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Transcript of Book of Abstracts - 2015
4th INTERNATIONAL EURASIAN CONFERENCE ON
MATHEMATICAL SCIENCES AND APPLICATIONS
BOOKS OF ABSTRACTS
31 August - 03 september 2015 Athens-greece
FOREWORD
The “4th International Eurasian Conference on Mathematical Sciences and Applications
(IECMSA-2015)’’ jointly organized by Sakarya University, Kocaeli University, Bilecik
Seyh Edebali University, Turkic World Mathematical Society, University of Hassan II,
University of Mohammed V and International Balkan University, will be hold on 1
August--3 September 2015 in Athens, Greece.
The series of the International Eurasian Conference on Mathematical Sciences and
Applications provide communication between the members of the mathematics community,
interdisciplinary researchers, educators, statisticians and engineers. These conferences are
held every year in different countries with distinguished participants from all over the
world and they build agelong Cultural Bridges.
After the following three very successful international conferences the IECMSA-2012,
Prishtine, Kosovo, IECMSA-2013, Sarajevo, Bosnia and Herzegovina, IECMSA-2014,
Vienna, Austria, now IECMSA-2015, Athens, Greece, hosts 300 esteemed participants
from 28 different countries .
IECMSA-2015 has taken a lot of applications all over the world. These big numbers of
applications have given us opportunity to choose the best ones to reach the higher scientific
level. After having been reviewed by the distinguished members of International Scientific
Committee; 187 oral and 43 poster presentations have been accepted and the abstracts of
them have been presented in this book. Moreover, five worldwide distinguished speakers
have been invited to the conference and the abstracts of the plenary talks have been
substituted in this book, too. The electronic version of the abstracts of all presentations can
be found in the Conference Abstracts Book at www.iecmsa.org
I wish to thank all members of scientific committee and sponsors for their continued
support to the IECMSA-2015. And finally, I would like to sincerely thank all the
participants of IECMSA-2015 for contributing to this great meeting in many different
ways. I believe and hope that each of them will get the maximum benefit from the
conference. Welcome to Athens!
Prof. Dr. Murat TOSUN
Chairman
On behalf of the Organizing Committee
HONORARY COMMITTEE Prof. Dr. Muzaffer ELMAS (Sakarya University Rector) Prof. Dr. Azmi ÖZCAN (Bilecik Seyh Edebali University Rector) Prof. Dr. Sadettin HÜLAGÜ (Kocaeli University Rector) Prof. Dr. Idriss MANSOURI (Hassan University Rector) Prof. Dr. Saaid AMZAZI (Mohammed University Rector) Prof. Dr. Ismail KOCAYUSUFOĞLU (Intenational Balkan Univeristy Rector)
SCIENTIFIC COMMITTEE
Prof. Dr. Abdullah Aziz ERGİN (Akdeniz University) Prof. Dr. Adnan TERCAN (Hacettepe University) Prof. Dr. Alexander ZLOTNIK (National Research University Higher School of Economics) Prof. Dr. Arif SALIMOV (Ataturk University) Prof. Dr. Arkadii KIM (Institute of Mathematics and Mechanics of Russian Academy of Sciences) Prof. Dr. Bayram ŞAHİN (Inonu University) Prof. Dr. Bo Wun HUANG (Cheng Shiu University) Prof. Dr. Claudio Rodrigo Cuevas HENRIQUEZ (Federal University of Pernambuco) Prof. Dr. Danal O’REGAN (National University of Ireland) Prof. Dr. Ekrem SAVAŞ (Istanbul Commerce University) Prof. Dr. Emin ÖZÇAĞ (Hacettepe University) Prof. Dr. Emine MISIRLI (Ege University) Prof. Dr. Fethi ÇALLIALP (Dogus University) Prof. Dr. Fikret ALİYEV (Baku State University) Prof. Dr. George BOGOSLOVSKY (Moscow State University) Prof. Dr. Gennady A. LEONOV (Saint-Petersburg State University) Prof. Dr. H. Hilmi HACISALİHOĞLU (Bilecik Seyh Edebali University) Prof. Dr. Halis AYGÜN (Kocaeli University) Prof. Dr. Hellmuth STACHEL (Vienna Technical University) Prof. Dr. Jan Van MILL (VU Amsterdam University)
Prof. Dr. Kadri ARSLAN (Uludag University) Prof. Dr. Kailash Chander MADAN (Ahlia University) Prof. Dr. Kil Hyun KWON (Korea Advanced Institute of Science and Technology) Prof. Dr. Khalil EZZENBI (Cadi Ayyad University) Prof. Dr. Laszlo LEMPERT (Purdue University) Prof. Dr. Mahir RESULOV (Beykent University) Prof. Dr. Mahmut ERGÜT (Firat University) Prof. Dr. Maksymilian DRYJA (Waraw University) Prof. Dr. Muhammad Rashid Kamal ANSARI (Sir Syed University of Engineering and Technology) Prof. Dr. Mukut Mani Tripathi (Banaras Hindu University) Prof. Dr. Murat ALTUN (Uludag University) Prof. Dr. Mustafa ÇALIŞKAN (Gazi University) Prof. Dr. Muvarskhan JENALIYEV (Institute of Mathematics and Mathematical Modeling) Prof. Dr. Müjgan TEZ (Marmara University) Prof. Dr. Naime EKİCİ (Cukurova University) Prof. Dr. Nuri KURUOĞLU (Bahcesehir University) Prof. Dr. Ömer AKIN (TOBB University of Economy and Technology) Prof. Dr. Pedro MACEDO (University of Aveiro) Prof. Dr. Prasanta SAHOO (Jadavpur University) Prof. Dr. Ram N. MOHAPATRA (University of Central Florida ) Prof. Dr. Reza LANGARI (Texas A&M University) Prof. Dr. Roberto BARRIO (University of Zaragoza) Prof. Dr. Sadık KELEŞ (Inonu University)
Prof. Dr. Saeid ABBASBANDY (Imam Khomeini International University) Prof. Dr. Sergeev Armen GLEBOVIC (Steklov Mathematical Institute) Prof. Dr. Snezhana HRISTOVA (Plovdiv University) Prof. Dr. Toka DIAGANA (Howard University) Prof. Dr. Walter RACUGNO (University of Cagliari) Prof. Dr. Wolfgang SPROESSING (Freiberg University of Mining and Technology) Prof. Dr. Varga KALANTAROV (Koc University)
ORGANIZING COMMITTEE
Prof. Dr. Murat TOSUN (Sakarya University) Prof. Dr. Cihan ÖZGÜR (Balikesir University) Prof. Dr. Kazım İLARSLAN (Kirikkale University) Prof. Dr. Ahmet KÜÇÜK (Kocaeli Univeristy) Assoc. Prof. Dr. Erdal ÖZÜSAĞLAM (Aksaray University) Assoc. Prof. Dr. Mehmet Ali GÜNGÖR (Sakarya University) Assoc. Prof. Dr. Melek MASAL (Sakarya University) Assoc. Prof. Dr. Nesip AKTAN (Necmettin Erbakan University) Assoc. Prof. Dr. Soley ERSOY (Sakarya University) Assoc. Prof. Dr. Şevket GÜR (Sakarya University) Assist. Prof. Dr. Ayşe Zeynep AZAK (Sakarya University) Assist. Prof. Dr. Mahmut AKYİĞİT (Sakarya University) Assist. Prof. Dr. Mehmet GÜNER (Sakarya University) Assist. Prof. Dr. Murat SARDUVAN (Sakarya University) Dr. Canay AYKOL YÜCE (Ankara University)
SECRETARIAT Prof. Dr. Murat TOSUN (Sakarya University) Research Assist. Hidayet Hüda KÖSAL (Sakarya University) Research Assist. Tülay ERİŞİR (Sakarya University) Researcher Selman HIZAL (Sakarya University)
CHAPTERS
INVITED TALKS………………………………………………………………….. 22
ALGEBRA…………………………………………………………………………... 28
ANALYSIS………………………………………………………………………….. 55
APPLIED MATHEMATICS……………………………………………………. 97
DISCRETE MATHEMATICS………………………………………………….. 168
GEOMETRY………………………………………………………………………... 173
MATHEMATICS EDUCATION………………………………………………. 221
STATISTICS………………………………………………………………………... 236
TOPOLOGY………………………………………………………………………… 245
THE OTHER AREAS…………………………………………………………….. 268
POSTERS……………………………………………………………………………. 281
PARTICIPIANTS………………………………………………………………….. 330
CONTENTS
INVITED TALKS
Authenticated Encryption Based on Prime Moduli
D. Giri..............................................................................................................................
22
An Algebraic Description of Gradient Descent Decoding
E. Martínez-Moro...........................................................................................................
23
Elliptic Diophantine Equations and the Elliptic Logarithm Method
N. Tzanakis.....................................................................................................................
24
Möbius Transformations and the Circle-Preserving Property
N. Yılmaz Özgür..............................................................................................................
26
Advances in Frames, Riesz Bases and Frames on Hilbert *C Modules
R. N. Mohapatra.............................................................................................................
27
- 1 -
ALGEBRA
Representations of a Leavitt Path Algebra with Coefficients in an Abelian Category
A. Koç, M. Özaydin.........................................................................................................
28
Some Relations via K -Balancing Numbers
A. Özkoç, A. Tekcan.......................................................................................................
29
Vertex-Decomposability and Depth Bounds of Independence Complexes
A. Ülker, T. Öner............................................................................................................
30
On Generalized FI-Extending Modules
C. Celep Yücel ................................................................................................................
31
An Explicit Homomorphism from the Complex Clifford Algebra 2nCl to the Cuntz
Algebra 2nO
D. Çelik............................................................................................................................
32
Circulant Matrices in Terms of Tetranacci Numbers
E. Ardıyok, A. Özkoç......................................................................................................
33
A Symmetric Key Fully Homomorphic Encryption Scheme using General Chinese
Remainder Theorem
E. Aygün, E. Lüy............................................................................................................
34
Soft Radicals
E. Aygün, A. O. Atagün, B. Erdal..................................................................................
36
Fixed Points on NG-Groups
F. A. Abdunabi................................................................................................................
38
Invariant Ring of ,Aut V H
F. Hussain.......................................................................................................................
39
Solution of a Matrix Completion Problem and Nil-Clean Companion Matrices
G. C. Modoi.....................................................................................................................
40
A Presentation of Free Lie Algebra '
3
FF
G. Kaya Gök, N. Ekici....................................................................................................
41
- 2 -
Centering Nonlinear Fuchsian Codes
M. Alsina .........................................................................................................................
42
On the Codes over the Ring 2
2
222 IFuuIFvIFIF
M. Özkan, F. Öke............................................................................................................
43
Some Special Codes Over 2
3 3 3 3v u u
M. Özkan, F. Öke............................................................................................................
44
Minimaxness and Cofiniteness Properties of Local Cohomology Modules
M. Sedghi........................................................................................................................
45
Embeddings of Solvable Productsi Matrix Algebras and Generalized Fox Derivatives
N. Ş. Öğüşlü, N. Ekici....................................................................................................
46
Rank Properties of the Direct Product of Finite Cyclic Groups
O. Kelekci........................................................................................................................
47
A Note on A Category of Cofinite Modules which is Abelian
R. Naghipour..................................................................................................................
48
The Values of a Class of Direchlet Series at the Non Positive Integers
S. Boualem......................................................................................................................
49
On Congruences with the Terms of Second order Sequence and Harmonic Numbers
S. Koparal, N. Ömür.......................................................................................................
50
On Commuting Automorphisms of Some Finite P-Groups
S. Singh...........................................................................................................................
51
Permuting n-f-Derivations on Lattices
Ş. Ceran, U. Pehlivan.....................................................................................................
52
Automorphisms of the Lie Algebras Related with 2x2 Generic Matrices
Ş. Fındık..........................................................................................................................
54
ANALYSIS
On the Ambarzumyan's Theorem for the Quasi-Periodic Boundary Conditions
A. A. Kıraç.......................................................................................................................
55
- 3 -
Quasi-Partial b-Metric Spaces and some Related Fixed Point Theorems
A. Gupta, P. Gautam......................................................................................................
56
On Transformation Types of the 0 N and the Suborbital Graph ,Fu N
A. H. Değer.....................................................................................................................
57
Boundedness of Fractional Maximal Operator Associated with Hankel Transform on
Weighted Lorentz Spaces
C. Aykol Yüce..................................................................................................................
58
Multipliers with Natural Spectra on Commutative Banach Algebras
C. Temel..........................................................................................................................
59
I -Double Statistically Convergent Sequences in Topological Groups
E. Savaş ..........................................................................................................................
61
Absolute Tauberian Constants
F. Aydın Akgün, B. Rhoades..........................................................................................
62
On Convex Meromorphic Functions
F. Uçar , Y. Avcı .............................................................................................................
63
The Relation between B-1
-Convex and Convex Functions
G. Adilov, I. Yeşilce, G. Tınaztepe.................................................................................
64
On Some Inequalities and Their Refinements
G. Tınaztepe, R. Tınaztepe, S. Kemali............................................................................
65
A Generalized Mountain Pass Theorem
H. J. Ruppen.......................................................................................................
66
A New Result on the almost Increasing Sequences
H. S. Özarslan, A. Karakaş...........................................................................................
67
Bi-Parametric Potentials and Their Inverses with the Aid of Wavelet-Type
Transforms
I. A. Aliev........................................................................................................................
69
Multi-Point Boundary Value Problems of Higher-Order Nonlinear Fractional
Differential Equations
İ. Yaslan.........................................................................................................................
71
- 4 -
Impulsive Boundary Value Problem for Nonlinear Differential Equations of
Fractional Order (2, 3]
İ. Yaslan Karaca, F. Tokmak Fen................................................................................
72
Some Coupled Fixed Point Theorems for Generalized Contractions
İ. Yıldırım........................................................................................................................
73
Characterizing - Limit Sets for Analytic Vector Fields in Open Subsets of the
Sphere
J. G. E. Buendía, V. J. López.........................................................................................
74
Weighted Composition Followed by Differentiation between Weighted Fréchet
Spaces of Holomorphic Functions
J. S. Manhas...................................................................................................................
75
The Effects of Privatisation and Competition on Malaysia Airlines Performance
K. F. See, A. A. Rashid ..................................................................................................
76
New Integral Inequalities via Ga-Convex Functions
M. Avcı Ardıç, A. Ocak Akdemir, E. Set........................................................................
77
On Multistep Iteration Method for Contractive Condition of Integral Type in Banach
Spaces
M. A. Maldar, V. Karakaya............................................................................................
78
An Inequality and Its Applications
M. A. Sarıgöl...................................................................................................................
79
Connectedness of Suborbital Graphs of Some Modular Groups
M. Beşenk........................................................................................................................
80
A New Perspective on Paranormed Riesz Sequence Space of Non-Absolute Type
M. Candan......................................................................................................................
81
Almost 𝐼-Convergent Sequence Spaces Defined by Orlicz Function
M. Daştan, M. İlkhan, E. E. Kara..................................................................................
82
On Exton's q -Exponential Function
M. J. S. Belaghi, N. Kuruoğlu........................................................................................
83
Some Properties of q -Exponential and q -Trigonometric Functions
M. J. S. Belaghi, N. Kuruoğlu........................................................................................
84
- 5 -
On a Volterra Equation of the Second Kind with Spectral Parameter
M. M. Amangaliyeva, M. T. Jenaliyev, M. T. Kosmakova, M. I. Ramazanov.............
85
On Some Fixed Point Results for Rational A -Contractive Mappings in 2-Metric
Spaces Endowed with Partial Order
M. Öztürk........................................................................................................................
86
An Approach to the Stability of Nonlinear Volterra Integral Equations
N. Eghbali.......................................................................................................................
87
Abel Convergence of Convolution Operators
Ö. Girgin Atlıhan, M. Ünver..........................................................................................
88
On Weighted Approximation by Singular Integral Operators Depending on Two
Parameters
Ö. Güller, G. Uysal, E. İbikli..........................................................................................
89
Lyapunov Functions and Strict Stability of Caputo Fractional Differential Equations
R. Agarwal, S. Hristova, D. O’Regan............................................................................
90
On a Caputo Type Fractional Operator
R. Almeida, A. Malinowska, T. Odzijewicz....................................................................
91
Chaotic Behavior of Logistic Maps in Mann Orbit
R. Chugh, Ashish............................................................................................................
92
Interpolation of Function Spaces Associated to a Vector Measure
R. del Campo, A. Fernández, F. Mayoral, F. Naranjo.................................................
94
Complex Interpolation Operators and Optimal Domains
R. del Campo, A. Fernández, O. Galdames, F. Mayoral, F. Naranjo..........................
95
q -Analogues of Some Classical Tauberian Theorems for Cesàro Integrability
S. A. Sezer, İ. Çanak, Ü. Totur.......................................................................................
96
APPLIED MATHEMATICS
Numerical Results of Extended Lane–Emden Type Equations
A. Akgül, A. Kılıçman, M. İnc........................................................................................
97
Inventory Model of Type (s,S) with Regularly Varying Demands Having Infinite
Variance
A. Bektaş Kamışlık, T. Kesemen, T. Khaniyev..............................................................
98
- 6 -
Inverse Spectral Problem for Sturm Liouville Operator with Spectral Parameter
Dependent Boundary Condition
A. Çöl...............................................................................................................................
100
Incorporating Views on Market Dynamics in Options Hedging
A. E. Zambelli.................................................................................................................
101
Numerical Identification of the Filtration Capacitive Parameters in Two-Phase
Petroleum Reservoirs
A. Sakabekov, D. Ahmed Zaki, Y. Auzhani...................................................................
102
Multiple Scales Analysis and Exact Solutions for KdV Type Nonlinear Differential
Equations
B. Ayhan, M. N. Özer, A. Bekir......................................................................................
104
Numerical Solution of the Rosenau KdV-RLW Equation by Using Collocation
Method
B. Korkmaz, Y. Dereli.....................................................................................................
105
Finite Difference Method for Film Equation in a Class of Discontinuous Functions
B. Sinsoysal, M. Rasulov, E. I. Şahin............................................................................
107
Analysis of Sound Diffraction from a Duct with Exterior Surface Impedance
B. Tiryakioğlu, A. Demir................................................................................................
108
The Comparison of the Eigenvalues of Sturm-Liouville Operators
B. Yılmaz.........................................................................................................................
109
Intuitionistic Fuzzy Optimization Technique in Multi-Commodity Solid
Transportation Models
D. Rani, T. R. Gulati.......................................................................................................
110
On a Disjoint Idempotent Decomposition for Linear Combinations Produced from n
Commutative Tripotent Matrices
E. Kişi, H. Özdemir.........................................................................................................
111
Respect to Two Spectra Stability of the Inverse Problem for Diffusion Equation
E. S. Panakhov, A. Ercan, T. Gülsen............................................................................
112
Spectral Problems for Regular Canonical Dirac Systems with More General
Separable Boundary Conditions
E. S. Panakhov, M. Babaoğlu........................................................................................
114
- 7 -
Generalized Second-Order Composed Radial Epiderivatives
G. İnceoğlu......................................................................................................................
116
Equations of Anisotropic Elastodynamics for 1 Dimensional Qcs as a Symmetric
Hyperbolic System: Deriving the Time-Dependent Fundamental Solution
H. Çerdik Yaslan.............................................................................................................
117
Equivalence Relations on Quaternion Matrices
I. Arda Kösal, H. H. Kösal.............................................................................................
118
The Fracture of the Elastic Matrix Containing Two Neighboring Co-phase
Periodically Curved Carbon Nanotubes
İ. Gülten, R. Köşker........................................................................................................
119
On Homoclinic Structure for 2D Coupled Nonlinear Schrödinger System
İ. Hacinliyan, C. Babaoğlu............................................................................................
121
Finite Difference Approximation for Solving Quasilinear Nonlocal Problem with
Boundary Layers
M. Çakır..........................................................................................................................
122
Inverse Spectral Problems for Energy-Dependent Sturm-Liouville Equations with
Finitely Many Point 𝛿 −Interactions
M. Dzh. Manafov............................................................................................................
123
Effect of a Surface Asperity at the Nanoscale
M. Grekov, S. Kostyrko...................................................................................................
124
Conservation Laws and Exact Solutions of Nonlinear Differential Equation
M. Kaplan, A. Akbulut, F. Taşcan.................................................................................
125
Stability of One Nonlinear System with Delay
M. L. Stanislavovich, P. A. Vital'evich..........................................................................
126
Control Problems of Nonlinear Phase Systems
M. N. Kalimoldayev, M. T. Jenaliyev.............................................................................
128
Interchange of Mass after a Close Encounter between Galaxies
M. Ollé, E. Barrabés, J.M. Cors.....................................................................................
129
One a Numerical Method of Riemann Type Problem for 2D Conservation Laws in a
Class of Discontinuous Functions
M. Rasulov, H. Bal, E. Boran........................................................................................
130
- 8 -
On Two-Dimensional Nonsteady Free Convection near Vertical Plate subject to
Stepped-up Surface Temperature
N. C. Sacheti, P. Chandran, A. K. Singh.......................................................................
131
An Alternative Technique for Solving Ordinary Differential Equations
N. Dernek, F. Aylıkçı, S. Kıvrak....................................................................................
132
New Identities for the Generalized Glasser Transform, the Generalized Laplace
Transform and the 2 ,1nE -Transform
N. Dernek, F. Aylıkçı, G. Balaban.................................................................................
135
Estimating Elasticity Modulus of the Piezo Ceramic Disc Using Basic Mathematical
Modelling
N. Elmas, L. Paralı, A. Sarı, J. Pechousek, F. Latal....................................................
139
Oscillation Results for a Class of Fourth-order Nonlinear Differential Equations with
Positive and Negative Coefficients
N. Kılınç Geçer, P. Temtek.............................................................................................
141
Different Methods for Nonlinear Fractional Differential Equation
Ö. Güner, A. Bekir, A. C. Çevikel, E. Aksoy..................................................................
143
On Solutions of Some Second-Order Ordinary Differential Equations by Jacobi Last
Multiplier Method
Ö. Orhan, T. Özer...........................................................................................................
145
A Gravitational Model with Y(R)F2-Type Coupling
Ö. Sert............................................................................................................................
146
Some Aspects of Natural Convection in a Hydrodynamically and Thermally
Anisotropic Porous Non-Rectangular Cavity
P. Chandran, N. C. Sacheti, A. K. Sing, B. S. Bhadauria............................................
147
Classic Metaheuristics and Evolutionary Optimization Algorithms for Routing
Problems: A Computational Study
P. Z. Lappas, M. N. Kritikos, G. D. Ioannou.................................................................
148
The pL Hardy Inequality with Two Weight Functions and Its Improved Versions
S. Ahmetolan, İ. Kombe..................................................................................................
150
Static Analysis of Euler-Bernouilli Beams Resting on Foundation of Pasternak and
Winkler using Differential Transformation Method
S. Bodur, K. B. Bozdoğan, G. Oturanç..........................................................................
151
- 9 -
Octonic Formulation of Dyons for Gravi-Electromagnetism
S. Demir, M. Tanışlı, M. E. Kansu................................................................................
152
Evaluation of Soybean Hydration Model with Volume Variation
S. Gülen, T. Öziş.............................................................................................................
154
Stabilization of Solutions of Linear Volterra Implicit Integro-Differential Equation of
the First Order
S. Iskandarov..................................................................................................................
155
Existence of a Solution for a General Class of Fredholm Integral Equations via F-
contractive Non-Self-Mappings
S. K. Shahkhali...............................................................................................................
157
Interval Oscillation Criteria for Functional Differential Equations of Fractional Order
S. Öğrekçi........................................................................................................................
158
Mathematical Aspects of Molecular Replacement: The Structure of Chiral Space
Groups Preferred by Proteins
S. Sajjadi, G. S. Chirikjian.............................................................................................
159
A New Extended Method for Solving Some Fractional Order Evolution Equations
S. M. Ege, E. Mısırlı.......................................................................................................
160
A Nonclassical and Nonautonomous Diffusion Equation with Infinite Delay
T. Caraballo, A. M. Márquez Durán.............................................................................
161
On a Spectral Expansion for Non-selfadjoint Boundary Value Problem
V. Ala, K. R. Mamedov...................................................................................................
163
Matrices with Null Columns in First Order Chemical Kinetics Mechanisms
V. Martinez-Luaces........................................................................................................
164
Stability of ODE Systems Associated to First order Chemical Kinetics Mechanisms
without Final Products
V. Martinez-Luaces........................................................................................................
165
A Two-Level Method for Emulating Parameterized Dynamic Partial Differential
Equation Models
W. Xing, V. Triantafyllidis, A. Shah, P. B. Nair, N. Zabaras.......................................
166
The Numerical Solution of the Symmetric RLW Equation by Using the Meshless
Kernel Based Method of Lines
Y. Dereli..........................................................................................................................
167
- 10 -
DISCRETE MATHEMATICS
One-Parameter Apostol-Genocchi Polynomials
B. Kurt.............................................................................................................................
168
The Computational Complexity of Some Domination Parameters
N. J. Rad..........................................................................................................................
169
Construction of 4-Connected Graphic Matroids with Essential Elements
P. P. Malavadkar, M. P. Gadiya, S. B. Dhotrey, M. M. Shikare..................................
171
Unified One-Parameter Apostol-Bernoulli, Euler and Genocchi Polynomials
V. Kurt.............................................................................................................................
172
GEOMETRY
Anti-Kahler-Codazzi Structures on Walker Manifolds
A. Salimov, S. Turanlı....................................................................................................
173
Classification of Geodesics on Sierpinski Gasket with the Intrinsic Metric
B. Demir, Y. Özdemir, M. Saltan ..................................................................................
175
On Ricci Solitons in Kenmotsu Manifolds with the Semi-Symmetric Non-Metric
Connection
C. Ekici, H. B. Çetin.......................................................................................................
176
Metallic Shaped Hypersurfaces in Lorentzian Space Forms
C. Özgür, N. Yılmaz Özgür.............................................................................................
177
Homothetic Cayley Formula for Homothetic Motions around a Timelike Axis and its
Applications in Lorentzian Space
D. Ünal, M. A. Güngör, M. Tosun.................................................................................
178
Screen Scalar Curvature in Screen Locally Conformal Coisotropic Lightlike
Submanifolds of a Semi-Euclidean Space
E. Kılıç, S. Keleş, M. Gülbahar......................................................................................
179
The Fermi-Walker Derivative on the Spherical Indicatrix
F. Karakuş, Y. Yaylı........................................................................................................
180
- 11 -
On Hypersurfaces of Indefinite Quaternionic Kaehler Manifolds
G. Aydın Şekerci, S. Sevinç, A. Ceylan Çöken..............................................................
181
On Holomorphically Projective Curvature Tensor in a Kahler-Weyl Space
G. Çivi..............................................................................................................................
182
Characterization of Null Scrolls in m n
v
G. Güner, F. N. Ekmekci................................................................................................
183
On Affine Translation Surfaces
H. Gün Bozok, M. Ergüt................................................................................................
184
On Quasi-Einstein Weyl Manifolds
İ. Gül, E. Özkara Canfes................................................................................................
185
On Constant Ratio Curves According to Bishop Frame in Minkowski 3-Space 3
1E
İ. Kişi, G. Öztürk.............................................................................................................
186
On Canal Surfaces According to Parallel Transport Frame in Euclidean Space 4E
İ. Kişi, G. Öztürk, K. Arslan...........................................................................................
188
Complete System of Polynomial Invariants of Vectors in the Pseudo-Euclidean
Geometry of Index 1
İ. Ören.............................................................................................................................
189
On Classification of Canal Surfaces in Minkowski 3-space
K. İlarslan, A. Uçum.......................................................................................................
190
A Note on Matrix Representations of Split Quaternions
M. Akyiğit, H. H. Kösal, M. Tosun................................................................................
191
On Integral Invariants of Parallel Ruled Surfaces with Darboux Frame
M. Çimdiker, Y. Ünlütürk, C. Ekici...............................................................................
192
Directional Bertrand Curves
M. Dede, C. Ekici............................................................................................................
193
On the Parallel Ruled Surfaces in Galilean Space
M. Dede, C. Ekici............................................................................................................
194
Directional Tubular Surfaces
M. Dede, C. Ekici, H. Tozak...........................................................................................
195
- 12 -
Special Proper Pointwise Slant Surfaces of an Almost Constant Curvature Manifold
M. Gülbahar, E. Kılıç, S. Saraçoğlu Çelik....................................................................
197
Self-Duality Operator on 2-Forms
N. Değirmenci, H. Zeybek..............................................................................................
198
Pointwise Slant Submersions from almost Contact Metric Manifolds
S. Aykurt Sepet, M. Ergüt...............................................................................................
199
On Constant Ratio Curves in Galilean Spaces
S. Büyükkütük, İ. Kişi, G. Öztürk..................................................................................
201
On Z-Projective Change of Kropina Spaces
S. Ceyhan, G. Çivi...........................................................................................................
202
Seiberg-Witten Equations on 8-Dimensional Manifolds
S. Eker, N. Değirmenci...................................................................................................
203
Bicomplex Fibonacci Numbers
S. Kaya Nurkan, İ. Arslan Güven..................................................................................
204
A New Version of Bishop Frame and Position Vector of a Timelike Curve in
Minkowski 3-Space
S. Yılmaz, Y. Ünlütürk, A. Mağden................................................................................
205
On f-Biharmonic Submanifolds in Space Forms
S. Yüksel Perktaş, E. Kılıç, S. Keleş...............................................................................
207
An Examination on the Frenet Ruled Surfaces along the Bertrand Pairs Α and Α∗,
according to Their Normal Vector Fields in Euclidean 3-Space
Ş. Kılıçoğlu, S. Şenyurt, H. H. Hacısalihoğlu...............................................................
208
The Steiner Formula and the Holditch Theorem for Homothetic Motions in the
Generalized Complex Plane
T. Erişir, M. Ali Güngör, M. Tosun...............................................................................
210
Relaxed Elastic Line of Second Kind on an Oriented Surface in the Galilean Space
T. Şahin...........................................................................................................................
