Abstract Book - 2018.icrapam.orgTania Luminita Costache 58. x Determination of a coefficient in a second-order hyperbolic problem Arzu Erdem Coúkun 60 On the Existence and Uniqueness

i

5th INTERNATIONAL CONFERENCE ON RECENT

ADVANCES IN PURE AND APPLIED

MATHEMATICS, TRABZON, TURKEY

23-27 JULY 2018

Abstract Book

ii

5th INTERNATIONAL CONFERENCE ON RECENT

ADVANCES IN PURE AND APPLIED

MATHEMATICS, TRABZON, TURKEY

23-27 JULY 2018

Abstract Book

List of Major Sponsors

• Karadeniz Technical University

• The Turkish Academy of Sciences (Türkiye Bilimler Akademisi – TÜBA)

iii

5th INTERNATIONAL CONFERENCE ON RECENT ADVANCES IN PURE AND APPLIED

MATHEMATICS, TRABZON, TURKEY, 23-27 JULY 2018

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, Uşak Commerce University Prof. Dr. Hamdhllah Şevli, Istanbul Commerce University Prof. Dr. Necip Şimşek, Istanbul Commerce University Prof. Dr. Ömer Pekşen, Karadeniz Technical 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. Rahmet Savas, Istanbul Medeniyet University Assoc. Prof. Dr. Erhan Deniz, Kafkas University Assoc. Prof. Dr. Mahpeyker Ozturk, Sakarya University Assoc. Prof. Dr. Bahadır Özgür Güler, Karadeniz Technical University Assoc. Prof. Dr. İdris Ören, Karadeniz Technical University ASsists. Prof. Dr. Ali Hikmet Değer, Karadeniz Technical 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 Assist Prof. Dr. Veli Capali, Usak University Dr. Lakhdar Ragoub, Alyammah University

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Local Organizing Committee

Abdurrahman Buyukkaya, Karadeniz Technical University Melek Eriş Büyükkaya, Karadeniz Technical University Ümmü Gülsüm AKBABA, Karadeniz Technical University Fatih Yilmaz, Karadeniz Technical University Rabia Savas, Sakarya University Sefa Anil Sezer, Istanbul Medeniyet University Tuğba Yıldırım, Istanbul Technical University Ekber Girgin, Sakarya University Neslihan Kaplan Kuru, Sakarya University Madiha Rashid, Pakistan Bouchra Ben AMMA, Morocco

Scientific Committee Prof. Dr. Huseyin Cakalli, Turkey Prof. Dr. Jeff Connor, USA Prof. Dr. Lubomira Softova, Italy Prof. Dr. Reza Langari, USA Prof. Dr. Mikail Et, Turkey Prof. Dr. S. A. Mohiuddine, S. Arabia Prof. Dr. Narendra Kumar Govil, USA Prof. Dr. T. A. Chishti, India Prof. Dr. Ayhan Serbetci, Turkey Prof. Dr. Bilal Altay, Turkey Prof. Dr. Ismail Ekincioglu, Turkey Prof. Dr. A. Sinan Cevik, Turkey Prof. Dr. Leiki Loone, Estonia Prof. Dr. Akbar B. Aliyev, Azerbaijan Prof. Dr. Vali M. Gurbanov, Azerbaijan Prof. Dr. Faqir M. Bhatti, Pakistan Prof. Dr. Said Melliani, Morocco Prof. Dr. Abdalah Rababah, Jordan Prof. Dr. Radouane Yafia, Morocco Prof. Dr. Sudarsan Nanda, India Prof. Dr. Seyit Temir, Turkey Prof. Dr. Halit Orhan, Turkey Prof. Dr. Vatan Karakaya, Turkey Prof. Dr. Amir Khosravi, Iran Prof. Dr. Seifedine Kadry, Kuwait Prof. Dr. Ali M. Akhmedov, Azerbaijan Prof. Dr. Ziyatkan Aliyev, Azerbaijan Prof. Dr. Poom Kumam, Thailand Prof. Dr. Agacik Zafer, Kuwait Prof. Dr. Tunay Bilgin, Turkey Prof. Dr. Gangaram S. Ladde, USA Prof. Dr. Claudio Cuevas, Brazil Prof. Dr. Abdelmejid Bayad Prof. Dr. Amiran Gogatishvili Prof. Dr. Oktay Duman Prof. Dr. Abdelmejid Bayad

Prof. Dr. Charles Swartz, USA Prof. Dr. Yagub A. Sharifov, Azerbaijan Prof. Dr. Niyazi A. Ilyasov, Azerbaijan Prof. Dr. Aref Jeribi, Tunisia Prof. Dr. Husamettin Coskun, Turkey Prof. Dr. Maria Zeltser, Estonia Prof. Dr. Kamalmani Baral, Nepal Prof. Dr. Ants Aasma, Estonia Prof. Dr. Ismail N. Cangul, Turkey Prof. Dr. Murat Tosun, Turkey Prof. Dr. Yilmaz Simsek, Turkey Prof. Dr. Harry Miller, Bosnia Prof. Dr. Ali Fares, France Prof. Dr. Ibrahim Canak, Turkey Prof. Dr. Naim Braha, Kosova Prof. Dr. Mustapha Cheggag, Algeria Prof. Dr. Fahrettin Abdullayev, Kırgizistan Prof. Dr. Praveen Agarwal, India Prof. Dr. P. D. Srivastava, India Prof. Dr. Mehmet Akbaş Prof. Dr. Erhan Coşkun Prof. Dr. Funda Karaçal Prof. Dr. Vasile Berinde Prof. Dr. Haskız Coşkun Prof. Dr. Erdal Karapınar Prof. Dr. Sultan Yamak Prof. Dr. Selçuk Han Aydın Assoc. Prof. Ayhan Aydın Assoc. Prof. Dr. Yasemin Sağıroğlu Assoc. Prof. Dr. Tülay Kesemen Assoc. Prof. Dr. Azhar Hussain Assoc. Prof. Dr. Bayram Ali Ersoy, Turkey Assist. Prof. Vishnu Narayan Mishra, India Assist. Prof. Dr. Hafize Gök Assist. Prof. Dr. Sukran Konca Dr. Dr.K.V.L.N. Acharyulu

v

Prof. Dr. Naila Rozi, Pakistan Prof. Dr. Emine Can, Turkey Prof. Dr. Hemen Dutta, India Prof. Dr. Reza Saadati, Iran Prof. Dr. Salih Aytar, Turkey

Dr. Lejla Miller Van-Wieren, Bosnia Dr. Mayssa Alqurashi, S. Arabia Dr. Fardous Taoufic, S. Arabia Dr. Ammar Edress Mohamed, Iraq

vi

Dear Collogues;

First of all I wish to offer you a warm welcome to the fifth International Conference on Recent

Advances in Pure and Applied Mathematics (ICRAPAM 2018).

The last conference of this series was organized in Kusadası, Turkey, during 11-15

May 2017 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

vii



INVITED TALKS

The Distance Between Two Sets and its Amazing Applications in Science and Technology Prof. Dr. Vasile Berinde

1 1

Totally boundedness and uniform continuity via quasi Cauchy sequences Prof. Dr. Huseyin Cakalli

3

Weighted norm inequalities for positive operators restricted on the cone of ρ-quasiconcave functions Prof. Dr. Amiran Gogatishvilili

4

Cofactor Matrix Theory Prof. Dr. Robin Harte

5

Indispensable remarks on some recent results in metric fixed point theory Prof. Dr. Erdal Karapınar

6

Symmetry Methods and Conservation Laws for Differential Equations Prof. Dr. Chaudry Masood Khalique

8

Best Uniform Approximation with 2n+1 Equioscillations Prof. Dr. Abedallah Rababah

9

viii



LIST OF TALKS

Multi-layer perceptrons with Levenberg-Marquardt training algorithm for predicting of heavy metals

concentrations in marine sediment

Abdelaziz Abdallaoui, Kaoutar Elazhari, Hicham El Badaoui

10

Quarterly prediction of water consumption using some statistical methods. A comparative study

Qais Mustafa Abdulqader

12

Extremal polynomials and their approximation properties in complex plane

Fahreddin Abdullayev, Gülnara Abdullayev

13

On Modular Happy Numbers

Raghib Abusaris

15

Distances from Unbounded Trajectories to their Limit-Strings on a Hadamard Kȁhler Manifold

Toshiaki Adachi

16

The Generating Functions of Recurrence Relation of the Path of Minimal Length on Suborbital Graphs

Ümmügülsün Akbaba, Ali Hikmet Değer, Tuğba Tuylu

17

Portfolio Selection by Fuzzy Linear Programming

Serkan Akbaş, Türkan Erbay Dalkılıç

19

Performance Comparison of Fuzzy Clustering Approaches on Burn Region Detection

Yeşim Akbaş, Tolga Berber

20

Fekete-Szego Functional Problem For A New Class Of M-Fold Symmetric Analytic Functions

Arzu Akgül

21

Continued Fractions and Exact Hermite matrices

Tülay Ayyıldız Akoğlu

22

The Arrowhead-Pell Sequences in Finite Groups

Yeşim Aküzüm, Taha Doğan, Ömür Deveci

23

Boundedness of potential operators in modified Morrey spaces defined on Carleson curves

Canay Aykol

24

Robust PCA-Algorithms for Detection of Metagenomic Biomarkers

Mustafa Alshawaqfeh, Ahmad Bashaireh, Jan Suchodolski, Erchin Serpedin, Khalid Qaraqe

26

Estimates of Faber polynomial coefficients for an unified class of bi-univalent functions defined through

the Jackson (p,q)-derivative operator

Şahsene Altınkaya, Sibel Yalçın

28

Total Lighting Level in the e-Studio and Polynomial Interpolation

Hamit Armağan, Tuncay Yiğit, Mevlüt Ersoy

29

Spherical fuzzy sets and its representation of spherical fuzzy ť-norms and ť-conorms

Shahzaib Ashraf, Saleem Abdullah

31

ix

On some spectral problems of boundary value problem with boundary condition containing spectral

parameter

Nigar Aslanova, Mamed Bayramoglu, Khalig Aslanov

32

Generalizing the order obtained from nullnorms

Emel Aşıcı

34

Numerical Treatment of Second Order Fuzzy Two-Point Boundary Value Problems

Basem S. Attili

35

On Fuzzy Metric Space of Continuous Function

Ebru Aydoğdu, Abdülkadir Aygünoğlu, Halis Aygün

37

Flip Bifurcation Analysis in a Discrete Bacteria Population Model

Ercan Balcı, Senol Kartal

38

Analysis of a Pneumonia Transmission Model with Random Components

Zafer Bekiryazici, Tulay Kesemen, Mehmet Merdan

39

Using Two-Layer Models for Flexible Pavement Sizing

Mouloud Belachia, Nadjette Bouacha

41

A Solution Approach for Multiobjective Probability Maximization Problems

Fatima Bellahcene

43

A Nonsmooth Newton Method for Solving a State Constrained Optimal Control Problem Governed by

Convection-Diffusion Equation

Boubakeur Benahmed, Kheira Mostepha Hanchour

44

Properties of The Bianchi Groups and Digraphs

Murat Beşenk, Serkan Kader, Bahadır Özgür Güler

46

A New Solution of the Einstein-Maxwell Equations with Linear Equation of State

Rashida Bibi, Azad A. Siddiqui

47

Characterizations of extremal graphs for some bounds on k-domination number

Ahmed Bouchou, Mohmed Zemir, Mostafa Blidia

48

Global dynamics of a Fractional model for viral infection with adaptive immunity

Adnane Boukhouima, Khalid Hattaf, Noura Yousfi

49

On BCB protocol

Sadek Bouroubi

51

Some properties of modified new Szasz-Mirakyan operators in polynomial weight spaces via power

summability method

Naim L. Braha

52

The Cardinal and Idempotent Number of Order-preserving and A-increasing Finite Transformations

Semigroup

Leyla Bugay

54

Best Proximity Point Theorems for Generalized Contractions

Abdurrahman Büyükkaya, Mahpeyker Öztürk

55

On The Estimator of Semi-Markovian Inventory Model with Subexponential Component

Melek Eriş Büyükkaya, Aslı Bektaş Kamışlık, Tülay Kesemen, Tahir Khaniyev

56

Projective covariant α-completely positive linear maps

Tania Luminita Costache

58

x

Determination of a coefficient in a second-order hyperbolic problem

Arzu Erdem Coşkun 60

On the Existence and Uniqueness of Solutions for a First And Second-Order Discrete Boundary Value

Problem

Arzu Erdem Coşkun, Ali Demir

62

Instability Intervals for Hill’s Equation with Symmetric Double Well Potential

Haskız Coşkun, Elif Başkaya

64

Analysis of Neutron Capture Cross Section Using Systematics of Different Level Density Model

Veli Çapalı

65

On the extensions of triangular norms and triangular conorms

Gül Deniz Çaylı

66

On Suborbital Graphs and Related Sequences

Ali Hikmet Değer, Ümmügülsün Akbaba, Tuğba Tuylu

68

Characterizations of Lipschitz spaces via commutators of fractional maximal operator in Orlicz spaces

Fatih Deringöz

69

Convergence Theorems on Measure Space of Exhaustive

Doris Doda

70

A New view of L-ideals on Semigroups

Emine Funda Ekinci, Sultan Yamak

71

Design and Application of Fractional Order Predictor for Unglazed Transpired Solar Collector

Köksal Erentürk, Saliha Erentürk

72

Determination of the Effective moisture Diffusivities in Convective Drying Systems of Banana using

Fractional Order Estimator

Saliha Erentürk, Köksal Erentürk

74

On Nullnorms On Bounded Lattices

Ümit Ertuğrul

76

About the Grundy coloring critical graphs

Noureddine Ikhlef-Eschouf

77

f - Statistical Boundedness of Order α

Mikail Et, Hüseyin Sönmez

78

The influence of fluid pressure on the displacement of the basilar membrane

Aboulkhouatem Fatima-Ezzahra 79

Change of the Cochlear Michromechanics model due to fluid presure

Kouilily Fatiha

80

Optimal scientific production over the life cycle

Gustav Feichtinger, Dieter Grass, Peter M. Kort

81

The Extended b-Metric Space and Related Fixed Point Results

Andreea Fulga

82

Suborbital Graphs For a Special Subgroup of the Normalizer

Nazlı Yazıcı Gözütok, Bahadır Özgür Güler

83

𝐺𝐿(𝑛,ℝ) Invariants For a Curve Family

Uğur Gözütok, Yasemin Sağıroğlu

85

xi

Conditions for Ruled Surfaces Obtained by Bending to be Developable

Uğur Gözütok, Hüsnü Anıl Çoban, Yasemin Sağıroğlu

86

Some Generalizations of Ricci Solitons Admitting Standard Static Spacetime Metric

Sinem Güler

87

Rough I - Statistical Convergence of Weight g

Hafize Gümüş

88

The concept of I-Asymptotically Lacunary Statistical Equivalence of Interval Numbers and A

Generalization With A Positive 𝑝 Sequence

Hafize Gümüş, Nihal Demir

89

Exact Solution of Some Nonlinear Partial Differential Equation

Hami Gündoğdu, Ömer Faruk Gözükızıl

90

On Invariance of Hyperstoneanness under Lattice Isomorphisms

Banu Güntürk, Bahaettin Cengiz

91

Nanofluid flow due to rotating disk with variable thickness andhomogeneous-heterogeneous reactions

Tasawar Hayat, Madiha Rashid, Maria Imtiaz, Ahmed Alsaedi

92

Spectral Theory of q,ω -Dirac System

Fatma Hıra

93

Self-Organizing Map of Artificial Neural Networks for superficial sediments classification of the Sidi

Chahed dam (NE of Meknes, Morocco)

Abdellah El Hmaidi, Mohamed Berrada, Abdelaziz Abdallaoui, Ali Essahlaoui, Abdelhadi El Ouali

94

Zero Operator Products on Hardy and Bergman spaces

Mualla Birgül Huban, Mübariz Garayev, Mehmet Gürdal

95

Compactness and Engliś Algebra

Mualla Birgül Huban, Mübariz Garayev, Mehmet Gürdal

96

Solvability of Infinite Systems of Differential Equations in Some BK Spaces

Merve İlkhan, Emrah Evren Kara

97

Dissipative Canonical Type Quasi-Differential Operators for First Order

Zameddin I. Ismailov, Pembe Ipek Al

98

Some Spectral Problems of Direct Sums of Operators

Zameddin I. Ismailov, Elif Otkun Çevik

99

New Exact Travelling Wave Solution of the Generalized Ito Integro-Differential Equation

Anwar Jaafar Mohamad Jawad

100

A Radon-Nikodym type theorem for completely positive multilinear ϕ-maps on Hilbert C*-modules

Maria Joita

101

Approximating Fixed Points of Certain Nonlinear mappings in CAT(0) Spaces

Amna Kalsoom, Hafiz Fukhar ud Din

103

A Parallel Multiobjective Evolutionary Algorithm with an application to the Knapsack Problem

Nedjmeddine Kantour, Sadek Bouroubi, Djamel Chaabane

105

Some Properties of t-closure Operators

Funda Karaçal, Mehmet Akif İnce

106

The Kinematic Model of Nonuniform Rotation of Liquid Medium

Zurab Kereselidze, Marina Chkhitunidze, Irine Khvedelidze, Nino Zhonzholadze

108

xii

Generalized Jensen-Mercer Inequality for Functions with Nondecreasing Increments

Asif R. Khan, Sumayyah Saadi

109

Approximating Min Graphic k-Path TSP with no Prespecified Path Endpoints

Rezika Kheffache

111

k- Generalized Pell Numbers and Its Properties

Havva Kırgız, Kemal Uslu

112

Exact and Nonstandard Finite Difference Schemes for the Generalized KdV-Burgers Equation

Canan Köroğlu, Ayhan Aydın

113

Some Studies on Congruence Equations using Suborbital Graphs

Tuncay Köroğlu, Zeynep Şanlı, Bahadır Özgür Güler, Murat Beşenk

114

On L1 - Convergence of Certain Cosine and Sine modified sums

Xhevat Krasniqi

115

Fixed Point Theorems for α-F-Suzuki Contraction Mapping with Rational Expressions in Branciari b-

Metric Spaces

Neslihan Kaplan Kuru, Mahpeyker Öztürk

116

About the packing chromatic number in graphs

Rachid Lemdani

117

Some characterizations of the homogeneous ruled real hypersurface in a complex hyperbolic space

Sadahiro Maeda

118

Precise long-term prediction of behavior in a 3-D chaotic map

M. Mammeri

119

Constructing 3-D maps with complex chaotic phenomena

M. Mammeri

120

Sequential structures of statistical manifolds on deformed exponential families

Hiroshi Matsuzoe

122

Approximate solutions of the inverse problem without a priori information on the exact solution

Yuri Menshikov

123

Modelling Asymmetric Magnetic Recording Heads With a Soft Underlayer Using Multiple Images Method

Ammar Edress Mohamed

124

Coefficient bound estimates for beta-starlike functions of order alpha

Nizami Mustafa, Veysel Nezir

126

On the coefficient bounds of gamma and beta starlike functions

Nizami Mustafa, Veysel Nezir, Mustafa Ateş

128

Application of the Chebyshev polynomials to coefficient estimates of analytic functions

Nizami Mustafa, Veysel Nezir, Erdi Akbulut

129

On the coefficient bounds of certain subclasses of analytic functions of complex order

Nizami Mustafa, Veysel Nezir, Tarkan Öztürk

130

Some Results on Classical 2-Absorbing Submodules

Osama A. Naji

131

Fixed point properties for a degenerate Lorentz-Marcinkiewicz space

Veysel Nezir, Nizami Mustafa

132

xiii

Renorming c0 and affine fixed point property

Veysel Nezir, Nizami Mustafa, Tahsin Ateş,

134

Infinite dimensional subspaces of a degenerate Lorentz-Marcinkiewicz space with the fixed point property

Veysel Nezir, Nizami Mustafa, Merve Delibaş

135

Applying Fourier Analysis to Computing New variables for Determining SNR

Sajedeh Norozpour, Cem Ergün

136

On Nordenian Properties of the Riemannian extension

Filiz Ocak

138

Numerical Solution of Nonlinear Singular Ordinary Differential Equations Arising in Biology via

Operational Matrix of Bernstein Polynomials

Sunday Adeyemi Olagunju, Peter Vanenchii Ayoo

139

On Ciric Type Contractions in Modular Spaces

Mahpeyker Öztürk

141

Some fixed point results concerning F-contractions in complete metric space

Alexandrina Maria Proca

142

Fixed point theorem for F-contraction type and applications to polynomial approximations

Abdalah Rababah, Hichem Ramoul, Nora Mahloul

144

Metric Properties of Fuzzy Curves and Fuzzy Invariants

Yasemin Sağıroğlu, Djavvat Khadjiev, Uğur Gözütok

145

Hamiltonicity and pancyclicity of superclasses of claw-free graphs

Abdelkader Sahraoui

146

On some evolution problems of degenerate type

Moulay Mohamed Said

147

Average approximate Sentinel for an Identification problem

S. Sandel, A.Berhail, A. Ayadi

148

λ – Double Statistical Convergence and Strongly Double Summable Functions of Weight g

Rabia Savaş, Mahpeyker Öztürk

150

On Infinite Matrices and Sequence Spaces

Rahmet Savaş

151

Numerical methods for certain classes of Markovian BSDEs and quasi-linear parabolic PDEs

Driss Seghir

152

Integral Type Darbo's Fixed Point Theorem via Sequences of Functions

Derya Sekman, Vatan Karakaya

153

Normal Differential Operators for Degenerate First-Order

Meltem Sertbaş, Fatih Yılmaz

154

Converse Theorems for the Discrete Power Series Methods of Summability

Sefa Anıl Sezer, İbrahim Çanak

155

α – Fuzzified Rough Set Model in Multigranulation Space

Tanzeela Shaheen

156

A Note on Generalized Bernstein-type Cheney-Sharma Operators

Dilek Söylemez, Fatma Taşdelen Yeşildal, Gülen Başcanbaz Tunca

157

xiv

Evaluating Gaussian Fuzzy System Reliability

Halil İbrahim Şahin, Melek Eriş Büyükkaya

158

Smoothed Auxiliary Function With Multiple Extrema In Global Optimization

Ahmed Şahiner, Shehab A. Ibrahem

159

Application of Modern Optimization Techniques in Determination of the Academic Achievement

Ahmet Şahiner, Sumeyya Koman, Raziye Akbay Uçar

160

Radiation Shielding Properties of Borided Fe-Ni Binary Alloys by Fuzzy Logic Modelling

Ahmet Şahiner , Nazım Uçar , Adnan Çalık, Gülden Kapusuz

161

Temperature and pressure based estimation of daily global solar radiation over Turkey using fuzzy logic

modeling

Ahmet Şahiner, Idris A. Masoud Abdulhamid, Murat Ozturk

162

Normalizer of Some Congruence Subgroups

Zeynep Şanlı, Tuncay Köroğlu, Bahadır Özgür Güler

164

Boundary Component of the Group Γ02(N)

Hatice Şengül, Mehmet Akbaş

165

GUI-EFG: Guided Element Free Galerkin Method

Süleyman Şengül, Erhan Coşkun

166

On the Matrix Nearness Problem of the Generalized Sylvester Quaternion Matrix Equations

Sinem Şimşek

167

Korovkin Type Theorems in Modular Spaces via Summability Methods

Emre Taş, Tuğba Yurdakadim

168

Pollution Detection for the Singular Linear Parabolic Equation

Laıb Teldja, Ayadi Abdelhamid

169

Relation of Some Special Vertices of Suborbital Graphs and Continued Fractions

Tuğba Tuylu, Ali Hikmet Değer, Ümmügülsün Akbaba

170

Bounds for Atom-bond Connectivity Index

Ahmet Uğur, Ayşe Dilek Maden

171

Travelling waves in diffusion, aggregation and diffusion-aggregation-diffusion in a single population

Judith Pérez Velázquez, Faustino Sanchez-Garduno

172

On a Integral-Transform to Differential Equations

Ervenila Xhaferraj

173

Aczél Type Inequalities for Hilbert Space Operators

Ulaş Yamancı, Mehmet Gürdal

174

On the Artinianness of Top Local Cohomology Modules

Tuğba Yıldırım

175

q-Quasinormal Operators and Its Extended Eginvalues

Fatih Yılmaz, Meltem Sertbaş

176

A New Algorithm Based on Penalty Function Approach for Inequality Constrained Global Optimization

Problems

Nurullah Yılmaz, Ahmet Şahiner

177

On Fuzzy Prime Hyperideals(Hyperfilters) in ∧ - Hyperlattices

Şerife Yılmaz, Dilek Bayrak

178

xv

Korovkin Theory for Extraordinary Test Functions via Power Series Method

Tuğba Yurdakadim, Emre Taş

179

Criticality indices of 2-independence of some graphs

Mohamed Zemir, Ahmed Bouchou

180

INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED

MATHEMATICS (ICRAPAM2018)

JULY 23-27 2018, Karadeniz Technical University, Prof. Dr. Osman Turan Convention Center, Trabzon, TURKEY

www.icrapam.org

______________________________________________________

1

The Distance Between Two Sets and its Amazing Applications

in Science and Technology

Vasile Berinde

Department of Mathematics and Computer Science, Technical University of Cluj-Napoca

Baia Mare, Romania

Abstract: The distance between two sets has been introduced at the beginning of

the XXth century by the successive contribution of D. Pompeiu (1873-1954), M.

Frechet (1878-1973) and F. Hausdorff (1878-1942). The importance of this

fundamental concept came rather late in mathematics (about 1940) and even later

in applied sciences and technology but nowadays it is widely used in almost all

research areas. The list of applications of what is generally known as Hausdorff

metric and less often as Pompeiu-Hausdorff metric is really impressive as it is

nowadays an indispensable mathematical tool in almost all fields of research, not

only in mathematics and computer science, as shown for example by a search in

Web of Science, where we find about 2800 articles covering more than 100

different research fields in which Pompeiu-Hausdorff metric is used. Amongst

the research fields with the most research articles found in WoS database we

mention: mathematics; mathematics applied; computer science artificial

intelligence; engineering electrical electronic; computer science theory methods;

computer science software engineering; imaging science photographic

technology; computer science information systems; radiology nuclear medicine

medical imaging; computer science interdisciplinary applications; engineering

biomedical; statistics probability; automation control systems;

telecommunications; operations research management science; mathematics

interdisciplinary applications; remote sensing; mathematical computational

biology; computer science cybernetics; multidisciplinary sciences etc. Amongst

other research fields covered, we also mention: neurosciences; psychology

mathematical; engineering civil; peripheral vascular disease; acoustics;

nanoscience nanotechnology; biochemical research methods; metallurgy

metallurgical engineering; engineering manufacturing; evolutionary biology;

geography physical; education scientific disciplines; clinical neurology;




www.icrapam.org

______________________________________________________

2

transportation science technology; transportation; biology; physics particles

fields; management; microscopy; engineering biomedical; economics; dentistry

oral surgery medicine; medical laboratory technology; water resources; green

sustainable science technology; physics fluids plasmas; engineering marine;

neuroimaging; electrochemistry; geochemistry geophysics; education

educational research; engineering aerospace; chemistry analytical; biotechnology

applied microbiology; astronomy astrophysics; social sciences mathematical

methods; ergonomics; health care sciences services; environmental studies;

environmental sciences; engineering petroleum; biochemistry molecular biology;

engineering ocean; energy fuels; engineering chemical; computer science

hardware architecture; construction building technology; endocrinology

metabolism; ecology; dermatology; oceanography; crystallography; information

science library science; cell biology; geography; business; oncology; genetics

heredity; architecture etc.

The main aim of this paper is to present some essential facts about Pompeiu-

Hausdorff metric and to illustrate its importance and use in various fields of

research by means of some relevant examples.




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3

Totally boundedness and uniform continuity via quasi Cauchy

sequences

Huseyin Cakalli

Maltepe University, Department of Mathematics, Graduate School

of Science and Engineering, Istanbul, Turkey

Abstract: A subset E of a metric space X is totally bounded if and only if

any sequence of points in E has a quasi Cauchy subsequence, where a

sequence (xn) is quasi-Cauchy if the sequence (d(xn+1, xn)) is a null se-

quence. A function f from a totally bounded subset of a metric space X into

a metric space Y preserves quasi Cauchy sequences if and only if it is

uniformly continuous, i.e. (f (xn)) is a quasi Cauchy sequence whenever (xn)

is a quasi Cauchy sequence of points of E.

Keywords: quasi cauchy sequence, totally boundedness.

References:

[1] D. Burton, and J. Coleman, Quasi-Cauchy Sequences, Amer. Math.

Monthly 117, 4, 328-333, (2010).

[2] H.C akallı, Sequential definitions of compactness, Appl. Math. Lett. 21

6, 594-598, (2008)

[3] H. C akallı, Forward continuity, J. Comput. Anal. Appl. 13, 2, 225-

230, (2011).

[4] H. C akallı, Statistical quasi-Cauchy sequences, Math. Comput.

Modelling, 54, no. 5-6, 1620-1624, (2011).

[5] H. C akallı, Statistical ward continuity, Appl. Math. Lett. 24, 10,

1724-1728, (2011).

[6] H. C akallı, On G-continuity, Comput. Math. Appl. 61, 2, 313-318,

(2011).

[7] H.C akallı, Sequential definitions of connectedness, Appl. Math. Lett. 25

3, 461-465(2012)

[8] O. Mucuk, T. Sahan On G-Sequential Continuity, Filomat, 28, 6,

1181-1189, (2014).

[9] R.W. Vallin, Creating slowly oscillating sequences and slowly

oscillating continuous functions, With an appendix by Vallin and H. Cakalli ,

Acta Math. Univ. Comenianae, 25, 1, 71-78, (2011).




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4

Weighted norm inequalities for positive operators restricted on

the cone of ρ-quasiconcave functions

Professor Amiran Gogatishvilili

Institute of Mathematics Czech Academy of Sciences

[email protected]

Abstract: In the talk I present a reduction theorems for weighted inequalities

restricted on the cone of quasiconcave functions to slightly different inequalities,

consider for all non-negative measurable functions. By using these reduction

theorems will present the characterization of the weighted Hardy-type inequality

restricted on the cone of quasiconcave functions.




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5

Cofactor Matrix Theory

Robin Harte

Trinity College, School of Mathematics

Dublin 2, Ireland

Abstract: The "cofactor" of a matrix is defined inductively, in the first instance

for block triangles. "Cofactor matrix theory" is not so much the theory of the

cofactor, but rather a version of matrix theory developed through the medium of

the cofactor.




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6

Indispensable remarks on some recent results in metric fixed

point theory

Erdal Karapınar

Department of Mathematics, Faculty of Art and Science

University of Atılım, Ankara, Turkey

erdalKarapı[email protected]

Abstract: The main goal of this talk is to indicate some indispensable notes on

the recent results in the frame of the metric fixed point theory. Starting with a

brief history of metric fixed point theory, we shall emphasize some basic

approaches on it. After then, we shall show the equivalence of the metric. Fixed

point results that were proved in the setting of different abstract spaces.

The talk will be largely expository.

References:

[1] H. Alsulami, S. Almezel, E. Karapınar, F. Khojasteh, “A note on fixed point

results in complex valued metric spaces”, Journal of Inequalities and

Applications, (2015), 2015:33 Doi:10.1186/s13660-015-0550-6.

[2] R.P. Agarwal, E. Karapınar, D. O'Regan, A.F. Roldan-Lopez-de-Hierro,

“Fixed Point Theory in Metric Type Spaces”, Springer International Publishing

(2015).

[3] R. P. Agarwal, H. Alsulami, E.Karapınar and F.Khojasteh, “Remarks on

some recent fixed point results in quaternion-valued metric spaces”, Abstract and

Applied Analysis, (2014) Article Id: 171624.

[4] S. Banach , “Sur les operations dans les ensembles abstraits et leur

applications aux equations integrales”, Fund. Math., 3, 133-181, (1922).

[5] R.H. Haghi, Sh. Rezapour, N. Shahzad, “Some fixed point generalizations are

not real generalizations”, Nonlinear Analysis: Theory, Methods &Applications,

Volume 74, Issue 5, 1 March 2011, 1799-1803.

[6] R.H. Haghi, Sh. Rezapour, N. Shahzad, “Be careful on partial metric fixed

point results”, Topology and its Applications, Volume 160, Issue 3, 15 February

2013, 450-454.




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7

[7]

J. Jachymski, “Equivalent conditions for generalized contractions on

(ordered) metric spaces “, Nonlinear Analysis: Theory, Methods and

Applications, Volume 74, Issue 3, 1 February 2011, 768-774.

[8] E. Karapınar, W.-S. Du, “A note on b-cone metric and its related results:

Generalizations or equivalence?” , Fixed Point Theory and Applications, (2013),

2013:210.

[9] A. Roldan, J. Martinez Moreno, C. Roldan, and E. Karapınar, “Some remarks

on multidimensional fixed point theorems”, Fixed Point Theory, FixedPoint

Theory, 15(2014), No. 2, 545-558.

[10] B. Samet, E. Karapınar, H. Aydi, V. Rajic, “Discussion on some coupled

fixed point theorems”, Fixed Point Theory and Applications, 2013, 2013:50 (10

March 2013).




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8

Symmetry Methods and Conservation Laws for Differential

Equations

Chaudry Masood Khalique

Department of Mathematics, North-West University,

Mmabatho, South Africa

Abstract: Symmetry methods were introduced in the latter part of the nineteenth

century by the Norwegian mathematician Sophus Lie (1842-1899). Lie was

inspired by the works of Galois and Abel. Lie’s work systematically unites the

well-known ad hoc methods to construct exact solutions for differential

equations. Conservation laws have numerous significant applications in the study

of differential equations, particularly for determining conserved quantities and

constants of motion. In this talk we present briefly the applications of Lie’s

theory and conservation laws to differential equations.




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9

Best Uniform Approximation with 2n+1 Equioscillations Abedallah Rababah

Department of Mathematics, Jordan University of Science and Technology,

Irbid, Jordan

[email protected]

Abstract: For a given continuous function f, the following classical best uniform

approximation problem is considered in the literature. Find a polynomial p of

degree n s.t. the uniform norm of the resulting error is minimized. The existence

is proved by Borel, and the uniqueness is proved by Chebyshev. The solution is

characterized by Chebyshev by showing that there exist Chebyshev alternant of

n+2 points. I.e. the error equioscillates n+2 times. However, there does not exist

a method to find these solutions so far.

