyad University, National School of Applied Sciences Marrakesh, Morocco [email protected]

Abstract: In this paper we consider the problem of estimating the quadratic loss of point estimators of a location parameter θ for family of symmetric distribution with known scale parameter, when its norm satisfies different constraints and when a residual vector U is available. We compare the robust and non robust estimators and condition on the distribution for the domination of competing estimators are given. In particular we show that it occurs for t-distributions when the dimension of the residual vector is sufficiently large. The main tools in the development are upper and lower bounds on the risk are exact at θ = 0.

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An overview on Fuzzy AHP and Its Priority Derivation

1Iftikhar, 2Musheer Ahmad

1,2Department of Applied Sciences and Humanities Faculty of Engineering and Technology Jamia Millia Islamia, New Delhi-110025, India [email protected], [email protected]

Abstract: Fuzzy AHP is a mathematical technique for multicriteria decision making (MCDM) in fuzzy environments. Priority derivation is one of the pivotal steps in Fuzzy AHP methods. In this work, we are introducing various methods for determining the priority vectors, 푤 = (푤1 , 푤2 … , 푤푛 ) 푇 . Each method is defined by a function, 휏: 푅 푛×푛 → , which synthesize pairwise comparisons into a rating. Several examples are solved for illustrating the working of each method.

Keywords: Fuzzy AHP; priority vectors; pairwise comparison matrices; multicriteria decision making. References: [1] Saaty T.L., Scaling method for priorities in hierarchical structures, J. Math. Psychol. 1977, 15, 3, 234-28. [2] Basak I., Comparison of statistical procedures in analytic hierarchy process using a ranking test, Math. Comp. Model. 1998, 28, 105-118. [3] Budescu D.V., Zwick R., Rapoport A., Comparison of the analytic hierarchy process and the geometric mean procedure for ratio scaling, Appl. Psychol. Meas. 1986, 10, 69-78. 4] Saaty T.L., Vargas L.G. Comparison of eigenvalue, logarithmic least square and least square methods in estimating ratio, J. Math. Model. 1984, 5, 309-324. [5] Koczkodaj W.W., A new definition of consistency of pairwise comparisons, Mathematical and Computer Modeling 1993, 18(7), 79-84.

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Solutions of odod(n)(n1) When n1 Has Three Distinct Odd Primes

Nazli Yildiz Ikikardes, Daeyeoul Kim and Lianrong Ma

Balikesir University, Turkey, National Institute for Mathematical Sciences, South Korea, Tsinghua University, China [email protected], [email protected], [email protected]

Abstract: In this study, we define  (n)  d . We investigate solutions of od d|n,2 |d the equation . We find all solutions of the equation up to n  240 . Also, we obtain the equation has no solution. od(n)  od (n  1)   od (n  2)  1(mod2)

Keywords: Congruence, Divisor Functions, Odd Divisor Functions First author was supported by The Research Fund of Balikesir University, Project No: 2016/44. References: [1] V. Annapurna, “Inequalities for  (n) and ( n ) ”, Math. Mag. 45(1972), 187-190. [2] B. Cho, D. Kim and H. Park, “The multinomial convolution sums of certain divisor functions”, J. Math. Anal. Appl, 448(2017), 1163-1174. [3] P. Erdös, “Some remarks on Eulers  function and some related problems”, Bull. Amer. Math. Soc., 51(1945), 540-544. [4] J. W. L. Glaisher, “On certain sums of products of quantities depending upon the divisors of a number”, Mess. Math., 15(1885), 1-20. [5] R. K. Guy, “Unsolved problems in number theory”, Springer, 2004. [6] D. Kim and A. Bayad, “Convolution identities for twisted Eisenstein Series and twisted divisor functions”, Fixed Point Theory and App., (2013), 2013:81. [7] Y. Li, L. Ma and J. Zhang “Odd solutions of (n) 2n 2 have at least six distinct prime factors”, Publicationes Mathematicae Debrecen, 82(2012), 1-12. [8] K. S. Williams, “Number theory in the spirit of Liouville”, Cambridge University Press, 2011 ______104

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Hypersurfaces of a Kenmotsu Space Form

Mohammad Ilmakchi

Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran [email protected]

Abstract: The purpose of this paper to introduce of hypersurfaces in Kenmotsu space form and study with some special conditions. In general, we have some relations about locally symmetric condition, recurrent and D -recurrent structure Jacobi operator, weakly constant holomorphic curvature and another inequality condition.

Keywords: Kenmotsu space form, locally symmetric, recurrent, Jacobi operator. References: [1] A. Bejancu, CR-submanifolds of Kaeher Manifold $I$, Proc. Amer.Math. Soc. 69, no.1, (1978) , 135-142. [2] A. Bejancu, Geometry of CR-submanifolds, D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, (1986). [3] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, (1976). [4] M. Djoric, M. Okumura, Certain CR submanifolds of maximal CR dimension of complex space forms, Differential Geometry and its Applications, 26(2), 208- 217, (2008). [5] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J.24, (1972) , 93-103.

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On Some Classes of Fractional Integrodifferential Equations in Hilbert Space Mamadsho Ilolov Center of Innovative Development of Science and New Technologies, Academy of Science of Republic of Tajikistan Dushanbe, Tajikistan [email protected]

1 Abstract: Let 푓 ∈ 퐿 (푅+, 퐻), 퐻 − separable Hillbert space and 0 ≤ 훼 < 1. The expression

푡 1 (풥훼푓)(푡) = ∫(푡 − 1)훼−1푓(푠)푑푠, 푡 > 0, 훼 > 0 푡 Γ(훼) 0 0 with 풥푡 푓(푡) = 푓(푡) is called Riemann-Liouville integral of order 훼 of 푓. For 푚−1 푓(푡) ∈ 퐶 (푅+퐻) the Caputo fractional derivalive of order 훼 of 푓 defined by 푚−1 푡푖 푑푖푓(0) 푑푚 ( 퐷훼)(푡) = 퐷푚풥푚−훼 [푓(푡) − ∑ ] (푡), 퐷푚 = . 퐶 푡 푡 푡 푖! 푑푡푘 푡 푑푡푚 푖=0 We are interested in studying the Cauchy problem for fractional integrodifferential equation in 퐻 of the type

푛 푡

훼 −훾푘(푡−푠) ( 퐶퐷푡 푢)(푡) + 퐴푢(푡) + ∑ ∫ 푒 퐴푘푢(푠)푑푠 = 푓(푡), 푘=0 0 푢(0) = 푢0, (1) where 퐴0, 퐴1, … , 퐴푛 –unbounded linear self-adjoint operators with 퐷(퐴푘) ⊃ −1 퐷(퐴0), 0 < 퐴푘 ∈ ℒ(퐻), 훾푘 –positive constants, 0 < 훾1 < ⋯ < 훾푛 < ∞, 푓 = 푓(푡): 푅+ → 퐻 given function, 푢0 ∈ 퐻.

Let 푢(푡) – strong solution of (1). We introduce new desired functions 푢푘(푡), 푘 = 0, … , 푛 with accordance with formulas 푡 1⁄ −훾푘(푡−푠) 2 푢0(푡) = 푢(푡), 푢푘(푡) = ∫ 푒 퐴푘 푢0(푠) 푑푠, 푘 = 1,2, … , 푛 0 and then we come to fractional differential equations ______106

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훼 ̃ ( 푐퐷 푢̃)(푡) + 풜̃0푢̃ = 푓(푡), 푢̃(0) = 퐻̃0, (2) in Hilbert space

푛 퐻̃ = 퐻0 + 퐻̂1, 퐻0 = 퐻, 퐻̂1 = ⨁푘=1퐻푘, 퐻푘 = 퐻, 푘 = 1,2, … , 푛, (3) where 푇 푇 ̃ 푢̃(푡) = (푢0(푡), 푢̂1(푡)) , 푢̂1(푡) = (푢1(푡), … , 푢푛(푡)) , 푓(푡) = (푓(푡), 0̂), 푇 푢̃(0) = (푢0, 0̂) and operator 풜̃0 have a following matrix presentation 1 푇 풜̃ = (퐴 ) = 0, 퐴 = 퐴 , 퐴 = (퐴1/2, … , 퐴1/2) , 0 푖푗 1푖푗 00 0 01 1 푛 1 1 푇 2 2 푛 퐴10 = − (퐴1, … , 퐴푛) , 퐴11 = 푑푖푎푔(훾푘퐼)푘=1.

Integer order of equation (1) was studied in [1]. References: [1] M. Ilolov, Kh. Kuchakshoev. “The classification of abstract integrodifferential equation of higher order”, 7th European Congress of Mathematics, (2016), 483.

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The Generators of Regular, Quasi-regular Representations and Casimir Operator

Yasemin Isik 1, Mehmet Sezgin 2

Department of Mathematics, Trakya University, Edirne, Turkey 1 [email protected] 2 [email protected]

Abstract: Special pseudo-orthogonal group is 푆푂(푝, 푞) = { 퐴 ∈ 푂(푝, 푞) | det 퐴 = 1 } . Dealing with regular and quasi-regular representstion of the group, infinitesimal operators 퐽0, 퐽1, 퐽2 can be found and Casimir operator of the group can be obtained using these operators. In this paper we shall obtain generators of regular, quasi-regular representations and the Casimir operator 퐶 of 푆푂(1,2) group. We also shall analyzed the solutions of 퐶 푓(푥) = 휎(휎 + 1) 푓(푥), where 휎(휎 + 1) is eigenvalue, 푓(푥) is eigenfunction.

Keywords: Lie group, Casimir operator, regular and quasi-regular representation. References: [1] Y. A. Verdiyev, “Harmonic Analysis on Homogeneous Spaces of SO(1,2)”, Hadronic Press. (1988) [2] A. Kirillov, Jr, “Introduction to Lie Groups and Lie Algebras”, Suny at Stony Brook. [3] A. M. Perelomov, V. S. Popov, “Casimir Operator for Semisimple Lie Groups”, Math USSR-Izvestija, 2(1968). [4] R. da Rocha, E. Capelas de Oliveria “The casimir Operator of SO(1,2) and the Pöschl-Teller Potential: an AdS Approach”, Analysis, 5(1985), 301-313. [5] S.A. Mohiuddine and Q.M. Danish Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space”, Rev. Mex. Fis., 51 (1) (2005), 1-4. ______108

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Some Properties of Cartan Null Curves in Semi-Euclidean 4-space with index 2

Esen Iyigün

Department of Mathematics, Uludag University, Görükle, Bursa, Turkey [email protected]

Abstract: In this paper, we give some properties by using Frenet equations which are given in [2] of Cartan null curves in Semi-Euclidean 4-space with index 2. Also, we discuss the conditions for Cartan null curves lying on some subspaces of these space. Finally, we obtain that the image of the curve lying on 3 the pseudohyperbolic space H 1 in the same space.

Keywords: Semi-Euclidean 4-space with index 2, Frenet frame, Cartan null curve. References: [1] B. O'Neill, "Semi-Riemannian geometry with applications to relativity", Academic Press, New York,1983. [2] A. Uçum, O. Keçilioğlu and K. İlarslan, "Generalized pseudo null Bertrand curves in Semi-Euclidean 4-space with index 2", Rend. Circ. Mat. Palermo, DOI 10.1007/s12215-016-0246-x, (2016), 1-14, 4 [3] M.A. Akgün and A.I. Sivridağ, "On the null Cartan curves of R 1", Global Journal of Mathematics, 1. January 26(2015), 41-50. [4] M.A. Akgün and A.I. Sivridağ, "On the characterizations of null Cartan 4 curves in R 1", International Journal of Mathematics, 1.June1(2015), 1-13. [5] K.L. Duggal and D.H. Jin, "Null curves and hypersurfaces of Semi- Riemannian manifolds" World Scientific, London, 2007. [6] M. Petrovic-Torgasev and E. Sucurovic, "Some characterizations of the spacelike, the timelike and the null curves on the pseudohyperbolic space H²0 in 3 E 0”, Kragujevac J. Math., 22(2000), 71-82. 4 [7] Z. Şanlı and Y. Yaylı, "On indicatrices of null Cartan curves in R 1", International Journal of Engineering Research & Technology (IJERT), 10.2(2013), 2567-2570. 4 [8] M. Sakaki, "Null Cartan curves in R 2 ", Toyama Math., 32 (2009), 31-39.

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Proximal Point Algorithms Involving Cesaro Type Mean of Total Asymptotically Nonexpansive Mappings in CAT(0) Spaces

Amna Kalsoom and Hafiz Futhar ud Din

Department of Mathematics& Statistics, International Islamic University, Islamabad, Paksitan [email protected]

Abstract: Fixed point theory in a CAT(0) space was first studied by Kirk [1]. Since then, fixed point theory for various types of mappings in CAT(0) spaces has been investigated rapidly. In 2008, Dhompongsa-Panyanak[2] studied the strong and ∆-convergence for the Mann Iteration process and Ishikawa iteration process for nonexpansive mappings in CAT(0) spaces.

Alber et al. [3] introduced a unified and generalized notion of a class of nonlinear mappings in Banach spaces which can be introduced in CAT(0) spaces. A modified proximal point algorithm involving fixed point of Cesaro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces is proposed. Under suitable conditions, the ∆-convergence and the strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the Cesaro type mean of total asymptotically nonexpansive mapping in CAT(0) space are proved.

Keywords: CAT (0) space, Cesaro type mean, Proximal point algorithm, Total asymptotically nonexpansive mapping. References: [1] W. A. Kirk, “Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003)”, Univ. Sevilla Secr. Publ., Seville, (2003), 195–225. [2] S. Dhompongsa, B. Panyanak, “On ∆-convergence theorems in CAT (0) spaces”, Comput. Math. Appl., 56 (2008), 2572–2579. [3] C.E. Chidume, E.U. Ofoedu, “Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings” J. Math. Anal. Appl., 333(2007), 128-141 ______110

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Semi–Empirical Systematic Development for Photon Induced Nuclear Reaction Cross–Section Calculations

Abdullah Kaplan2, Hasan Ozdogan1,2, Mert Sekerci2, Veli Capali2

1Department of Biophysics, Akdeniz University, Antalya, Turkey 2Department of Physics, Süleyman Demirel University, Isparta, Turkey [email protected]

Abstract: In this study, photon-neutron cross-section calculations have been performed using pre-equilibrium nuclear reaction models. For pre-equilibrium model calculation comparisons, some computation codes have been employed that include theoretical nuclear reaction models. In addition, by obtaining giant dipole resonance parameters in standard Lorentzian mode, semi-empirical cross sections have been calculated. Obtained results have been compared with the experimental values exist in the literature and nuclear reaction models’ computation results.

Keywords: Semi–Empirical Formula, Nuclear Cross–Section, Asymmetry Parameter.

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Positive Solutions for Fractional-Order Boundary Value Problems

Ilkay Yaslan Karaca

Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey [email protected] [email protected]

Abstract: In this study, six functionals fixed point theorem is used to research the existence of positive solutions for fractional-order nonlinear boundary value problems. As an applications, examples are presented to illustrate the main results.

Keywords: Four functionals fixed point theorems, impulsive dynamic equations, positive solutions, boundary value problems, time scale. References: [1] Avery, R. Henderson, J. and O’Regan, D., 2008, Six functional fixed point theorem, Communications in Applied Mathematics, 12(1):69-81p. [2] Dalir, M. and Bashour, M., 2010, Applications of fractional calculus, Mathematical Sciences, Vol. 4, 1021-1032. [3] Guo D., Lakshmikkantham V., Ames W. F., Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. [4] Kilbas, A.A. Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, 69- 79p. [5] Liu, X., Jia,M., Ge, W., 2013, Multiple Solutions of a p- Laplacian Model Involving a Fractional Derivatives, Advances in Difference Equations.

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The Existence of Positive Solutions of Boundary Value Problems with P-Laplacian on the Half-Line

Ilkay Yaslan Karaca and Aycan Sinanoglu

Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey [email protected] [email protected]

Abstract: In this study, four functionals fixed point theorem is used to investigate the existence of positive solutions for second-order time-scale boundary value problem of dynamic equations on the half-line.

Keywords: Four functionals fixed point theorems, impulsive dynamic equation, positive solutions, boundary value problems, time scale. References: [1] G. Chai, Existence of positive solutions of boundary value problem for second-order functional differential equations on infinite intervals, Fixed Point Theory, 13 (2012), 423-437. [2] X. Chen, X. Zhang, Existence of positive solutions for nonlinear systems of second-order differential equations with inregral boundary conditions on an infinite interval in Banach Space, Electron. J. Differential Equations, 2011 (2011), no. 154 19 pp. [3] X. Chen, X. Zhang, Existence of positive solutions for singular impulsive differential equations with integral boundary conditions on an infinite interval in Banach Space, Electron. J. Qual.Differ. Equ., 2011 (2011), no. 28 18 pp. [4] Y. Guo, C. Yu, J. Wang, Existence of three positive solutions for m-point boundary value problem in infinite intervals , Nonlinear Anal., 71 (2009),717- 722. [5] Z. Hao, L. Ma, Existence of positive solutions for multi-point boundary value problem on infinite intervals in Banach Space, Abstr. Appl. Anal., 2012 (2012), Art. ID 107276, 18pp. [6] I. Y. Karaca, F. Tokmak, Existence of three positive solutions for m-point time scale boundary value problem on infinite intervals, Dynam. Systems Appl., 20 (2011), 355-367. [7] X. Zhao, W. Ge, Multiple positive solutions for time scale boundary value problem on infinite interval, Acta Appl. Math., 106 (2009), 265-273. [8] X. Zhao, W. Ge, Unbounded positive solutions for m-point time-scale boundary value problem on infinite intervals, J. Appl. Math. Comput., 33 (2010), 103-123. ______113

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The Dimension of Digital Khalimsky Manifolds

Ismet Karaca1, Gokhan Temizel2

1Ege University Faculty of Science, Department of Mathematics, Izmir,Turkey [email protected] 2Ege University Graduate School of Natural and Applied Sciences, Izmir, Turkey [email protected]

Abstract: The digital counterpart of Euclidean topology which is defined on the real line, has been studied by Efim Khalimsky. Khalimsky has defined this topology on the integers and it is called Khalimsky’s topology. Digital Khalimsky manifolds, digital manifolds with respect to Khalimsky topology, is a digital response of real manifolds. These manifolds are used in order to identifying to digital counterpart of the images on the Euclidean geometry and they play an important role in image processing and computer graphics. In this poster, we will explain the dimension of digital Khalimsky manifolds.

Keywords: Khalimsky Topology, Digital Manifold, Join Operator, Topological Embedding References: [1] L. Boxer, “A classical construction for the digital fundamental group”, Journal of Mathematical Imaging and Vision, 10:51-62, 1999. [2] E. Melin, “How the find a Khalimsky-continuous approximation of a real- valued function”, IWCIA 2004, LNSC 3322:351-365, 2004. [3] E. Melin, “Extension of continuous function in digital spaces with the Khalimsky topology”, Topology and Its Applications, 153:52-65, 2005. [4] E. Melin, “Continuous Extension in topological digital spaces”, Applied General Topology, 9:51-61, 2008. [5] E. Melin, “Digital Khalimsky Manifolds”, Journal of Mathematical Imaging and Vision, 33:267-280 [6]L.W. Tu, “An introduction to manifolds”, Springer Science+ Business Media, LLC, 2008.

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Some Properties of Persistent Homology Groups

Ismet Karaca and Hatice Sevde Denizalti

Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected]; [email protected]

Abstract: Simplicial homology is a corner stone of algebraic topology in terms of extracting topological features of spaces and persistent homology is occured as an extension to simplicial homology. Persistence is investigated in [6] and then generalized in [4]. Persistent homology is generally used for solving the problem of unveiling global topological informations obtained from a sample of a high-dimensional data sets of the space. To find the persistent homology of a space, the space must be represented as a simplicial complex filtrations. In this poster, we give some information about history of persistent homology and offer its fundamental notions.