211
Centro-Equiaffine Differential Invariants of Curve Families
Y. Sağıroğlu....................................................................................................................
212
- 13 -
On Parallel Ruled Surfaces with Darboux Frame
Y. Ünlütürk, M. Çimdiker, C. Ekici...............................................................................
213
On the Metrics of Some Archimedean and Catalan Solids
Z. Can, Z. Çolak..............................................................................................................
214
Tetrakis Hexahedron Space Isometry Group
Z. Çolak, Z. Can..............................................................................................................
216
Conformal Weyl-Euler-Lagrange Equations on Lorentzian Trans and Para Sasakian
Manifolds
Z. Kasap..........................................................................................................................
218
Euler-Lagrange Equations on Almost Paracontact Metric Manifolds
Z. Kasap..........................................................................................................................
220
MATHEMATICS EDUCATION
Examining Pre-Service Mathematics Teachers’ Math Literacy and Their Attitudes
towards Mathematics Education Courses
D. Çağırgan Gülten........................................................................................................
221
The Inclusion of Geometric Thinking in Elementary School Mathematics Textbooks
D. Özen, N. Yavuzsoy Köse.............................................................................................
222
Examination of 6th
Students’ Quantitative Reasoning Skills and Developments in
Their Problem Solving Process
D. Tanışlı, M. Dur..........................................................................................................
223
The Investigation of Using Mathematical Language of 7th
Graders when Identifying
Circle and It’s Elements
E. Akarsu, S. Yılmaz.......................................................................................................
225
Linear Algebra with SAGE
E. Özüsağlam..................................................................................................................
226
Views of High School Mathematics Teachers and Students on Computer-Assisted
Mathematics Instruction: Mathematica Case
M. A. Ardıç, T. İşleyen....................................................................................................
227
- 14 -
Mental Components of Mathematical Literacy Success of Secondary School Students
M. Altun, I. Bozkurt........................................................................................................
229
The Competence of Students in Understanding the Properties of a Function from Its
Graph
N. Mahir..........................................................................................................................
230
Reasoning in Mathematics Education
N. Yavuzsoy Köse............................................................................................................
231
Concept Association of Prospective Elementary and Secondary Mathematics
Teachers
Ş. Güray, Ş. Kılıçoğlu, M. Koştur .....................................................................................
232
Investigating the Concept Knowledge of Prospective Elementary and Secondary
Mathematics Teachers
Ş. Kılıçoğlu, Ş. Güray, M. Koştur.....................................................................................
234
STATISTICS
Asymptotic Results for a Semi-Markovian Random Walk with Generalized Beta
Interferences
C. Aksop, T. Khaniyev....................................................................................................
236
A Queueing System Equipped with Two Components Subject to Random Failures
and Heterogeneous General Service with Fluctuating Rates Depending on State of the
Components
K. C. Madan....................................................................................................................
237
Higher-Order Adjustments of the Signed Scoring Rule Root Statistic
L. Ventura, V. Mameli, M. Musio...................................................................................
238
In the Ridge Regression Method, ‘k Point Estimation Method’ (Approach) in the
Estimation of the Parameter ‘‘k’’
M. Kurtuluş.....................................................................................................................
239
A Comparison of Information Criteria in Clustering based on Mixture of Multivariate
Normal Distributions
S. Akoğul, M. Erişoğlu...................................................................................................
240
The Performance of the Optimal Extended Balanced Loss Function Estimators and
Predictors
S. Kaçıranlar, I. Dawoud................................................................................................
241
- 15 -
Stein-Rule Restricted Ridge Regression Estimator
S. Sakallıoğlu, S. Kaçıranlar..........................................................................................
242
On the Stationary Characteristics of a Renewal Reward Processes with Generalized
Reflecting Barrier
T. Khaniyev, B. Gever, Z. Mammadova.........................................................................
243
TOPOLOGY
Ideal Rothberger Spaces
A. Güldürdek...................................................................................................................
245
A New Almost Continuity for -b-Open Sets
A. Keskin Kaymakcı........................................................................................................
246
Cardinal invariants of the Vietoris Topology on the Space of Minimal CUSCO Maps
B. Novotný, L. Holá........................................................................................................
248
Fg Morphisms and Their Some Properties
C. S. Elmalı, T. Uğur......................................................................................................
249
Simplicial Cohomology Rings of a Connected Sum of Minimal Simple Closed
Surfaces
G. Burak, İ. Karaca........................................................................................................
250
Introduction to Disoriented Knot Theory
İ. Altıntaş.........................................................................................................................
251
Digital Homotopy Fixed Point Theory
İ. Karaca, Ö. Ege............................................................................................................
253
Some Classes of Functions between Continuous and Quasicontinuous Functions
J. Borsík..........................................................................................................................
254
HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials
K. Taşköprü, İ. Altıntaş..................................................................................................
255
Minimal Usco and Minimal Cusco Maps and Compactness
L. Holá, D. Holý.......................................................................................................
257
On -P-Open Sets in Bitopological Spaces
M. İlkhan, M. Akyiğit, E. E. Kara..................................................................................
258
- 16 -
Invariant Measures and Controllability of Affine Control Systems
M. Kule............................................................................................................................
259
Soft Topological Questions and Answers
M. Matejdes.....................................................................................................................
260
A Fixed Point Criterion for 𝑝-adic Actions
M. Onat...........................................................................................................................
262
Complex valued dislocated metric spaces and an application to differential equations
Ö. Ege, İ. Karaca............................................................................................................
263
Principal P-adic Bundles over Circle Groups
S. Eren.............................................................................................................................
264
Topological Group-2-Groupoids and Topological 2G-Crossed Modules
S. Temel, N. Alemdar......................................................................................................
265
Textures and Approximation Spaces
Ş. Dost .............................................................................................................................
266
THE OTHER AREAS
A Numerical Analysis of Non Newtonian Flow through Microchannels
A. Chanda.......................................................................................................................
268
Beautiful number 6174
A. Rezafadaei..................................................................................................................
269
Density Functional Theory in the Solid State Science
E. Deligöz, H. Özışık.......................................................................................................
270
Mathematical Ratio in Painting
E. Gürsoy, S. Kaya Nurkan............................................................................................
271
Non-linear Analysis of Rotating Cracked FGM Beams
G. Pohit...........................................................................................................................
272
Different Viewpoint for Puzzle Problems as Artificial Intelligence Toy Problems:
Heuristic Angular Metric Approach
İ. Ö. Bucak, M. Tatlılıoğlu.............................................................................................
273
- 17 -
Development of Mathematical Model using Group Contribution Method to Predict
Exposure Limit Values in Air for Safeguarding Health
M. El-Harbawi, P. T. K. Trang......................................................................................
274
The Way Ahead for Bug-fix time Prediction
M. Sharma......................................................................................................................
275
Construction of Lossless Broadband Matching Networks with Lumped Elements
M. Şengül........................................................................................................................
277
American options with regime-switching uncertainty
S. D. Jacka, A. Ocejo......................................................................................................
278
Designed Filter with the New Generation Current Conveyor and Analysis of Ecg
Ş. Kitiş, E. Köklükaya, R. Güntürkün............................................................................
279
A Study on Mathematical Modelling for Open Source Software Optimal Release
Planning
V. B. Singh......................................................................................................................
280
POSTERS
Equivariantly Formal Solenoidal Actions
A. A. Özkurt....................................................................................................................
281
An Extension of Srivastava's Triple Hypergeometric Function CH
A. Çetinkaya, M. B. Yağbasan, İ. O. Kıymaz.................................................................
282
Construction of Independent Spanning Trees for Bi-Rotator Graphs
C. J. Lee, C. C. Hsu, Y. T. Tsai, Y.-C. Chu....................................................................
283
Pre-service Mathematics Teachers’ Views of Language in Mathematics Teaching and
Their Math Literacy Self-Efficacies
D. Çağırgan Gülten, Y. Yaman......................................................................................
285
Fixed Points of Involutions in a Lie Algebra of the Form 𝐹/𝑅
D. Ersalan, N. Ekici........................................................................................................
286
Stability of vertical Cylindrical Cavity with Circular Gross –SectionViev of Elastic-
Plastic Deformations at Small Homogenous Subcritical States
E. A. Hazar, M. K. Cerrahoğlu, E. Cerrahoğlu.............................................................
287
- 18 -
The Mathematical Equations of a New Five Phases Segmental Switched Reluctance
Motor
E. Büyükbıçakcı, A. F. Boz............................................................................................
288
Dynamic Simulation Results of a New Five Phases Segmental Switched Reluctance
Motor
E. Büyükbıçakcı, A. F. Boz, Z. Büyükbıçakcı...............................................................
289
Investigation of the Effect of Project – Based Learning Method on Academic Success
for Mathematic Course in Higher Education
E. Büyükbıçakcı, Z. Büyükbıçakcı.................................................................................
290
On the Existence of Bertrand Curves in Dual Space 4D
E. Karaca, M. Çalışkan..................................................................................................
291
Surface Growth Kinematics in Minkowski 3 Space
G. Güner, Z. Özdemir, F. N. Ekmekci...........................................................................
292
On the Codimension-Two and -Three Bifurcations of a Food Web of Four Species
H. C. Wei, Y. Y. Chen, S. F. Hwang, J. T. Lin..............................................................
293
Algorithms for Calculating the Limits of Convergent Infinite Series
H. L. García, L. M. G. Cruz, A. E. García.....................................................................
294
An Extension of Srivastava's Triple Hypergeometric Function BH
İ. O. Kıymaz, M. B. Yağbasan, A. Çetinkaya................................................................
295
Canonical transformations to the Schrodinger equation: Hypergeometric solutions
and exponential-type potentials
J. J. Peña, J. Morales, J. García-Ravelo.......................................................................
296
Siacci’s Resolution of the Acceleration Vector for a Non-Null Space Curve in
Minkowski Space
K. E. Özen, M. Tosun, M. Akyiğit..................................................................................
297
Two Parameter Homothetic Motions on the Galilean Plane
M. Çelik, M. A. Güngör..................................................................................................
298
Exact Traveling Wave Solutions of some Nonlinear Evolution Equations
M. Odabaşı, E. Mısırlı....................................................................................................
299
On Reflections and Cartan-Dieudonné Theorem in Minkowski 3-space
M. Özdemir, M. Erdoğdu................................................................................................
300
- 19 -
Analytical Problem Solution about Initial Step of Pressing Powder Material Tube
M. Ya. Flax, A. V. Bochkov, V. A. Goloveshkin, A. V. Ponomarev.............................
301
X-ray and Computational Studies of (2Z, 3E)-3-(((E)-3-Ethoxy-2-
Hydroxybenzylidene) Hydrazono)Butan-2-One Oxime
N. Çalışkan, Ç. Yüksektepe Ataol, H. Batı, N. Kurban, P. Kurnaz, G. Ekmekçi.........
303
Fixed Points of Certain Automorphisms of Free Solvable Lie Algebras
N. Ekici............................................................................................................................
304
Generalizations of Sherman's inequality by Lidstone's interpolating polynomial
R. P. Agarwal, S. I. Bradanović, J. Pečarić...................................................................
305
A simple State Observer Design for Linear Dynamic Systems by Using Taylor Series
Approximation
S. Aksoy, H. Kızmaz........................................................................................................
306
On the Classical Zariski Topology over Prime Spectrum of a Module
S. Çeken, M. Alkan.........................................................................................................
308
Strongly k -Spaces
S. Ersoy, İ. İnce, M. Bilgin.............................................................................................
309
On the Study of the Matrix in a Model of Economic Dynamics
S. I. Hamidov..................................................................................................................
311
Numerical Solutions of Steady Incompressible Dilatant Flow in an Enclosed Cavity
Region
S. Şahin, H. Demir..........................................................................................................
312
Smarandache Curves of Mannheim Curve Couple According to Frenet Frame
S. Şenyurt, A. Çalışkan...................................................................................................
313
Smarandache Curves of Involute-Evolute Curve Couple According to Frenet Frame
S. Şenyurt, S. Sivas, A. Çalışkan....................................................................................
315
The Analysis of the Mathematical Modelling Activities in the Ninth Grade
Mathematics Coursebook
S. Urhan, Ş. Dost............................................................................................................
317
On Geodesic Paracontact CR-Lightlike Submanifolds
S. Yüksel Perktaş, B. Eftal Acet.....................................................................................
319
- 20 -
A Generalized Method for Centres of Trajectories in Kinematics
T. Erişir, M. A. Güngör, M. Tosun................................................................................
320
A New Generalization of the Steiner Formula and the Holditch Theorem
T. Erişir, M. A. Güngör, M. Tosun................................................................................
321
On the Construction of Generalized Bobillier Formula
T. Erişir, M. A. Güngör, S. Ersoy.................................................................................
322
Generalized 𝐹-Expansion Method and Application to Nonlinear Fractional
Differential Equation
Y. A. Tandoğan, Y. Pandır.............................................................................................
323
Views of Pre-service Elementary Mathematics Teachers toward the Reasons for
Student’s Mistakes about Fractions and Preventing These Mistakes
Y. Kıymaz........................................................................................................................
324
Interface Design for Comparative Solution of Mathematical Equations by Classical
Interpolation Methods and Artificial Nerve Network Approaches
Z. Batık, E. Büyükbıçakcı..............................................................................................
325
Interface Design for Genetic Algorithm Based Solution of Polynomial Equations
Z. Batık, E. Büyükbıçakcı..............................................................................................
326
Fixed Point of Automorphisms Permuting Free Generators of Free Metabelian Lie
Algebras
Z. Esmerligil, N. Ekici....................................................................................................
327
Mechanics Equations of Frenet-Serret Frame on Minkowski Space
Z. Kasap, E. Özyılmaz.....................................................................................................
328
Test Elements of Free Metabelian Leibniz Algebras
Z. Özkurt.........................................................................................................................
329
- 21 -
Authenticated Encryption Based on Prime Moduli
Debasis Giri1
Abstract. In an authenticated encryption scheme, a signer signs a message for a
particular verifier using signer’s own private key and the public key of the verifier. The verifier
recovers the original message from the signcrypted message using the signer’s public key and the
verifier’s own private key. Significant work is done in this direction by the authors Zheng and
others. We first describe an authenticated encryption scheme for signing fixed block length
message, which is based upon a variant of the ElGamal encryption scheme and a variant of the
ElGamal signature scheme over large prime moduli. We then also present an another
authenticated encryption scheme that can be adapted for signature generation of arbitrary length
of message.
1 Haldia Institute of Technology, Haldia, India, [email protected]
- 22 -
An Algebraic Description of Gradient Descent Decoding
Edgar Martínez-Moro 1
Abstract. Several constructions in (binary) linear block codes are also related to matroid
theory topics or integer programming. These constructions rely on a given order in the ground set
of the matroid/code. In this talk we will show how basic coding theory and gradient descent
coding methods can be seen as instances of some problems associated to reductions and ideal
membership to some binomial ideals. We will give a short introduction to the Gröbner
representation of a binary matroid/code and we show how it can be used for studying different
sets bases, cycles, minimal code words, etc..
1 Universidad de Valladolid, Soria, Spain, [email protected]
- 23 -
Elliptic Diophantine Equations and the Elliptic Logarithm Method
Nikos Tzanakis1
Abstract. By an Elliptic Diophantine equation we mean one of the form ( ), 0f x y = ,
where:
• f is a polynomial with integer coefficients.
• The plane curve defined by the equation 0f = is of genus one and has a (projective)
point with rational coordinates.
• The sought for solutions ( ),x y are rational integers or S-integers.
The standard elliptic Diophantine equation example comes from
( ) 3 2, ,f x y x ax b y= + + − where ,a b are integers and 3 24 27 0a b+ ≠ ; this is the Weierstrass
equation.
Around 1994, with R.J. Stroeker, we elaborated the ideas of S. Lang [3], [4] concerning
these equations, and published in [6] a practical method-the Elliptic Logarithm Method; Ellog
for short-for solving explicitly such equations over the integers. A little latter, a more or less
analogous method was independently developed in [2].
In the years to follow, many interesting applications of the Ellog to Diophantine equations
over the integers (for example, to ( ), 0f x y = where ( ) 4 3 2 2 2, ,f x y ax bx cx dx e y= + + + + −
and to the resolution of simultaneous Pell equations) have been published. Moreover, the
method was implemented by the MAGMA group [1] into a routine of the package.
In the turning of the century, with R.J. Stroeker, we developed a general method for
solving
over the integers an elliptic equation of any shape, like, for example, ( ), 0f x y = with
( ) 5 4 3 6 3 2 7 4 2, 2 , 2 , 2f x y x x y x x y x x y= + − + − + − or with much more “complicated”
( ),f x y ; see [7]. Generalizing the method in an other direction, Pethö, Zimmer, Gebel and
Herrmann, in [5], extended the method for solving the Weierstarass elliptic equation from the
integers to the S-integers for an explicitly given (finite) set S of primes. The combination of
1 University of Crete, Heraklion, Greece, [email protected]
- 24 -
these two generalizations which would make possible a practical method for solving the
general elliptic Diophantine equation over the S-integers, has not been accomplished yet.
In my lecture I will try to give an overview of the Ellog and the elegant mathematical
tools and techniques behind it. I do not intend to address to specialists of Diophantine
equations, but to a general mathematical audience.
AMS 2010. 11G05; 11D25; 11J86; 11Y50.
References
[1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J.
Symbolic Comput., 24, 235-265, 1997.
[2] J. Gebel, A. Pethö and H.G. Zimmer, Computing integral points on elliptic curves, Acta
Arith. 68, 171-192, 1994.
[3] S. Lang, Diophantine approximation on toruses, Amer. j. Math. 86, 521-533, 1964.
[4] S. Lang, Elliptic Curves: Diophantine Analysis, Grundlehren Math. Wiss. 231, Springer,
Berlin, 1978.
[5] A. Petho, H.G. Zimmer, J. Gebel and E. Herrmann, Computing all S-integral points on
elliptic curves, Math. Proc. Camb. Phil. Soc. 127, 383-402, 1999.
[6] R.J. Stroeker, N. Tzanakis, Solving elliptic diophantine equations by estimating linear
forms in elliptic logarithms, Acta Arithm. 67, 177-196, 1994.
[7] R.J. Stroeker, N. Tzanakis, Computing all integer solutions of a genus 1 equation, Math.
Comp. 72, 1917-1933, 2003.
[8] N. Tzanakis, Elliptic Diophantine Equations - A concrete approach via the elliptic
logarithm, Series in Discrete Mathematics and Applications 2, De Gruyter, Berlin/Boston,
2013.
- 25 -
Möbius Transformations and the Circle-Preserving Property
Nihal Yılmaz Özgür1
Abstract. Let n be the real n dimensional space and let ˆ = n n ∪ ∞ . A map
ˆ ˆ: n nf → is said to be r-sphere preserving if f maps every r-dimensional sphere onto an r-
dimensional sphere. When 1r = it is called the corresponding map f as a circle-preserving
map in ˆ . n In this talk we give a survey on the circle-preserving property of Möbius
transformations acting on ˆ = n n ∪ ∞ .
Keywords. Möbius Transformation, Circle-Preserving Property.
AMS 2010. 30C35, 51F15.
References
[1] Carathéodory, C., The most general transformations of plane regions which transform circles into circles, Bull. Amer. Math. Soc., 43, 573-579, 1937.
[2] Gibbons, J., Webb, C., Circle-preserving functions of spheres, Trans. Amer. Math. Soc., 248, 1, 67-83, 1979.
[3] Haruki, H., Rassias, T. M., A new invariant characteristic property of Möbius transformations from the standpoint of conformal mapping, J. Math. Anal. Appl., 181, 2, 320-327, 1994.
[4] Jeffers, J., Lost theorems of geometry, Amer. Math. Monthly, 107, 800-812, 2000.
[5] Li, B., Wang, Y., Transformations and non-degenerate maps, Sci. China Ser. A, 48, 195-205, 2005.
[6] Yao, G., On existence of degenerate circle-preserving maps, J. Math. Anal. Appl., 334, 2, 950-953, 2007.
[7] Yılmaz Özgür, N., On some mapping properties of Möbius transformations, Aust. J. Math. Anal. Appl., 6, Art. 13, 8 pp., 2010.
[8] Yılmaz Özgür, N., On the circle preserving property of Möbius transformations, Invited contribution for the volume: Mathematics without Boundaries: Surveys in Pure Mathematics. Themistocles M. Rassias and Panos M. Pardalos eds., Springer Verlag, 2014.
1 Balikesir University, Balikesir, Turkey, [email protected]
- 26 -
Advances in Frames, Riesz Bases and Frames on Hilbert C* Modules
Ram N. Mohapatra1
Abstract. Frames have been of interest for past several years after the success of
wavelets for image processing and the introduction of wavelet frames. There has been
considerable work on Frames in Hilbert spaces, Banach spaces and Hilbert C* module. There is
also a lot of work on optimal Frames for Erasers, gframes, semi-frames and frames in semi-inner
product spaces. In this talk we shall consider some of the developments in these areas and some
open problems relating to unbiased frames.
1 University of Central Florida, Orlando, USA, [email protected]
- 27 -
Representations of a Leavitt Path Algebra with Coefficients in an Abelian Category
Ayten Koç and Murad Özaydin
Abstract. We define and study the module category of a Leavitt path algebra of
finitely separated digraph with coefficients in an abelian category. When the abelian
category is that of F -vector spaces then we recover the module category of ( )FL Γ , the
Leavitt path algebra with coefficients in F . In the case of an arbitrary abelian category the
module category is defined without specifying the algebra (which may not even exist and
when it exits is unique only up to Morita equivalence).
Keywords. Noncommutative Rings, Representations, Morita Theory, Abelian
Categories.
AMS 2010. 16G20, 18E35.
References
[1] G. Abrams, Leavitt Path Algebras: The First Decade, Bull. Math. Sci. 5:59-120, 2015.
[2] A. Koç, M. Özaydın, Representations of Leavitt Path Algebras, Submitted.
[3] M. Tomforde, Leavitt Path Algebras with Coefficients in a Commutative Ring, J. Pure
Appl. Algebra 215, 471-484, 2011.
- 28 -
Some Relations via K-Balancing Numbers
Arzu Özkoç1 and Ahmet Tekcan2
Abstract. In this work, we derive some relations between the k-balancing numbers
and the sums of k-balancing numbers. Also we consider some formulas for the greatest
common divisor of k-balancing numbers. Further we deduce the simple continued fraction
expansion of k-balancing numbers.
Keywords. Balancing Number, Pell Number, Binary Linear Recurrences.
AMS 2010. 11B37, 11B39.
References
[1] Dash K.K., Ota R.S. and Dash S., t-Balancing Numbers. Int. J. Contemp. Math. Sciences,
7(41), 1999-2012, 2012.
[2] Kovacs T., Liptai L. and Olajos P., On (a,b)-Balancing Numbers. Publ. Math. Debrecen
77/3-4, 485-498, 2010.
[3] Liptai K., Luca F., Pinter Á. and Szalay L., Generalized Balancing Numbers. Indag.
Mathem., N.S., 20(1), 87-100, 2009.
[4] Olajos P., Properties of Balancing, Cobalancing and Generalized Balancing Numbers.
Annales Mathematicae et Informaticae 37, 125-138, 2010.
[5] Panda G.K., Some Fascinating Properties of Balancing Numbers. Proceedings of the
Eleventh International Conference on Fibonacci Numbers and their Applications, Cong.
Numer. 194, 185-189, 2009.
[6] Panda G.K. and Ray P.K., Some Links of Balancing and Cobalancing Numbers with Pell
and Associated Pell Numbers. Bul. of Inst. of Math. Acad. Sinica 6(1), 41-72, 2011.
[7] Tekcan A., Tayat M. and Özbek M., The Diophantine Equation 8x²-
y²+8x(1+t)+(2t+1)²=0 and t-Balancing Numbers. ISR Combinatorics 2014, Article ID
897834, 5 pages, 2014.
1 Duzce University, Duzce, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
- 29 -
Vertex-Decomposability and Depth Bounds of Independence Complexes
Alper Ülker1 and Tahsin Öner2
Abstract. Let G be a simple undirected graph. Let ( )Ind G be the independence
complex of G whose faces correspond to the independent sets of G. We call the graph G
Cohen-Macaulay if and only if ( )( ) ( )dim Ind depth(Ind )G G= [1]. A simplicial complex ∆ is
said to be vertex-decomposable if it is either a simplex or else has some vertex v so that both
∆ and link ( )v∆ are vertex-decomposable and no face of link ( )v∆ is a facet of \v∆ [3]. We
call a simplicial complex pure if all its facets have the same dimension. If ∆ is pure and
vertex-decomposable complex then ∆ is Cohen-Macaulay complex. Let skeld∆ be the
d skeleton− of complex ∆ whose all faces have dimension ≤d [4]. depth( ) d∆ ≥ if and only
if skeld∆ is Cohen-Macaulay [2]. In this talk, we give some results about vertex-
decomposability of some particular dimensional skeletons of ( )Ind G .
Keywords. Cohen-Macaulay Graphs, Vertex-Decomposability.
AMS 2010. 13F55, 05E45, 05E40.
References
[1] Woodroofe, R., Vertex-Decomposable graphs and Obstruction to Shellability, Proc. Amer.
Math. Soc., 137 (10), 3235-3246, 2009.
[2] Woodroofe, R., Chains of Modular Elements and Shellability, J. Combin. Theory Ser. A
119, 1315–1327, 2012.
[3] Provan, J. S., Billera, L. J., Decompositions of simplicial complexes related to diameters
of convex polyhedra, Math. Oper. Res. 5, no. 4, 576–594, 1980.
[4] Villarreal, R.H., Monomial Algebras 2nd Ed., Monographs and Research Notes in
Mathematics, Chapman and Hall/CRC, 704 p, 2015.
1 Ege University, Izmir, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
- 30 -
On Generalized FI-Extending Modules
Canan Celep Yücel1
Abstract. A module M is called FI-extending if every fully invariant submodule of M
is essential in a direct summand of M. In this work, we define generalized FI-extending (GFI-
extending) module as for any fully invariant submodule N of M there exists a direct summand
D of M such that N≤D and that D/N is singular. We give some characterizations of this class
of modules. To this end we focus on direct sums and summands of the former class.
Keywords. Extending Module, Fully Invariant Submodule, FI-Extending, GFI-
Extending.
AMS 2010. 16D50; 16D70.
References
[1] F.W. Anderson, F.W., Fuller K.R., Ring and Categories of Modules, New York, USA,
Springer-Verlag, 1974.
[2] Birkenmeier, G.F., Tercan, A., When some complement of a submodule is a direct
summand, Communications in Algebra, 35, 597-611, 2007.
[3] Birkenmeier, G.F., Tercan, A., Yücel, C.C., The extending condition relative to sets of
submodules, Communications in Algebra, 42, 764-778, 2014.
[4] Birkenmeier, G.F., Müller, B.J., Rizvi, S.T., Modules in which every Fully Invariant
Submodule is essential in a direct summand, Communications in Algebra, 30 (3), 1395-1415,
2002.
[5] Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R., Extending Modules, Longman
Scientific and Technical, England, 1994.
[6] Quing-yi, Z., On generalized extending modules, Journal of Zhejiang Univ. Sci. A., 86,
939-945, 2007.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 31 -
An Explicit Homomorphism from the Complex Clifford Algebra to the
Cuntz Algebra
Derya Çelik1
Abstract. It is well known that there exists an isomorphism from the complex
Clifford algebra to the matrix algebra Moreover, the matrix algebra is
homomorphic (with homomorphism ) to the Cuntz algebra (see [1]). Thus, the
composition gives naturally a homomorphism from to . Since we do not have
an explicit expression of this composition yet, it would be desirable to give a direct
homomorphism from to explicitly. In this work, we construct such an explicit
homomorphism from the complex Clifford algebra to the Cuntz algebra .
Keywords. Clifford Algebra, Cuntz Algebra, Homomorphism.
AMS 2010. 15A66.
References
[1] J. Lawrynowicz and O. Suzuki, A Fractal Method for Infinite-Dimensional Clifford
Algebras and The Related Wavelet Bundles, Bull. Soc. Sci. Lettres Lodz 53, 53—70, 2003.
1 Anadolu University, Eskisehir, Turkey, [email protected]
- 32 -
Circulant Matrices in Terms of Tetranacci Numbers
Elif Ardıyok1 and Arzu Özkoç2
Abstract. In this work, the eigenvalues and determinants of the circulant matrices
involving tetranacci sequence nM and companion-tetranacci sequence nK are denoted by
tetranacci and companion-tetranacci numbers. Also Euclidean norms, spectral norms of
circulant and negacyclic matrices are obtained.
Keywords. Circulant Matrix, Negacyclic Matrix, Tetranacci Numbers, Determinant.
AMS 2010. 11B83, 15A60, 05A15, 11C20.
References
[1] Davis P. J., Circulant Matrices , John Wiley &Sons, New York, 1979.
[2] Karner H., Schneid J.C. and Ueberhuber W., Spectral decomposition of real circulant
matrices, Linear Algebra and its Applications 367, 301-311, 2003.
[3] Kocer E.G., Circulant, Negacyclic and Semicirculant Matrices with the Modified Pell,
Jacobsthal-Lucas numbers. Hacettepe Journal of Mathematics and Statistics 36(2), 133-142,
2007.
[4] Ribenboim P. My Numbers, My Friends, Popular Lectures on Number Theory, Springer-
Verlag, New York, Inc. 2000.
[5] Shen S.Q., Cen J.M., On the determinants and inverses of circulant matrices with the
Fibonacci and Lucas numbers, Applied Mathematics and Computation, 217, 9790-9797,
2011.
[6] Spickerman W. R., Joyner R. N., "Binet's Formula for the Recursive Sequence of Order k"
Fibonacci Quarterly 22.4 :327-31, 1984.