We consider a smooth curve C rather than only a continuous function f. We ask

the following question: Is it possible to find a polynomial p of degree n s.t. the

uniform norm of the resulting error is minimized and the error function

equioscillates m times, where m>n+2, rather than n+2 times? I.e does there exist

an alternant m points s.t. the error equioscillates m times.

In this talk the following conjecture is shown to be valid for a class of curves of

measure not equal zero:

Conjecture: For every smooth curve C, there exists a polynomial p of degree n

s.t. the uniform norm of the resulting error is minimized and the error function

equioscillates 2n+1 times rather than n+2 times. We further show that there exist

alternant of 2n points.

Keywords: Best uniform approximation, Chebyshev alternant.




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10

Multi-layer perceptrons with Levenberg-Marquardt training

algorithm for predicting of heavy metals concentrations in

marine sediments

Abdelaziz ABDALLAOUI, Kaoutar ELAZHARI, Hicham EL BADAOUI

Department of Chemistry, Analytical Chemistry and Environment Team

Moulay Ismail University, Faculty of Science

Meknes, Morocco.

[email protected]

Abstract: With the ability for classification and generalization, artificial neural

networks (ANN) are typically used of treating mathematical problems, in

particular in the statistical problems where the variables are non-linear. Today,

these neural networks, which are methods of calculation whose design, are

inspired schematically the operation of biological neurons, found many

applications in several areas, such as image processing, signal processing,

optimization and also in the areas of forecasts and predictions.

In the present work, we will develop a network model of artificial neurons

(ANN) Multilayer Perceptron (MLP) type for the prediction of heavy metals (Fe,

Mn and Zn) concentrations in red sea sediments. To determine the architecture of

the network to use, we varied the number of hidden layers, the number of

neurons in the hidden layer, the transfer functions and learning algorithms.

Several statistical criteria such as mean square error (MSE) and the correlation

coefficient (R) were evaluated to test the effectiveness of the developed models.

The purpose of the study of these statistical criteria is to choose the most relevant

neural model and most successful in predicting the heavy metals concentrations

from the physicochemical parameters.

The originality of this work lies in the application of ANN method for predicting

the concentrations of heavy metals, in slices of sediment cores of the Red Sea

from the physicochemical parameters. These cores were taken at the axial zone

of the Red Sea between 19 ° and 23 ° N.

This study showed that predictive models established by the recent method,

which is based on the principle of artificial neural network multilayer perceptron




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11

type (PMC) with learning algorithm of Levenberg-Marquardt, perform much

better compared to those established by the method based on the multiple linear

regression. This performance demonstrates the existence of a nonlinear

relationship between physicochemical characteristics (independent variables) and

the concentrations of heavy metals (dependent variables) of the Red Sea

sediment.

Keywords: Prediction, ANN, MLP, Levenberg-Marquardt algorithm, Heavy

metals, Red Sea.




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12

Quarterly prediction of water consumption using some

statistical methods. A comparative study

Qais Mustafa Abdulqader

Department of Geology of Petroleum and Sediments, Duhok Polytechnic University,

Duhok, Zakho, Iraq

[email protected]

Abstract: Time-series analyses of water consumption data are important for

investigating the amount of water variation and predicting consumption change.

Many applications have been done for estimating and predicting water

consumption. In this paper, some statistical analysis methods such as Box –

Jenkins methodology, wavelet analysis, and artificial neural network have been

used for obtaining a suitable model for making quarterly prediction of water

consumption in Erbil city, Kurdistan Region of Iraq. The comparison procedure

was depending on some statistical criteria. The results of the analysis have been

discussed and mentioned.

Keywords: Time Series, Box-Jenkins, Wavelets, Artificial Neural Networks.

References:

[1] A. H. Nury, K. Hasan, and J. B. Alan "

Comparative study of wavelet-

ARIMA and wavelet-ANN models for temperature time series data in

northeastern Bangladesh", Journal of King Saud University. 29(2017), 47–61.

[2] A. Altunkaynak, M. Ozger, and M. Cakmakci "

Water Consumption

Prediction of Istanbul City by Using Fuzzy Logic Approach ", Water Resources

Management 19(2005), 641–654.

[3] J. Adamowski, H. F. Chan, S. O. Prasher, B. O. Zielinski, and A.

Sluisarieva"Comparison of multiple linear and nonlinear regression,

autoregressive integrated moving average, artificial neural network, and wavelet

artificial neural network methods for urban water demand forecasting in

Montreal, Canada", Water Resources Research, 48(2012), 1-14.




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13

Extremal polynomials and their approximation properties in

complex plane

2ülnara Abdullayev, G1,2Fahreddin Abdullayev

¹Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan

²Mersin University , Mersin, Turkey,

[email protected] [email protected],

Abstract: Let CG be a finite Jordan region, 0 G ; :L G= ,

: C \ G = . For 0p let 1 ( )pA G denote the set of analytic in G

functions ( )f z normalized by (0) 0,f = (0) 1f = and such that

1 ( ): ( ) ,

p

p p

A G

G

f f z d=

where is a two dimensional Lebesgue measure. Let ( )w z= be the

conformal mapping of G onto the disk 0 0(0, ) : { : }B w w =

normalized by (0) 0 = , (0) 1 = . Let us consider the following extremal

problem for 0p :

1 ( ), inf .

pn n nA G

P P − →

In this work, we investigated some approximation properties of the extremal

polynomials in different regions of the complex plane. Analogous questions was

investigated in [1]-[6] and others.

Keywords: Complex approximation, Coformal mapping, Quasiconformal curve

References:

[1] F.G.Abdullayev, Uniform convergence of the generalized Bieberbach

polynomials in regions with non zero angles. Acta Math. Hung. Vol.77, No:3,

(1997), pp. 223-246.




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14

[2] F.G.Abdullayev, N.P.Özkartepe, Uniform convergence of the p-Bieberbach

polynomials in domains with zero angles, Science China Math, 58 (5), (2015),

1063-1078.

[3] V.V. Andrievskii, D. Uniform convergence of Bieberbach polynomials in

domains with piecewise quasiconformal boundary (Russian). In: Theory of

Mappings and Approximation of Functions. Kiev, Naukova Dumka (1983), pp.3-

18.

[4] D.M.Israfilov, Uniform Convergence of the Bieberbach Polynomials in

closed smooth domains of bounded boundary rotation, Journal of Approximation

Theory, 125, (2003), 116-130.

[5] D.Gaier, On the convergence of the Bieberbach polynomials in regions with

corners. Constructive Approximation, 4 (1988), pp.289-305.

[6] M.V.Keldych, Sur l'approximation en moyenne quadratique des fonctions

analytiques. Math.Sb., 5(47), (1939), pp.391-401.




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15

On Modular Happy Numbers

Raghib Abusaris

Department of Epidemiology and Biostatistics,

College of Public Health and Health Informatics,

King Saud bin Abdelaziz University for Health Science,

Riyadh, Saudi Arabia.

[email protected]

Abstract: In this paper, we investigate the asymptotic behavior of the sequences

generated by iterating the process of summing the modular powers of the

decimal digits of a number. In particular, we identify all modular happy

numbers. A number is called modular happy if the sequence obtained by iterating

the process of summing the modular powers of the decimal digits of the number

ends with 1.

Keywords: happy numbers, sequences, recurrence relations, difference

equations, modular arithmetic.




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16

Distances from Unbounded Trajectories to their Limit-Strings

on a Hadamard Kȁhler Manifold

Toshiaki Adachi

Department of Mathematics, Nagoya Institute of Technology,

Nagoya, Japan

[email protected]

Abstract: A smooth curve γ parameterized by its arclength on a Kȁhler manifold

M is said to be a trajectory for a Kȁhler magnetic field Bk = kBJ if it satisfies

∇𝛾′𝛾′ = 𝑘𝐽𝛾′. Since it is a geodesic when k = 0, we may regard trajectories as

generalizations of geodesics which are deeply concerned with complex structure.

By using trajectories, we can define magnetic exponential maps by just the same

way as we defined exponential maps by using geodesics. In [1] by studying

trajectory-harps which consist of trajectories and variations of geodesics, we

showed that they are diffeomorphisms when M is a Hadamard Kȁhler manifold

of sectional curvature not greater than a negative constant c and the strength of

Kȁhler magnetic field satisfies k2 ≤ |c|. For a Hadamard manifold, it has a

geometric compactification with the ideal boundary ∂M which consists of all

asymptotic classes of geodesic half-lines. In [2], it was shown that each

magnetic exponential map extends to a homeomorphism of a unit tangent sphere

to the ideal boundary. Thus, for each trajectory-half line γ, we have a unique

geodesic σγ satisfying γ(0)= σ(0) and γ(∞)= σ(∞) in ∂M . In this paper, we study

the distance from each point γ(t) on γ to σγ and show that when k2 < |c| it is

bounded and when k2 = |c| it grows at most linear order.

Keywords: Hadamard Kȁhler manifold, ideal boundary, trajectory-harps, limit-

strings.

References:

[1] T. Adachi, “A theorem of Hadamard-Cartan type for Kähler magnetic fields”,

J. Math. Soc. Japan, 64(2012), 969-984.

[2] Q. Shi & T. Adachi, “Trajectory-harps and horns applied to the study of the

ideal boundary of a Hadamard Kähler manifold”, Tokyo J. Math., 40(2017), 223-

236.




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17

The Generating Functions of Recurrence Relation of the Path

of Minimal Length on Suborbital Graphs

Ümmügülsün AKBABA, Ali Hikmet DEĞER and Tuğba TUYLU

Karadeniz Technical University Faculty of Science, Department of Mathematics,61080,

Trabzon, Turkey.

[email protected]

[email protected]

[email protected]

Abstract: Jones, Singerman and Wicks [1] used the idea of suborbital graphs for

finite permutation groups, described by Sims [2], for the congruence subgroup

𝛤0(𝑢, 𝑁) of the modular group 𝛤. They studied the action of 𝛤 on the rational

projective line by using suborbital graphs. These are 𝛤 invariant directed graphs

with vertex set ℚ and their edge-sets being the orbits of 𝛤 on the cartesian

square ℚ2 . For each integer which is bigger than 1 and for each of the ф(𝑁)

units (𝑁) , there is one suborbital graph 𝐺𝑢,𝑁 with edge-set which is the orbit

containing the pair (∞,𝑢/𝑁). Then they extend the properties for 𝐺1,1. Farey

graphs to suborbital graphs of 𝐹𝑢,𝑁, where 𝑢 and 𝑁 are relatively prime and 𝑁 is

bigger than 1.

Relation between continued fractions and vertices of the paths of minimal length

on the subgraphs were established and valued of the farthest vertex which a

vertex can be found on this path of the suborbital graph 𝐹𝑢,𝑁 .

In this work we study the generating function of the recurrence relation of

continued fractions. In previous works we showed that every vertex on the path

of minimal length of suborbital graph 𝐹𝑢,𝑁 has a continued fraction. So this

continued fraction generate a recurrence relation. Koshy [5] studied relation

between Fibonacci numbers and generating functions. By this motivation we

investigate relation between Fibonacci numbers and special vertices.

Keywords: Minimal Length, Suborbital Graphs, Generating Functions

References:

[1] G.A. Jones, D. Singerman and K. Wicks,” The Modular Group and

Generalized Farey Graphs”, London Mathematical Society Lecture Note Series

160 (1991), 316-338.




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18

[2] C.C. Sims, “Graphs and Finite Permutation Group”, Mathematische

Zeitschrift 95 (1967), 76-86.

[3] A.H. Deger, “Vertices of Paths of Minimal Lengths on Suborbital Graphs”,

Filomat 31:4(2017), 913-923

[4] A.H. Deger and Ü. Akbaba,”Some Special Values of Vertices of Trees on

the Suborbital Graphs”, American Institute of Physics 1926(2018), 020013. [5] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-

İnterscience Publication, Canada, 2001.




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19

Portfolio Selection by Fuzzy Linear Programming

Serkan AKBAŞ Türkan ERBAY DALKILIÇ

[email protected] [email protected]

Department of Statistics and Computer Science, Karadeniz Technical University,

Trabzon, Turkey

Abstract: In this study, a new portfolio selection model based on the analytic

hierarchy process was proposed, which would help investors who plans to make

the most accurate investment in the face of uncertainties in financial markets. In

this context, the weights of the criteria in the study were obtained by the

Constrained Fuzzy AHP method proposed by Enea and Piazza. A fuzzy linear

programming model is proposed in which the cost variables in the objective

function is fuzzy. A portfolio selection problem existing in the literature is

discussed to examine the effectiveness of the proposed model. The results

obtained from the existing methods and the results obtained from the proposed

model were compared.

Keywords: Fuzzy linear programming, Fuzzy AHP, Portfolio selection

References:

[1] Y.J. Lai and C.L. Hwang, “A new approach to some possibilistic linear

programming problems”, Fuzzy Sets Syst., 49, 2, (1992), 121-133.

[2] M. Enea and T. Piazza, “Project selection by constrained fuzzy AHP”, Fuzzy

Optim. Decis. Mak. 3, 1, (2004), 39-62.

[3] F. Tiryaki and B. Ahlatcioglu “Fuzzy portfolio selection using fuzzy analytic

hierarchy process”, Inf. Sci. (Ny)., 179, 1-2, (2009), 53-69.

[4] J. Krejčí, O. Pavlačka and J. Talašová, “A fuzzy extension of Analytic

Hieararchy Process based on the constrained fuzzy arithmetic”, Fuzzy Optim.

Decis. Mak. 16, 1, (2017), 89-110.




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20

Performance Comparison of Fuzzy Clustering Approaches

on Burn Region Detection

Yeşim AKBAŞ Tolga BERBER


Department of Statistics and Computer Science, Karadeniz Technical University,

Trabzon, Turkey

Abstract: The World Health Organization reported that the annual number of

deaths caused by burn wounds was 180.000 in 2017. Hence, it is important to

develop utilities to help diagnosis of burn wounds. There are many parameters

needed to be determined in the planning phase of burn treatment. Percentage of

burn wound is the one of the most important parameters. In this study,

performance of fuzzy clustering methods was assessed to identify burnt and

normal skin lesions in burn images, which is required to determine the actual

burn percentage. We selected 19 sample images from the burn wound image

dataset, which is being developed using images of the patients who applied to the

burn unit of the Karadeniz Technical University Faculty of Medicine Farabi

Hospital. Fuzzy C-Means (FCM) and Gustafson Kessel (G-K) algorithms were

applied to sample images for this task. We performed exhaustive series of

experiments to find the optimal number of clusters for the selected sample

images, with the help of fuzzy cluster validity indices for all clusters within C =

[3, 9]. In addition, we conducted experiments to measure effects of Gauss filter

with different levels of sigma on burn region identification task. Finally, results

of our experiments are presented and discussed in detail.

Keywords: Burn images, Fuzzy Clustering Methods

References:

[1] Shankar, A. S., Asokan, A. ve Sivakumar, D., Brain Tumor Classification

Using Gustafson-Kessel (G-K) Fuzzy Clustering Algorithm Int. J. Latest Eng.

Res. Appl., 1, 5 (2016) 68–72.

[2] Dunn, J. C., A Fuzzy Relative of the ISODATA Process and Its Use in

Detecting Compact Well-Separated Clusters J. Cybern., 3, 3 (1974) 32–57.

[3] Bezdek, J. C., A Convergence Theorem for the Fuzzy ISODATA Clustering

Algorithms IEEE Trans. Pattern Anal. Mach. Intell., PAMI-2, 1 (1980) 1–8.

[4] Moertini, V. S., Introduction to Five Data Clustering Algorithms Integral, 7,

2 (2002) 87–96.




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21

Fekete-Szego Functional Problem For A New Class Of M-Fold

Symmetric Analytic Functions

Arzu Akgül

Department of Mathematics, Kocaeli University,

Kocaeli, Turkey

[email protected]

Abstract: In this paper, we investigate a new subclass consisting of analytic and

m-fold symmetric bi-univalent functions satisfying subordination in the open

unit disk U. We consider the Fekete-Szegö inequalities for this class. Also, we

establish estimates for the coefficients for this subclas and several related classes

are also considered and connections to earlier known results are made.

Keywords: Analytic functions, m-fold symmetric bi-univalent functions,

Coefficient bounds.

References:

[1] A. Akgül, On the coefficient estimates of analytic and bi-univalent m-fold

symmetric functions, Mathematica Aeterna, 7 (3), 2017, 253 – 260

[2] S. Altınkaya and S. Yalçın, Coefficient Bounds for Certain Subclasses of m-

Fold Symmetric Biunivalent Functions, Journal of Mathematics , Volume 2015,

(2015) , 5 pages.

[3] D. A. Brannan and J. Clunie, Aspects of contemporary complex analysis,

Proceedings of the NATO Advanced Study Instute Held at University of

Durham, New York: Academic Press, (1979).

[4] D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, in

Mathematical Analysis and Its Applications (S. M. Mazhar, A. Hamoui and N. S.

Faour, Editors) (Kuwait; February 18.21, 1985), KFAS Proceedings Series, Vol.

3, Pergamon Press (Elsevier Science Limited), Oxford, 1988, pp. 53.60; see also

Studia Univ. Babes¸-Bolyai Math. 31 (2) (1986), 70.77.

[5] J.H.Choi, Y. C. Kim, and T. Sugawa, .Ageneral approach to the Fekete-

Szegö problem, Journal of the Mathematical Society of Japan, vol. 59,no. 3, pp.

707.727, 2007.




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22

Continued Fractions and Exact Hermite matrices

Tülay Ayyıldız Akoğlu

Department of Mathematics, Karadeniz Technical University,

Trabzon, Turkey

[email protected]

Abstract: The aim of this study is to present a symbolic-numeric method to

construct exact rational Hermite matrices using continued fractions. Efficient

numerical methods can solve polynomial systems if the given system is well-

posed. However, in most applications, only the knowledge of the location of the

real roots are desired. The signature of Hermite matrices is defined by Hermite in

1856 [1], and this signature can provide important information on the location of

the real roots [2]. In this study, we use approximate solutions and continued

fractions to construct the exact Hermite matrix of the given polynomial system.

This construction is used for root certification purposes [3,4], and it works even

if the given system is not well-posed.

This is joint work with Agnes Szanto (North Carolina State University).

Keywords: Symbolic Computation, Continued Fractions, Polynomial Systems.

References:

[1] C. Hermite, "Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt

de Berlin sur le nombre des racines d'une équation algébrique comprises entre

des limites données." Journal für die reine und angewandte Mathematik 52

(1856): 39-51

[2] S. Basu, R. Pollack and M. Roy, “Algorithms in real algebraic geometry”,

Springer (2007).

[3] T. Ayyildiz Akoglu, J. D. Hauenstein, A. Szanto, “Certifying solutions to

overdetermined and singular polynomial systems over Q”, Journal of Symbolic

Computation (2018), 147--171.

[4] J.D. Hauenstein and F. Sottile, “Algorithm 921: alphaCertified: certifying

solutions to

polynomial systems”, ACM Trans. Math. Software 38 (2012), 28:1--28:20




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23

The Arrowhead-Pell Sequences in Finite Groups

¹ Yeşim Aküzüm, 2 Taha Doğan and 3Ömür Deveci 1,2,3 Department of Mathematics, Faculty of Science and letters,

Kafkas University, Kars-Turkey

[email protected], [email protected],

[email protected]

Abstract: In [1], Akuzum and Deveci defined the arrowhead-Pell sequences and

obtain miscellaneous properties of these sequences. In this work, we redefine the

arrowhead-Pell sequences by means of the elements of groups which is called

arrowhead-Pell-orbit. Then we examine the arrowhead-Pell-orbit of the finite

groups in detail. Finally, we obtain the periods of the arrowhead-Pell orbit of the

semidihedral group 2mSD as applications of the results obtained.

Keywords: Sequence, Orbit, Group.

Acknowledgement:This Project was supported by the Commission for the

Scientific Research Projects of Kafkas University. The Project number. 2017-

FM-21 References:

[1] Y. Akuzum and O. Deveci, “The arrowhead-Pell sequences”, is submitted.




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24

Boundedness of potential operators in modified Morrey spaces

defined on Carleson curves

Canay Aykol

Department of Mathematics, Ankara University,

Ankara, Turkey

Abstract: In this talk we study the potential operator

I , 10 in the

modified Morrey space )(,

~

pL and the spaces )(BMO defined on

Carleson curves . We prove that for

−

11 p the potential operator

I is bounded from the modified Morrey space )(,

~

pL to )(,

~

qL if and in

the case of infinite curve only if

−−

1

11

qp , and from the spaces

)(,1

~

L to )(,

~

qWL if and in the case of infinite curve only if

−−

1

11

q. Furthermore, for the limiting case

11

−p we

show that if is an infinite Carleson curve, then the modified potential operator

I

~is bounded from )(,

~

pL to )(BMO , and if is a finite Carleson

curve, then the operator

I is bounded from )(,

~

pL to )(BMO ,

Keywords:

Carleson curves, modified Morrey spaces, BMO spaces, potential operators,

Sobolev-Morrey inequality.

References:

[1] V.S. Guliyev, M.N. Omarova, M.A. Ragusa, A. Scapellato, ‘’Commutators

and generalized local Morrey spaces,’’ Journal of Mathematical Analysis and

Applications, 457 n.2, doi: 10.1016/j.jmaa.2016.09.070, (2018), 1388-1402.




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25

[2] V. Kokilashvili, ‘’Fractional integrals on curves’’, (Russian) Trudy Tbliss.

Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 95 (1990), 56-70.

[3] V. Kokilashvili, S. Samko, ‘’Sobolev theorem for potentials on Carleson

curves in variable Lebesgue spaces,’’ Mem. Differential Equations Math. Phys.

33 (2004), 157-158.

[4] N. Samko, ‘’Weighted Hardy and singular operators in Morrey spaces’’, J.

Math. Anal. Appl., 350 (2009), no. 1, 56-72.




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26

Robust PCA-Algorithms for Detection of Metagenomic

Biomarkers

Mustafa Alshawaqfeh, Ahmad Bashaireh,

Jan Suchodolski, Erchin Serpedin and Khalid Qaraqe

ECEN Department, Texas A&M University, College Station, USA

[email protected]

Abstract: Recent progress in sequencing technologies have enabled the

characterization of the microbial communities inhabiting the human body. In the

same time, many metagenomic studies have suggested using microbial taxa as

potential biomarkers for certain diseases. There are two major challenges

associated with the design and the performance assessment of biomarker

detection algorithms. The first challenge is to assess the reproducibility

performance of an algorithm and to design algorithms that exhibit consistent

results. The second challenge is due to the fact that microbial communities

exhibit complex interactions among their constituent members. This renders the

univariate algorithms that treat these features individually ineffective in

capturing such complex dependencies. In this work, we propose a new metric to

determine the reproducibility performance of the biomarker detection algorithms.

Such a metric is referred to as the normalized common-feature consistency. To

capture the reproducibility performance probabilistically, we propose also a

framework to estimate its distribution based on random resampling. We also

propose a robust principal component analysis (RPCA) based algorithm for

biomarker detection. This is achieved by modeling the metagenomic data matrix

as a superposition of a low-rank matrix depicting the abundance levels of the

irrelevant bacteria and a sparse matrix capturing the abundance levels of the

bacteria that are differentially abundant between different phenotypes.

Comprehensive comparisons of RPCA with the state-of-the-art algorithms on

two realistic datasets are conducted. Our preliminary results in [1]-[2] show that

RPCA consistently outperforms the other algorithms in terms of classification

accuracy and reproducibility performance.

Keywords: bioinformatics, PCA, biomarker, detection




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27

References:

[1] M. Alshawaqfeh, A. Bashaireh, E. Serpedin and J. Suchodolski, “Consistent

metagenomic biomarker detection via robust PCA”, Biology Direct, 12.4(2017).

[2] M. Alshawaqfeh, A. Bashaireh, E. Serpedin and J. Suchodolski, “Reliable

biomarker discovery from metagenomics data via RegLRSD algorithm,” BMC

Bioinformatics, 18:328(2017).




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28

Estimates of Faber polynomial coefficients for an unified class

of bi-univalent functions defined through the Jackson (p,q)-

derivative operator

Şahsene Altınkaya1 and Sibel Yalçın2

1Department of Mathematics, Uludag University,

[email protected] 1Department of Mathematics, Uludag University,

[email protected]

Abstract: In this work, we introduce and investigate an unified class of bi-

univalent functions defined through the Jackson (p,q)-derivative operator.

Furthermore, by using the Faber polynomial expansions, we obtain upper bounds

for the coefficients of functions belonging to this analytic and bi-univalent

function class.

Keywords: Bi-univalent functions, subordination, Faber polynomials, Jackson

(p,q)-derivative operator.

References:

[1] H. Airault, “Remarks on Faber polynomials”, Int. Math. Forum, 3 (2008),

449-456.

[2] H. Airault and H. Bouali, “Differential calculus on the Faber polynomials”,

Bulletin des Sciences Mathematiques, (2006), 179-222.

[3] R. Chakrabarti and R. Jagannathan, A (p,q)-oscillator realization of two-

parameter quantum algebras, J. Phys. A: Math. Gen, 24 (1991) L711–L718.

[4] P.L. Duren, “Univalent Functions”, New York: Grundlehren der

Mathematischen Wissenschaften, Springer, 1983.

[5] Ş. Altınkaya and S. Yalçın, “Faber polynomial coefficient estimates for Bi-

univalent functions of complex order based on subordinate conditions involving

of the Jackson (p,q)-derivative, Miskolc Mathematical Notes, 18 (2017), 555-

572.




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29

Total Lighting Level in the e-Studio and Polynomial

Interpolation

Hamit ARMAĞAN, Tuncay YİĞİT, Mevlüt ERSOY

1 Suleyman Demirel University, Dept. of Informatics, Isparta, Türkiye,

[email protected]

2 Süleyman Demirel Üniversitesi, Engineering Faculty, Dept. of Computer Engineering,

Isparta, Türkiye

3 Süleyman Demirel Üniversitesi, Engineering Faculty, Dept. of Computer Engineering,

Isparta, Türkiye

Abstract: e-Studio is a platform that provides active participation of students

who take lessons from universities/institutions that provide online (synchronous)

training for distance education. This is a model designed and developed to

increase the success of distance education by using and adapting educational

technology at the highest potential in this environment (Turkish Patent Institute,

Design Registration Number: 201100256). In this study, the lighting level

generated by the lighting sources in the e-studio was calculated with a software

developed and an interpolation polynomial model was created. The lighting level

in the e-studio is optimized and modeled with the interpolation function. Keywords: e-Studio, Lighting Level, polynomial interpolation.

References:

[1] İşman, A., Uzaktan Eğitim. Geliştirilmiş 4.Baskı. Ankara: Pegem Akademi,

2011.

[2] Bittinger, M.L., Basic College Mathematics. 11th Edition, International

Edition. Boston: Pearson/Addison Wesley, 2010.

[3] https://www.mathworks.com/help/curvefit/polynomial.html#d119e6504

(30.03.2018)

[4] https://www.mathworks.com/help/curvefit/fit-comparison-in-curve-fitting-

app.html (30.03.2018)

[5] Carnicer, J. M. (2008). Interpolation and reconstruction of curves and

surfaces. Rev. Real Academia de Ciencias. Zaragoza, 63, 7-40.




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30

[6] Chapra, S. C., Canale, R. P., Heperkan, H., & Kesgin, U. (2008). Yazılım

ve programlama uygulamalarıyla: mühendisler için sayısal yöntemler.

Literatür. [7] MathWorks Inc, 2018. Erişim Tarihi: 24.04.2018.

https://www.mathworks.com/help/curvefit/evaluating-goodness-of-

fit.html#bq_5kwr-3




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31

Spherical fuzzy sets and its representation of spherical fuzzy ť-

norms and ť-conorms

𝑆ℎ𝑎ℎ𝑧𝑎𝑖𝑏 𝐴𝑠ℎ𝑟𝑎𝑓1 and 𝑆𝑎𝑙𝑒𝑒𝑚 𝐴𝑏𝑑𝑢𝑙𝑙𝑎ℎ1

𝐷𝑒𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 𝑜𝑓 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠, 𝐴𝑏𝑑𝑢𝑙 𝑊𝑎𝑙𝑖 𝐾ℎ𝑎𝑛 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦,1

Mardan, Pakistan

[email protected] and [email protected]

Abstract: In this paper, we introduce spherical fuzzy sets (SFSs) which is an

advanced tool of the fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets.

We investigate the basic properties of SFSs and compare the idea SFSs with

picture fuzzy sets. The spaces of spherical membership grades and picture

membership grades are examined by graphically. Some useful operation such as

spherical fuzzy ť-norms and spherical fuzzy ť-conorms are introduced. Also

introduced spherical fuzzy negator and some classifications of spherical fuzzy ť-

norms and spherical fuzzy ť-conorms which are useful for development the

aggregation operator to aggregate the spherical fuzzy information.

Keywords: Spherical fuzzy set, Spherical fuzzy ť-norm, Spherical fuzzy ť-

conorm.

References:

[1] L. A. Zadeh, “Fuzzy sets”, Information and Control. 8(3) (1965) 338- 356.

[2] K. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20 (1986)

87-96.

[3] B. C. Cuong, “Picture Fuzzy Sets- a new concept for computational

intelligence problems”, In Proceedings of The World Congress on Information

and Communication Technologies WIICT 2013, 1-6, December (2013)

[4] B. C. Cuong, V. Kreinovitch, R. T. Ngan, “A classification of representable t-

norm operators for picture fuzzy sets”, Eighth international conference on

knowledge and systems engineering, (2016).




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32

On some spectral problems of boundary value problem with

boundary condition containing spectral parameter

Nigar Aslanova

Department of Higher Mathematics, Azerbaijan University of Architecture and

Construction,

Baku,Azerbaijan

[email protected],

Mamed Bayramoglu,

Department of differential Equations, Institute of Mthematics and Mechanics of NAS of

Azerbaijan

Baku, Azerbaijan

[email protected]

Khalig Aslanov

Department of Mathematics, Azerbaijan State Economics University (UNEC) ,

[email protected]

Abstract: In present work our aim is to define self-adjoint operator associated

with boundary value problem for second order differential operator equation with

boundary condition depending rationally on spectral parameter. First we

introduce the direct sum of Hilbert spaces and show self-adjointness of operator

defined there. Further prove its discreteness and find asymptotics of spectrum.

Finally, formula for regularized trace of the studied operator is driven. Results

obtained for studied abstract problem might be applied to boundary value

problems for partial differential equations.

Consider in 𝐿2(𝐻, (0,1)), where H is abstract separable Hilbert space,the

following problem

−𝑦′′(𝑥) + 𝐴𝑦(𝑥) + 𝑞(𝑥)𝑦(𝑥) = 𝜆𝑦(𝑥) = 0,

𝑦(0) = 0,𝑦′(1)

𝑦(1)= (𝑎𝜆 −

𝑏1

𝜆−𝑐) .

Here coefficients of differential operator equation 𝐴 and 𝑞(𝑥) are operators. 𝐴 is

discrete bounded from below operator, while 𝑞(𝑥) is bounded operator valued

function for each 𝑥 acting in space 𝐻.Boundary condition constants 𝑎, 𝑏, 𝑐 are

nonnegative.




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33

Some spectral properties of second differential equation in scalar case and with

more general boundary conditions are studied by Binding, Browne and Watson

[1,2].

Our results relate operator 𝐿 defined in 𝐿2(𝐻, (0,1)),⨁𝐻2 by

𝐿𝑌 = (−𝑦′′(𝑥) + 𝐴𝑦(𝑥) + 𝑞(𝑥)𝑦(𝑥), 𝑐𝑦1 − 𝑏1𝑦(1), 𝑦′(1) +

𝑏1𝜆 − 𝑐

𝑦(1)

with domain 𝐷(𝑌) = (𝑌: 𝑌 = (𝑦(𝑥), 𝑦1, 𝑦2) = (𝑦(𝑥), −𝑦(1)𝑏1

𝜆−𝑐, 𝑎𝑦(1)).

Keywords: differential operator equation, spectrum, regularized trace.

References:

[1] P.Binding, P.Browne, B.Watson, “Sturm Liouville problems with boundary

conditions rational;ly dependent on eigenparameter.” Proceedings of Edinburgh

Math. Soc.45(2002), , 631-645.

[2] P.Binding, P.Browne, B.Watson, “Sturm Liouville problems with boundary

conditions rational;ly dependent on eigenparameter.” Journal of Computational

and Appl. Analysis,148(2002), 147-168.




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34

Generalizing the order obtained from nullnorms Emel Aşıcı

Faculty of Technology, Department of Software Engineering,

Karadeniz Technical University, Trabzon, Turkey

[email protected]

Abstract: Nullnorms and t-operators were introduced by Calvo, De Baets, Fodor

[2] and Mas, Mayor, Torrens by [5] to generalize the concepts of t-norms and t-

conorms. Mas, Mayor and Torrens [6] have shown that nullnorms and t-

operators are equivalent since they have the same block structures in [0,1]².

Nullnorms and t-operators were generalized and semi t-operators were

introduced by eliminating commutativity from the axiom of t-operators by

Dryga�� [3]. These operators are interesting not only from a theoretical point of

view, but also for their applications, since they have proved to be useful in

several fields like fuzzy logic framework and fuzzy quantifiers [4], expert system

and neural networks [7]. In this paper, we survey some properties of an order

induced by nullnorms, called the 𝐹-partial order. We show that an order induced

by nullnorms and an order induced by uninorms do not coincide. We also obtain

some important and critical results for 𝐹-partial order. Keywords: Nullnorm, Partial Order.

References:

[1] E. Aşıcı, “An order induced by nullnorms and its properties”, Fuzzy Sets

Syst. 325(2017), 35-46.

[2] 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.

[3] P. Drygas, “Distributivity between semi t-operators and semi nullnorms ”,

Fuzzy Sets Syst. 264(2015), 100-109.

[4] G. J. Klir and B. Yuan “Fuzzy Sets and Fuzzy Logic, Theory and

Application ”, Prentice-Hall PTR, Upper Saddle River, New Jersey 1995.

[5] M. Mas, G. Mayor and J. Torrens “t-operators”, Int. J. Uncertain. Fuzziness

Knowl.-Based Syst. 7(1999), 31-50.

[6] M. Mas, G. Mayor and J. Torrens “The distributivity condition for uninorms

and t-operators”, Fuzzy Sets Syst. 128(2002), 209-225.