Keywords: Simplicial homology, simplicial complex filtration, persistent homology. References: [1] A. Hatcher, “Algebraic Topology”, cambridge University Press, (2002). [2] G. Carlsson, T. Ishkhanov, V. de Silva and A. Zomorodian “On the local behaviour of spaces of natural images”, International Journal of Computer Vision, 76.1(2008), 1-12. [3] H. Edelsbrunner and E. P. Mcke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13.1(1994), 43-72. [4] H. Edelsbrunner, D. Letscher and A. Zomorodian “Topological persistence and simplification”, Discrete Computational Geometry, 33(2005), 249-274. [5] R. Ghrist, “Barcodes: the persistent topology of data”, Bulletin of the American Mathematical Society, 45.1(2008), 61-75. [6] A. Zomorodian and G. Carlsson, “Computing persistent homology”, Discrete Computational Geometry, 28.4(2002), 511-533

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On Digital Cohomology Groups

Ismet Karaca1, Ozgur Ege2

1Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected] 2Department of Mathematics, Manisa Celal Bayar University, Yunusemre, Manisa, Turkey [email protected]

Abstract: Digital topology, introduced by Rosenfeld [8], is an area of great theoretical interest having the additional bonus of significant applications in imaging science and related areas. It continues to rise in many fields of science and engineering such as mathematics, image processing, biology, information system and computer science with a great number of applications. Important developments have been made in this area after the works of Boxer[1,2]. Some results on digital simplicial homology groups were introduced in [4] and [5]. In [6], simplicial cohomology theory is given for digital images. Digital cohomology operations were introduced in [7]. In this paper, we compute simplicial cohomology groups of connected sum of certain minimal simple surfaces by using the Universal Coefficient Theorem for digital cohomology groups. We also prove some theorems related to degree properties of a map on digital spheres.

Keywords: Digital image, Universal Coefficient Theorem, digital cohomology group. References: [1] L. Boxer, “Digital continuous functions”, Pattern Recognition Letters, 15(1994), 833-839. [2] L. Boxer, “A classical construction for the digital fundamental group”, Journal of Mathematical Imaging and Vision, 10(1999), 51-62. [3] L. Boxer, I. Karaca and A. Oztel, “Topological invariants in digital images”, Journal of Mathematical Sciences: Advances and Applications, 11.2(2011), 109- 140.

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[4] O. Ege and I. Karaca, “Some results on simplicial homology groups of 2D digital images”, International Journal of Information and Computer Science, 1.8(2012), 198-203. [5] O. Ege and I. Karaca, “Fundamental properties of simplicial homology groups for digital images”, American Journal of Computer Technology and Application, 1.2(2013), 25-42. [6] O. Ege and I. Karaca, “Cohomology theory for digital images”, Romanian Journal of Information Science and Technology, 16.1(2013), 10-28. [7] O. Ege and I. Karaca, “Digital cohomology operations”, Applied Mathematics and Information Sciences, 9.4(2015), 1953-1960. [8] A. Rosenfeld, “Digital topology”, American Mathematical Monthly, 86(1979), 76-87.

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Some Common Fixed Point Theorems on Complex Valued Gb-Metric Spaces

Ismet Karaca1, Ozgur Ege2

1Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected] 2Department of Mathematics, Manisa Celal Bayar University, Yunusemre, Manisa, Turkey [email protected]

Abstract: Fixed point theory is very active area with various applications in mathematics, biology, image processing [2] and computer science. In this theory, the Banach contraction principle plays a key role to solve many problems. After the Banach’s work [1], researchers have obtained important results in various metric spaces. The notion of complex valued Gb-metric space was introduced in [3]. Then some new fixed point results have been given by several authors [4,5] in this space. The concept of the common fixed point of mappings satisfying contractive type condition has generally been used to prove existence problems. In this study, we prove some common fixed point theorems in complex valued Gb-metric spaces.

Keywords: Fixed point, common fixed point theorem. References: [1] S. Banach, “Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales”, Fundamenta Mathematicae, 3(1922), 133- 181. [2] O. Ege and I. Karaca, “Banach fixed point theorem for digital images”, Journal of Nonlinear Science and Applications, 8.6(2015), 1014-1021. [3] O. Ege, “Complex valued Gb--metric spaces”, Journal of Computational Analysis and Applications, 21.2(2016), 363-368.[4] O. Ege, “Some fixed point theorems in complex valued Gb--metric spaces”, to appear in Journal of Nonlinear and Convex Analysis, (2017). [5] A.H. Ansari, O. Ege and S. Radenovic, “Some fixed point results on complex valued Gb-metric spaces”, to appear in Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Seria A. Matematicas, Doi: 10.1007/s13398-017- 0391-x, (2017), 1-10

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On Some Deddens Subspaces of Banach Algebras

Mubariz Karaev 1, Mehmet Gurdal2, Havva Tilki2

1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey [email protected]; [email protected]; [email protected]

Abstract: Let A be a Banach algebra with a unit e , and let a A be an invertible element. We define the following algebra: loc n n x Ba : x A : a xa  cxn for some x 0 and cx  0. In this article we study some properties of this algebra; in particular, we prove loc that Be p  x A : pxe  p  0, where p is an idempotent in A . We also investigate the following Deddens subspace. Let a,b A be two elements. Fix any number , 0  1, and consider the following subspace of A :

 n n  Da,b : x  A : a xb  On  as n  .

  Here we study some properties of the subspaces Da,b and Db,a .

Keywords: Banach algebra, Deddens subspace, Deddens algebra, Idempotent element, Nilpotent element Acknowledgement. This work is supported by Suleyman Demirel University with Project 4799-YL1-16. References: [1] W.Arveson, Interpolation problems in nest algebras, J.Funct. Anal., 20(1975), 208-233. [2] J.A.Deddens, Another description of nest algebras, Lecture Notes in Math., 693(1978), 77-86. [3] D.Drissi and M.Mbekhta, Operators with bounded conjugation orbits, Proc. Amer. Math. Soc., 128(2000), 2687-2691. [4] J.A.Erdos, Unitary invariants for nests, Pacific J. Math., 23(1967), 229-256. [5] M.Gürdal, Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra, Expo. Math., 27(2009), 153-160. ______119

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Duhamel Operator and Existence of Invariant Subspace

Mubariz Karaev 1, Mehmet Gurdal2, Mualla Birgul Huban2

1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey [email protected]; [email protected]; [email protected]

Abstract: In this work, we gives some sufficient conditions in terms of reproducing kernels and Duhamel operators for the existence of nontrivial invariant subspace in Hardy-Hilbert Space.

Keywords: Reproducing kernel, Berezin symbol, Invariant subspace, Duhamel operator, Hardy space Acknowledgement This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) with Project 115F265. References: [1] P. Ahern, M. Flores and W. Rudin, “An invariant volume-mean-value property”, J. Funct. Anal., 111 (2) (1993), 380-397. [2] M. Engliš, “Functions invariant under the Berezin transform”, J. Funct. Anal., 121 (1) (1994), 233-254. [3] M. Lacruz, “Invariant subspaces and Deddens algebras”, Expo. Math., (33) (1) (2015), 116-120. [4] A. Lambert, “Hyperinvariant subspaces and extended eigenvalues”, New York J. Math., 10 (2004), 83-88. [5] M.M. Malamud, “Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators”, Oper. Theory Adv. Appl., 102 (1998), 143-167. [6] N. M. Wigley, “The Duhamel product of analytic functions”, Duke Math. J., 41 (1974), 211-217. [7] N.M. Wigley, “A Banach algebra structure for H p ”, Canad. Math. Bull., 18 (4) (1975), 597-603. [8] K. Zhu, “Operator Theory in Function Spaces”, Marcel Dekker, New York, 1990.

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On Extended Eigenvalues and Extended Eigenvectors of Toeplitz Operators

Mubariz Karaev 1, Mehmet Gurdal2, Mualla Birgul Huban2

1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey [email protected]; [email protected]; [email protected]

Abstract: In this paper, we consider the operators from Englis algebras, in particular, the Toeplitz operators on the Hardy space, and give some results on the commutant, extended eigenvalues and extended eigenvectors of operators.

Keywords: English algebra, Hardy space, Toeplitz operator, extended eigenvalue Acknowledgement: This work is supported by TUBA through Young Scientist Award Program (TUBA-GEBIP/2015). References: [1] H. Alkanjo, “On extended eigenvalues and extended eigenvectors of truncated shift”, Concrete Operators, 1 (2013), 19-27. [2] A. Biswas, A. Lambert and S. Petrovic, “Extended eigenvalues and the Volterra operator”, Glasgow Math. J., 44 (2002), 521-534. [3] M. Gürdal, “On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra”, Appl. Math. Lett., 22 (2009), 1727-1729. [4] M. T. Karaev, “On extended eigenvalues and extended eigenvectors of some operator classes”, Proc. Amer. Math. Soc., 134 (2006), 2383-2392.

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A Generalization on the Incidence Energy and Laplacian- Energy-Like Invariant

Ezgi Kaya and A. Dilek Maden

Department of Mathematics-Computer, Igdir University, Igdir, Turkey [email protected] Department of Mathematics, Selcuk University, Konya, Turkey [email protected]

Abstract: For a graph G and a real number α, the graph invariant 푠훼(퐺) is the sum of the powers of signless Laplacian eigenvalues and 휎훼(퐺) is equal to the sum of powers of Laplacian eigenvalues of G. In this study, considering these sum for some special cases of α, we give some bounds for incidence energy and Laplacian-energy-like invariant of graphs.

Keywords: Laplacian matrix, signless Laplacian matrix, Incidence energy, Laplacian-Energy-Like invariant References: [1] S. Akbari, E. Ghorbani, J. H. Koolen and M. R. Oboudi, “On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs”, Electron. J. Combin. 17 (2010), R115. [2] S. Furuichi, “On refined Young inequalities and reverse inequalities”, J. Math. Inequal., 5(2011), 21-31. [3] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz 103 (1978), 1-22. [4] I. Gutman, D. Kiani, M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009), 573-580. [5] M. Jooyandeh, D. Kiani, M. Mirzakhah, “Incidence energy of a graph”, MATCH Commun. Math. Comput. Chem. 62 (2009), 561-572. [6] H. Kober, “On the arithmetic and geometric means and the Hölder inequality”, Proc. Amer. Math. Soc., 59(1958), 452-459. [7] J. Liu, B. Liu, A Laplacian-energy like invariant of a graph, MATCH Commun. Math. Comput. Chem. 59 (2008), 355-372. [8] B. Liu, Y. Huang, Z. You, A survey on the Laplacian–energy like invariant, MATCH Commun. Math. Comput. Chem. 66 (2011), 713–730. [9] B. Zhou, “On sum of powers of the Laplacian eigenvalues of graphs”, Linear Algebra Appl. 429(2008), 2239–2246. ______122

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A Data Mining Approach: Application to the Extraction of the Characteristics of IARD Products in the Insurance Sector

Sadi Khadidja1,2, Lounici Mosbah Nora1,2

1. Higher National School of Statistics and Applied Economics 2. Laboratory of Applied Statistics ENSSEA, University of Kolea, Algeriers, Algeria [email protected] , [email protected]

Abstract: In this study, we are interested in IARD property and Multi-risk insurance products, which covers fire, accidents and various risks [1] of an Algerian insurance company. We want to know the variables that best characterize each of the two products. To solve this problem, we have combined three methods borrowed from Data Mining and decisional statistics [2][3]. The use of data mining tools to realize a classification and to bring out the most informative variables is usual tasks of this discipline. The aim of this work is to extract knowledge from an actuarial database by adopting an approach to data mining [4] and to compare the results obtained by querying tools and techniques, implemented in data mining software.

Keywords: Insurance IARD, data mining approach, classification, machine Learning References : [1] F. Noël. La gestion des sinistres IARD incendies et risques divers, édition séfi, Canada 2014. [2] S. Tuffery. Data mining et statistique décisionnelle 4ième ed, Broché – 21 août 2012. [3] C. Wesphal & T. Blaxton. "Data Mining Solutions ", John Wiley, New York, 1998. [4] M. Boullé. Recherche d’une représentation des données efficace pour la fouille des grandes bases de données. Ph. D. thesis, ENST. 2007.

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The Le Corbusier Approach in the Relationship between Architecture and Mathematics

Murat Kilic and Melih Kurnali

Faculty of Fine Arts, Deparment of Interior Architechure & Environment Design, Kırıkkale University, Kırıkkale, Turkey [email protected]

Abstract: In defining the relationship between architecture and mathematics, dealing with and evaluating different points of view is important and necessary in terms of both creating new horizons and a new perception for these two disciplines. In general, architecture and mathematics are dealt with according to their technical characteristics. Architecture makes use of mathematics as a technique and a tool. However, these two disciplines should be used together to create art and allow art reach top level aesthetic values as well. In the period of Modernism, Le Corbusier who has been in search of this purpose and mathematical systems which could meet all conditions for architecture is in fact an important architect who has internalized mathematics to reach artistic aesthetics. In Le Corbusier’s and contemporary mathematicians’ point of view on mathematics, it can be seen that there is a deep opposition. This is caused by the contradictive comparisons made between scientistd and artists. Although architecture is regarded as an art, it is actually a discipline which is formed by the perfect synthesis of science and art [1]. Le Corbusier can be considered as being closer to the artistic side of this synthesis. Although his modular system seems mathematical, or scientific, we can say that his artistic side gains more importance due to the fact that he has not been educated on building construction techniques and the technical complaints he has received about his structures from his clients. In particular, the Ronchamp Chapel embodies a sculptural artistry and he is firstly a painter. Although Le Corbusier’s architectural and mathematical approach has been criticized, it is apparent that he had an intellectual foundation which gave inspiration. So much so that, the proportional system called modulor which he developed for architecture has inspired the composition of a piece

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called “Metastassis” [2]. Le Corbusier is at a critical point in the art, mathematics and architecture triangle and this study is in search of knowledge about Le Corbusier and his mathematical understanding to allow these disciplines to be evaluated with new approaches.

Keywords: Architecture, Mathematics, Le Corbusier, Modulor. References: [1] J. Loach, “Le Corbusier and the Creative Use of Mathematics”, The British Society for the History of Science, (1998), 185-215. [2] Le Corbusier. “Modulor 2”, Yem Press (2011), 328-329.

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The Absolute Möbius Divisor Function and Euler   function

Daeyeoul Kim1, Umit Sarp2 and Sebahattin Ikikardes2

1National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, South Korea [email protected] 2Department of Mathematics, Balikesir University, 10100 Balikesir, Turkey [email protected] [email protected]

Abstract: In this study,we introduce the absolute Mobius divisor function Un(). Also we investigate the sequences Un , which concerns the iteration of  m  m the absolute Mobius divisor function . According to some numerical computational evidence, we consider integer pairs nn;1  satisfying; n  n 1  U n  U n  1 .

Furthermore, we give some examples and proofs for our results.

Keywords: Mobius divisor function, Fermat pirmes, Euler function, n-gonal number. References: [1] A. Bayad and D. Kim, Polygon Numbers Associated with the Sum of Odd Divisors Function, to appear of Exp. Math. http://dx.doi.org/10.1080/10586458.2016.1162231. [2] V. Annapurna, Inequalities for (n) and '(n), Math. Mag. 45 (1972), 187-190. [3] L. E. Dickson, History of the theory of numbers. Vol I: Divisibility and Primality. Chelsea Publishing Co., New York 1966. [4] P. Erdös, Some remarks on Eulers   function and some related problems, Bull. Amer. Math. Soc. 51 (1945), 540-544. [5] R. K. Guy, Unsolved Problems in Number Theory, Springer, 2004.

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S-Generalized Lauricella’s Hypergeometric Functions

I. Onur Kiymaz, M. Baki Yagbasan, Aysegul Cetinkaya

Department of Mathematics, Ahi Evran University, Kırşehir, Turkey [email protected]

Abstract: In this study, we introduced new generalizations of Lauricella’s hypergeometric functions by using S- generalized beta function. Furthermore, we investigated some of their properties such as integral representations and transformation formulas.

Keywords: S-Generalized Beta function, Lauricella’s hypergeometric functions. Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.D1.16.001 References: [1]Bailey W.N., “Generalized Hypergeometric Series”, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 32, Cambridge University Press, Cambridge, (1935). [2]Chaudhry M. A., Qadir A., Rafique M., Zubair S. M., “Extension of Euler's beta function”, J. Comput.Appl. Math., 78, (1997): 19-32. [3]Hasanov, A., Srivastava, H. M., “Some decomposition formulas associated with the Lauricella function and other multiple hypergeometric functions”, Appl. Math. Lett., 19.2, (2006): 113-121. [4]Luo, Min-Jie, Milovanovic, G. V., Agarwal, P., “Some results on the extended beta and extended hypergeometric functions”, Applied Mathematics and Computation, 248, (2014): 631-651. [5]Padmanabham, P. A., Srivastava, H. M., “Summation formulas associated with the Lauricella function Appl. Math. Lett., 13.1, (2000): 65-70. [6] Srivastava H. M., Karlsson P. W., “Multiple Gaussian Hypergeometric Series”, Ellis Horwood Limited, (1985). [7] Srivastava, H. M., Agarwal, P., Jain, S., “Generating functions for the generalized Gauss hypergeometric functions”, Applied Mathematics and Computation, 247, (2014): 348-352. [8]Srivastava, H. M., Jain, R., Bansal, M. K., “A Study of the S-Generalized Gauss Hypergeometric Function and Its Associated Integral Transforms”,Turkish Journal of Analysis and Number Theory, 3.5, (2015): 116-119. ______127

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Some Remarks on Fuzzy Anti-Normed Spaces

Ljubiša D.R. Kočinac

Faculty of Sciences and Mathematics, University of Niš, Serbia [email protected]

Abstract: Let E be a real linear space with a fuzzy anti-norm with respect to a t-conorm. We discuss some (statistical) convergence and covering properties of such spaces. We also consider norms on E generated by .

Keywords: Statistical convergence, fuzzy anti-normed space, boundedness.

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On Asymptotically f-Statistical Equivalent Sequences

Sukran Konca and Mehmet Kucukaslan

Department of Mathematics, Bitlis Eren University, 13000, Bitlis, Turkey Department of Mathematics, Mersin University, 33343, Mersin, Turkey [email protected]; [email protected]

Abstract: By using modulus functions, we have obtained a generalization of statistical convergence of asymptotically equivalent sequences, a new non-matrix convergence method, which is intermediate between the ordinary convergence and the statistical convergence. Further, we have examined some inclusion relations related to asymptotically f-statistical equivalence of real sequences in the light of a partial order.

Keywords: Statistical Convergence, strong Cesaro summability; f- asymptotically equivalent sequences. References: [1] M. Marouf, “Asymptotic equivalence and summability”, Internat. J. Math. Math. Sci. 16 (4) (1993), 755-762. [2] R. F. Patterson, “On asymptotically statistically equivalent sequences”, Demonstratio Math, 36 (1) (2003), 149-153. [3] A. Aizpuru, M. C. Listan-Garcia and F. Rambla-Barreno, “Density by moduli and statistical convergence”, Quaestiones Mathematicae., 37 (4) (2014), 525–- 530. [4] V. K. Bhardwaj, S. Dhawan, and S. Gupta, “Density by moduli and statistical boundedness”, Abst. Appl. Analysis, 2016(2016), 6 pages. [5] H. Nakano, “Concave modulars”, J. Math. Soc. Japan, 5(1953), 29–-49.

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Some Permanents of Hessenberg Matrices

Sibel Koparal, Nese Omur and Cemile Duygu Sener

Department of Mathematics, Kocaeli University, Kocaeli, Turkey [email protected] ,[email protected], [email protected]

Abstract: In this study, we define a sequence {푅푛(푎, 푏, 푐)} and represent relationships between this sequence and permanents of certain matrices. Some special cases for permanents are given.