[7] Waddill M.E., The Tetranacci Sequence and Generalizations. Fibonacci Quarterly 30.1:9-
20, 1992.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected]
- 33 -
A Symmetric Key Fully Homomorphic Encryption Scheme using General
Chinese Remainder Theorem
Emin Aygün1 and Erkam Lüy2
Abstract. The Fully Homomorphic Encryption was an open problem up to 2009. In
2009, Gentry solved the problem. After Gentry's solution, a lot of work have made on Fully
Homomorphic Encryption. However these works have stil too high time complexities for
pratical use. In [7], authors suggested a symmetric key homomorphic encryption scheme.
Authors proved that scheme is faster and the security of scheme is equivalent to the large
integer factorization problem. Scheme is based on matrix operations which are
computationally fully homomorphic. After that, in [8] and [9] authors extended the ideas and
suggested a new scheme. In [7] authors used 2m prime numbers in keygen algorithm and in
[8] and [9] authors extended this to 2m mutually prime numbers. In [7], [8] and [9] authors
used Chinese Remainder Theorem in Encryption algorithm for security of their encryption
scheme is equivalent to the large integer factorization problem.
We extend the approach used in [7], [8] and [9]. We use General Chinese Remainder
Theorem in Encryption algorithm and obtained a new fully homomorphic encryption scheme
by using General Chinese Remainder Theorem.
Keywords. Fully Homomorphic Encryption, Large Integer Factorization, General
Chinese Remainder Theorem.
AMS 2010. 53A40, 20M15.
References
[1] R.Rivest, L.Adleman ve M.L.Dertouzos, On data banks and privacy homomorphisms,
Foundations of Secure Computation, 169-170, 1978.
[2] Alice Silverberg, Fully Homomorph_c Encrypt_on for Mathemat_c_ans, sponsored by
DARPA under agreement numbers FA8750-11-1-0248 and FA8750- 13-2-0054. 2013.
[3] S. Goldwasser ve S. Micali, Probabilistic encryption and how to play mental poker
keeping secret all partial information, in proceedings of the 14th ACM Symposium on
Theory of Computing, 365-377, 1982.
1 Erciyes University, Kayseri, Turkey, [email protected] 2 Erciyes University, Kayseri, Turkey, [email protected]
- 34 -
[4] P.Pailler, Public-Key Cryptosystems Based on Composite degree Residuosity Classes, in
Advances in Cryptology, EUROCRYPT, 223-238, 1999.
[5] Craig Gentry, A Fully Homomorph_c Encrypt_on Scheme, phd thesis, Stanford University,
2009.
[6] Vinod Vaikuntanathan, Computing Blindfolded: New Developments in Fully
Homomorphic Encryption, 52nd Annual Symposium on Foundations of Computer Science, 5-
16, 2011.
[7] Liangliang Xiao, Osbert Bastani, I-Ling Yen, An E_cent Homomorphic Encryption
Protocol for Multi-User Systems, iacr.org, 2012.
[8] C. P. Gupta, Iti Sharma, Fully Homomorphic Encryption Scheme with Symmetric Keys,
Master of Technology in Department of Computer Science & Engineering, Rajasthan
Technical University, Kota, August – 2013.
[9] C P Gupta and Iti Sharma, A Fully Homomorphic Encryption scheme with Symmetric Keys
with Application to Private Data Processing in Clouds, Network of the Future (NOF), 2013
Fourth International Conference on the Digital Object Identi_er: 10.1109/NOF.2013.6724526
Publication Year: 2013 , Page(s): 1 - 4 IEEE CONFERENCE PUBLICATIONS
[10] Voting Resources http://theory.lcs.mit.edu/ rivest/voting
[11] Marten van Dijk; Craig Gentry, Shai Halevi, and Vinod Vaikuntanathan "Fully
Homomorphic Encryption over the Integers". International Association for Cryptologic
Research., 2010.
[12] H. E. Rose, A Course n Number Theory, School of Mathematics , Üniversity of Bristol,
1988.
[13] W. J. Leveque, Topics in Number Theory, Addison-Wesley Publishing Company,
University of Michigan, 35-35, 1965.
- 35 -
Soft Radicals
Emin Aygün1, Akın Osman Atagün2 and Betül Erdal1
Abstract. Soft set theory, proposed by Molodtsov, has been regarded as an effective
mathematical tool to deal with uncertain objects. In this paper, we define a radical of an ideal
in soft set theory by using two different soft ideal concepts: a soft ideal of a ring and a soft
ideal of a soft ring. We give some results and illustrate with several examples.
Keywords. Soft sets, Radicals, soft radicals, Ideals.
AMS 2010. 16Y30, 03G25.
References
[1] H. Aktaş and N. Çagman, Soft sets and soft groups, Inform. Sci. 177, 2726 - 2735. 2007
[2] M.I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set
theory, Comput. Math. Appl. 57 (9), 1547 - 1553, 2009.
[3] A.O. Atagün and A. Sezgin, Soft substructures of rings, Fields and modules, Comput.
Math. Appl. 61 (3), 592 - 601, 2011.
[4] A. Sezgin, A. O. Atagun, On operations of soft sets, Computers and Mathematics with
Applications,61 (5), 1457-1467, 2011
[5] K. Atanassov, Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and
Systems 64, 159174, 1994.
[6] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87- 96.
[7] D. Chen, E.C.C. Tsang, D. S. Yeung and X. Wang, The parametrization reduc-
tion of soft sets and its applications, Comput. Math. Appl. 49 (2005), 757-763.
[8] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56(2008), 2621 -
2628.
[9] W. L. Gau and D. J. Buehrer,Vague sets, IEEE Tran. Syst. Man. Cybern. 23(1993), 610 -
614.
[10] M. B. Gorzalzany, A method of inference in approximate reasoning based on interval-
valued fuzzy sets, Fuzzy Sets and Systems 21 (1987), 1 - 17.
[11] Z. Kong, L. Gao, L. Wang and S. Li, The normal parameter reduction of soft sets and its
algorithm, Comput. Math. Appl. 56 (12)(2008), 3029-3037.
1 Erciyes University, Kayseri, Turkey, [email protected], [email protected] 2 Bozok University, Yozgat, Turkey, [email protected]
- 36 -
[12] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003),
55562.
[13] P. K. Maji, A. R. Roy and R.Biswas, An application of soft sets in a decision making
problem, Comput. Math. Appl. 44 (2002), 1077-1083.
[14] D. Molodtsov, Soft set theory-¯rst results, Comput. Math. Appl. 37 (1999), 19-31.
[15] Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci. 11 (1982), 341-356.
[16] Z. Pawlak and A. Skowron, Rudiments of soft sets, Inform. Sci. 177 (2007), 3-27.
[17] G. Pilz, Near-rings, North Holland Publishing Company, Amsterdam-New York-Oxford,
1983.
[18] A. Sezgin and A.O. Atagun, On operations of soft sets, Comput. Math. Appl.
61 (5) (2011), 14571467.
[19] A. Sezgin and A.O. Atagun, Soft groups and normalistic soft groups, Comput. Math. ppl.
(submitted).
[20] C. F. Yang, A note on soft set theory, Comput. Math. Appl. 56 (2008), 1899-1900.
[Comput. Math. Appl. 45 (4 - 5) (2003), 555-562.
[21] L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (1965), 338-353.
[22] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline, Inform. Sci.
172 (2005), 140.
[23] L. Zhou and W. Z. Wu, On generalized intuitionistic fuzzy rough approximation
[24] A. Sezgin, A. O. Atagun and E. Aygun, A note on soft near-rings and idealistic soft near-
rings, Filomat,25 (1) (2011) 53 - 68.
[25] A. Sezgin, A. O. Atagun, N. Cagman, Union soft substructures of near-rings and N-
groups, Neural Computing and Applications, 21 (Issue 1-Supplement) (2012) 133 - 143.
[26] A. Sezgin, A. O. Atag¨un, On operations of soft sets, Computers and Mathematics with
Applications,61 (5) (2011) 1457 - 1467.
[27] A. Sezgin Sezer, A new view to ring theory via soft union rings, ideals and bi-ideals,
Knowledge-Based Systems, 36 (2012) 300 - 314.
- 37 -
Fixed Points on NG-Groups
Faraj A. Abdunabi1
Abstract. We consider the NG-group which consisting of transformations on a non-empty
set A and the group has no bijection as its element. Recall a permutation group on A is a group
consisting of bijections from A to A with respect to compositions of mappings. It is well known
that any permutation group on a set A with cardinality n has order not greater than n!. There are
some authors, [10], [9], Problem 1.4 in [1], considering groups which consists of non-bijective
transformations on A where the binary operation is the composition of mappings. Our result is on
the orders of such groups are mapping on a non-empty set A with respect to function
compositions which are not subsets of symmetric groups by using the fixed points.
Keywords. Symmetic Groups, Fixed Points.
AMS 2010. 53A40, 20M15.
References
[1] Isaacs, I. Martin, Algebra (A Graduate Course), Wadsworth, Inc: 2003.
[2] Isaacs, I. Martin, Finite Group Theory Graduate Studies in Mathematics, Vol. 92, American, Mathematical Society Providence, Rhode Island, 2008.
[3] Isaacs, I. Martin, U. Meierfrankenfeld, Repeated and Final Commutators in Group Actions, Proceedings of AMS, 140, 3777-3783, 2012.
[4] N. Jacobson, Basic Algebra I, Deover Publications, Inc: Mineola, New York, 1985.
[5] O. Bogopolski, Intreudction to Group Theory, European Mathematical Society, Printed in Germany, 2008.
[6] Y. L. Wu, X. Y. Wei, Condition of the groups generated by nonbijective transformations on a set (In Chinese), Journal of Hubei University (Natural Science), 27, 1-3, 2005.
[7] Y.X. Zhu, J.K.Hai, Groups Generated by Non -bijective Transformations on a Set and Their Representations in the Matrix Algebra (In Chinese), Journal of Capital Normal University (Natural Science), 24, 5-14, 2003.
- 38 -
Invariant Ring of Aut(V, H)
Fawad Hussain1
Abstract. Let V be a finite dimensional vector space over the finite field Fq with
basis e1,…,en. Suppose x1,...,xn is the dual basis of the dual vector space V∗.
Let G ≤ GL(V) and consider the polynomial ring in the n indeterminates Fq[x1,…,xn].
Invariant theory over finite fields is a branch of abstract algebra. The theory deals with those
elements of Fq[x1,…, xn] which do not change under the action of the group G. These
elements form a ring structure which is called the ring of invariants of the group G.
For a long time there has been interest in finding the ring of invariants of the group G ≤ GL(V)
The rings of invariants of the general linear and the special linear groups were computed early
in the 20th century by Dickson in [2]. These were found to be a graded polynomial algebras in
both cases. In 1992, Carlisle and Kropholler calculated the ring of invariants of the symplectic
group in [4] and their result showed that this ring of invariants is a graded complete
intersection. In 2005 Kropholler, Mohseni Rajaei and Segal [4] found explicit generators and
relations for the rings of invariants of orthogonal groups over F2 but the general case is still
open. In 2006, Chu and Jow [3] computed rings of invariants of unitary groups. In the last two
cases it was found that the rings of invariants are graded complete intersections. In this talk I
will discuss the ring of invariants of the following group:
Aut (V, H) = g ∈ GL(V) : H(gv, gu) = H(v, u) for all v, u V
where V is a vector space over the finite field Fq2 and H is a singular hermitian form on V.
Keywords. Aut (V, H), Polynomial Invariants, Graded Complete Intersection.
References
[1] Dickson, L.E., A fundamental system of invariants of the general modular linear group
with a solution of the form problem, Trans. Amer. Math. Soc., 12, 1, 75-98, 1911.
[2] Carlisle. D., Kropholler, P.H., Modular invariants of finite symplectic group, Preprint.
1992.
[3] Chu, H., Jow, S.Y., Polynomial invariants of finite unitary groups, J. Algebra., 302, 2,
686-719, 2006.
[4] Kropholler, P.H., Rajaei, S.M., Segal, J., Invariant rings of orthogonal groups over F2,
Glasg. Math. J., 47, 1, 7-54, 2005.
1 Hazara University, Mansehra, Pakistan, [email protected]
- 39 -
Solution of a Matrix Completion Problem and Nil-Clean Companion Matrices
George Ciprian Modoi1
Abstract. We solve a matrix completion problem, more precisely we write any
companion matrix as a sum between an idempotent matrix and a matrix with prescribed
characteristic polynomial. The number of applications of the matrix completion problem to
information theory, electrical engineering etc. is huge (see for example [2], [3] or [4]). Our
application is a more theoretical one: We describe nil-clean companion matrices over fields.
Recall that an element of a ring is called clean or nil-clean if it decomposes as a sum between
an idempotent and a unit, respectively a nilpotent. It turns out that decompositions like these
are important in the study of some properties of rings, see [5].
This presentation is based on a joint work with Simion Breaz, [1].
Keywords. Matrix Completion, Companion Matrix, Nil-Clean Matrix
AMS 2010. 15A24, 15A83, 16U99.
References
[1] S. Breaz, G. C. Modoi, Nil-clean companion matrices, arXiv: 1503.06668v2 [math.RA],
preprint 2015.
[2] E. Candès, Y. Plan, Matrix Completion with Noise, Proc. IEEE, vol. 98, no. 6, 925-936,
2010.
[3] E. Candès, T. Tao, The Power of Convex Relaxation: Near-Optimal Matrix Completion, in
IEEE Transactions on Information Theory, vol. 56, 2053-2080, 2010.
[4] B. Eriksson, L. Bolzano, R. Novak, High-Rank Matrix Completion, in Proceedings of the
15th International Conference on Artificial Intelligence and Statistics (AISTATS) 2012;
appeared in J. Machine Learning Research, W&CP 22, 373-381, 2012.
[5] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229,
269-278, 1977.
1 Babeş-Bolyai University, Romania, [email protected]. This research is supported by UEFISCDI grant PN-II-ID-PCE-2012-3-0100.
- 40 -
A Presentation of Free Lie Algebra 𝐹 𝛾3(𝐹)′
Gülistan Kaya Gök1 and Naime Ekici2
Abstract. Let 𝐹 be free Lie algebra generated by the free generators 𝑥 and 𝑦. By using
the technique of Gröbner- Shirshov bases we show that the Lie algebra 𝐹 𝛾3(𝐹)′ has the
presentation ⟨𝑥,𝑦 | ∆⟩, where ∆ is the minimal Gröbner basis of the algebra 𝛾3(𝐹)′.
Keywords. Free Lie Algebra, Presentation, Gröbner Basis.
AMS 2010. 17B01.
References
[1] G. M. Bergman, The Diamond lemma for ring theory, Adv. in Math. 29, 178-218, 1978.
[2] L. A. Bokut, Unsolvability of the word problem and subalgebras of finitely presented Lie
algebras, Izv. Akad. Nauk. SSSR Ser. Math. 36, 1173-1219, 1972.
[3] L. A. Bokut, Embeddings into simple associative algebras, Algebrai Logika, 15, 117-142,
1976.
[4] L. A. Bokut, Algorithmic and Combinatorial Algebra, Kluwer, Dordrecht, 1994.
[5] L. A. Bokut, P. Kolesnikov, Gröbner-Shirshov bases: from incipient to nowadays,
Proceedings of the POMI 272, 26-67, 1994.
[6] L. A. Bokut, P. Kolesnikov, Gröbner-Shirshov bases: from their incipiency to the present,
J. Math. Sci. 116, 1, 2894-2916, 2003.
[7] L. A. Bokut, Y. Chen, Gröbner-Shirshov bases: some new results, Proc. Second Int.
Congress in Algebra and Combinatorics, World Scientific, 35-56, 2008.
[8] B. Buchberger, An algorithm for finding a basis for the residue class Ring of a zero-
dimensional polynomial ideal, Phd. Thesis, Univ. of Innsbruck, Austria, 1965.
[9] B. Buchberger, An algorithmical criteria for the solvability of algebraic system of
equations, Aequationes Math., 4, 374-383, 1970.
[10] V. Drensky, Defining relations of noncommutative algebras, Institue of Mathematics and
Informatics Bulgarian Academy of Sciences.
[11] A. I. Shirshov, Some algorithmic problems for Lie algebras, Sibirsk. Mat. Z., 3, 292-296,
1962; English translation in SIGSAM Bull, 33(2) 3-6, 1999.
1 Hakkari University, Hakkari, Turkey, [email protected] 2 Cukurova University, Adana, Turkey, [email protected]
- 41 -
Centering nonlinear Fuchsian Codes
M. Alsina 1
Abstract. Algebraic structures as Fuchsian groups have been applied to information
theory to construct nonlinear codes. A new transmission scheme for additive white Gaussian
noisy (AWGN) channels based on Fuchsian groups was presented at [1], and their
generalization to higher rate was developped at [2]. In this talk we review the general
construction and focus on the study of the center of those codes. In order to do that we
use the geometry of the hyperbolic uniformizations of Shimura curves attached to the
Fuchsian groups developped at [3]. This work is also related with a series of papers by
Palazzo et al. among others (see for example [4]).
Keywords. Nonlinear Codes, Fuchsian Groups, Hyperbolic Geometry.
AMS 2010. 94B60, 94B35, 11F06, 20H10, 51M10.
References
[1] Blanco, I.; Remón, D.; Hollanti, C.; Alsina, M., Nonuniform Fuchsian codes for
noisy channels, Journal of the Franklin Institute. 351, 11. 5076-5098, 2014.
[2] Blanco, I.; Hollanti, C.; Alsina, M.; Remón, D., Fuchsian codes with arbitrarily
High code rates. Accepted at Journal of Pure and Applied Algebra, 2015.
[3] Alsina, M., Bayer, P., Quaternion orders, quadratic forms and Shimura curves,
CRM Monograph Series 22, American Mathematical Society, 210 pag, 2004.
[4] Carvalho, E. Andrade, A., Palazzo, R. Filho, J.V., Yaglom, I. M., Aarithmetic
Fuchsian groups ans space-time block codes, Comput. Appl.Math. 30, 485-498, 2011.
1 Universitat Politècnica de Catalunya, Manresa, Catalunya, [email protected]
- 42 -
On the Codes over the Ring 22
222 IFuuIFvIFIF +++
Mustafa Özkan1 and Figen Öke2
Abstract. In this paper the ring 22
222 IFuuIFvIFIF +++ where 03 =u , 02 =v ,
0== vuuv is defined. Thus it is shown that the Gray image of −+ )1( v constacyclic codes
with odd order over the ring 22
222 IFuuIFvIFIF +++ is a cyclic code over the ring
22
22 IFuuIFIF ++ . Also it is shown that there exist a quasicyclic codes of index 2 over the
ring 22
222 IFuuIFvIFIF +++ .
Keywords. Cylic Codes, Gray Map, Codes Over Rings.
AMS 2010. 94B05 , 94B15 , 94B60.
References
[1] Roman S, Coding and Information Theory, Graduate Texts in Mathematics, Springer
Verlag, 1992.
[2] Ling S. and Blackford J. T., _1+Ζ kp Lineer Codes, IEEE Trans. On Inform. Theory
Vol.45,No.9, 2592-2605, 2002.
[3] Udomkavanich P. and Jitman S., On the Gray Image of ( mu−1 )-Cyclic Codes
kkk pm
pp IFuIFuIF +++ ... , Int. J. Contemp. Math. Sciences, Vol. 4, No. 26, 1265-1272,
2009.
[4] B. Yıldız, S. Karadeniz, Linear codes over 2222 uvIFvIFuIFIF +++ ,Des. Codes
Cryptgr., 54, 61-71, 2010.
[5] S. Karadeniz, B. Yıldız, On )1( v+ -constacyclic codes over 2222 uvIFvIFuIFIF +++ ,
Journal of the Franklin Institude , 348, 2625-2632, 2011.
[6] Xu Xioafang, )1( v+ -constacyclic codes over 222 vIFuIFIF ++ , Computer Engineering
and Applications, 49,12,77-79, 2013.
1 Trakya University, Edirne, Turkey, [email protected] 2 Trakya University, Edirne, Turkey, [email protected]
- 43 -
Some Special Codes Over 23 3 3 3v u u+ + +
Mustafa Özkan1 and Figen Öke2
Abstract. In this paper the structure of the ring 33 2
[ , ], ,
u vu v uv< >
where 03 =u ,
02 =v and 0== vuuv is described. The distance function on this ring which is isomorphic to
the ring 23 3 3 3R v u u= + + + is defined. This means that linear codes over the ring R can
be written. Then it’s shown that the Gray images of cyclic codes over the ring R are quasi-
cyclic codes of index 2 over the ring 3 3v+ . Then another Gray map from the ring 3 3v+
to 3 is described. Also the relation between the cyclic codes over the ring
23 3 3 3R v u u= + + + and quasi-cyclic codes over the field 3 is established.
Keywords. Linear Codes, Quasi-Cylic Codes, Gray Map.
AMS 2010. 94B05 , 94B15 , 94B60.
References
[1] J. Wolfmann, Negacyclic and cyclic codes over 𝑍4, IEEE Trans. Inf. Theory, 45, 2527-
2532, 1999.
[2] J. F. Qian, L. N. Zhang, Shixin Zhu, Constacyclic and cyclic codes over
22
22 IFuuIFIF ++ , IEICE Trans. Fundamentals, E89-a, no 6, 1863-1865, 2006.
[3] M. C. Amarra, F. R. Nemenzo, On (1 − 𝑢 )-cyclic codes over 𝐼𝐼𝑝𝑘 + 𝑢𝐼𝐼𝑝𝑘 , Applied
Mathematics Letters 21, 1129-1133, 2008.
[4] B. Yıldız, S. Karadeniz, Linear codes over 2222 uvIFvIFuIFIF +++ , Des. Codes
Cryptgr., 54, 61-71, 2010.
[5] S. Karadeniz, B. Yıldız, On )1( v+ -constacyclic codes over 2 2 2 2 ,IF uIF vIF uvIF+ + +
Journal of the Franklin Institude, 348, 2625-2632, 2011.
1 Trakya University, Edirne, Turkey, [email protected] 2 Trakya University, Edirne, Turkey, [email protected]
- 44 -
Minimaxness and Cofiniteness Properties of Local Cohomology Modules
Monireh Sedghi1
Abstract. R be a commutative Noetherian ring, I and ideal of R and M a non-zero
R − module. The Purpose of this paper is to introduce the notation of the
( ) ( ) ( ) dim : inf : / dim / ,p
nI IR pn th finiteness ension f M f M p Supp M IM and R p n− = ∈ ≥
for all 0 ,n∈ and to prove the following results:
(i) ( ) ( ) 10: inf : minimax .i
I If M i H M is not= ∈
(ii) The ( )modules iIR H M− are I − cofinite for all ( )2
Ii f M< and for all minimax
submodules N of ( ) ( )2
,If MIH M the R − modules
( )( )2
/ , /IfR IHom R I H M N and ( )( )21 / , /If
R IExt R I H M N
are finitely generated, whenever ( )2If M is finite. This implies that if I has dimension one,
then ( )iIH M is I − cofinite for every 0,i ≥ which is generalization of the main results of
Defino-Marley, yoshide and Bahmanpour-Naghipour.
(iii) ( ) ( ) 20inf : is not weakly Laskerian ,i
I If M i H M= ∈ whenever R is semi-
local.
(iv) The R − modules ( )( )/ ,j iR IExt R I H M are weakly laskerian for all 0j ≥ and all
( )3 ,Ii f M< whenever ( ),R m is complete Noetherian local ring.
Keywords. Cofinite Modüle, Minimax Modüle, Eakly Laskerian Modüle.
AMS 2010. 13D45, 14B15, 13E05.
1Azarbijan Shahid Madani University, Tabriz, Iran, [email protected]
- 45 -
Embeddings of Solvable Products Into Matrix Algebras and Generalized Fox
Derivatives
Nazar Şahin Öğüşlü1 and Naime Ekici2
Abstract. In this work we give a survey of the generalized Shmel’kin embedding. Let
F be a free product of finitely generated free abelian Lie algebras, R be an ideal of F and
V be a variety of Lie algebras. Shmel’kin has found an embedding for a Lie algebra ( )F
V R
, where ( )V R is the verbal ideal of R corresponding to variety V . In the case of ( )F
V R is a
solvable product, we apply Shmel’kin embedding for the algebra ( )F
V R . Using this
embedding we describe generalized Fox derivatives in a solvable product of finitely generated
free abelian Lie algebras.
Keywords. Free Lie Algebras, Solvable Product, Embedding.
AMS 2010. 17B01, 17B40.
References
[1] Gupta, C. K., Timoshenko, E. I., The test rank of a soluble product of free abelian groups,
Sbornik Math., 199, 4, 495-510, 2008.
[2] Romonovsky, N. S., On Shmel’kin embeddings for abstract and pro-finite groups, Algebra
Logika, 38, 5, 598-612, 1999.
[3] Shmel’kin, A. L., Embeddings into wreath products of some factor algebras of free sums
of Lie algebras, Notes of the Chair of Higher Algebra of MSU (in Russian).
[4] Shmel’kin, A. L., Syrtsov, A. V., On embeddings of some factor algebras of free sums of
Lie algebras, J. Math. Sci., 131, 6, 6148-6152, 2005.
1 Cukurova University, Adana, Turkey, [email protected] 2 Cukurova University, Adana, Turkey, [email protected]
- 46 -
Rank Properties of the Direct Product of Finite Cyclic Groups
Osman Kelekci1
Abstract. There are five different ranks defined as small rank, lower rank,
intermediate rank, upper rank and large rank, respectively ([1], [2]). In previous studies, these
ranks are calculated for certain semigroups and groups ([3], [4]). All the ranks of the direct
product of finite cyclic groups are given in this work.
Keywords. Cyclic Group, Direct Product, Ranks.
AMS 2010. 20F05, 20K25.
References
[1] Howie, J.M., Marques Ribeiro, M.I., Rank properties in finite semigroups, Comm.
Algebra, 27, 5333-5347, 1999.
[2] Howie, J.M., Marques Ribeiro, M.I., Rank properties in finite semigroups II: the small
rank and the large rank, Southeast Asian Bull. Mat., 24, 231-237, 2000.
[3] Minisker, M., Rank properties of certain semigroups, Semigroup Forum, 78, 1, 99-105,
2009.
[4] Kelekci, O., Upper rank of full transformation semigroups on a finite set, Int. J.
Algebra, 5, 29-32, 1527–1532, 2011.
1 Niğde University, Nigde, Turkey, [email protected]
- 47 -
A Note on a Category of Cofinite Modules which is Abelian
Reza Naghipour1
Abstract. Let $R$ denote a commutative Noetherian (not necessarily local) ring, and
I an ideal of R of dimension one. The main purpose of this paper is to generalize, and to
provide a short proof of, K. I. Kawasaki's Theorem that the category ( , )cofR I of I -
cofinite modules over a commutative Noetherian local ring R forms an Abelian subcategory
of the category of all $R$-modules. Consequently, this assertion answers affirmatively the
question
raised by R. Hartshorne in it Affine duality and cofiniteness, Invent. Math. (1970), 145-
164], for an ideal of dimension one in a commutative Noetherian ring R .
Keywords. Abelian Category, Arithmetic Rank, Cofinite Module, Noetherian Rings.
AMS 2010. 13D45, 14B15, 13E05.
1 University of Tabriz and IPM, Tabriz, Iran, [email protected]
- 48 -
The Values of a Class of Direchlet Series at the Non Positive Integers
Sadaoui Boualem1
Abstract. We relate the special value at a non positive integer s=(s1,…, sr)=(-N1,…, -Nr)=-N,
obtained by meromorphic continuation of the multiple Dirichlet series
𝑍(𝑃, 𝐬) = 1
∏ 𝑃𝑖𝑠𝑖(𝑚)𝑟
𝑖=1𝑚=(𝑚1,….,𝑚𝑛)∈𝑁∗𝑛
to special values of the function
𝑌(𝑃, 𝑠) = 1
∏ 𝑃𝑖𝑠𝑖(𝑥)𝑟
𝑖=1 𝑑𝑥
[1,+∞[𝑛
Where 𝑥 = (𝑥1, … , 𝑥𝑛) and 𝑃 = (𝑃1, … ,𝑃𝑟) are polynomials of several variables which
verified a certain conditions.
We prove a simple relation between 𝑍(𝑃𝑎,−𝑁) and 𝑌(𝑃𝑎,−𝑁), such that for all
𝑎 = (𝑎1, … ,𝑎𝑛) ∈ 𝑅𝑛, with the notation 𝑃𝑎 = (𝑃1,𝑎, … ,𝑃𝑟,𝑎), where
𝑃𝑖,𝑎(𝑥) = 𝑃𝑖,𝑎(𝑥 + 𝑎) = 𝑃𝑖,𝑎(𝑥1 + 𝑎1, … , 𝑥𝑛 + 𝑎𝑛) is the shifted polynomial.
Keywords. Meromorphic Continuation; Integral Representation; Special Values.
AMS 2010. 11M32; 11M41.
References
[1] S. Akiyama and S. Egami and Y. Tanigawa, “Analytic continuation of multiple zeta-
functions and their values at non-positive integers”, Acta Arith. 98, (2), 107-116, 2001.
[2] F. V. Atkinson, “The mean-value of the Riemann zeta function”, Acta Math., 81, 353-
376, 1949.
[3] E. Friedman and A. Pereira, “Special Values of Dirichlet Series and Zeta Integrals”, Int. J.
Number Theory, 08, (3), 697-714, 2012.
[4] D. Zagier, “Values of zeta functions and their applications”, First European Congress of
Mathematics (Paris, 1992), Vol. II, A. Joseph et. al.(eds), Birkhauser, Basel, 497-512, 1994.
1 Khemis Miliana University, Algeria, [email protected]
- 49 -
On Congruences with the Terms of Second order Sequence and Harmonic Numbers
Sibel Koparal1 and Neşe Ömür2
Abstract. In this study, we give the congruences involving the terms of second order
sequence 𝑢𝑛(𝐴,𝐵) and harmonic numbers. For example, for any prime 𝑝 > 2𝑏 + 1 ,
𝑢𝑘−1−𝑝𝛥/2
(1, 𝑏2)
𝑏𝑘≡ 0(𝑚𝑚𝑚𝑝)
𝑝−1
𝑘=0
,
and for a prime 𝑝 > 3,
𝐻𝑘𝑃𝑘−1
2𝑘≡ 0(𝑚𝑚𝑚𝑝),
𝑝−1
𝑘=1
𝐻𝑘𝑄𝑘−1
2𝑘≡
2𝑝
𝑄𝑘−12𝑘
𝑝−1
𝑘=1
(𝑚𝑚𝑚𝑝),𝑝−1
𝑘=1
where 𝛥 = 4𝐵 − 𝐴², 𝑃𝑛 and 𝑄𝑛 are 𝑛th Pell and Pell-Lucas numbers.