[7] M. Mas, G. Mayor and J. Torrens “The modularity condition for uninorms

and t-operators”, Fuzzy Sets Syst. 126(2002), 207-218.




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35

Numerical Treatment of Second Order Fuzzy Two-Point

Boundary Value Problems

Basem S. Attili

Department of Mathematics, University of Sharjah,

Sharjah, United Arab Emirates

[email protected]

Abstract: Many problems in applied sciences and engineering require the

solution of fuzzy differential equations that are subject to fuzzy initial or

boundary conditions, see O’Regana and Lakshmikantham[4]. This gives rise to

fuzzy initial value problems and two-point boundary value problems. It is too

complicated to obtain a closed form solution for such problems. This leads to

numerical approximations.

We consider the fuzzy two-point boundary value problem of the form, see Guo,

et al.[2] and. Li, et al.[3].

𝒚′′(𝒙) = 𝒇(𝒙,𝒚(𝒙), 𝒚′(𝒙)), 𝒂 ≤ 𝒙 ≤ 𝒃

subject to the fuzzy boundary conditions

𝒚(𝒂) = 𝜶, 𝒚(𝒃) = 𝜷,

where 𝒚: [𝒂, 𝒃] × 𝑹𝑭 × 𝑹𝑭 → 𝑹𝑭 is continuous fuzzy valued function and 𝜶, 𝜷

are fuzzy numbers.

Numerically, we start by transforming the two-point boundary value problem

into a system of fuzzy initial value problems. The missing initial conditions at

𝒙 = 𝒂 will then be assumed and we solve the initial value problem along the

interval. The assumed initial values will then be corrected using the shooting

method, Attili[1], so that the numerical solution at 𝒙 = 𝒃 match the given

boundary condition at the point 𝒙 = 𝒃 up to a certain tolerance.

Some numerical results will be presented to give the numerical details and to

show efficiency of the shooting method.

Keywords: Two-point BVP’s, Fuzzy Differential Equations, Shooting Method.

References:

[1] B. S. Attili, "On the Numerical Implementation of the Shooting Methods to

the one-dimensional Singular Boundary Value Problems", Intern. J. Computer

Math., 47 (1993), 65-75.




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36

[2] Guo, X., Shang, D. and Lu, X., “Fuzzy Approximate Solutions of Second

Order Fuzzy Linear Boundary Value Problems”, Boundary Value Problems

(2013), 212.

http://dx.doi.org/10.1186/1687-2770-2013-212.

[3] Li, D., Chen, M., Xue, X., “Two Point Boundary Value Problems of

Uncertain Dynamical Systems”, Fuzzy Sets and Systems, 179(2011), 50-61.

[4] O’Regana, D. and Lakshmikantham, V., “Initial and Boundary Value

Problems for Fuzzy Differential Equations”, Nonlinear Analysis, 54(2003), 405-

415.




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37

On Fuzzy Metric Space of Continuous Function

Ebru Aydoğdu, Abdülkadir Aygünoğlu, Halis Aygün


Kocaeli, Turkey


[email protected]

Abstract: In this study, a fuzzy metric on continuous function was obtained.

Then several properties of this function spaces were investigated such as

completeness and principal.

Keywords: Continuous Function Spaces, Fuzzy Metric Space.

References:

[1] A. George and P. Veeramani, "Some theorems in fuzzy metric spaces."

Journal of Fuzzy Mathematics 3 (1995), 933-940.

[2] A. George and P. Veeramani, “On some results in fuzzy metric spaces”,

Fuzzy Sets and Systems, 64 (1994), 395-399.

[3] A. George and P. Veeramani, "On some results of analysis for fuzzy metric

spaces", Fuzzy Sets and Systems 90.3 (1997), 365-368.

[4] V. Gregori, A. L opez-Crevill en, S. Morillas and A. Sapena, “On

convergence in fuzzy metric spaces”, Topology and its Applications, 156 (2009),

3002-3006.

[5] V. Gregori and S. Romaguera, "Some properties of fuzzy metric spaces",

Fuzzy sets and systems 115.3 (2000), 485-489.

[6] D. Mihet, “On fuzzy contractive mappings in fuzzy metric spaces”, Fuzzy

Sets and Systems,158 (2007), 915-921.

[7] V. Gregori, S. Morillas, A. López-Crevillén, "On continuity and uniform

continuity in fuzzy metric spaces." Proc. Work. App. Topology WIAT'09, 85-91.




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38

Flip Bifurcation Analysis in a Discrete Bacteria Population

Model

Ercan Balcı1 and Senol Kartal2

1Department of Mathematics, Erciyes University,

Kayseri, Turkey

[email protected] 2Department of Mathematics, Nevsehir Hacı Bektas Veli University

Nevsehir, Turkey

[email protected]

Abstract: In discrete time systems, it is very important to study bifurcation

analysis for a better understanding mechanism of the biological models. Many

interesting research works on bifurcation theory can be found in [1-4]. In this

paper, we shall study Flip bifurcation in a discrete bacteria population model by

using the center manifold theorem and bifurcation theory. We also obtain

bifurcation diagrams, phase portraits and Lyapunov exponents by the numerical

simulations.

Keywords: piecewise constant arguments, difference equation, stability,

bifurcation.

References:

[1] B. Xin, J. Ma, Q. Gao, “The complexity of an investment competition

dynamical model with imperfect information in a security market”, Chaos.

Soliton. Fract, 42(2009), 2425-2438.

[2] M. Peng, “Multiple bifurcations and periodic “bubbling” in a delay

population model”, Chaos. Soliton. Fract, 25(2005), 1123-1130.

[3] S. M. Sohel Rana, “Bifurcation and complex dynamics of a discrete-time

predator-prey system”, Comput. Ecol. Softw, 5(2015), 187-200.

[4] Z. He, X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-

prey system”, Nonlinear Anal. Real, 12(2011), 403-417.




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39

Analysis of a Pneumonia Transmission Model with Random

Components

Zafer Bekiryazici

Department of Mathematics, Recep Tayyip Erdogan University,

Rize, Turkey

[email protected]

Tulay Kesemen


Trabzon, Turkey

[email protected]

Mehmet Merdan

Department of Mathematical Engineering, Gumushane University,

Gumushane, Turkey

[email protected]

Abstract: Recently, a SCIR-type compartmental model has been given by Kizito

and Tumwiine to model Pneumonia infection dynamics with treatment and

vaccination [1]. In our study, this model is used to investigate the random

behavior of Pneumania transmission. Random approximation methods are used

to obtain random numerical characteristics of the model such as the expected

value and variance. The results are compared with the simulation results to

comment on the randomness of the model components.

Keywords: Random Differential Equation, Pneumonia, Mean Square Calculus,

Expected Value, Simulation.

References:

[1] M. Kizito and J. Tumwiine, "A Mathematical Model of Treatment and

Vaccination Interventions of Pneumococcal Pneumonia Infection Dynamics."

Journal of Applied Mathematics 2018(2018), Article ID 2539465.

[2] C. J. Carlos, L. Jódar, and L. Villafuerte, "Numerical solution of random

differential equations: a mean square approach." Mathematical and Computer

Modelling 45.7-8(2007): 757-765.

[3] M. Mehmet, Z. Bekiryazici, T. Kesemen and T. Khaniyev, "Comparison of

stochastic and random models for bacterial resistance." Advances in Difference

Equations 2017.1(2017), 133.




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40

[4] T. T. Soong, Random differential equations in science and engineering.

Elsevier, 1973.

[5] World Health Organization, Pneumonia Fact Sheet, September 2016.

[6] L. Villafuerte, C. A. Braumann, J. C. Cortés and L. Jódar, "Random

differential operational calculus: theory and applications." Computers &

mathematics with applications 59.1 (2010): 115-125.

[7] M. A. El-Tawil and A. H. TOLBA, "On the application of mean square

calculus for solving random differential equations." Electronic Journal of

Mathematical Analysis and Applications 1.2 (2013): 202-211.




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41

Using Two-Layer Models for Flexible Pavement Sizing

Mouloud BELACHIA, Nadjette BOUACHA

Laboratory of Materials, Geotechnics, Habitat and Urban

Department of Civil engineering, University of 20 Aout 1955 - Skikda ALGERIA

[email protected]

Abstract: The construction of roads and highways has long been based on

experience and empiricism and it is only since the Second World War that

pavement structures are calculated so that they can meet certain traffic

constraints. Taking into account the carrying capacity of infrastructure soils [1].

Two schools of thought, empiricism and mechanism have developed in parallel

to meet the ever-increasing needs of pavements until their relatively recent

merger [2], [3].

In general, all these methods follow the following thread:

• Determine the soil-bearing index

• Appreciate the expected composite traffic and reduce it to a fictitious number

of repetitions of a reference load.

• Use graphs, based on the number of repetitions of the reference load, either the

total thickness of the pavement "old methods" or a structural index that takes into

account the qualities of the different layers of the roadway.

In general, the rules for pavement sizing are based on geotechnical knowledge at

break and the characteristics of deformability of materials determined by

laboratory tests and findings on the behavior of structures and, on the other hand

[4], Mathematical methods: rational approaches through a suitable mathematical

model and finally, confrontation, between technological knowledge and

theoretical results [5].

The examination of a set of analytical models which succeeded one another in

time from the simplest model such as that of Boussinesq to the most complicated

model like that of Burmister, led us to conclude that the main fault is to introduce

a lot Simplifying hypotheses, it is necessary to introduce the finite element

method because the development of the latter makes it possible to envisage the

use of more realistic models which have the possibility to study the problems of




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42

loading on board and the possibility of To consider materials not exhibiting

elastic behavior [6].

Keywords: Mathematical model, analytical model, pavement design, finite

element methodspace.

References:

[1] P. Autret, A. Boissoudy, J. Baucheron Marchand and P. Alize iii practice.

Proc. 5th Int. Conf. Structural Design of Asphalt Pavements, 1982.

[2] J. Boussinesq Potential application to the study of equilibrium and motion of

elastic cops. Gauthier Villars, Paris, 1985.

[3] N. Bouacha. Dimensioningand Choiceofthe Bodyof Flexible Pavements. PhD

Thesis, University of Skikda, Algeria, 2010.

[4] C. Peyronie, G. Caroff and C. Gilbert, Route - Design of pavements, edited

by G. Jeuffroy and R. Sauterey, Presses de l'EcoleNationale des Pontset

Chaussées, 1991.

[5] LCPC. Design and measurements of pavement structures. Technical Guide,

LCPC-SETRA, 1994.

[6] S. Salasc, Contact phenomena in concrete pavements: theoretical modeling

and experimental validation. Thesis: Ecole Centrale de Nantes, 1998.




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43

A Solution Approach for Multiobjective Probability

Maximization Problems

Fatima Bellahcene

Department of Mathematics, Faculty of Sciences, Mouloud Mammeri University,

Tizi-Ouzou, Algeria

[email protected]

Abstract: We suggest a method which enables the decision maker to freely

search for a compromise solution to stochastic multiobjective problems where

the objective functions depend on parameters which are continuous random

variables with normal multivariate distributions. The developed algorithm is

based on the combined use of the bisection method and the probabilities of

achieving goals. The main advantage of this approach is to allow the decision-

maker to act, at any moment, on the results and to revise, if necessary, the

proposed choices made by the mathematical model in order to insure that they

reflect his preferences in the best possible way.

Keywords: Multiobjective programming; Stochastic programming; Nonlinear

programming; Minimum-risk criterion.

References:

[1] F. Bellahcene, M. Abbas, Cutting plane method for multiple objective

stochastic integer linear programming problem. Eur. J. of oper. Res. 168 (3),

(2006), 967-984.

[2] R. Caballero, E.Cerda, M. Del Mar, L. Rey, “Efficient Solution Concepts and

their Relations in Stochastic Multiobjective programming”. J. of Opt. Theo. and

Appl. 110 (1) (2001), 53-74.

[3] K.M. Miettinen,”Nonlinear Multiobjective Optimization”. Kluw. Inter Series

(1999).




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44

A Nonsmooth Newton Method for Solving a State Constrained

Optimal Control Problem Governed by Convection-Diffusion

Equation

Boubakeur Benahmed

Department of Mathematics and Computer Sciences, National Polytechnic School of

Oran-Maurice Audin, Oran, Algeria

[email protected]

Kheira Mostepha Hanchour

Ecole Supérieure d'Economie d'Oran, Oran, Algeria

[email protected]

Abstract: We are interesting in solving numerically the optimal control problem

governed by convection-diffusion equation with state inequality constraint [1].

First the infinite dimensional problem is discretized by application of the

Upwind Symmetric Interior Penalty Galerkin (SIPG) method [2]. Then, we write

the optimality conditions for the discretised problem. We reformulate these

conditions into an equivalent system of nonlinear and nonsmooth equations by

use of the Fischer-Burmeister function. A nonsmooth Newton’s method ([4],[5])

is applied to solve this system of non linear and nonsmooth equations, the

resulting matrix system for each Newton iteration will be of saddle point

structure [3]. We give sufficient conditions to ensure the nonsingularity of such

matrix.

Keywords:

PDE-Constrained Optimisation, Convection-Diffusion Equation, Fisher-

Burmeister Function,Smoothing Method, Newton Iteration, Saddle Point System.

References:

[1] A. Quarteroni, G. Rozza, and A. Quaini. Reduced basis methods for optimal

control of advection-diffusion problems. In Advances in Numerical

Mathematics. pages 193-216, Moscow, Russia and Houston, USA, 2007.

[2] H. Yucel, P. Benner. Adaptive discontinuous Galerkin methods for state

constrained optimal control problems governed by convection diffusion

equations. Comput Optim. Appl (2015).




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45

[3] M. Benzi, Gene H. Golub, and Jörg Liesen. Numerical solution of saddle

point problems. Acta Numer., 14, pp. 1-137, 2005.

[4] M. Gerdts. A Nonsmooth Newton’s Method for discretized Optimal Control

problems with State and Control Constraints. Journal of industrial and

management optimization, volume 4, Number 2, pp. 247- 270, May 2008.

[5] L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical

Programming, 58 (1993) 353-367.




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46

Properties of The Bianchi Groups and Digraphs

Murat Beşenk1, Serkan Kader2, Bahadır Özgür Güler3

1Department of Mathematics, Pamukkale University, Denizli, Turkey,

2Department of Mathematics, Ömer Halisdemir University, Niğde, Turkey, 2Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey,

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

Abstract: The group PSL(2, ℂ) has a natural action on 3-dimensional hyperbolic

space ℍ3 which may be described in geometric terms. Any element of PSL(2, ℂ)

induces a biholomorphic map of the Riemann sphere ℂ∞. Jules Henri Poincare

(1854-1912) observed that PSL(2, ℂ) can be identified with the group of

orientation preserving isometries of ℍ3. As in the study of the hyperbolic plane,

the importance of finding discrete groups of hyperbolic isometries of ℍ3 was

emphasized. Starting with an example by Emile Picard (1856-1941) and pursued

by Luigi Bianchi (1856-1928) one particularly interesting class of discrete

subgroups of PSL(2, ℂ) was considered. Denote by 𝑂𝑑 the ring of integers of an

imaginary quadratic number field ℚ(√𝑑) with 𝑑 < 0 and square free. Then 𝑂𝑑

gives rise to a discrete group which is known Bianchi group Γ𝑑 ≔ 𝑃𝑆𝐿(2, 𝑂𝑑) <

𝑃𝑆𝐿(2, ℂ) consisting of fractional linear transformations with coefficients in 𝑂𝑑 .

In this study we discuss properties of the Bianchi groups and give an extension

of suborbital digraphs on 3-dimensional upper half space ℍ3.

Keywords: Bianchi groups, Quadratic number field, Suborbital digraphs,

Circuits.

References:

[1] Grunewald F., Schwermer J., “Subgroups of Bianchi groups and arithmetic

quotients of

hyperbolic 3-space”, American Mathematical Society, Vol:335, 47-78, 1993.

[2] Sims C.C., “Graphs and finite permutation groups”, Math. Zeitschrift,

Vol:95, 76-86, 1967.

[3] Maclachlan C., Reid A.W., “The arithmetic of hyperbolic 3-manifolds”,

Springer, 2003.




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47

A New Solution of the Einstein-Maxwell Equations with Linear

Equation of State

Rashida Bibi, and Azad A. Siddiqui

National University of Science and Technology,

Islamabad, Pakistan

[email protected]

Abstract: In the literature, different equations of state have been assumed to

obtain solutions for charged static spherically symmetric spacetimes [1,9].

Theses equations are highly non-linear partial differential equations. Here we

obtain a new solution of the Einstein-Maxwell equations with linear equation of

state, p = αρ + β. The solution fulfils all the requirements to be a physically valid

solution and can be considered as a model of a compact star.

Keywords: Non-linear PDEs, Einstein-Maxwell equation, Equation of state, and

Physically valid solution.

References:

[1] S. Thirukkanesh and S. D. Maharaj, Charged Anisotropic Matter with Linear

Equation of State, Class. Quant. Grav., 25 (2008)

[2] T. Feroze, Exact Solutions of the Einstein-Maxwell Equations with Linear

Equation of State, Can. J. Phys., 90 (2012).

[3] M. K. Mak and T. Harko, Quark Stars Admitting a One Parameter Group of

Conformal Motions, Int. J. Mod. Phys. D, 13 (2004).

[4] J. M. Sunzu, S. D. Maharaj and S. Ray, Quark Star Model with Charged

Anisotropic Matter, Astrophys. Space Sci., 354 (2014).

[5] M. Malaver, Strange Quark Star Model with Quadratic Equation of State,

Frontiers of Mathematics and its Applications, 1 (2014).

[6] T. Feroze and A. A. Siddiqui, Some Exact Solutions of the Einstein-Maxwell

Equations with a Quadratic Equation of State, J. Korean Phys. Soc., 65 (2014).

[7] T. Feroze and Azad A. Siddiqui, Charged Anisoptropic Matter with

Quadratic Equation of State, Gen. Rel. Grav., 43 (2011).

[8] S. D. Maharaj and P. M. Takisa, Regular Models with Quadratic Equation of

State, Gen. Rel. Grav., 44 (2012).

[9] P. M. Takisa, S. D. Maharaj and S. Ray, Stellar Objects in the Quadratic

Regime, Astrophys. Space Sci., 354 (2014).




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48

Characterizations of extremal graphs for some

bounds on k-domination number

Ahmed Bouchou 1, Mohmed Zemir1 and Mostafa Blidia 2, 1Dept. of Mathematics and Computer Science

University of Médéa, Algeria 2Dept. of Maths, University of Blida, B.P. 270, Blida, Algeria

E-mail: [email protected], [email protected], m_blidia @yahoo.fr,

Abstract: Let k 1 be an integer and let G be a graph with vertex set V(G). A

subset SV(G) is called a k-dominating set if every vertex not in S has at least k

neighbors in S. The k-domination number k(G) of G is the minimum cardinality

of a k-dominating set in G. We know that for any graph G of order n, k(G) ≤ n.

Obviously the upper bound n is sharp

for a graph with maximum degree at most k-1. In this work, we give

characterizations of graphs satisfying k(G) = 𝑘𝑛

𝑘+∆ and k(G) =

2𝑚−(𝑘−)𝑛

2−𝑘 with

maximum degree k. Also, we give characterizations of graphs with k(G) =

n-1 and k(G) = n-2. Finally, we give a new upper bound for k(G). And, we

characterize graphs achieving this bound, which

generalize the result of Cockayne, Gamble and Shepherd [1].

Keywords: k-domination number.

References

[1] E. J. Cockayne, B. Gamble and B. Shepherd. An upper bound for the k-

domination number of a graph. J . Graph Theory 9 (1985) 533-534.

[2] O. Favaron, A. Hansberg and L. Volkmann, On k-dominationt and minimum

degree in graphs, J. Graph Theory 57 (2008) 33-40.

[3] J. F. Fink and M. S. Jacobson, n-domination in graphs, in : Graph Theory

with Applications to Algorithms and Computer. John Wiley and Sons, New York

(1985) 283-300.




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49

Global dynamics of a Fractional model for viral infection

with adaptive immunity

Adnane Boukhouima1, Khalid Hattaf1,2 and Noura Yousfi1

1Laboratory of Analysis, Modeling and Simulation (LAMS), Hassan II University,

Casablanca, Morroco 2Centre Régional des Métiers de l’Education et de la Formation (CRMEF),

Casablanca, Morroco

[email protected]

Abstract: Fractional order differential equations (FDEs) are a generalization of

ordinary differential equations (ODEs) and recently, they become an interesting

tool to model real process in various fields of sciences and engineering due to the

long-range memory property whish exist in most biological systems. In this

work, we propose a fractional mathematical model in order to describe the role

of the adaptive immunity in viral infections and take into account the memory

effect. In the proposed model, the infection transmission is modeled by a general

functional response which covers various types existing in the literature. We first

show that the model is mathematically and biologically well-posed. By using the

fractional Lyapunov method, the global stability of the equilibria is established

and characterized by five treshold parameters. Some applications are presented

to prove our theoretical results.

Keywords: Viral infection, adaptive immunity, Caputo fractional derivative,

global stability.

References:

[1] A. Boukhouima, K. Hattaf and N. Yousfi: "Dynamics of a Fractional Order

HIV Infection Model with Specific Functional Response and Cure Rate."

International Journal of Differential Equations 2017 (2017): 1-8.

[2] K. Hattaf and N. Yousfi: "A class of delayed viral infection models with

general incidence rate and adaptive immune response." International Journal

of Dynamics and Control 4.3 (2016): 254-265.

[3] K. Hattaf, M. Khabouz and N. Yousfi: "Dynamics of a generalized viral

infection with adaptive immune response." International Journal of

Dynamics and Control 3.3 (2015): 253-261




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50

[4] M. Maziane, E.M. Lotfi, K. Hattaf and N. Yousfi: "Dynamics of a class of

HIV Infection Models with Cure of Infected Cells in Eclipse Stage." Acta

Biotheor 63 (2015): 363-380.




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51

On BCB protocol

Sadek BOUROUBI

USTHB University, Faculty of Mathematics,

L’IFORCE Laboratory

Algiers, Algeria

[email protected]

Abstract: Vernam cipher, also known the One Time Pad Cipher, provides

perfect security, despite its simplicity. The certificate that Vernam cipher is

unconditionally secure, has transformed the problem to ensure the confidentiality

of information to a problem of the secret key distribution used in the encryption

process between two parties. Until the eighties, one way to distribute the secret

key, apart from hand to hand, was to use algorithms whose security is based on

the computational complexity. Keys generated by such algorithms are reasonably

secret but not unconditionally secret. In this presentation, we developed a new

cryptographic protocol, called BCB (Bouroubi-Charchali-Benyahia), which will

be used to provide random fresh secret keys shared via a classical channel, using

the set partitions.

Keywords: Cryptography, Vernam Cipher, BCB protocol, Set Partitions.

References:

1. C.E. Shannon. Communication Theory of Secrecy Systems. Bell System

Technical Journal, Volume: 28 , Page: 657-715, 1949.

2. G. Brassard. A Bibliography of Quantum Cryptography. Département IRO,

Université de Montréal, Canada, 3 December 1993.

3. G. S. Vernam. Cipher Printing Systems for Secret Wire and Radio Telegraphic

Communications. J. AIEE 45, Pages: 109-115, 1926.




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52

Some properties of modified new Szasz-Mirakyan operators in

polynomial weight spaces via power summability method

Naim L. Braha

Department of Mathematics and Computer Sciences, University of Prishtina

Avenue Mother Teresa, No-5, Prishtine, 10000,Kosova

[email protected]

Abstract: In this paper we will prove the Korovkin type theorem for new

modified Szasz-Mirakyan operators via A-statistical convergence and power

summability method. Also we give the rate of the convergence related to the

above summability methods and in the last chapter we give a kind of Voronoi

type theorem for A-statistical convergence.

Keywords: A- statistical convergence, new modified Szasz-Mirakyan operators,

power summability method, Korovkin type theorem, Voronoi type theorem, rate

of convergence.

References:

Atlihan, Ozlem Girgin; Unver, Mehmet; Duman, Oktay, Korovkin theorems on

weighted spaces: revisited. Period. Math. Hungar. 75 (2017), no. 2, 201-209.

F.Basar, Summability theory and its applications, Istanbul, 2011.

J.Boos, Classical and Modern Methods in Summability. OxfordUniversity Press,

Oxford (2000).

N.L.Braha, Some weighted Equi-Statistical convergence and Korovkin type-

theorem, Results Math. 70 (2016), no. 3-4, 433-446.

N.L. Braha, Valdete Loku, H.M. Srivastava, Lambda^2-Weighted statistical

convergence and Korovkin and Voronovskaya type theorems, Appl. Math.

Comput. 266 (2015), 675-686.

Naim L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin Type

Approximation Theorem for Periodic Functions via the Summability of the

Modified de la Vallee Poussin Mean, Appl. Math. Comput. 228 (2014), 162-169.

N. L. Braha, Some properties of Baskakov-Schurer-Szasz operators via power

summability method,(submitted for publication).

Campiti, Michele; Metafune, Giorgio. L^p-convergence of Bernstein-

Kantorovich-type operators. Ann. Polon. Math. 63 (1996), no. 3, 273-280.




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53

O.Duman, M.K.Khan, C.Orhan, A-Statistical convergence of approximating

operators. Math. Inequal. Appl. 6, 689-699 (2003)

H.Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241-244.

J.A.Fridy, H.I.Miller, A matrix characterization of statistical convergence.

Analysis 11, 59-66 (1991)

Ugur Kadak, Naim L. Braha and H. M. Srivastava, Statistical Weighted B-

Summability and Its Applications to Approximation Theorems, Appl. Math.

Comput. 302 (2017), 80-96.

Kirisci, Murat; Karaisa, Ali. Fibonacci statistical convergence and Korovkin type

approximation theorems. J. Inequal. Appl. 2017, Paper No. 229, 15 pp.

W.Kratz, U.Stadtmuller,Tauberian theorems for J_p-summability. J.Math.Anal.

Appl. 139, 362-371 (1989)

Loku, Valdete; Braha, N. L. Some weighted statistical convergence and

Korovkin type-theorem. J. Inequal. Spec. Funct. 8 (2017), no. 3, 139-150.

Soylemez, Dilek; Unver, Mehmet. Korovkin Type Theorems for Cheney-Sharma

Operators via Summability Methods. Results Math. 72 (2017), no. 3, 1601-1612.

U.Stadtmuller, A.Tali, On certain families of generalized N\"orlund methods and

power series methods. J. Math. Anal. Appl. 238, 44-66 (1999)

E.Tas, T.Yurdakadim, Approximation by positive linear operators in modular

spaces by power series method. Positivity 21 (2017), no. 4, 1293-1306.

E.Tas, Some results concerning Mastroianni operators by power series method.

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 63(1), 187-195 (2016)

M.Unver, Abel transforms of positive linear operators. In: ICNAAM 2013. AIP

Conference Proceedings, vol. 1558, pp. 1148-1151 (2013)

Walczak, Zbigniew. Error estimates and the Voronovskaja theorem for modified

Szász-Mirakyan operators. Math. Slovaca 55 (2005), no. 4, 465-476.




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54

The Cardinal and Idempotent Number of Order-preserving

and A-increasing Finite Transformations Semigroup

Leyla Bugay

Department of Mathematics, Çukurova University,

Adana, Turkey

[email protected]

Abstract: Let Tn be the (full) transformation semigroup on Xn={1,...,n}, under

its natural order. A transformation f is called order-preserving if

(for each , )x y xf yf x y Xn

Moreover, for any non-empty subset A of Xn, a transformation f is called A-

increasing if

(for each ).x xf x A

Then the subsemigroup of all order-preserving and A-increasing transformations

in Tn is denoted by On(A+). In this talk we give the cardinal and idempotent (the

element whose square is itself) number of On(A+) connection with Fibonacci

numbers.

Keywords: Order-preserving/Increasing transformation, Idempotent, Fibonacci

numbers.

References:

[1] V.H. Fernandes, G.M.S. Gomes, M.M. Jesus, “The cardinal and the

idempotent number of various monoids of transformations on a finite chain”

Bull. Malays. Math. Sci. Soc. 34(2011), 79-85.

[2] P.M. Higgins, “Combinatorial results for semigroups of order-preserving

mappings”, Math. Proc. Cambridge Phil. Soc. 113(1993), 281-296.

[3] J.M. Howie, “Products of idempotents in certain semigroups of

transformations”, Proc. Edinburgh Math. Soc. 17(1970/71), 223-236.

[4] A. Laradji, A. Umar, “Combinatorial results for semigroups of order-

preserving full transformations”, Semigroup Forum 72(2006), 51-62.

[5] P. Zhao, “On the semigroups of order-preserving and A-decreasing finite

transformations”, Algebra Colloq. 21(2014), 653-662.




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55

Best Proximity Point Theorems for Generalized

Contractions

Abdurrahman Büyükkaya1 , Mahpeyker Öztürk2

1Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey 2Department of Mathematics, Sakarya University, Sakarya, Turkey

[email protected] , [email protected]

Abstract: In this paper, we investigate the existence of best proximity points of

of generalized A-contraction mappings in the frame of complete metric spaces

endowed with a partial order. We obtain some consequences which extend and

generalize existing literature.

Keywords: Fixed Point, Best Proximity Point, Generalized A-Conrtaction,

Metric Spaces

References:

[1] M. Akram, A. A. Zafar, A. A. Siddiqui, A general class of contractions: A-

contractions, Novi Sad J. Math., 38(1), 25-33, 2002.

[2] K. Fan, Extensions of two fixed point theorems of F. E. Browder. Math. Z.

112, 234–240, 1969.

[3] Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity

pair theorems. Numer. Funct. Anal. Optim. 24,851-862, 2003.

[4] Sadiq Basha, S., Veeramani, P., Best proximity pair theorems for

multifunctions with open _bres J. Approx Theory, 103, 119-129, 2000.

[5] A. Amini-Harandi, Best proximity points for proximal generalized

contractions in metric spaces, Optim Lett, (7), 913-921, 2013.

[6] S Sadiq Basha, N. Shahzad, Best proximity point theorems for generalized

proximal contractions, Fixed Point Theory and Applications, 42, 1-9, 2012.

[7] H.K. Nashine, P. Kumam, C. Vetro, Best proximity point theorems for

rational proximal contractions, Fixed Point Theory and Applications, 95, 1-11,

2013.

A−




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56

On The Estimator of Semi-Markovian Inventory Model with

Subexponential Component

Melek Eriş Büyükkaya

Department of Statistics and Computer Sciences, Karadeniz Technical University,

[email protected]

Aslı Bektaş Kamışlık

Department of Mathematics, RecepTayyipErdoğan University,

[email protected]

Tülay Kesemen

Department of Mathematics, Karadeniz Technical University, [email protected]

Tahir Khaniyev

Department of Industrial Engineering, TOBB University of Economics and Technology,

[email protected]

Abstract:In this study, a semi Markovian inventory model with subexponential

demands has been investigated. New statistical estimators for ergodic

distribution and moments for ergodic distribution of the process which define the

model have been obtained, using renewal process. A statistical estimation

problem for some characteristics of a semi Markovian inventory model of type

(s,S) has been studied comprehensively in the literature [2],[3],[4],[5]. In all of

the current studies in the literature, it is assumed that random variables

representing demands are light tailed.Unlike other studies in the literature, in this

study a semi Markovian inventory model will be considered with the assumption

that the random variables expressing the demands have heavy tail distribution

and the statistical estimators for the characteristics of the process will be

investigated under this assumption.Semi Markovian inventory model of type

(s,S) with subexponentialWeibull distributed demand quantities has been studied

by Kesemen et. al. [6] and two term asymptotic expansions have been obtained

for some probabilistic characteristics of this model. It is well known that heavy

tailed distributions tend to produce extreme values and using heavy-tailed

distributions when modeling datas related to unusual events gives more realistic

results.The main purpose of this study is proposing statistical estimators for

some characteristics of the stochastic process expressing the model based on the

main results of the study by Kesemen et. al. [6] and using statistical estimators




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57

presented by Frees [1]. Thus, the effect of the heavy tailed demands, which tend

to produce extreme values on the semi-Markovian inventory models, will be

examined statistically.

Keywords: Statistical estimation, Semi Markovian inventory model with heavy

tailed component, Subexponential distributions.

References:

E. W. Frees, “Nonparametric renewal function estimation, Annals of Statistics”,

14.4(1986), 1366-1378.

E. Gökpınar, T. Khanıyev, H. Gamgam, F. Gökpınar, “Estimators of the

moments for the inventory model of type (s,S), Iranian Journal of Science and

Technology”, 42(2018), 5-12.

E. Gökpınar, T. Khanıyev, H. Gamgam, F. Gökpınar, “A new estimators for

stationary distribution of the inventory model of type (s,S)”, Gazi University

Journal of Science, 28.1(2015), 87-101.

[4]E. Gökpınar, T. Khanıyev, H. Gamgam, “Asymptotic properties of the straight

line estimator for a renewal function, SainsMalaysiana”, (2015), 1041-1051.

[5]E. Gökpınar, T. Khanıyev, H. Gamgam, F. Gökpınar, “Estimators of the

moments for the inventory model of type (s,S) with a discrete interference of

chance, Journal of the Turkish Statistical Association”, 9.1(2016), 17-28.

[6] T. Kesemen, A. B. Kamışlık, Z. Küçük, E. Şenol, “On the asymptotic

expansion for the moments of an inventory model of type (s,S) with

subexponentialWeibull distributed demand” Journal of the Tuekish Statistical

Association, 3(2016), 81-92.




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58

Projective covariant α-completely positive linear maps

Tania Luminita Costache

Department of Mathematical Methods and Models,

Faculty of Applied Sciences,

University Politehnica of Bucharest,

Bucharest, Romania

[email protected]

Abstract: In this paper we construct a Krein space, a J-representation and a

projective J-representation associated with a unital projective J-covariant

completely positive linear map. We also find a Krein space representation for a

unital projective (θ,u)-covariant α-completely positive map. For a given

projective unital (θ,u)-covariant α-completely positive map ρ and a projective

(Ƭ,σ,u)-covariant ρ-map we construct a projective covariant J-representation of a

C*-dynamical system. We form projective J-unitary representations associated to

a projective (u,u’)-covariant completely positive ρ-map. Also, we prove that

there is a projective covariant representation associated with a projective

covariant α-completely positive map on an S-module.

Keywords: Hilbert space, Krein space, completely positive map, projective

representation, C*-dynamical system

Acknowledgment : This work has been financed by University Politehnica of

Bucharest, through the “Excellence Research Grants” Program, UPB – GEX

2017, “Aplicatii complet pozitive”, Id 5, No. SA 54-17-08.