Keywords: Hessenberg matrix, Relation recurrence, Permanent. References: [1]R.A. Brualdi and P.M. Gibson, “Convex Polyhedra of Doubly Stochastic Matrices: Applications of the permanents Function”, J.Combin Theory A, 22(1977), 194-230. [2]E. Kılıç and D.Taşcı, “On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers”, Rocky Mountain Journal of Mathematics, 37.6(2007), 203-219. [3]E. Kılıç, Tribonacci sequences with certain indices and their sums”, Ars Combinatoria, 86(2008), 31-40. [4]E.Kılıç and D. Taşcı, “Negatively subscripted Fibonacci and Lucas numbers and their complex factorizations”, Ars Combinatoria, 96(2010), 275-288. [5] H. Mine, “Permanents of (0,1)-circulants”, Canad. Math. Bull., 7(1964), 253- 263. [6]J.L. Ramnrez, “Hessenberh matrices and the generalized Fibonacci-Narayana sequence”, Filomat, 29.7(2015), 1557-1563.

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Best Proximity Points for Generalized Geraghty Proximal Contraction Mapping in Elliptic Valued Metric Space

Isil Arda Kosal1, Hidayet Huda Kosal2, Mahpeyker Ozturk3

1,2,3 Department of Mathematics, Sakarya Universitesi, Sakarya, TURKEY [email protected], [email protected], [email protected]

Abstract: In this study, we introduce the concept of best proximity points for the generalized Geraghty proximal contraction mappings between two subsets of elliptic valued metric space. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known fixed and best proximity point results from the literature.

Keywords: Contraction mapping, Best proximity, Ellipitc valued metric spaces. References: [1] A. Harkin and J. Harkin, “Geometry of generalized complex numbers”, Mathematics Magazine, 77.2(2004), 118–129. [2] A. Azam, B. Fisher and M Khan, “Common fixed point theorems in complex valued metrik spaces”, Numerical Functional Analysis and Optimization. An International Journal, 32.3(2011), 243–253. [3] M. Ozturk and N. Kaplan,” Common fixed points of f-contraction mappings in complex valued metric spaces”, Mathematical Sciences, 8.129(2014). [4] M. Ozturk, “Common fixed points theorems satisfying contractive type conditions in complex valued metrik spaces”, Abstract and Applied Analysis, 7(2014). [5] J. Hamzehnejadi and R. Lashkaripour, “Best proximity points for generalized Geraghty proximal contraction mapping and its applications”, Fixed Point Theory and Applications, 72.1(2016), 1-13. [6] E. Karapinar, “A Discussion on Geraghty contraction type mapping”, Filomat, 28.4(2014), 761-766. [7] N. Bilgili, E. Karapinar and K. Sadarangani, “A generalization for the best proximity point of Geraghty contraction”, Journal of Inequalities and Applications, 286(2013), 1-9

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Characterizations of  pq,,  - Convex Sequences

Xhevat Z. Krasniqi

Faculty of Education, University of Prishtina “Hasan Prishtina”, 10000 Prishtina, Republic of Kosovo [email protected]

Abstract: The class of convex sequences has important applications in several branches of mathematics as well as their generalizations. In this paper, we have introduced a new class of convex sequences, the so-called  pq,,  -convex sequences. Moreover, the characterizations of such sequences belonging to this class has been shown.

Keywords: Sequence, Convexity,$(p;r)-$monotonicity, $(p,q;r)-$convexity, $p$-starshaped sequence, $(p,q;\alpha )$-convexity. References: [1] H. Bor, A new application of convex sequences. J. Class. Anal. 1 (2012), no. 1, 31--34. [2] H. Bor, Xh. Z. Krasniqi, A note on absolute Cesàro summability factors. Adv. Pure Appl. Math. 3 (2012), no. 3, 259--264. [3] Xh. Z. Krasniqi, Some properties of $(p,q;r)$-convex sequences. Appl. Math. E-Notes 15 (2015), 38--45. [4] Xh. Z. Krasniqi, Characterizations of $(p,\alpha )$-convex sequences. Appl. Math. E-Notes, accepted. [5] L. M. Koci\'c, I. Z. Milovanovi\'c, A property of $(p,q)$-convex sequences, Period. Math. Hungar. Vol. 17 (1) (1986), pp. 25--26. [6] I. B. Lackovi\'c, M. R. Jovanovi\'c, On a class of real sequences which satisfy a difference inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. No. 678 (1980), 99--104. [7] B. Makarov, A. Podkorytov, Real Analysis: Measures, Integrals and Applications, Springer--Verlag London, 2013. [8] J. E. Pe\u{c}ari\'c, On some inequalities for convex sequences. Publ. Inst. Math. (Beograd) (N.S.) 33(47) (1983), 173--178. [9] F. Qi, B.-N. Guo, Monotonicity of sequences involving convex function and sequence. Math. Inequal. Appl. 9 (2006), no. 2, 247--254. ______132

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A Note on the Numbers Yn(Λ) and the Polynomials Yn(X;Λ) and Their Generating Functions

Irem Kucukoglu1,a and Yilmaz Simsek2,b

1Department of Software Engineering, Faculty of Engineering and Architecture, Antalya AKEV University Antalya, Turkey. 2Department of Mathematics, Faculty of Science University of Akdeniz TR- 07058, Antalya, Turkey. [email protected], [email protected]

Abstract: In this talk, we study on the numbers Yn(λ) and the polynomials Yn(x;λ) which have been recently introduced by Simsek in [7]. We give partial derivative formulas including the generating functions of these numbers and polynomials. By using these formulas and functional equations for the generating functions, we give some recurrrence relations and identities for these numbers and polynomials related to some special numbers such as the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers.

Keywords: Generating functions, Functional equations, Partial differential equations, Daehee numbers, Changhee numbers, Stirling numbers. References: [1] T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), pp. 161- 167. [2] N. P. Cakic and G. V. Milovanovic, On generalized Stirling numbers and polynomials, Mathematica Balkanica 2004; 18: 241-248. [3] G. B. Djordjevic and G. V. Milovanovic, Special classes of polynomials, University of Nis, Faculty of Technology Leskovac, 2014. [4] D. S. Kim, T. Kim and J. Seo, A note on Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 7 (2013), 993-1003. [5] D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. (Ruse) 7 (120) (2013), 5969-5976. [6] T. Kim, D. V. Dolgy, D. S. Kim and J. J. Seo, Differential equations for Changhee polynomials and their applications, J. Nonlinear Sci. Appl. 9 (2016), 2857-2864. [7] Y. Simsek, Generating Functions for family of (q-) generalized Apostol-type Numbers And Polynomials: Analysis of the p-adic q-integrals, (preprint). ______133

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Modelling Worldwide CO2 Emissions and Oil Consumption based on the 푳ퟏ, 푳ퟐ 퐚퐧퐝 푳∞-norm Regressions 1Pranesh Kumar and 2Mohamadtaghi Rahimi

1Department of Mathematics and Statistics, University Northern British Columbia, Prince George, BC V2N 4Z9, Canada [email protected] 2Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran. [email protected]

Abstract. Regression models are commonly used to model the relationship among random variables in statistical data analysis with an aim to use them to make future predictions. The regression model estimated by the method of least squares is 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 least squares error linear regression models may fail to provide best results in non-Gaussian situations especially when the errors follow distributions with fat tails and error terms possess a finite variance. Historically, in the eighteenth century, mathematicians, notably, Mayer, Boscovich, Laplace, Legendre, Simpson, Gauss, and Euler did pioneering work to develop procedures for fitting functions. The most significant research was the development of methods of least absolute deviations, least squares deviations and minimax absolute deviations. The theory of function fitting methods of least squares is credited to the published works of Legendre (1805) and Gauss (1809). Alternatively, we may use 퐿푝-norm to estimate the linear regression model parameters to search for an appropriate regression model. In this paper, we provide a perspective on the 퐿1, 퐿2 and 퐿∞-norm based regression models. Worldwide oil consumption and CO2 emissions are an on-going environmental worldwide concern and often result in both immediate and long-term environmental damage. We discuss the results on modeling of worldwide CO2 emissions and oil production from these models. We conclude the paper with relevant concerns in the applications of regression-model methodology and further research of interest.

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Some Symmerty Identities for Modified Degenerate Apostol- Bernoulli and Modified Degenerate Apostol-Euler Polynomials Related to Multiplier Sums

Burak Kurt

Department of Mathematics, Faculty of Educations University of Akdeniz TR-07058 Antalya, Turkey [email protected]

Abstract: Dogly et. al. in [2] defined and investigated modified degenerate Bernoulli polynomials and numbers. They proved some identities and recurrence relations for these polynomials. H.-In Known et. al. in [3] defined the modified degenerate Euler polynomials and numbers. They investigated some properties for these polynomials. Also, they gave some identities arising from the fermonic p-adic integral on ℤ _{p}. In this article, we define the modified degenerate Apostol-Bernoulli polynomials and numbers. Also, we give modified degenerate Euler-polynomials and numbers. We give some symmetric relations between for these polynomials. Also, we generalize to Srivastava-Pintér summation formulea for the modified degenerate Apostol-Bernoulli polynomials and the modified degenerate Apostol- Euler polynomials

Keywords: Bernoulli polynomials and numbers, Euler polynomials and numbers, Apostol-Bernoulli polynomials and numbers, Apostol-Euler polynomials and numbers, Degenerate Bernoulli polynomials and numbers, Degenerate Euler polynomials and numbers, Modified Degenerate Apostol- Bernoulli polynomials, Modified Degenerate Apostol-Euler polynomials. References: [1] L. Carlitz, “Degenerate Stirling Bernoulli and Eulerian numbers”, Utilitas Math., 15 (1979), 51-88. [2] D. V. Dolgy, T. Kim, Known H.-In and J. J. Seo, “On the modified degenerate Bernoulli polynomials”, Adv. Stu. In Comtep. Math., 26(2016), 203- 209. [3] Q.-M. Luo, “Multiplication formulas for the Apostol-Bernoulli and Apostol- Euler polynomials of higher order”, Integral Transforms, Spec. Func., 20(2009), 377-391. ______135

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Univalency Conditions of a General Nonlinear Integral Operator of Analytic Functions with Different Domains

Shuhai Li, Huo Tang

School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, China [email protected]; [email protected]

Abstract: In this paper, we give some new univalence conditions of a general nonlinear integral operator of analytic functions defined by subordination with different domains. The results present here improve and generalize some known results.

Keywords: Analytic function, Univalent functions, Nonlinear integral operator, Subordination. References: [1] P. L. Duren. Univalent functions. Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983. [2] W. Janowski. Some extremal problems for certain families of analytic functions,I, Ann. Polon. Math. 28(1973), 297-326. [3] J. Stankiewicz and J. Waniurski. Some classes of functions subordinate to linear transformation and their applications, Ann. Univ, Mariae Curie- Sklodowska, Section A. 28(1974), 85-94. [4] B. A. Frasin. Univalency of a nonlinear integral operator of analytic functions, Journal of Mathematical Inequalities. 9(2015), 763-771. [5] Adriana Oprea, Daniel Breaz and H. M. Srivastava, Univalence conditions for a new family of integral operators, Filomat. 30(2016), 1243-1251. [6] Liangpeng Xiong, Properties of certain nonlinear integral operator associated with Janowski Starlike and convex functions, Journal of Mathematical Research with Applications. 36(2016), 432-440. [7] Sh. Najafzadeh, A. Ebadian and H. Rahmatan, Univalency conditions for a new integral operator, Journal of Computer Science and Applied Mathematics. 1(2015), 35-37.

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Certain Subclasses of Harmonic Univalent Functions Defined By Convolution and Subordination

Shuhai Li, Huo Tang

School of Mathematics and Statistics, Chifeng University, Chifeng , Inner Mongolia, China [email protected]; [email protected]

Abstract: Let S H be the class of functions f  h  g that are harmonic univalent and sense-preserving in the open unit disk U  z : z 1 for which f (0)  f ' (0) 1  0. In the present paper, we introduce some new subclasses of consisting of univalent and sense-preserving functions defined by convolution and subordination. Sufficient coefficient conditions, distortion bounds, extreme points and convolution properties for functions of these classes are obtained. Also, we discuss the radius of starlikeness and convexity.

Keywords: Harmonic univalent functions, subordination, convolution, sufficient coefficient condition, radius. References: [1] J. Clunie and T. Sheil Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 39(1) (1984), 3–25. [2] J. M. Jahangiri, Coefficient bounds and univalent criteria for harmonic functions with negative coefficients,Ann. Univ. Marie-Curie Sklodowska Sect. A. 52 (1998), 57–66. [3] J. M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math. Anal. Appl. 235 (1999), 470–477. [4] H. Silverman and E. M. Silvia, Subclasses of harmonic univalent functions, New Zealand J. Math. 28(1999), 275–284. [5] S. Nagpal and V. Ravichandran, A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions, Stud. Univ. Babes Bolyai-Math. 59(1) (2014), 41–55. [6] S. Nagpal and V. Ravichandran, Fully starlike and fully convex harmonic mappings of order α, Ann. Polon.Math. 108 (2013), 85–107. [7] R. M. El-Ashwah and B. A. Frasin, Hadamard product of certain harmonic univalent meromorphic functions, Theory and Applications of Mathematics Computer Science. 5 (2) (2015), 126–131.

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Refinement of Some Inequalities Concerning to Bn-Operator of Polynomials with Restricted Zeros

A. Liman

Department of Mathematics, National Institute of Technology Srinagar Jammu and Kashmir, India [email protected]

Abstract: Let Pn be the class of polynomials of degree at most n. Rahman introduced the class Bn of operators B that map Pn into itself. We present the correct proof of the result of Rather and Gulzar (Adv Inequal Appl 2:16–30, 2013). Moreover our result improves many prior results involving Bn operators and a number of polynomial inequalities can also be deduced by a uniform procedure.

Keywords: Polynomials, B-operator, inequalities. References: [1] Abdul Aziz, A refinement of an inequality of S. Bernstein, J. Math. Anal. Appl., 142(1989), 1-10. [2] N. A. Rather, S. Gulzar, On an operator preserving inequalities between polynomials. Adv. Inequal. Appl. 2, 16–30 (2013) [3] S. Bernstein, Sur la limitation des derivees des polynomes, C. R. Acad. Sci. Paris., 190(1930), 338-340.

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Gaussian Approximation to the Estimator of the Mean of a Heavy-Tailed Distribution under Random Censoring

Djamel Mearghni

Department of Mathematics, Mohamed Khider University, Biskra, Algeria [email protected]

Abstract: In many real life applications, the observations may not be available in their entirety: they are usually randomly censored. This happens quite often with, for instance, lifetime, reliability or insurance data. We model this situation by introducing a non-negative random variable (rv), called censoring rv, independent of the rv of interest. Then, we focus on minimum of the two rv's and an indicator rv which determines whether or not there has been censorship. In this work, we first apply the extreme value theory results to define an estimator for the mean of a heavy-tailed distribution under random censoring. Then, we make use of the empirical process theory to provide a Gaussian approximation to the proposed estimator, which would lead to its asymptotic normality.

Keywords: Empirical process; Gaussian approximation; Heavy-tailed distribution; Mean estimator; Order statistics; Random censoring.

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Industrial Application of Fuzzy Logic Control for Torque-ripple Minimization in Electricals Machines

Zineb Mekrini, Seddik Bri

Materials and Instrumentation (MIM), Electrical Engineering Department, High School of Technology, Moulay Ismail University Meknes-Morocco [email protected]

Abstract: This paper deals with fuzzy system application in the industry. Fuzzy logic use linguistic descriptions of variable and linguistic for the input and output behavior.Numerical input quantities are mapped to numerical output quatities by using, fuzzification,inference ,and defuzzification procedures.As a consequence,fuzzy systems can be based as nonlinear systems .The asynchronous machine constitute a theoretically interesting and practically important class of nonlinear systems. They are described by nonlinear differential equation.The major problem that is usually associated with control of this machine is the high torque ripple as it is not directly controlled. The high torque ripple causes vibrations to the motor which may lead to component lose, bearing failure or resonance. The fuzzy logic controller is applied to reduce electromagnetic torque ripple. Keywords: Fuzzy logic, Asynchronous machine ,Torque –Electromagnetic flux . References: [1] F.Korkmaz, I.Topaloğlu, H.Mamur,“Fuzzy Logic Based Direct Torque Control Of Induction Motor With Space Vector Modulation ”, International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI),Vol.2, No. 5/6, pp.32–40, December 2013. [2]Rajendra .S. Soni, S. Dhamal, “ Direct Torque Control Of Three Phase Induction Motor Using Fuzzy Logic”, International Conference on Electrical, Electronics and Computer Engineering , pp.34–38, 25th August 2013, [3] F.Korkmaz, I.Topaloğlu, H.Mamur,“ Direct Torque Control of Induction Motor With Fuzzy Logic for Minimization of Torque Ripples”, International Journal of Engineering Research and General Science , Vol 3, Issue 1, pp.361– 367, January-February, 2015. [4] D. SUN, HE ,Yikang, I.Topaloğlu, H.Mamur. Fuzzy Logic Direct Torque Control for Permanent Magnet Synchronous Motors”, IEEE Transactions, the 5Ih World Congress on Intelligent Control and Automation, pp.4401-4405, June 15-19,2004,Hangzhou.

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On Optimal Control of Stochastic Mean Field Systems

Brahim Mezerdi

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria [email protected]

Abstract: In this talk, we deal with optimal control of systems driven by mean- field stochastic differential equations. These equations are obtained as limits of interacting particle systems, as the number of particle tends to infinity. This kind of approximation result is called "propagation of chaos", which says that when the number of particles (players) tends to infinity, the equations defining the evolution of the particles could be replaced by a single equation, called the McKean-Vlasov equation. This mean-field equation, represents in some sense the average behavior of the infinite number of particles. Since the earlier papers by Lasry-Lions and Huang-Malhamé-Caines, mean-field control theory and mean-field game theory has raised a lot of interest, motivated by applications to various fields such as game theory, mathematical finance, communications networks, management of oil ressources. Mean-field control problems occur in many applications, such as in a continuous-time Markowitz's mean--variance portfolio selection model where the variance term involves a quadratic function of the expectation. We are interested in relaxed controls which are measure valued processes. We prove that the strict and relaxed control problems have the same value function and that an optimal relaxed control exists. In a second step, we establish necessary conditions for optimality in the form of a relaxed stochastic maximum principle, obtained via the first and second order adjoint processes, see [1,2].

Keywords: Mean-field stochastic differential equation; relaxed control; martingale measure; adjoint process; stochastic maximum principle; variational principle. References: [1] K. Bahlali, M. Mezerdi, B. Mezerdi, Existence of optimal controls for systems governed by mean-field stochastic differential equations, Afrika Statistika, Vol. 9 (2014), No 1, 627-645. [2] K. Bahlali, M. Mezerdi, B. Mezerdi, Existence and optimality conditions for relaxed mean-field stochastic control problems, Systems and Control Letters, Issue in progress 2017 http://dx.doi.org/10.1016/j.sysconle.2016.12.009.

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Ring Theory Approaches to Solve Cauchy-Euler Differential Equations of Several Variables

Assal Miloud

Department of Mathematics, university of Jeddah, Jeddah, KSA Department of Mathematics, Carthage University, Tunisia

Abstract: In this paper we introduce a new ring R of ponderation functions and we study a class of modules over R and prove that Laplace transform and Fourier transform generate some free modules over the ring R. Moreover we characterize the projective modules and simple modules and we prove that the socle of this ring is not an injective module. As an application we use the ring properies to give a new methode to solve equations of the form Div(Xf)= g in several variables. Furthermore, we give a general solution of the Cauchy-Euler Equations in high dimensions.