Keywords. Congruences, Fibonacci Numbers, Pell Numbers.
AMS 2010. 11B39, 05A19.
References
[1] J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5, 35-39, 1862.
[2] Z.H. Sun, Values of Lucas sequences modulo primes, Rocky Mount. J. Math. 33, 1123-
1145, 2003.
[3] Z.W.Sun, On harmonic numbers and Lucas sequences, Pub. Math. Debrecen 8081-2, 25-
41, 2012.
1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected]
- 50 -
On Commuting Automorphisms of Some Finite P-Groups
Sandeep Singh1
Abstract: Let G be a finite p-group. In this article, under some certain conditions, we
prove that if G is nilpotent group of either maximal or co-class 2 or co-class 3, then Set of all
commuting automorphisms of G form the subgroup. Also some positive answers of questions
raised in the article of Deaconescu, Silberg and Walls [2] are given.
Keywords. Nilpotent Groups, Commuting Automorphism.
References
[1] J. E. Adney and T. Yen, Automorphisms of a p-group, Illionois J. Math. 9 (1965), 137-143.
[2] M. Deaconescu, G. Silberberg and G. L. Walls, On commuting automorphisms of groups,
Arch. Math., 79(2002), 479-484.
[3] N. Herstein, Problem proposal, Amer. Math. Monthly, 91, 203, 1984.
[4] R. James, The groups of order p6 (p an odd prime), Math. Comp., 94, 613-637, 1980.
[5] T. J. Laey, Solution of problem E 3039, Amer. Math. Monthly, 93 (1986), 816.
[6] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, Oxford
University Press, Oxford, 2002.
[7] M. Pettet, personal communication.
1 Thapar University, Patiala, India, [email protected]
- 51 -
Permuting n-f-Derivations on Lattices
Şahin Ceran1 and Utku Pehlivan2
Abstract. In this paper as a generalization of permuting tri-f-derivation on a lattice we
introduced the notion of permuting n-f- derivation of a lattice. We defined the isotone
permuting n-f-derivation and got some interesting results about isotoneness. We characterized
the distributive and isotone lattices by permuting n-f-derivation.
Keywords. Lattice, Derivation, Permuting n-f-Derivation.
AMS 2010. 06B35, 06B99, 16B70.
References
[1] A. Honda and M. Grabisch, Entropy of capacities on lattices and set systems, Inform. Sci.
176, no. 23, 3472-3489, 2006.
[2] Balbes, R. and Dwinger, P. Distributive Lattices, University of Missouri Press, Columbia,
Mo., 1974.
[3] Bell, A. J., The co-information lattice, 4th Int. Symposium on Independent Componentv-
Analysis and Blind Signal Seperation (ICA2003), Nara, Japan, 921-926, 2003.
[4] Carpineto, C. and Romano, G., Information retrieval through hybrid navigation of lattice
representations, International Journal of Human-Computers Studies 45, 553-578, 1996.
[5] Çeven, Y. Ozturk, M. A., On f-derivations of lattices. Bull. Korean Math. Soc. 45, no.4,
701-707, 2008.
[6] Çeven, Y., Symmetric bi derivations of Lattices, Quaestiones Mathematicae, 32, 1-5, 2009.
[7] Çeven, Y. Ozturk, M. A., Some properties of symmetric bi-( ,σ τ )-derivations in near-
rings. Commun. Korean Math. Soc. 22, no. 4, 487-491, 2007.
[8] Davey, B. A., Priestley, H. A. Introduction to lattices and order. Second edition.
Cambridge University Press, New York, xii+298 pp. ISBN: 0-521-78451-4, 2002.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Pamukkale University, Denizli, Turkey, [email protected]
- 52 -
[9] Degang, C., Wenxiu, Z., Yeung, D. and Tsang, E. C. C., Rough approximations on a
complete completely distributive lattice with applications to generalized rough sets, Inform.
Sci. 176, no. 13, 1829-1848, 2006.
[10] Durfee, G., Cryptanalysis of RSA using algebraic and lattice methods, A dissertation
submitted to the department of computer sciences and the committe on graduate studies of
Stanford University, 1-114, 2002.
[11] Ferrari Luca, On derivations of lattices. Pure Math. Appl. 12, no. 4, 365-382, 2001.
[12] G. Birkhoof, Lattice Theory, American Mathematical Society, New York, 1940.
[13] Jun, Y. B., and Xin, X. L., On derivations of BCI-algebras, Inform. Sci. 159, no. 3-4,
167-176, 2004.
[14] Ozbal, S. A, Firat, A., Symmetric f bi Derivations of Lattices. Ars Combin. 97, in press,
2010.
[15] Ozturk, Mehmet Ali; Yazarlı, Hasret; Kim, Kyung Ho, Permuting tri-derivations in lattices. Quaest. Math. 32, no. 3, 415-425, 2009.
- 53 -
Automorphisms of the Lie Algebras Related with 2x2 Generic Matrices
Şehmus Fındık1
Abstract. Let Fm be the relatively free algebra of rank m in the variety of Lie algebras
generated by the algebra sl2(K) over a field K of characteristic 0. We describe the inner and
outer automorphisms of the completion of Fm with respect to the formal power series
topology ([1], [2]). As a consequence we obtain the description of the inner and outer
automorphisms of the factor algebra of Fm modulo the members of its lower central series.
Some of the results were obtained with Vesselin Drensky2.
Keywords. Free Lie Algebras, Generic Matrices, Inner and Outer Automorphisms.
AMS 2010. 17B01, 17B30, 17B40, 16R30.
References
[1] Fındık, Ş., Outer automorphisms of Lie algebras related with generic 2×2 matrices,
Serdica Math. Journal, 38, 273-296, 2012.
[2] Drensky, V., Fındık, Ş., Inner automorphisms of Lie algebras related with generic 2×2
matrices, Algebra Discrete Math., 14, 49-70, 2012.
1 Cukurova University, Adana, Turkey, [email protected] 2 Bulgarian Academy of Sciences, Sofia, Bulgaria, [email protected]
- 54 -
On the Ambarzumyan's Theorem for the Quasi-Periodic Boundary Conditions
Alp Arslan Kıraç 1
Abstract. We obtain the classical Ambarzumyan’s theorem for the Sturm-Liouville
operators with a summable potential and quasi-periodic boundary conditions, when there is
not any additional condition on the potential q.
Keywords. Ambarzumyan Theorem, Inverse Spectral Theory, Hill Operator, Quasi-
periodic
AMS 2010. 34A55, 34B30, 34L05, 47E05, 34B09
References
[1] V. Ambarzumian, Über eine Frage der Eigenwerttheorie, Zeitschrift für Physik, 53, 690–695, 1929.
[2] Y. H. Cheng, T. E. Wang and C. J. Wu, A note on eigenvalue asymptotics for Hill’s equation., Appl. Math. Lett., 23, 1013–1015, 2010.
[3] H. H. Chern, C. K. Lawb and H. J. Wang, Corrigendum to Extension of Ambarzumyan’s theorem to general boundary conditions, J. Math. Anal. Appl., 309, 764–768, 2005.
[4] H. H. Chern and C. L. Shen, On the n-dimensional Ambarzumyan’s theorem, Inverse Problems, 13, 15–18, 1997.
[5] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, Scottish Academic Press, Edinburgh, 1973.
[6] H. Hochstadt and B. Lieberman, An inverse sturm-liouville problem with mixed given data, SIAM J. Appl. Math., 34, 676–680, 1978.
[7] B. M. Levitan and M. G. Gasymov, Determination of a differential equation by two of its spectra, Usp. Mat. Nauk, 19, 3–63, 1964.
[8] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987.
[9] O. A. Veliev and M. Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl., 265, 76–90, 2002.
[10] O. A. Veliev and A. A. Kıraç, On the nonself-adjoint differential operators with the quasiperiodic boundary conditions, International Mathematical Forum, 2, 1703–1715, 2007.
[11] C. F. Yang, Z. Y. Huang and X. P. Yang, Ambarzumyan’s theorems for vectorial sturm-liouville systems with coupled boundary conditions., Taiwanese J. Math., 14, 1429–1437, 2010.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 55 -
Quasi-Partial b-Metric Spaces and Some Related Fixed Point Theorems
Anuradha Gupta1 and Pragati Gautam 2
Abstract: In 1989, Bakhtin [1] introduced the concept of a b-metric space as a
generalization of metric spaces which was further extended by Czerwik [2].In 1994, Matthews
[3] introduced the notion of partial metric spaces and extended the Banach Contraction Principle
from metric spaces to partial metric spaces. Later, Shukla [4] generalized both the concept of b-
metric and partial metric spaces by introducing the partial b-metric spaces. Karapinar [5]
introduced the concept of quasi –partial metric space and studied some fixed point theorems on
these spaces. In this paper, the concept of quasi-partial b-metric space is introduced which is a
generalization of quasi-partial metric space and general fixed point theorems are proved in setting
of such spaces. Some examples are also given to verify the effectiveness of the main results.
Keywords. Quasi-partial Metric, T-orbitally Lower Semi-continuous, Quasi-partial b-
metric Space, Fixed Point Theorem.
AMS 2010. 47H10,54H25.
References:
[1] I.A. Bakhtin, The Contraction Principle in Quasi metric spaces, It. Funct. Anal., 3026-37.
1989.
[2] S. Czerwick, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1,
5-11, 1993.
[3] S.G. Matthews, Partial Metric Topology, Research Report 212, Department of Computer
Science, University of Warwick, 1992.
[4] Satish Shukla, Partial b –metric spaces and Fixed point theorems, Mediterr.Jour. of
mathematics, 2013.
[5] Erdal Karapinar, I.M. Erhan, Ali Ozturk, Fixed point theorems on quasi partial metric spaces,
Math. Comput. Model, 2442-2448, 2013.
1 University of Delhi, India , [email protected] 2 University of Delhi, India , [email protected]
- 56 -
On Transformation Types of the ( )0 NΓ and the Suborbital Graph ,Fu N
Ali Hikmet Değer1
Abstract. In [1] authors studied the action of Γ Modular group on the extended
rational set ℚ ≔ ℚ ∪ ∞ by using suborbital graphs introduced in 1967 by Sims [2] for finite
permutation groups. The simplest example of the suborbital graph is the well-known Farey
graph ℱ ≔ 𝐺1,1 which is related to the Farey sequence 𝐹𝑚. In [1] authors generalized the
properties of ℱ to suborbital graphs 𝐹𝑢,𝑁 which are isomorphic to a subgraph of ℱ.
In this paper we give the 𝐹𝑢,𝑁 -distance by using the Farey distance from [1]. We
derive the result how a shortest path in 𝐹𝑢,𝑁 from vertices ∞ to 𝑤 can be found by expressing
𝑤 as a continued fraction. Also we examine the transformation types of the congruence
subgroup Γ0(𝑁) of Γ which gives paths of minimal length of the suborbital graph 𝐹𝑢,𝑁.
Keywords. Suborbital Graphs, Modular Group, Paths of Minimal Length.
AMS 2010. 20H10, 20H05, 05C05, 05C20.
References
[1] Jones, G. A., Singerman, D., Wicks, K., The Modular Group and Generalized Farey
Graphs, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, Vol. 160, pp. 316-338,
1991.
[2] Sims, C.C., Graphs and finite permutation groups, Math. Zeitsch., Vol. 95, pp. 76-86,
1967.
[3] Akbas, M., On Suborbital Graphs for the Modular Group, Bull. London Math. Soc. 33,
647-652, 2001.
[4] Değer, A. H., Beşenk, M., Güler, B. Ö., On Suborbital Graphs and Related Continued
Fractions, Applied Mathematics and Computation, 218, 3, 746-750, 2011.
[5] Sarma, R., Kushwaha, S., Krishnan, R., Continued fractions arising from 𝐹1,2, Journal of
Number Theory, Vol. 154, pp. 179-200, 2015.
1 Karadeniz Technical University, Trabzon, Turkey, [email protected]
- 57 -
Boundedness of Fractional Maximal Operator Associated with Hankel Transform on
Weighted Lorentz Spaces
Canay Aykol Yüce1
Abstract. In this paper we characterize the boundedness of fractional maximal
operator associated with Hankel (Fourier-Bessel) transform on weighted Lorentz spaces .
Keywords. Fractional Maximal Operator; Hankel Transform; Weighted Lorentz
Spaces; Weighted Norm Inequalities
AMS 2010. 42B25, 46E30, 46F12, 47G10.
References
[1] Gogatishvili, A., & Pick, L., A reduction theorem for supremum operators, Journal of
Computational and Applied Mathematics (208), 270-279, 2007.
[2] Guliyev, V.S., Serbetci A., & Ekincioglu, I., On the boundedness of the generalized B-
potential integral operators in the Lorentz spaces. Integral Transforms Spec. Funct., 18 (12),
885-895, 2007.
[3] Stempak, K., Almost everywhere summability of Laguerre series. Studia Math. (100), 129-
147, 1991.
[4] Trimeche, K., Transformation integrale de Weyl et theoreme de Paley-Wiener associes a
un operateur differentiel sur, J. Math. Pures Appl., (60), 51-98, 1981.
[5] Trimeche, K., Inversion of Lions transmutation operators using generalized wavelets,
Appl. Comput. Harmonic Anal., (4), 1-16, 1997.
1Ankara University, Ankara, Turkey, [email protected]
- 58 -
Multipliers with Natural Spectra on Commutative Banach Algebras
Cesim Temel1
Abstract. Let X be a Banach space and let )(XB be the algebra of all bounded linear
operators on .X G will be denoted a locally compact abelian group with the dual group .Γ
Let )(1 GL and )(GM , respectively be the group algebra and the measure algebra of .G By
)(KC we denote the space of all continuous functions on a compact .K Let )(: XBGU →
be a strongly continuous bounded representation of G on X . For arbitrary )(GM∈µ , we
can define )(XBU ∈µ , by )(1 gxdUxUG g µµ ∫ −= , .Xx∈ Then, the mapping
)()(: XBGMh → defined by µµ U→ is a bounded unital algebra homomorphism. The
Arveson spectrum )(Usp [1] of U is defined as the hull of the closed ideal
.0:)(: 1 =∈= fU UGLfI Said that )(GM∈µ has the spectral mapping property if
,))(()( UspU µs µ= where µ is the Fourier-Stieltjes transform of µ .
Recall that a compact set K in Γ is a Helson set for )(1 GL if for every )(KCg ∈
there is a function )(1 GLf ∈ such that gf =
on K [4, Section 5.6.1]. If Γ⊂K is a Helson
set, then there is a constant 0>C with the following property: If )(KCg ∈ , there exists
)(1 GLf ∈ such that gf =
on K and .)(sup1
χχ
gCfK∈
≤ If K is a Helson set and is a set of
synthesis for )(1 GL , then K is said to be WTP (without true pseudomeasures) set for )(1 GL .
For example, compact countable independent set in Γ is a WTP set for )(1 GL [4, Theorem
5.6.7]. By )(0 GM we will denote the set of all )(GM∈µ such that µ vanishes at infinity.
We have the following
Theorem. If )(Usp is a WTP set for )(1 GL , then every )(0 GM∈µ has the spectral
mapping property.
Keywords. Banach Algebra, Multiplier, (Local) Spectrum, Spectral Mapping
Property.
AMS-2010. 47B07, 30H05.
1 Yuzuncu Yil University, Van, Turkey, [email protected]
- 59 -
References
[1] Arveson, W., The harmonic analysis of automorphism groups, Proc. Symp. Pure Math.,
38, 199-269, 1982.
[2] Larsen, R., Banach algebras, Marcel Dekker, New York, 1973.
[3] Kahane, J. P., Series de Fourier absolument convergentes, Springer-Verlag, New York, 1970.
[4] Rudin, W., Fourier Analysis on Groups, Wiley-Interscience, New York, 1962.
- 60 -
Iλ - Double Statistically Convergent Sequences in Topological Groups
Ekrem Savaş 1
Abstract. In this paper, we introduce and study Iλ -statistical convergence for double
sequences in topological groups and we shall also present some inclusion theorems.
Keywords. Ideal Convergence, Ideal Statistical Convergence, Double Statistical
Convergence, Topological Groups.
AMS 2010. 42B15; 40C05.
References
[1] E. Savas, Iλ -statistically convergent sequences in topological groups, international
conference "Kangro-100. Methods of Analysis and Algebra", dedicated to the Centennial of
Professor Gunnar Kangro. Tartu, Estonia, on September 1-6, 2013.
[2] E. Savas, On generalized double statistical convergence via ideals, The Fifth Saudi
Science Conference, 16-18, April, 2012.
[3] E. Savas, and Pratulananda Das, A generalized statistical convergence via ideals, Appl.
Math. Lett. 24, 826-830, 2011.
[4] H. Çakalli, On statistical convergence in topological groups, Pure and Appl. Math. Sci.
43, no:1-2, 27-31, 1996.
1 Istanbul Commerce University, Istanbul, Turkey, [email protected]
- 61 -
Absolute Tauberian Constants
Fatma Aydın Akgün1 and Billy Rhoades2
Abstract. Sherif [2] obtained estimates of the form ∑|𝜏𝑛 − 𝑎𝑛| ≤ 𝐾∑|∆(𝑛𝑎𝑛)| and
∑|𝜏𝑛 − 𝑎𝑛| ≤ 𝐾′ ∑𝑛|∆𝜏𝑛−1| under the assumption that ∑𝑛|∆𝜏𝑛−1| is finite where ∆ is the
forward diference operator and 𝜏𝑛 = 𝐶𝑛𝑘 − 𝐶𝑛−1𝑘 . The constants K and 𝐾′ are called Tauberian
constants. Sherif [3] also obtained analogous results for regular Haudorff matrices. In a recent
paper [1] the authors obtained absolute Tauberian constants for the H-J generalized Hausdorff
transformations, which generalized the corresponding results, obtained earlier by Sherif, for
ordinary Hausdorff matrices. In this paper we obtain absolute Tauberian constants for regular
lower triangular matrices with row sums one. As corollaries we obtain the corresponding
results for factorable and weighted mean matrices.
Keywords. Factorable Matrices, Hausdorff Matrices, Tauberian Constants, Weighted
Mean Matrices.
AMS 2010. 47H10.
References
[1] F. Aydin Akgun and B. E. Rhoades, Absolute Tauberian constants for H−J Hausdorff
matrices, Appl. Math. and Information Sci. 7(4), 1405-1413, 2013.
[2] Soroya Sherif, Absolute Tauberian constants for Ces´aro means, Trans. Amer. Math. Soc.
168, 23-241, 1972.
[3] Soroya Sherif, Absolute Tauberian constants for Hausdorff transformations, Canadian J.
Math. 26(1), 19-26, 1974.
1 Yildiz Technical University, Istanbul, Turkey, [email protected] 2 Indiana University, Bloomington, [email protected]
- 62 -
On Convex Meromorphic Functions
Faruk Uçar 1 and Yusuf Avcı2
Abstract. Let us denote by S(p) the set of univalent functions in the unit disc
: 1D z z= ∈ < such that 𝑓(0) = 0,𝑓′(0) = 1 and f(p) = ∞ . We denote by K the subset
of functions in S(p) which omits a convex set in the extended plane = ∪ ∞ , that is,
𝑓 ∈ 𝐾 if and only if the set ( ) ( ) \ :f D w f z w= ∈ ≠ is convex. Functions in K are
called as convex meromorphic functions. In this paper, we define a function
:p D D D× × → in terms of f and show that Re p > 12 for all ζ, z, w ∈ 𝐷 if and only if f
belongs to K.
Keywords. Univalent Function, Convex Meromorphic Function, Starlike Function.
AMS 2010. 30C45, 30D30.
References
[1] Duren, P.L., Univalent Functions, Die Grundlehren der mathematischen
Wiesseschaften 259, Springer-Verlag, Berlin-Heidelberg-New York, 1983.
[2] Miller, J.E., Convex and starlike meromorphic functions, Proc. Am. Math. Soc, 80, 607-
613, 1980.
[3] Ruscheweyh, St. and Sheill-Small, T., Hadamard Products of schlicht functions and the
Polya-Schoenberg conjecture, Comment. Math. Helv., 48, 119-135, 1973.
[4] Schober, G., Univalent Functions - Selected Topics, Lecture Notes in Math., 478,
Springer- Verlag, New York/Berlin, 1975.
[5] Sheil-Small, T., On convex univalent functions, J. London Math. Soc. (2) 1, 483-492,
1969.
[6] Yulin, Z. and Owa, S., Some remarks on a class of meromorphic starlike functions, Indian
J. pure appl. Math., 21(9), 833-840, 1990.
1 Marmara University, Istanbul, Turkey, [email protected] 2 Bahcesehir University, Istanbul, Turkey, [email protected]
- 63 -
The Relation between B-1-Convex and Convex Functions
Gabil Adilov1, Ilknur Yesilce2 and Gultekin Tinaztepe3
Abstract. Convexity has generalized different types ([4]). B-1-convexity is of these
abstract convexity classes ([1]). B-1-convex sets and functions are studied in [2,3,5].
In this work, some examples of convex and B-1-convex functions are discussed and the
relationship between convex and B-1-convex functions is examined. Finally, taking into
account given examples, it is shown that B-1convex functions and convex functions do not
include each other.
Keywords. Abstract Convexity, B-1-Convexity, B-1-Convex Functions.
AMS 2010. 26B25, 52A41.
References
[1] Adilov, G., Yesilce, I., B-1-convex set and B-1-measurable maps, Numerical Functional
Analysis and Optimization, 33, 2, 131-141, 2012.
[2] Briec, W., Liang Q. B., On Some Semilattice Structures for Production Technologies,
European Journal of Operational Research, 215, 740-749, 2011.
[3] Kemali, S., Yesilce, I., Adilov, G., B-convexity, B-1-convexity and their comparison,
Numerical Functional Analysis and Optimization, 36, 2, 133-146, 2015.
[4] Rubinov, A., Abstract convexity and global optimization, Kluwer Academic Publishers,
Boston-Dordrecht-London, 2000.
[5] Tinaztepe, G., Yesilce, I., Adilov, G., Separation of B-1-convex sets by B-1-measurable
maps, Journal of Convex Analysis, 21, 2, 571-580, 2014.
1 Akdeniz University, Antalya, Turkey, [email protected] 2 Mersin University, Mersin, Turkey, [email protected] 2 Akdeniz University, Antalya, Turkey, [email protected]
- 64 -
On Some Inequalities and Their Refinements
Gültekin Tınaztepe1, Ramazan Tınaztepe2 and Serap Kemali 3
Abstract. In this work, the results derived in [1] is summarized, that is, generalized
mean inequalities and some important inequalities is sharpened by using a theorem related to
minimization of a function in the framework of abstract convexity. Apart from that, by using
same ideas, sharpened inequality for Hölder inequality is given.
Keywords. Abstract Convexity, Functional Inequalities, Harmonic-Geometric-
Arithmetic Means, Hölder Inequality.
AMS 2010. 26D07
References
[1] Adilov, G. and Tınaztepe, G, The sharpening of some inequalities via abstract convexity,
Mathematical Inequalities and Applications. 12, 1, 33-51, 2009.
[2] Beckenbach, E. and Bellman, R., Inequalities, Springer-Verlag, 1961.
[3] Rubinov A., Abstract convexity and global optimization, Kluwer Academic Publishers,
2000.
[4] Rubinov, A., and Wu, Z., Optimality conditions in global optimization and their
applications, Mathematical Programming B, 120, 101–123, 2009.
1 Akdeniz University, Antalya, Turkey, [email protected] 2 Zirve University, Gaziantep, Turkey, [email protected] 3 Akdeniz University, Antalya, Turkey, [email protected]
- 65 -
A Generalized Mountain Pass Theorem
Hans-Jörg Ruppen1
Abstract. We present a new variational characterization of multiple critical points for
even energy functionals functionals corresponding to nonlinear Schrödinger equations of the
following type:
We assume , , a.e. with and . Our
results cover the following 3 cases in a uniform way:
1. ;
2. is a Coulomb potential and
3. with for all .
The eigenvalue thereby may or may not lie inside a spectral gap.
Our variational characterization is “simple” and well suited for discussing multiple
bifurcation of solutions.
Keywords. Variational Principles, Critical Points
AMS 2010. 58E05, 58E30
References
[1] Antonio Ambrosetti and Paul H. Rabinowitz. Dual variational methods in critical point
theory and applications. J. Functional Analysis, 14:349–381, 1973.
[2] Wojciech Kryszewski and Andrzej Szulkin. Generalized linking theorem with an
application to a semilinear Schrödinger equation. Adv. Differential Equations, 3(3):441–472,
1998.
[3] Hans-Jörg Ruppen. A generalized mountain pass theorem. to appear.
[4] Hans-Jörg Ruppen. Odd linking and bifurcation in gaps: the weakly indefinite case. Proc.
Roy. Soc. Edinburgh Sect. A, 143(5):1061–1088, 2013.
[5] Hans-Jörg Ruppen. A generalized min-max theorem for functionals of strongly indefinite
sign. Calc. Var. Partial Differential Equations, 50(1-2):231–255, 2014.
[6] C. A. Stuart. Bifurcation into spectral gaps. Bull. Belg. Math. Soc. Simon Stevin,
(suppl.):59, 1995.
1 Ecole polytechnique fédérale de Lausanne EPFL, Lausanne, Switzerland, [email protected]
- 66 -
A New Result on the Almost Increasing Sequences
Hikmet S. Özarslan1 and Ahmet Karakaş2
Abstract. In this paper, we have generalized a known theorem on , n kN p
summability factors of infinite series to the , n kA pϕ − summability by using an almost
increasing sequence. This new theorem also includes several new results.
Keywords. Summability Factors, Absolute Matrix Summability, Almost Increasing
Sequences, Infinite Series.
AMS 2010. 40D15, 40D25, 40F05, 40G99.
References
[1] Bari, N. K., Stečkin, S. B., Best approximations and differential properties of two
conjugate functions, (Russian) Trudy Moskov. Mat. Obšč ., 5 , 483-522, 1956.
[2] Bor, H., On two summability methods, Math. Proc. Camb. Philos Soc., 97, 147-149, 1985.
[3] Bor, H., A note on , n kN p summability factors of infinite series, Indian J. Pure Appl.
Math., 18 , 330-336, 1987.
[4] Bor, H., Absolute summability factors of infinite series, Indian J. Pure Appl. Math., 19 ,
664-671, 1988.
[5] Bor, H., On the relative strength of two absolute summability methods, Proc. Amer. Math.
Soc., 113, 1009-1012, 1991.
[6] Bor, H., A note on absolute Riesz summability factors, Math. Inequal. Appl., 10, 619-625,
2007.
[7] Flett, T.M., On an extension of absolute summability and some theorems of Littlewood
and Paley, Proc. London Math. Soc., 7, 113-141, 1957.
[8] Hardy, G. H., Divergent series, Oxford Univ. Press, Oxford, 1949.
1 Erciyes University, Kayseri, Turkey, [email protected] 2 Erciyes University, Kayseri, Turkey, [email protected]
- 67 -
[9] Mazhar, S. M., A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica, 25 ,
233-242, 1997.
[10] Özarslan, H. S., A new application of almost increasing sequences, Miskolc Math. Notes,
14 , 201-208, 2013.
[11] Özarslan, H. S., Keten, A., A new application of almost increasing sequences, An. Ştiinţ.
Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 61, 153-160, 2015.
[12] Sulaiman, W. T., Inclusion theorems for absolute matrix summability methods of an
infinite series (IV), Indian J. Pure Appl. Math., 34 (11), 1547-1557, 2003.
[13] Tanovič-Miller, N., On strong summability, Glas. Mat., 34(14), 87-97, 1979.
- 68 -
Bi-Parametric Potentials and Their Inverses with The Aid of Wavelet-Type
Transforms
İlham A. Aliev1
Abstract. Given 0>α and 0>β , we introduce the following integral transform
(bi-parametric potential) of a function ( )npL R∈ϕ :
( )( ) ( )( )( ) ( ).,1=1
0
nt
t xdtxWetxJ R∈
Γ
−−∞
∫ ϕ
βα
ϕ ββα
αβ
Here,
( )( )( ) ( ) ( ) ( ) ( ),<<0,;= ∞−∫ tdytyyxxWn
tββ ωϕϕ
R
( )( ) ( )
−−ytGtty
nββββω1
=;
and
( )( ) ( ) .=,2=1=
⋅ ∑∫ −⋅−
kk
n
k
iy
n
n yydeyG ξξξπβξξβ
R
The integral ( )ϕβtW coincides with the classical Gauss-Weierstrass integral when 2=β and
Poisson integral when 1=β . The bi-parametric potentials ϕαβJ reduce to classical Bessel
potentials for 2=β and to Flett potentials for 1=β .
We introduce relevant wavelet-type transforms and by making use of these transforms we
obtain explicit inversion formulas for bi-parametric potentials ϕαβJ , ( )npL R∈ϕ .
Keywords. Bessel Potentials, Wavelet Transform, Gauss-Weierstrass Integral.
AMS 2010. 26A33, 46E35, 42C40.
1Akdeniz University, Antalya, Turkey, [email protected]
- 69 -
References
[1] I. A. Aliev and B. Rubin, Wavelet-like transforms for admissible semi-groups; Inversion
formulas for potentials and Radon transforms, J. of Fourier Anal. and Appl., 11, 333-352, 2015.
[2] I. A. Aliev, B. Rubin, S. Sezer and S. Uyhan, Composite wavelet transforms: Applications and
Perspectives, Contemporary Mathematics, Vol. 464, 1-25, Amer. Math. Soc. Providence, RI, 2008.