References:

[1] B.V. Rajarama Bhat, G. Ramesh, K. Sumesh, Stinespring's theorem for maps

on Hilbert C*-modules, J. Operator Theory 68, No. 1 (2012), 173-178.

[2] T.-L. Costache, Projective version of Stinespring type theorems, Bulletin

Mathematique de la Societe des Sciences Mathematiques de Roumanie, Vol. 53,

No.2 (2010), 109-124.

[3] T.-L. Costache, KSGNS type representations on Krein C*-modules

associated with projective J-covariant - completely positive maps, UPB Sci.

Bull., Series A, Vol. 77, Iss.2 (2015), 67-78.

[4] J. Heo, Hilbert C*-modules and projective representations associated with

multipliers, J. Math. Anal. Appl. 331 (2007), 499-505.




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59

[5] J. Heo, J. P. Hong, U. C. Ji, On KSGNS Representations on Krein C*-

modules, J. Math. Phys. (5), 51 (2010), 1-17.

[6] J. Heo, U. C. Ji, Y. Y. Kim, Projective covariant representations of locally

C*-dynamical systems, Taiwanese J. Of Math., vol.17, No.2 (2013), 529-544.

[7] J. Heo, Un Cig Ji, Y. Y. Kim, -completely positive maps on locally C*-

algebras, Krein modules an Radon-Nicodym theorems, J. Korean Math. Soc.

(50), No.1 (2013), 61-80.

[8] J. Heo, Covariant representations on Krein C*-modules associated to pairs of

two maps, J of Math. Anal. and Appl. 398 (2013), 35-45.

[9] J. Heo, -completely positive maps of groups systems and Krein module

representations, J. of Math. Anal. and Appl., vol. 409, Issue 1 (2014), 544-555.

[10] U. C. Ji, M. Joita, M. Sal Moslehian, KSGNS type construction for -

completely positive maps on Krein C*-modules, Complex Analysis and

Operator Theory, 10 (2016), 3, 617-638. [11] M. Joita, Covariant version of the

Stinespring type theorem for Hilbert C*-modules, Cent. Eur. J. Math. 9 (2011),

4, 803-813.

[12] H. Trivedi, A covariant Stinespring type theorem for -maps, Surveys in

Mathematics and its Applications, Vol. 9 (2014), 149-166.




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60

Determination of a coefficient in a second-order hyperbolic

problem

Arzu Erdem Coşkun


Kocaeli,Turkey

[email protected]

Abstract: We consider an inverse problem of determination of the coefficient

𝑘(𝑥), 𝑥 ∈ Ω in the second-order hyperbolic equation 𝑢𝑡𝑡 = ∇(𝑘(𝑥)∇𝑢) in ΩT =

Ω × (0,T) from the initial and boundary information 𝑢(𝑥, 0), 𝑢𝑡(𝑥, 0) and

𝑘(𝑥)∇𝑢𝒏|𝜕Ω×(0,T). Here Ω ⊂ ℝn is a rectangularly-shaped bounded domain and

𝒏 is the outward unit normal to the boundary of Ω at the point 𝑥 ∈ 𝜕Ω. This

problem is playing an important role in ground-water hydrology and contaminant

transport in porous media, vibrations of a membrane, acoustic vibrations of a

gas, and electromagnetic processes in non-conducting media by (1) The inverse

problem of identifying unknown coefficients has already been investigated by

some authors (2; 3; 4). We prove an existence and uniqueness theorem for the

associated inverse coefficient hyperbolic problem. Our approach consists of

monotonicity method (5; 6) based on integral relationships between the input and

output data. Finally, we present various numerical results that show feasibility of

our approach.

Keywords: Inverse coefficient problem, the second order hyperbolic PDEs, the

adjoint problem, monotonicity, existence, uniqueness.

References:

1. Petrovsky, I.G. Lectures on Partial Differential Equations. New York :

Interscience, 1966.

2. Determination of a coefficient in an acoustic equation with a s ingle m

easurement. Imanuvilov, O.Y. and Yamamoto, M. s.l. : Inverse Problems, 2003,

Vol. 19, pp. 157-171.

3. A posteriori error estimates in a globally convergent FEM for a hyperbolic

coefficient inverse problem. Asadzadeh, M. and Beilina, L. s.l. : Inverse

Problems, 2010, Vol. 26, p. 34.




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61

4. Uniqueness for a hyperbolic inverse problem with angular control on the

coefficients. Rakesh and Sacks, P. s.l. : J. Inv. Ill-Posed Problems, 2011, Vol. 19,

pp. 107 - 126.

5. Monotonicity and invertibility of coefficient-to-data mappings for parabolic

inverse problems. DuChateau, P. 6, s.l. : SIAM J. Math. Anal., 1005, Vol. 26, pp.

1473-1487.

6. Monotonicity of input-output mappings in inverse coefficient and source

problems for parabolic equations. Hasanov, A., Demir, A. and Erdem, A. 2, s.l. :

Journal of Mathematical Analysis and Applications, 2007, Vol. 35, pp. 1434-

1451.




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62

On the Existence and Uniqueness of Solutions for a First And

Second-Order Discrete Boundary Value Problem

Arzu Erdem Coşkun, Ali Demir


Kocaeli,Turkey

[email protected], [email protected]

Abstract: In recent years, we witness that the discrete boundary value problems

serve as mathematical models in diverse fields, such as economy, biology,

physics, mechanics, computer science, finance. Several approaches such as fixed

point theorem, critical point theory and global bifurcation techniques have been

proposed for solving for the discrete boundary value problems, (1; 2; 3; 4; 5).

In this paper.there are two parts In the first part, subjected to two-point boundary

condition, we consider the nonlinear first order discrete boundary value problem.

A new sufficient condition for the existence and uniqueness of the nonlinear first

order difference evolution equation are established . We define an iteration

procedure by means of Banach contraction principle and apply our results to

show that iterated solutions converge to the solutions of the discrete problem.

When the residual is acceptably small we prove the iteration procedure is

terminated. To illustrate the existence of a unique solution of the contraction

mapping theorem some examples are included.

In the second part of the study, we consider the following second-order

difference boundary value problem

{

𝛥2𝑥𝑖𝜏2

= 𝑓(𝑡𝑖 , 𝑥𝑖), 𝑖 = 0,1,2, … , 𝑛 − 1,

𝑢𝑥0 + 𝑣𝑥𝑛 = 0,

��𝛥𝑥0𝜏+ ��

𝛥𝑥𝑛𝜏= 0,

where f: [0, T] × ℝ → ℝ, and the difference operator is defined as follows

𝛥²𝑥𝑖 = {𝑥𝑖+1 − 2𝑥𝑖 + 𝑥𝑖−1, 𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑛 − 2,0, 𝑓𝑜𝑟 𝑖 = 0 𝑜𝑟 𝑖 = 𝑛 − 1,

.

By using Banach fixed point theorem we establish some sufficient conditions for

the existence and uniqueness of the above second-order difference boundary

value problem.




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63

Keywords:

Existence of solutions; First-order Discrete Boundary Value

Problems; Second-order Discrete Boundary Value Problems; Banach

contraction; Fixed Point Iteration.

References:

1. On first-order discrete boundary value problems. Tisdell, C.C. 12, s.l. : Journal

of Difference Equations and Applications, 2006, Vol. 12, pp. 1213-1223.

2. A note on improved contraction methods for discrete boundary value

problems. Tisdell, C.C. 10, s.l. : Journal of Difference Equations and

Applications, 2012, Vol. 18, pp. 1773-1777.

3. Discrete First-Order Three-Point Boundary Value Problem. Mohamed, M. 2,

s.l. : Journal of Mathematics Research, 2009, Vol. 1, pp. 207-215.

4. Initial-boundary-value problems for discrete linear evolution equations.

Biondini, G. and Wang, D. s.l. : IMA Journal of Applied Mathematics, 2010,

Vol. 75, pp. 968-997.

5. On positive periodic solutions of second-order difference equations with

attractive-repulsive singularities. Lu, Yanqiong and Ma, Ruyun. s.l. : Advances

in Difference Equations, 2012, Vol. 2012, p. 186.




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64

Instability Intervals for Hill’s Equation with Symmetric Double

Well Potential

Haskız Coşkun and Elif Başkaya


Trabzon, Turkey


Abstract: In this paper, we derive some estimates explicitly for the periodic and

the semi-periodic eigenvalues and the length of the instability intervals of Hill’s

equation with symmetric double well potential by using an auxiliary eigenvalue

problem. We also give bounds for the gaps of Dirichlet and Neumann

eigenvalues.

Keywords: Hill’s equation, symmetric double well potential, instability

intervals.

References:

[1] H. Coşkun and B. J. Harris, Estimates for the periodic and semi-periodic

eigenvalues of Hill’s equations, Proc. Roy. Soc. Edinburgh Sec. A 130 (2000),

991-998.

[2] H. Coşkun, Some inverse results for Hill’s Equation, J. Math. Anal. Appl.

276 (2002), 833-844.

[3] H. Coşkun, On the spectrum of a second order periodic differential equation,

Rocky Mountain J. Math. 33 (2003), 1261-1277.

[4] M. S. P. Eastham, The spectral theory of periodic differential equations,

Scottish Academic

Press, Edinburgh and London, 1973.

[5] M. J. Huang, The first instability interval for Hill equations with symmetric

single well potentials, Proc. Amer. Math. Soc. 125 (1997), 775-778.

[6] M. J. Huang and T. M. Tsai, The eigenvalue gap for one-dimensional

Schrodinger operators with symmetric potentials, Proc. Roy. Soc. Edinburgh

Sec. A 139 (2009), 359-366.




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65

Analysis of Neutron Capture Cross Section Using Systematics

of Different Level Density Model

Veli Çapalı

Usak University,

[email protected]

Abstract: The neutron capture cross–section reactions for some isotopes have

been calculated with different level density models by using TALYS 1.6 and

EMPIRE 3.2 codes for incident neutron energies up to 300 keV. The obtained

results have been analyzed with each level density model parameters exists in

Reference Input Parameter Library (RIPL–3) and available experimental data. As

an outcome, to obtain more precise and coherent calculations with the

experimental values, level density model and its parameters have been

determined.

Keywords: Cross section; Reactor materials; Level density models; Hadronic

interactions




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66

On the extensions of triangular norms and triangular conorms

Gül Deniz Çaylı

Department of Mathematics, Faculty of Sciences,

Karadeniz Technical University,

61080, Trabzon, Turkey

[email protected]

Abstract: Triangular norms (t-norms) and the corresponding triangular conorms

(t-conorms), introduced by Schweizer and Sklar in [8], are used in several

branches of mathematics in different manners such as in probabilistic metric

spaces, many-valued logic, fuzzy logic, fuzzy sets, decomposable measures and

their applications. A t-norm (t-conorm) on a lattice 𝐿 is a function 𝑇 ∶ 𝐿2 → 𝐿

(𝑆 ∶ 𝐿2 → 𝐿) which is commutative, associative, nondecreasing in both variables

and fulfills, for all 𝑥 ∈ 𝐿, the boundary condition 𝑇(1, 𝑥) = 𝑥 (𝑆(0, 𝑥) = 𝑥).

In this contribution, we consider the extensions of t-norms and t-conorms on

lattices and investigate their characteristics. Moreover, some examples are

provided to illustrate the feasibility of the proposed extensions.

Keywords: Fuzzy logic, Triangular norm, Triangular conorm, Lattice.

References:

[1] D. Butnariu, E.P. Klement, “Triangular Norm-Based Measures and Games

with Fuzzy Coalitions”, Kluwer, Dordrecht, 1993.

[2] G.D. Çaylı, “On a new class of t-norms and t-conorms on bounded lattices”,

Fuzzy Sets Syst. 332(2018), 129-143.

[3] G.D. Çaylı, “Characterizing ordinal sum for t-norms and t-conorms on

bounded lattices”, In Advances in Fuzzy Logic and Technology 2017. IWIFSGN

2017, EUSFLAT 2017. Advances in Intelligent Systems and Computing;

Kacprzyk, J. et al. (eds), vol.641 Springer, Cham, 2018.

[4] Ü. Ertuğrul, F. Karaçal, R. Mesiar, “Modified ordinal sums of triangular

norms and triangular conorms on bounded lattices”, Int. J. Intell. Syst. 30(2015),

807-817.

[5] E.P. Klement, “Construction of fuzzy a-algebras using triangular norms”, J.

Math. Anal. Appl. 85(1982), 543-565

[6] J. Medina, “Characterizing when an ordinal sum of t-norms is a t-norm on

bounded lattices”, Fuzzy Sets Syst. 182(2012), 75-88.




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67

[7] S. Saminger “On ordinal sums of triangular norms on bounded lattices”

Fuzzy Sets Syst. 157.10(2006), 1403-1416.

[8] B. Schweizer, A. Sklar, “Probabilistic Metric Spaces”, In North-Holland,

New York, 1983.

[9] H.J. Zimmermann, “Fuzzy Set Theory and Its Applications”, Kluwer,

Dordrecht, 1991.




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68

On Suborbital Graphs and Related Sequences

Ali Hikmet DEĞER1, Ümmügülsün AKBABA1, Tuğba TUYLU1

1Department of Mathematics, Karadeniz Technical University,

Ortahisar, Trabzon, Turkey


Abstract: The two number sequences Fibonacci and Lucas, which have a very

popular place in the literature and are important applications in various fields. On

the other hand, the action of a group on a set is also used in different areas. In

particular, the Möbius transformation is an important transformation of the

Complex Analysis field of theory of Analysis and Functions. The modular group

Γ and its congruence subgroups, which are a group of linear fractional

transformations obtained from these Möbius transformations, also yield results in

different fields. The suborbital graphs were studied based on the action of the

congruence subgroup Γ0(N) < Γ on ℚ ≔ ℚ ∪ {∞} and the results related to the

vertex which can be joined with the farthest vertex on these suborbital graphs

were obtained. Taking these results into account, significant and different results

were obtained in relation to the Fibonacci and Lucas numbers from the minimal

length problem of the respective vertices. It is expected that different results will

be reached by taking all the features of the Fibonacci and Lucas number

sequences by going out from these results obtained in this work. Generally, on

the minimal-length paths in the suborbital graphs, the relevant vertices will be

related to these number sequences.

Keywords: Suborbital graphs, Continued fraction, Fibonacci and Lucas

sequences

References:

[1] A.H. Değer, M. Beşenk and B.Ö. Güler, “On suborbital graphs and related

continued fractions”, Applied Mathematics and Computation, 218 (2011), 746-

750.

[2] G. A. Jones, D. Singerman and K. Wicks, “The modular group and

generalized Farey Graphs”, London Math. Soc. Lecture Notes, CUP, Cambridge,

160 (1991), 316-318.

[3] A.H. Değer, “Vertices of Paths of Minimal Lengths on Suborbital Graphs”,

Filomat, 31:4 (2017), 913-923.




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69

Characterizations of Lipschitz spaces via commutators of

fractional maximal operator in Orlicz spaces

Fatih Deringöz

Department of Mathematics, Ahi Evran University,

Kırşehir, Turkey

[email protected]

Abstract: In this talk, we give necessary and sufficient conditions for the

boundedness of fractional maximal operator M on Orlicz spaces. As an

application of this result we consider the boundedness of fractional maximal

commutator ,bM and nonlinear commutator of fractional maximal operator

[ , ]b M on Orlicz spaces, when b belongs to the Lipschitz space, by which

some new characterizations of the Lipschitz spaces are given.

This work is supported by Ahi Evran University Scientific Research Projects

Coordination

Unit (Project Number : FEF.A4.17.009).

This contribution is based on recent joint work with V.S Guliyev and S.H.

Hasanov.

Keywords: Orlicz spaces, Lipschitz spaces, fractional maximal operator,

commutator.

References:

[1] V.S. Guliyev, F. Deringoz, S. H. Hasanov, “Fractional maximal function and

its commutators on Orlicz spaces”, Anal. Math. Phys., DOI 10.1007/s13324-

017-0189-1.




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70

Convergence Theorems on Measure Space of Exhaustive

Doris Doda

Wisdom University

Economics Faculty

Business Management

[email protected]

Abstract:In this paper we propose some propositions in reference to notion of

exhaustivness which applies both for families and sequences of functions. This

new notion given for the first time from Gregoriades and Papanastassiou in 2008

and has a good progress and in mathematical community is followed by many

papers. We are concerned in study these families in framework of measure

theory and Henstok-Kurzweil integration.We investigate the properties of new

convergence which we named -convergence and compare it with other known

convergencies.

Keywords: Exhaustivness, alfa-convergenece, exhaustiv sequence,

equicontinuity, -convergence, Henstok-Kurzweil integral.




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71

A New view of L-ideals on Semigroups

Emine Funda Ekinci, Sultan Yamak


Trabzon, Turkey


Abstract: In this paper, we first introduce the notion of L-left (resp. L-right)

ideals of a TL-subsemigroup in a semigroup, and develop some properties of

them. We also build the relation between L-ideals of a semigroups and L-ideals

of a TL-subsemigroups. Furthermore, the lattice structure of L-left (resp. L-right)

ideals of a TL-subsemigroups are studied.

Keywords: t-norm, semigroups, TL-subsemigroups, L-ideals.

References:

[1] Chon, I., On fuzzy bi-ideals in semigroups, Korean J. Math, 19, (2011), 321-

330.

[2] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174.

[3] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul.

28 (1979), 17-21.

[4] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets

Syst.

5 (1981), 203-215.

[5] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets Syst. 871-80

(1982).

[6] N. Kuroki, On fuzzy semigroups, Inform. Sci. 53:203-236 (1991).

[7] Mordeson, J.N., Malik, D.S., Kuroki, N., Fuzzy semigroups. Springer-Verlag,

2003.

[8] N. Thillaigovindan, V. Chinnadurai, S. Coumaressane, T-fuzzy

subsemigroups and T-fuzzy ideals of regularΓ-semigroups, Ann. of Fuzzy

Math. And Inf., 11. 4 (2016), 669–680.

[9] Yandong Yu, Zhudeng Wan, TL-subrings and TL-ideals. Part 1. Basic

concepts, Fuzzy Sets and Systems, 68-1, 28(1994), 93-103.

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




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72

Design and Application of Fractional Order Predictor for

Unglazed Transpired Solar Collector

Köksal ERENTÜRK, Saliha ERENTÜRK

Department of Computer Engineering, Ataturk University,

Erzurum, Turkey

[email protected]

Abstract: In order to evaluate different modeling techniques for Unglazed

Transpired Collectors (UTC), not only mathematical modeling method for UTC

based on heat transfer expressions to estimate the various heat transfer

coefficients for the UTC components and empirical relationship, but also

fractional order predicting approach have been designed and introduced, in this

study. Thermal performance experiments of UTC have been carried out on an

optimized experimental setup. Firstly, obtained experimental results have been

compared with the mathematical model. To constitute a common point, output

temperature of the UTC has been selected as the output variable. Then, a

fractional order predictor has been used to forecast the output temperature with

higher accuracy with the aid of simple mathematical equations. Finally, obtained

results have been compared and comparison results have been illustrated in both

graphical and tabular form. Fractional order predictor is the simplest method to

forecast the output temperature with high accuracy.

Keywords: Solar heating, fractional order modeling, fractional order predictor,

unglazed transpired collectors.

References:

[1] Sozer H. Improving energy efficiency through the design of the building

envelope. Building and Environment 2010; 45 (12): 2581–93.

[2] SOLARWALL, 2016. The SOLARWALL Solar Heating System. Conserval

Engineering Inc., Canada. Available from: <http://www.solarwall.com/>.

Accessed in August, 2016.

[3] Rakesh K, Lei C. An experimental investigation of an inclined passive wall

solar chimney for natural ventilation. Solar Energy 2014; 107: 461–74.




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73

[4] Feng J, Xin W, Yinping Z. A new method to estimate optimal phase change

material characteristics in a passive solar room. Energy

Conversion&Management 2011; 52(6): 2437–41.

[5] Wei S, Jie J, Chenglong L, Wei H. Performance of PV-trombe wall in winter

correlated with south facade design. Applied Energy 2011; 88(1): 224–31.

[6] Yanfeng L, Dengjia W, Chao M, Jiaping L. A numerical and experimental

analysis of the air vent management and heat storage characteristics of a

trombe wall. Solar Energy 2013; 91: 1–10.

[7] KUDAKA Technical Report, 2016. Available from:

<http://www.kudaka.org.tr/e-dergi/Erzurum_Gunes_Enerjisi_Potansiyeli/>.

Accessed in August, 2016.

[8] Podlubny, I., Fractional Differential equations, San Diego: Academic Press,

1999.

[9] Podlubny, I., Fractional-order systems and PIλDμ-controllers. IEEE Trans.

Automatic Cont., vol. 44, no. 1, 1999, pp. 208-214.

[10] Dumlu, A., Erenturk, K. Trajectory Tracking Control for a 3-DOF Parallel

Manipulator Using Fractional-Order PIλDμ Control, IEEE Transactions on

Industrial Electronics 2014, 61(7): 3417-3426.




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74

Determination of the Effective moisture Diffusivities in

Convective Drying Systems of Banana using Fractional Order

Estimator

Saliha ERENTÜRK, Köksal ERENTÜRK

Department of Chemical Engineering, Ataturk University,

Erzurum, Turkey

[email protected]

Abstract: Drying techniques have been used for the food preservation for

centuries as it caused the minimize biochemical, chemical and microbiological

deterioration. Despite high energy consumption and decreasing of quality,

obtained product gains longer life by drying. The moisture diffusivity of a

foodstuff is a material property and it changes with many factors including

moisture content, process variables, water holding capacity and physiochemical

attributes of food. Effective moisture diffusivity describes all possible

mechanisms of moisture movement within food, which is combination of liquid

diffusion, vapor diffusion, surface diffusion, capillary flow and hydrodynamic

flow. In this work, the effective moisture diffusivity of banana was determined in

air convective drying system. In the system three different air velocities, air

temperatures and solid thickness were investigated. Effective moisture

diffusivities varied between 4.75-6.48 10-10m2/s for temperature investigated,

3.66-9.53 10-10m2/s for air velocities investigated and 6.48-3.12 10-10m2/s for

solid thickness investigated.

In this study, amount of effective moisture diffusivity of banana has been

determined using Fractional Order Estimator. Based on obtained results, an

effective approach has been developed and better results have been observed

compared with classical mathematical modeling approaches and methods.

Keywords: Fractional order estimator, drying, banana, effective moisture

diffusivity.

References:

S.Erenturk, M. S. Gulaboglu, S. Gultekin. “Experimental determination of

effective moisture diffusivities of whole- and cut-rosehips in convective drying”

Food and Biop. Processing, 88(2010), 99–104.




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75

G.P. Sharma, S. Prasad,”Effective moisture diffusivity of garlic cloves

undergoing microwave-convective drying” Journal of Food Engineering, 65,

(2004), 609–617.

M. Vasic, Z. Grbavcic, Z. Radojevic, “Determination of the moisture diffusivity

coefficient andmathematical modeling of drying”, Chem. Eng. and Processing,

76, (2014), 33– 44.

A. Fabbri, C. Cevoli, R. Troncoso,”Moisture diffusivity coefficient estimation in

solid food by inversion of a numerical model”, Food Research International, 56,

(2014),63-67.

B. D.A. Porciuncula, M. F. Zotarelli, B. A.M. Carciofi, J. B.

Laurindo,“Determining the effective diffusion coefficient of water in banana

(Prata variety) during osmotic dehydration and its use in predictive

models”,Journal of Food Engineering, 119, 3, (2013),490-496.




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76

On Nullnorms On Bounded Lattices

Ümit Ertuğrul


Trabzon, Turkey

[email protected]

Abstract: Special aggregation functions nullnorms are generalization of t-norms

and t-conorms ([6]). Because of this feature, they are being studied extensively

by researchers ([2],[4]). Nullnorms are useful not only in theory but also in

many applications. Nullnorms are first defined on the unit real interval [0,1].

Subsequently, the work has been detailed, taking into account the bounded lattice

case ([3],[4],[6]). In this study, some properties of nullnorms on bounded lattices

have been investigated.

Keywords: Nullnorm, T-norm, T-conorm, Bounded Lattices.

References:

[1] G. Birkhoff, “Lattice Theory”, American Mathematical Society Colloquium

Publishers, Providence, RI, 1967.

[2] P. Drygás, “A characterization of idempotent nullnorms”, Fuzzy Sets Syst.

145 (2004), 455-461.

[3] M. Grabish, J.-L. Marichal, R. Mesiar, E. Pap, “Aggregation Functions”,

Cambridge University Press, 2009.

[4] F. Karaçal, M.A.İnce, R. Mesiar, “Nullnorms on bounded lattices”, Inf. Sci.

325 (2015), 227-236.

[4] F. Karaçal, M.N. Kesicioğlu, “A T-partial order obtained from t-norms”,

Kybernetika 47 (2011), 300-314.

[5] E.P. Klement, R. Mesiar, E. Pap, “Triangular norms”, Kluwer Academic

Publishers, Dordrecht, 2000.

[6] M. Mas, G. Mayor, J. Torens, “t-Operators”, Int. J. Uncertain. Fuzziness

Knowl.-Based Syst. (1999) 31-50.




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77

About the Grundy coloring critical graphs

Noureddine Ikhlef-Eschouf

Department of Mathematics and Computer Sciences, University of Yahia Farès

Médéa, Algeria

[email protected]

Abstract:Let G be a graph with a vertex-set V. A proper coloring of a graph G

with k colors is a partition {V1,V2,…,Vk} of V into k independent sets (color

classes). A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex

in each color class Vj for every j < i. A Grundy coloring is a proper coloring such

that every vertex is a Grundy vertex, and the Grundy number (G) of a graph G

is the maximum number of colors in a Grundy coloring. A graph G is called g-

edge critical if the removal of any edge of G decreases its Grundy number. In

this paper, we explain some properties in g-edge critical graphs, and as

consequence, we characterize trees, P4-sparse graphs and d-regular graphs that

are g-edge critical.

Keywords:Coloring, Grundy coloring, Grundy number, edge-critical graphs

References:

[1] C.A. Christen and S. M. Selkow, Some Perfect coloring properties of

graphs, H. Combin. Theory Ser. B27 (1979) 49-59.

[2] Results on the Grundy chromatic number of graphs, Discrete Appl. V. 306,

23(2006), 3166-3173.




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78

f - Statistical Boundedness of Order α

Mikail Et and Hüseyin Sönmez

Department of Mathematics, Firat University,

Elazığ, Turkey


Abstract: In this paper, using an unbounded modulus function, we introduce the

notion of f-statistical boundedness of order α for scalar sequences and give some

relations about this concept.

Keywords: Statistical Convergence, Statistical boundedness, Modulus function.

References:

[1] Bhardwaj V. K. and Gupta, S. On some generalizations of statistical

boundedness, J. Inequal. Appl. 2014, 2014:12

[2] Çolak R. Statistical Convergence of Order α. Modern Methods in Analysis

and Its Applications, NewDelhi, India: Anamaya Pub., 2010: 121-129.

[3] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2(1951), 241-244.

[4] J. A. Fridy, “On statistical convergence”, Analysis, 5(1985), 301-313.

[5] I. J. Schoenberg, “The integrability methods”, Amer. Math. Monthly,

66(1959), 361-375.

[6] E. Savaş, “Asymptotically J-lacunary statistical equivalent of

order $\alpha$ for sequence of sets” J. Nonlinear Sci. Appl. 10(6) (2017), 2860--

2867

[7] T. Salat, “On statistically convergent sequences of real numbers”, Math

Slovaca, 30.2(1980), 139-150.




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79

The influence of fluid pressure on the displacement of the

basilar membrane

Aboulkhouatem Fatima-Ezzahra

Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik,

Hassan II University, Casablanca, Morocco

[email protected]

Abstract: An increase of pressure on the structure of cochlea causes hearing

loss. In this paper, we established the relationship between the pressure and the

amplitude of displacement of Basilar membrane. Therefore, our goal is to clarify

the mechanisms of hearing loss caused by increasing of pressure. So, a

mathematical cochlear model was formulated using finite difference method in

order to explain and demonstrate this malfunction in passive model. Numerical

simulations are helpful tools which may extend and complete the understanding

of the cochlea physiology.

Keywords: Mathematical model, hearing loss, Basilar membrane, Finite

difference method, Fluid pressure.

References:

[1] K. Kamieniecki, J. Piechna & P. Borkowski, “Basilar Membrane Vibration

in Time Domain Predicted by Fluid–Structure Interaction Model in Pre-and Post-

stapedotomy State”, Procedia IUTAM, 24(2017), 48-63.

[2] M. Jimenez-Hernandez, “Power analysis of the basilar membrane in the

cochlea by mechanical resonance”, Rev. Mex. Ing. Bioméd , 38(2017), 1.

[3] S.T. Neely , “Mathematical modeling of cochlear mechanics”, J. Acoust. Soc.

Am., 78(1985), 345-352.

[4] L.C. Peterson, B.P. Bogert, “A dynamical theory of the Cochlea”. J. Acoust.

Soc. Am. 22(1952), 175-84.




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80

Change of the Cochlear Michromechanics model due to fluid

presure

Kouilily Fatiha

Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik,

Hassan II University, Casablanca, Morocco

[email protected]

Abstract: Abnormality in the disposition of the structure of the inner ear leads to

hearing loss. In this paper, we studied the effect of fluid pressure on the

displacement of Basilar Membrane (BM) model included the amplification

mechanism of Outer hair cell (OHC) using finite difference method. As result the

distribution of the characteristic frenquency was observed and the displacement

of the BM was decreased by applying a fluid pressure to BM. These changes are

considered to be the reasons for hearing loss caused by fluid pressure.

Keywords: Basilar membrane, Outer hair cell, hearing loss, Fluid pressure.

References:

[1] Lee, S., & Koike, T. (2017). Simulation of the Basilar Membrane Vibration

of Endolymphatic hydrops. Procedia IUTAM, 24, 64-71.

[2] Kouilily, F., Aboulkhouatem, F. E., Yousfi, N., & Achtaich, N. Predicting the

Effect of Physical Parameters on the Amplitude of the Passive Cochlear Model.

[3] S.T. Neely, “Mathematical modeling of cochlear mechanics”, J. Acoust. Soc.

Am., 78(1985), 345-352.




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81

Optimal scientific production over the life cycle

Gustav Feichtinger

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna

University of Technology,

Vienna, Austria.

[email protected]

Dieter Grass

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna

University of Technology,

Vienna, Austria.

Peter M. Kort

Econometrics and Operations Research, Tilburg School of Economics and Management

Tilburg, Netherlands

Abstract: The paper develops an optimization model of the career of a scientist.

Recognizing that research efforts and networking get more efficient if the

scientist is more knowledgeable, history dependent solutions are developed. We

give a theoretical underpinning of the four different research patterns detected in

[1]. If the scientist does not bother about his reputation at the end of his career,

we show that a sufficient education level is needed for the scientist to develop a

typical research pattern where productivity increases in the beginning of his

career, while it declines towards retirement. If the education level is not

sufficient, a fading research pattern will result where productivity declines over

time. On the other hand, when the scientist appreciates to have a good reputation

at the end of his career, sufficient education will result in increasing productivity

over the career lifetime, preventing a midlife slump.

Keywords: Optimal control, research patterns, Skiba, age-dependent scientific

production, networking: initial knowledge

References:

[1] Way, S. F., Morgan, A. C., Clauset, A., Larremore, D. B., 2017. The

misleading narrative of the, canonical faculty productivity trajectory.

Proceedings of the National Academy of Sciences 114 (44), E9216–E9223




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82

The Extended b-Metric Space and Related Fixed Point

Results

Andreea Fulga

Department of Mathematics and Computer Science,

Transilvania University of Brasov, Romania

[email protected]

Abstract. Very recently, Kamran et al. [7] replaced the modified triangle

inequality 𝑑(𝑥, 𝑧) ≤ 𝑠(𝑑(𝑥, 𝑦) + 𝑑(𝑦, 𝑧)) by 𝑑(𝑥, 𝑧) ≤ 𝜃(𝑥, 𝑧)(𝑑(𝑥, 𝑦) +

𝑑(𝑦, 𝑧)) where 𝜃: 𝑋 × 𝑋 → [1,∞) and they proved a Banach type fixed point

theorem. In this paper, continuing in this direction, we investigate the existence

of fixed points of certain mapping in the frame of extended b-metric space. We

state some examples to illustrate our results.

References

[1] H. Afshari, H. Aydi, E. Karapinar, “Existence of Fixed Points of Set-Valued

Mappings in b-Metric Spaces”, East Asian mathematical journal, volume 32,

issue 3, 2016, Pages 319 332

[2] U. Aksoy, E. Karapınar and Y. M. Erhan, “Fixed points of generalized alpha-

admissible contractions on b-metric spaces with an application to boundary value

problems”, Journal of Nonlinear and Convex Analysis, (2016), Volume 17,

Number 6, 1095-1108

[3] M. Bota, C. Chifu, E. Karapinar, “Fixed point theorems for generalized

(alpha-psi)-Ciric-type contractive multivalued operators in b-metric spaces” J.

Nonlinear Sci. Appl. 9 (2016), Issue: 3 pages 1165-1177

[4] L.B. Ciric, “On some maps with a nonunique fixed point.”, Publ. Inst. Math.,

17, 52–58 (1974).

[5] M. Geraghty, “On contractive mappings”, Proc. Amer. Math. Soc. 40(1973),

604-608.

[6] E. Karapinar, A New Non-Unique Fixed Point Theorem, J. Appl. Funct.

Anal. 7 (2012), no:1-2, 92-97

[7] T. Kamran, M. Samreen, O. U. Ain, “A Generalization of b−Metric Space

and Some Fixed Point Theorems”




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83

Suborbital Graphs For a Special Subgroup of the Normalizer

Nazlı Yazıcı Gözütok and Bahadır Özgür Güler


Trabzon, Turkey


Abstract: Let 𝑃𝑆𝐿(2,ℝ) denote the group of all linear fractional transformation

𝑇: 𝑧 →𝑎𝑧 + 𝑏

𝑐𝑧 + 𝑑,𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑑 𝑎𝑟𝑒 𝑖𝑛 ℝ 𝑎𝑛𝑑 𝑎𝑑 − 𝑏𝑐 = 1.