Keywords: Ring, Cauchy-Euler differential Equations. References: [1] Assal M., Zeyada N, “New ring of a class of Bessel integral operators. Integral Transforms Spec Funct. 2016;27(8):611-619.. [2] Atiyah, MF, Macdonald, IG. “Introduction to Commutative Algebra”. Westview Press: New York; 1969. [3] T. Pierce RS. “Associative Algebras”. Graduate Texts in Mathematics

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Subgradient Method of Solving the Problem of Linear Stochastic Programming with Bifurcation Effect

Fakhriddin Mirzoahmedov

Center of Innovative Development of Science and New Technologies, Academy of Science of Republic of Tajikistan Dushanbe, Tajikistan [email protected]

Abstract: In modern conditions from innovation point of view necessarily of the mathematical modeling is the presence of uncertainty, nonlinearity, ambiguities, critical points, branching processes, etc. Hereinafter, the bifurcation theory has become a source and a part of the mathematical theory of catastrophes and description of the various systems associated with the presence of nonlinearity or piecewise linearity. Examples of bifurcation may be a split in the experiments and processes, when after the critical point, can be observed in case either one way or the other. One of the sections developed by mathematical modeling is a linear programming problem, which is widely used in technological processes of production systems, where the main objective is to optimize the costs. However, products manufactured with such a criterion in this period cannot be implemented (in the case of proficiently) or insufficient to meet the demand (deficiency case), which by nature is random and creates the effect of bifurcation. This report will explore the stochastic analogue of the general problem of linear programming considering bifurcation effect and subgradient algorithm to solve it. We give also the results for numerical experiments.

References: 1. Mirzoahmedov F. Conflicting models of linear programming in production and economic systems. -Dushanbe: Tajik State National University, 2002. 2. Ermoliev YM Methods of stochastic programming. - M .: Nauka, 1976. 3. Mirzoahmedov F. Mathematical models and methods of production management, taking into account the random faktorov. Kiev, "Naukova Dumka," 1991.

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Modelling Asymmetric Magnetic Recording Heads with an Underlayer Using Superposition

Ammar Edress Mohamed

Duhok Polytechnic University, Zakho Technical Institute, Kurdistan Region - Iraq [email protected]

Abstract: This paper analyses and calculates the head fields of asymmetrical 2D magnetic recording heads when the soft-underlayer is present using the appropriate Green's function to derive the surface potential/field by utilising the surface potential for asymmetrical head without underlayer. The results follow closely the corners, while the gap region shows a linear behaviour for d/g < 0.5 compared with the calculated fields from finite-element.

Keywords: Magnetic recording heads, Laplace’s equation, Karlqvis head, Finite-element. References: [1] G. Fan, “A study of the playback process of a magnetic ring head,” IBM J. Res. Dev., vol. 5, pp. 321–325, 1961. [2] 0. Karlqvist, “Calculation of the magnetic field in the ferromagnetic layer of a magnetic drum,” Trans. Roy. Inst. Technol. Stock. Sweden, vol. 86, no. 1, pp. 1–27, 1954. [3] N. Curland and J. Judy, “Calculation of exact ring head fields using conformal mapping,” Magnetics, IEEE Transactions on, vol. 22, no. 6. pp. 1901–1903, 1986. [4] J. S. Yang and H. L. Huang, “Calculation of exact head and image fields of recording heads by conformal mapping,” IEEE Trans. Magn., vol. 25, no. 3, pp. 2761–2768, 1989. [5] T. J. Szczech, D. M. Perry, and K. E. Palmquist, “Improved Field Equations For Ring Heads,” IEEE Trans. Magn., vol. M, no. 5, pp. 5–9, 1983. [6] S. Iwasaki, “Perpendicular Magnetic Recording,” IEEE Trans. Magn., vol. 16, no. 1, pp. 71–76, 1980.

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G-compactness for Topological Groups with Operations

1Osman Mucuk and 2Huseyin Cakalli

1Department of Mathematics, Erciyes University, Kayseri, Turkey [email protected] 2Maltepe University, Graduate School of Science and Engineering, Maltepe, Istanbul-Turkey [email protected]

It is well known that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by lim from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann in [7] for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. Some authors have recently extended the concept to the topological group setting and introduced the concepts of G-sequential continuity [2], [10], G-sequential connectedness [3], [5] and G-sequential compactness [4]. On the other hand in [12] Orzech introduced a certain algebraic category C called category of groups with operations including groups, rings without identity, R-modules, Lie algebras, Jordan algebras, and many others. Mucuk and Çakallı in [6] extended the connectedness of topological groups to more general topological groups with operations. In this paper we prove some results on the different types of G- compactness for topological group with operations.

Keywords: Sequences, G-continuity, G-conpactness, topological group with operations References: [1] H. F. Akız, N. Alemdar, O. Mucuk, T. Şahan, Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J., 20-2 (2013) 223-238. [2] H. Çakallı, On G-continuity, Comput. Math. Appl. , Vol. 61, No.2, (2011) 313-318. [3] H. Çakallı, Sequential definitions of connectedness, Appl. Math. Lett., Vol. 25, No.3, , (2012) 461-465. ______145

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[4] H. Çakallı Sequential definitions of compactness, Appl. Math. Lett., 21 , 6, (2008) 594-598. [5] H. Çakallı, O. Mucuk, On connectedness via a sequential method, Revista de la Uniòn Matemática Argentina, Vol.54, No.2, (2013) 101-109. [6] O. Mucuk, H. Cakalli, G-connectedness for topological groups with operations, Filomat (ICAAM 2016). [7] J.Connor, K.-G. Grosse-Erdmann,Sequential definitions of continuity for real functions, Rocky Mountain J. Math. , Vol. 33, No.1, (2003)93-121. [8] S. Lin and L. Liu, G-methods, G-sequential spaces and G-continuity in topological spaces, Top. App.212 (2016) 29-48. [9] G. D. Maio, L.D.R. Kocinac, Statistical convergence in topology, Topology Appl. 156 (2008) 28-45. [10] O. Mucuk, T. Şahan, On G-sequential Continuity, Filomat} Vol.28, No.6, (2014) 1181-1189. [11] O. Mucuk, T. Şahan, N. Alemdar, Normality and quotients in crossed modules and group-groupoids, Appl. Categ. Structures, 23-3 (2015) 415-428. [12] G. Orzech, Obstruction theory in algebraic categories I and II, J. Pure. Appl. Algebra}, Vol.2, (1972) 287-314 and 315-340.

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Topological Aspects of Monodromy Groupoids for Group- Groupoids

1Osman Mucuk and 2Serap Demir

1Department of Mathematics, Erciyes University, Kayseri, Turkey [email protected] 2Department of Mathematics, Erciyes University, Kayseri, Turkey [email protected]

Abstract: One form of the monodromy principle was enunciated by Chevalley in [4]. The general idea is that of extending a local morphism f on a topological structure G, or extending a restriction of f, not to G itself but to some simply connected cover of G. A different approach was indicated by J. Pradines in [8] to generalise the standard construction of a simply connected Lie group from a Lie algebra to a corresponding construction of a Lie groupoid from a Lie algebroid.

Let G be a topological groupoid such that the stars G_x the fibres of initial point map of the groupoid are path connected and have universal covers. Let Mon(G) be the disjoint union of the universal covers of the stars G_x's at the base points identities of the groupoid G. Then there is a groupoid structure on Mon(G) defined by the concatenation composition of the paths in the stars G_x. We call Mon(G) as the monodromy groupoid of G. In [3] Brown and Mucuk in the smooth groupoid case including topological groupoids, the star topological groupoid and topological groupoid structures of Mon(G) were studied under some suitable local conditions. Then the group-groupoid structure of Mon(G) was recently developed in [7] and internal groupoid structure of Mon(G) was given in [1] . In this paper we will develop the topological aspect of Mon(G) and prove that if G is a topological group-groupoid, then Mon(G) becomes a topological group-groupoid; and give a monodromy principle for topological group-groupoids.

Keywords: Group-groupoid, monodromy groupoid, topological group- grouppoid

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References: [1] H. F. Akız, N. Alemdar, O. Mucuk, T. Şahan, Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J., 20-2 (2013) 223-238. [2] M.E.-S.A.-F., R. Brown, The holonomy groupoid of a locally topological groupoid, Top. Appl., 47 (1992)7-113. [3] R. Brown, O. Mucuk, The monodromy groupoid of a Lie groupoid, Cah. Top. Géom.Diff. Cat. 36 (1995) 345-370. [4] C. Chevalley, Theory of Lie groups, Princeton University Press, 1946. [5] L. Douady, M. Lazard, Espaces fibrés en algébres de Lie et en groupes, Invent. Math. 1 (1966) 133-151. [6] K.C.H Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc.Lecture Note Series 124, Cambridge University Press, 1987. [7] O. Mucuk, B. Kılıçarslan, T. Şahan, N. Alemdar, Group-groupoid and monodromy groupoid, Topology Appl. 158 (2011) 2034-2042. [8] J. Pradines, J., Théorie de Lie pour les groupoïdes différentiables, relation entre propriétés locales et globales, Comptes Rendus Acad. Sci. Paris, Sér A, 263 (1966), 907-910.

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A Characterization of the Two-Weight Inequality for Riesz Potentials on Cones of Radially Decreasing Functions

Ghulam Murtaza

Department of Mathematics, GC University, Faisalabad, Pakistan. [email protected]

Abstract: We establish necessary and sufficient conditions on a weight pair

(v,w) governing the boundedness of the Riesz potential operator Iα defined on a p q p homogeneous group G from L dec,r(w,G) to L (v,G), where L dec,r(w,G) is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels Iα1,α2 defined on a product of two homogeneous groups G1 × G2. In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and G1 × G2, respectively.

Keywords: Riesz potential; multiple Riesz potential; homogeneous group; cone of decreasing functions; two-weight inequality; Sawyer’s duality theorem

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On The Fourth Geometric-Arithmetic Index of Graphs

Y. Nacaroglu and A. Dilek Maden

Department of Mathematics, Selcuk University, Konya, Istanbul, Turkey [email protected] and [email protected]

Abstract: Topological indices are the numerical value associated with chemical constitution professes for correlation of chemical structure with various physical properties, chemical and biological activity [1]. In 2010, M. Ghorbani and A. Khaki defined the eccentricity based geometric arithmetic index named as eccentric version of Geometric arithmetic index [2]. In this paper, we present some upper and lower bounds on the fourth geometric arithmetic index. . Keywords: Topological index, Eccentricity, Geometric-arithmetic index, Fourth geometric arithmetic index References: [1] M.K. Jamil, M.R. Farahani, M.R.R. Kanna, “Fourth Geometric Arithmetic Index of Polycyclic Aromatic Hydrocarbons (PAHk)”, Pharm. and Chem. J. 3(2016), 94-99. [2] M. Ghorbani and A. Khaki, “A note on the fourth version of Geometric – Arithmetic index”, Optoelectron. Adv. Mater-Rapid Commun., 4(2010), 2212- 2215. [3] K. Ch. Das, I. Gutman, B. Furtula, “Survey on Geometric – Arithmetic indices of graphs”, Match. Commun. Math. Comput. Chem., 65(2011), 595-644. [4] J. M. Rodríguez, J. M. Sigarreta, “On the Geometric- Arithmetic Index”, Match. Commun. Math. Comput. Chem., 74(2015), 103-120. [5] I. Gutman, N. Trinajstić, “Graph theory and molecular orbitals. Total φ- electron energy of alternant hydrocarbons”, Chemical Physics Letters, 17(4) (1972), 535-538.

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Laplace Transform of Fractional Differential Equations

Khaled I. Nawafleh Physics Department, Mu,tah University, Al-Karak, Jordan; [email protected]

Abstract: The aim of this work is to study the possible extension of applying the Laplace transform for solving fractional differential equations. The Laplace transform allow us to transform fractional differential equations into algebraic equations and then by solving this algebraic equations, we can obtain the unknown function by using the inverse Laplace transform. Two illustrative examples are included to demonstrate the validity and applicability of the presented technique, the first one is the LC circuit as a conservative system, and the second one is the RC circuit as a non-conservative system.

Keywords: Fractional Differential Equations, Laplace Transform, LC Circuit, RC Circuit. References: [1] Om P. Agrawal (2001), Formulation of Euler-Lagrange Equations for Fractional Variational Problems, J. Math. Anal. Appl. 272(2002) 386-379. [2] Eltayeb. A. M. Yousif and Fatima. A. Alawad. Laplace Transform Method Solution of Fractional Ordinary Differential Equations, University of Africa Journal of Sciences. (2012) 2, 139-160. [3] Saeed Kazem. Exact Solution of Some Linear Fractional Differential Equations by Laplace Transform, International Journal of.Nonlinear Science, (2013) 16 (1) 3-11. [4] Shy-Der Lin and Chia-Hunglu. Laplace transform for solving some families of fractional differential equations and its applications, a springer open journal, (2013) 137. [5] Joseph M. Kimeu (2009), Fractional Calculus: Definitions and Applications, Masters Theses & Specialist Projects. [6] Arfken, G., & Weber, H. J. Mathematical Method for Physicists Academic. New York, (1985) 309. [7] Saxena R. K., and Nishimoto, K. On a fractional integral formula of .Saigo operator, J. Fract. Calc., (2002) 22, 57-58. ______151

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Gaussian Approximation of a New Tail Index Estimator for Right-Censored Pareto-Type Distributions

Abdelhakim Necir

Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria [email protected]

Abstract: A new consistent and an asymptotically normal estimator for the positive tail index of right-censored data is proposed. In this context, a tail empirical process is introduced and its Gaussian approximation is established. A simulation study is carried out to compare the proposed estimator with the existing ones in terms of bias and mean squared error. An application to survival time of Australian male Aids patients are provided.

Keywords: Extreme value index; Gaussian approximation; Random censoring, Tail empirical process. References: [1] Einmahl, J.H.J., Fils-Villetard, A. and Guillou, A., 2008. Statistics of extremes under random censoring. Bernoulli 14, 207-227. [2] de Haan, L. and Ferreira, A., 2006. Extreme Value Theory: An Introduction. Springer. [3] Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163-1174. [4] Stupfler, G., 2016. Estimating the conditional extreme-value index under random right-censoring. J. Multivariate Anal. 144, 1-24. [5] Worms, J. and Worms, R., 2014. New estimators of the extreme value index under random right censoring, for heavy-tailed distributions. Extremes 17, 337- 358.

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On Residual Algebraic Free Extensions of Valuations

Figen Oke

Trakya University Department of Mathematics Edirne,Turkey [email protected]

Abstract: If v is a valuaton on a field K with rankv=2 then there exist three kind residual algebraic free extensions of v to the rational function field K(x) with one variable over K. In this study the first kind residual algebraic free extension of v to K(x) is studied.

Keywords: extensions of valuations, residual algebraic free extensions, valued fields

References: [1] V. Alexandru - N. Popescu - A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ., 28 (1988), 579-592. [2] V. Alexandru - N. Popescu - A. Zaharescu, Minimal pair of definition of a residual transcendental extension of a valuation, J. Math. Kyoto Univ. 30 (1990), no. 2, 207-225 [3] V. Alexandru - N. Popescu - A. Zaharescu, All valuations on K(X), J. Math. Kyoto Univ. 30 (1990), no. 2, 281-296. [4] N. Bourbaki, Algebre Commutative, Ch. V: Entiers, Ch. VI: Valuations, Hermann, Paris (1964). [5] O. Endler, Valuation Theory, Springer, Berlin -Heidelberg-New York (1972). [6] N. Popescu, C. Vraciu, On the extension of valuations on a field K to K(x)-I, Ren. Sem. Mat. Univ. Padova, 87 (1992), 151-168 [7] N. Popescu, C. Vraciu, On the extension of valuations on a field K to K(x)-II, Ren. Sem. Mat. Univ. Padova, 96(1996), 1-14 [8] O.F.G. Schilling, The Theory of Valuations, A.M.S. Surveys, no. 4, Providence, Rhode Island (1950).

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Statistically (C,1,1) Summable Double Sequences of Fuzzy Numbers and a Tauberian Theorem

Zerrin Onder, Ibrahim Canak, Umit Totur

Department of Mathematics, Ege University, İzmir, Turkey Department of Mathematics, Ege University, İzmir, Turkey Department of Mathematics, Adnan Menderes University, Aydın, Turkey [email protected], [email protected], [email protected]

Abstract: Developed based on the concept of fuzzy sets which was discovered and introduced by Zadeh, fuzzy set theory have received more and more attention from researchers who have intended to apply the concept of fuzziness to individual works with different aspects from theoretical to practical in almost all scientific areas. One of the areas which the concept of fuzziness was practised is the summability theory, as well. In this talk, we recall some notations, basic definitions and theorems with respect to fuzzy numbers and its double sequences and define the concepts of slow oscillation for the double sequences of fuzzy numbers in certain senses. In the sequel, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. Finally, we indicate that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same fuzzy number in Pringsheim's sense under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.

Keywords: Double sequences of fuzzy numbers, ,Slowly oscillating sequences, (C,1,1) summability, Statistical convergence, Tauberian theorems. References: [1] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2(1951), 241-244. [2] I. J. Schoenberg, “The integrability of certain functions and related summability methods”, Am. Math. Mon. 66(1959), 361-375. [3] B. C. Tripathy and A. J. Dutta, “On fuzzy real-valued double sequence spaces”, Soochow J. Math. 32(2006), 509-520. [4] B. C. Tripathy and A. J. Dutta, “Statistically convergent and Cesaro summable double sequences of fuzzy real numbers ”, Soochow J. Math. 33(2007), 835-848. [5] L. A. Zadeh, “ Fuzzy sets”, Inform. Control 8(1965), 338-353. ______154

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The Sheffer Stroke Basic Algebras on the Intervals

Tahsin Oner1, Tugce Katican2

Department of Mathematics, Ege University1,2, Bornova, Izmir, Turkey [email protected], [email protected]

Abstract: The axiom system of Sheffer Stroke operation was introduced by Henry Sheffer in 1913 [6], and the improvements of this operation have beeen given in [2] and [7]. The notions about basic algebras were mentioned in [1], [3] and [4]. Afterwards, Oner and Senturk introduce some definitions and notions about the Sheffer Stroke basic algebras in [5]. In this work, giving basic concepts about the Sheffer Stroke operation and Sheffer stroke basic algebras 풜 = (퐴; ∣), 푏 푏 we define the operations ∣푎, ∣ and ∣푎 for any elements 푎, 푏 ∈ 퐴 such that 푏 푏 ([푎, 1]; ∣푎), ([0, 푏]; ∣ ) and ([푎, 푏]; ∣푎) are Sheffer Stroke basic algebras, respectively. We also show that these interval Sheffer Stroke basic algebras on a given Sheffer Stroke basic algebra 풜 = (퐴; ∣) verify the patchwork condition.

Keywords: Sheffer Stroke basic algebra, interval sheffer stroke basic algebra, patchwork condition. References: [1] Chajda, Ivan. "Basic algebras, logics, trends and applications." Asian- European Journal of Mathematics, Vol.8, No 31550040, (2015). [2] Chajda, Ivan. "Sheffer operation in ortholattices." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Vol. 44, no.1, pp. 19-23, (2005). [3] Chajda I, Kolařík M. “Independence of the axiomatic system of basic algebras.” Soft Computing, Vol. 13, pp. 41-43, (2009). [4] Chajda I, Kolařík M. "Interval Basic Algebras." NOVI SAD J. MATH., Vol. 39, 2, (2009). [5] Oner, T., Senturk I. "The Sheffer Stroke Operation Reducts of Basic Algebras", submitted, (2017). [6] Sheffer, H. M.. “A set of five independent postulates for Boolean algebras, with application to logical constants.” Transactions of the American Mathematical Society, 14(4), 481-488, (1913). [7] Whitehead, A. N., Russell, B. Principia Mathematica. New York, Cambridge University Press, (1927).

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A Reduction of Basic Algebras: Sheffer Stroke Basic Algebras

Tahsin Oner1, Ibrahım Senturk2

Department of Mathematics, Ege University1,2, Bornova, Izmir, Turkey [email protected] [email protected]

Abstract: The concept of basic algebras was introduced in Chajda and Emanovský [1], see also Chajda [2] and Chajda et al.[3] for the further information. Basic algebras are an important notions of algebras used in diff erent non-classical logics since they not only contain orthomodular lattices 퐿 = (L; ∨, ∧, ⊥, 0, 1), where x ⊕ y = (x ∧ 푦⊥) ∨ y and ¬ x = 푦⊥, but they also constitute an axiomatization of the logic of quantum mechanics along with MV-algebras [4], which get an axiomatization of many-valued Łukasiewicz logics; see Chajda [5] and Chajda et al. [6]. In this study, we present a term operation Sheffer stroke in a given basic algebra A and examine properties of the Sheffer stroke reduct of A. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between basic algebras and Boolean algebras.