[3] I. A. Aliev, Bi-parametric potentials, relevant function spaces and wavelet-like transforms,
Integral Equations and Operator Theory, 65, 151-167, 2009.
- 70 -
Multi-Point Boundary Value Problems of Higher-Order Nonlinear Fractional
Differential Equations
İsmail Yaslan1
Abstract. We investigate the existence and uniqueness of solutions for multi-point
nonlocal boundary value problems of higher-order nonlinear fractional differential equations
by using some well-known fixed point theorems.
Keywords. Boundary Value Problems, Fixed Point Theorems, Fractional Derivative.
AMS 2010. 26A33, 34B10, 34B15.
References
[1] Das, S., Functional fractional calculus for system identification and controls, Springer,
New York, 2008.
[2] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional
differential equations, Elsevier, 2006.
[3] Podlubny, J., Fractional differential equations, Academic Press, New York, 1999.
[4] Ahmad, B., Nieto, J. J., Existence of solutions for nonlocal boundary value problems of
higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., Art. ID 494720,
1-9, 2009.
[5] El-Shahed, M., Nieto, J., Nontrivial solutions for a nonlinear multi-point boundary value
problem of fractional order, Comput. Math. Appl., 59, 3438-3443, 2010.
[6] Salem, H. A. H., On the fractional order m-point boundary value problem in reflexive
Banach spaces and weak topologies, J. Comput. Appl. Math., 224, 565-572, 2009.
[7] Guo, D., Lakshmikantham, V., Nonlinear problems in abstract cones, Academic Press,
Orlando, 1988.
[8] Krassnoselskii, M. A., Positive solutions of operator equations, Noordhoff, Groningen,
1964.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 71 -
Impulsive Boundary Value Problem for Nonlinear Differential
Equations of Fractional Order α ∈ (2, 3]
İlkay Yaslan Karaca1 and Fatma Tokmak Fen2
Abstract. In this paper, we investigate the existence of solutions for the boundary
value problem of nonlinear impulsive differential equations of fractional order α ∈ (2, 3]. By
using some well-known fixed point theorems, sufficient conditions for the existence of
solutions are established. Some examples are presented to illustrate the main results.
Keywords. Impulsive Boundary Value Problem, Fractional Order, Fixed-Point
Theorem.
AMS 2010. 26A33, 34B15, 34B37.
References
[1] Kilbas, A. A., Srivastava H. M., Trujillo, J. J., Theory and applications of fractional
differential equations, Elsevier, Amsterdam, 2006.
[2] Samoilenko, A. M., Perestyuk, N. A., Impulsive Differential Equations, World Scientific,
Singapore, 1995.
[3] Wang, X., Impulsive boundary value problem for nonlinear differential equations of
fractional order, Computers and Mathematics with Applications 62, 2383-2391, 2011.
[4] Wang, G., Ahmad B., Zhang, L., Impulsive anti-periodic boundary value problem for
nonlinear differential equations of fractional order, Nonlinear Analysis, 74, 792-804, 2011.
[5] Zhang, L., Wang, G., Song, G., Existence of solutions for nonlinear impulsive fractional
differential equations of order α ∈ (2, 3] with nonlocal boundary conditions, Abstract and
Applied Analysis, 2012, Article ID 717235, 26 pages, 2012.
1 Ege University, Izmir, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected]
- 72 -
Some Coupled Fixed Point Theorems for Generalized Contractions
İsa Yıldırım1
Abstract. In this work, we prove new coupled fixed point theorems for mapping
having the mixed monotone property in partially ordered metric space. Here the mappings are
assumed to satisfy certain contractive type inequalities. These new results generalize and
improve several related results in coupled fixed point theory.
Keywords. Coupled Fixed Point, Partially Ordered Set, Mixed Monotone Mappings.
AMS 2010. 47H10, 34B15.
References
[1] Agarwal, R.P., El-Gebeily, M. A., Oregano, D., Generalized contractions in partially ordered metric spaces, Appl. Anal. 87, 1-8, 2008.
[2] Banach, S., Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3, 133-181, 1922.
[3] Berinde, V., Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Analysis 74, 7347-7355, 2011.
[4] Bhaskar, T. G., Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65, 1379-1393, 2006.
[5] O.Regan, D., Petrutel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J.Math. Anal. Appl. 341, 241-1252, 2008.
[6] Zamfirescu, T., Fix point theorems in metric spaces, Arch. Math., 23, 292-298, 1972.
[7] Kannan, R., Some results on fixed points, Bull. Calcutta Math. Soc. 60, 71-76, 1968.
[8] Chatterjea, S. K., Fixed point theorems, C.R. Acad. Bulgare Sci. 25, 727-730, 1972.
1 Ataturk University, Erzurum, Turkey, [email protected]
- 73 -
Characterizing -Limit Sets for Analytic Vector Fields in Open Subsets of the Sphere
José Ginés Espín Buendía1 and Víctor Jiménez López2
Abstract. In [1] Jiménez López and Llibre gave a topological characterization, up to
homeomorphisms, of -limit sets of (real) analytic vector fields on the plane, the sphere and
the projective plane. Here, recall that if f is a smooth vector field on a surface S, and x(t) is a
solution of the differential equation x’=f(x), then the -limit set of (the orbit associated to the
solution) x(t) is the set of limit points of x(t) as t goes to +∞.
In the last section of the paper an argument is outlined to extend these results to
analytic vector fields just defined on arbitrary subsets of these surfaces, but this argument has
a gap and, as a consequence, the proposed characterizations are incomplete. In this work, still
in progress, we partially fill this gap by proving the following result:
Theorem. If K is a totally disconnected compact subset of the sphere S2 and is an -
limit set for some analytic vector field defined on S2\K, then is the boundary of a shrub.
Conversely, if S2 is the boundary of a shrub, then there are a homeomorphism h: S2→ S2
and a C∞ vector field f on the sphere, which is analytic except for a totally disconnected
compact set of points, such that h( is an -limit set of some orbit of the equation x’=f(x).
In the second statement of the theorem, the set of points of non-analyticity can be
“minimized”: in particular, if a point x of is locally the vertex of a 2n-star in , then f is
analytic at h(x). By a shrub we mean a compact, connected, locally connected subset T of S2
containing a (possibly empty) family Dnn of closed topological disks with the following
property: if C is a topological circle in T, then there is exactly one disk Dn containing C.
Keywords. Analytic Vector Field, -Limit Set, Sphere.
AMS 2010. 37E35, 37B99, 37C10
References
[1] Jiménez López, V., Llibre, J., A topological characterization of the -limit sets for
analytic flows on the plane, the sphere and the projective plane, Adv. Math, 216, 677-710,
2007.
1 Universidad de Murcia, Murcia, Spain, [email protected] 2 Universidad de Murcia, Murcia, Spain, [email protected]
- 74 -
Weighted Composition Followed by Differentiation between Weighted Fréchet Spaces of Holomorphic Functions
Jasbir S. Manhas1
Abstract. Let U and V be two countable families of weights on the unit disc D and let
HU(D) and HV(D) be the weighted Fréchet spaces of holomorphic functions. In this paper, we
investigate the holomorphic mappings φ: D →D and ψ: D→₵ which characterize
boundedness and compactness of products of weighted composition operators and
differentiation operators 𝐷𝐷𝜓,φ and 𝐷𝜓,φ𝐷 between weighted Fréchet spaces of holomorphic
functions HU(D) and HV(D).
Keywords. Weighted Composition Operators, Differentiation Operators, Weighted
Frechet Spaces.
AMS 2010. 47B38, 47B33.
1 Sultan Qaboos University, Muscat, Oman, [email protected]
- 75 -
The Effects of Privatisation and Competition on Malaysia Airlines Performance
Kok Fong See 1 and Azwan Abdul Rashid 2
Abstract. A commercially sustainable flag carrier airline is central to the broader geo-
political and macroeconomic national objectives of global connectivity and trade linkages for
Malaysia. However, Malaysia Airlines is in crisis. The survival of Malaysia Airlines is at
stake given the internal and external factors combining to create the perfect storm. Malaysia
Airlines reported a loss of RM1.17 billion in the full year 2013 and has had a negative
operating cash flow for the past three years, which means that it is not generating enough cash
to meet its day-to-day operating costs. As the national flagship air carrier, Malaysia Airlines
has to strike a balance between its commercial, political and social obligations and will
always be under close scrutiny. The study examines the total factor productivity (TFP) growth
of Malaysia Airlines over a 33 year period from 1981 to 2013 using the Törnqvist index
method. The privatisation of Malaysia Airlines is observed to coincide with lower TFP
growth rates. Furthermore, our results could suggest that the introduction of competition was
insufficient to produce improved TFP performance.
Keywords. Productivity Growth, Malaysia Airlines, Törnqvist Index.
AMS 2010. 91B82, 91B38.
1 Universiti Sains Malaysia, Malaysia, [email protected] 2 Universiti Tenaga Nasional, Malaysia, [email protected]
- 76 -
New Integral Inequalities Via Ga-Convex Functions
Merve Avcı Ardıç1, Ahmet Ocak Akdemir2 and Erhan Set3
Abstract. In this paper, we prove a new integral identity and based on this equality,
we established some integral inequalities for functions whose derivatives of absolute values
are GA-convex functions.
Keywords. GA-Convex Functions, Logarithmic Mean, Hölder Inequality.
AMS 2010. 26D15, 26A51, 26E60, 41A55
References
[1] Anderson, G.D., Vamanamurthy, M.K. and Vuorinen, M., Generalized convexity and inequalities, J. Math. Anal. Appl. 335, 1294-1308, 2007.
[2] İşcan, İ., Hermite-Hadamard type inequalities for s-GA-convex functions, arXiv:1306.1960v1, 2013.
[3] İşcan, İ., New general integral inequalities for some GA-convex and quasi-geometrically convex functions via fractional integrals, arXiv:1307.3265v1, 2013.
[4] İşcan, İ., Some Generalized Hermite-Hadamard Type Inequalities for Quasi-Geometrically Convex Functions, American Journal of Mathematical Analysis 1, no. 3, 48-52. doi: 10.12691/ajma-1-3-5, 2013.
[5] Latif, M.A., New Hermite-Hadamard type integral inequalities for GA-convex functions with applications, Analysis, Volume 34, Issue 4, 379-389, 2014.
[6] Niculescu, C.P., Convexity according to the geometric mean, Math. Inequal. Appl., 3 (2), 155-167. Available online at http://dx.doi.org/10.7153/mia-03-19, 2000.
[7] Niculescu, C.P., Convexity according to means, Math. Inequal. Appl. 6 (4), 571-579. Available online at http://dx.doi.org/10.7153/mia-06-53, 2003.
[8] Satnoianu, R.A., Improved GA-convexity inequalities, Journal of Inequalities in Pure and Applied Mathematics, Volume 3, Issue 5, Article 82, 2002.
[9] Zhang, T-Y., Ji, A-P. and Qi, F., Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means, Le Matematiche, Vol. LXVIII - Fasc. I, pp. 229-239, doi: 10.4418/2013.68.1.17, 2013.
[10] Zhang, X-M., Chu, Y-M. and Zhang, X-H., The Hermite-Hadamard type inequality of GA-convex functions and its application, Journal of Inequalities and Applications, Volume 2010, Article ID 507560, 11 pages, doi:10.1155/2010/507560, 2010.
1 Adiyaman University, Adiyaman, Turkey, [email protected] 2 Agri Ibrahim Cecen University, Agri, Turkey, [email protected] 3 Ordu University, Ordu, Turkey, [email protected]
- 77 -
On Multistep Iteration Method for Contractive Condition of Integral Type in
Banach Spaces
M. Abdussamed Maldar1 and Vatan Karakaya2
Abstract. In this work, we show that the new multistep iteration process converges to
unique fixed point of contractive condition of integral type. Also, we prove some stability
results for new multistep iteration process by using this mapping in normed linear space.
Furthermore, this iteration method is equivalent to Mann iterative scheme.
Keywords. Contractive Condition of Integral Type, Multistep Iteration, Normed
Linear Space.
AMS 2010. 47H10, 54H25
References
[1] Berinde, V., On the stability of some fixed point procedures, Bul. Ştiint. Univ.Baia Mare,
Ser. B, Matematica-Informatica, vol. 18, No:1,7-14, 2002.
[2] Branciari, A., A fixed point theorem for mappings satisfying a general contractive
condition of integral type, Int. J. Math. Math. Sci., no 9,521-536. 1.1, 2002.
[3] Gürsoy, F., Karakaya, V., and Rhoades, B. E., Data dependence results of new multistep
and S iterative schemes for contractive like operators, Fixed Point Theory an Applications,
article 76, 2013.
[4] Harder, A. M., and Hicks, T. L., Stability results for fixed point iteration procedures,
Math. Japonica, vol. 33, no. 5, 693-706, 1988.
[5] Mann, W.R. Mean value methods in iteration, Proceedings of the American Mathematical
Society, Vol. 4, No. 3, pp. 506-510, 1953.
[6] Olatinwo, M. O., Some stability results for Picard and Mann iteration processes using
contractive condition of integral type, creative Math. and inf.,vol. 19, pp. 57-64, 2010.
[7] Osilike, M. O., Stability results for the Ishikawa fixed point iteration procedure, Indian J.
Pure Appl. Math., vol. 26, pp. 937-945, 1995.
1 Aksaray University, Aksaray, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected]
- 78 -
An Inequality and Its Applications
Mehmet Ali Sarıgöl1
Abstract. In this paper, we prove a simple inequality which plays important role in the
summability theory, matrix operators theory, approximation theory, and also provides great
convenience in computations. As a corollary, we give the well-known results of [1], [2], [5]
under some simpler conditions, and a very short and different proofs of results in [6],[7].
Keywords. Inequality, Matrix Transformations, Sequence Spaces
AMS 2010. 26D15, 40C05, 46A045
References
[1] Stieglitz, M. und Tietz, M., Matrixtransformationen von Folgenraumen Eine
Ergebnisübersicht, Math Z., 154, 1-16, 1977.
[2] Wilansky, A., Summability Throught Functional Analysis, Elsevher Science Publishing
Company, New York, 1984.
[3] Maddox, I. J., Elements of Functional Analysis, Cambridge University Press, London
1970.
[4] Bor, H., A note on two summability methods, Proc. Amer Math. Soc. 98, 81-84, 1986.
[5] Malkowsky, E., Rakocevic, V., Zivkovic, S., Matrix transformations between the
sequence space bv^pand certain BK spaces. Bull. Cl. Sci. Math. Nat. Sci. Math. 27, 33-46.
2002.
[6] Sarigol, M.A., Characterization of absolute summability factors, J. Math. Anal. Appl. 195,
537-545, 1995.
[7] Sarigol, M.A., On two absolute Riesz summability factors of infinite series, Proc. Amer.
Math. Soc. 118, 485-488, 1993.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 79 -
Connectedness of Suborbital Graphs of Some Modular Groups
Murat Beşenk1
Abstract. The matrix group 𝑆𝑆(2,ℤ) is generated by two elements 0 1−1 0 and
1 1−1 0. Identifying elements in 𝑆𝑆(2,ℤ) which differ by a sign, one obtains the modular
group 𝑃𝑆𝑆(2,ℤ). We use Γ to denote the image of Γ/±𝐼 in 𝑃𝑆𝑆(2,ℤ) if there is no confusion.
A complete classification of the normal congruence subgroups of the modular group Γ is
given by Newman [4]. Congruence subgroups are a class of arithmetic subgroups which are
easy to describe. For example, given positive integers 𝑚 and 𝑛 the following are some well-
known congruence subgroups: Γ(𝑛),Γ0(𝑛),Γ1(𝑛),Γ(𝑚, 𝑛),Γ𝜃.
In this paper, we investigate connectedness of a suborbital graph for some congruence
subgroups. In order to make graph connected, we examine necessary and sufficient conditions
for the 𝐹𝑢,𝑛, whose vertices form the block [∞]. Furthermore the structure of Farey graph in
Poincare disk 𝔻 = 𝑧 ∈ ℂ ∶ |𝑧| < 1 is analyzed.
Keywords. Congruence Subgroups, Suborbital Graph, Connectedness.
AMS 2010. 05C20, 11F06, 20H10.
References
[1] Akbaş, M., On suborbital graphs for the modular group, Bulletin London Mathematical
Society, 33, 647-652, 2001.
[2] Kesicioğlu, Y., Akbaş, M., Beşenk, M., Connectedness of a suborbital graph for the
congruence subgroups, Journal of Inequalities and Applications, 1, 117-124, 2013.
[3] Jones, G. A., Singerman, D., Wicks, K., The modular group and generalized Farey
graphs, London Mathematical Society Lecture Note Series, 160, 316-338, 1991.
[4] Newman, M., Classification of normal subgroups of the modular group, American Journal
of Mathematics,126, 267-277, 1967.
[5] Sims, C. C., Graphs and finite permutation groups, Math. Zeitschrift, 95, 76-86, 1967.
1 Karadeniz Technical University, Trabzon, Turkey, mbesenktu.edu.tr
- 80 -
A New Perspective on Paranormed Riesz Sequence Space of Non-Absolute Type
Murat Candan1
Abstract. The current article mainly dwells on introducing Riesz sequence space
)~( pu
q Br that consists of all sequences whose −BRqu
~ transforms are in the space ),( p where
( )nn srBB ,~ = stands for double sequential band matrix and ( )∞=0nnr and ( )∞=0nns are given
convergent sequences of positive real numbers. Some topological properties of the new brand
sequence space have been investigated as well as −α , −β and −γ duals. Additionally, we
have also constructed the basis of ).~( pu
q Br Eventually, we characterize a matrix class on the
sequence space.
Keywords. Sequence Spaces, Double Sequential Band Matrix, Alpha-, Beta-,
Gamma-Duals, Matrix Transformations.
AMS 2010. 44A45, 40C05, 46J05.
References
[1] B. Altay and F. Başar, On the paranormed Riesz sequence spaces of non-absolute type,
Southeast Asian Bull. Math. 26, 701—715, 2002.
[2] M. Candan and G. Kılınç, A different look for paranormed Riesz sequence space of
derived by Fibonacci Matrix, under press.
[3] M. Başarır, On the generalized Riesz B −difference sequence spaces, Filomat 24(4), 35-
52, 2010.
[4] M. Başarır, M. Öztürk, On the Riesz diference sequence space, Rend. Circ. Mat. Palermo
57, 377-389, 2008.
[5] M. Başarır, E.E. Kara, On compact operators on the Riesz mB − difference sequence
space-II, Iran. J. Sci. Technol. Trans. 36A (3), 371—376, 2012.
[6] N.A. Sheikh, A.H. Ganie, A new paranormed sequence space and some matrix
transformations, Acta Math. Acad. Paedago. Nyregy., 28, 47-58, 2012.
[7] A.H. Ganie, N.A. Sheikh, New type of paranormed sequence space of non-absolute type
and a matrix transformation, Int. J. of Mod. Math. Sci, 8 (2), 196-211, 2013.
[8] M. Candan and A. Güneş, Paranormed sequence space of non-absolute type founded
using generalized difference matrix, Proc. Nat. Acad. Sci. India Sect. A., 85 (2), 269-276,
2015.
1 Inonu University, Malatya, Turkey, [email protected]
- 81 -
Almost 𝑰-Convergent Sequence Spaces Defined by Orlicz Function
Mahmut Daştan1, Merve İlkhan2 and Emrah Evren Kara3
Abstract. The purpose of this presentation is to introduce and study some sequence
spaces which are defined by combining the concepts of Orlicz function, an infinite matrix and
ideal convergence. We establish some inclusion relations between the resulting spaces and
examine some properties of these spaces.
Keywords. Sequence Spaces, Orlicz Function, 𝐼-Convergence.
AMS 2010. 46A45, 40A35
References
[1] Kara, E. E., İlkhan, M., On some paranormed A-ideal convergent sequence spaces defined
by Orlicz function, Asian J. Math. Comput. Res., (in press).
[2] Güngör, M., Et, M., Altın, Y., Strongly (𝑉𝜎; 𝜆; 𝑞)-summable sequences defined by Orlicz
functions, Appl. Math. Comput., 157, 561-571, 2004.
[3] Tripathy, B. C., Hazarika, B., Some I-convergent sequence spaces defined by Orlicz
functions, Acta Math. Appl. Sin. 271, 149-154, 2011.
[4] Aiyub, M., Some Lacunary sequence spaces of invariant means defined by Musielak-
Orlicz functions, British J. Math. Comput. Sci., 4, 12, 1682-1692, 2014.
1 Duzce University, Duzce, Turkey, [email protected] 2 Duzce University, Duzce, Turkey, [email protected] 3 Duzce University, Duzce, Turkey, [email protected]
- 82 -
On Exton's q-Exponential Function
Mahmoud Jafari Shah Belaghi1 and Nuri Kuruoğlu2
Abstract. In this paper, we study about the q-exponential function which was
introduced by Exton and addition theorem for Exton’s q-exponential function was proposed.
Keywords. Exton's q-Exponential Function, Symmetric q-Binomial.
AMS 2010. 11B65, 33D05.
References
[1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012.
[2] Exton, H.: Q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester; Halsted Press [Wiley, Inc.], New York, 1983.
[3] Gasper, G., Rahman, M., Basic hypergeometric series. Vol. 96, Cambridge university
press, 2004.
1Bahcesehir University, Istanbul, Turkey, [email protected] 2Bahcesehir University, Istanbul, Turkey, [email protected]
- 83 -
Some Properties of q-Exponential and q-Trigonometric Functions
Mahmoud Jafari Shah Belaghi1 and Nuri Kuruoğlu2
Abstract. In this paper, we study about the q-exponential functions and some useful
properties of these q-exponential functions are proposed.
Keywords. q-Exponential Functions, q-Trigonometric Function, q-Hypergeometric
Functions.
AMS 2010. 11B65, 33D05.
References
[1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012.
[2] Jackson F.H., A basic-sine and cosine with symbolical solution of certain differential
equations. Proc. Edinburgh math. Soc. 22, 28-39, 1904.
[3] Gasper, G., Rahman, M., Basic hypergeometric series. Vol. 96, Cambridge university
press, 2004.
[4] Hahn W., Beiträge zur Theorie der Heineschen Reihen. Mathematische Nachrichten 2,
340-379, 1949.
1Bahcesehir University, Istanbul, Turkey, [email protected] 2Bahcesehir University, Istanbul, Turkey, [email protected]
- 84 -
On a Volterra Equation of the Second Kind with Spectral Parameter
Amangaliyeva M.M.,1 Jenaliyev M.T.,2 Kosmakova M.T,3 and Ramazanov M.I.4
Abstract. Solving the boundary value problems of the heat equation in noncylindrical
domains degenerating at the initial moment leads to the necessity of research of the singular
Volterra integral equations of the second kind, when the norm of the integral operator is equal
to 1. The paper deals with the singular Volterra integral equation of the second kind, to which
by virtue of 'the incompressibility' of the kernel the classical method of successive
approximations is not applicable. It is shown that the corresponding homogeneous equation
when |𝜆| > 1 has a continuous spectrum, and the multiplicity of the characteristic numbers
increases depending on the growth of the modulus of the spectral parameter |𝜆|. By the
Carleman-Vekua regularization method [1] the initial equation is reduced to the Abel
equation. The eigenfunctions of the equation are found explicitly. Similar integral equations
also arise in the study of spectral-loaded heat equations [2], [3].
Keywords. Volterra Integral Equation, Abel Equation, Spectrum.
AMS 2010. 45D05, 45C05, 45E10.
References
[1] Vekua, I.N., Generalized analytic functions (In Russian), Glavnaya redaktsiya fiz.-mat.lit.,
Moscow, 1988.
[2] Amangaliyeva, M.M., Akhmanova, D.M., Jenaliyev, M.T., Ramazanov, M.I., Boundary
value problems for a spectrally loaded heat operator with load line approaching the time axis
at zero or infinity (In Russian), Differential Equations [Differetialniye uravneniya], 47:2, 231-
243, 2011.
[3] Amangaliyeva, M.M., Akhmanova, D.M., Jenaliyev, M.T., Ramazanov, M.I., About
Dirichlet boundary value problem for the heat equation in the infinite angular domain,
Boundary Value Problems, 2014:213, 1-21, 2014.
1 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, [email protected] 2 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, [email protected] 3 Al-Farabi Kazakh National University, Almaty, Kazakhstan, [email protected] 4 E.A. Buketov Karaganda State University, Karaganda, Kazakhstan, [email protected]
- 85 -
On Some Fixed Point Results for Rational 𝑨𝝋-Contractive Mappings in 2-Metric Spaces
Endowed with Partial Order
Mahpeyker Öztürk1
Abstract. In this paper some new fixed point results have been established in the
context of 2-metric spaces endowed with partial order for rational 𝐴𝜑-contractive mappings.
The results given in this study generalize, extend and unify the fixed point theorems existing
in the literatüre.
Keywords. Fixed Point, Rational 𝐴𝜑-Contraction, Ordered 2-Metric Space.
AMS 2010. 47H10, 54H25.
References
[1] Shahkoohi, R.J. Razani, A., Some fixed point theorems for rational Geraghty contractive
mappings in ordered b-metric spaces, Journal of Inequalities and Applications, 2014, 373, 23
pages, 2014.
[2] Gupta, V. Kaur, R., Some common fixed point theorems for a class of A-contractions on a
2-metric space, Inter. J. Pure and Applied Mathematics, 78, 6, 909-916, 2012.
[3] Rhaodes, B. E., A comparison of various definitions of contractive mappings, Trans.
Amer. Math. Soc., 26, 257-290, 1977.
[4] Sharma, P. L. Bajaj, N., Fixed point theorems for mappings satisfying rational
inequalities, Jnanabha, 13, 107-112, 1983.
1 Sakarya University, Sakarya, Turkey, [email protected]
- 86 -
An Approach to the Stability of Nonlinear Volterra Integral Equations
Nasrin Eghbali P0F
1
Abstract. Fractional differential and integral equations can serve as an excellent tool
for the description of mathematical modelling of systems and processes in the fields of
economics, physics, chemistry, aerodynamics, and polymer rheology. It also serves as an
excellent tool for the description of hereditary properties of various materials and processes.
The purpose of this talk is to investigate the Hyers-Ulam stability for a class of
nonlinear Volterra integral equations under some natural conditions which corresponds to
numerous related outcomes in [1,2,3,4,5,6]. In fact, we presented and studied two types of
stability of this equation.
Keywords. Fractional Differential Equation, Fractional Derivative, Global Existence,
Hyers-Ulam Stability, Hyers-Ulam-Rassias Stability.
AMS 2010. Primary 46S40; Secondary 39B52, 39B82, 26E50, 46S50.
References:
[1] M. Akkouchi, Hyers--Ulam--Rassias stability of nonlinear Volterra integral equations
via a fixed point approach, Acta Univ. Apulensis Math. Inform. 26, 257—266, 2011.
[2] A. Bahyrycz, J. Brzdek and Z. Lesniak, On approximate solutions of the generalized
Volterra integral equation, Nonlinear Analysis: Real World Applications, 20 , 59—66, 2014.
[3] N. Brillouet-Belluot, J. Brzdek and K. Cieplinski, On some recent developments in Ulam's
type stability, Abstr. Appl. Anal. 2012, Article ID 716936, 41 pp, 2012.
[4] J. Brzdek, L. Cadariu and K. Cieplinski, Fixed point theory and the Ulam stability, Journal
of Function Spaces, 2014, Article ID 829419, 16 pp 2014.
[5] L. P. Castro and A. Ramos, Hyers--Ulam--Rassias stability for a class of nonlinear
Volterra integral equations, Banach J. Math. Anal. 3, 36—43, 2009.
[6] L. P. Castro and A. Ramos, Hyers--Ulam and Hyers--Ulam--Rassias stability of Volterra
integral equations with delay, Integral Methods in Science and Engineering, 1, Birkhäuser,
Boston, 85—94, 2010.
1 University of Mohaghegh Ardabili, Ardabil, Iran, [email protected];[email protected]
- 87 -
Abel Convergence of Convolution Operators
Özlem Girgin Atlıhan1 and Mehmet Ünver2
Abstract. The classical Korovkin approximation theory deals with the convergence of
a given sequence ( )nL of positive linear operators on C[a,b]. When the sequence of positive
linear operators does not converge to the identity operator then it may be usefull to use some
summability methods. In this paper, we study some Korovkin type approximation theorems
for the sequences of convolution operators via Abel method which is a sequence-to-function
transformation. We also deal with the rate of Abel convergence.
Keywords. Abel Convergence, Convolution Operator, Korovkin Approximation
Theorem.
AMS 2010. 41A25, 41A36, 40A05
References
[1] F. Altomare, Korovkin-type theorems and approximation by positive linear operators,
Surveys in Approximation Theory 5, 92-164 , 2010.
[2] J. Boos, Classical and Modern Methods in Summability, Oxford Univ. Press , 2000.
[3] O. Duman, A-statistical convergence of sequences of convolution operators. Taiwanese J.
Math. 12, no. 2, 523-536, 2008.
[4] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence,
Rocky Mountain J. Math. 32 ,129-138, 2002.
[5] H. Hacısalihoğlu and A. D. Gadjiev, Lineer pozitif operatör dizilerinin yakınsaklığı,
AÜFF Döner Sermaye İşletmesi Yayınları, no: 31, Ankara, 1995.
[6] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi,
1960.
[7] M. Unver, Abel transforms of positive linear operators, AIP Conference Proceedings
1558, 1148, 2013.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected]
- 88 -
On Weighted Approximation by Singular Integral Operators Depending on
Two Parameters
Özge Güller1, Gümrah Uysal2 and Ertan İbikli3
Abstract. In this talk, we give some theorems about pointwise approximation to the
functions which belong to weighted Lebesgue space Lp,w (D), where D=<a,b> is closed
semi-closed or open interval in R, at their characteristic points, by family of singular
integral operators depending on two parameters.
Keywords. Singular Integral, Pointwise Approximations.