In terms of matrix representation, the elements of 𝑃𝑆𝐿(2, ℝ) correspond to the

matrices

±(𝑎 𝑏𝑐 𝑑

) ; 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑑 𝑎𝑟𝑒 𝑖𝑛 ℝ 𝑎𝑛𝑑 𝑎𝑑 − 𝑏𝑐 = 1.

This is the automorphism group of the upper half plane

ℍ ≔ {𝑧 ∈ ℂ: 𝐼𝑚(𝑧) > 0}.

The modular group Γ is the subgroup of 𝑃𝑆𝐿(2, ℝ) such that 𝑎, 𝑏, 𝑐, 𝑑 are

integers. Γ0(𝑁) = {𝑔 ∈ 𝛤: 𝑐 ≡ 0(𝑚𝑜𝑑 𝑁)} is a well known congruence subgroup

of the classical modular group Γ for a positive integer 𝑁. The normalizer of

Γ0(𝑁) in 𝑃𝑆𝐿(2,ℝ) turns to be a very important group in the study of moonshine

and for this reason has been studied by many authors. It consists exactly of the

matrices

(𝑎𝑒 𝑏/ℎ𝑐𝑁/ℎ 𝑑𝑒

)

where all parameters are integers, 𝑒||𝑁

ℎ2 and ℎ is the largest divisor of 24 for

which ℎ2|𝑁 with understandings that the eterminant 𝑒 of the matrix is positive,

and that 𝑟||𝑠 means that 𝑟|𝑠 and (𝑟,𝑠

𝑟) = 1 (𝑟 is called an exact divisor of s). We

denote the normalizer by Γ𝐵(𝑁). Jones, Singerman and Wicks examined

suborbital graphs o the modular group Γ [1]. After this, similar studies were done

for related finitely generated groups. Since the group structure of 𝛤𝐵(𝑁) is much

more complex than the modular group, its suborbital graphs have been studied

under various restriction in many studies [2,3]. Two important subgroups of

Γ𝐵(𝑁) are Γ𝑊(𝑁) and Γ𝐶(𝑁). Suborbital graphs of the Atkin-Lehner group




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84

Γ𝑊(𝑁) is also studied in [4]. The aim of this paper fills the gap in literature

examining suborbital graphs of Γ𝐶(𝑁).

Keywords: Normalizer, Congruence subgroup, Suborbital graphs.

References:

[1] G. A. Jones, D. Singerman, K. Wicks, “The modular group and generalized

Farey graphs”, London Math. Soc. Lecture Note Series, 160(1991), 316-338.

[2] B.Ö. Güler, M. Beşenk, A.H. Değer, “Elliptic Elements and Circuits In

Suborbital Graphs”, Hacettepe Journal of Mathematics and Statistics, 40(2011),

203-210.

[3] R. Keskin, “Suborbital graphs for the normalizer of 𝚪𝟎(𝒎)”, European J

Combin, 27(2)(2006), 193-206.

[4]T. Köroğlu, B.Ö. Güler, Z. Şanlı, “Suborbital graphs for the Atkin-Lehner

group”, Turk J Math., 41(2017), 235-243.




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85

𝑮𝑳(𝒏,ℝ) Invariants For a Curve Family

Uğur Gözütok, Yasemin Sağıroğlu


Trabzon, Turkey


Abstract: Differential geometry of curves has been studied in many aspects in

the groups 𝑆𝐿(𝑛, ℝ),𝐸𝐴(𝑛, ℝ),𝐺𝐿(𝑛, ℝ) which are the subgroups of the affine

group. In some of these studies, invariants such as arc-length, curvature have

been examined. In [1] centro-affine invariants, arc length and curvature

functions, of a curve in affine 𝑛 −space are obtained. In addition, several authors

studied the affine curves and their invariants in several works [2,3]. Invariants of

𝑛 curves and equivalence of 𝑛 curves in 𝑆𝐿(𝑛, ℝ) are investigated in [4]. In

𝑆𝐴𝑓𝑓(𝑛, ℝ), the equivalence problem of two curves is studied in [5]. In this

study, the system of differential invariants for three curves in 𝐺𝐿(𝑛, ℝ) is studied

and by using this system, equivalence of two curve families which consist of

three curves is obtained. Also, it is shown that the system of differential

invariants of this curve family is minimal. It should be noted that we study the

problem under a fixed parametrization of the curves.

Keywords: Differential invariants, equivalence problem, curves

References:

[1] H. Liu, “Curves in affine and semi Euclidean spaces”, Results. Math.,

65(2014), 235-249.

[2] P.J. Giblin, T. Sano, “Generic equi-centro-affine differential geometry of

plane curves”, Topology Appl., 159(2012), 476-483.

[3] D. Khadjiev, Ö. Pekşen, “The complete system of global differential and

integral invariants for equi-affine curves”, Dif. Geom. Appl., 20(2004), 167-175.

[4] Y. Sağıroğlu, “Centro-equiaffine differential invariants of curve families”,

IEJG, 9(2016), 23-29.

[5] Y. Sağıroğlu, “Equi-affine differential invariants of a pair of curves”, TWMS

J. Pure Appl. Math., 6(2015), 238-245.




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86

Conditions for Ruled Surfaces Obtained by Bending to be

Developable

Uğur Gözütok1, Hüsnü Anıl Çoban2, Yasemin Sağıroğlu3

Department of Mathematics, Karadeniz TechnicalUniversity,

Trabzon, Turkey

[email protected]

[email protected]

sagı[email protected]

Abstract: Ruled surfaces, being a special type of surface, have an important

place in the study of surfaces, because ruled surfaces are the easiest of all to

parametrize. In the papers [1,2], authors study some properties of ruled surfaces.

The ruled surfaces with vanishing Gaussian curvature, which can be transformed

into the plane without any deformation and distortion, are called developable

surfaces. These surfaces are related to various applications from contemporary

architacture to manufacturing of clothing, since they are suitable to the modeling

of surfaces which can be made out of leather, paper, fiber, and sheet metal [3,4].

In this study, we consider the infinitesimal bending of a curve to obtain a ruled

surface. We reveal the relation between them and give conditions to get a

developable surface from a ruled surface obtained by bending.

Keywords: Ruled Surfaces, Bending of Curves, Developable Surfaces.

References:

[1] Izumiya, S., Takeuchi, N., “Special curves and ruled surfaces”, Contributions

to Algebra and Geometry 44(1), 203-212, 2003.

[2] Ali, A.T., Aziz, H.S.A., Sorour, A.H., “Ruled surfaces generated by some

special curves in Euclidean 3-space”, Journal of the Egyptian Mathematical

Society 21(3), 285-294, 2013.

[3] Glaeser, G., Gruber, F., “Developable surfaces in contemporary architecture”,

Journal of Mathematics and the Arts 1(1), 59-71, 2007.

[4] Liu, Y., Pottmann, H., Wallner, J., Yang, Y.L., Wang, W., “Geometric

modeling with conical meshes and developable surfaces”, ACM Transactions on

Graphics 25(3), 681-689, 2006.




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87

Some Generalizations of Ricci Solitons Admitting Standard

Static Spacetime Metric

Sinem Güler

Department of Mathematics, Istanbul Technical University,

Istambul, Turkey

[email protected]

Abstract: In this talk, we focus on certain generalizations of Ricci solitons

endowed with the standard static spacetime metric and we give some

characterizations of such spacetimes via their potential functions. Also, some

necessary and sufficient conditions have been obtained regarding the questions

of the existence of certain vector fields on such standard static spacetimes. In the

final part of this presentation, by taking into account the Walker metric, we

construct some explicit examples of the spacetimes we are working on.

Keywords: Standard Static Spacetime, Ricci Soliton, Walker metric.

References:

[1] F. Dobarro, B. Ünal, “Special standard static spacetimes”, Nonlinear

Analysis: Theory, Methods and Applications. 59(2004), 759-770.

[2] G. Catino, “Generalized quasi Einstein manifols with harmonic Weyl tensor”,

Math. Z., 271(2012), 751-756.

[3] F. Dobarro, B. Ünal, “Characterizing killing vector fields on standard static

spacetimes”, J. Geo. Phys., 62(2012), 1070-1087.

[4] B. O’Neill, “Semi-Riemannian geometry with applications to Relativity”,

Academic Press, NewYork, (1983), 336-341.




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88

Rough I - Statistical Convergence of Weight g

Hafize Gümüş

Department of Mathematics Education, Necmettin Erbakan University,

Konya, Turkey

[email protected]

Abstract: We know from previous studies, it has been shown that we can extend

the concept of natural or asymptotic density by considering natural density of

weight g. The function 𝑔:𝑁 → [0,∞) mentioned here satisfies lim𝑛→∞

𝑔(𝑛) = ∞

and lim𝑛→∞

𝑛

𝑔(𝑛)≠ 0 properties. By using this extension, we define rough I-

statistical convergence of weight g which is a natural combination of rough

convergence and I-statistical convergence of weight g. We also investigate some

observations about these new classes. Keywords: Rough Convergence, I- convergence, I- statistical convergence.

References:

[1] Aytar, S., 2008, Rough statistical convergence, Numerical Functional

Analysis and Optimization, 29, Issue 3-4, 291-303.

[2] Dündar, E. and Çakan, C., 2014, Rough I-convergence, Demonstratio

Mathematica, Vol. XLVII, No.3.

[3] Pal, S. K., Chandra, D. and Dutta, S., 2013, Rough ideal convergence,

Hacettepe Jour. of Math.and stat., Vol. 42(6), 633-640.

[4] Phu, H. X., 2002, Rough convergence in normed linear spaces, Numer.

Funct. Anal. Optmiz. 22:201-224.

[5] Savaş, E., 2017, On I-lacunary statistical convergence of weight g of

sequences of sets, Filomat 31:16, 5315-5322.




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89

The concept of I-Asymptotically Lacunary Statistical

Equivalence of Interval Numbers and A Generalization With A

Positive 𝑝 Sequence

Hafize Gümüş* and Nihal Demir**

*Department of Mathematics Education, Necmettin Erbakan University,

Konya, Turkey

[email protected]

**İkizli Middle School, Emirgazi, Konya, Turkey

[email protected]

Abstract: Previous studies for statistical convergence and I-convergence have

recently begun to study for interval numbers. One of these studies is

asymptotically lacunary statistical equivalence concept defined by Esi [4]. In our

study, we define the concept of I-asymptotically lacunary statistical equivalence

of interval numbers and we extend it for any sequence of positive numbers

𝑝 = (𝑝𝑘). We will examine our studies separately according to whether 𝑝 is

constant or not.

Keywords: I-statistical convergence, lacunary I-statistical convergence,

asymptotic equivalence, interval numbers

References:

[1] P. Kostyrko, T. Salát and W. Wilezyński, “I-Convergence”, Real Analysis

Exchange Vol. 26(2), (2000), 669-686.

[2] M. S. Marouf, “Asymptotic Equivalence And Summability”. Internat. Jour.

Math & Math. Sci. Vol. 16 No.4 (1993), 755-162.

[3] E. Savas and R. F. Patterson, “An Extension Asymptotically Lacunary

Statistical Equivalent Sequences”, Aligarh Bull. Math. 27(2),(2008), 109-113.

[4] A. Esi and A. Esi, “Asymptotically Lacunary Statistically Equivalent

Sequences of Interval Numbers”, Int. Jour. of Math. and its Appl.1 (1),(2013),

43-48.




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90

Exact Solution of Some Nonlinear Partial Differential Equation

Hami Gündoğdu, Ömer Faruk Gözükızıl

Department of Mathematics, Sakarya University,

Sakarya, Turkey


Abstract: In this paper, we are interested in obtaining the approximate analytical

solutions of nonlinear partial differential equations. In this regard, the Laplace

decomposition method is utilized. Nonlinear Gardner equation and Benjamin-

Bona-Mahony-Peregrine-Burgers equation are studied. The method is applied to

these equations with their initial values. Then, the solutions satisfying the given

initial condition are gained.

Keywords: Laplace decomposition method, Gardner equation, Benjamin-Bona-

Mahony-Peregrine-Burgers equation .

References:

[1] Damil N., Potier-Ferry M., Najah A., Chari R., Lahmam H., An

iterative method based upon Pade approximamants, Communications In

Numerical Methods In Engineering, 15, (1999), 701-708.

[2] He J.-H., A coupling method of homotopy technique and perturbation

technique for nonlinear problems, International Journal Of Non-Linear

Mechanics, 35, (2000), 37-43

[3] He J.-H., An approximate solution technique depending upon artificial

parameter, Communications in Nonlinear Science and Numerical Simulaton, 3,

(1998), 92-97.

[4] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves

in nonlinear dispersive systems, Philos TransR Soc London, Ser A, 272 (1972)

47-78.

[5] Peregrine D. H., Calculations of the development of an undular bore, J.

Fluid Mech., 25, (1996), 321-330.




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91

On Invariance of Hyperstoneanness under Lattice

Isomorphisms

Banu Güntürk 1,Bahaettin Cengiz 2 1Faculty of Engineering, Baskent University,

Ankara, Turkey

[email protected] 2Faculty of Engineering, Baskent University,

Ankara, Turkey

[email protected]

Abstract: Let 𝑋 and 𝑌 be compact Hausdorff spaces with 𝑌 hyperstonean. In

this paper we prove that if 𝐶(𝑋, ℝ) and 𝐶(𝑌, ℝ) are lattice isomorphic then these

Banach spaces are linearly isometric and, consequently, 𝑋 and 𝑌 are

homeomorphic, which in turn implies that 𝑋 is also hyperstonean. Actually we

prove more than what is announced in the headline above. This result, in some

ways, is a generalization of the well-known Banach-Stone theorem.

Keywords: Hyperstonean space, Banach-Stone theorem, lattice isomorphism.




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92

Nanofluid flow due to rotating disk with variable thickness and

homogeneous-heterogeneous reactions

Tasawar Hayat, Madiha Rashid, Maria Imtiaz and Ahmed Alsaedi,

Department of Mathematics, Department of Mathematics, Quaid-I-Azam University

45320,Islamabad 44000, Pakistan

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of

Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

[email protected]

Abstract: An analysis has been carried out to examine the water based nanofluid

flow with silver nanoparticles. Flow due to a rotating disk with variable

thickness is considered. Heat and mass transfer subject to volume fraction of

nanoparticles and homogeneous-heterogeneous reactions are examined.

Homotopy concept is utilized for the development of solutions. Special emphasis

is given to the non-dimensional velocity, temperature, concentration, skin

friction coefficient and Nusselt numbers. It is observed that with an increase in

disk thickness the radial, axial and azimuthal velocities are enhanced. Here

surface shear stress rates increase for larger thickness coefficient of disk and

volume fraction of silver based nanoparticles while magnitude of heat transfer

rate decreases for higher thickness coefficient of disk.

Keywords: Variable thickness; Rotating disk; Nanofluid; Homogeneous-

heterogeneous reaction.

References:

[1] S. U. S. Choi and J. A. Eastman, Enhancing thermal conductivity of fluids

with nanoparticles ASME Int. Mech. Eng. Cong. Expo. 66 (1995) 99-105.

[2] T. Fang, J. Zhang and Y. Zhong, Boundary layer flow over a stretching sheet

with variable thickness, Appl. Math. Comp. 218 (2012) 7241-7252.

[3] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk,

Comp. Fluids 94 (2014) 139-146.

[4] T. Hayat, M. Imtiaz, A. Alsaedi and F. Alzahrani, Effects of homogeneous-

heterogeneous reactions in flow of magnetite-Fe₃O₄ nanoparticles by a rotating

disk, J. Mol. Liq. 216 (2016) 845-855.

[5] S.AS. Liao, On the homotopy analysis method for nonlinear problems, App.

Math. & Comput. 147 (2004) 499-513.




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93

Spectral Theory of 𝒒,𝝎 -Dirac System

Fatma Hıra

Department of Mathematics, Hitit University,

Çorum, Turkey

[email protected]

Abstract: Hahn [1] introduced the difference operator

, where and are fixed real

number. This operator extends the classical difference operator

as well as Jackson [2] - difference operator

. In this paper, we introduce -analogue of

Dirac system and obtain some spectral properties of this -Dirac system

similar to -Sturm-Liouville problems in [3].

Keywords: q,w-Dirac system, Hahn difference operator.

References:

[1] W. Hahn,“Über orthogonalpolynome, die q-differenzenlgleichungen

genügen”, Math. Nachr. 2(1949), 4-34.

[2] F.H. Jackson, “Basic integration”, Quart. J. Math. 2(1951), 1-16.

[3] M. H. Annaby, A. E. Hamza, and S. D. Makharesh, “A Sturm–Liouville

Theory for Hahn Difference Operator”, Frontiers in Orthogonal Polynomials and

q-Series: (2018) pp. 35-83.

( )( ) ( )( )

,q

f qt f tD f t

qt t

+ −=

+ −0 1q 0

( )( ) ( )f t f t

f t

+ − = q

( )( ) ( )( )1

q

f qt f tD f t

q t

−=

−,q

,q

,q




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94

Self-Organizing Map of Artificial Neural Networks for

superficial sediments classification of the Sidi Chahed dam (NE

of Meknes, Morocco)

Abdellah El HMAIDI1, Mohamed BERRADA2, Abdelaziz ABDALLAOUI3, Ali

ESSAHLAOUI1 and Abdelhadi El OUALI1 1 University of Moulay Ismail, Faculty of Science, Department of Geology, Research

Team "Water Sciences and Environmental Engineering", BP 11201, Zitoune, Meknes,

Morocco. Email: [email protected] 2University of Moulay Ismail, ENSAM, Laboratory of Mathematical and Computational

Modeling, Marjane II, BP 15290, Al Mansour, 50000, Meknes, Morocco. 3University of Moulay Ismail, Faculty of Science, Department of Chemistry, Research

Team "Analytical Chemistry and Environment", BP 11201, Zitoune, Meknes, Morocco.

Abstract: The dam of Sidi Chahed is located downstream of the Mikkes and

Lmaleh rivers. It is located about 30 km NE of Meknes city in north of Morocco.

The classification approach proposed in this work is based on a network of

artificial neurons type of self-organizing map (SOM). It has been applied on the

physico-chemical parameters of surface sediments of the dam Sidi Chahed. The

use of ArcGIS software enabled the realization of thematic maps related to the

distribution in time with the seasons, and space from upstream to downstream of

the dam.

The principal component analysis (PCA) and hierarchical classification by SOM

(SOM-HC) validated classification and detected seasonal variations in physical

and chemical parameters of the sediment dam Sidi Chahed. Dependencies

between different parameters have been identified and their spatial and temporal

distribution. The classes obtained were represented by season. The studied

sediments generally have slight physicochemical changes that are related to the

importance of the geological formations, anthropogenic activity and particularly

strong surrounding farming.

Keywords: Dam Sidi Chahed, Self-organizing map (SOM), Artificial Neural

Networks, Physico-chemical parameters, Superficial sediments, ACP, AHC,

GIS.




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95

Zero Operator Products on Hardy and Bergman spaces

Mualla Birgül Huban, Mübariz Garayev and Mehmet Gürdal

Department of Mathematics, Süleyman Demirel University,

Isparta,Turkey

[email protected]

Abstract:In this paper, we study zero operator products, in particular, zero

Toeplitz products. It is proved that if )(,...,1 T Ln are n functions and

nHH ,...,1 are operators from the Engliš algebra 2HF such that

( ) ( ) ,0...11=++ nHTHT

n then .0...21 =n This generalizes the known

Douglas' lemma. In particular, we obtain a new proof of Brown-Halmos

theorem. The similar results are proved for zero Toeplitz products on the

Bergman space .2aL In particular, we partially solve a conjecture due to

Čučkovič [5] (see Corollary 4.7).

Keywords:Reproducing kernel, Berezin symbol, Toeplitz operator, Hardy space,

Bergman space.

References:

[1] A. Aleman and D. Vukotic, "Zero products of Toeplitz operators", Duke

Math. J. 148(3)(2009),373-403.

[2]S. Axler, S.-Y. A. Chang and D. Sarason, "Products of Toeplitz operators”,

Integral Equations Operator Theory, 1(1978), 285-309.

[3] Ž, Čućković, "Berezin Versus Mellin", J. Math. Anal. Appl. 287 (2003), 234-

243.

[4]K. Zhu, “Operator Theory in Function Spaces”, Marcel Dekker, New York,

1990.




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96

Compactness and Engliś Algebra

Mualla Birgül Huban, Mübariz Garayev and Mehmet Gürdal

Department of Mathematics, SüleymanDemirelUniversity,

Isparta,Turkey

[email protected]

Abstract:In this paper, we mainly interested with the following question: if

CBA ,, belong to Engliš algebra ,0HA then under which conditions CAB− is a

compact operator on a reproducing kernel Hilbert space (RKHS) ( )?= HH Here

we characterize in terms of Berezin symbols compactness of the operator

.CAB−

Keywords: Reproducing kernel, Berezin symbol, Compact operator, Engliś

algebra.

References:

transform”, [1] S. Axler and D. Zheng, “Compact operators via the Berezin

400.-Indiana Uni. Math. 47(1998), 387

”, Linear 2[2] M. Engliś, “Toeplitz operators and the Berezin transform on H

204.-Algebra Appl., 223/224 (1995), 171

[3] K. Zhu, “Operator Theory in Function Spaces”, Marcel Dekker, New York,

1990.




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97

Solvability of Infinite Systems of Differential Equations in

Some BK Spaces

Merve İlkhan and Emrah Evren Kara

Department of Mathematics, Düzce University,

Düzce, Turkey

[email protected]

[email protected]

Abstract: Infinite systems of ordinary differential equations define numerous

real world problems which can be seen in the theory of branching processes,

neural nets, mechanics etc. The theory of measure of noncompactness has been

used in determining the classes of compact operators on some sequence spaces

(cf. [1,2]). In [3,4], the authors have studied the solvability of infinite systems of

differential equations in classical Banach sequence spaces by using the measure

of noncompactness. In this presentation, we investigate solvability of infinite

sysytems of differential equations in some BK spaces by applying the technique

of measure of noncompactness.

Keywords: Sequence spaces, BK spaces, Measure of noncompactness, Infinite

system of differential equations.

References:

[1] M. Mursaleen and A.K. Noman, Applications of the Hausdorff measure of

noncompactness in some sequence spaces of weighted means. Comput. Math.

Appl., 60 (2010) 1245-1258.

[2] M. Başarır, E.E. Kara, On the B-difference sequence space derived by

generalized weighted mean and compact operators, J. Math. Anal. Appl., 391

(2012) 67-81.

[3] J. Banas, M. Lecko, Solvability of infinite systems of di

erential equations in Banach sequence spaces, J. Comput. Appl. Math., 137(2)

(2001) 363-375.

[4] M. Mursaleen, S.A. Mohiuddine, Applications of measure of

noncompactness to the infinite system of differential equations in l_p spaces.

Nonlinear Anal., 75 (2012) 2111-2115.




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98

Dissipative Canonical Type Quasi-Differential Operators for

First Order

Zameddin I. Ismailov and Pembe Ipek Al


Trabzon, Turkey

[email protected]

Institute of Naturel Sciences, Karadeniz Technical University,

Trabzon, Turkey

[email protected]

Abstract: In this work, based on Calkin-Gorbachuk method a description of all

maximally dissipative extensions of the minimal operator generated by first order

linear symmetric singular canonical type quasi-differential expression in the

weighted Hilbert space of vector-functions is given. Finally, the spectrum set of

such extensions is studied.

Keywords: Dissipative operator, quasi-differential operator, spectrum.

References:

[1] V. I. Gorbachuk and M. I. Gorbachuk, “Boundary Value Problems for

Operator Differential Equations”,Kluwer, Dordrecht, (1999).

[2] J. von Neumann, “Allgemeine eigenwerttheorie hermitescher

funktionaloperatoren”, Math. Ann., 102(1929), 49-131.

[3] C. Fischbacher, “On the Theory of Dissipative Extensions”, PhD Thesis,

University of Kent School of Mathematics, Statistic and Actuarial Science,

Canterbury, 2017.

[4] B. Sz.-Nagy and C. Foias, “Analyse harmonique des operateurs de L’ espace

de Hilbert”, Masson, Paris and Akad Kiodo, Budapest, (1997) (English transl,

North-Holland, Amsterdam and Akad Kiado, Budapest: 1970).

[5] F. S. Rofe-Beketov and A. M. Kholkin, “Spectral analysis of differential

operators.”, World Scientific Monograph Series in Mathematics, 7(2005).




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99

Some Spectral Problems of Direct Sums of Operators

Zameddin I. ISMAILOV 1, Elif OTKUN ÇEVIK 2

1 Department of Mathematics, Karadeniz Technical

University,

Trabzon/TURKEY

[email protected]

2Department of International Finance and Banking,

Avrasya University,

Trabzon/TURKEY

[email protected]

Abstract: The connection between some spectral properties (as power and

polynomial boundedness, nilpotentity, quasinilpotentity) of direct sum of

operators in the direct sum of Hilbert spaces and its coordinate operators has

been investigated.

Keywords: Power bounded operator, polynomial bounded operator, nilpotent

operator, quasinilpotent operator.

References:

[1] N. Dunford, “A Spectral Theory for Certain Operators on a Direct Sum of

Hilbert Spaces”, Math. Ann., 162(1966), 294-330.

[2] L.A. Fialkow, “A Note of Direct Sums of Quasinilpotent Operators”, Proc.

Amer. Math. Soc., 48(1975), 125-131.

[3] E.J. Ionascu, “On Power Bounded Operators”, Proc. Amer. Math. Soc.,

125(1997), 1435-1441.

[4] M.A. Naimark and S.V.Fomin, “Continuous Direct Sums of Hilbert Spaces

and Some of Their Applications”,Uspehi Mat.Nauk.64 (1955), 111-142 (In

Russian).




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100

New Exact Travelling Wave Solution of the Generalized Ito

Integro-Differential Equation

Anwar Jaafar Mohamad Jawad

Al -Rafidain University College, Computer Engineering Department

Baghdad,10014 Iraq.

Abstract: Three different methods are applied to construct new types of

solutions of nonlinear evolution equations. First, the Csch method is used to

carry out the solutions then the Extended Tanh-Coth method and the modified

simple equation method are used to obtain the soliton solutions. The

effectiveness of these methods is demonstrated by applications to the generalized

Ito integro-differential equation. The solitary waves from the Csch and Tanh-

Coth methods illustrate a significant similarity in behavior. The modified simple

equation method does not produce the soliton solution in general case. The

obtained solutions are very useful in the nonlinear evolution equations.




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101

A Radon-Nikodym type theorem for completely positive

multilinear ϕ-maps on Hilbert C*-modules

Maria Joita

Department of Mathematics Methods and Models, University Politehnica of Bucharest

Bucharest Romania

[email protected]

Abstract: In theory of quantum information, quantum operations are modeled by

completely positive linear maps between C*-algebras. Stinespring [7] showed

that an operator valued completely positive linear map on a C*-algebra A is of

the forma V*π(⋅)V, where π is a ∗-representation of A and V is a bounded linear

operator. A Stinespring type construction for a pair of completely positive linear

maps on Hilbert C*-modules and C*-algebras was obtained by Bhat, Ramesh

and Sumesh [1].

Christensen and Sinclair [5], were the first to formulate the notion of

completely bounded (respectively, completely positive) multilinear maps from a

C*-algebra into L(H). Using Wittstock's matricial Hahn-Banach theorem, they

gave representations for completely bounded symmetric multilinear maps in

terms of ∗-representations of the algebra and suitable bridging operators between

the Hilbert spaces. As a generalization of invariant sesquilinear forms, Heo [2]

introduced the notion of invariant multilinear maps and construct a family of

commuting representations associated with a completely bounded and

completely positive invariant multilinear map which extend the Stinespring's

construction. This representation is different from the multilinear representation

of Christensen and Sinclair, it does not depend on the concept of the matricial

sublinearity as developed by Wittstock. In [4] it is proved a Stinespring type

construction for a pair of symmetric invariant completely positive and

completely bounded multilinear maps on Hilbert C*-modules and C*-algebras.

Heo [3] introduced a partial order relation on the set of all operator valued

symmetric invariant completely positive and completely bounded multilinear

maps on C*-algebras. In this talk we will discuss about a Radon Nikodym type

theorem for symmetric invariant completely positive and completely bounded

multilinear maps on Hilbert C*-modules and C*-algebras [6].




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102

Acknowledgement. The research was supported by "Excellence Research

Grants" Program, UPB -- GEX 2017, No. SA 54-17-08 /2017.

Keywords: completely positive multilinear map; invariant multilinear map;

Hilbert C*-module; ϕ-map

References:

[1] B.V.R. Bhat, G. Ramesh, K. Sumesh, "Stinespring's theorem for maps on

Hilbert C*-modules," J. Operator Theory, 68(2012), 173--178.

[2] J. Heo, "Representations of invariant multilinear maps and Hilbert C*-

modules", Israel J. Math. 118(2000), 125--146.

[3] J. Heo, "A Radon-Nikodym theorem for completely positive invariant

multilinear maps and its applications", Math. Proc. Camb. Phil. Soc. 132(2002),

181--192.

[4] J. Heo, M. Joiţa, "A Stinespring type theorem for completely positive

multilinear maps on Hilbert C*-modules", To appear in Linear and Multilinear

Algebra.

[5] E. Christensen, A. Sinclair, "Representations of completely bounded

multilinear operators", J. Funct. Anal. 72(1987), 151--181.

[6] J. Heo, M. Joiţa, "A Radon-Nikodym type theorem for symmetric invariant

completely positive and completely bounded maps on Hilbert C*-modules",

preprint.

[7] W. Stinespring, "Positive functions on C*-algebras", Proc. Amer. Math.

Soc.,6(1955),211-216.




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103

Approximating Fixed Points of Certain Nonlinear mappings in

CAT(0) Spaces

Amna Kalsoom and Hafiz Fukhar ud Din

Department of Mathematics and Statistics, International Islamic University,

Islamabad, Pakistan

[email protected]

Abstract: CAT(0) space was firstly introduced by Alexandrov in 1950's [1].

Fixed point theory in CAT(0) spaces was firstly studied by Kirk [2]. He also

assured the existence of fixed point in nonexpansive mapping defined on a

bounded closed and convex subset of a complete CAT(0) space. Nonexpansive

mappings are continuous on their domain. In 2016 Rajendra Pant & Rahul

Shukla introduce a new type of nonexpansive mappings, α-nonexpansive

mapping, Suzuki type generalized nonexpansive mapping , generalized α-

nonexpansive mapping which don’t need to be continuous. They also presented

some existence and convergence theorem for generalized nonexpansive mapping

in Banach space. In 1976, Lim [4] presented the concept of Δ-convergence in

general metric spaces and in Kirk and Panyank [5] specialized this concept to

CAT(0) space. They show that many weak convergence theorems in a Banach

space setting can be extended to a CAT(0) space. In this paper we extend the

above mappings on CAT(0) space. Some convergence results including Δ-

convergence and strong convergence for generalized α-nonexpansive mapping

on CAT(0) space using some iterative schemes faster than Mann and Ishikawa

are also discussed. Some useful examples are given to illustrate the importance

of generalized α-nonexpansive mapping.

Keywords: generalized α-nonexpansive mapping, CAT(0) Space, Δ-

convergence,

References:

1 W. A. Kirk, Geodesic geometry and fixed point theory, In Seminar of

Mathematical Analysis 64(2003)195-225.

2 W. A. Kirk, A fixed point theorem for mappings which do not increase

distances, Am. Math. Monthly. 72(1965)1004-1006.




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104

3 R. Pant, R. Shukla, Approximating fixed points of generalized α-nonexpansive

mappings in Banach spaces, Numerical Functional Analysis and Optimization.

38(2017)248-266

4 T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc.

60(1976)179-182.

5 W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces,

Nonlinear Anal. 68(2008)3689-3696




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105

A Parallel Multiobjective Evolutionary Algorithm with an

application to the Knapsack Problem

Nedjmeddine Kantour 1, Sadek Bouroubi 2, Djamel Chaabane 3.

Faculty of Mathematics,

University of Sciences and Technology Houari Boumedien,

Algiers, Algeria


Abstract: In this presentation, we propose a parallel multiobjective evolutionary

algorithm, with an application to the Mutliobjective Knapsack Problem. The

suggested search strategy is based on a periodic decomposition of potentially

efficient solutions, while combining both criterion-based and dominance-based

approaches. It addresses two main sub-objectives: (1) minimizing the distance

between the current non-dominated solutions and the ideal point, (2) ensuring the

spread of potentially efficient solutions. Experimental results are included, where

we compare the suggested algorithm against some of well-known multiobjective

Metaheuristics.

Keywords: Multiobjective optimization, Evolutionary Algorithms, Knapsack

Problem.

References:

[1] Srinivas, N. and Deb, K. (1995) Multi-Objective function optimization using

non-dominated sorting genetic algorithms, Evolutionary Computation, 2(3): 221-

248.

[2] K Deb, A Pratap, S Agarwal, T Meyarivan. A fast and elitist multiobjective

genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation,

2002.

[3] E. Zitzler and L. Thiele. Multiobjective Evolutionary Algorithms: A

Comparative Case Study and the Strength Pareto Approach. IEEE Transactions

on Evolutionary Computation, 3(4): 257-271, November 1999.

[4] E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the Strength

Pareto Evolutionary Algorithm. EUROGEN 2001. Evolutionary Methods for

Design, Optimization and Control with Applications to Industrial Problems,

pages 95-100, Athens, Greece, 2001.




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106

Some Properties of t-closure Operators

Funda Karaçal 1 and Mehmet Akif İnce 2 1 Department of Mathematics, Karadeniz Technical University,

Trabzon, Turkey

[email protected] 2 Department of Mathematics, Recep Tayyip Erdoğan University,

Rize, Turkey

[email protected]

Abstract: Closure operators have influenced many areas of mathematics and

have been studied on a variety of structures (on lattices, on topological spaces,

on fuzzy sets, on graphs, in categories, etc.). In this regard, closure operators are

not only an abstract concept that works in the theoretical sense but also important

maps which have many applications on various areas ( [3,4,7]).

Schweizer and Sklar ([8]) introduced triangular norms (or just t-norm) in the

framework of probabilistic metric spaces and their definition is based on a notion

used by Menger in order to extend the triangle inequality in the definition of

metric spaces towards probabilistic metric spaces.