Keywords: Basic algebras, Sheffer Sroke operation, Reduction References: [1] I. Chajda and P. Emanovský, “Bounded lattices with antitone involutions and properties of MV-algebras”, Discussiones Mathematicae, General Algebra and Applications, 24.1 (2004), 32-42. [2] I. Chajda, “Lattices and semilattices having an antitone involution in every upper interval”, Comment. Math. Univ. Carolin, 44.4 (2003), 577-585. [3] I. Chajda, “Basic algebras”, Clone Theory and Discrete MathematicsAlgebra and Logic Related to Computer Science, (2013). [4] R. L. Cignoli, I. M. d’Ottaviano and D. Munduci, Algebraic foundations of many valued reasoning (Vol. 7), Springer Science & Business Media. [5] I. Chajda, “Basic algebras and their applications”, an overview. Contr Gen Algebra, 20, (2011). [6] I. Chajda, R. Halaš and J. Kühr, “Many-valued quantum algebras”, AlgebraUniversalis,60.1(2009),63-90.

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Applications on Weak and Strong Forms of Fuzzy α-Open

(Closed) Sets

Hakeem A. Othman*

*Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, KSA. *Department of Mathematics, Rada'a College of Education and Science, Albaydaa University, Albaydaa, Yemen. [email protected] and [email protected]

Abstract: The present paper tries to introduce some applications on fuzzy (supra-) infra- α -open (closed) sets, likely, fuzzy (supra-) infra- α –continuous mappings, fuzzy (supra-) infra- α -open (closed) mappings, fuzzy supra- α - irresolute mapping and fuzzy supra- α -connected space. Moreover, The relations and converse relations between these mappings are highlighted. Important results about these mappings and fuzzy supra- α –connected space are investigated and presented.

Keywords: fuzzy infra- α -open set; fuzzy infra- α -closed set; fuzzy supra- α - continuous mapping; fuzzy infra- α -continuous mapping; Fuzzy supra- α - irresolute mapping; Fuzzy infra- α - irresolute mapping; Fuzzy infra- α - connected space. References [1] K. K. Azad, "On fuzzy semi continuity, fuzzy almost continuity and weakly continuity", J. Math. Anal. Appl., 82 (1981), pp.14-32. [2] A. S. Bin Shahna, "On fuzzy strongly semi continuity and fuzzy precontinuity", Fuzzy Sets and Systems, 44 (1991), pp.330-308. [3] M. H. Ghanim, E. E. Kerre and A. S. Mashhour, "Separation Axioms, Subspace and Sums in Fuzzy Topology ", J. Math. Anal Appl, 102(1984), pp. 189-202. [4] Hakeem A. Othman, "On fuzzy sp-open sets", Hindawi Publishing Corporation, Advances in Fuzzy Systems Volume 2011, Article ID 768028, 5 pages,doi:10.1155/2011/768028. [5] Hakeem A. Othman and Md.Hanif.Page, "ON An Infra-_-Open Sets ", Global Journal of Mathematical Analysis, 4(3) (2016) pp. 12-16. ______157

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Code Verification Using Method of Manufactured Solutions for CFD Problems

Hatice Ozcan

Department of Mathematics, Ahi Evran University, Kırsehir, Turkey [email protected]

Abstract: To find numerical solutions of non-linear hyperbolic, parabolic systems of partial differential equations (PDEs), which predominantly appear in computational fluid dynamics (CFD) field, are fundamental parts of discovering the physical properties of flows [1]. The essential aim of CFD is to utilize numerical methods and algorithms to solve and analyze fluid dynamic problems. To this end, the motivation of this work is to ensure that the computer code designed to solve a set of PDEs is accurate; in another words, it is bug free. Therefore, Method of Manufactured Solutions (MMS) is used to check the accuracy of existing code and to estimate order of accuracy of the developed numerical method [2,3,4]. An example for Euler equations of gas dynamics is provided to demonstrate the effective use of this method.

Keywords: manufactured solution, code verification, order of accuracy, Euler equations of gas dynamics. Acknowledgement : This work is supported by Ahi Evran University Scientific Research Projects Coordination Unit (Project Number : FEF.E2.17.029). References: [1] C. B. Laney, “Computational gas dynamics”, Cambridge University Press (1998). [2] B. W. Boehm, “Verifying and validating software requirements and design specifications”, IEEE Software 1.1(1984), 75-88. [3] P. J. Roache, “Fundamentals of computational fluid dynamics”, Albuquerque, NM, Hermosa Publishers (1999). [4] P. J. Roache, “Verification and validation in computational science and engineering”, Albuquerque, NM, Hermosa Publishers (1998).

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On the Existence of Einstein Weyl Manifold with a Special Metric Connection

F. Ozdemir, M. D. Turkoglu

Faculty of Science and Letters, Department of Mathematics, İstanbul Technical University Maslak-Istanbul,Turkey [email protected] , [email protected]

Abstract: On a Weyl manifold, the existence of a semi-symmetric recurrent metric connection is proved and curvature invariants and their characteristics of manifolds having this connection are studied. Also it is introduced and examined the necessary and sufficient condition for an Einstein Weyl (E푊푛) manifold to be an Einstein Weyl manifold with semi-symmetric special metric connection (EW푆 ). 푛 Keywords: Einstein Weyl manifold, metric connection, semi-symmetric recurrent metric connection. References: [1] L. P. Eisenhart, “Non-Riemannian Geometry” , New York: The American Mathematical Society Publishing (1927). [2] V. Hlavaty, “Theorie d'immersion d'une W_m dans W_n“, Ann. Soc. Polon. Math., 21 (1949), 196-206. [3] K. Yano, “On semi-symmetric metric connection”, Rev. Roumaine Math. Pures Appl., 15(1970), 1579-1586. [4] Y. X. Liang, “On semi-symmetric recurrent-metric connection”, Tensor (N.S.), 55 (1994), 107-112. [5] N. Rosen, “Weyl's geometry and physics”, Foundations Of Physics, 12/3 (1982), 213-248. [6] E. Scholz, “Weyl geometric gravity and breaking of electroweak symmetry”, Annalen De Physik, 523(2011), 507-530. [7] J. T. Wheeler, “Weyl gravity as general relativity”, Phys. Rev. D 90 (2014), 025027.

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Repeat Codes, Even Codes, Odd Codes and Their Equivalence

Mustafa Ozkan and Figen Oke

Department of Mathematics, Trakya University, Edirne, Turkey [email protected]

Abstract: Codes over the chain ring are obtained by writting special matrices. Gray images of these codes are binary codes. It is shown that first repeat code, second repeat code, even code and odd code are either equivalent or equal to these codes. The definitions of direct sum and direct product of these codes were given. Moreover dual codes were classed. Self dual codes and self orthogonal codes were established.

Keywords: Codes over rings, Lee distance, Even codes, Odd codes, Dual codes. References: [1] M. Ozkan and F. Oke, “A relation between Hadamard codes and some special codes over 22 u ”, App.Mathematics and Inf. Sci, 10.2 (2016), 701- 704. [2] M. Ozkan and F. Oke, “ Results On Hadamard Codes and Codes Over Rings”, 3rd International Conference On Recent Advances In Pure and Applied Mathematics (2016)

[3] S. Zhu, Y. Wang, M. Shi, Some Result On Cylic Codes Over 22 v , IEEE Trans. Inf. Theory, 56. 4 (2010),1680-1684.

[4] A. Bonnecaze and P. Udaya, Cyclic codes and self dual codes 22 u , IEEE Trans. Inf. Theory, 45, (1999),1250-1255. [5] Krotov, D. S. Z4-linear perfect codes ,Diskretn. Anal. Issled. Oper.7.4 (2000), 78–90.

[6] J.Wolfmann, Negacyclic and cyclic codes over 푍4, IEEE Trans. Inf. Theory, 45, (1999),2527-2532.

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Tauberian Theorems for the Weighted Mean Summability Methods of Integrals

Firat Ozsarac, Ibrahim Canak

Department of Mathematics, Ege University, Bornova, Izmir, Turkey [email protected]; [email protected] Abstract: Let px 0 be a nondecreasing real-valued differentiable function   on 0, such that p 00 and px as x . Given a real-       x valued function fxwhich is continuous on 0, and sxftdt          0 We define the weighted mean of sx as   1 x  xptstdt  , p        px  0 where pt is the derivative of pt.     We give some classical type Tauberian theorems to retrieve convergence of sx out of the weighted mean integrability of  x under some Tauberian   p   conditions. Key words: Tauberian theorem, Tauberian condition, Weighted mean method of integrals, slow decrease, slow oscillation. References: [1]F. Moricz, ‘‘Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences’’, Colloq. Math. , 99(2), (2004), 207-219. [2] İ. Çanak and Ü.Totur, ‘‘A Tauberian theorem for Cesaro summability of integrals’’, Appl. Math. Lett. , 24(3), (2011), 391-395. [3] İ. Çanak and Ü.Totur, ‘‘Tauberian conditions for Cesaro summability of integrals’’, Appl. Math. Lett. , 24(6), (2011), 891-896. [4] İ. Çanak and Ü.Totur, ‘‘Altenative proofs of some classical type Tauberian theorems for Cesaro summability of integrals’’, Math. Comput. Modell. , 55(3), (2012), 1558-1561. [5] A. Fekete and F. Moricz, ‘‘ Necessary and sufficient Tauberian conditions in the case of weighted mean summable integrals over R+’’ , II. Publ. Math. , 67(1- 2), (2005) , 65-78. ______161

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On Lifting Polynomials and Distinguished Pairs

Burcu Ozturk, Figen Oke

Department of Mathematics,TrakyaUniversity, Edirne, Turkey [email protected] , [email protected]

Abstract: Let푣 be a valuation of a field 퐾 and 푢 be a residual algebraic free extension of 푣 to the rational function field 퐾(푥). So푢 can be defined by using a residual transcendental extension of 푣 to 퐾(푥) and a lifting polynomial. In this study relations between lifting polynomials and distinguihed pairs are given bykeeping in view of valuation 푢.

Keywords:Valuations, Lifting Polynomials, Distinguished Pairs, Transcendental Extensions References: [1] N. Popescu, A. Zaharescu, “ On the Structure of Irreducible Polynomials over Local Fields”, J. Number Theory, 52, (1995), 98-118. [2] V. Alexandru, N. Popescu, A. Zaharescu, “ A Theorem of Characterization of Residual Transcendental extensions of a Valuation”, J. Math. Kyoto Univ., 28, (1988), 579-592 [3] S. Bhatia, S.K. Khanduja, “On Extensions Generated by Roots of Lifting Polynomials”, Mathematika, 49 (1-2), (2002), 107-118 [4] A. Zaharescu, “Lifting Polynomials over a Local Field”, Bol. Soc. Mat. Mexicana (3) vol.10 (2004), 15-27 [5] S.K. Khanduja, U. Grag, “Rank 2 Valuations of K(x)”, Mathematika, 37, (1990), 97-105 [6] K. Aghigh, S.K. Khanduja, “ On Chains Associated wirh Elements Algebraic over a Henselian Valued Field”, Algebra Colloquium 12:4 (2005), 607-616 [7] K. Aghigh, S.K. Khanduja, “ On The Main Invariant of Elements Algebraic over a Henselian Valued Field”, Proceedings of the Edinburgh Mathematical Society, 45, (2002), 219-227 [8] P. J. McCarthy, “Algebraic Extensions of Fields” (Blaisdell Publishing Company, Waltham, Massachusetts, Toronto, London), (1966)

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On Some Fixed Point and Common Fixed Theorems in b- Metric-Like Spaces

Mahpeyker Ozturk

Department of Mathematics, Sakarya University, Sakarya, TURKEY [email protected]

Abstract: In this study, we establish some existence and uniqueness theorems for mappings via the concept of admissibility in the settings of b-metric-like and generalized b-metric-like spaces.

Keywords: b-metric-like spaces, fixed point, admissible mappings. References: [1] H. Aydi, E. Karapınar, B.Samet, Fixed points for generalized alpha, psi- contractions on generalized metric spaces., J. Inequal. Appl., (2014), 16 pages. [2] E. Karapınar, P.Kumam, P. Salimi, On alpha-psi-Meir-Keeler contractive mappings, Fixed Point Theory and Appl., 2013, (2013), 12 pages. [3] A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory and Appl., 2012, (2012), 10 pages. [4] H. Aydi, A. Felhi, S. Sahmim, Common fixed points via implicit contractions on b-metric-like spaces, J. Nonlinear Sci. Appl., 10,1524--1537, (2017). [5] M.A.Alghamdi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl., 2013, (2013), 25 pages.

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Experimental Evidence of Landau Damping in a Fluid at a Macroscopic Scale

Eric Padilla, William Cody Wilson, Andrei Ludu

Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, Florida, USA [email protected] , [email protected] , [email protected]

Abstract: The Landau damping effect occurs mainly in collisionless plasmas [1- 3] as a microscopic resonant mechanism of the excitation of collective modes of oscillations. In our paper, we demonstrate the occurrence of this effect at the macroscopic scale in a fluid populated by a system of freely moving repelling probes (magnetic buoys) which simulate the free electrons in a plasma. The electrons’ Coulombian interaction is replaced by magnetic dipole repulsion, and instead of the electromagnetic wave we use a free, nonlinear, liquid surface wave traveling through the fluid. We couple the Green-Naghdi Hamiltonian for the incompressible fluid with the kinetic and magnetic potential of the buoys. The dynamical resulting equation is a Vlasov-Poisson type equation which includes at the same time the Benjamin-Feir instability and the Landau damping phenomenon. We derive analytically and numerically the values of the Phillips’ constant and of the enhancement factor, and we compare the results with experimental analysis of the dynamics of buoys and waves. We find that the effect of Landau damping is present in our system, and manifests by suppressing the formation of coherent structures. In addition, we will study the phase transitions of the buoy network under nonlinear wave perturbations.

Keywords: Landau Damping, resonance, damping, Benjamin-Feir instability, Green-Naghdi Hamiltonian, Vlasov-Poisson equation, nonlinear waves, fluid dynamics, magnetic dipole, collective modes. References: [1] L. D. Landau, J. Physics USSR, 10 (1946) 26. [2] I. Langmuir and L. Tonks, Phys. Rev., 33 (1929) 195. [3] D. D. Ryutov, “Landau damping: half a century with the great discovery”, Plasma Phys. Control Fusion, 41 (1999) A1-A12. ______164

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A Boundary Value Problem for an Irrational Order Partial Equation

A.A. Pashavand, N.A. Aliyev, A.Y. Delshad Gharegheshlaghi

Institute of Mathematics and Mechanics of NAS of Azerbaijan 9, B.Vahabzade str., AZ 1141, Baku, Azerbaijan [email protected]

Abstract: It is known that as the investigation of the solutions of problems stated for partial equations is difficult than the investigation of problems stated for ordinary equations, the solution of problems stated for fractional order equations is more difficult than the investigation of the solutions of problems stated for irrational order equations. Here we will be engaged in finding the solution of a boundary value problem for a partial equation whose order is an irrational number

Keywords irrational order partial discrete equation, a boundary value problem for such an equation, finding of the solution in the form of unknown coefficient series corresponding to Mittag-Loffler functions. References: [1] Tricomi F.G. Differential Equations. Blackie & Son Limited 1961, pp. 349. [2]Petrovsky L.G. Lectures on theory of ordinary differented equations. Gostichizdat. M.-L, 1952, 232 p. (Russian). [3]Petrovsky I.G., Lectures on partial equations, “Nauka” Moscow, 1974 (Russian) [4]Aliyev N. Jahanshahi M. Investiqation of BVP and IVP including Differential Equations with real orders. Second Joint Seminar On Applied Mathematics. Zanjan University and Baku State University, 2000, p. 92.

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Generalized Close to Convex Functions with q-Properties

Yasar Polatoglu 1, Oya Mert 2, 3 and Asena Cetinkaya 1

1Department of Mathematics and Computer Sciences, Istanbul Kultur University, Turkey 2Department of Mathematics Duzce University, Turkey 3Department of Mechanical Engineering, Istanbul Kemerburgaz University, Turkey 1 [email protected], 2, 3 [email protected] 1 [email protected]

Abstract: q-property of analytic functions can be described as a property of any class of these functions which use methods of quantum calculus.The main idea of the present work is inspired on the R.J.Libera [3]. This study uniquely aims to obtain a generalized close to convex functions that include q-properties.

Keywords: Growth theorem, distortion theorem. References: [1] G.E.Andrews, Application of basic Hypergeometric Functions, SIAM Rew,(16) 1974, 441-484. [2] M.K. Aouf and M.A.Nasr, On convex functions of complex order, Bull.Fac.Sci.Univ.Mansoura, (9), 1982, 566-581 [3] V. Paatero, Uber Gebiete von beschrankter randdrehung Ann. Acad. Sci. Fenn. Ser. A, 37(1933), 1-20. [4] K. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31(1975), 311-323. [5] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math., 10(1971), 7-16. [6] M. S. Roberston, On the theory of univalent functions, Ann. Math., 37(1936), 374-408. 02010 Mathematics Subject Classification: 30C45 Key words and phrases: Growth theorem, distortion theorem.

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Steady-State Modeling of the Biological Network via Long-tailed Symetric Distribution

Vilda Purutcuoglu1 and Melih Agraz2

1, 2Department of Statistics, Middle East Technical University, Ankara, Turkey [email protected] ; [email protected]

Abstract: The steady-state modeling of the biological systems presents the acivation of the systems’ elements without the randomness. This model is the most common modeling approach of the complex systems since the majority of the available data is more suitable in this type of description. In the mathematical representation of the underlying activation, the Gaussian graphical model (GGM) is the most well-known probabilistic model which is depedent on the conditional independence of the nodes (i.e., proteins, genes or other species of the system in this context), under the multivariate normality assumption of the measurements [1]. Althougt GGM is successful in the description of the small and moderately large systems, its inference is computationally demanding particularly for large networks and it is powerful under the strick normality assumption [2]. Hereby, in this study, we extend this model by denoting the measurements via the long- tailed symmetric distribution family which covers a wide range of densities from Cauchy, student-t to normal. By this way, we can obtain a new graphical explaination of the system having more realistic assumption abou the data and we call it the long-tailed graphical model (LTGM). Then, we derive the explicit form of the model parameteres via the modified maximum likelihood estimators (MMLE) [3] which is obtained from the order statistics and is the asymptotic equivalence of the maximum likelihood estimator. From our simulation studies, we show that LTGM with MMLE is computationally more efficient and accurate than GGM with its inference algorithms. Hence, we believe that our suggested approach can be a promising alternative of the steady-state description of the biological systems.

Keywords: Gaussian graphical model, modified maximum likelihood estimators, biological networks.

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Acknowledgement: The authors thank the BAP project (no: BAP-01-09-2017- 002) at Middle East Technical University for its support.

References: [1] J. Whittaker, “Graphical models in applied multivariate statistics”, John Wiley and Sons, (1990). [2]E. Ayyıldız, M. Ağraz and V. Purutçuoğlu, “MARS as an alternative approach of Gaussian graphical model for biochemical networks”, Journal of Applied Statistics, (2016), 1-19. [3] M.L. Tiku, “Estimating the mean and standard deviation from a censored normal sample”, Biometrika, 54 (1967), 155-165.

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Quadrature Formula with High Degree of Exactness

Abedallah Rababah

Department of Mathematics and Statistics, Jordan University of Science andTechnology, 22110 Irbid, Jordan [email protected]

Abstract: In this article, a quadrature formula of degree 2 is given that has degree of accuracy 3 and order 5. The formula is valid for any planar curve given in parametric form unlike existing Gaussian quadrature formulas that are valid only for functions.