MSC 2010. 41A35, 41A25.
References
[1] Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation, Academic Press,
New York, London, 1971.
[2] Gadjiev, A. D., On convergence of integral operators depending on two parameters, Dokl.
Acad. Nauk. Azerb. SSR, XIX, 12, 3-7, 1963.
[3] Gadjiev, A. D., On the order of convergence of singular integrals depending on two
parameters, Special questions of functional analysis and its applications to the theory of
differential equations and functions theory, Baku, 40-44, 1968.
[4] Karslı, H. and İbikli, E., On convergence of convolution type singular integral operators
depending on two parameters, Fasc. Math., 38, 25-39, 2007.
[5] Taberski, R., Singular integrals depending on two parameters, Rocznicki Polskiego
towarzystwa matematycznego, Seria I. Prace matematyczne, VII, 173-179, 1962.
1 Ankara University, Ankara, Turkey, [email protected] 2 Karabuk University, Karabuk, Turkey, [email protected] 3 Ankara University, Ankara, Turkey, [email protected]
- 89 -
Lyapunov Functions and Strict Stability of Caputo Fractional Differential Equations
R. Agarwal1, S. Hristova2 and D. O’Regan3
Abstract. One of the main properties studied in the qualitative theory of differential
equations is the stability of solutions. Many authors defined and studied various types of
stability for different classes of differential equations but the investigation of the stability of
fractional order systems is quite recent. There are several approaches in the literature to study
stability, one of which is the Lyapunov approach. Note application of Lyapunov techniques to
fractional differential equations causes many difficulties. We will present the approach using
Lyapunov functions for studying some stability properties of nonlinear Caputo fractional
differential equations. A new definition for the derivative of Lyapunov functions among the
studied equation is introduced. It is based on the Caputo fractional Dini derivative of a
function. Comparison results using this new definition and scalar fractional differential
equations are presented and sufficient conditions for some types of stability are given. Several
examples will illustrate the advantages and the usefulness of the introduced definition
comparatively to the known ones in the literature.
Keywords. Luapunov Functions, Caputo Fractional Derivative, Stability, Fractional
Differential Equations.
AMS 2010. 34A34, 34A08, 34D20.
Acnowldgements. The research was partially supported by the Fund NPD, Plovdiv
University, No. NI15FMIIT002.
References
[1] R. Agarwal, D. O'Regan, S. Hristova, Stability of Caputo fractional differential equations
by Lyapunov functions, Appl. Math. (accepted)
[2] R. Agarwal, S. Hristova, Strict stability in terms of two measures for impulsive differential
equations with “supremum”, Applicable Analysis, 91, 7, 1379-1392, 2012.
1 Texas A&M University, Kingsville, USA, [email protected] 2 Plovdiv University, Plovdiv, Bulgaria, [email protected] 3 National University of Ireland, Galway, Ireland, [email protected]
- 90 -
On a Caputo type fractional operator
Ricardo Almeida1 Agnieszka Malinowska2 and Tatiana Odzijewicz3
Abstract. We study a Caputo-Katugampola fractional derivative, which is a
generalization of the Caputo and the Caputo--Hadamard fractional derivatives [1,2]. After
presenting some important results about the fractional operator, we study variational problems
with dependence on this operator. We present sufficient and necessary conditions of first and
second order to determine the extremizers of a functional. The cases of integral and
holomonic constraints are also considered. An existence and uniqueness theorem for a
fractional Caputo type problem, with dependence on the Caputo-Katugampola derivative, is
proven. A decomposition formula for the Caputo-Katugampola derivative is obtained. This
formula allows us to provide a simple numerical procedure to solve the fractional differential
equation.
Keywords. Fractional Calculus, Fractional Differential Equations, Caputo-
Katugampola Derivative, Numerical Methods
AMS 2010. 26A33, 34A08, 34K28
References
[1] Katugampola, U.N., New approach to a generalized fractional integral, Appl. Math.
Comput. 218, 860-865, 2011.
[1] Katugampola, U.N., A new approach to generalized fractional derivatives, Bull. Math.
Anal. App. 6, 1-15, 2014.
1 University of Aveiro, Aveiro, Portugal, [email protected] 2 Bialystok University of Technology, Bialystok, Poland, [email protected] 3 University of Aveiro, Aveiro, Portugal, [email protected]
- 91 -
Chaotic Behavior of Logistic Maps in Mann Orbit
Renu Chugh 1 and Ashish 2
Abstract. The idea of logistic map rx(1-x) was given by the Belgian mathematician Pierre
Francois Verhulst around 1845 and worked as basic model to study the discrete dynamical
system. It is a model of population growth that exhibits different types of behavior depending on
the value of a few parameters. Above a certain parameter value, the logistic map shows the
chaotic behavior. For choosing x between 0 and 1 and 0 < r ≤ 4, the logistic map has found a
celebrated place in chaos, fractal and discrete dynamics. In 1994, Holmgren Richard A., [5] has
studied that the logistic map in Picard orbit becomes chaotic in nature for r > 4 (see also [1], [2],
[4]). In recent decades, Rani et. al. [3, 6, 7] has shown that in Mann, Ishikawa and Noor iterative
methods the logistic map is stable for the larger values of r than that of Picard orbit. Our goal in
this paper is to study the chaotic behavior of logistic map in Mann iterative method using
bifurcation representation. We see that the range of chaotic behavior of logistic map increases
drastically. Also, the chaotic behavior disappears in some cases.
Keywords. Logistic Map, Mann Iterative Procedure, Periodicity, Chaos, Bifurcation.
AMS 2010. 28A80, 34H10.
References
[1] Alligood Kathleen T., Sauer Tim D., Yorke James A., Chaos: An Introduction to
Dynamical Systems, New York, Springer-Verlag, 2006.
[2] Ausloos M., Dirickx M., The Logistic Map and the Route to Chaos: from the Beginnings
to Modern Applications, New York, Springer-Verlag, 2006.
[3] Chugh R., Rani M. and Ashish, Logistic map in Noor orbit, Chaos and Complexity Letter,
6, 3, 167-175, 2012.
1 Maharshi Dayanand University, Rohtak, INDIA, [email protected] 2 Central University of Haryana, Mahendergarh, [email protected]
- 92 -
[4] Devaney Robert L., A First Course in Chaotic Dynamical Systems: Theory and
Experiment, Addison-Wesley, 1992.
[5] Holmgren R. A., A First Course in Discrete Dynamical Systems, Springer-Verlag, 1994.
[6] Rani M. and Agarwal R., A new experimental approach to study the stability of logistic
map, Chaos Solitons Fractals, 41, 4, 2062-2066, 2009.
[7] Rani M. and Kumar V., A new experiment with the logistic map, J. Indian Acad. Math. 27,
1, 143-156, 2005.
- 93 -
Interpolation of Function Spaces Associated to a Vector Measure
R. del Campo1 , A. Fernández2, F. Mayoral3 and F. Naranjo4
Abstract. A basic problem in interpolation theory is to describe the spaces obtained by
applying an interpolation method to concrete compatible couples of spaces. In this talk we
analyze the results obtained in [1], [2], [3] and [4] by applying different interpolation methods
(real and complex) to different couples of Banach and quasi-Banach spaces associated to a
vector measure.
Let ν be a countably additive vector measure with values in a Banach space defined
on a δ -ring (or on a σ − algebra) of subsets of a set Ω . Associated with ν , for 1 ,p≤ < ∞ are
the real function Banach spaces ( ),pL ν and ( ),pwL ν of equivalence classes of scalar measurable
functions p -integrable, weak p -integrable, with respect to ν . These two spaces are
equipped with the topology of convergence in p -mean given by the norm
1
( ): sup | | | , |: 1 , ( ).p
w
pp pwL
f f d x x f Lν
ν ν′ ′
Ω = ⟨ ⟩ ≤ ∈ ∫
We are going to point out that the interpolation results for vector measures defined on δ -
rings can be very different from those on the context of σ − algebras.
Keywords. Complex Interpolation Methods, Real Interpolation Methods, Lions-Peetre
K-Functional, Integrable Function, Vector Measure, Locally Strongly Additive Measure.
AMS 2010. 46B70, 46G10, 46E30
References [1] del Campo R., Fernández A., Mayoral F., Naranjo F., Complex interpolation of pL -spaces of vector measures on δ -rings, J. Math. Anal. Appl., 405, 2, 518-529, 2013. [2] del Campo R., Fernández A., Mayoral F., Naranjo F., A note on real interpolation of pL -spacesof vector measures on δ -rings, J. Math. Anal. Appl., 419, 2, 995-1003, 2014. [3] Fernández A., Mayoral F., Naranjo F., Real interpolation method on spaces of scalar integrable functions with respect to vector measures, J. Math. Anal. Appl., 376, 1, 203-211, 2011. [4] Fernández, A., Mayoral, F., Naranjo, F., Sánchez-Perez, E. A., Complex interpolation of spaces of integrable functions with respect to a vector measure, Collect. Math., 61, 3, 241-252, 2010.
1 University of Seville, Spain, [email protected] 2 University of Seville, Spain, [email protected] 3 University of Seville, Spain, [email protected] 4 University of Seville, Spain, [email protected]
- 94 -
Complex Interpolation Operators and Optimal Domains
R. del Campo1, A. Fernández2, O. Galdames3, F. Mayoral4 and F. Naranjo5
Abstract. Let X be an order continuous Banach function space on a finite measure
space ( , , )µΩ Σ and let E a Banach space. Given a continuous linear operator :T X E we
consider the vector measure ( ) : ( )T Am A T χ= . Then the space 1( )TL m of scalar integrable
functions with respect to the vector measure Tm is the optimal domain of the operator T . In
this talk we consider two order continuous Banach function spaces 0X , 1X a Banach space
interpolation pair ( )0 1,E E and admissible operator between the pairs ( )0 1,X X and ( )0 1,E E .
If 0 1 [ ] 0 1 [ ]:[ , ] [ , ]T X X E Eθ θ θ is the interpolated operator by the first complex method
Calderon and 0m , 1m and mθ are the vector measure coming from the operators 0XT ,
1XT and
Tθ respectively, we study the relationship between the optimal domain 1( )L mθ of Tθ and the
interpolated space 1 10 1 [ ][ ( ), ( )]L m L m θ of the optimal domains 1
0( )L m and 11( ).L m Next we
apply the obtained result to study interpolation of p -th power factorable operators.
Keywords. Complex Interpolation, Vector Measure, Factorable Operator.
1 University of Seville, Spain, [email protected] 2 University of Seville, Spain, [email protected] 3 University of Valencia, Spain, [email protected] 4 University of Seville, Spain, [email protected] 5 University of Seville, Spain, [email protected]
- 95 -
q-Analogues of Some Classical Tauberian Theorems for Cesàro Integrability
Sefa Anıl Sezer1,2, İbrahim Çanak2, Ümit Totur3
Abstract. Let ( )f x be a function defined on [0,∞) satisfying
0
( )x
qf t d t < ∞∫
for all ,qx +∈ , and let 0
( ) ( ) .x
qs x f t d t= ∫ The q-Cesàro means of ( )s x are defined by
0
1( ( )) ( ) .x
qs x s t d tx
s = ∫
The function ( )s x is said to be q-Cesàro integrable to A if
lim ( ( )) .x
s x As→∞
=
In this talk we introduce the concept of q-Cesàro integrability and develop q-analogues of
some results in summability theory. We also present sufficient Tauberian conditions under
which the convergence of ( )s x follows from its q-Cesàro integrability.
Keywords. Cesàro Integrability, Tauberian Theorems, Quantum Calculus.
AMS 2010. 40E05, 05A30, 26A03.
References
[1] Çanak, İ., Totur Ü., A Tauberian theorem for Cesàro summability of integrals, Appl.
Math. Lett. 24, 3, 391–395, 2011.
[2] Çanak, İ., Totur Ü., Alternative proofs of some classical type Tauberian theorems for
Cesàro summability of integrals, Math.Comput. Modelling 55, 3, 1558–1561, 2012.
[3] Fitouhi A., Brahim, K., Tauberian theorems in quantum calculus, J. Nonlinear Math.
Phys., 14, 3, 324–340, 2007.
[4] Hardy, G. H., Divergent series, Clarendon Press, Oxford, 1949.
[5] Kac, V., Cheung, P., Quantum calculus, Universitext, Springer-Verlag, New York, 2002.
[6] Tauber, A., Ein satz der Theorie der unendlichen Reihen, Monatsh. f. Math. 8, 273–277,
1897.
1 Istanbul Medeniyet University, Istanbul, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected] 3 Adnan Menderes University, Aydin, Turkey, [email protected]
- 96 -
Numerical Results of Extended Lane–Emden Type Equations
Ali Akgül1, Adem Kılıçman2 and Mustafa Inc3
Abstract. We use the reproducing kernel Hilbert space method [1], [2], [3] for solving
extended Lane–Emden type equations [4]. A powerful method is shown in the reproducing
kernel Hilbert spaces. We define many useful reproducing kernel functions and we find the
best kernel function for approximate solutions.
Keywords. Lane–Emden Type Equations, Reproducing Kernel Functions and White-
Dwarf Equation.
AMS 2010. 47B32, 46E22 and 74S30.
References
[1] Aronszajn, N., Theory of reproducing kernels. Transactions of the America Mathematical
Society, 68:337–404, 1950.
[2] Cui, M., Lin, Y., Nonlinear numerical analysis in the reproducing kernel space. Nova
Science Publishers, Inc., New York, 2009.
[3] Akgül, A, A new method for approximate solutions of fractional order boundary value
problems. Neural Parallel and Sciecntific Computations, 22(1-2):223–237, 2014.
[4] Turkyilmazoglu, M., Effective computation of exact and analytic approximate solutions to
singular nonlinear equations of Lane-Emden-Fowler type. Applied Mathematical Modelling,
37(14-15):7539–7548, 2013.
1 Siirt University, Siirt, Turkey, [email protected] 2 University of Putra Malasia, Kuala Lumpur, Malaysia, [email protected] 3 Firat University, Elazıg, Turkey, [email protected]
- 97 -
Inventory Model of Type (s,S) with Regularly Varying Demands Having
Infinite Variance
Aslı Bektaş Kamışlık1,Tülay Kesemen2 and Tahir Khaniyev3
Abstract. In this study an inventory model of type (s,S) with regularly varying
demand having infinite variance is considered. Pareto distribution with infinite variance when
𝐹(𝑥) = 𝑏𝑥𝛼
, 𝑥 ≥ 𝑏, 𝑏 > 0, 1 < 𝛼 < 2 is used as a repsentative of regularly varying
distributions. This work is motivated by a study of Geluk (1997). The results from Geluk’s
paper are implemented to the stochastic process X(t) which represents the model that we
consider. Based on the main result of the study Geluk (1997) two term asymptotic expansion
for the ergodic distribution of the process X(t) is obtained as follows:
𝑄𝑌(𝜐) = 𝐹(𝜐) +1
2𝜇1𝛽𝛼−1𝐺(𝜐) + 𝑂
1𝛽1−(2−𝛼)2
where;
𝐹(𝜐) =4𝜐 − 𝜐2
4
𝐺(𝜐) =(21−𝛼(2 − 𝜐)2 − (2 − 𝜐)3−𝛼)𝑏𝛼
(𝛼 − 1)(2 − 𝛼)(3 − 𝛼)
𝜐 ∈ [0,2), 𝑥 ≥ 𝑏, 𝑏 > 0, 1 < 𝛼 < 2, 𝛽 = 𝑆 − 𝑠
2→ ∞.
Moreover weak convergence theorem is proved for the ergodic distribution.
Keywords. Inventory Model of Type (s,S), Heavy Tailed Distributions with Infinite
Variance, Regular Variation, Renewal Reward Process, Asymptotic Expansion.
AMS 2010. 60K05, 41A60.
References
[1] Asmussen, S., Ruin probabilities ,Singapore: Advanced Series on Statistical Science and Applied Probability, World Scientific Publishing 2000.
[2] Bekar N., Aliyev R., Khaniyev T., Asymptotic expansions for a renewal-reward process with Weibull distributed interference of chance, Contemporary Analysis and Applied Mathematics, 1(2), 200-211 2013.
1 Recep Tayyip Erdogan University, Rize, Turkey, [email protected] 2 Karadeniz Technical University, Trabzon, Turkey, [email protected] 3 TOBB University of Economics and Technology, Ankara, Turkey, [email protected]
- 98 -
[3] Bingham N.H, Goldie, C.M. Teugels, J.L., Encyclopedia of Mathematics and its Applications-Regular Variation, Cambridge University Press New-York Vol-2, 1987. [4] Borovkov A.A., Asymptotic Methods in Queuing Theory, John Wiley, New York, 1984. [5] Borokov,A.A., Stochastic Process in Queuing Theory, Springer-Verlag, New York , 1976. [6] Embrechts, P., Klüppelberg,C., Mikosh, T., Modelling Extremal Events, Springer Verlag, 1997. [7] Feller W., Introduction to Probability Theory and Its Applications II, John Wiley, New York, 1971. [8] Geluk, J. L., A Renewal Theorem in the finite-mean case., Proceedings of the American Mathematical Society, 125(11), 3407-3413, 1997. [9] Gihman I. I., Skorohod A. V., Theory of Stochastic Processes II, Springer, Berlin, 1975. [10] Khaniyev T., Kokangül A., Aliyev R., An asymptotic approach for a semi-Markovian inventory model of type (s, S), Applied Stochastic Models in Business and Industry, 29(5), 439-453, 2013. [11] Seneta, E., Lecture Notes in Mathematics-Regularly Varying Functions Springer-Verlag New York, 1976.
- 99 -
Inverse Spectral Problem for Sturm Liouville Operator with Spectral Parameter
Dependent Boundary Condition
Aynur Çöl 1
Abstract. Consider the boundary value problem generated by the equation
−𝑦" + 𝑞(𝑥)𝑦 = 𝜆2𝑦, 0 < 𝑥 < ∞
with the boundary condition
(𝛼0 + 𝑖𝛼1𝜆 − 𝛼2𝜆2−𝑖𝛼3𝜆3)𝑦′(0) − (𝛽0 + 𝑖𝛽1𝜆 − 𝛽2𝜆2−𝑖𝛽3𝜆3)𝑦(0) = 0
where λ is a spectral parameter, 𝑞(𝑥) is real valued function with the condition
∫ (1 + 𝑥)|𝑞(𝑥)|𝑑𝑥∞0 < ∞.
Here the polynomials in boundary condition satisfy the relations for 𝛼𝑖 ,𝛽𝑖 ∈ ℝ (𝑖 = 1,2,3)
𝛼𝑖+1𝛽𝑖 − 𝛼𝑖𝛽𝑖+1 > 0, 𝛼𝑖+2𝛽𝑖 − 𝛼𝑖𝛽𝑖+2 < 0, 𝛼𝑖+3𝛽𝑖 − 𝛼𝑖𝛽𝑖+3 = 0.
In this work, it is considered the Sturm Liouville operator with eigenvalue parameter
dependent boundary condition, and the corresponding inverse scattering problem is analyzed.
Scattering data are defined, some properties of the scattering data are examined, the main
equation is obtained and it is shown that the potential is uniquely recovered by the scattering
data. When the boundary condition doesn’t contain spectral parameter, inverse problem for
equation on the half line was solved in [1]. The similar problem for Sturm Liouville operators
with discontinuous coefficient was examined in [2].
Keywords. Sturm-Liouville Operator, Inverse Problem, Scattering Data, Spectral
Parameter.
AMS 2010. 34A55, 34B40, 34B07, 34L25, 81U40.
References
[1] Marchenko, V. A. Sturm-Liouville Operators and Applications. Birkhauser, Basel, 1986.
[2] Çöl, A. Inverse spectral problem for Sturm-Liouville operator with discontinuous
coefficient and cubic polynomials of spectral parameter in boundary condition, Advances in
Difference Equations 2015:132, doi:10.1186/s13662-015-0478-7, 2015.
1 Sinop University, Sinop, Turkey, [email protected]
- 100 -
Incorporating Views on Market Dynamics in Options Hedging
Antoine E. Zambelli1
Abstract. We examine the possibility of incorporating information or views of market
movements during the holding period of a portfolio, in the hedging of European options with
respect to the underlying. Given a fixed holding period interval, we explore whether it is
possible to adjust the number of shares needed to effectively hedge our position to account for
views on market dynamics from present until the end of our interval, to account for the time-
dependence of the options' sensitivity to the underlying. We derive an analytical expression
for the number of shares needed by adjusting the standard Black-Scholes-Merton Δ quantity,
in the case of an arbitrary process for implied volatility, and we present numerical results.
Keywords. Derivatives, Δ-hedging, Black-Scholes-Merton.
AMS 2010. 91G20.
References
[1] Black, F., Scholes, M., The Pricing of Options and Corporate Liabilities, Journal of
Political Economy, 81, 3, 637-654, 1973.
[2] Cox, J., Ingersoll, J., Ross, S., A Theory of the Term Structure of Interest Rates,
Econometrica, 53, 2, 385-408, 1985.
[3] Heston, S., A Closed-Form Solution for Options with Stochastic Volatility with
Applications to Bond and Currency Options, The Review of Financial Studies, 6, 2, 327-343,
1993.
[4] Hull, J. J., Options, Futures and Other Derivatives, Pearson/Pretice-Hall, 8th Edition,
2009.
[5] Mastinsek, M., Charm-Adjusted Delta and Delta Gamma Hedging, The Journal of
Derivatives, 19, 3, 69-76, 2012.
[6] Primbs, J., Yamada, Y., A moment computation algorithm for the error in discrete
dynamic hedging, Journal of Banking and Finance, 30, 2, 519-540, 2006.
1 Imperial College of Science, Technology and Medicine, London, UK, [email protected]
- 101 -
Numerical Identification of the Filtration Capacitive Parameters in Two-Phase
Petroleum Reservoirs
A. Sakabekov1, D. Ahmed Zaki2 and Y. Auzhani3
Abstract. Modeling method is used to control a reservoir [1]. Modeling allows us to
understand the geology of a reservoir and predict its behavior under different scenarios of
development. It is necessary to predict the behavior of reservoir and technological
development indicators at all stages of the development of field. Reliable geological and
technological models are necessary to make decisions about the further study and
optimization of the development of hydrocarbon deposits. It is necessary to note that a
permanent operating geological and technological model is a central object for fields under
development [2]-[3]. Adaptation of the model to the exploration history is an important and
current problem of the field development. The identification of filtration capacitive
parameters of a reservoir, that are inherent in the model, is done by solving the inverse
problem. This is the process of history matching. Usually parameters with the highest
uncertainty, that strongly influence the solution, are corrected. And usually production and
injection of each component are known when exploration history is reproduced.
In this paper was considered non-linear three-dimensional two-phase filtration
problem in a bounded three-dimensional region under the corresponding initial and boundary
conditions. The algorithm for solving the inverse problem of parameter identification of
inhomogeneous oil reservoir is made. It was build full implicit scheme for numerical solution
for an imitation problem. Was introduced the target function [4]-[5], which is the measure of
disparity between observed values and system response. The elements of Jacobian matrix for
three-dimensional two-phase filtration problem in case of three-dimensional rectangular grid
are calculated. The iterative process for determining coefficients of model sensitivity and
coefficients of filtration parameters of inhomogeneous oil reservoir was built. It was build
algorithm for numerical solution of the parameter identification problem.
Keywords. Two-phase Petroleum Reservoirs, Filtration Capacitive Parameters,
Geological and Technological Model.
AMS 2010. 49M37.
1 KBTU, Almaty, Kazakhstan, [email protected] 2 KAZ NU, Almaty, Kazakhstan, [email protected] 3 KBTU, Almaty, Kazakhstan, [email protected]
- 102 -
References
[1] Reservoir Simulation, Heriot Watt University, 565 p., 2005.
[2] ECLIPSE, Technical Description, Schlumberger, 2, 25, 2008.
[3] ECLIPSE, Reference Manual, Schlumberger, 2, 2552 p., 2008.
[4] Auzhani E. The identification of petroleum reservoir parameters. Synopsis of the
dissertation for the degree of candidate of physical and mathematical sciences, Almaty, 2010.
[5] Danayev N.T., Mukhambetzhanov S.T., Akhmed-Zaki D.Zh. Data Interpretation of
electromagnetic sounding at thermal exposure to oil reservoir // 6th International Conference
on 'Inverse Problems: Modeling and Simulation', May 21st-26th, Antalya, Turkey, p. 112.,
2012.
- 103 -
Multiple Scales Analysis and Exact Solutions for KdV Type
Nonlinear Differential Equations
Burcu Ayhan1, M. Naci Özer2 and Ahmet Bekir3
Abstract. In this paper we apply multiple scale analysis for Korteweg-de Vries (KdV)
type equations and we derive Nonlinear Schrödinger (NLS) equation. So we get a relation
between KdV Type equations and NLS equations. Also exact solutions are found for KdV
type equations. The (G′/G)-expansion methods and the (G′/G,1/G)-expansion methods are
proposed to establish new exact solutions for KdV type differential equations. We obtain
periodic and hyperbolic function solutions for these equations. These methods are very
effective for getting travelling wave solutions of nonlinear differential equations.
Keywords. KdV Type Equations, The (G′/G)-Expansion Method, The (G′/G,1/G)-
Expansion Method.
AMS 2010. 35Q53, 83C15, 35C07.
References
[1] Fordy, A. P., Özer, M. N., A new Integrable Reduction Of The Matrix NLS Equation,
Hadronic Journal, 21, 387-404, 1998.
[2] Zayed, E. M. E., Abdelaziz M. A. M., The two variables ((G′)/G,1/G)-expansion method
for solving the nonlinear KdV-mKdV, Math. Probl. Eng. 2012, 725061, 2012.
[3] Li X. L., Li Q. E. and Wang L.M., The -(G′/G,1/G) expansion method and its application
to travelling wave solutions of the Zakharov equations, Appl. Math. J. Chinese Univ. 25, 454-
462, 2010.
[4] Wang, M.L., Li, X.Z., Zhang, J.L., The-(G′/G )Expansion Method and Traveling Wave
Solutions of Nonlinear Evolution Equations in Mathematical Physics Phys. Lett., A 372-417,
2008.
1 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 3 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected]
- 104 -
Numerical Solution of the Rosenau KdV-RLW Equation by Using Collocation Method
Bahar Korkmaz1 and Yılmaz Dereli2
Abstract. In this study, Rosenau KdV-RLW equation is solved numerically by a
meshless method based on RBF collocation method. Linear stability analysis is applied to
analyze the stability of the proposed method for Rosenau-KdV-RLW equation. Accuracy of
the method is discussed by computing the numerical conserved laws, error norms 𝐿2 and 𝐿∞.
Moreover, convergence rate of the solution is investigated. Comparisons are made between
the results of this method and some other earlier works.
Keywords. RBF Collocation Method, Rosenau-KdV Equation, Rosenau-RLW
Equation, Rosenau-KdV-RLW Equation.
AMS 2010. 34A37, 34A60, 34K45.
References
[1] Chung, S. K.; Finite difference approximate solutions for the Rosenau equation, Appl. Anal., 69, 149-156, 1998.
[2] Cui,Y.; Mao, D. K.; Numerical method satisfying the rst two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys. 227, 1, 376-399, 2007.
[3] Doğan, A.; Numerical solution of RLW equation using linear finite elements within Galerkin method, Appl. Math. Model. 26, 771-783, 2002.
[4] Hu, J.; Xu, Y.; Hu, B.; Conservative linear dierence scheme for Rosenau-KdV equation, Adv. Math. Phys. 2013, Article ID 423718, 2013.
[5] Pan, X.; Zheng, K.; Zhang, L.; Finite difference discretization of the Rosenau-RLW equation, Appl. Anal. 92, 12, 2578-2589, 2013.
[6] Park, M. A.; On the Rosenau equation, Mat. Aplicada e Comput. 9, 2, 145-152, 1990.
[7] Park, M. A.; Pointwise decay estimate of solutions of the generalized Rosenau equation, J.
Korean Math. Soc. 29, 261-280, 1992.
[8] Peregrine, D.H.; Calculations of the development of an undular bore, J. Fluid Mech. 25, 321-330, 1966.
[9] Peregrine, D. H.; Long waves on a beach, J. Fluid Mech. 27, 815-827, 1967.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected]
- 105 -
[10] Rosenau, P.; A quasi continuous description of a nonlinear transmission line, Phys. Scr. 34, 827-829, 1986.
[11] Rosenau, P.; Dynamics of dense discrete systems, Prog. Theor. Phys. 79, 1028-1042, 1988.
[12] Rubin, S. G.; Graves, R. A.; A Cubic Spline Approximation for Problems in Fluid Mechanics, NASA TR R-436, Washington, DC, 1975.
[13] Zuo, J. M.; On the Solitons and periodic solutions for the Rosenau-KdV and Rosenau-Kawahara equations, Appl. Math. and Comput. 215, 2, 835-840, 2009.
[14] Wongsaijai, B.; Poochinapan, K.; A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and the Rosenau-RLW equation, Appl. Math. and Comput. 245, 289-304, 2014.
- 106 -
Finite Difference Method for Film Equation in a Class of Discontinuous Functions
Bahaddin Sinsoysal1, Mahir Rasulov2 and Ethem Ilhan Sahin3
Abstract. In this study, an original method has been suggested to find a numerical
solution of initial-boundary value problem for a fourth order degenerate diffusion equation
which describes the thin film flow. For this goal, a special auxiliary problem which has some
advantages over the main problem has been introduced and using the solution of it, a new
method has been suggested for finding the weak solution of the main problem. Thus, the
numerical solution of the main problem can be calculated by using the solution of the
auxiliary problem obtained before, which expresses all of the physical properties accurately.
In the study also some computer experiments are carried out.
Keywords. Thin Film Equation, Weak Solution, Numerical Solution in a Class of
Discontinuous Functions.
AMS 2010. 35K35, 65M06.
References
[1] Ansini, L., Giacomelli, L., Doubly nonlinear thin-film equations in one space dimension,
Arch. Ration. Mech. Anal., 173 (1), 89–131, 2004.