In this study, based on the knowledge of the existence of t-norms on an arbitrary

given bounded lattice, we introduce t-closure operators with the help of a t-norm

on the lattice and a subset of the lattice including the top element. We define two

equivalence relations by using t-closure operators. The first one is on the set of

all t-norms on a bounded lattice. An important class is obtained according to this

relation. We define a partially order on the set of all equivalent relations given as

secondly. Lastly, we define a set, denoted by 𝐿∗, and by using this set, we

investigate under which conditions 𝐿 can be embedded into 𝐿𝐶𝐴,𝑇 .

Keywords: Closure operator, triangular norm, bounded lattice.

References:

[1] G. Birkhoff, Lattice Theory, 3 rd edition, Providence, 1967.

[2] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Springer-

Verlag, 1981.

[3] K. Denecke, J. Koppitz, Closure Operators and Lattices, in: M-Solid

Varieties of Algebras. Advances in Mathematics, vol 10. Springer, Boston, MA.




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107

[4] V. Even, M. Gran, Closure operators in the category of quandles, Topology

and its Applications, 200 (2016), 237-250.

[5] M.A. İnce, F. Karaçal, R. Mesiar, Medians and nullnorms on bounded

lattices, Fuzzy Sets and Systems, 289 (2016), 74-81.

[6] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic

Publishers, Dordrecht, 2000.

[7] B. Pang, Y. Zhao, Z. Xiu, A new definition of order relation for the

introduction of algebraic fuzzy closure operators, International Journal of

Approximate Reasoning, 92 (2018), 87-96.

[8] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier, Amsterdam,

1983.




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108

The Kinematic Model of Nonuniform Rotation of Liquid

Medium

Zurab Kereselidze1, Marina Chkhitunidze1, Irine Khvedelidze1, Nino Zhonzholadze2

1M.Nodia institute of Geophysics, Ivane Javakhishvili Tbilisi State University, Tbilisi,

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

[email protected] 2Telavi Iakob Gogebashvili State University, Telavi, Georgia, [email protected]

Abstract: Development of perturbances in a liquid medium may be formally

considered as a failure of uniform rotation, the result of which in most cases is

initiation and development of nonuniform motion . Perfect modeling of such

process, regarding the solution of motion equation, may appear either very

difficult or completely impossible due to specific mathematical problems. In this

case the matter is the approximation of the rotation velocity field, which must be

sufficiently adequate for the real motion. In some cases such goal is reached by

means of some kinematic model of velocities. By strict standards it must satisfy

certain criteria, so called conditions of motion dynamics.

We consider the above criteria are fulfilled by following plane

kinematic model [1]

𝑉𝑟 =1

2𝑢0 (

𝑟

𝑅0) (𝑐𝑜𝑠𝜑 + 𝑠𝑖𝑛𝜑), 𝑉𝜑 = 𝑢0 (

𝑟

𝑅0) (𝑐𝑜𝑠𝜑 − 𝑠𝑖𝑛𝜑),

where 𝑅0- the characteristic radius, 0< 𝑟 ≤ 𝑅0, 0 < 𝜑 < 2𝜋, 𝑢0 - the

characteristic velocity of rotational noncompressible liquid. It is obvious that

the restrictions imposed by the probable dynamics reduce the class of the flows,

which may be modeled in kinematic approximation. However, at the same time

the restrictions raise the value of the kinematic models, which are sufficiently

adequate for some types of motions realized in some cases.

Keywords: Kinematic model, velocity.

References:

[1] V.Kirtskalia, Z. Kereselidze, N. Dzonzoladze and M. Chkhitunidze. An

Analytical Model of an Asymmetrical Temperature Field in the Polar and

Auroral Ionosphere. Georgian International Journal Of Science and Technology,

Volume3,Issue4,pp.381-390.2012

https://www.novapublishers.com/catalog/product_info.php?products_id=32804




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109

Generalized Jensen-Mercer Inequality for Functions with

Nondecreasing Increments

ASIF R. KHAN AND SUMAYYAH SAADI Department of Mathematics, University of Karachi,

Karachi, Pakistan

[email protected]

Abstract: In year 2003, A. McD. Mercer established an interesting variation of

Jensens Inequality and later in 2009 Mercer's result was generalized to higher

dimensions by M. Niezgoda.

Recently, Asif et al. has stated an integral version of Niezgoda's result for convex

functions. We further generalize Niezgoda's integral result for functions with

nondecreasing increments, and give some refinements with applications. In the

way we generalize an important result, Jensen-Boas inequality, using functions

with nondecreasing increments. These results would constitute a valuable

addition to Jensen-type inequalities in the literature.

Keywords: Convex Functions, Functions with non-decreasring increments,

Jensen’s Ineqaulity, Niezgoda’s Inequaltiy, Majorization and refinements.

References:

[1] H. D. Brunk, Integral inequalities for functions with nondecreasing

increments, Pacific J. Math. 14(1964),

783-793.

[2] W. S. Cheung, A. Matkovic, and J. Pecaric, A variant of Jessen's inequality

and generalized means, J. Ineq. Pure Appl. Math., 7 (1), Article 10, 2006.

[3] Jan Haluska and Ondrej Hutnik, Some Inequalities involving Integral Means,

Tatra Mountains Mathematical Publications, 35,131-146

[4] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge

University Press, Cambridge 1978.

[5] Milica Klaricic Bakula, ANita Matkovic, and J. Pecaric, On a Variant of

Jensen's Inequality for functions with nondecreasing increments, J. Korean Math.

Soc.,45 (2008), pp.821{834

[6] M. Adil Khan, Asif R. Khan and J. Pecaric, On the refinements of Jensen-

Mercer's Inequality, Rev. Anal. Number. Theor. Approx., 41 (2012), 62-81.




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110

[7] Asif R. Khan, J. Pecaric, Marjan Praljak, A note on generalized Mercer's

Inequality, Bulletin of Malaysian Maths Science Society (2015).

[8] A. McD. Mercer, An “error term” for the Ky Fan inequality, J. Math. Anal.

Appl., 220, no. 2 (1998) 774-777.

[9] Malika Klari_ci_c, Anita Matkovic, J. Pecaric, On A Variant Of Jensen's

Inequality For Functions with Nondecreasing Increments,J'Korean Maths.Soc.,

45 (2008), 821{834.

[10] A. McD. Mercer, A variant on Jensen's Inequality, J. Ineq. Pure and Appl.

Maths4(2003), Article 73

[11] J. Pecaric, On some inequalities for functions with nondecreasing

increments, J. Math. Anal. Appl., 98 (1984) 188-197.




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111

Approximating Min Graphic k-Path TSP with no Prespecified

Path Endpoints

Rezika Kheffache

Department of Mathematics, Mouloud Mammeri University,

AMCD & RO,USTHB, Algeria

[email protected]

Abstract:

We consider a generalized Path Traveling Salesman Problem where the distances

are defined by a 2-edge-connected graph metric and a number of salesmen have

to cover all the destinations by traveling along paths without prespecified

endpoints, of minimum total length. We show that for this problem there is an

algorithm with asymptotic approximation ratio of 3/2 that runs in polynomial

time whenever the number of paths is bounded by a fixed constant.

Keywords: T-joins, Graphic Path TSP, Multiple Path TSP , Approximation

Algorithms , 2-edge connected graph.

References:

[1] H.C. An, R. Kleinberg, D.B. Shmoys, “ Improving Christofides' algorithm for

the s-t path TSP”, In Proc. of the 44th ACM STOC,(2012), 875-886.

[2] J. Bae, S. Rathinam, “Approximation algorithms for multiple terminal,

Hamiltonian path problems”, Optimization Letters 6 (p) (2012): 69-85.

[3] A. Sebo, “ Eight-Fifth Approximation for the Path TSP”, In Proc. IPCO

(2013), LNCS 7801: 362-374.

[4] N. Christofides, “Worst-case analysis of a new heuristic for the traveling

salesman problem”, Report 388 (1976), Graduate School of industrial

Administration. Carnegie-Mellon University, Pittsburgh .




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112

k −Generalized Pell Numbers and Its Properties

Havva Kırgız

Kemal Uslu

Department of Mathematics, Selcuk University,

Konya, Turkey

[email protected]

Abstract: The aim of this study is to introduce generalized new family of k-Pell

numbers. The recurrence relations and the relation between new k-Pell numbers

and generalized numbers are also considered.

Keywords: Pell numbers, Pell-Lucas numbers, generalized numbers.

References:

[1] M. El-Mikkawy, T. Sogabe, A New Family of k-Fibonacci Numbers, Applied

Mathematics and Computation, 215(2010), 4456-4461.

[2] P. Catarino, “On some identities and generating functions for k-Pell

numbers”, Int. Journal of Math. Analysis, 7(2013), 1877-1884.

[3] Y. Taşyurdu, N. Çobanoğlu, Z. Dilmen, “On the a new family of k-Fibonacci

numbers”, Erzincan Üniversitesi Fen Bİlimleri Enstitüsü Dergisi, 9.1 (2016), 95-

101.

[4] N. Yılmaz, Y. Yazlık, N. Taşkara, “On the k-Generalized Fibonacci

Numbers”, Selcuk Journal of Applied Mathematics, 13.1(2012), 83-88.

[5] A. F. Horadam and Bro. J. M. Mahon, “Pell and Pell-Lucas Polynomials”,

Fibonacci Quarterly, 23.1(1985), 17-20.

[6] E. Özkan, A. Aydoğdu, İ. Altun, “Some identities for a family of Fibonacci

and Lucas numbers”, Journal of Mathematics and Statistical Science, 3.10

(2017), 295-303.




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113

Exact and Nonstandard Finite Difference Schemes for the

Generalized KdV-Burgers Equation

Canan Köroğlu1 and Ayhan Aydın2

1 Department of Mathematics, Hacettepe University, Ankara, Turkey 2 Department of Mathematics, Atilim University, Ankara, Turkey

[email protected]

[email protected]

Abstract: In this study, we propose a nonstandard finite difference scheme

(NSFD) for the numerical solutions of the generalized KdV-Burgers equation.

Moreover, exact finite difference schemes are constructed for KdV-Burgers

equation. Some numerical examples are given to demonstrate the accuracy of the

schemes.

Keywords: Nonstandard finite difference scheme, exact finite difference

scheme, generalized KdV-Burgers equation.

References:

[1] R. Mickens, “Nonstandard Finite difference models of differential

equations”, World Scientific, 1994

[2] R. Mickens, “Advances in applications of Nonstandard Finite Difference

Schemes”, Wiley-Interscience, Singapore, 2005.

[3] L. Zhang, L. Wang and X. Ding, “Exact Finite Difference Scheme and

Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher

Equations”, Journal of Applied Mathematics, 2014(2014), 1-12.

[4] S. Zibaei, M. Zeinadini and M. Namjoo, “Numerical solutions of Burgers-

Huxley equation by exact finite difference and NSFD schemes ”, Journal of

Difference Equations and Applications, 22.8(2016), 1098-1113.




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114

Some Studies on Congruence Equations using Suborbital

Graphs

Tuncay Köroğlu1, Zeynep Şanlı2, Bahadır Özgür Güler3, Murat Beşenk4,

1,2,3Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey, 4Department of Mathematics, Pamukkale University, Denizli, Turkey,

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

Abstract: It was shown that suborbital graph is useful tool in the study of the

modular group which is a finitely generated Fuchsian group, some other finitely

generated groups have been studied by suborbital graphs [1, 2]. Interesting

number theoretic results arise from suborbital graphs are as follows:

• A shortest path in subgraphs can be expressed as a continued fraction;

• A shortest path in trees of suborbital graphs is a special case of

Pringsheim continued

fraction;

• The subgraph F1,2 can be defined as a new kind of continued fraction

and any irrational numbers has a unique F1,2 expansion;

• The set of vertices of some suborbital graphs is strongly connected to

the Fibonacci

sequence.

The aim of this paper is to examine the action of the normalizer of which

produce some congruence equations with solutions.

Keywords: Bianchi groups, Quadratic number field, Suborbital digraphs,

Circuits.

References:

[1] Jones G.A., Singerman D., Wicks K., “The modular group and generalized

Farey graphs”, Lond Math Soc Lect Note Ser 160(1991), 316–338.

[2] Sims C.C., “Graphs and finite permutation groups”, Math. Zeitschrift,

Vol:95, 76-86, 1967.




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115

On 𝑳𝟏 - Convergence of Certain Cosine and Sine modified sums

Xhevat Krasniqi

Faculty of Education, University of Prishtina “Hasan Prishtina”

10000 Prishtina, Republic of Kosovo

[email protected]

Abstract: In this paper we have introduced some new modified cosine and sine

sums. We have studied their pointwise convergence as well as their 𝑳𝟏 –

convergence whose coefficients belong to the well-known Sidon’s class 𝑺. As a

consequence of these results is that a cosine or sine series converges in 𝑳𝟏-norm

to the its sum-function.

Keywords: L1 - convergence, modified sums, Sidon’s class, trigonometric series.

References:

[1] S. Y. Sheng, Extension of the theorems of Č. V. Stanojević and B. V.

Stanojević. Chinese Sci. Bull. 35 (1990), no. 23, 2014 - 2016.

[2] S. A Teljakovskiĭ, A certain sufficient condition of Sidon for the integrability

of trigonometric series. (Russian) Mat. Zametki 14 (1973), 317 - 328.

[3] J. Kaur; S. S. Bhatia, Convergence of new modified trigonometric sums in

the metric space L. J. Nonlinear Sci. Appl. 1 (2008), no. 3, 179 - 188.

[4] A. N. Kolmogorov, Sur l’ordere de grandeur des coefficients de la series de

Fourier-Lebesgue. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys., (1923),

83-86.

[5] B. Ram; S. Kumari, On L1-convergence of certain trigonometric sums. Indian

J. Pure Appl. Math. 20 (1989), no. 9, 908 - 914.

[6] B. Ram, Convergence of certain cosine sums in the metric spaces L. Proc.

Amer. Math. Soc. 66 (1977), no. 2, 258–260.




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116

Fixed Point Theorems for α-F-Suzuki Contraction Mapping

with Rational Expressions in Branciari b-Metric Spaces

Neslihan Kaplan Kuru, Mahpeyker Öztürk


Sakarya, Turkey

[email protected]


Sakarya, Turkey

[email protected]

Abstract: Recently, Piri and Kumam [Fixed Point Theory and Applications,

2014: 2014:210] improved concept of F-contraction and proved some

Wardowski and Suzuki type fixed point results in metric space. The aim of this

work is to establish some new fixed point theorems for α-F-Suzuki contraction

mappings satisfying rational expressions in Branciari b-metric space.

Keywords: Fixed point, α-admissible, F-Suzuki contraction, Branciari b-metric

space

References:

[1] P. Hossein, P. Kumam, “Fixed point theorems for generalized F-Suzuki-

contraction mappings in complete b-metric spaces” , Fixed Point Theory and

Applications, 2016:90(2016), 13 pages.

[2] S. Gülyaz Özyurt, “On some α-admissible contraction mappings on Branciari

b-metric spaces”, Advances in the Theory of Nonlinear Analysis and its

Applications, 1.1(2017), 13 pages.

[3] A. Latif, V.Parvaneh, P.Salimi, A.E. Al-Mazrooei,”Various Suzuki type

theorems in b-metric spaces”, Journal of Nonlinear Science and Applications,

8(2015), 363-377.




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117

About the packing chromatic number in graphs

Rachid Lemdani

Department of MI, Faculty of sciences, Medea University,Algeria

[email protected]

Moncef Abbas

USTHB, Algiers, Algeria

[email protected]

Abstract: Let 𝐺 = (𝑉, 𝐸) be a simple graph with no isolated vertices. A packing

coloring of order 𝑘 in 𝐺 is a function 𝜋: 𝑉 → {1,… , 𝑘} that is if 𝜋(𝑢) = 𝜋(𝑣)

implies that 𝑑(𝑢, 𝑣) ≻ 𝜋(𝑢). The minimum order of a packing coloring is called

the packing chromatic number of 𝐺 and is denoted by 𝜒𝑝(𝐺). It was introduced

by Goddard et al. in [2] under the name Broadcast chromatic number of graphs

and study its properties. They showed that it is NP-hard to determine if 𝜒𝑝(𝐺) ≤

4 and they also determined the packing chromatic number of all trees of diameter

at most 4. In this paper we will use the following results on 𝜒𝑝(𝐺) in [2] and

others to determine the packing chromatic number of the finite super subdivision

of the classes of graphs: tree, cycle, neighborhood corona of cycle with path.

Keywords:

Graph theory, Finite super subdivision of graph, Neighborhood

corona of graph.

References:

[1] B. Brear, S. Klavsar, D. F. Rall, On the packing chromatic number of

cartesian products, Hexagonal lattice, and trees, May 24, 2007.

[2] W. Goddard, S.M. Hedetniemi, J. M. Harris, D.F. Rall, Broadcast chromatic

numbers of graphs, Ars Combin., 86:33-49, 2008.

[3] G. Indulal, The spectrum of neighborhood corona of graphs, Kragujevac J.

Math., 35(3)(2011),493-500.

[4] A. William and S. Roy. Packing chromatic number of certain trees and cycle

related graphs. International Conference on Mathematical Computer Engineering

(ICMCE), 2013.




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118

Some characterizations of the homogeneous ruled real

hypersurface in a complex hyperbolic space

Sadahiro Maeda

Department of Mathematics, Saga University,

Saga, Japan

[email protected]

Abstract: Motivated by the notion of ruled surfaces in Euclidean 3-space, we

consider ruled real hypersurfaces isometrically immersed into an n-dimensional

complex hyperbolic space of constant holomorphic sectional curvature c(<0).

The purpose of this expository paper is to give some geometric characterizations

of the homogeneous ruled real hypersurface, that is, this ruled real hypersurface

is an orbit of some subgroup of the full isometry group of the ambient space (see

[1, 2, 3, 4]).

Keywords: ruled real hypersurfaces, the homogeneous ruled real hypersurface,

complex hyperbolic space, characteristic vector filed, integral curves, circles, the

first curvature.

References:

[1] T. Adachi, Y.H. Kim and S. Maeda, “A characterization of the homogeneous

minimal uled real hypersurface in a complex hyperbolic space”, J. Math. Soc.

Japan 61(2009), 315-325.

[2] M. Kimura, S. Maeda and H. Tanabe, “Integral curves of the characteristic

vector field of minimal ruled real hypersurfaces in non-flat complex space

forms’’, to appear in Hokkaido Math. J..

[3] M. Kimura, S. Maeda and H. Tanabe, “The homogeneous ruled real

hypersurface in a complex hyperbolic space”, to appear in J. Geometry.

[4] S. Maeda and H. Tanabe, “A characterization of the homogeneous ruled real

hypersurface in a complex hyperbolic space in terms of the first curvature of

some integral curves”, Arch. Math. (Basel) 105(2015), 593-599.




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119

Precise long-term prediction of behavior in a 3-D chaotic map

M. Mammeri

Department of Mathematics, University of Kasdi Merbah, Ouargla, Algeria

E-mail: [email protected]


Abstract: In this paper, the prediction accuracy on long-term of future behavior

in a 3-D chaotic map is possible. This paper introduces and justifies by numerical

simulation a new phenomenon, shown by the 3-D map which is the behavior in

the 3-D map is always regular and periodic and bounded on large time, i.e., that

the behavior repeats itself regularly after cycles on large time intervals. The new

3-D map produces new several chaotic attractors obtained via quasi-periodic

route to chaos.

Keywords: Precise prediction, 3-D map, future behavior, symmetric intervals,

regular behavior.

References:

[1] Lorenz EN. Deterministic non-periodic flow., J Atmos Sci, 1963, 20, 130-41

[2] K. Mandal, S. Banerjee, C. Chakraborty., "Symmetry-Breaking Bifurcation

in Series-Parallel Load Resonant DC-DC Converters.," IEEE Trans. Circuits

Syst. I., vol.60, no.3, 2013, 778-787.

[3] S. V. Gonchenko, J. D. Meiss, I. I. Ovsyannikov., Chaotic Dynamics of

Three-Dimensional Hénon Maps that Originate from a Homoclinic Bifurcation.,

Regular and Chaotic Dynamics, 11 (2), 2006, 191-212.

[4] S. V. Gonchenko, I. I. Ovsyannikov, C. Simó, D. Turaev., Three-

Dimensional Hénon-like Maps and Wild Lorenz-like Attractors., International

Journal of Bifurcation and Chaos, 15 (11), 2005, 3493-3508.

[5] S. V. Gonchenko, I. I. Ovsyannikov, C. Simó, D. Turaev., Three-

Dimensional Hénon-like Maps and Wild Lorenz-like Attractors., International

Journal of Bifurcation and Chaos, 15 (11), 2005, 3493-3508.

[6] Arrow, K. J.et al., The promise of prediction markets. Science, 320, 2008,

877-878.




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120

Constructing 3-D maps with complex chaotic phenomena

M. Mammeri

Department of Mathematics, University of Kasdi Merbah, Ouargla, Algeria



Abstract: Any dynamical phenomena has higher unpredictability and more

randomness than simple chaotic phenomena is called complex chaotic

phenomena. The essential motivation of this letter is to proposed 3D maps with

the necessaries conditions for the existence of the chaotic phenomena, and is to

detecting the complex chaotic phenomena of the maps. The direct method of

analysis used here is the analytic calculation of the biggest Lyapunov exponents

(BLEs).

Keywords: 3-D map, complex chaotic phenomena, biggest Lyapunov exponents

(BLEs).

References:

[7] Chen, G & Lai, D, Making a discrete dynamical system chaotic: feedback

control of lyapunouv exponents for discrete-time dynamical system, IEEE Trans.

Circ. Syst.-I, 44, 1997, 250-253.




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121

[8] Wang, X. F., Chen, G, On feedback anticontrol of discrete chaos. Int.J. Bifur.

Chaos, 9, 1999, 1435-1441.

[9] S. V. Gonchenko, J. D. Meiss, I. I. Ovsyannikov., Chaotic Dynamics of Three-

Dimensional Hénon Maps that Originate from a Homoclinic Bifurcation.,

Regular and Chaotic Dynamics, 11 (2), 2006, 191-212.

[10] G. Baier, M. Klein, Maximum hyperchaos in generalized Hénon maps, Phys.

Lett. A, 151, (6-7), 1990, 281--284.

[11] S. V. Gonchenko, I. I. Ovsyannikov, C. Simó, D. Turaev., Three-Dimensional

Hénon-like Maps and Wild Lorenz-like Attractors., International Journal of

Bifurcation and Chaos, 15 (11), 2005, 3493-3508.

[12] Marotto, F. R, Snap-back repellers imply chaos in ℝⁿ. J. Math. Anal. Appl.,3,

1978, 199-223.

[13] M. Mammeri., A Large Chaotic Region in a 3-D Sinusoid Discrete Map.

International Journal of Applied Mathematics and Statistics. Vol. 55, no. 3, 133-

144, 2016.




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122

Sequential structures of statistical manifolds

on deformed exponential families

Hiroshi Matsuzoe

Department of Computer Science and Engineering,

Nagoya Institute of Technology, Nagoya, Japan

[email protected]

Abstract: Information geometry is one of differential geometrical methods of

statistical sciences. In information geometry, a statistical model which is a set of

probability density distributions is regarded as a Riemannian manifold with dual

affine connections (cf. [1]), In the case of exponential family, a statistical

manifold structure is reduced to a Hessian structure, and such a manifold

structure is applied various fields of statistical sciences.

A deformed exponential family is a generalization of an ordinary exponential

family, and it naturally admits at least three kinds of statistical manifold

structures [2]. In this talk, we show that such statistical manifold structures are

obtained as a sequence (cf. [3]). The alpha-divergence and the beta-divergence

are closely related to the sequence of statistical manifolds.

Keywords: information geometry, statistical manifold, anomalous statistics.

References:

[1] S. Amari, “Information Geometry and Its Applications”, Springer, Tokyo,

2016.

[2] H. Matsuzoe and M. Henmi, “Hessian Structures and Divergence Functions

on Deformed Exponential Families”, Geometric Theory of Information, Signals

and Communication Technology, Springer, (2014), 57-80.

[3] H. Matsuzoe, “A sequence of escort distributions and generalizations of

expectations on q-exponential family”, Entropy, 19(2017), no. 1, 7.




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123

Approximate solutions of the inverse problem without a priori

information on the exact solution

Yuri Menshikov

Department of Mechanics & Mathematics, Dnepro University,

Dnepro, Ukraine

[email protected]

Abstract: One of the main assumptions in the theory of inverse problems is the

hypothesis that the exact solution of the inverse problem belongs to the set of

possible solutions Q which is determined by the inaccuracy of the original data

[1]. This hypothesis makes it possible to construct approximate solutions that

converge to the exactness of the solution with a decrease in the error of the

initinal data. However, in the practice of solving inverse problems, problems

have recently appeared in which the indicated property of convergence does not

make sense [2]. These are the problems of constructing (synthesis) an adequate

mathematical description of physical processes and they play an important role

in the use of methods of mathematical simulation [3]. In this paper it is proposed

to abandon the hypothesis of an exact solution. This will allow us to proceed to a

more reliable formulation of certain types of inverse problems. The examples

were given [2].

Keywords: Inverse problems, exact solutions, adeguate mathematical

descriptions.

References:

[1] A.N. Tikhonov, V.Y. Arsenin,”Methods of incorrect problems solution”,

Science, Moscow (1979), 288p.

[2] Yu. L. Menshikov,“Synthesis of Adequate Mathematical Description as

Solution of Special Inverse Problems“, European Journal of Mathematical

Sciences, 2, n.3,(2013), 256-271.




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124

Modelling Asymmetric Magnetic Recording Heads With a Soft

Underlayer Using Multiple Images Method

Ammar Edress Mohamed

Dohuk Polytechnic University, Technical College of Petroleum and Minerals / Zakho,

Iraq

[email protected]

Abstract: For magnetic recording heads, the case when the SUL is present, the

head fields are calculated using two methods; Firstly, Green functions are

employed along with evaluating the integration numerically. The results follow

closely the corners, while the gap region shows linear behaviour for d/g < 0.5

compared with the calculated fields from finite-element the reason of the

disagreement caused by using the potential for head without underlayer case. The

second method is superposition technique, which utilising the principle of the

single image method for d=g. It was found that for the head with the underlayer

present, the superposition (single image method) results follow closely the shape

of the calculated fields from finite-element modelling. The results shows that y

field components based on the superposition (single image method) are in very

good agreement with finite-element calculations for angles up to 45 for

y/g≤0.1.

Keywords: Magnetic Recording Heads, Multiple Images Method, Magnetic

field.

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.




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125

[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.

[7] 0. Lopez, W. P. Wood, N. H. Yeh, and M. Jursich, “Interaction of a ring

head and double layer media-field calculations,” IEEE Trans. Magn., vol. MAG-

18, no. 10, pp. 1179–1181, 1982.

[8] D. Bloomberg, “Spectral response from perpendicular media with

gapped head and underlayer,” IEEE Trans. Magn., vol. 19, no. 4, pp. 1493–1502,

1983.

[9] S. I. Iwasaki and K. Ouchi, “On design of Co-Cr perpendicular

recording media for usage of a ring-type head,” IEEE Trans. Magn., vol. 26, no.

1, pp. 97–99, 1990.

[10] D. T. Wilton, H. A. Shute, and D. J. Mapps, “Accurate approximation

of fields and spectral response functions for perpendicular recording heads,”

IEEE Trans. Magn., vol. 35, no. 4, pp. 2172–2179, Jul. 1999.




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126

Coefficient bound estimates for beta-starlike functions of order

alpha

Nizami Mustafa1,a, Veysel Nezir1,b

1Department of Mathematics, Kafkas University,

Kars, Turkey a [email protected]

[email protected]

Abstract: It is well-known that the coefficient problems of analytic functions is

one of the frequently studied mathematical problems. There are many studies on

this subject (see for example [1-5] and references therein). In this paper, we

investigate coefficients problems for the so-called beta starlike functions of order

alpha ( )f z , which is analytic and univalent in the open unit disk

: 1U z z= in the complex plane, and normalized so that

( ) ( )0 0 1 0f f = − = , and satisfying the condition

( )

( )

( )

( )

1

Re 1zf z zf z

f z f z

− +

, )0,1 . We give upper bound

estimates for the first three coefficients of the functions belonging to this class.

Fekete-Szegö problem for this mentioned function class is also examined. The

main results obtained here for the coefficients bounds estimates extend the

results obtained by Thomas in [4].

Keywords: starlike function, convex function, beta-starlike function, coefficient

bound

References:

[1] S. Prema, B. S. Keerthi, “Coefficients bounds for certain subclasses of

analytic functions”, J. Math. Anal., 4.1(2013), 22-27.

[2] N. Mustafa, “Fekete-Szegö problem for certain subclass of analytic and bi-

univalent functions”, Journal of Scientific and Engineering Research, 4.8(2017),

390-400.




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127

[3] S. Topkaya and N. Mustafa, “The General Sublasses of the Analytic

Functions and their Various Properties”, Asian Research Journal of Mathematics,

7.4(2017), 1-12.

[4] H. Orhan and P. Zaprawa, “Third Hankel Determinants for Starlike and

Convex Functions of Order Alpha”, Bull. Korean Math. Soc. 55.1(2018), 165-

173.

[5] N. Mustafa, G. Murugusundaramoorthy, T. Janani, “Second Hankel

determinant for a certain subclass of bi-univalent functions”, Mediterranean

Journal of Mathematics (accepted).

[6] Derek K. Thomas, “On the coefficients of gamma-starlike functions”, J.

Korean Math. Soc., 55.1(2018), 175-184.




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128

On the coefficient bounds of gamma and beta starlike functions

Nizami Mustafa1,a, Veysel Nezir1,b, Mustafa Ateş1,c



[email protected] c [email protected]

Abstract: In this paper, we introduce and investigate a new subclass of the

analytic functions in the open unit disk. Here, we obtain upper bound estimates

for the first three coefficients for the functions belonging to this class. Keywords: analytic function, coefficient bound estimate, gamma and beta

starlike functions

References:

[1] O. Altıntaş, “On a subclass of certain starlike functions with negative

coefficient”, Math. Japon., 36(1991), 489-495.

[2] O. Altıntaş, H. Irmak and H. M. Srivastava, “Fractional calculus and certain

starlike functions with negative coefficients”, Comput. Math. Appl., 30.2(1995),

9-16.

[3] O. Altıntaş and S. Owa, “On subclasses of univalent functions with negative

coefficients”, Pusan Kyongnam Mathematical Journal, 4(1988), 41-56.

[4] O. Altıntaş, H. Irmak, S. Owa and H. M. Srivastava, “Coefficients bounds for

some families of starlike and convex functions of complex order”, Applied

Mahematics Letters, 20(2007), 1218-1222.

[5] O. Altıntaş, Ö. Özkan and H. M. Srivastava, “Neighborhoods of a Certain

Family of Multivalent Functions with Negative Coefficients”, Comput. Math.

Appl., 47(2004), 1667-1672.

[6] P. L. Duren, “Univalent Functions”, Grundlehrean der Mathematischen

Wissenshaften, Bd. 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo,

1983.




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129

Application of the Chebyshev polynomials to coefficient

estimates of analytic functions

Nizami Mustafa1,a, Veysel Nezir1,b, Erdi Akbulut,1,c




Abstract: In this paper, making use of the Chebyshev polynomials, we introduce

and investigate new subclasses of the analytic functions in the open unit disk in

the complex plane. Here, upper bound estimates for the initial second

coefficients of the functions belonging to these classes are obtained.

Keywords: analytic function, coefficient bound, Chebyshev polynomials

References:

[1] E. H. Doha, “The first and second kind Chebyshev coefficients of the

moments of the general order derivative of an infinitely differentiable function”,

Intern. J. Comput. Math., 51(1994), 21-35.

[2] P. L. Duren, “Univalent Functions”, Grundlehren der Mathematischen

Wissenschaften 259, Springer-Verlag, New York, 1983.

[3] A. W. Goodman, “Univalent Functions”, Volume I, Polygonal, Wasington,

1983.

[4] E. Netanyahu, “The minimal distance of the image boundary from the origin

and the second coefficient of a univalent function in |z|<1”, Arch. Rational Mech.

Anal. 32(1969),100-112.

[5] M. Lewin, “On a coefficient problem for bi-univalent functions”, Proc. Amer.

Math. Soc. 18(1967), 63-68.

[6] H. M. Srivastava and S. Owa, “Current Topics in Analytic Function Theory”,

World Scientific, Singapore, 1992.




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130

On the coefficient bounds of certain subclasses of analytic

functions of complex order

Nizami Mustafa1,a, Veysel Nezir1,b, Tarkan Öztürk1,c




Abstract: In this paper, we introduce and investigate a new subclass

( ) * *, , 0, 0S C = − of analytic functions of complex order

in the open unit disk. Here, we obtain upper bound estimates for the first three

coefficients for the functions belonging to this class. Keywords: analytic function, coefficient bound, starlike function, convex

function, analytic function of complex order

References:

[1] P. L. Duren, “Univalent Functions”, Grundlehren der Mathematischen

Wissenschaften 259, Springer-Verlag, New York, 1983.

[2] A. W. Goodman, “Univalent Functions”, Volume I, Polygonal, Wasington,

1983.

[3] C. H. Pommarenke, “Univalent Functions”, Vandenhoeck and Rupercht,

Göttingen, 1975.

[4] H. M. Srivastava and S. Owa, “Editors, Current Topics in Analytic Function

Theory”, World Scientific, Singapore, 1992.




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131

Some Results on Classical 2-Absorbing Submodules

Osama A. Naji


Sakarya, Turkey

[email protected]

Abstract: Let 𝑅 be a commutative ring with identity and 𝑀 be a unitary 𝑅 –

module. In this paper, we investigate some properties of classical 2-absorbing

submodules of M as a new generalization of 2-absorbing. We also give some

characterizations of classical 2-absorbing submodules under some conditions.

Keywords: 2-absorbing submodules, classical 2-absorbing submodules.

References:

[1] Ashour, A.E., Al-Ashker, M.M. and Naji, O.A., On Almost 2-Absorbing

Primary Sub-modules. IUG Journal of Natural Studies. (2017).

[2] Ashour, A.E., Al-Ashker, M.M. and Naji, O.A., Some results on almost 2-

absorbing sub-modules. Journal of Al Azhar University-Gaza (Natural

Sciences).Vol.18, pp.1-13, (2016).

[3] Azizi, A. On prime and weakly prime submodules, Vietnam Journal of

mathematics, Vol. 36, No. 3 pp. 315-325, (2008).

[4] Darani, A.; Soheilnia, F. 2-Absorbing and Weakly 2-Absorbing Submodules,

Thai Journal of Mathematics. Vol. 9, No. 3, pp. 577-584, (2011).