Key words: quadrature formula; quadratic formula; parametric curves; degree of accuracy three; fifth order. References: 1. G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Boston (1988). 2. K. Hollig, J. Horner, Approximation and Modeling with B-Splines, SIAM, Titles in Applied Mathematics 132, (2013). 3. A. Rababah, Taylor theorem for planar curves, Proc. Amer. Math. Soc. Vol 119 No. 3, (1993), 803-810. 4. A. Rababah, Approximation von Kurven mit Polynomen und Splines, Ph. Dissertation, Stuttgart Universitat, (1992). 5. A. Rababah, High accuracy Hermite approximation for space curves in

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Best Cubic Spline Interpolation Based on Minimizing the Error

Abedallah Rababah, Mohammed Bani Khalid

Department of Mathematics and Statistics Jordan University of Science and Technology Irbid, Jordan [email protected] ; [email protected]

Abstract: In this talk, a modified cubic spline is introduced. The proposed spline can be obtained by considering some flexible boundary conditions that involve some additional parameters. These parameters are incorporated to improve the accuracy of the classical cubic spline. Moreover, these parameters are estimated by minimizing the error function in L1 - norm. The boundary conditions are adjusted to get new parameters that are used to get better approximation. Some numerical examples are given to demonstrate the advantages in accuracy and efficiency of the proposed method over traditional cubic methods.

Keywords: cubic spline; minimizing the error; best cubic approximation References: K. Hӧllig, J. Hӧrner: Approximation and Modeling with B-Splines, SIAM, Titles in Applied Mathematics 132, 2013. I. J. Schoenberg, On equidistant cubic spline interpolation. Bull. Amer. Math. Soc. 77 (1971), no. 6, 1039-1044. Rababah, High accuracy Hermite approximation for space curves in Rd, J. Math. Anal. Appl. 325, Iss. 2 (2007), 920-931. J. Rice, The approximation of functions, Vol. 1: linear theory. Addison-Wesley, (1964).

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On Chebyshev Collocation Method and Applications to Nonlinear Integral Equations

Abdalah Rababah1, Benferhat Leila2, Hichem Ramoul3, Nora Mahloul4

1Department of Mathematics and Statistics, Jordan University of Science and Technology Irbid, Jordantage [email protected] 2Department of Algebra and Numbers Theory, University of Science and Technology Houari Boumediene, Algiers, Algeria [email protected] 3ICOSI laboratory, Abbes Laghrour University-Khenchela, Khenchela, Algérie [email protected] 4Department of Algebra and Numbers Theory, University of Science and Technology Houari Boumediene, Algiers, Algeria [email protected]

Abstract: This paper is devoted to solve some classes of one-dimensional non- linear Volterra-Hammerstein integral equations with singular kernels. The key ingredient is to transform integral equation to algebraic equation with unknown Chebyshev coefficients. We use here a suitable weighted Chebyshev polynomial to approximate the solution of the non-linear integral equation.

Keywords: Chebyshev polynomials, integral equations, collocation method. References: [1] Dardery, S. M., & Allan, M. M. “Chebyshev polynomials for solving a class of singular integral equations”, Applied Mathematics, 5.04 (2014), 753. [2] Mason, John C., and David C. Handscomb. “Chebyshev polynomials”. CRC Press, 2002. [3] Danaei, R., Molaei, H., & Khazili, M. “Solving Fredholm integral equations using with Chebyshev polynomials”, International Journal of Innovative Science, Engineering & Technology, 2.5(2015), 297-300.

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The Treatment of Fractional Singular Lagrangian

Eqab Rabei

Al al-Bayt University [email protected]

Abstract: The singular Lagrangian with fractional derivative will be investigated. The Euler-Lagrangian will be derived and some example will be given.

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MHD Convective Flow due to a Curved Surface with Thermal Radiation and Chemical Reaction

Madiha Rashid

Quaid-i-azam University, Islamabad [email protected]

Abstract: Present work is devoted to the convection flow of viscous fluid by a curved stretching sheet. Fluid is electrically conducting with the presence of uniform magnetic field. Heat and mass transfer characteristics are studied by using heat and mass convective conditions. Thermal radiation and chemical reaction are also taken into consideration. With the help of allocated transformations the presented nonlinear partial differential systems are reduced into the nonlinear ordinary differential system. Impact of pertinent parameters on physical quantities like fluid velocity, temperature and concentrations fields are observed. Computations for surface shear stress and heat and mass transfer rates are carried out. It is observed that pressure inside the boundary layer flow due to curved stretching plate cannot be neglected

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On The Isolated Points of the Spectrum of M-Paranormal Operators

Mohammad Rashid

Department of Mathematics, Alkarak, Mutah University Alkarak, Jordan [email protected]

Abstract: For 푀 −paranormal operator 푇 on a separable complex Hilbert space

퐻 we show that (1) If 휎푤 = {0}, then 푇 is compact and normal and (2) every Riesz idempotent 퐸 with respect to a non-zero isolated point 휆 of 휎(푇) is self- adjoint (i.e., it is an orthogonal projection) and satisfies that 푅(푇) = ker(푇 − 휆) = ker(푇∗ − 휆̅).

Keywords: paranormal operators, M-paranormal operators, Riesz idempotent. References: [1] P. Aiena, Fredholm and local spectral theory with applications to multipliers, Kluwer, 2004. [2] S.C. Arora, Ramesh Kumar, M-Paranormal operators, Publ. Inst. Math., Nouvelle serie 29 (43) (1981), 5-13. [3] M. Ch_o and S. ^ Ota, On n-paranormal operators, J. Math. Research 5 (2013), no. 2, 107-114. [4] R.E. Harte, Invertibility and singularity for bounded linear operators, Dekker, New York, 1988. [5] P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton,1967. [6] J.K. Han, H.Y Lee, W.Y Lee, Invertible completions of 2 _ 2 upper triangular operator matrices, Proc. Am. Math. Soc. 128(2000),no.1, 119-123. [7] H. Heuser, Functional Analysis, Dekker, New York, 1982.

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Existence of Homoclinic Orbit in Generalized Planar System of Lienard Type

Vahid Roomi

Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran [email protected]

Abstract: The object of this paper is to study the orbit structure of a generalized Li_enard type system in a neighborhood of a trajectory which is doubly asymptotic to an equilibrium solution, i.e., an orbit which lies in the intersection of the stable and unstable manifolds of a critical point. Such an orbit is called a homoclinic orbit.

Keywords: Lienard System, Homoclinic orbit.

Consider the planar system ẋ = P(Q(y) − F(x) { (1) ẏ = −g(x) which is a generalized Lienard type system, where P, Q, F and g are continuous functions which ensure the existence of a unique solution to the initial value problem. System (1.1) includes the classical Lienard system as a special case, which is of great importance in various applications (see [1-15] and the references cited therein). In system (1), a trajectory is said to be a homoclinic orbit if its α − and ω −limit sets are the origin. The purpose of this paper is to give an implicit necessary and sufficient condition and some explicit sufficient conditions on F(x), g(x), P(u) and Q(y) under which system (1) has homoclinic orbits.

References: [1] R. P. Agarwal, A. Aghajani, V. Roomi, Existence of homoclinic orbits for generalized planar Dynamical System of Lienard Type”, Dynamic of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 81 (2012), 271-284. [2] A. Aghajani, A. Moradifam, “”Some sufficient Conditions for the Intersection with the Vertical Isocline”, Appl. Math. Lett, 19(2006), 491-497. ______175

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Some Results of the Picard-Krasnoselskii Hybrid Iterative Process

Aynur Sahin and Metin Basarir

Department of Mathematics, Sakarya University, Sakarya, 54050, Turkey [email protected], [email protected]

Abstract: In this study, we establish the strong convergence and stability results of Picard-Krasnoselskii hybrid iterative process for a general class of contractive-like operators in hyperbolic spaces. Furthermore, we give an example to support our results. Finally, we apply this iterative process to obtain the solution of a functional equation in a Banach space.

Keywords: Fixed point, Picard-Krasnoselskii hybrid iterative process, Stability, Functional equations, Contractive-like operators. References: [1] V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, 2007. [2] V. Berinde and A. R. Khan, “On a functional equation arising in mathematical biology and theory of learning”, Creath. Math. Inform. 24.1(2015), 9-16. [3] A. O. Bosede and B. E. Rhoades, “Stability of Picard and Mann iteration for a general class of functions”, J. Adv. Math. Stud. 3.2(2010), 23-25. [4] C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and Mann iteration processes”, Carpath. J. Math. 19(2003), 155-160. [5] G. A. Okeke and M. Abbas, “A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process”, Arab. J. Math. 6(2017), 21-29. [6] M. O. Osilike, “Stability results for Ishikawa fixed point iteration procedure”, Indian J. Pure Appl. Math. 26.10(1995), 937-941. [7] I. Timiş, “On the weak stability of Picard iteration for some contractive type mapping”, Annal. Uni. Craiova, Math. Comput. Sci. Series, 37.2(2010), 106- 114.

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Optimal Coincidence Best Proximity Point Results in Fuzzy Metric Spaces

Naeem Saleem

Department of Mathematics, University of Management and Technology, C-II Johar Town, Lahore,Pakistan [email protected]

Abstract: In this paper, we disscused some sufficient conditions for existence and uniqueness of the best proximity points and optimal coincidence point results for non-self mapping in fuzzy metric space. We also mentioned some interesting aspects of best proximity point theory in the setup of fuzzy metric spaces. We provided some examples to explain the obtained results, which also shows that obtained results are generalization of already existing results in literature. This article could be viewed as a discussion on extension of recent development on proximal contraction mappings in such spaces.

Keywords: Fuzzy metric Space, Best Proximity point, Optimal Coincidence best proximity point References: [1] K. Fan, “Extensions of two fixed point theorems of F. E. Browder”. Math. Z. 112, 234-240 (1969). [2] S. Sadiq Basha, N. Shahzad, R. Jeyaraj.“Optimal approximate solutions of fixed point equations”. Abstr. Appl. Anal. 2011, 174560 (2011). [3] N. Saleem, B. Ali,M. Abbas, Z. Raza.“Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications”. Fixed Point Theory Appl. 2015, 36 (2015). [4] LA. Zadeh. “Fuzzy sets”. Inf. Control 8, 103-112 (1965). [5] M. Abbas, N. Saleem and M. De la Sen, “Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces”, Fixed Point Theory and Applications, 2016 (1), 1-18. [6] N. Saleem, M. Abbas and Z. Raza, “Optimal coincidence best approximation solution in non-Archimedean Fuzzy Metric Spaces”, Iranian Journal of Fuzzy Systems, 13(3), 113-124, 2016. [7] Z. Raza, N. Saleem and M. Abbas, “Optimal coincidence points of proximal quasi-contraction mappings in non-Archimedean fuzzy metric spaces”, Journal of nonlinear science and application, 9 (2016), 3787- 3801 ______177

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New Concept of Determinants with Three Indexes (3D Determinants) and Possibilities of Use

Armend Salihu

Department of Computer Science, University for Business and Technology Prishtine, Kosovo [email protected]

Abstract: In this paper will will present a new concept of determinants with three indexes (3D Determinants), we will present how we will try to solve the 3D determinants and what are possibilities of usage of those determinants.

Keywords: Determinants, 3D, Determinants Calculation. References: [1] Cramer, Gabriel (1750). "Introduction à l'Analyse des lignes Courbes algébriques" (in French). Geneva: Europeana. pp. 656–659. Retrieved 2012-05- 18 [2] Thomas S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Springer Science & Business Media. p. 132. ISBN 978-0-387-48947-6. [3] Leon, Steven J. (2006), Linear Algebra with Applications (7th ed.), Pearson Prentice Hall [4] Hedman, Bruce A. (1999). "An Earlier Date for "Cramer's Rule"". Historia Mathematica 26 (1999), 365–368

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Boundedness Properties of Some Operators on M P ,Q  d 

Ayse Sandikci

Department of Mathematics, Ondokuz Mayis University, Samsun, Turkey [email protected]

Abstract: The Lorentz mixed normed modulation space is the set of all tempered distributions fS  d  such that the short-time Fourier 2d transform Vfg of f is in the Lorentz mixed norm space LP,Q   , where d gS   is a non-zero window function, 1Pp,p  12 and

1Qq,q  12 . In this work we investigate the boundedness properties of Wigner distribution and time-frequency localization operator on . Some key references are given below.

Keywords: Lorentz mixed normed modulation space, Wigner distribution, time- frequency localization operator. References: [1] P. Boggiatto, Localization operators with L^{p} symbols on the modulation spaces, In Advances in Pseudo-Differential Operators, vol. 155 of Oper. Theory Adv. Appl., 149--163, Birkhäuser, Basel, 2004. [2] E. Cordero, K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal., 205(1) (2003), 107-131. [3] D.L. Fernandez, Lorentz spaces, with mixed norms, J. Funct. Anal., 25 (1977), 128-146. [4] K. Gröchenig, Foundation of Time-Frequency Analysis. Birkhäuser, Boston 2001, ISBN 0-8176-4022-3. [5] R.A. Hunt, On L(p,q) spaces, Extrait de L'Enseignement Mathematique, T.XII, fasc.4 (1966), 249-276. [6] A. Sandıkçı, On Lorentz mixed normed modulation spaces, J. Pseudo-Differ. Oper. Appl., 3 (2012), 263-281. [7] A. Sandıkçı, Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces, Turk J. Math., 38 (2014), 728-745.

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On Approximate Biprojectivity of Banach Algebras

M. H. Sattari

Department of Mathematics Azarbaijan Shahid Madani University, Tabriz, Iran [email protected]

Abstract: The notion of approximate biprojectivity of Banach algebras was introduced by Pourmahmood-Aghababa [4], O. Yu. Aristov [1] and Y. Zhang

[5] in different ways. Here some hereditary properties of this concept is investigated, according to definaion of Pourmahmood-Aghababa. Among other things is shown that approximate biprojectivity of the second dual of A implies approximate biprojectivity of , where A ** is equipped with the first Arens product [2].

Keywords: Approximate biprojective, Banach Algebra, Second dual. References: [1] O. Yu. Aristov, “On approximation of flat Banach modules by free modules”, Sbornik. Math. 196(11) (2005), 1553–1583. [2] H. G. DALES. Banach Algebras and Automatic Continuity (Oxford, 2000). [3] F. Ghahramani and Y. Zhang, “Pseudo-amenable and pseudo-contractible Banach algebras”, Math. Proc. Cambridge Philos. Soc. 142 (2007), 111–123. [4] H. Pourmahmood-Aghababa, “Approximately biprojective Banach algebras and nilpotent ideals”, Bull. Aust. Math. Soc. 87 (2013) 158-173. [5] Y. Zhang, “Nilpotent ideals in a class of Banach algebras”, Proc. Amer. Math. Soc. 127(11) (1999), 3237–3242.

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On Some New Sequence Spaces Defined By Almost Lacunary Bounded Variation

Ekrem Savas

Department of Mathematics, Istanbul Ticaret University, Sütlüce, Istanbul, Turkey [email protected]

Abstract: In this paper we introduce and examine a new sequence space by using Orlicz function and almost lacunary bounded variation. We also study some basic topological and algebraic properties for these spaces. Furthermore we shall established inclusion theorems between these sequence spaces.

Keywords: almost convergence, almost lacunary convergence, lacunary sequence; bounded variation ; Orlicz function . References: [1] S. Banach, Theorie des Operations linearies, Warszawa, 1932. [2] G. Das and S. K. Mishra, Banach limits and lacunary strong almost convergence, J. Orissa Math. Soc. 2(2) (1983), [3] V. Karakaya and E. Savas, On almost p-bounded variation of lacunary sequences , Computer and Math. with Appl., 61(2011), 1502-1506 [4] S. D. Parashar, and B. Choudhury, Sequence space defined by Orlicz functions, Indian J. Pure Appl. Math., 25(14) (1994)419-428. [5] E. Savas, and V. Karakaya, Some new sequence spaces defined by lacunary sequences ,Math. Slovaca, 57(2007), 393-399.

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On Filter Convergence of Nets in Uniform Spaces

Ekrem Savas1 and Ulas Yamanci2

1Department of Mathematics, Istanbul Ticaret University, İstanbul, Turkey 2Department of Statistics, Süleyman Demirel University, Isparta, Turkey [email protected]; [email protected];

Abstract: In this paper, we introduce F -convergence and Fst -fundamental of nets in uniform space and study some of its consequences

Keywords: Ideal, filter, net, filter-convergence. References: [1] B.T. Bilalov, T.Y. Nazarova, “On statistical type convergence in uniform spaces”, Bull. Iranian Math. Soc., 42(2016), 975-986. [2] J.A. Fridy, On statistical convergence, Analysis, 5 (1985) 301-313. [3] P. Das, E. Savaş, “On I -convergence of nets in locally solid Riesz spaces”, Filomat, 27(2013), 89-94. [4] B.K. Lahiri, P. Das, “I and I -convergence of nets”, Real Anal. Exchange, 33 (2007-2008), 431-442. [5] E. Savaş, P. Das, “A generalized statistical convergence via ideals”, Appl. Math. Lett., 24 (2011), 826-830.

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On E-J Hausdorff Transformations for Double Sequences

Rabia Savas and Hamdullah Sevli

Sakarya University, Sakarya, Turkey [email protected] Istanbul Commerce University, Istanbul, Turkey [email protected]

Abstract: In 1933, Adams [1] developed Hausdorff transformations for double sequences. Şevli and Savaş [2] proved some result for double Endl- Jakimovski (E-J) generalization. In this study, we consider some further results for E-J Hausdorff transformations for double sequences.

Keywords: Hausdorff matrices, double series, absolute summability. References: [1] Adams, C. R., Hausdorff transformations for double sequences, Bull. Amer. Math. Soc. 39, 303-312, 1933. [2] Şevli, H. and Savaş, R., On double Hausdorff summability method, Journal of Inequalities and Applications, 240, 1-10, 2014.

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Double Lacunary Statistical Boundedness of Order α

Rabia Savas and Mahpeyker Ozturk

Department of Mathematics, Sakarya University, Sakarya, Turkey

[email protected] and [email protected]

Abstract: In this paper we introduce and the study the concept of the double lacunary statistical boundedness of order α and also give a general descriptions of inclusion between double statistical boundedness and double lacunary boundedness of order α.

Keywords: Double lacunary sequences, double statistical boundedness, double lacunary statistical boundedness

References: [1] V. K. Bhardwaj, S. Gupta, S. A. Mohiuddine and A. Kılıçman, On Lacunary Statistical Boundedness, J.Ineq. Applications, 2014, 2014:311. [2] M. Et, S.A. Mohiuddine and H. Şengul, On Lacunary Statistical Boundedness of order α, Facta Universitatis, Ser. Math. Inform. Vol. 31, No 3, 2016, 707-716. [3] R. F. Patterson and E. Savas, Lacunary Statistical Convergence of Double Sequences, Mathematical Com. 10, 2005, 55-61.

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Statistıcal Convergent Functions Via Ideals With Respect To The Intuitionistic Fuzzy 2-Normed Spaces

Rahmet Savas

Department of Mathematics, Istanbul Medeniyet University, Istanbul, Turkey [email protected]

Abstract: The main objective of this paper is to introduce and study the notion of ideal  -statistical convergence of a non-negative real-valued Lebesque- measurable function in the interval (1,  ) with respect to the intuitionistic fuzzy 2-normed ( , ) . Investigate their relationship, and make some observations about these classes. Further, we prove some inclusion theorems.