[2] Bowen,M., King, J.R., Asymptotic Behavior of the Thin Film Equation in Bounded
Domains, European J. Appl. Math., 2 (3), 321-356, 2001.
[3] Gonoskov, I., Cyclic operator decomposition for solving the differential equations, Adv.
Pure Math., 3, 178-182, 2013.
[4] O’Brien, S.B.G., Schwartz, L.W., Theory and modeling of thin film flows, In:
Encyclopedia of Surface and Colloid Science, Copyright © by Marcel Dekker, Inc., 2002.
[5] Rasulov, M. A., Ragimova, T. A., A numerical method for solving a nonlinear first-order
equation of hyperbolic type, Differential Equations, 28, No. 7, 1016–1023, 1992.
[6] Sinsoysal, B., The analytical and a higher-accuracy numerical solution of a free boundary
problem in a class of discontinuous functions, Math. Probl. Eng., doi:10.1155/2012/791026,
2012.
1 Beykent University, Istanbul, Turkey, [email protected] 2 Beykent University, Istanbul, Turkey, [email protected] 3 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 107 -
Analysis of Sound Diffraction from a Duct with Exterior Surface Impedance
Burhan Tiryakioğlu1 and Ahmet Demir2
Abstract. In the present work, a rigorous solution is presented for the problem of
diffraction of sound waves emanating from a ring source by a semi-infinite circular
cylindrical pipe whose exterior surface is lined by an acoustically absorbent material. This
boundary-value problem is investigated using Wiener-Hopf technique. The solution is
obtained analytically by using fourier transform and saddle point technique. At the end of the
analysis, numerical results illustrating the effects of the acoustic absorbent lining on the
exterior surface of the cylinder on the sound spread are presented.
Keywords. Diffraction, Wiener-Hopf, Fourier Transform, Saddle Point Technique.
AMS 2010. 78A45, 47A68, 34B30.
References
[1] A.D. Rawlins, Radiation of sound from an unflanged rigid cylindrical duct with an
acoustically absorbing internal surface, Proc. Roy. Soc. Lond. A361,6591, 1978.
[2] A. Büyükaksoy and B. Polat, Diffraction of acoustic waves by a semi-infinite cylindrical
impedance pipe of certain wall thickness, J. Engin. Math. 33, 333-352, 1998.
[3] A.D. Rawlins, A bifurcated circular waveguide problem, IMA J. Appl. Math., 54, pp. 59-
81, 1995.
[4] A. Demir and S.W. Rienstra, Sound Radiation from a Lined Exhaust Duct with Lined
Afterbody, paper AIAA-2010-3947 of the 16th AIAA/CEAS Aeroacoustics Conference, 7-9
June 2010, Stockholm, Sweden, 2010.
[5] B. Noble, Methods Based On The Wiener-Hopf Technique For The Solution Partial
Differential Equation, Pergamon Press, New York, 1958.
1 Marmara University, Istanbul, Turkey, [email protected] 2 Karabuk University, Karabuk, Turkey, [email protected]
- 108 -
The Comparision of the Eigenvalues of Sturm-Liouville Operators
Bülent Yılmaz1
Abstract. In this paper the eigenvalues obtained by the asymptotic method and the
eigenvalues obtained by the finite difference method followed by a numerical correction, are
compared. These eigenvalues are relevant to Sturm-Liouville problems having singular
potential function, with Dirichlet boundary conditions.
Keywords. Eigenvalues, Sturm-Liouville Problems, Asymptotic Method, Numerical
Method
AMS 2010. 34L16, 34L20
References
[1] B. Yilmaz and O. A. Veliev, Asymptotic formulas for Dirichlet boundary value problems, Studia Scientiarum Mathematicarum Hungarica, vol. 42, no. 2, pp. 153-171, 2005.
[2] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 76-90, 2002.
[3] W. N. Everitt, J. Gunson, and A. Zettl, Some comments on Sturm-Liouville eigenvalue problems with interior singularities, Journal of Applied Mathematics and Physics, vol. 38, no. 6, pp. 813-838, 1987.
[4] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkh¨auser, Basel, Switzerland, 1986.
[5] M. A.Naimark, Linear Differential Operators, GeorgeG. Harrap and Company, 4th edition, 1967.
[6] A. L. Andrew, Correction of finite difference eigenvalues of periodic Sturm-Liouville problems, Australian Mathematical Society Journal Series B, vol. 30, no. 4, pp. 460-469, 1989.
[7] J. W. Paine, F. R. de Hoog, and R. S. Anderssen, On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems, Computing, vol. 26, no. 2, pp. 123-139, 1981.
[8] A. L. Andrew and J. W. Paine, Correction of finite element estimates for Sturm-Liouville eigenvalues, Numerische Mathematik, vol. 50, no. 2, pp. 205-215, 1986.
1 Marmara University, Istanbul, Turkey, [email protected]
- 109 -
Intuitionistic Fuzzy Optimization Technique in Multi-Commodity Solid
Transportation Models
Deepika Rani1 and T. R. Gulati1
Abstract. While traditional as well as solid transportation problems have received
enough attention in the literature, same is not the case with multi-commodity solid
transportation problems. But, the occurrence of such problems is not unusual in industrial
problems. This paper studies a solid transportation problem with heterogeneous products to be
transported from supply points to demand points taking into consideration the safety factors
while transporting. There always exist some risk in transporting the products from supply
points to the demand points due to bad road conditions, insurgency etc. in some routes
specially in the developing countries. In view of this, desired total safety factor is being
introduced as well as an additional constraint on the budget at each destination is also
considered. To reflect the uncertainty in real life situations, transportation parameters are
taken as fuzzy numbers. The proposed algorithm proceeds by transforming the problem to
deterministic model using the interval approximation of the fuzzy numbers and applying the
intuitionistic fuzzy programming technique with different type of membership and non-
membership functions. LINGO 15.0 software is then used to solve the final model and get the
optimal solution. The performance of the proposed approach is shown with a numerical
example. The obtained results are discussed and conclusions are drawn on the use of kind of
membership and non-membership functions.
Keywords. Solid Transportation Problem, Fuzzy Optimal Solution, LR Flat Fuzzy
Number.
AMS 2010. 90B05, 90C05, 90C70.
1 Indian Institute of Technology, Roorkee, India [email protected] (Deepika Rani), [email protected] (T. R. Gulati)
- 110 -
On a Disjoint Idempotent Decomposition for Linear Combinations Produced from
Commutative Tripotent Matrices
Emre Kişi1 and Halim Özdemir2
Abstract. It has been established a 3n -term disjoint idempotent decomposition (DID)
for the linear combinations produced from ( 2)n ≥ commutative tripotent matrices, their
products and their products of power 2 at most. The results obtained in this way generalize
those in [1]. Moreover, an algorithm to get a DID has been provided. Finally, a numerical
example has been given to exemplify the results.
Keywords. Idempotent Matrix, Tripotent Matrix, Involutory Matrix, Linear
Combination, Disjoint Idempotent Decomposition.
AMS 2010. 15A09, 15A24.
References
[1] Tian, Y., A disjoint idempotent decomposition for linear combinations produced from two
commutative tripotent matrices and its applications, Linear Multilinear Algebra 59, 1237–
1246, 2011.
[2] Baksalary, J.K., Baksalary, O.M., Özdemir, H., A note on linear combinations of
commuting tripotent matrices, Linear Algebra Appl. 388, 45–51, 2004.
[3] Coll, C., Thome, N., Oblique projectors and group involutory matrices, Appl. Math.
Comput. 140, 522–571, 2003.
[4] Baksalary, J.K., Baksalary, O.M., Idempotency of linear combinations of two idempotent
matrices, Linear Algebra Appl. 321, 3–7, 2000.
[5] Benitez, J., Thome, N., Idempotency of linear combinations of an idempotent matrix and t-
potent matrix that commute, Linear Algebra Appl. 403, 414–418, 2005.
[6] Benitez, J., Thome, N., Idempotency of linear combinations of an idempotent matrix and t-
potent matrix that do not commute, Linear Multilinear Algebra 56, 679–687, 2008.
[7] Özdemir, H., Sarduvan, M., Özban, A.Y., Güler, N., On idempotency and tripotency of
linear combinations of two commuting tripotent matrices, Appl. Math. Comput. 207, 197–
201, 2009.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 111 -
Respect to Two Spectra Stability of the Inverse Problem for Diffusion Equation
Etibar S. Panakhov,1 Ahu Ercan2 and Tuba Gulsen3
Abstract. Let consider the diffusion equation
( ) ( ) ( )( ) ( ) ( )212 ,0y x p x q x y x y x xλ λ p′′− + + = ≤ ≤ (1)
with the following boundary conditions
( ) ( ) ( ) ( )10 0 0, 0y h y y Hyp p′ ′− = + = (2)
( ) ( ) ( ) ( )20 0 0, 0y h y y Hyp p′ ′− = + = (3)
where λ is the spectral parameter, ),(0,)( 22 pWxp ∈ 1
1 2( ) (0, ), q x W p∈ 1 2,h h ( )1 2h h≠ and H
are some real constants. Let indicate the spectra of the boundary value problems (1)-(2) and
(1)-(3) by 1,nλ∞
−∞ and 1,nµ
∞
−∞respectively.
Now consider the equation
( ) ( ) ( )( ) ( ) ( )222 ,0y x p x q x y x y x xλ λ p′′− + + = ≤ ≤ (4)
with the boundary conditions (2) and (3). Here 12 2( ) (0, )q x W p∈ and corresponding spectra to
these conditions, respectively, are 2,nλ and 2, ,n nµ ∈Ζ .
Theorem 1. If the spectra ( ), , 1, 2j k jλ = of the problems (1)-(2) and (1)-(3) coincide
the numbers of 2 2N + on the interval ( )1, 1N N− − + for 1 ,2Nk N n> + < and 1k N< − −
, 2Nn > − then the difference of the spectral functions is
1 Firat University, Elazig, Turkey, [email protected] 2 Firat University, Elazig, Turkey, [email protected] 3 Firat University, Elazig, Turkey, [email protected]
- 112 -
( ) ( )
2
2
34 12
1 31
1
2
1 2 54 122
6 5122
2
34 12 , 0
1 3 21
54 12 , 0
6 5 21
ANAN
N
N AN
N ANN
ANN e
ANN
Var
ANN e
ANN
λ
λ
r λ
r λ r λ
r λ
+
− +
+−∞< <
− − < <∞ +
+ > − + − ≤ + − < − +
where ( ) ( )( ) 2, 1,2 1
0
12
k kc cA q t q t dt
k k
p
p−
= − +∫ .
Keywords. Stability, Spectral Function, Eigenvalues, Diffusion Equation
AMS 2010. 34B09, 34K20, 34L05.
References
[1] B. M. Levitan and I. S. Sagsjan, Introduction to Spectral Theory, American Mathematical
Society, Providence, RI, USA,1975.
[2] Guseinov, G. Sh., On construction of a quadratic SturmLiouville operator pencil from
spectral data, Proceeding of IMM of NAS of Azerbaijan, 40, Special issue:203-214, 2014.
[3] V. A. Marchenko and K. V. Maslov, “Stability of the problem of recovering the Sturm-
Liouville operator from the spectral function,” Mathematics of the USSR Sbornik, 81(123),
pp. 475-502, 1970.
- 113 -
Spectral Problems for Regular Canonical Dirac Systems with More General Separable
Boundary Conditions
Etibar S. Panakhov 1 and Mine Babaoğlu 2
Abstract. In this work, we examined Dirac system and succeeded in performing our approach for regular canonical Dirac systems. Thus, we obtained satisfactory spectral results by using the Paley-Wiener spaces [1]:
( ) ( ) 2Im, , .R
PW f entire f Ce f dπ mπ m m m= ≤ < ∞∫
Let
( ) ( ) ( ) ( ) ( ) ( )2 2 211 1, , cos sinv x y x x x x xm m α m α β m α= − − − − (1)
and
( ) ( ) ( ) ( ) ( ) ( )2 2 221 2, , sin cosv x y x x x x xm m α m α β m α= − − + − (2)
we claim the subsequent results.
Theorem 1. ( ) ( )11 21, , , xv x v x PWm m ∈ are functions of m for each x and the following estimates hold:
( ) ( )( ) ( )
2
2
Im11 1 0 2
Im21 3 0 4
,
,
x
x
v x c c c e
v x c c c e
m
m
m
m
≤ +
≤ +
(3)
Theorem 2. ( ) ( )12 22, , , xv x v x PWm m ∈ are functions of m for each x and the following estimates hold:
( )
( )
2
2
Im12 0 2
Im22 0 4
,
,
x
x
v x c c e
v x c c e
m
m
m
m
≤
≤ (4)
Hence, functions ( ) ( ) ( ) ( )11 21 12 22, , , , , , ,v x v x v x v xm m m m are entirely of type x order 1 and square integrable on the real line as a function of m for each x .
The boundary function(characteristic equation) ( )B m is not necessarily in PWπ as in the Dirichlet-Dirichlet case. However, we have the following theorem.
Theorem 3. ( ) ( ) ( )21 11 22 21, , ,B a v a v PWππ m π m π m= + ∈ is a function of m and the following
estimate holds:
1 Firat University, Elazig, Turkey, [email protected] 2 Firat University, Elazig, Turkey, [email protected]
- 114 -
( ) ( )2Im
5 6, xB e c cmπ m ≤ + (5)
Keywords. Dirac System, Paley-Wiener Space, Sampling Theory.
AMS 2010. 34B09, 34L05, 34L10.
References
[1] Chanane, B., Computing eigenvalues of regular Sturm-Liouville problems, Appl. Math.
Lett., 12, 119-125, 1999.
[2] Levitan, B. M., Sargsjan, I. S., Introduction to Spectral Theory: Selfadjoint Ordinary
Differential Operators, American Mathematical Society, Providence, Rhode Island, 1975.
[3] Zayed, A. I., Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton, 1993.
[4] Chadan, K., Sabatier, P. C., Inverse Problems in Quantum Scattering Theory, Second
edition, Springer-Verlag, 1989.
- 115 -
Generalized Second-Order Composed Radial Epiderivatives
Gonca İnceoğlu1
Abstract. In this study, we introduce the concept of the generalized second-order
composed radial epiderivative for set-valued maps and investigated some of its properties. A
unifed necessary and sufficient condition is derived in terms of the generalized second-order
composed radial epiderivative.
Keywords. Radial Cone, Generalized Radial Epiderivative, Generalized Second-Order
Composed Radial Epiderivative.
AMS 2010. 90C26, 90C30, 49J52.
References
[1] Kasımbeyli, R., Radial epiderivatives and set-valued optimizations , Optimzation, 58, 5,
519-53, 2009.
[2] Kasımbeyli, R., İnceoğlu G., Optimality conditions via generalized radial epiderivatives
in nonconvex set-valued optimization, In: R. Kasımbeyli, C. Dinçer, S. Özpeynirci and L.
Sakalauskas (eds) selected papers. 24th Mini Euro Conference on Continuous Optimization
and Information-Based Technologies in Financial Sector ( 24th MEC EurOPT 2010) June23-
26, 2010, Izmir Univrsyt of Economics, Izmir, Turkey, 148-154, ISBN:978-9955-28-597-7,
Vilnus “technika” 2010.
[3] Li, S.J., Zhu, S.K., Teo, K.L., New generalized second-order rdial epiderivatives and set-
valued optimization problems, , Journal of Optimization Theory , 152, 3, 587-604, 2012.
1 Anadou University, Eskisehir, Turkey, [email protected]
- 116 -
Equations of Anisotropic Elastodynamics for 1 Dimensional Qcs as a Symmetric
Hyperbolic System: Deriving the Time-Dependent Fundamental Solution
H. Çerdik Yaslan1
Abstract. In this study dynamic elasticity equations for 1D quasicrystals (QCs) with
arbitrary system of anisotropy are considered. For these equations the phonon-phason
displacements, displacement speeds and stresses arising from pulse point source are
computed. Firstly, definition of the fundamental solution (FS) for the time-dependent
differential equations of anisotropic elasticity in 1D QCs is given. These equations are written
in the form of a symmetric hyperbolic system of the first order. Using the Fourier transform
with respect to the space variables and matrix transformations we obtain formulae of the FS
columns. As a computational example FS components are computed for orthorhombic and
triclinic structures in 1D QCs.
Keywords. Anisotropic Dynamic Elasticity(3D) ,One-Dimensional Quasicrystals,
Symmetric Hyperbolic System, Triclinic QCs, Fundamental Solution.
AMS 2010. 53A40, 20M15.
References
[1] W. Q. Chen, Y. L. Ma, Y.L., H. J. Ding, On three-dimensional elastic problems of one-
dimensional hexagonal quasicrystal bodies. Mech. Res. Commun. 31, 633-641, 2004.
[2] V. G. Yakhno, H. C. Yaslan, Computation of the time-dependent fundamental solution for
equations of elastodynamics in general anisotropic media, Comput. Struct. 89, 646-655,
2011.
[3] Y. Gao, S. P. Xu, B. S. Zhao, General solutions of equilibrium equations for 1D
hexagonal quasicrystals, Mech. Res. Commun. 36, 302-308, 2009.
[4] T. Y. Fan, Mathematical Theory of Elasticity of Quasicrystals and its Applications,
Science Press - Springer, Beijing, 2011.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 117 -
Equivalence Relations on Quaternion Matrices
Işıl Arda Kösal1 and Hidayet Hüda Kösal2
Abstract. In this paper, semi-similairty relation, semi-consimilarity relation, pseudo-
similarity relation and pseudo-consimilarity relation for quaternion matrices are defined. Also,
Relationships between these equivalence relations are studied.
Keywords.Quaternion matrices, Involutory automorphism, Equivalence relation.
AMS 2010. 15B33, 15A18.
References
[1] R. E. Hartwig and M. S. Putcha, Semisimilarity for matrices over a division ring, Linear
Algebra Appl. 39, 125-132, 1981.
[2] R. E. Hartwig and F. J. Hall, Pseudosimilarity for matrices over a field, Proc. Amer. Math.
Sot. 71, 6-10, 1978.
[3] J, Bevis, F. Hall, and R. E. Harhvig, Pseudoconsimilarity and semiconsimilarity of
complex matrices, Linearr Algebra App. 90, 73-80, 1987.
[4] R. E. Hartwig, The resultant and the matrix equation AX XB= , SIAM J. Appl. 22, 538-
544, 1972.
[5] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., vol. 251, pp.
21–57, 1997.
[6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York,
1985.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 118 -
The Fracture of the Elastic Matrix Containing Two Neighboring Co-phase Periodically
Curved Carbon Nanotubes
İ. Gülten 1 and R. Köşker 2
Abstract. It is known that one of the major mechanisms of the fracture of the
unidirectional composites under uniaxial compression along the reinforcing elements is the
stability loss in the material structure (structural or internal instability). According to this
mechanism, the theoretical investigations of the fracture of the unidirectional composites
under uniaxial compression along the reinforcing elements are reduced to the investigations of
the stability loss in the material structure, and the value of the external critical forces is
accepted as the value of failure forces in compression. At present, numerous theoretical and
experimental investigations have been carried out in this field. In [1,2,3] within the
framework of the piecewise homogeneous body model with the use of the Three-Dimensional
Geometrically Nonlinear Exact Equations of the Theory of Viscoelasticity the approach for
the investigation of the internal stability loss (microbuckling) in the structure of the
viscoelastic unidirected fibrous composites under compression along the fibers is developed.
However, in [1,2,3] the composite material was modelled as an infinite viscoelastic body
containing fibers. In the paper [4] the attempt is made for development of the internal stability
loss problems in the structure of the unidirectional fibrous composites for the case where the
reinforcing element in the composite is the double-walled carbon nanotube (DWCNT). In the
paper [5], the case is considered where a single periodical curved carbon nanotube (CNT)
with an infinite length is contained by an infinite body with low concentration of CNT and the
stability loss problem in that is investigated. The investigation on how the critical values
related to the internal stability loss of composites is affected by the CNTs reciprocally as the
volume ratio of CNTs gets bigger in composites, is very important. In the present
investigation the approaches [5] is developed for solution of these stability loss problems and
investigates the influence of the interaction between two neighboring CNTs on the values of
the critical values. It is assumed that in the initial state the CNTs have the same initial
imperfection in the periodical curving form. Within the framework of the piecewise
homogeneous body model with the use of the Three-Dimensional Geometrically Nonlinear
Exact Equations of the Theory of Elasticity the growing of the initial imperfection is
investigated. The corresponding non-linear boundary-value problem is solved by employing
1 Yildiz Technical University, Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected]
- 119 -
boundary-shape perturbation method. In this case a small parameter characterizing the degree
of the initial imperfection is introduced and the solution of the mentioned problem is
presented in power series form in this small parameter. It is proven that the zeroth and first
approximations are enough for investigation of the development of the initial imperfection.
The case where the mentioned imperfection starts to increase and becomes indefinitely is
taken as a fracture (stability loss) criterion.
Acknowledgement. This research has been supported by Yıldız Technical University
Scientific Research Projects Coordination Department. Project Number: 2014-07-03-DOP01.
Keywords. Fracture in compression, Nanocomposite, Nanotube, Internal stability loss
References
[1] Akbarov, S.D., Kosker, R., Fiber buckling in a viscoelastic matrix. Mechanics of
Composite Materials 37(4), 299-306, 2001.
[2] Akbarov, S.D., Kosker, R., Internal Stability Loss of Two Neighbouring Fibers in a
Viscoelastic Matrix. International Journal of Engineering Science 42, 1847–1873, 2004.
[3] Akbarov, S.D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized
Approach for Elastic and Viscoelastic Composites, Springer, 2012.
[4] Akbarov, S.D., Microbuckling of a Double-Walled Carbon Nanotube Embedded in an
Elastic Matrix, International Journal of Solids and Structures 50, 2584- 2596, 2013.
[5] Kosker R. and Gulten İ., Internal Stability Loss of a Periodical Curved Carbon Nanotube
in a Viscoelastic Matrix, 3rd Internatıonal Symposıum On Innovatıve Technologies In
Engineering And Scıence, ISITES2015, Valencia, Spain, Proceeding Book, pp:2307-2317,
2015.
- 120 -
On Homoclinic Structure for 2D Coupled Nonlinear Schrödinger System
Irma Hacinliyan1 and Ceni Babaoğlu2
Abstract. It has been demonstrated that an analytic description of the homoclinic
structure for 1D nonlinear Schrödinger equation is obtained via soliton type solutions [1]. In a
further work, these analytic expressions of homoclinic orbits are used for 1D coupled
nonlinear Schrödinger system and long-short wave equations [2]. In this study, the homoclinic
structure for 2D coupled nonlinear Schrödinger (CNLS) system is investigated by following a
similar approach. It is observed that the fixed point in the CNLS system is hyperbolic. Then,
the soliton type solutions which form homoclinic orbits are found by using Hirota's method.
Some consequences are also supported by numerical computations.
Keywords. Homoclinic Orbits, Nonlinear Schrödinger-Like Equations.
AMS 2010. 35Q55, 37C29.
References
[1] Ablowitz, M.J., Herbst, B.M., On homoclinic structure and numerically induced chaos for
the nonlinear Schrödinger equation, SIAM J. Appl. Math., 50, 339-351, 1990.
[2] Gao, P., Guo, B., Homoclinic orbits for the coupled nonlinear Schrödinger system and
long-short wave equation, Phys. Lett. A, 340, 209-211, 2005.
1 Istanbul Technical University, Istanbul, Turkey, [email protected] 2 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 121 -
Finite Difference Approximation for Solving Quasilinear Nonlocal Problem
with Boundary Layers
Musa Çakır1
Abstract. In this paper, we consider the following singularly perturbed quasilinear
nonlocal problem:
( ) ( ) ( )( ) <<=−′+′′≡ xxuxfxuxuLu 0 ,0,2 εε (1.1)
( ) Au =0 (1.2)
( ) ( ) ( )∫ ≤<≤=−≡ 1
0100 0 ,
BdxxuxguuL (1.3)
where 10 <<< ε is the perturbation parameter, A and B are given constants, the functions
( ) 0≥xa and ( )uxf , are sufficiently smooth on [ ],0 and [ ] R×,0 , respectively, and ( )xg is
a continuous function on [ ]10 , , moreover .0 *<≤
∂∂
≤< ββuf
First we discuss the nature of the continuous solution of singularly perturbed differential
problem before presenting method for its numerical solution. The numerical method is
constructed on piecewise uniform Shishkin type mesh. We show that the method is first-order
convergent in the discrete maximum norm, independently of singular perturbation parameter
except for a logarithmic factor. We give effective iterative algorithm for solving the nonlinear
difference problem. Numerical results which support the given estimates are presented.
Keywords. Singular Perturbation, Fitted Difference Scheme, Uniformly Convergence
AMS 2010. 65L12, 65L70, 34B10, 34D15.
References
[1] Roos, H. G., Stynes, M., Tobiska, L., Robust numerical methods for singularly perturbed
differential equations, Springer-Verlag, Berlin, Heidelberg, 2008.
[2] Cakir, M., Amiraliyev, G.M., A finite difference method for the singularly perturbed
problem with nonlocal boundary condition, Appl. Math. and Coput.160, 539-549, 2005.
1 Yuzuncu Yil University, Van, Turkey, [email protected]
- 122 -
Inverse Spectral Problems for Energy-Dependent Sturm-Liouville Equations with
Finitely Many Point 𝜹 −Interactions
Manaf Dzh. Manafov1
Abstract. We study inverse spectral problems for energy-dependent Sturm-Liouville
equations with finitely many point 𝛿 −interactions a finite interval. Spectral problems of
differential operators are studied in two main branches, namely, direct and inverse problems.
Direct problems of spectral analysis consist in investigating the spectral properties of an
operator. On the other hand, inverse problems aim at recovering operators from their spectral
characteristics. One takes for the main spectral data, for instance, one, two, or more spectra,
the spectral function, the spectrum, and the normalized constants, the Weyl function. Direct
and inverse problems for the classical Sturm-Liouville operators have been extensively
studied (refer to [1-4]). In this study, various uniqueness results are proved, and a constructive
procedure for the solution is provided.
Keywords. Energy-Dependent Sturm-Liouville Equations, Inverse Spectral Problems,
Point 𝛿 −Interactions.
AMS 2010. 34A55, 34B24, 34L05
References
[1] Marchenko, V. A., Sturm-Liouville Operators and Their Applications, Operator Theory:
Advanced and Application, Birkhauser: Basel, 22, 1986.
[2] Levitan, B. M., Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.
[3] Pöschel, J., Trubowitz, E., Inverse Spectral Theory, Academic Press, New York, 1987.
[4] Frelling, G., Yurko, V. A., Inverse Sturm-Liouville Problems and Their Applications,
NOVA Science Publ., New York, 2001.
1 Adiyaman University, Adiyaman, Turkey, [email protected]
- 123 -
Effect of a Surface Asperity at the Nanoscale
M. Grekov1 and S. Kostyrko2
Abstract. Unlike bulk material elements, the nanostructures have elastic properties
which are highly depend on their size. This size dependency of properties at the nanoscale can
be understood by incorporating the effect of surface stress. The intent of this work is to
examine the effect of surface and bulk elastic parameters, surface stress and nanosized
asperity shape on stress concentration and local stress distribution at a solid surface. We
consider the 2-D model of semi-infinite elastic solid having a nanosized surface roughness
and subjected to remote tensile loading. It is assumed that, according to Young-Laplace law
[1], the traction at the boundary is expressed in terms of surface stress which is intrinsic to
nanometer size structures. In order to find the surface stress we take into consideration the
condition of surface and bulk inseparability. To solve the boundary value problem, we use
Gurtin-Murdoch surface elasticity model [1] containing constitutive equations for the surface
linear elasticity with two elastic parameters and residual surface stress [2]. The way of
deriving an analytical solution is based on Goursat-Kolosov complex potentials,
Muskhelishvili representations and the boundary perturbation technique. This technique leads
to the singular integro-differential equation in expansion coefficients of surface stress in each-
order approximation. The solution of this equation and numerical results are presented in the
first-order approximation.
Acknowledgements. The work was supported by RFBR, grant 14-01-00260.
Keywords. Boundary Perturbation Method, Singular Integral Equations, Surface
Nanodefects, Surface Stress, Stress Concentration, Size Effect.
AMS 2010. 74G10, 74M25, 74S70.
References
[1] Gurtin, M., Murdoch, A., A continuum theory of elastic material surfaces, Arch. Rat.
Mech. Anal., 57, 291-323, 1975.
[2] Duan, H.L., Wang, J., Karihaloo, B.L., Theory of elasticity at the nano-scale, Adv. Appl.
Mech., 42, 1-68, 2009.
1 St. Petersburg State University, St. Petersburg, Russia, [email protected] 2 St. Petersburg State University, St. Petersburg, Russia, [email protected]
- 124 -
Conservation Laws and Exact Solutions of Nonlinear Differential Equation
Melike Kaplan1, Arzu Akbulut2 and Filiz Tascan3
Abstract. The concept of conservation laws has a long and profound history in
physics. Whatever the physical laws considered: classical mechanics, fluid mechanics, solid
state physics, as well as quantum mechanics, quantum field theory or general relativity ,
whatever constituents of the theory and the intricate dynamic processes involved, quantities
left dynamically invariant have always been essential ingredients to describe nature. At the
mathematical level conservation laws are deeply connected with the existence of a variational
principle which admits symmetry transformations [1], [2].
In this study we have used conservation theorem approach to construct conservation
laws of an equation [3]. Also by using the (G’/G,1/G)- expansion method exact solutions have
been verified [4].
Keywords. Conservation Law, Exact Solution, (G’/G,1/G)-Expansion Method .
AMS 2010. 70S10, 83C15.
References
[1] Compére, G., Symmetries and conservation laws in Lagrangian gauge theories with
applications to the mechanics of black holes and to Gravity in three dimensions, Ph.D. thesis,
Université Libre de Bruxells Faculté des Sciences, 2007.