[5] Dubey, M.; Aggarwal, P. On 2-absorbing primary Submodules of modules

over commutative rings, Asian-European Journal of Mathematics. Vol. 8, No. 4,

(2015).

[6] Mostafanasab, H.; Tekir, U.; Oral, K. Classical 2-Absorbing Submodules of

Modules over Commutative Rings ,European Journal of Pure Mathematics

Vol.8,No. 3, pp. 417-430, (2015).

[7] Payrovi, Sh.; Babaei, S. On 2-Absorbing Submodules, Algebra Colloquium

19 (Spec 1), pp 913-920, (2012).




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132

Fixed point properties for a degenerate Lorentz-Marcinkiewicz

space

Veysel Nezir1,a and Nizami Mustafa1,b



b [email protected]

Abstract: We construct an equivalent renorming of ℓ1 which turns out to

produce a degenerate ℓ1-analog Lorentz-Marcinkiewicz space ℓ𝛿,1, where the

weight sequence 𝛿 = (𝛿𝑛)𝑛∈𝑁 = (2,1,1,1,⋯ ) is a decreasing positive sequence

in ℓ∞\𝑐0, rather than in 𝑐0\ℓ1 (the usual Lorentz situation). Then, we obtain its

isometrically isomorphic predual ℓ𝛿,∞0 and dual ℓ𝛿,∞, corresponding degenerate

𝑐0-analog and ℓ∞-analog Lorentz-Marcinkiewicz spaces, respectively. We prove

that both spaces ℓ𝛿,1 and ℓ𝛿,∞0 enjoy the weak fixed point property (w-fpp) for

nonexpansive mappings; besides, ℓ𝛿,1 has the weak* fixed point property (w*-

fpp) with respect to its predual ℓ𝛿,∞0 yet they fail to have the fixed point property

(fpp) for nonexpansive mappings since they contain an asymptotically isometric

copy of ℓ1 and 𝑐0, respectively. In fact, we prove for both spaces that there exist

non-empty, closed, bounded and convex subsets with invariant fixed point-free

affine, nonexpansive mappings on them and so they fail to have fpp for affine

nonexpansive mappings. Also, we show that any non-reflexive subspace of 𝑙𝛿,∞0

contains an isomorphic copy of 𝑐0 and so fails fpp for strongly asymptotically

nonexpansive maps. Finally, we get a Goebel and Kuczumow analogy by

proving that there exists an infinite dimensional subspace of ℓ𝛿,1 with fpp for

nonexpansive mappings.

Keywords: nonexpansive mapping, non-reflexive Banach space, fixed point

property, weak fixed point property, asymptotically isometric copy of c0,

asymptotically isometric copy of l1, Lorentz-Marcinkiewicz spaces

References:

[1] D. E. Alspach, “A fixed point free nonexpansive map”, Proceedings of the

American Mathematical Society, 82(1981), 423–424.




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133

[2] N. L. Carothers, S. J. Dilworth and C. J. Lennard, “On a localization of the

UKK property and the fixed point property in L(w,1)”, Lecture notes in Pure and

Applied Mathematics, (1996), 111–124.

[3] P. N. Dowling, C. J. Lennard and B. Turett, “Renormings of l1 and c0 and

Fixed Point Properties”, in Handbook of Metric Fixed Point Theory, Springer

Netherlands, (2001), 269–297.

[4] K. Goebel and W. Kirk, “Topics in metric fixed point theory”, Cambridge

University Press, 1990.

[5] K. Goebel and T. Kuczumow, “Irregular convex sets with fixed-point

property for nonexpansive mappings”, Colloq. Math., 40(1979), 259–264.

[6] P. K. Lin, “There is an equivalent norm on l1 that has the fixed point

property”, Nonlinear Analysis 68(2008), 2303–2308.

[7] J. Lindenstrauss and L. Tzafriri, “Classical Banach spaces I: Sequence

Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete”, 92(1977),

Springer-Verlag.

[8] G. G. Lorentz, “Some new functional spaces”, Annals of Mathematics,

(1950), 37-55.

[9] V. Nezir, “Fixed Point Properties for c0-like Spaces”, Ph.D. thesis, University

of Pittsburgh, 2012.

[10] V. Nezir and N. Mustafa, “c0 can be renormed to have the fixed point

property for affine nonexpansive mappings”, Filomat, to appear.




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134

Renorming c0 and affine fixed point property

Veysel Nezir1,a, Nizami Mustafa1,b, Tahsin Ateş,c



b [email protected] c [email protected]

Abstract: Reflexivity is strongly related to the fixed point property for

nonexpansive mappings [FPP(n.e)]. It was a long time open question as to

whether or not non-reflexive spaces can be renormed to have FPP(n.e). Recently,

Lin provided the first example of a non-reflexive Banach space with FPP(n.e) by

verifying this fact for 𝑙1. It is natural to ask whether there is an analogue of Lin's

theorem for another non-reflexive space that shares many common properties

while it may not offer nice tools like Schur property. 𝑐0, another non-reflexive

Banach space of great importance in Banach space theory, is one of the most

well-known such these spaces. Recently, Nezir and Mustafa gave positive

answer for the 𝑐0-analogue of P. K. Lin’s theorem under affinity condition. In

this study, we prove that there is another renorming of 𝑐0 with the fixed point

property and a family of equivalent norms on 𝑐0 that has FPP(n.e) if the

mappings are affine.


property, renorming, affine mapping

References:



Netherlands, (2001), 269–297.



[3] P. K. Lin, “There is an equivalent norm on l1 that has the fixed point

property”, Nonlinear Analysis 68(2008), 2303–2308.

[4] V. Nezir and N. Mustafa, “c0 can be renormed to have the fixed point

property for affine nonexpansive mappings”, Filomat, to appear.




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135

Infinite dimensional subspaces of a degenerate Lorentz-

Marcinkiewicz space with the fixed point property

Veysel Nezir1,a, Nizami Mustafa1,b, Merve Delibaş1,c


Kars, Turkey a [email protected], b [email protected],b c [email protected]

Abstract: We work on some infitine dimensional subpaces of a degenerate ℓ1-

analog Lorentz-Marcinkiewicz space ℓ𝛿,1, where the weight sequence 𝛿 =

(𝛿𝑛)𝑛∈𝑁 = (2,1,1,1,⋯ ) is a decreasing positive sequence in ℓ∞\𝑐0, rather than

in 𝑐0\ℓ1 (the usual Lorentz situation). We show that a Goebel and Kuczumow

analogy can be obtained by proving that there exist large classes of non-weak*,

closed, bounded and convex subsets of ℓ𝛿,1 with the fixed point property for

nonexpansive mappings.


property, closed bounded convex subset, Lorentz-Marcinkiewicz spaces

References:



Netherlands, (2001), 269–297.



[3] J. Lindenstrauss and L. Tzafriri, “Classical Banach spaces I: Sequence

Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete”, 92(1977),

Springer-Verlag.

[4] G. G. Lorentz, “Some new functional spaces”, Annals of Mathematics,

(1950), 37-55.




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136

Applying Fourier Analysis to Computing New variables for

Determining SNR

Sajedeh Norozpour

Department of Mathematics and computer science, Eastern Mediterranean University,

Gazimagusa, Cyprus

[email protected]

Cem Ergün

Department of computer Engineering, Eastern Mediterranean University,

Gazimagusa, Cyprus

[email protected]

Abstract: In this paper the quantitive description of signal to noise ratio through

novel computational parameters will be debated and computed using Fourier

analysis method. The significance of this study is when the geomagnetic

observed field has high transient noise and high energy content and it is going to

be more important to work on, if signal analysing action leans to the quasi-

punctiform, low energy and kinetic magnetic source recognition. The following

study demonstrates the consequences which are obtained by defining two new

informative spectral parameters: the informative capability “C” and the enhanced

informative capability “eC”. These variables are depending on the assessment of

the signal target energy with total field energy. C and eC lead to compute the

value of the “visibility” of the informative signals in high energy geomagnetic

field or spectrum. The problem of the visibility becomes quite complex when the

target signal is coming from a quasi-punctiform and low amplitude source, inside

a high energy (noise) geomagnetic volume and also the low visibility problem

becomes practically unsolvable when these classes of sources are in motion and

their signals are recorded by a fixed sensor. In order to quantify the phenomena

of visibility the parameter “C” and its derivate “eC” has been used.

Keywords: Signal to noice ratio (SNR), Fourier transforms, computational

parameter.

References:

[1]Cooley, J.W. and Tukey, J.W. (1965), An algorithm for the machine

calculation of complex Fourier series, Mathematics of Computation, 19, 297.




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137

[2]Faggioni, O., Soldani, M., Gabellone, A., Hollett, R.D., and Kessel, R.T.

(2010) Undersea harbour defence: A new choice in magnetic networks. Journal of

Applied Geophysics, 72, n. 1, 46-56.

[3]De Vuyst, A., and De Meyer, F. (1973) Spectral analysis of geomagnetic data

from one station (Dourbes 1960-1970). Institut Royal Meteorologique de

Belgique, Publ., A, n° 80, 32.




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138

On Nordenian Properties of the Riemannian extension

Filiz OCAK


Trabzon, Turkey

[email protected]

Abstract: In this study, we investigate the Nordenian properties of the

Riemannian extension by using compatible almost complex structure in the

cotangent bundle. Then, we get the conditions under which the cotangent bundle

endowed with the almost complex structure and the Riemannian extension is a

Kähler-Norden manifold and anti-Kähler-Codazzi manifold.

Keywords: Riemann extension, cotangent bundle, almost complex structure,

Norden metric.

References:

[1] S. Aslanci, S. Kazimova and A.A Salimov, “Some notes concerning

Riemannian extensions”, Ukrainian Math. J., 62(2010), 661–675.

[2] G. T. Ganchev and A. V. Borisov, “ Note on the almost complex manifolds

with a Norden metric”, C. R. Acad. Bulg. Sci., 39 (1986), 31–34 .

[3] E.M Patterson and A.G. Walker, “ Riemann Extensions”, Quart. J. Math.

Oxford Ser., 3 (1952), 19-28.

[4] A. Salimov, “ Tensor Operators and Their Applications”, Nova Science

Publishers, New York, 2012.

[5] A. Salimov, S. Turanli, “Curvature properties of anti-Kähler-Codazzi

manifolds”, C. R. Math. Acad. Sci. Paris, 351(2013), 225-227.

[6] K. Yano and S. Ishihara , “Tangent and Cotangent Bundles”, Pure and

Applied Mathematics, Marcel Dekker, Inc., New York, 1973.




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139

Numerical Solution of Nonlinear Singular Ordinary

Differential Equations Arising in Biology via Operational

Matrix of Bernstein Polynomials

Sunday Adeyemi Olagunju and Peter Vanenchii Ayoo

Department of Mathematics, Federal University Lafia,

PMB 146 Lafia, Nigeria


Abstract: Many authors including Akyuz-Dascioglu and Acar[1] and Ordokhani

and Far[2] have studied Bernstein polynomials, their properties and applications

to linear ordinary differential equations. The Bernstein method since its

introduction, has undergone improvements including that by Olagunju[3].

Maleknejad and Hashemicadeh[4] and Singh et.al[5] studied singular boundary

value problems (SBVPs) that arise in biology and physiology which are of linear

or nonlinear functions. Some of these arise in the study of steady-state oxygen

diffusion in a spherical cell with Michaelis-Menten uptake kinetics according to

Lin[6] and in heat conduction model in human head according to Gray [7]. In

this paper, we deploy Bernstein polynomials to find solution to nonlinear SBVPs

that arise in biology and physiology. With the aid of operational matrix of

derivatives, the entire nonlinear differential equation is written in algebraic

equations that are handled by an algebraic solver- MATLAB.

Keywords: Nonlinear Singular Boundary Value Problem, Bernstein

Polynomials, Physiology, Biology.

References:

[1] A. Akyuz-Dascioglu and N. I. Acar, “Bernstein collocation method for

solving linear differential equations”, GU J Sci. 26.4(2013), 527-534.

[2] Y. Ordokhani and S. D. Far, “Approximate solutions of differential equations

by using the bernstein polynomials”, ISRN Applied Mathematics, 2011(2011), 1-

15.

[3] A. S. Olagunju, “Chebyshev series representation for product of chebyshev

polynomials and some notable functions”, IOSR Journal of Mathematics,

2.2(2012), 9-13.




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140

[4] K. Maleknejad and E. Hashemicadeh, “Numerical solution of nonlinear

singular ordinary differential equations arising in biology via operational matrix

of shifted Legendre polynomials”, AJCM, 1.1(2011), 15-19.

[5] R. Singh, J. Kumar and G. Nelakanti, “Singular boundary value problems

arising in physiology” The Scientific World Journal, 2014(2014), 1-10.

[6] S. H. Lin, “Oxygen diffusion in a spherical cell with nonlinear oxygen uptake

kinetics” Journal of Theoretical Biology, 60.2(1976), 449-457.

[7] B. F. Gray, “The distribution of heat sources in the human head theoretical

consideration” Journal of Theoretical Biology, 82.3(1980), 473-476.




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141

On Ciric Type Contractions in Modular Spaces

Mahpeyker Öztürk


Sakarya, Turkey

[email protected]

Abstract: In this study, common fixed point theorems for some mappings

satisfying Ciric type contractions in modular spaces are established. The

theorems proved herein extend, generalize and improve some results of the

existing literature. An illustrative example to support the given results is

constructed.

Keywords: Ciric type contraction, common fixed point, modular space.

References:

[1] Ljubomir B. Ćirić, On some maps with a nonunique fixed point, Publications

de l Institut Mathematique, 17 (31), (1974).

[2] Memudu Olaposi Olatinwo, Some non-unique fixed point theorems of Ćirić

type using rational-type contractive conditions, Georgian Math. J., (2016)

[3] Erdal Karapınar, Ravi P Agarwal,

A note on Ćirić type nonunique fixed

point theorems, Fixed Point Theory and Applications, (2017)

[4]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in

Mathematics, 1034, Springer, Berlin (1983)

[5] J. Musielak, W. Orlicz, On modular spaces, Studia Mathematica, 18, (1959)




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142

Some fixed point results concerning F-contractions in complete

metric space

Alexandrina Maria Proca

Department of Mathematics and Computer Science, Transilvania University of Brasov,

Brasov, Roumania

[email protected]

Abstract: In this paper, we extend the results of Wardowski by applying some

new conditions on the self map on a complete metric space, concerning the F-

contractions defined by Wardowski. We present some fixed point results of

Wardowski type. An example is given to demonstrate the novelty of our work.

Keywords: Wardowski type contraction, fixed point, metric space

References:

[1] S.Banach, Sur les operations dans les ensembles abstraits et leur application

aux equations integrales, Fundamenta Mathematicae 3(1922), 133-181.

[2] V.Berinde, F.Vetro, Common fixed points of mappings satisfying implicit

contractive conditions, Fixed Point Theory and Applications 2012:105

(2012).

[3] M.Concertino, P.Vetro, Fixed point results for F-Contractive Map- pings of

Hardy-Rogers type, Filomat 28:4(2014), 715-722; doi:

10.2298/FIL1404715C.

[4] N.Van Dung, V.L.Hang, A fixed point theorem for generalised F-

contractions on complete metric space, Vietnam J.Math, 43, 743-753(2015).

[5] A.Fulga, A.M.Proca, A new Generalization of Wardowski Fixed Point

Theorem in Complete Metric Spaces, Advances in the Theory of Nonlinear

Analysis and its Applications 1 (2017) No. 1, 57–63 DOI:10.31197\atnaa.379119

[6] O.Popescu, Some new fixed point theorems for -Geraghty contraction type

maps in metric spaces, Fixed Point Theory Appl.2013, Article ID 329(2013).

[7] O.Popescu, A new type of contractive mappings in complete metric spaces,

submitted.

[8] H.Piri, P.Kumam, Some fixed point theorems concerning F-contraction in

complete metric spaces, Fixed Point Theory Appl., 2014, 210(2014)

doi:10.1186/1687-1812-2014-210.




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143

[9] N.A.Secelean, Iterated function sistems consisting of F-contractions, Fixed

Point Theory Appl.2013, Article ID 277(2013).doi:10.1186/1687- 1812-2013-

277.

[10] T.Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear

Anal 71, 5313-5317 (2009).

[11] T.Suzuki, Generalized distance and existence theorems in complete metric

spaces, J.Math.Appl.253, 440-458 (2001).

[12] T.Suzuki, Several fixed point theorems concerning -distance, Fixed Point

Theory Appl.2004, 195-209 (2004).

[13] D.Wardowski, Fixed Points of a new type of contractive mappings in com-

plete metric spaces, Fixed Point Theory and Applications 2012, 2012:94.

[14] D.Wardowski, N. Van Dung, Fixed points of F-weak contractions on com-

plete metric space, Demonstr.Math. 1, 146-155(2014).




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144

Fixed point theorem for F-contraction type and applications to

polynomial approximations

Abdalah Rababah1, Hichem Ramoul2, Nora Mahloul3 1Department of Mathematics and Statistics, University of Science and

Technology Irbid, Jordan

[email protected] 2ICOSI laboratory, Abbes Laghrour University-Khenchela,

Khenchela, Algeria

[email protected] 3USTHB, Faculty of Mathematics

Algebra and Number Theory Department

[email protected]

Abstract: The purpose of this work is to prove a new result concerning a fixed

point for F-contraction type in b-metric space. We establish existence and

uniqueness without the continuity of the function F. As an application, we give

some results about polynomial operators.

Keywords: Orthogonals polynomials, fixed point.

References:

[1] Wardowski, D: Fixed point theory of a new type of contractive mappings in

complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)

[2] S. Ostrovska, q-Bernstein polynomials and their iterates, Journal of

Approximation Theory, 123 (2003) 232-255.

[3] I. Altun, N. AL Arifi, M. Jleli, A. Lashin, B. Samet, A fixed point theorem

for JS-contraction type with applications to polynomial approximation, Filomat

31 :15 (2017), 4969-4978.




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145

Metric Properties of Fuzzy Curves and Fuzzy Invariants

Yasemin Sağıroğlua, Djavvat Khadjievb, Uğur Gözütoka aDepartment of Mathematics, Karadeniz Technical University,

Trabzon, Turkey

[email protected], [email protected] bInstitute of Mathematics, National University of Uzbekistan,

Tashkent, Uzbekistan

[email protected]

Abstract: Geometric properties of fuzzy sets and fuzzy geometric concepts are

introduced in the papers [1,2,3]. Metric properties of fuzzy sets are considered in

papers [4,5]. Our study is devoted to an investigation of metric properties of

fuzzy paths and fuzzy curves in the 3-dimensional Euclidean space. In this paper,

𝐸(3)-equivalence of fuzzy paths and 𝐸(3)-equivalence of fuzzy curves are

introduced. Definition of the fuzzy type of a fuzzy curve and an invariant

parametrization of a fuzzy curve are given. We show that the fuzzy type of a

fuzzy curve is an 𝐸(3)-invariant. Also, the conditions of the global 𝐸(3)-

equivalence of fuzzy curves are given in terms of the fuzzy type and the three

differential invariants (curvatures) of fuzzy curves. Finally, the description of the

complete system of correlations between these invariants of a fuzzy curve is

given.

Keywords: Differential invariant, Fuzzy path, Fuzzy curve.

References:

[1] X. Yuan, Z. Shen, “Notes on fuzzy plane geometry i, ii”, Fuzzy Sets and

Systems, 121(2001), 545-547.

[2] J. J. Buckley, E. Eslami, “Fuzzy plane geometry i: Points and lines”, Fuzzy

Sets and Systems, 86(1997), 179-187.

[3] A. Rozenfeld, “Fuzzy geometry: An updated overview”, Information

Sciences, 110(1998), 127-133.

[4] A. Bogomolny, “On the perimeter and area of fuzzy sets”, Fuzzy Sets and

Systems, 23(1987), 257-269.

[5] L. Boxer, “On Housdorff-like metrics for fuzzy sets”, Pattern Recognition

Letters, 18(1997), 115-118.




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146

Hamiltonicity and pancyclicity of superclasses of claw-free

graphs

Abdelkader Sahraoui

Department of Mathematics, Medea University,

Medea, Algeria

[email protected]

Abstract : The class of claw-free graphs was a _eld of research interesting for

many researchers. Many of the known results for claw-free graphs are extended

to superclasses of claw-free graphs. The superclass of partly claw-free graphs,

defined by Z. Benmeziane ([1]), contains the two superclass of quasi claw-free

graphs, defined by A. Ainouche ([2]) , and almost claw-free graphs, defined by

Z. Ryjàĉek ([3]). A graph G is parlay claw-free graph if for each vertex v of A,

the set of centers of claw of G, there exists two vertices x and y of V (G) - A

such that NG(v) subset of N[x] [ N[y]. Here, we will extend the fully extendable

cycle graphs to the partly claw-free graphs.

Keywords: Nearly Claw-Free Graphs, partly claw-free graphs, quasi claw-free

graphs, et fully extendable cycle graphs. Ryjàĉek

References:

[1] M. Abbas and Z. Benmeziane. Hamiltonicity in partly claw-free graphs.

RAIRO Oper. Res., 43 1 (2009) 103-113.

[2] A. Ainouche. Quasi claw-free graphs. Discrete Math. 179 (1998) 13-26.

[3] Z. Ryjàĉek, Almost claw-free graphs. J. Graph Theory. 18(1994), 469-477.

[4] M. Zhan, Full cycle extendability of triangulary connected almost claw-free

graphs. ABS Combinatoria, vol. 96 (2010).

[5] M. Zhan et Shuxin Zhan, Full cycle extendability of Nearly Claw-Free

Graphs. April 6,2012.




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147

On some evolution problems of degenerate type

MOULAY Mohamed Said

Faculty of Mathematics, USTHB, Algiers

[email protected]

Abstract: We consider some nonlinear problems of degenerate type. Some class

of them are of parabolic kind and arise in the modeling of oil in the porous

medium, or in biological reaction diffusion systems. In this case we look

especially for the regularity of solutions and for that purpose we use a

generalized method of De Giorgi iterations type in some appropriate functional

spaces to show the optimal regularity.

Other class of our interest arises in physical area and it is of anisotropic elliptic

or parabolic type. For this purpose, we use the anisotropic singular perturbation

method to study the existence of solutions.

Keywords: degenerate problems, nonlocal term, porous media, anisotropic,

singular perturbation, regularity, reaction diffusion system




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148

Average approximate Sentinel for an Identification problem

S. Sandel

University of Abbse Laghrour, Algeria..

A.Berhail

University 08 May 1945, Guelma, Algeria

A. Ayadi

University Larbi Ben M'hidi, Oum El Bouaghi, Algeria

.Abstract: In this paper, we intoduce the problem of identification of the

Pollution term in a heat system when the dynamics of the state is gouverned by a

parametrized operator. In this way, we introduce a new notion called average

approximate sentinel. First, we prove the existence of such average approximate

sentinels by solving a problem of approximate average controllability introduced

by Zuazua. Secondly, we give information for the polution term by the average

obsevation.

Keywords: Average approximate sentinel, Average approximate controllability,

pollution term

References:

[1] A. El JAI, E. ZERRIK, K. ZTOT, Systèmes dynamiques, Analyse et contrôle

des systèmes localisés. Presses Universitaires de Perpignan, 2008.

[2] J.P. KERNEVEZ, The sentinel method and its aplications to envirenmental

pollution problems, CRC Methematical modeling series, CRC Press, Boca

Raton, 1997.

[3] J. L. LIONS, Contrôle optimal des systèmes gouvernés par des équations aux

dérivées partielles, Dunod. Paris, 1968.

[4] J. L. LIONS, Contrôlabilité exacte, stabilisation et perturbation des systèmes

distribuès, Vol 1. Masson, 1988.

[5] J. L. LIONS, Controllability, penalty and stiffness, Anal, Scie, Nor, Serie 4,

Tome 25, N 3-

[6] J .L. LIONS, Sentinelle pour les systèmes distribués à données

incomplètes.Vol 21, Masson, RMA, Paris, 1992.

[7] J. L. LIONS, Remarks on systems with incompletely given initial data and

incompletely given part of boundary, pp. 239-250, 1990.

[8] E. Zuazua, Averaged Control. Preprint submitted to Automatica, 2014.




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149

[9] E. Zuazua,Q.Lu, Averaged controllability for parameter-dependent Evolution

Partial Differential Equations. Preprint submitted to Automatica, 2014.

[10] E. Zuazua, M. Lazar, Averaged control and observation of

parameterdepending wave equations. C. R. Acad. Sci. Paris, Ser.JUNE 2014.




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150

λ – Double Statistical Convergence and Strongly Double

Summable Functions of Weight g

Rabia Savaş and Mahpeyker Öztürk


Sakarya, Turkey

[email protected] and [email protected]

Abstract: In this paper, by using a nonnegative real valued measurable functions

of two variables in the interval (1,∞) × (1,∞) we present the concepts of

strongly g

pW -summable and

gS

-statistically convergence of weight g:

[0, ∞) → [0,∞) where g(𝑥𝑛) → ∞ for any sequences (𝑥𝑛) in [0,∞) with 𝑥𝑛 →

∞. Moreover, using these definitions we examine a series of basic results, and

inclusions of theorem, extensions as well as variations are proved.

Keywords: Lebesgue measurable functions of two variables, weight g,

statistically convergence, strong summability.

References:

[1] M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Generalized kinds of

density and the associated ideald, Acta. Math. Hungar. 147 (2015), no:1, 97-115.

[2] H. Fast, “Sur la convergence statistique”, Colloq. Math. 2(1951), 241-244.

[3] M. Mursaleen and O. H. Edely, Statistical Convergence of Double Sequence,

J. Math. Anal. Appl. 288 (2003), no:1, 223-231.

[4] F. Nuray, \lambda-Strongly Summable and \lambda- Statistically

Convergence Functions, İranian Jour. Science and Tech. Trans.A, Vol. 34,

No:A4, (2010), 335-339.




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151

On Infinite Matrices and Sequence Spaces

Rahmet SAVAS

Department of Mathematics, Istanbul Medeniyet University,

Istanbul, Turkey

[email protected]

Abstract: The purpose of this paper is to study the spaces 0( )V p

and ( )V p

by using de la Valée poussin and generalized almost convergence. Furthermore

we characterize certain matrices in ( )V

which will up a gap in the existing

literature. Also we prove that 0( )V p

and ( )V p

are complete linear

topological spaces.

Keywords: Infinite matrices, Sequence Spaces, de la Val’ee Poussin

References:

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

[2] G. Das and S. K. Mishra, Banach limits and lacunary strong almost

convergence, J. Orissa Math. Soc. 2(2)(1983), 61-70.

[3] V. Karakaya, θσ-sumable sequences and some matrix transformations,

Tamkang J. Math. 35(4)(2004), 313-320.

[4] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta.

Math. 80(1948),167-190.

[5] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math.

18(1967), 345-355.

[6] M. Mursaleen, A. M. Jarrah and S. A. Mohiuddine, Bounded linear operators

for some new matrix transformations, Iran. J. Sci. Technol. Trans. A Sci. 33

(2009), no. 2, 169177.




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152

Numerical methods for certain classes of Markovian BSDEs

and quasi-linear parabolic PDEs

Driss Seghir

Department of Mathematics, Faculty of sciences in Meknes,

Moulay Ismail University,

Meknes, Morocco.

[email protected]

Joint work with :

Aissa Sghir and Soukaina Hadiri

Abstract: In this paper, we use the Girsanov's theorem of change of probability

measure to give an approximation result and numerical simulation for certain

classes of Markovian coupled forward-backward stochastic differential

equations. As application, by using the Feynman-Kac's formula, we give a time-

space discretization scheme for certain classes of quasi-linear parabolic partial

differential equations.

Keywords: Backward stochastic differential equation, Girsanov's theorem,

parabolic partial differential equation, discrete time approximation, Monte Carlo

simulation.

References:

[1] B. Bouchard and N. Touzi (2004). Discrete-time approximation and Monte-

Carlo simulation of backward stochastic differential equations. Stochastic

Processes and their Applications, 111(2), p. 175- 206.

[2] N. El Karoui, S. Peng and M.C. Quenez (1997). Backward stochastic

differential equations in finance. Mathematical Finance, 7 (1), p. 1-71.

[3] E. Pardoux and S. Peng (1992). Backward stochastic differential equations

and quasilinear parabolic partial differential equations. Lecture Notes in CIS,

176, Springer Berlin, p. 200-217.




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153

Integral Type Darbo's Fixed Point Theorem via Sequences of

Functions

Derya Sekman1, Vatan Karakaya

2

1,2Department of Mathematics, Kirsehir Ahi Evran University,

Kirsehir, Turkey

2Department of Mathematical Engineering, Yildiz Technical University,

Istanbul, Turkey


Abstract: In this paper, we will give a generalization of the integral type Darbo's

fixed point theorem with the help of sequences of functions having shifting

distance property and an application of integral equation.

This work was supported by the Ahi Evran University Scientific Research

Projects Coordination Unit. Project Number: RKT.A3.17.001.

Keywords: Fixed point, Integral type mapping, Measure of noncompactness,

Sequences of functions, Shifting distance property.

References:

[1] A. Samadi and M. B. Ghaemi, “An extension of Darbo's theorem and its

application”, Abstract and Applied Analysis 2014 (2014), Article ID 852324, 11

pages.

[2] L. Cai, J. Liang, “New generalization of Darbo's fixed point theorem”, Fixed

Point Theory and Applications, 2015:156 (2015), DOI: 10.1186/s13663-015-

0406-2.

[3] M. Berzig, “Generalization of the Banach contraction principle” (2013),

arXiv:1310.0995 [math.CA].




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154

Normal Differential Operators for Degenerate First-Order

Meltem Sertbaş & Fatih Yılmaz


Trabzon, Turkey


Abstract: In this paper, in terms of boundary values all normal extensions of a

class of formally normal minimal operators, generated by a linear differential-

operator expression of degenerate first-order with smooth operator coefficients

are described in Hilbert space of vector-functions in a finite interval. The

structure of the spectrum of the normal extensions is examined.

Keywords: Degenarete differential operator, Formally normal and normal

operator, Minimal and maximal operators, Extension, Spectrum.

References:

[1] E. A. Coddington, ‘‘Extension theory of formally normal and symmetric

Subspaces’’, Mem. Amer. Math. Soc., 134(1973), 1-80.

[2] G. Biriuk and E. A. Coddington, ‘‘Normal extensions of unbounded formally

normal operators’’, J. Math. and Mech., 13(1964), 617-638.

[3] J. Stochel and F. H. Szafraniec, ‘‘On normal extensions of unbounded

operators’’, I, Oper. Theory, 14(1985), 31-55.

[4] N. Dunford and J.T. Schwartz, ‘‘Linear Operators’’, vol. II, Interscience,

New York, 1963.

[5] V. I. Gorbachuk and M. L. Gorbachuk, ‘‘Boundary value problems for

operator differential equations’’, Kluwer Academic, Dordrecht, 1991.

[6] Z. I. Ismailov, ‘‘Compact inverses of first-order normal differential

operators’’, J. Math. Anal. Appl. 320(2006), 266-278.




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155

Converse Theorems for the Discrete Power Series Methods of

Summability

Sefa Anıl Sezer 1, İbrahim Çanak 2 1 Department of Mathematics, İstanbul Medeniyet University,

İstanbul, Turkey

[email protected] 2 Department of Mathematics, Ege University,

İzmir, Turkey

[email protected]

Abstract: The discrete type power series methods were introduced by Watson

[4]. In this study, under certain conditions we recover convergence of an infinite

series from its discrete power series summability. Our results both extend and

generalize Ishiguro's [1] theorems.

Keywords: Tauberian theorems, power series methods of summability.

References:

[1] K. Ishiguro, Two Tauberian theorems for (J, p) summability, Proc. Japan

Acad. 41 (1965) 40-45.

[2] S. A. Sezer, İ. Çanak, On converse theorems for the discrete Bürmann power

series method of summability, Maejo Int. J. Sci.Technol. 10 (2016) 346-353.

[3] S. A. Sezer, İ. Çanak, Conditions for the equivalence of power series and

discrete power series methods of summability, Filomat 29 (2015) 2275-2280.

[4] B.Watson, Discrete power series methods, Analysis (Munich) 18 (1998) 97-

102.

[5] B.Watson, A Tauberian theorem for discrete power series methods, Analysis

(Munich) 22 (2002) 361-365.




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156

α – Fuzzified Rough Set Model in Multigranulation Space

Tanzeela Shaheen

Department of Mathematics, Air University,

Islamabad, Pakistan

[email protected]

Abstract: Multigranulation rough set structure has been developed in recent

years [2] by extending the Pawlak’s rough set [1] (single granularity) approach.

Multigranulation approach is getting attention of many researchers as it has

widened the possible application areas of rough set technique. Recently, a new

approach has been presented for defining the rough approximations of α-

indiscernible objects (indiscernible up to a certain degree α) under fuzzy

environment [3]. In this paper, we extend that approach under the framework of

multigranulation over two different universes of discourse, that is,

multigranulation α-fuzzified rough set model over two universes. Structural

properties have been discussed and an algorithm has been presented to deal with

multi criteria group decision making problems. The proposed model not only

extends α-fuzzified rough set approach but also provides a new perspective for

multiple criteria group decision making with uncertainty.

Keywords: Rough sets, α-fuzzified rough sets, rough sets over two universes,

multigranulation α-fuzzified rough sets, multiple criteria group decision making.

References:

[1] Z. Pawlak, “Rough Sets” International Journal of Computer and Information

Sciences 11(5) (1982) 341-356.

[2] Y. Qian, J. Liang, Y. Yao, C. Dang, “MGRS: A multi-granulation rough set”,

Information Sciences, 180(6) (2010), 949-970.

[3] M. Shabir, T. Shaheen, A new methodology for fuzzification of rough sets

based on α-indiscernibility. Fuzzy Sets and Systems, 312, (2017)1-16.




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157

A Note on Generalized Bernstein-type Cheney-Sharma

Operators

Dilek Söylemez,

Department of Computer Programming of Elmadag Vocational School, Ankara

University,

Ankara, Turkey

[email protected]

Fatma Taşdelen Yeşildal


Ankara, Turkey

[email protected]

Gülen Başcanbaz Tunca


Ankara, Turkey

[email protected]

Abstract: In this paper, we deal with the Korovkin type approximation

properties of Cheney-Sharma Chlodovsky operators by using A-statistical

convergence . We also study rate of convergence. Furthermore, we show that

these operators preserve properties of the modulus of continuity function when f

is a modulus of continuity function and Lipschitz condition of a given Lipschitz

continuous function f. Finally, we give a result for the new operators when f is a

convex functions and we add a graphical example for these operators.