Keywords: Ideal, filter, I-Statistical convergence of function, I - statistical convergence of functions, intuitionistic fuzzy normed space, intuitionistic fuzzy 2− normed space. References: [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986) 87–96. [2] K. Atanassov, G. Pasi, R. Yager, Intuitionistic fuzzy interpretations of multi- person multicriteria decision making, in: Proceedings of 2002 First International IEEE Symposium Intelligent Systems, 1(200) ,115–119. [3] R. Colak, Statistical convergence of order α, Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Pub., (2010), 121-129. [4] R. Colak, C. A. Bektas, λ-statistical convergence of order α, Acta Math. Scientia, 31B (3) (2011), 953-959. [5] J. Connor, The statistical and strong p - Cesaro convergence of sequences, Analysis, 8(1988), 47-63. [6] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 1951, 241-244. [7] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301-313. [8] S. G¨ahler, Linear 2-normietre Raume, Math. Nachr. 28(1965). [9] P. Kostyrko, T. ˇSal´at, W. Wilczynki, I-convergence, Real Anal. Exchange 26 (2) (2000-2001) 669-685. [10] Mohiuddine, S. A., Lohani, Q. M. D. On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos, Solitons and Fractals, 42, 3, 2009 [11] M. Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000) 111–115. [12] M. Mursaleen and Q. M. Danish Lohani, Intuitionistic fuzzy 2- normed space and some relates concepts Chaos, Solitons and Fractals, 42,(2009), 224- 234. ______185

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Submanifolds in m1 with Finite Type Pseudo- H (2 1)  Hyperbolic Gauss Map

Ruya Yegin Sen* and Ugur Dursun**

* Department of Mathematics, Istanbul Medeniyet University, Uskudar, Istanbul, Turkey [email protected] ** Department of Mathematics, Isik University, Sile, Istanbul, Turkey [email protected]

Abstract: The notion of finite type submanifold of a Euclidean space has been introduced in late seventies. Since then the finite type submanifolds of Euclidean or pseudo-Euclidean spaces have been studied extensively. In this work, we mention the pseudo-hyperbolic Gauss map in the Obata’s sense. We study pseudo-Riemannian submanifolds of with finite pseudo-hyperbolic Gauss map. We classify the pseudo-Riemmanian submanifolds of with 1-type pseudo-hyperbolic Gauss map. Then, we investigate the maximal surface fully lying in 41m with 2-type pseudo-hyperbolic H(1)H(1)22 Gauss map.

Keywords: Finite type map, pseudo-hyperbolic Gauss map, pseudo-hyperbolic space. References: [1] Chen, B.-Y., Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J., 8 (1985), 358-374. [2] Chen, B.-Y., Finite-type pseudo-Riemannian submanifolds, Tamkang Journal [3] Dursun, U. and Yeğin, R., Hyperbolic submanifolds with finite type Hyperbolic Gauss map, Int. J. Math., 26(2015). [4] Ishihara, T., Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature, Michigan Math. J., 35 (1988), 345-352. [5] Sakaki, M., Spacelike maximal surfaces in 4-dimensional space form of index 2, Tokyo J. Math , 25(2002), 295-306. ______186

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On the Determination of Validity of Categorical Syllogisms by Using a Mathematical Model

Ibrahım Senturk1, Tahsin Oner2

Department of Mathematics, Ege University1,2, Bornova, Izmir, Turkey [email protected] [email protected]

Abstract: The earliest systematic approachment for determination of validity of syllogisms was introduced by Aristotle [1] . In the 19th. and 20th. centuries, Lewis Caroll [2] and Jan Łukasiewicz [3] made a significant contribution to convert categorical syllogisms into modern logical structures. In this work, we examine the validity of categorical syllogisms with the help of Caroll Diagrammatic Method (SLCD) by constructing a logical calculus system [4] . In the sequel, we indicate that any categorical syllogism is valid if and only if it is provable in this system. In addition to these, we tackle with sorites validity problems in categorical syllogisms. This enables us to construct a more general solution technique for determination of their conclusions and validity.

Keywords: Categorical Syllogisms, Syllogistic systems, Carrolls' Diagrammatic Method References: [1] Aristotle, “Prior Analytics” translation and commentary by Robin Smith, Hackett Publishing, (1989). [2] L. Carroll, “Symbolic Logic”, Clarkson N. Potter, (1896). [3] J. Łukasiewicz, “Aristotle's Syllogistic From the Standpoint of Modern Formal Logic”, Oxford University Press (1957). [4] I. Senturk and T. Oner, “A Construction of Heyting Algebra on Categorical Syllogisms”, Matematichki Bilten, 40.4 (2016), 5-12.

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Converse Theorems for Statistical Convergence

Sefa Anil Sezer 1, Rahmet Savas 1, Ibrahim Canak 2

1 Department of Mathematics, Istanbul Medeniyet University, Istanbul, Turkey 2 Department of Mathematics, Ege University, Izmir, Turkey [email protected]; [email protected]; [email protected]

Abstract: A real or complex sequence ()sn is said to be statistically convergent to a finite  if for all   0 , 1 lim:0, nNs n  n N 1 where the vertical bars indicate the cardinality of the enclosed set. It is known that the statistical convergence can be studied as a regular summability method, and also it is not included by any matrix method. In this study we investigate relations between statistical convergence and a certain class of matrix summability methods. Besides, we establish conditions needed for a statistically convergent sequence to be ordinary convergent.

Keywords: Statistical convergence, converse theorems, Tauberian theorems, matrix summability methods. References: [1] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2(1951), 241-244. [2] I. J. Schoenberg, “The integrability of certain functions and related summability methods”, Amer. Math. Monthly, 66(1959), 361-375. [3] J. A. Fridy, “On statistical convergence”, Analysis, 5(1985), 301-313. [4] J. A. Fridy, H. I. Miller, “A matrix characterization of statistical convergence”, Analysis, 11(1991), 59-66. [5] F. Mòricz, “Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences”, Analysis, 24(2004), 127–145. [6] S. A. Sezer, İ. Çanak, “Tauberian theorems for the summability methods of logarithmic type”, Bull. Malays. Math. Sci. Soc. (2016), doi:10.1007/s40840- 016-0437-9 [7] S. A. Sezer, İ. Çanak, “Power series methods of summability for series of fuzzy numbers and related Tauberian Theorems”, Soft Comput. 21(2017), 1057– 1064

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Cesàro Summability of Sequences in 2-Normed Spaces

Sefa Anil Sezer, Rahmet Savas

Department of Mathematics, Istanbul Medeniyet University, Istanbul, Turkey [email protected]; [email protected]

Abstract. The concept of 2-normed spaces was introduced by Gähler [7] in 1964. This notion recieved the attention of a wider audience after the study of White [1] in 1969. He defined convergent sequences and Cauchy sequences in 2- normed spaces which lead the birth of theory of 2-Banach spaces. In the last decade, 2-normed spaces has attained noticeable importance and popularity from the researchers working on summability theory. Firstly, Gürdal and Pehlivan [4,5] presented the statistical convergence of sequences in 2-normed spaces. Meanwhile, Şahiner et al. [2] defined the ideal convergence of sequences in such spaces. Later, Savaş [3] and Das et al. [6] determined certain new sequence spaces using ideal convergence and Orlicz functions in 2-normed spaces and examined some of their properties. In this study we have introduced the concept of Cesàro summability for sequences in 2-normed spaces and obtained necessary and sufficient conditions under which convergence follows from Cesàro summability.

Keywords: Tauberian conditions, 2-normed spaces, Cesàro summability. References: [1] A. G. White Jr., “2-Banach spaces”, Math. Nachr. 42(1969), 43–60. [2] A. Şahiner, M. Gürdal, S. Saltan, H. Gunawan, “Ideal convergence in 2- normed spaces”, Taiwanese J. Math., 11(2007), 1477-1484. [3] E. Savaş, “Δm-strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function”, Appl. Math. Comput., 217(2010), 271-276. [4] M. Gürdal, S. Pehlivan, “The statistical convergence in 2-Banach spaces”, Thai J. Math., 2(2004), 107-113. [5] M. Gürdal, S. Pehlivan, “Statistical convergence in 2-normed spaces”, Southeast Asian Bull. Math., 33(2009), 257-264. [6] P. Das, E. Savaş, S. Bhunia, “Two valued measure and some new double sequence spaces in 2-normed spaces”, Czechoslovak Math. J., 61(2011), 809- 825. [7] S. Gähler, “Lineare 2-normierte Räume”, Math. Nachr., 28(1964), 1–43. ______189

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Applications of the Schwarz Lemma to Inequalities for Polynomials with Restricted Zeros

Lubna Wali Shah

Department of Mathematics, National Institute of Technology, Srinagar, Jammu and Kashmir India [email protected]

Abstract: By using a boundary refinement of the classical Schwarz Lemma some results for the polynomial inequalities with restricted zeros have been proved.The estimates obtained strengthen some known results earlier proved by Lax, Turan, Aziz and others.

Keywords: Schwarz Lemma, Inequalities in the complex domain, s-fold zeros

References: [1] Abdul Aziz, A refinement of an inequality of S. Bernstein, J. Math. Anal.

Appl., 142(1989), 1-10. [2] A. Aziz and W. M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl., 7(3) (2004), 379-391. [3] S. Bernstein, Sur la limitation des derivees des polynomes, C. R. Acad. Sci.Paris., 190(1930), 338-340. [4] V. N. Dubinin, Applications of the Schwarz Lemma to inequalities for entire functions with constraints on zeros. J. Math. Sci.,(N.Y) 143(3)(2007), 3069- 3076. [5] S. G. Krantz, The Schwarz Lemma at the Boundary, Complex Var. EllipticEqu., 56(5)(2011), 455-468.

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Growth of Maximum Modulus of Polynomials and Rational Functions in the Complex Domain

Wali Mohammad Shah

Department of Higher Education University of Kashmir, Srinagar, India. [email protected]

Abstract: In this talk we discuss the latest developments concerning the extremal properties and growth of maximum modulus of polynomials and rational functions. Our main objective will be to discuss Bernstein [4] type of inequalities for rational functions with prescribed poles and the use of a boundary refinement of Schwarz lemma [2,3] and a lemma due to Dubinin [1]. We [5] also show how the inequalities for the derivative and polar derivative besides the growth of polynomials can be deduced as special cases from these inequalities concerning the rational functions. Key words : Polynomials, Rational functions, Inequalities , zeros References: [1] V.N. Dubinin, Distortion theorems for polynomials on the circle, Sb. Math., 191(12) (2000) 1797-1807. [2] S. G. Krantz, The Schwarz lemma at the boundary, Complex Var. Elliptic Equ., 56 (5) (2011), 455-468. [3] R. Osserman, A sharp Schwarz inequality on the boundary for functions regular in the disk, Proc. Amer . Math. Soc., 12(2000), 3513-3517 . [4] Q. I. Rahman and G Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York, 2002. [5] S. L. Wali and W. M. Shah, Some applications of Dubinin's lemma to rational functions with prescribed poles., J. Math. Anal. Appl., 450(2017), 769- 779.

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Trace Formula for Witt Vector Rings

Mokhfi Siham

Department of Mathematics, Saad DahlabUniversity, Blida, Algérie [email protected]

Abstract: We commence by giving a generalisation of Pulita exponential series. We thenuse series to establish an analog of the trace formula for Witt vector rings.

Keywords: Trace formula, Pulita series, Witt vector rings. References: [1] E. Artin, “Algebraic Numbers and Algebraic Functions”, Gordon and Breach, New York. Math (1967). [2] D. Barsky, “On Morita's p-adic gamma Functions”, Math. Proc.Camb.phil. Soc, 89, Fasc 1(1981), 23-27. [3] B. Benzaghou, “Algèbres de Hadamard ”, Soc. Math. France Slovaca, 98(1970), 209-252. [4] R. Blache, “Stickelberger Theorem for p-adic Gauss sums”, Acta Arithmetica, vol 18 n°1(2005), 11-26. [5] B. Dwork , “The Rationality of the zeta Function of an Algebraic Variety ”, Amer. Math 82 (1960) 631-648. [6] M. Hazewinkel, “Witt vectors ”, Part 1 . Handbook of Algebra , Vol 6 section 4H, 319-472. Elsiever , North Holland. (1960) 631-648 [7] J. Lubin and J.Tate , “Formal Complex Multiplication in Local Fields ”, Ann of Math (2) 81 (1965) 380-387. [8]B. Benzaghou and S. Mokhfi , “Trace formula for Witt Vector Rings”, Comptes Rendus Mthématiques (2016) [9] A. Pulita , “Rank one solvable p-adic Differential Equations and Finite Abelian Characters via Lubin-Tate groups ”, Math. Ann. 337 (2007) n°03 489- 555.

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On Parametrization of the q-Bernstein Basis Functions

Yilmaz Simsek

Department of Mathematics, Faculty of Science University of Akdeniz TR-07058, Antalya, Turkey [email protected]

Abstract: In this talk, we study main properties of the q-Bernstein basis functions with their generating functions. We show that these q-Bernstein basis functions are parametrizations of the standard Bernstein basis functions. We define q-Bezier type curves. Moreover, we plot graphs of these basis functions, their generating funtions and q-Bezier type curves. Finally, we give remarks and observations on the q-Bezier type curves related to parametrizations of the q- Bernstein basis functions.

Keywords: q-Bernstein basis functions, Bezier curves, Generating function. References: [1] S. N. Bernstein, “Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités”, Comm. Soc. Math. Charkow Sér. 2 t. 13, 1-2 (1912- 1913). [2] R. Goldman, “Identities for the Univariate and Bivariate Bernstein Basis Functions”, Graphics Gems V, edited by Alan Paeth, Academic Press, (1995), 149-162. [3] T. Kim, L.-C. Jang, and H. Yi, “A note on the modified q-Bernstein polynomials”, Discrete Dyn. Nat. Soc. 2010, Article ID 706483, 12 pages, 2010. [4] T. Kim, “Some identities on the q-integral representation of product of several q-Bernstein-type polynomials”, Abst. Appl. Anal., 2011, Article ID: 634675, 1-11. [5] M.-S. Kim, D. Kim, T. Kim, “On q-Euler numbers related to the modified q- Bernstein polynomials”, Abst. Appl. Anal. 2010, Article ID 952384, 15 pages. [6] G. M.Phillips, “Bernstein polynomials based on the q-integers”, The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, Annals of Numerical Math. 1-4 (1997), 511-518. [7] Y. Simsek, “Interpolation function of generalized q-Bernstein type polynomials and their application, Curve and Surface”, Springer Verlag Berlin Heidelberg 2011, LNCS 6920, (2011), 647-662.

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Control of the Performance of the Panel of Judge in Sensory Analysis by a Functinal Principal Component Analysis of Probability Densities Function

Yousfi Smail

Mouloud Mammeri University [email protected]

Abstract: In this work, the Functional Principal Component Analysis (FPCA) of a set of probability density functions is used to study the performance of a panel of 14 judges in sensory analysis context. For this, we integrate the FPCA of densities in the context of Multiblock data analysis where, each block contain the scores assigned by judges for 10 rosebushes according two sensory variables at three different occasions. The planes of representations of densities gives a firt global appreciation of variations contained in the data, one part of these variations summarizes the repetability of the scores assigned by the judges and another part, the discrimination of products tested.

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Optimization of an Execution Time for Parallel Matrix Multiplication by adding a New Set of Processors on the Array

1Halil Snopce, 2Sadri Alija, 3Azir Aliu and 4Artan Luma

1,3,4 Faculty of Contemporary Sciences and Technologies, SEE-University Tetovo, Macedonia 2Faculty of Business and Economics, SEEU, Ilindenska n.335, Tetovo, Macedonia [email protected], , [email protected], [email protected], [email protected]

Abstract: In this paper we investigate the possible methods of optimization of the execution time of some parallel algorithms for matrix-matrix multiplication. Actually, there is known that the time complexity of execution the algorithm of matrix multiplication by using just one processor element is O(n3). If one uses the hexagonal array of n2 processor the optimized execution time is proved to be 3n-2. In this research, among others we show that if the array of processors is increased by n new processors (putting new column of processors), the execution time will be shortened for one unit of time. If we continue adding new columns of processors, the execution time of the algorithm is shortened consecutively per one unit of time after each new step. Assuming that there are put k columns of n processors, then we show that the time complexity of an algorithm is optimized to the order of O(N). Actually, we show that this value at the worst case is equal to 2n-k where n is the number of processors per column and k is the number of new added columns of processors to the hexagonal array of processors. For confirmation of obtained results, there are used some methods of linear algebra. . Keywords: Matrix multiplication, optimization, execution time of algorithm, time complexity, hexagonal array of processors, linear methods.

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From Dido to Morrey: Variational Problems and Regularity Theory!

Lubomira G. Softova

Second University of Naples Department of Civil Engineering, Design, Construction and Environment, Italy [email protected]

Abstract: Although ancient Greek and Roman sources report that Dido, the founder and first queen of Carthage was the first person who formulated a problem in Calculus of Variations (CV), the classical existence theory is connected mainly with the names of Euler, Lagrange and Ostrogradskij. The notorious Euler-Lagrange equation is a second order Partial Differential Equation (PDE), the solvability of which ensures the existence of a minimizer of a given functional. The question of regularity for the solutions of this PDE was firstly posed by David Hilbert in his 19th and 20th problem, presented during a celebrated lecture at the International Congress of Mathematics (ICM) 1900 in Paris. During the last century, these two problems gave a strong impulse to the development of the regularity theory for problems from CV and PDE. Our goal is to present some classical and new results concerning regularity properties of the solutions to the Dirichlet problem for elliptic equations and systems. We obtain essential boundedness of the solution to a class of nonlinear elliptic systems. In addition, we establish estimates in

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A Publicly Verifiable Authenticated Encryption Scheme Based on Chaotic Maps and Factoring Problems

Nedal Tahat

Department of Mathematics, Faculty of Sciences The Hashemite University, Zarqa 13133, Jordan [email protected]

Abstract: In this study, an authenticated encryption scheme with public verifiability based on chaotic maps and factoring problems is proposed. The main aim of deploying a chaos-based cryptosystem is to provide encryption with several advantages over traditional encryption algorithms such as high security, speed, and reasonable computational overheads and computational power requirements. Therefore, to enhance system security, we explore the implementation of a cryptosystem algorithm based on both cryptographic and chaotic system characteristics. We also provide security against known cryptographic attacks and discuss the performance analysis of the developed system.

Keywords: Authenticated encryption scheme, chaotic maps, factoring problem. References: [1] C. Tsai, C. Liu, S. Tsaur and M. Hwang, A publicly verifiable authenticated encryption scheme based on factoring and discrete logarithms. International Journal of Network Security, 2017, 19(3): 443-4480 [2] P. Horster, M. Michel and H. Peterson, “Authenticated encryption schemes with low communication costs,” Electronics letters, 1994, 30(15): 1212-1213. [3] W.-B. Lee and C.-C. Chang, “Authenticated Encryption without Using a One Way Function,” Electronics Letters,1995,31(19):1656-1657. [4] Y. Zheng, “Digital Signcryption or How to Achieve Cost (Signature & Encryption) << Cost (Signature) + Cost (Encryption),” Proc. CRYPTO’97, LNCS 1294, Springer Verlag,1997: 165-179. [5] Y. Zheng, “Signcryption and Its Application in Efficient Public Key Solution,” Proc. Information Security Workshop (ISW’97), LNCS 1397, 1998.Springer-Verlad: 291-312. [6] C. Ma and K. Chen, “Publicly Verifiable Authenticated Encryption,” Electronics Letters, 2003. 39(3): 281-282. [7] S. F. Tzeng, Y. L. Tang, and M. S. Hwang, \A new convertible authenticated encryption scheme with message linkages," Computers and Electrical Engineering, vol. 33, no. 2, pp. 133-138, 2007. ______197

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Third-Order Differential Sandwich-type Results Involving the Liu-Owa Operator

1 2 3 1 Huo Tang , M. K. Aouf , Shigeyoshi Owa and Shu-Hai Li

1. School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, People's Republic of China [email protected]; [email protected] 2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt [email protected] 3. Department of Mathematics, Faculty of Education, Yamato University, [email protected]

Abstract: In this paper, we derive some third-order differential subordination and superordination results for multivalent analytic functions in the open unit disk, which are defined by using the Liu-Owa operator. The results are obtained by investigating appropriate classes of admissible functions. New third-order differential sandwich-type results for the Liu-Owa operator are also obtained.

Keywords: Differential subordination; Differential superordination; Multivalent analytic functions; Sandwich-type results; Liu-Owa operator. References: [1] R. M. Ali, V. Ravichandran and N. Seenivasagan, Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava operator, J. Franklin Inst. 347 (2010), 1762-1781. [2] R. M. Ali, V. Ravichandran and N. Seenivasagan, On subordination and superordination of the multiplier transformation for meromorphic functions, Bull. Malaysian Math. Sci. Soc. 33 (2010), 311-324. [3] J. A. Antonino and S. S. Miller, Third-order differential inequalities and subordinations in the complex plane, Complex Var. Elliptic Equ. 56 (2011), 439-454. [4] I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993), 138-147.

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Second-Order Differential Superordination for Analytic Functions in the Upper Half-Plane

1 2,3 4 1 Huo Tang , H. M. Srivastava , Guan-Tie Deng and Shu-Hai Li

1. School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, People's Republic of China [email protected]; [email protected] 2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China [email protected] 4. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China [email protected]

Abstract: There are many articles in the literature dealing with the second-order differential subordination and differential superordination problems for analytic functions in the unit diskUzzC andz:1 , but there are only a few articles dealing with the above problems in the upper half-plane  zzC:Im()0 andz  . The concept of second-order differential subordination in the upper half-plane was introduced by Raducanu and Pascu in [1], and studied recently by Tang et al. in [2]. Let  be a set in the complex plane C . Also let pz() be analytic in the upper half-plane  and suppose that  :CC3   . In this paper, we investigete the problem of determining properties of functions that satisfy the following second-order differential superordination:   (p (), z p (), z p  ();): z z z  C . Applications of these results to second-order differential superordination for analytic functions in are also presented.

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Keywords: Differential subordination; Differential superordination; Analytic functions; Admissible functions; Upper half-plane. References: [1] D. Raducanu and N. N. Pascu, Differential subordinations for holomorphic functions in the upper half-plane, Mathematica (Cluj). 36(1994), 215-217. [2] H. Tang, M. K. Aouf, G.-T. Deng and S.-H. Li, Differential subordination results for analytic functions in the upper half-plane, Abstr. Appl. Anal. 2014 (2014), Article ID 565727, 1-6.

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Capacity Sizing and Pricing with Heterogeneous Products and Flexible Resources

Salih Tekin

Industrial Engineering Department, TOBB University of Economics and Technology, Ankara, Turkey [email protected]

Abstract: We consider the capacity and pricing decisions made by a monopolistic firm producing three heterogeneous products under demand uncertainty. The objective is to maximize profit. Our model includes dedicated and flexible resources, product substitutability, and processing rates that may depend on the product and on the resource type. We provide the optimum prices and production quantities as functions of resource capacities and demand intercepts. We also show that investment in flexible capacity is only desirable when it is optimal to invest in dedicated capacities for both products, and obtain upper bounds for the costs of the dedicated capacities that need to be satisfied for investment in the flexible resource. We conclude with numerical examples that illustrate the points discussed and provide insights into how the optimal capacities and expected production quantities, prices, and profit depend on various model parameters.

Keywords: Revenue Management, Stochastic Programming, Pricing.

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A New Class of the r-Stirling Numbers and the Generalized Bernoulli Polynomials

Meriem Tiachachat

Faculty of Mathematics, RECITS's laboratory, USTHB Algiers, Algeria. [email protected]

Abstract: The r-Stirling number of the second kind, counts the number of partitions of an n-set into k non-empty subsets such that the r-first elements are in different subsets [3]. In [11], the authors expressed the n-th high order polynomial [7, 14] in terms of the r-Stirling numbers of both kinds by some formulas. In 2010, Kurt [6] introduced an extension of the generalized Bernoulli polynomials. The main object of this paper is to express the values at non negative integers of the generalized Bernoulli polynomials on using a new class of the Stirling numbers of the second kind.

Keywords: The r-stirling numbers, the quasi associated r-Stirling numbers, the generalized Bernoulli polynomials References: [1] E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934) 258-277. [2] G. Bretti, P. Natalini, P. E. Ricci, Generalizations of the Bernoulli and Appell polynomials, Abst. Appl. Anal., 7 (2004) 613–623. [3] A. Z. Broder, The r-Stirling numbers, Discrete Math., 49 (1984) 241-259. [4] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht-Holland / Boston-U.S.A, (1974). [5] H. W. Gould, Higher order extensions of Melzaks formula, Util. Math., 72 (2007) 23-32. [6] B. Kurt, A further generalization of the Bernoulli polynomials and on the 2D- Bernoulli polynomials B 2 n (x; y); Appl. Math. Sci., 4 (2010) 2315-2322. [7] Y. L. Luke, The special functions and their approximations, vol. I, Academic Press, New York, London, 1969.

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Degree Sequences and Inverse Problem on Graphs

Muge Togan, Aysun Yurttas, Ismail Naci Cangul

Department of Mathematics, Uludag University, Bursa, Turkey [email protected], [email protected], [email protected]

Abstract: Degree sequence of a graph is the set of all vertex degrees of the given graph in non-decreasing order. There are algorithms to determine which sets of positive integers can be the degree sequence of a graph, but yet no complete solution is given. Topological graph indices are getting more and more interest every day due to their applications in Chemistry and Pharmacology and most of them are calculated for many graph types. Molecular graphs are the graphs with vertex degree at most 4. A recent problem in graph theory is called inverse problem. This problem deals with what integer values can be taken by these topological graph indices. In this talk, by means of degree sequences, we give the answer for several topological indices.

Keywords: Degree sequence, inverse problem, topological graph index References: [1] H. Wang, G. Yu, All but 49 numbers are Wiener indices of trees, Acta Appl Math 92 (2006) 15-20 [2] I. Gutman, Y.-N. Yeh, The sum of all distances in bipartite graphs, Math. Slovaca 45 (4) (1995) 327-334 [3] S. M. Cioaba, Sums of powers of the degrees of a graph, Discrete Mathematics 306 (2006) 1959-1964

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Notes on Permuting tri-derivations on Prime and Semi-prime Rings

Seda Oguz Unal, Hasret Durna

Cumhuriyet University, Sivas, Turkey [email protected], [email protected]

Abstract: Derivations in prime rings firstly initiated by Posner [1] and it is considered a fundamental construction in the theory of centralizing maps on prime rings.. A great deal of work in this context are available in the literature (see, for example [2] and [3]). In this sense, in [4], Ozturk presented permuting tri-derivations in prime and semi-prime rings.The aim of this talk is to show that a ring is a prime and semiprime ring admitting the trace satisfying several conditions of permuting tri-derivation.

Keywords: Permuting tri-derivations, traces, prime rings, semi-prime rings, Lie ideals. References: [1] E. C. Posner,"Derivations in prime rings", Proc. Amer. Math. Soc., 8(1957), 1093–1100. [2] N. Argac and M. S. Yenigul,"Lie ideals and symmetric bi-derivation on prime and semi-prime rings", Pure and Applied Mat. Sci., 44(1996), 17–21. [3]M. A. Ozturk, D. Ozden and Y. B. Jun,"Permuting tri-derivations in prime and semi-prime gamma rings",Kyungpook Math. J., 46(2011), 153–167.

[4] M. A. Ozturk, " Permuting tri-derivations in prime and semi-prime rings", East Asian Math. J., 15(1999), 177–190.

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Weakly Invariant Subspaces for Multivalued Linear Operators on Banach Spaces

Gerald Wanjala

Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman [email protected]

Abstract: Losomonov [1] showed that if a bounded linear operator T on a Banach space X commutes with a non-zero compact operator, then T has a nontrivial invariant subspace. This result was generalized by Peter Saveliev [2] to the case of multivalued linear operators by applying fixed point techniques. In particular, he proved that if S and K are multilivalued linear operators defined on a Banach space X and having finite dimensional multivalued parts, and if K is compact and right commutes with S, that is SK ⊂ KS, then S has a nontrivial weakly invariant subspace. However, the case of left commutativity remained open. In this talk, we address the case of left commutativity. We apply the operator representation techniques developed in [3] to acheive the desired result. Keywords: Multivalued linear operator, Weakly invariant subspace.

References: [1] V. I. Lomonosov, “Invariant subspaces for the family of operators which commute with a completely continuous operator”, Func. Anal. Appl., 7(1973), 213-214. [2] P. Savelieve, “Lomonosov’s invariant subspace theorem for multivalued linear operators”, Proc. AMS, 131(2002), 825-8344. [3] G. Wanjala, “Operator representation of sectorial linear relations and applications”, J. Ineq. Appl., (2015), 2015:60.

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Speech Quality Analysis with Respect to Noise Corruption by a Kalman Filter to Estimation the Parameters of the SWLP Method

Ervenila Xhaferraj (Musta)

Department of Mathematics, Faculty of Mathematics and Physics Engineering, Polytechnic University of Tirana Tirane, Albania [email protected] Abstract: Revolutions Applications such as telecommunications, hands-free communications, recording, etc. which need at least one microphone, the signal is usually infected by noise and echo. The important application is the speech enhancement, which is done to remove suppressed noises and echoes taken by a microphone, beside preferred speech. Accordingly, the microphone signal has to be cleaned using digital signal processing DSP tools before it is played out, transmitted, or stored. Engineers have so far tried different approaches to improving the speech by get back the desired speech signal from the noisy observations. Especially Mobile communication, so in this paper will do reconstruction of the speech signal, observed in additive background noise, using the Kalman filter technique to estimate the parameters of the Autoregressive Process (AR) in the state space model and the output speech signal obtained by the MATLAB. The accurate estimation by Kalman filter on speech would enhance and reduce the noise then compare and discuss the results between actual values and estimated values which produce the reconstructed signals.

References: [1]. J. R. Deller, J. H. L. Hanson and J. G.Proakis. Diccrere-The Processing of Speech Signal, EEE PRESS, NewYork 2000. [2]. K. K. Paliwal and A.Basu, “A speech enhancement method based on Kahan filtering, Proceedings of ICASSP’87,pp.177-180. Dallas, TX, USA, 1987. [3]. T.K.Mmn, “The Expectation-Maxlmizaion Algorithm”,IEEE Signof Processing Magazine, pp. 47-60, Nov. 1996. [4] Shumway, Robert H., Stoffer, David S., Time Series Analysis and Its Applications with R Examples, second edtition, Springer Texts in Statistics Series, 2006.

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S-Generalized Srivastava’s Triple Hypergeometric Functions

M. Baki Yagbasan, Aysegul Cetinkaya, I. Onur Kiymaz

Department of Mathematics, Ahi Evran University, Kırşehir, Turkey [email protected]

Abstract: In this study, we introduced new generalizations of Srivastava's triple hypergeometric functions by using S-generalized beta function. Furthermore, we investigated some properties of these new functions.

Keywords: S-Generalized Beta function, Srivastava's triple hypergeometric functions. Acknowledgement: This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.A4.17.002 References: [1] Çetinkaya, A., Yagbasan, M. B., Kıymaz İ. O., “The extended Srivastava’s triple hypergeometric functions and their integral representations”, Journal of Nonlinear Sciences and Applications, 9.6, (2016): 4860-4866. [2] Bailey W.N., “Generalized Hypergeometric Series”, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 32, Cambridge University Press, Cambridge, (1935). [3]Chaudhry M. A., Qadir A., Rafique M., Zubair S. M., “Extension of Euler's beta function”, J. Comput. Appl. Math., 78, (1997): 19-32. [4] Luo, Min-Jie, Milovanovic, G. V., Agarwal, P. “Some results on the extended beta and extended hypergeometric functions”, Applied Mathematics and Computation, 248, (2014): 631-651. [5] Srivastava H. M., Karlsson P. W., “Multiple Gaussian Hypergeometric Series”, Ellis Horwood Limited, (1985). [6] Srivastava, H. M., Agarwal, P., Jain, S. “Generating functions for the generalized Gauss hypergeometric functions”, Applied Mathematics and Computation, 247, (2014): 348-352. [7] Srivastava, H. M., Jain, R., Bansal, M. K., “A Study of the S-Generalized Gauss Hypergeometric Function and Its Associated Integral Transforms”, Turkish Journal of Analysis and Number Theory, 3.5, (2015): 116-119.

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Berezin Number Inequality for Convex Function in

Reproducing Kernel Hilbert Space

Ulas Yamanci1, Mehmet Gurdal2, Mubariz T. Garayev3

1Department of Statistics, Süleyman Demirel University, Isparta, Turkey 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey 3Department of Mathematics, King Saud University, Riyadh, Saudi Arabia [email protected]; [email protected]; [email protected]

Abstract: By using Hardy-Hilbert's inequality, some power inequalities for the Berezin number of a self-adjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs) with applications for convex functions are given.

Acknowledgement: The second author is supported by TUBA through Young Scientist Award Program (TUBA-GEBIP/2015). Keywords: Berezin number, Hardy-Hilbert’s inequality, convex function, self- adjoint operator. References: [1] G. Hardy, J.E. Littlewood, G., Polya, “Inequalities”, 2 nd ed. Cambridge University Press, Cambridge, 1967. [2] F.A. Berezin, “Covariant and contravariant symbols for operators”, Math. USSR-Izv., 6 (1972), 1117-1151. [3] M. Kian, “Hardy-Hilbert type inequalities for Hilbert space operators”, Ann. Funct. Anal., 3(2)(2012), 128-134. [4] M.T. Garayev, M. Gürdal, M., A. Okudan, “Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators”, Math. Inequal. Appl., (3)(19)(2016), 883-891. [5] B. Yang, “Discrete Hilbert-type inequalities”, Bentham Science Publishers Ltd., 2011.

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On Power Inequalities for Berezin Number of Operators and Convex Functions

Ulas Yamanci1, Mehmet Gurdal2, Ceren Celik2

1Department of Statistics, Süleyman Demirel University, 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey [email protected]; [email protected]; [email protected]

Abstract: In this paper, some power inequalities are given for Berezin numbers, defined by reproducing kernel, on reproducing kernel Hilbert spaces.

Acknowledgement. This work is supported by Suleyman Demirel University with Project 4903-YL1-17. Keywords: Berezin number, Hardy-Hilbert’s inequality, convex function, reproducing kernel Hilbert space. References: [1] G. Hardy, J.E. Littlewood, G., Polya, “Inequalities”, 2 nd ed. Cambridge University Press, Cambridge, 1967. [2] F.A. Berezin, “Covariant and contravariant symbols for operators”, Math. USSR-Izv., 6 (1972), 1117-1151. [3] M. Kian, “Hardy-Hilbert type inequalities for Hilbert space operators”, Ann. Funct. Anal., 3(2)(2012), 128-134. [4 M.T. Karaev, “Berezin symbol and invertibility of operators on the functional Hilbert spaces”, J. Funct. Anal., 238(2006), 181-192. [5] S. Saitoh, Y. Sawano, “Theory of reproducing kernels and applications”, Springer, 2016.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Spectral Properties of Discrete Klein-Gordon Equations

Nihal Yokus, Nimet Coskun

Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, Turkey [email protected] ; [email protected]

Abstract: Spectral analysis of Sturm-Liouville boundary value problem was investigated by Naimark[1]. Bairamov and Celebi[2] studied the Klein-Gordon s- wave equation which includes the Sturm-Liouville equation as a special case in their paper. As a result of developments in discrete calculus, discrete analogues of the Sturm-Liouville and Klein-Gordon equation have gained a prominent attention. Spectral theory of discrete Klein-Gordon equation has been treated by Adivar[3]. In this study, we shall present Jost solution, discrete spectrum and spectral singularities of the discrete Klein-Gordon equation under certain conditions.

Keywords: Discrete equations, spectral analysis, Klein-Gordon equation. References: [1] M.A. Naimark, ‘‘Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis’’, AMS Transl. 2(1960), 103-193. [2] E. Bairamov and A.O. Celebi, ‘‘Spectral properties of the Klein-Gordon s- wave equation with complex potential’’, Indian J. Pure Appl. Math. 28(1997), 813-824. [3] M. Adivar, ‘‘Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors’’, Quaestiones Mathematicae, 33(2010), 305- 323.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

On the Inverse Problem on Graphs

Aysun Yurttas, Muge Togan, Ismail Naci Cangul

Department of Mathematics, Uludag University, Bursa, Turkey [email protected], [email protected], [email protected]

Abstract: Topological graph indices are getting more and more interest every day due to their applications in Chemistry and Pharmacology. Their values are calculated for many graph types and relations with molecular graphs are established. Molecular graphs are the graphs with vertex degree at most 4. A recent problem in graph theory is called inverse problem. This problem deals with what integer values can be taken by these topological graph indices. In this talk, we give the answer for several topological indices.

Keywords: Molecular graph, inverse problem, topological graph index References: [1] H. Wang, G. Yu, All but 49 numbers are Wiener indices of trees, Acta Appl Math 92 (2006) 15-20 [2] I. Gutman, Y.-N. Yeh, The sum of all distances in bipartite graphs, Math. Slovaca 45 (4) (1995) 327-334 [3] S. M. Cioaba, Sums of powers of the degrees of a graph, Discrete Mathematics 306 (2006) 1959-1964

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Summability of Subsequences of Divergent Sequences

Maria Zeltser with Johann Boss

School of Digital Technologies, Mathematics and Didactics of Mathematics, Tallinn University, Tallinn, Estonia [email protected]

Abstract: C. Stuart proved in [1] that the Cesàro matrix 퐶1 cannot sum almost every subsequence of a bounded divergent sequence 푥. At the end of the paper he stated the problem whether this proposition could be generalized for any regular matrix. We confirm Stuart's conjecture, and even we extend it to the more general case of divergent sequences x. Moreover we determine a class 푄 of subsequences (푥푛푖) of 푥 such that a given regular matrix does not sum sum all elements of Q.

Keywords: Regular matrices, summability of subsequences. References: [1] C. Stuart. Summability of subsequences of a divergent sequence. Rocky Mountain J. Math., 44(1): 289-295, 2014.

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INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2017) May 11-15, 2017, Palm Wings Ephesus Resort Hotel, Kusadasi - Aydin, TURKEY www.icrapam.org

Many to one Embedding Crossed Cube into Pancake

Mohamed Faouzi Zerarka

Abstract: Among Cayley graphs on the symmetric group, the pancake graph is one as a viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computers. Some good and attractive properties of this interconnection network include: Small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing, and regularity of topology, fault tolerance, extensibility and embeddability of other topologies. A graph embedding has been known as a powerful tool for implementation of parallel algorithms and simulation of interconnection networks. In this paper, we present the many-to- one model of mapping nodes and edges of crossed cubes into nodes and paths of pancake respectively and to minimize the dilation and the expansion costs.

Keywords: Embedding; dilation, expansion, crossed cubes, pancake. References: 1. X. Yang, Q. Dong and Y.Y. Tan, Embedding meshes/tori in faulty crossed cubes, Information Processing Letters, Vol. 110, no. 14-15, 559 - 564, 2010. 2. P.-L. Lai and C.-H. Tsai, Embedding of tori and grids into twisted cubes, Theoretical Computer Science, Vol. 411, no. 40-42, 3763 - 3773, 2010. 11. 3. Y. Han, J. Fan, S. Zhang, J. Yang and P. Qian, Embedding meshes into locally twisted cubes, Information Sciences, Vol. 180, no. 19, 3794 - 3805, 2010. 12. 4. J. Fan and X. Jia, Embedding meshes into crossed cubes, Information Sciences, Vol. 177, no. 15, 3151 - 3160, 2007. 5. Y. Saad and M.H. Schultz, Topological properties of hypercubes, IEEE Transactions on Computers,Vol. 37, no. 7, 867 - 872, 1988. 6. K. Efe, The crossed cube architecture for parallel computing,IEEE Transactions Parallel and Distruibuted Systems, Vol. 3, no. 5, 513 -524, 1992 8. Akers S. and Krishnamurthy B., “A Group Theoretic Model for Symmetric Interconnection Networks,” IEEE Transactions on Computers, vol. 38, no. 4, pp. 555-566, 1989. 9. Asai S., Kounoike Y., Shinano Y., and Kaneko K., “Computing the Diameter of 17-Pancake Graph Using a PC Cluster,” Euro-Par 2006 Parallel Processing, Dresden, Germany, pp. 1114-1124, 2006 10. Christian Lavault. Embeddings into the Pancake Interconnection Network. Parallel Processing Letters, World Scientific Publishing, 2002, 12 (3-4), pp.297- 310 ______213