Keywords: Chlodovsky operators, Lipschitz continuous function, modulus of

continuity function.

References:

[1] F. Altomare, M. Campiti, Korovkin-type approximaton theory and its

applications, Walter de Gruyter, Berlin-New York, 1994.

[2] B.M. Brown, D. Elliott, D.F. Paget, Lipschitz constants for the Bernstein

polynomials of a Lipschitz continuous function, J. Approx. Theory, 49(1987),

no.2, 196-199.

[3] E. W. Cheney, A. Sharma, On a generalization of Bernstein polynomials,

Riv. Mat. Univ.Parma, 2(5)(1964), 77-84.

[4] D. D. Stancu, C. Cisma¸siu,On an approximating linear positive operator of

Cheney-Sharma, Rev. Anal. Numer. Theory Approx., 26(1997),no.1-2, 221-227.




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158

Evaluating Gaussian Fuzzy System Reliability

Halil İbrahim Şahin


[email protected]

Melek Eriş Büyükkaya


[email protected]

Abstract: Gaussian fuzzy membership functions are quite popular in the fuzzy

logic literature. Therefore, if we accept components fuzzy reliability as a

Gaussian membership, it is important to calculate fuzzy system reliability. It is

difficult to find Gaussian membership system reliability, because of erf function.

Therefore, in this study, we approximate Gaussian membership to triangular

membership function. As a results, we obtained fuzzy system reliability as a

function of mean and standard deviation .

Keywords: Fuzzy System Reliability, Gaussian Fuzzy Membership, Triangular

Membership Function.

References:

[1]A., K., Shaw and T., K., Roy ,Generalized Trapezoidal Fuzzy Number with its

Arithmetic Operations and its Application in Fuzzy System Reliability Analysis,

International Journal of Pure and Applied Sciences and Technology ISSN 2229-

6107 Int. J. Pure Appl. Sci. Technol., 5(2), pp. 60-76, 2011.

[2]E., A., Elsayed, Reliability Enginering, Addison Wesley Longman, Inc., USA,

1996.

[3]I. M. Aliyev and Z. Kara, Fuzzy system reliability analysis using time

dependent fuzzy set, Control and cybernetics vol.33, 2004.

[4]J. Yao, J. Su and T. Shih , Fuzzy System Reliability Analysis Using

Triangular Fuzzy Numbers Based on Statistical Data, Journal of information

science and engineering 24, 1521- 1535 ,2008.




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159

Smoothed Auxiliary Function with Multiple Extrema in Global

Optimization

Ahmed SAHINER and Shehab A. IBRAHEM

Department of Mathematics, Suleyman Demirel University,

Isparta, Turkey


Abstract: In this study, we present a new auxiliary function method to solve

unconstrained global optimization. First, a new smoothing auxiliary function

with multiple extrema is constructed by using local smoothing approximations

and then the algorithm and its properties are demonstrated. Second, we apply the

algorithm to the test problems in order to explain the efficiency and effectiveness

of the method.

Keywords: Global optimization, Smoothing technique, Auxiliary function.

References:

[1] Z. Y. Wu and A. M. Rubinov, “Global Optimality Conditions for Some

Classes of Optimization Problems”, J. Optimiz. Theory App., 145(2010), 164-

185.

[2] Y. Lin and Y. Yang, “A new filled function method for constrained nonlinear

equations”, Appl. Math. Comput, 219(2012), 3100-3112.

[3] N. Yilmaz and A. Sahiner, “A new smoothing approximation to piecewise

smooth functions and applications”, International Conference on Analysis and

Application, (2016), 226-226.

[4] B. W. K. Ling, C. Z. Wu, K. L. Teo and V. Rehbock, “Global Optimal

Design of IIR Filters via Constraint Transcription and Filled Function Methods”,

Circ. Syst. Signal Pr., 32(2013) 1313-1334.




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160

Application of Modern Optimization Techniques in

Determination of the Academic Achievement

Ahmet Sahiner, Sumeyya Koman and Raziye Akbay Uçar


Isparta, Turkey


Abstract: In the study, we develop an attitude scale with the aim of measuring

students’ interest towards the lesson and the teacher. The prepared scales are

applied to the students. The obtained data are tabulated. The data is modeled

using fuzzy logic and neural networks techniques separately. Obtained results are

compared with each other and the real data.

Keywords: Fuzzy logic, Neural network, Attitude scale, Academic achievement.

References:

[1] R. Akbay, A. Şahiner, & N. Yılmaz, “Determining of The Achievement of

Students by Using Classical and Modern Optimization Techniques” International

Journal of Physics and Chemistry Education 8(1) (2016), 3–13.

[2] S. Wagner, “Conservation of equation and function under transformations of

variable”, Journal For Research in Mathematics Education, 12(2) (1981), 107-

118.

[3] M. Saltan, S. Saltan S. & A. Sahiner , Fuzzy logic modelling of deflection

behavior against dynamic loading in flexible pavements, Construction and

Building Materials, 21(2007), 1406-1414.




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161

Radiation Shielding Properties of Borided Fe-Ni Binary Alloys

by Fuzzy Logic Modelling

Ahmet Şahiner1, Nazım Uçar2, Adnan Çalık3 and Gülden Kapusuz1 1 Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

2Department of Physics, Suleyman Demirel University, Isparta, Turkey 3Department of Machine and Manufacturing, Suleyman Demirel University,

Isparta,Turkey



Abstract: In this study, firstly the radiation shielding properties of borided Fe-Ni

binary alloys (Ni-60 wt.% and Ni-80 wt.%) have been investigated. For this

purpose, linear attenuation coffecients as a function of boronizing parameters

(time and temperature) have been measured at photon energy levels of 662 keV,

1173 keV and 1332 keV. According to experimental results, the fuzzy logic

model (FLM) is established. The results showed that the fuzzy logic model can

be used to predict the radiation shielding properties of borided Fe-Ni alloys for

untested conditions.

Keywords: Fuzzy Logic Modelling, Boriding, Radiation Shielding, Binary

Alloys.

References:

[1] A. Sahiner, F. Ucun, G. Kapusuz, N. Yilmaz, “Completed Optimized

Structure of Threonine Moleculeby Fuzzy Logic Modeling”, Z. Naturforsch A,

71( 4),(2016), 381–386.

[2] A. Sahiner, H. Gokkaya, N. Uçar, “Nonlinear Modelling of the Layer

Microhardness of Fe-Mn Binary Alloys” , Journal of the Balkan Tribological

Association, 19(4),(2013), 537-548.

[3] A. Calik, S. Akbunar, N. Ucar, N. Yilmaz, M. S. Karakas and I. Akkurt, “A

Comparison of Radiation Shielding of Stainless Steel with Different Magnetic

Properties”, Nuclear Technology & Radiation Protection 29(2014), 186-189.




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162

Temperature and pressure based estimation of daily global

solar radiation over Turkey using fuzzy logic modeling

Ahmet Şahiner


Isparta, Turkey

[email protected]

Idris A. Masoud Abdulhamid


Isparta, Turkey

[email protected]

Murat Ozturk

Department of Mechatronics Engineering, Süleyman Demirel University,

Isparta, Turkey

[email protected]

Abstract: In this paper, a fuzzy logic model (FLM) for predicting and modeling

of the daily average solar radiation for Turkey is developed. The daily values of

ambient temperature and relative pressure duration as meteorological data are

utilized for FLM. These data for estimating solar radiation are generally obtained

easily. Also, the calculated daily solar radiation values by using the FLM for

selected twelve cities are compared with the actual data and the national database

of selected cities. For similarity tests, the most common statistical procedures

(MAE, RMSE, MSE, MAPE and R2) are utilized. The fuzzy logic based models

show the stable processor different meteorological data introduced to the

operation for modeling of daily average solar radiation.

Keywords: Fuzzy system modeling, optimization modeling, artificial

intelligence.

References:

[1] V. Gomez and A. Casanovas, “Fuzzy modeling of solar irradiance on

inclined surfaces”, Solar Energy 75(4) (2003), 307–315.

[2] O., “Modeling of solar radiation using remote sensing and artificial neural

network in Turkey”, Energy 35(12) (2010), 4795–4801.




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163

[3] M. Ozturk, “An Evaluation of Global Solar Radiation Empirical

Formulations in Isparta, Turkey”, Energy Sources, Part A: Recovery, Utilization,

and Environmental Effects 37(22) (2015), 2474–2486.

[4] M. Iqbal, “An introduction to solar radiation”, Elsevier, 2012.

[5] Ö. Çelik, A. Teke and H.B. Y, “The optimized artificial neural network

model with Levenberg-Marquardt algorithm for global solar radiation estimation

in Eastern Mediterranean Region of Turkey”, Journal of cleaner production 116

(2016), 1–12.




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164

Normalizer of Some Congruence Subgroups

Zeynep Şanlı, Tuncay Köroğlu, Bahadır Özgür Güler


Trabzon, Turkey


Abstract: The purpose of the this paper is to obtain the normalizers of Γ0𝑘(𝑛) in

the groups Γ𝑘. The groups we consider in here are analogous to some

congruence subgroups of Γ. Our method is based on Ivrissimtzis in [6]. The

paper is organised as follows: In section 2, we give the definition of a H axes of

X which defining an equivalence relation on X, and give some preliminary

results. In section 3, we obtained the normalizers of two groups Γ02(𝑛) and Γ0

3(𝑛)

by using the H axes notion.

Keywords: Modular groups; congruence subgroup; group action

References:

[1] M. Akbaş, D. Singerman, The normalizer of Γ0 (𝑛) in PSL(2,R), Glasgow

Math. 32 (1990), 317–327.

[2] A. O. L. Atkin, J. Lehner, Hecke operators on Γ0 (𝑛), Math. Ann. 185

(1970), 134–160.

[3] F. Bars, The Group Structure of the Normalizer of Γ0 (𝑛) after Atkin-Lehner,

Communications in Algebra 36(6) (2008) 2160–2170.

[4] K. S. Chua, M. L. Lang, Congruence subgroups associated to the monster,

Experimental Mathematics 13(3) (2004) 343–360.

[5] C. Conway, S. Norton, Monstrous moonshine, Bull. London Math. Soc. 11

(1979), 308–339.

[6] I. Ivrissimtzis, Congruence subgroups of Hecke groups and regular

dessins,Ph.D. Thesis, University of Southampton, 1998.




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165

Boundary Component of the Group ��𝟎𝟐(𝐍)

Hatice Şengül*, Mehmet Akbaş


Trabzon, Turkey


Abstract: In this paper, the number of boundary components in the signature of

the subgroup Γ02(N) obtained by taking squares of the elements of the extended

modular group Γ and the numbers of cusps and finite link periods in each

boundary components are obtained by using Hoare-Uzzel Theorem.

Keywords: Signature, Period cycles, Boundary components.

References:

[1] M. Akbaş, The Normalizer of Modular Subgroups, PhD Thesis, University of

Southampton, 1989.

[2] D. Singerman, “On the Sturucture of non-Euclidean Crystallographic

Groups”, Proc. Camb. Phil. Soc., 76(1974), 233-240.

[3] B. Uzzell, Groups Acting on Hyperbolic 3-Space, PhD Thesis, University of

Birmingham, 1981.

[4] H. C. Wilkie, “On non-Euclidean Crystallographic Groups”, Math. Zeit.,

91(1966), 87-102.




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166

GUI-EFG: Guided Element Free Galerkin Method

Süleyman ŞENGÜLa , Erhan COŞKUNb

aDepartment of Mathematics, Recep Tayyip Erdoğan University,

Rize, Turkey

[email protected] bDepartment of Mathematics, Karadeniz Technical University,

Trabzon, Turkey

[email protected]

Abstract: An interactive tool, called GUI-EFG, using MATLAB GUI[1] feature

is developed that allows for interactive solution of boundary value problems on

finite intervals using Element Free(meshless) Galerkin(EFG) method[2]. GUI-

EFG allows for using user defined coefficient functions including discontinuous

ones, as well as various types of boundary conditions and values. The numerical

results are displayed graphically and compared with analytical solution, if

obtainable by analytical methods. Dependence of solutions on the use of weight

functions are investigated using two types of weight functions. Furthermore,

computational load of shape functions is reduced by using the so-called

translation-invariance property of shape functions[3,4]. We observe that EFG

whose shape functions are computed with translation invariant property becomes

compatible with the Finite Element Method(FEM). GUI-EFG allows for

numerical solution of various boundary value problems, including singular ones

and also leads to a comparitive study of various approaches.

Keywords: Element Free Galerkin method, MATLAB GUI, Shape functions.

References:

[1] MATLAB, www.mathworks.com, The MathWorks .

[2] Y. Belytschko, Y.Y. Lu, and L. Gu, “Element Free Galerkin methods”,

International Journal for Numerical Methods in Engineering, 37(1994), 229-256.

[3] Şengül, S. ve Coşkun, E., Optimized EFG Method and its Applications on

one and two dimensional Problems, 14th Mathematics Symposium, MATDER,

Niğde, May 14-16, 2015.

[4] Şengül, S., Optimization of Meshless Galerkin Method and it’s Applications

with some New Adaptive Algorithms, Dissertation, The Graduate School of

Nature and Applied Science, Karadeniz Technical University, 2016.




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167

On the Matrix Nearness Problem of the Generalized Sylvester

Quaternion Matrix Equations

Sinem ŞİMŞEK

Department of Mathematics, Kırklareli University,

Kırklareli, Turkey

[email protected]

Abstract: The generalized Sylvester quaternion matrix equations have a great

number of applications in control theory, signal and color image processing. So,

investigating solutions of this matrix equations should be significant and

interesting. In this work, it is established the precise solution for the matrix

nearness problem of the generalized Sylvester quaternion matrix equations by

Moore Penrose Inverse.

Keywords: Quaterion Matrices, Matrix nearess problem, Moore Penrose

Inverse.

References:

[1] S. F. Yuan, Q. W. Wang, “L-structured quaternion matrices and quaternion

linear matrix equations”, Linear and Multilinear Algebra, 64.2(2016), 321-339.

[2] S. F. Yuan, Q. W. Wang, “Two Special Kinds of Least Squares Solutions for

the Quaternion Matrix Equation”, Electron J. Linear Algebra, 23(2012), 257-274.

[3] Z. H. He, O. M. Agudelo, Q. W. Wang ad B. D. Moor, “Two-sided coupled

generalized Sylvester matrix equations solving using a simultaneous

decomposition for fifteen matrices”, Linear Algebra Appl., 496(2016), 549-593.

[4] S. Şimşek, M. Sarduvan and H. Özdemir, “Centrohermitian and Skew-

Centrohermitian Solutions to the Minimum Residual and Matrix Nearness

Problems of the Quaternion Matrix Equation (AXB,DXE) = (C,F)”, Adv. Appl.

Clifford Algebras, 27.3(2017), 2201-2214.




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168

Korovkin Type Theorems in Modular Spaces via Summability

Methods

Emre Taş, Tuğba Yurdakadim


Kırşehir, Turkey


Çorum, Turkey


Abstract: In this talk , we examine the problem of approximation to a function

by means of positive linear operators in modular spaces via summability

methods. Indeed, in order to get stronger results than the classical cases, we use

the power series method which also includes both Abel and Borel methods. An

application that satisfies our theorem is also provided.

Keywords: Power series method, Modular spaces, Positive linear operators

References:

[1] F. Altomare and S. Diomede, “Contractive Korovkin subsets in weighted

spaces of continuous functions”, Rend. Circ. Mat. Palermo 50(2001), 547-568.

[2] C. Bardaro and I. Mantellini, “Approximation properties in abstract modular

spaces for a class of general sampling-type operators”, Appl. Anal. 85(2006),

383-413.

[3] C. Bardaro and I. Mantellini, “Multivariate moment type operators:

approximation properties in Orlicz spaces”, J. Math. Ineq. 2(2008), 247-259.

[4] P. P. Korovkin, “Linear Operators and Approximation Theory”, Hindustan

Publ. Co., Delhi, (1960) 37.




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169

Pollution Detection for the Singular Linear Parabolic Equation

LAIB TELDJA AND AYADI ABDELHAMID

Laboratory of Dynamical Systems and Control, Larbi Ben Mhidi University,

OEB, Algéria,


Abstract: In this article we are concerned by the problem of identification of

pollution terms which arise in singular problems as for remote sensing problems,

and which are modeled by a linear singular parabolic equation. Some data are

missing, then the sentinel method of J.-L. Lions (1992) is changed and a new

sentinel is used; it is a particular least square method which permits to

distinguish between the missing terms and the pollution terms. In particular, the

new sentinel is given here in its more general and more realistic setting for

singular parabolic problems: in this case, the observation and the control have

their supports in different open sets. The problem of finding a new sentinel is

equivalent to finding singular optimality system of the least square control for

the parabolic equation that we solve.

Keywords: Singular heat equation, controlability, optimal control, sentinels

References:

[1] Lions J. L., Control of distributed singular systems, Bordas, Paris, (1985).

[2] Lions J. L., Contrôlabilité exacte, stabilisation et perturbations des

systèmes distribués, Vol. 1. Controlabilité exacte, Masson, Paris (1988).

[3] Lions J. L., Sentinelles pour les systèmes distribués à données

incomplètes, Masson, Paris (1992).

[4] Mizohata .S, Mem.coll.Sc.Univ.Kyoto,A 31,(3),1958,p.219-239.




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170

Relation of Some Special Vertices of Suborbital Graphs and

Continued Fractions Tuğba TUYLU, Ali Hikmet DEĞER, Ümmügülsün AKBABA


Trabzon, Turkey

[email protected]

[email protected]

[email protected]

Abstract: In this work, suborbital graphs for congruence subgroups Γ0(𝑛) of the

moduler group Γ and some properties have been studied. Some connections were

made between the special vertex values on the paths in these graphs and the

continued fractions. Moreover, this has been viewed as a special case of

continued fractions from Pringsheim’s theorem.

Keywords: Suborbital Graphs, Modular Group, Möbius Transformations.

References:

[1] G. A. Jones, D. Singerman, and K. Wicks, “The Modular Group and

Generalized Farey Graphs”, London Math. Soc. Lecture Note Ser., 1991, pp.

316-338.

[2] A. H. Değer, M. Beşenk, and B. Ö. Güler, “On Suborbital Graphs and

Related Continued Fractions”, Applied Math. and Computation, 2011, 746-750.

[3] C. C. Sims, “Graphs and Finite Permutation Groups”, Math. Zeitschr., 1967,

pp. 76-86.

[4] M. Akbaş, “On Suborbital Graphs for the Modular Group”, Bull. London

Math. Soc., 2001, pp. 647-652.

[5] P. M. Neumann, “Finite Permutation Groups, Edge-Coloured Graphs and

Matrices”, Topics in Group Theory and Computation, Academic Press, New

York, 1977.

[6] T. Tsukuzu, “Finite Groups and Finite Geometries”, Cambridge University

Press, 1982.

[7] N. L. Biggs, and A. T. White, “Permutation Groups and Combinatorial

Structures”, London Math. Soc. Lecture Note Ser., Cambridge, CUP, 33, 1982.

[8] A. Cuty and V. B. Petersen and B. Verdonk and H. Waadeland and W. B.

Jones, “Handbook of Continued Fractions for Special Functions”, Springer

Science + Business Media B. V., 2008.




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171

Bounds for Atom-bond Connectivity Index

Ahmet Uğur, Ayşe Dilek Maden

Department of Mathematics, Selcuk University,

Konya, Turkey

[email protected]

[email protected]

Abstract: Atom-bond connectivity (ABC) index, introduced by Ernesto Estrada

et al.[2], attracted a great deal of interest in mathematical chemistry. Kinkar Ch.

Das[1] gave some lower and upper bounds on ABC index of graphs. In this

work, we give new bounds for this summation index by using a different method.

Since ABC index is a function of two variables, its terms have monotone

behaviour on certain domains. Then, we can give possible upper and lower

bounds with the help of Comparison Theorem for Integrals.

Keywords: Molecular graph, Atom-bond connectivity index.

References:

[1] K. C. Das, “Atom-bond connectivity index of graphs”, Discrete Applied

Mathematics, 158(2010),1181-1188.

[2] E. Estrada, L. Torres, L. Rodriguez, I. Gutman, “An atom-bond connectivity

index:

modelling the entalpy of formation of alkanes”, Indian J. Chem., 37A (1998),

849-855.

[3] L. Gan, H. Hou, B. Liu, “Some results on atom-bond connectivity index of

graphs”, MATCH Commun.Math Comput. Chem., 66(2011), 669-680.

[4] W. Rudin, “Principal of Mathematical Analysis”, McGraw-Hill,Newyork,

1976.

[5] R. Xing, B. Zhou, Z. Du, “Further results on atom-bond connectivity index of

trees”, Discrete Applied Mathematics, 158(2010), 1536-1545.




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172

Travelling waves in diffusion, aggregation and diffusion-

aggregation-diffusion in a single population Judith Pérez Velázquez

Institute of Computational Biology,

Helmholtz Zentrum München, Germany

[email protected]

Faustino Sanchez-Garduno

Universidad Nacional Autonoma de Mexico

Faculty of Science, Department of Mathematics

Abstract: Reaction-diffusion equation (RDE) are, under specific conditions,

known to possess travelling wave solutions (TWS). These equations are

sometimes employed to model dispersion of individuals of one specie. Certain

species however, rather present aggregative behavior, which is also modeled

through a density dependent equation but with a negative term, also proved to

have TWS, see [1]. Furthermore, in diffusion-aggregation, one single population

is capable to present both modes. Here we revise these concepts in the context of

TWS and also present an example of a RDE where the diffusive part is a

nonlinear density dependent function and the growth rate is given by Fisher-KPP

type also call monostable reaction term. By using a biased random walks

approach, the diffusive part was derived by Turchin (see [2] and revisited by us

in [1])). By choosing the appropriate parameter values, the diffusive term is

capable of describing different types of population movements including: strong

diffusion, diffusion-aggregation-diffusion and strong aggregation. Because of

this, the RDE would have up to two points of degeneracy which, combined with

the negativity of the diffusion density-dependent coefficient, give rise to

interesting problems from both, mathematical and interpretative, points of view.

Keywords: travelling wave solutions, degeneracy, reaction/diffusion equations.

References:

[1] F Sanchez-Garduno, P.K. Maini and J. Perez-Velazquez. A nonlinear

degenerate equation for direct aggregation and traveling wave dynamics. Disc.

Cont. Dyn. Syst. Series B, Vol. 13, No. 2, pp. 455-487.2010.

[2] P Turchin (1989): Population consequences of aggregative movement. J. An.

Ecol., 58 , pp. 75-1




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173

On a Integral-Transform to Differential Equations

Ervenila Xhaferraj

Department of Mathematics, Faculty of Mathematics and Physics Engineering,

Polytechnic University of Tirana , Albania,

[email protected]

Abstract: In this paper we presented a bilateral integral transform which results

in the new Fourier-like integral transform of the function f(t) . In addition, some

properties of the NITF are explored . We present the new application of the

NITF integral transform to solve a steady heat transfer problem equation in this

work. The fundamental solutions of this problem with the initial conditions were

obtained with the use of the NITF integral transform . Keywords: NIFT integral transform, analytical solution, heat equation,

References:

[1] Beerends, R. J., et al., Fourier and Laplace Transforms, Cambridge

University Press, Cambridge, UK, 2003

[2] Dyke, P., An Introduction to Laplace Transforms and Fourier Series.

Springer, New York, USA, 2014

[3] John, W., Integral Transforms in Applied Mathematics, Cambridge

University Press, Cambridge, UK, 1971

[4] I. J. Schoenberg, “The integrability methods”, Amer. Math. Monthly,

66(1959), 361-375.

[5] A. Kashuri & A. Fundo & M. Kreku, “Mixture of a New Integral Transform

and Homotopy Perturbation Method for Solving Nonlinear Partial Differential

Equations”, Advances in Pure Mathematics, Vol. 3, No. 3 (2013), pp. 317 − 323..

[6] Yang, X. J., A New Integral Transform with an Application in Heat-Transfer

Problem, Thermal Science, 20 (2016), Suppl. 3, pp. S677-S681

[7] Yang, X. J. A New Integral Transform Method for Solving Steady Heat-

Transfer Problem, Thermal Science, 20 (2016), Suppl. 3, pp. S639-S642.




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174

Aczél Type Inequalities for Hilbert Space Operators

Ulaş Yamancı and Mehmet Gürdal


Isparta, Turkey

[email protected]; [email protected]

Abstract: Some Aczél type inequalities for Hilbert space operators are

established. Several particular cases of interest are given as well.

Keywords: Berezin symbol, Reproducing kernel Hilbert space, Aczel-type

inequalities.

References:

[1] J. Aczél, “Some general methods in the theory of functional equations in one

variable”, New applications of functional equations, Uspehi. Mat. Nauk (N. S.),

11(3)(1956), 3-68 (in Russian).

[2] J. Tian, “Reversed version of a generalized Aczél's inequality and its

application”, J. Inequal. Appl., 2012, 2012: 202.

[3] J. Tian and M.H HA, “Properties and refinemets of Aczél-type inequalities”,

Journal of Mathematical Inequalities, 12(1) (2018), 175-189.




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175

On the Artinianness of Top Local Cohomology Modules

Tuğba Yıldırım

Department of Mathematical Engineering, Istanbul Technical University,

Istanbul, Turkey

[email protected]

Abstract: Throughout, let R be a Noetherian ring, I an ideal of R, M an R-

module and let 𝐻 (𝑀)𝐼𝑖 denote the i-th local cohomology module of M with

support in I.

Local cohomology theory is an active area of research in commutative algebra,

algebraic geometry and related fields. One of the classical and long-standing

problem in this theory is to determine whether a given local cohomology module

is Artinian, or not [Third Problem, [1]]. It has been studied by several authors,

see eg. [2, 3, 4]. In this talk, we collect some results related to this problem,

particularly for the top local cohomology modules, 𝐻 (𝑀)𝐼 𝑐𝑑(𝐼,𝑀)

, where

𝑐𝑑(𝐼,𝑀) = 𝑠𝑢𝑝{𝑖 ∈ ℕ|𝐻 (𝑀)𝐼𝑖 ≠ 0}.

Keywords: Local cohomology, cohomological dimension, Artinian module.

References:

[1] C. Huneke, “Problems in local cohomology”, Free Resolutions in

Commutative Algebra and Algebraic Geometry Sundance 90, Res. Notes in

Math. Jones and Barlett 2 (1992) 93–108.

[2] M. Aghapournahr and L. Melkersson, “Artinianness of local cohomology

modules”, Ark. Mat. 52 (2014) 1–10.

[3] M. T. Dibaei and A. Vahidi, “Artinian and Non-Artinian Local Cohomology

Modules”, Canad. Math. Bull. 54 (2011) 619–629.

[4] M. Hellus and J. Stückrad, “Artinianness of local cohomology”. J. Commut.

Algebra, 1 (2009), no. 2, 269–274.




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176

q-Quasinormal Operators and Its Extended Eginvalues

Fatih YILMAZ, Meltem SERTBAŞ


Trabzon, Turkey


Abstract: In this paper, the relation between q-deformed quasinormal operators

and q-quasinormal operator classes is investigated. Moreover, we proof that

these are same. Also, we consider some extended eginvaule problems for q-

quasinormal operators.

Keywords: q-Quasinormal operator, q-Deformed operators, Extended

eigenvalue.

References:

[1] S. Ota, “Some classes of q-deformed operators”, J. Operator Theory.

48(2002), 151-186.

[2] S. Lohaj, “Quasi-normal operators”, Int. J. of Math. Analysis, 4(2010), no.

47, 2311-2320.

[3] S. Ota, F. H. Szafraniec, “q-Positive definiteness and related operators”, J.

Math. Anal. Appl., 329(2007), 987-997.

[4] A. Biswas, A. Lambert, S. Petrovic “Extended eigenvalues and the Volterra

operator”, Glasg. Math. J., 44.3(2002), 521-534.

[5] M.T. Karaev, “On extended eigenvalues and extended eigenvectors of some

operator classes”, Proc. Amer. Math. Soc., 134.8(2006), 2383-2392.




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177

A New Algorithm Based on Penalty Function Approach for

Inequality Constrained Global Optimization Problems

Nurullah Yılmaz and Ahmet Şahiner

Department of Mathematics, Süleyman Demirel University

Isparta, Turkey


Abstract: In this study, the nonlinear constrained global optimization ploblems

are handled. A new smoothing for exact penalty functions in constrained

optimization problems is invetigated. The new exact penalty function is

combined with the auxiliary function and an adapted algorithm for solving

nonlinear inequality constrained global optimization problems is presented.

Finally, the efficiency of the algorithm on some numerical examples is

illustrated.

Keywords: Constrained optimization, penalty function, smoothing approach.

References:

[1] S. S. Rao, “Engineering Optimization 4th ed.” John Wiley & Sons, New

Jersey (2009).

[2] M. C. Pinar, & S. Zenios, “On smoothing exact penalty functions for convex

constrained optimization”, SIAM Journal on Optimization, 4(1994), 468-511.

[3] S.J. Lian, “Smoothing approximation to l1 exact penalty for inequality

constrained optimization”, Appl. Math. Comput., 219 (2012), 3113-3121.

[4] A. Sahiner, G. Kapusuz, & N. Yilmaz, “A new smoothing approach to exact

penalty functions for inequality constrained optimization problems”, Numerical

Algebra Control and Optimimization, 6(2) (2016), 161-173.




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178

On Fuzzy Prime Hyperideals(Hyperfilters) in ∧ - Hyperlattices

Şerife YILMAZ1, Dilek BAYRAK2

1Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey

[email protected] 2Department of Mathematics, Tekirdağ Namık Kemal University, Tekirdağ, Turkey

[email protected]

Abstract: The purpose of this paper is the study of fuzzy prime hyperideals and

fuzzy prime hyperfilters in ∧-hyperlattices. Some properties of such hyperideals

are investigated. The relationship between fuzzy prime hyperideals and fuzzy

prime filters is discussed.

Keywords: ∧ - hyperlattice; hyperideal; hyperfilter.

References:

[1] O. Kazancı, B. Davvaz, “On the structure of rough prime (primary) ideals

and rough fuzzy prime (primary) ideals in commutative rings”, Information

Sciences, 178(2008), 1343-1354.

[2] R. Ameri, M. Amiri-Bideshki, A. B. Saeid, S. Hoskova-Mayerova, “Prime

filters of hyperlattices”, An. Şt. Univ. Ovidius Constanta. 24(2)(2016), 15-26.

[3] M. Amiri-Bideshki, R. Ameri, A. Borumand Saeid, “On Prime Hyperfilters

(Hyperideals) in ∧ - Hyperlattices”, European Journal of Pure and Applied

Mathematics, 11(1)(2018), 169-188.




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179

Korovkin Theory for Extraordinary Test Functions via Power

Series Method

Tuğba Yurdakadim, Emre Taş


Çorum, Turkey


Kırşehir, Turkey


Abstract: Korovkin subsets are the subsets of functions satisfying the same

property as {1, 𝑥, 𝑥2} and have been studied by many authors. In this talk we

introduce Korovkin subsets in the sense of power series method for T and

present a characterization theorem that a subset of 𝐶0(𝑋) is Korovkin subset in

the sense of power series method for T.

Keywords: Power series method, Korovkin subset.

References:

[1] F. Altomare and S. Diomede, “Contractive Korovkin subsets in weighted

spaces of continuous functions”, Rend. Circ. Mat. Palermo 50(2001), 547-568.

[2] J. Boos, “Classical and Modern Methods in Summability”, Oxford University

Press (2000).

[3] S. E. Takahasi, “Bohman-Korovkin-Wulbert operators on normed spaces”, J.

Approx. Theory 72(1993), 174-184.

[4] P. P. Korovkin, “Linear Operators and Approximation Theory”, Hindustan

Publ. Co., Delhi, (1960) 37.




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180

Criticality indices of 2-independence of some graphs

Mohamed Zemir and Ahmed Bouchou

Department of Mathematics and computer science

Yahia Fares University of medea, Algeria

[email protected]

[email protected]

Abstract: Let 𝐺 = (𝑉, 𝐸) be a simple graph of order n, and of sise m with vertex

set 𝑉(𝐺) and edge set 𝐸(𝐺), A subset 𝑆 of 𝑉 is 2-independent set if ∆(𝑆) < 2 ,

and the 2-independence number 𝛽2(𝐺) is the maximum cardinality of a 2-

independent set .

For a graphe 𝐺, we define the criticality index of an edge 𝑒 ∈ 𝐸(𝐺) as 𝑐𝑖𝛽2− (𝑒) =

𝛽2(𝐺) − 𝛽2(𝐺 − 𝑒), and the removal criticality index of a graph 𝐺 as 𝑐𝑖𝛽2− (𝐺) =

(∑ 𝑐𝑖𝛽2− (𝑒)��

𝑖=1 ) /𝑚(𝐺) .

Similarity we define the criticality index of an edge 𝒆 ∈ 𝑬(��) as 𝒄𝒊𝜷𝟐+ (𝒆) =

𝜷𝟐(𝑮) − 𝜷𝟐(𝑮 + 𝒆), and the adding criticality index of a graph 𝑮 as 𝒄𝒊𝜷𝟐+ (𝑮) =

(∑ 𝒄𝒊𝜷𝟐+ (𝒆)��

𝒊=𝟏 ) /𝒎(��) and �� is the complementary of the graph 𝑮.

In this work, now we determine the removal and the adding criticality indices of

2-independence of cycles and paths.

Keywords: 2-independence number, Criticality index.

References:

[1] J. F. Fink and M.S. Jacobson, “n-domination, n-dependence and forbidden

subgraphs in: Graph Theory with Applications to Algorithms and Computer

(kalamazoo, Mich.1984)eds. Alavi, Chartrand, Lesniak, Lick and wall,

Wiley, New-York (1985), 283–300.

[2] J. F. Fink and M.S. Jacobson, “n-domination, n-dependence and forbidden

subgraphs in: Graph Theory with Applications to Algorithms and Computer

(kalamazoo, Mich.1984)eds. Alavi, Chartrand, Lesniak, Lick and wall,

Wiley, New-York (1985), 301–311.

[3] M. Chellali and N. J. Rad, “On k-independence critical graphs,” Australasian

Journal of Combinatorics 53(2012), 289–298.

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