[2] Naz, R. , Mahomed, F.M., Mason, D.P., Conservation laws via the partial Lagrangian
and invariant solutions for radial and two-dimensional free jets, Non. Analy.:Real World
Appl., 10, 3457-3465, 2009.
[3] Noether, E., Invariante Variationsprobleme, Nachr. Konig. Gesell. Wiss. Gottingen
Math.-Phys. Kl. Heft 2, 235-257, 1918, English translation in Transport Theory statist. Phys.
1, 3, 186-207, 1971.
[4] Li, L.X., Li, E.Q., Wang, M.L., The (G′/G,1/G)-expansion method and its application to
travelling wave solutions of the Zakharov equations, Appl. Math. J. Chinese Univ., 25, 454-
462, 2010.
1 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 3 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected]
- 125 -
Stability of One Nonlinear System with Delay
Mikhlin Leonid Stanislavovich 1 and Prasolov Alexander Vital'evich 2
Abstract. In the book [1] Lotka-Volterra mathematical model is widely used in economic
dynamics. The model for two economic agents and in the multidimensional case has a second
order polynomial nonlinearity in the right-hand side. Furthermore, the delay is entered in its
reproduction mechanism. Thereby, the analysis of the qualitative behavior of solutions requires
apparatus of nonlinear systems with concentrated delay. Many papers are focused on these
problems. In particular, the recognized authoritative textbooks and monographs by N. N.
Krasovsky [2], R. Bellman and K. Cooke [3], J. Hale [4], etc.
In the paper [1] (see also [5]) basic model is analyzed in point of asymptotic stability of
the equilibrium states and the presence of oscillations about stationary points when asymptotic
stability is absent. Analysis algorithm is generally accepted: calculating stationary points, building
the linear part of the system in deviations and the problem of asymptotic stability of the
equilibrium solutions of the nonlinear system is solved by the location of the roots of the
characteristic equation. Herewith nonlinear part must satisfy the conditions of the theorem about
stability in first approximation by N. N. Krasovsky, which we'll give for the sake of completeness
by [2] in the case of the same constant delay in all system's equations.
The novelty of this study is selection of the class of mathematical models with a special
introducted delay. Let us recall that the characteristic equation of a linear system with
concentrated delay is not polynomial as in systems without delay but quasipolynomial, and this
brings difficulties as a countable number of roots [3]. Selected class reduces the problem to the
analysis of a finite number of roots, for each of which are known conditions of asymptotic
stability. In this article were obtained three theorems which allow us to find the critical value of
delay for a sufficiently broad class of systems arbitrary dimension by a simple algorithm. Lotka-
Volterra systems are widely used in the simulation of economic processes, and examples confirm
the applicability of these theorems.
Keywords. Lotka-Volterra Model, Delay, Stability.
AMS 2010. 34A34, 34D05.
References
[1] Prasolov A.V., Mathematical models of economic dynamics, Lan', Saint-Petersburg, 2008.
1 Saint Petersburg State University, Saint-Petersburg, Russia, [email protected] 2 Saint Petersburg State University, Saint-Petersburg, Russia, [email protected]
- 126 -
[2] Krasovsky N.N., Some problems in the theory of stability of motion, Fizmatgiz,Moscow,
1959.
[3] Bellman R., Cooke K., Differential-Difference Equations, Moscow, Mir., 1967.
[4] Hale J., Theory of Functional Differential Equations, Moscow, Mir, 1984.
[5] Prasolov A.V., Dynamic models with delay and their applications in economics and
engineering, Saint-Petersburg, Lan', 2010.
- 127 -
Control Problems of Nonlinear Phase Systems
M. N. Kalimoldayev1 and M. T. Jenaliyev2
Abstract. Let us consider the problem of functional minimization:
))(),((exp)(21),...,(
1 0
221 TSTdttwSwJ
l
i
T
iiisl iidγννν ν Λ++= ∑∫
=
, (1)
ii S
dtd
=d
, iiiiiiii
i NfSDdt
dSH νdd +−−−= )()( , ),...,( 1 lddd = , ),...,( 1 lSSS = , (2)
where isw ,
iwν are weight coefficients, correspondingly positive functions and constants; iH
is an inertial constants; )( iif d are π2 -periodical continuously differentiated functions;
)(diN are π2 -periodical continuous differentiated function srelative to ldd ,...,1 ; for
summands )(diN the conditions of integrability are carried out; +∞<T .
Theorem 1. For optimality of controls iiii StwtSi
exp][),( 10 γν ν −−= − , li ,...,1= ,
and their corresponding solution )(),( 00 tStd of system (2)–(3), it is necessary and
sufficient, that TtTt SKS == =Λ || ),(),( dd , 02exp][exp2)( 1 >−+−= − twtDtw iiis iiγγ ν ,
,,...,1 li = where
∑ ∫∑ ∫>
==+−
=
+
+=
l
jji
iliii
l
iiiiii
j
ii
dNdfSHSK,0,1 0
1111 0
2 ),...,,,....,()(21),(
d
dd
ξddddddd -
Bellman-Krotov function [1] and besides, ),()(min)( 000 SKJJ dυυυ
== .
In this work while solving the control synthesis problem for the considered electric
power system, the constructions of the method of Bellman-Krotov function in the form of
necessary and sufficient optimality conditions were used.
Keywords. Optimal Control, Phase System, Bellman-Krotov Function.
AMS 2010. 53A40, 20M15.
References
[1] Krotov V.F. Global methods in optimal control theory, M.Dekker, New York, 1996.
1 Institute of Informatic and Computing Technologies, Almaty, Kazakhstan, [email protected] 2 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, [email protected]
- 128 -
Interchange of Mass after a Close Encounter between Galaxies
M. Ollé 1, E. Barrabés and J.M. Cors
Abstract. The talk's context is Celestial Mechanics and the application of dynamical
systems tools to study the motion of the parabolic restricted three body problem (PRTBP).
The goal of this problem is to study the motion of a massless body attracted, under the
Newton's law of gravitation, by two equal masses moving in parabolic orbits all over in the
same plane. The PRTBP may be regarded as a simplified model for the motion of two
galaxies, taken as the primaries, and an infinitesimal mass. In order to discuss possible
motions for the particle, first we consider a rotating and pulsating frame where the equal
masses (primaries) remain at rest. The obtained system of ODE is gradient-like and has
exactly ten hyperbolic equilibrium points lying on the boundary invariant manifolds
corresponding to escape of the primaries in past and future time. The invariant manifolds of
the equilibrium points play a key role in the dynamics and we study some trajectories
described by the particle before and after a close encounter between the primaries. Finally
some numerical simulations are done, paying special attention to capture and escape orbits.
Keywords. Parabolic Restricted Three-Body Problem, Equilibrium Points, Invariant
Manifolds, Numerical Simulations, Global Dynamics
AMS 2010. 70F07, 70F15.
1 Universitat Politecnica De Catalunya, Spain, [email protected]
- 129 -
One a Numerical Method of Riemann Type Problem for 2D Conservation Laws in a
Class of Discontinuous Functions
Mahir Rasulov1, Hakan Bal2 and Emine Boran3
Abstract. In this study, a new method for obtaining a numerical solution of first order
nonlinear equation describing conservation laws is suggested. In order to realize it, a special
auxiliary problem is introduced. This problem permits us to find the solution of the
investigated problem with higher accuracy, such that the obtained solution expresses all of the
physical properties of the considered problem.
Keywords. Riemann-Type Problem, 2D-Burgers Equation, Numerical Solution in a
Class of Discontinuous Functions, Numerical Weak Solution
AMS 2010. 35L65, 65M06.
References
[1] Guckenheimer, J., Shocks and rarefactions in two space dimensions, Arch. Rational Mech.
Anal., Vol. 59, No. 3, 281-291, 1975.
[2] Pang, Y., Tian, J.P., Yang, H., Two-dimensional Riemann problem for a hyperbolic system
of conservation laws in three pieces, Appl. Math. Comput., Vol. 219, No:4, 1695-1711, 2012.
[3] Rasulov, M.A., Identification of the saturation jump in the process of oil displacement by
water in a two-dimensional domain, Dokl RAN, Vol. 319, No.4, pp. 943-947, 1991.
[4] Rasulov, M.A, Coskun, E., Sinsoysal, B., A finite differences method for a two-
dimensional nonlinear hyperbolic equations in a class of discontinuous functions, Appl.
Math. Comput., Vol.140, No: 2-3, 279-295, 2003.
[5] Yoon, D., Hwang, W., Two-dimensional Riemann problems for Burger's equation, Bull.
Korean Math. Soc., Vol. 45, No. 1, 191-205, 2008.
1 Beykent University, Istanbul, Turkey, [email protected] 2 Beykent University, Istanbul, Turkey, hakanbal@ beykent.edu.tr 3 Beykent University, Istanbul, Turkey, [email protected]
- 130 -
On Two-Dimensional Nonsteady Free Convection near Vertical Plate subject to
Stepped-up Surface Temperature
Nirmal C. Sacheti 1, Pallath Chandran2 and Ashok K. Singh 3
Abstract. Theoretical studies involving buoyancy-driven flows of viscous
incompressible fluids near stationary or moving vertical flat plates have been a subject of
intense investigations in the literature, apparently due to numerous applications in a number
of areas of engineering and technology. In such convection problems, the role of momentum
as well as thermal boundary conditions is known to have a significant bearing on the
developing as well as fully developed flows. For instance, in certain applications, the nature
of the initial thermal profile of the bounding surface could play an important role in the design
processes. We have thus investigated a specific unsteady free convective flow near a semi-
infinite vertical plate assuming that bounding plate has been subjected to stepped-up time
dependent temperature distribution. This leads to the development of boundary layer near the
surface. We have solved the governing momentum and thermal boundary layer equations
numerically using an appropriate implicit finite difference technique, and exhibited the
profiles of the developing and steady state velocity and temperature at various cross-sections.
Furthermore, we have also included the plots of isotherms to analyze the thermal changes in
the boundary layer. The comparison of some results with infinite vertical wall case as well as
variations of two quantities of engineering interest, namely, wall skin friction and Nusselt
number, will also be reported during the presentation.
Keywords. Free Convection, Unsteady Flow, Stepped-Up Temperature, Boundary
Layer.
AMS 2010. 76R10, 76D09, 80A20.
1 Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected] 2 Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected] 3 Banaras Hindu University, Varanasi, India, [email protected]
- 131 -
An Alternative Technique for Solving Ordinary Differential Equations
Neşe Dernek1, Fatih Aylıkçı2 and Sevil Kıvrak3
Abstract. In the present paper, a new method for solving ordinary differential
equations is given by using the generalized Laplace transform Ln:
The Ln-transform and the complex inverse generalized Laplace transform Ln-1 were
introduced by Dernek and Aylıkcı in [4]. The Ln-1-transform is defined by
where Lnf(x),y1/n has a finite number of singularities in the left-half plane Re(y) ≤ c.
The object of this paper is first to introduce a differentiation operator δ for the Ln-transform
that we call the δ-derivative and define as ([4],[11])
We note
The δ-derivative operator can be successively applied in a similar fashion for any positive
integer power. A relation between the Ln-transform of the δ-derivative of a function and the
Ln-transform of the function itself is derived by the following theorem.
Theorem1.If f is piecewise continuous function on the interval [0,∞) and is exponential order
exp(xnyn) as x → ∞, for some constant a, then the following relation
holds true for k ≥ 1, k is a positive integer.
The theorem for the inverse transform Ln-1 is given as follows.
1Marmara University, Istanbul, Turkey, [email protected] 2Yildiz Technical University, Istanbul, Turkey, [email protected] 3Marmara University, Istanbul, Turkey, [email protected]
- 132 -
Theorem2. Let Lnf(x),y1/nbe an analytic function of y except at singular points, each of
which lies to the left of the vertical line Re(y)=c and they are finite numbers. Suppose that
y=0 is not a branch point and limy→∞Lnf(x),y1/n=0 in the left-plane Re(y) ≤ c, then the
following identity
holds true for m singular points.
Definition: The convolution of two integrable functions f(x) and g(x) is defined by
Theorem3. If Lnf(x),y=F(y) and Lng(x),y=G(y), then the relation for the convolution
off(x) and g(x),
holds true.
In the last part of this paper using above theorems, some initial-value problems are solved as
examples.
Keywords. The Laplace Transform, The Ln-Transform, The Ln-1-Transform, Linear
Ordinary Differential Equations.
AMS 2010.44A10, 44A15, 44A20, 34A30.
References
[1] Aghili, A., Ansari, A., Sedghi A., An inversion technique for the Ln-transform with
applications, Int. J. Contemp. Math. Sciences, 2.28, 1387-1394, 2007.
[2] Aghili, A., Ansari, A., A new approach to solving SIEs and system of PFDEs using the L2-
transform, Differential Equations and Control Processes, N3, 1817-2172, 2010.
[3] Dernek, N., Aylıkcı, A., Identities for the Ln-transform, The L2n-transform and the P2n-
transform and their applications, Journal of Inequality and Special Functions, 5.4, 1-16, 2014.
- 133 -
[4] Dernek, N., Aylıkcı, F., Laplace veL2-dönüşümleriylekısmitürevlidenklemlerinçözümleri,
Marmara University, Master Thesis, 2014.
[5] Duffy, D.G., Transform methods for solving partial differential equations. Symbolic &
Numeric Computation, 2004.
[6] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms
Vol. 1, New York,NY,USA, McGraw-Hill, 1954.
[7] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms
Vol. 2, New York,NY,USA, McGraw-Hill, 1954.
[8] Yürekli, O., Theorems on L2-transform and its applications, Vol.13 of Physical Sciences
Data, Elsevier Scientific Publishing Co., Amsterdam, 1983.
[9] Yürekli, O., Wilson, S., A new method of solving Bessel's differential equation using the
L2-transform, Applied Mathematics and Computation, 130.2, 587-591, 2002.
[10] Yürekli, O., Wilson, S., A new method of solving Hermite's differential equation using
the L2-transform, Applied Mathematics and Computation, 145.2, 495-500, 2003.
[11] Zauderer, E., Partial differential equations of applied mathematics., John Wiley & Sons,
Inc. 2004.
- 134 -
New Identities for the Generalized Glasser Transform, the Generalized Laplace
Transform and the E2n,1-Transform
Neşe Dernek1, Fatih Aylıkçı2 and Gülesin Balaban3
Abstract. In the present paper the authors defined the generalized Glasser transform
G2nand gave the following iteration identity for the generalized Laplace transform L2n[5] and
the generalized Glasser transform G2n:
Using this identity, a Parseval-Goldstein type theorem for the L2n-transform and the G2n-
transform is given as follows:
and
By making use of these results, new Parseval-Goldstein type identities are obtained for G2n-
transform and many other well-known integral transforms. Some of them are:
1 Marmara University, Istanbul, Turkey, [email protected] 2 Yildiz Technical University, Istanbul, Turkey, [email protected] 3 Marmara University, Istanbul, Turkey, [email protected]
- 135 -
where the integrals involved converge absolutely. Kv,n andHv,nare generalized Bessel and
Hankel transforms, respectively [6].
The identities that are proven in this paper are shown to give rise to useful corollaries for
evaluating infinite integrals of special functions. Illustrative examples are also given. Some of
them are:
etc.
We conclude by remarking that, many other infinite integrals can be evaluated by applying
theorems and corollaries that are given in our paper.
Keywords. Laplace Transforms, L2n-Transforms, G2n-Transforms, Fs,n-Transforms,
Fc,n-Transforms, Hv,n-Transforms, Kv,n-Transforms, E2n,1-Transforms, Parseval-Goldstein
Type Theorems.
AMS 2010. Primary 44A10, 44A15, Secondary 33C10, 44A35.
References
[1] Adawi, A., Alawneh, A., A Parseval-type theorem applied to certain integral transforms
on generalized functions, IMA Journal of Applied Mathematics,68 (6), 587-593, 2003.
- 136 -
[2] Apelblat, A., Table of definite and infinite integrals, Vol.13 of Physical Sciences Data,
Elsevier Scientific Publishing Co., Amsterdam, 1983.
[3] Brown, D., Dernek, N., Yürekli, O., Identities for the exponential integral and the
complementary error transforms, Applied Mathematics and Computation, 182 (2), 1377-
1384, 2006.
[4] Brown, D., Dernek, N., Yürekli, O., Identities for the E2,1-transform and their
applications, Applied Mathematics and Computation, 187 (2), 1557-1566, 2007.
[5] Dernek, N., Aylıkçı, F., Identities for the Ln-transform, the L2n-transform and the P2n-
transform and their applications, Journal of Inequalitiesand Special Functions, V5 Issue 4,
2014.
[6] Dernek, N., Aylıkçı F., On the generalizations of the Widder potential transform, the K-
transform, the Hankel transform and their applications, To appear, 2015.
[7] Dernek, N., Kurt, V., Şimşek, Y., Yürekli, O., A generalization of the Widder potential
transform and applications, Integral Transforms and Special Functions Vol22, No 6, 391-401,
2011.
[8] Dernek, N., Srivastava, H.M. and Yürekli, O., Parseval-Goldstein type identities involving
the FS,2-transform, the FC,2-transform and the P4-transform and their applications, Applied
Mathematics and Computation, 202.1, 327-337, 2008.
[9] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms
Vol. 1, New York,NY,USA, McGraw-Hill, 1954.
[10] Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of integral transforms
Vol. 2, New York,NY,USA, McGraw-Hill, 1954.
[11] Glasser, M.L., Some Bessel function integrals, Kyungpook Math. J.13, 171-174, 1973.
[12] Gradshteyn, I.S., Rhyzik, I.M., Table of integrals, Series and Products, Academic Press,
Inc. 4. Edition, 1980.
- 137 -
[13] Kahramaner, Y., Srivastava, H.M., Yürekli, O., A theorem on the Glasser transform and
its applications, Complex Variables Theory and Application. An International Journal, 27 (1),
7-15, 1995.
[14] Prudnikov, A.P., Bryckov, Y.A., Marichev, D.I., Integrals and series, Academic Press,
Vol.4, Gardon and Breach Science Publishers, New York, 1992.
[15] Rainville, E.D., Special functions, The Macmillan Company, New York, 1960.
[16] Srivastava, H.M. and Yürekli, O., A theorem on a Stieltjes-type integral transform and its
applications, Complex Variables Theory Appl., 28, 159-168, 1998.
[17] Spanier, J., Oldham, K.B., An atlas of functions, Hemisphere Pub. Corp., Washington,
1987.
[18] Yürekli, O. and Sadek, I., A Parseval-Goldstein type theorem on the Widder potential
transform and its applications, Internat, J. Math. and Math. Sci., 14, 517-524, 1991.
[19] Yürekli, O., Identities, inequalities, Parseval-type relations for integral transforms and
fractional integrals, Ph.D.thesis, University of California, Santa Barbara, 1988.
[20] Yürekli, O., New identities involving the Laplace and L2-transforms and their
applications, Applied Mathematics and Computation, 99 (2-3), 141-151, 1999.
[21] Widder, D.V., A transform related to the Poisson integral for a half-plane, Duke
Math.J.33, p.355-362, 1966.
- 138 -
Estimating Elasticity Modulus of the Piezo Ceramic Disc Using Basic Mathematical
Modelling
Nedret Elmas 1, Levent Paralı, Ali Sarı, Jiri Pechousek and Frantisek Latal
Abstract. The objective of this paper is to determine a mathematical modelling of
piezoceramic disc vibration using a single degree freedom mechanical model, with estimation of
its elasticity modulus. The experimental vibration displacement values of piezo ceramic disc have
been achieved utilizing the swept-sine signal excitation following the peak values in the signal
response measured by the laser Doppler vibrometer. Consistency between the mathematical
modelling and experimental values have been observed from 97 to 80 % between excitation
amplitudes of 0.5 and 3.5 V when the mathematical modeling of piezo ceramic disc is normally
taken into consideration with a linear working range. The results obtained from experimental
studies on resonance frequency are in a compliance with reference value declared by producer of
the piezo ceramic disc.
Keywords. Elasticity Modulus, Piezo Ceramic, Resonance Frequency, Vibration,
Displacement.
References
[1] Ajitsaria J, Choe S Y, Shen D and Kim D J, Modelling and analysis of a bimorph piezo
electric cantilever beam for voltage generation, Smart Mater.and Struc. 16, 447–454, 2007.
[2] Dakua I and Afzulpurkar N, Piezoelectric Energy Generation and Harvesting at the Nano-
Scale: Materials and Devices, Nanomater. and Nanotechn. 3, 21, 2013.
[3] Gupta M N, Suman and Yadav S K, Electricity Generation Due to Vibration of Moving
Vehicles Using Piezoelectric Effect, Adv. in Electron. and Electric Engin. 4, 313-318, 2014.
[4] Vijaya M S, Piezoelectric Materials and Devices: Applications in Engineering and Medical
Sciences, CRC Press, 2013.
[5] Jalili N, Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems,
(Springer: New York), 2010.
[6] Staworko M,Uhl T, Modeling and simulations of piezoelectric elementscomparison of
available methods and tools, Mechanics, 27, 4, 2008.
1 Celal Bayar University, Manisa, Turkey, [email protected]
- 139 -
[7] Umeda M, Nakamura K and Ueha S, Analysis of the transformation of mechanical impact energy to electric energy using piezoelectric vibrator, Japan J. Appl. Phys. 35, 32, 67–73, 1996.
[8] Kasyap A, Lim J, Ngo K, Kurdila A, Nishida T, Sheplak M and Cattafesta L, Energy reclamation from a vibrating piezoceramic composite beam, 9th Int. Congr.on Sound and Vibr. 9, 2002.
[9] Hwang W S and Park H C, Finite element modeling of piezoelectric sensors and actuators AIAA J. 31 93 0–7, 1993.
[10] Williams C B and Yates R B, Analysis of a micro-electric generator for microsystems Sensors Actuators A 52 8–11, 1996.
[11] Roark R J, Budynas R G and Young W C, Roark's Formulas for Stress and Strain, McGraw-Hill, 2001.
[12] Santhosh K V and Dutta T, Measurement of Elasticity Modulus Using Image Processing, Int. Conf. Comp. Commun. and Infor, 2013.
[13] Francis S T, Morse E I, Hinkle R T, Mechanical Vibrations: Theory and Applications, Allyn& Bacon, 1978.
[14] Den Hartog J P, Mechanical Vibrations, Dover Publications, 1985.
[15] Timoshenko S, Young D H and Weaver W, Vibration Problems in Engineering, Wiley, 1990.
[16] Tongue B, Principles of Vibration, Oxford University Press, 2001.
[17] Seto W W, Mechanical Vibrations, (Schaum's Outline), McGraw-Hill, 1964.
[18] Tong K N, Theory of Mechanical Vibration, Literary Licensing, 2013.
[19] Meirovitch L, Elements of Vibration Analysis, McGraw-Hill, 1986.
[20] Murata Manufacturing Co., Ltd. Piezoelectric Sound Components, available on http://www.murata.com/en-global/products/sound, 02-12-2015.
[21] LK-G3000 series High-speed, High-accuracy CCD Laser Displacement Sensor, available on http://www.keyence.eu/products/measure/laser-1d/lk-g3000/index.jsp, 02-12-2015.
[22] LK-G3001 series Controller, available on
http://cincinnatiautomation.com/pdf/LKG.LJG%20Brochure.pdf , 02-12-2015.
[23] Witten I H and Frank E, Data Mining: practical machine learning tools and techniques - 2nd ed., Morgan Kaufmann series in data management systems, 2005.
[24] Lehmann E L and Casella G, Theory of Point Estimation-2nd ed., New York: Springer, 1998.
- 140 -
Oscillation Results for a Class of Fourth-order Nonlinear Differential Equations with
Positive and Negative Coefficients
Nagehan Kılınç Geçer1 and Pakize Temtek2
Abstract. Over the past few years, there has been a strong interest in the study of the
oscillatory behavior of solutions of delay differential equations with positive and negative
coefficients of the first and second orders. However, very few results are available on the
study of oscillatory and asymptotic behavior of solutions of the fourth-order equations, which
is explained by significant technical difficulties arising in the analysis of these equations.
Thus we are concerned with oscillation of the fourth-order differential equations with positive
and negative coefficients and obtained oscillation results by using some techniques for these
differential equations.
Keywords. Oscillation, Fourth Order Nonlinear Differential Equation.
AMS 2010. 34C10, 35C15.
References
[1] Chuanxi, Q., Ladas, G., Oscillation in differential equations with positive and negative
coefficients, Can. Math. Bull., 33, 442-450, 1990.
[2] Li, W. T., Quan, H. S., Oscillation of higher order neutral differential equations with
positive and negative coefficients, Ann. Different. Equat., 2, 70-76, 1995.
[3] Li, W. T., Yan, J., Oscillation of first-order neutral differential equations with positive
and negative coefficients, Collect. Math., 50, 199-209, 1999.
[4] O’calan, O., Oscillation of forced neutral differential equations with positive and negative
coefficients, Comput. Math. and Appl., 54, 1411-1421, 2007.
[5] O’calan, O., Oscillation of neutral differential equation with positive and negative
coefficients, J. Math. Anal. and Appl., 331, 644-654, 2007.
1 Ahi Evran University, Kırşehir, Turkey, [email protected] 2 Erciyes University, Kayseri, Turkey, [email protected]
- 141 -
[6] Parhi, N., Chand, S., On forced first-order neutral differential equations with positive and
negative coefficients, Math. Slovaca, 50, 183-202, 2000.
[7] Tripathy, A. K., Oscillation properties of a class of neutral differential equations with
positive and negative coefficients, Fasc. Math., 45, 133-155, 2010.
[8] Parhi, N., Tripathy, A. K., On oscillatory fourth-order nonlinear neutral differential
equations, Math. Slovaca, 54, 389-410, 2004.
[9] Tripathy, A. K., Panigrahi, S., Basu, R., Oscillation results for fourth-order nonlinear
neutral differential equations with positive and negative coefficients, Journal of Mathematical
Sciences, 194(4), 453-471, 2013.
[10] Edwards, R. E., Functional analysis, Holt, Rinehart and Winston Inc., New York, 1965.
[11] Gyori, I., Ladas, G., Oscillation theory of delay differential equation with application,
Clarendon Press, Oxford, 1991.
- 142 -
Different Methods for Nonlinear Fractional Differential Equation
Özkan Guner1, Ahmet Bekir2, Adem C. Cevikel3 and Esin Aksoy4
Abstract. In this study, the nonlinear fractional partial differential equation have been
defined by the modified Riemann-Liouville fractional derivative [1,2]. By using this fractional
derivative and the fractional complex transform [3-5], the nonlinear fractional partial
differential equations have been converted into nonlinear ordinary differential equations. Then
the reduced equations can be solved by symbolic computation.
The different methods [6-14] are implemented to get exact solutions of the nonlinear
fractional equation. As a result, some exact solutions including hyperbolic solutions of this
equation have been successfully obtained.
Keywords. Exact Solution, Modified Riemann-Liouville Derivative, Fractional
Complex Transform.
AMS 2010. 34A08, 26A33, 83C15, 35R11.
Acknowledgements: This research has been supported by Yıldız Technical University
Scientific Research Projects Coordination Department. Project Number: 2014-09-04-DOP01
and Eskişehir Osmangazi University Scientific Research Projects (Grant No: 201519F03)
References
[1] Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of
nondifferentiable functions further results, Comput. Math. Appl., 51, 1367-1376, 2006.
[2] Jumarie, G., Table of some basic fractional calculus formulae derived from a modified
Riemann-Liouvillie derivative for nondifferentiable functions, Appl. Maths. Lett., 22, 378-
385, 2009.
[3] He, J H., Elegan, S.K., Li, Z.B., Geometrical explanation of the fractional complex
transform and derivative chain rule for fractional calculus, Physics Letters A, 376, 257-259,
2012.
1 Cankiri Karatekin University, Cankiri, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 3 Yildiz Technical University, İstanbul, Turkey, [email protected] 4 Yildiz Technical University, İstanbul, Turkey, [email protected]
- 143 -
[4] Saad, M., Elagan, S.K., Hamed, Y.S., Sayed, M., Using a complex transformation to get
an exact solutions for fractional generalized coupled MKDV and KDV equations, Int. Journal
of Basic & Applied Sciences, 13, 01, 23-25, 2013.
[5] Elghareb, T., Elagan, S.K., Hamed, Y.S., Sayed, M., An Exact Solutions for the
Generalized Fractional Kolmogrove-Petrovskii Piskunov Equation by Using the Generalized
(G′/G)-expansion Method, Int. Journal of Basic & Applied Sciences, 13, 01, 19-22, 2013.
[6] Bekir, A., Güner, Ö., Exact solutions of nonlinear fractional differential equations by
(G′/G)-expansion method, Chin. Phys. B, 22, 11, 110202, 2013.
[7] Bekir, A., Güner, Ö., Cevikel, A.C., The fractional complex transform and exp-function
methods for fractional differential equations, Abstract and Applied Analysis, 426462, 2013.
[8] Zhang, S., Zhang, H-Q., Fractional sub-equation method and its applications to nonlinear
fractional PDEs, Physics Letters A, 375, 1069-1073, 2011.
[9] Bekir, A., Guner, O., Aksoy, E., Pandir, Y., Functional variable method for the nonlinear
fractional differential equations, AIP Conf. Proc., 1648, 730001, 2015.
[10] Pandir, Y., Gurefe, Y., Misirli, E., The Extended Trial Equation Method for Some Time
Fractional Differential Equations, Discrete Dynamics in Nature and Society, 2013, 491359,
2013.
[11] Bekir, A., Guner, O., Bhrawy, A.H., Biswas, A., Solving nonlinear fractional differential
equations using exp-function and (G′/G)-expansion methods, Rom. Journ. Phys., 60, 3-4,
360-378, 2015.
[12] Demiray, S.T., Pandir, Y., Bulut, H., Generalized Kudryashov Method for Time
Fractional Differential Equations, Abstract and Applied Analysis, 2014, 901540, 2014.
[13] Bekir, A., Aksoy, E., Guner, O., A generalized fractional sub-equation method for
nonlinear fractional differential equations, AIP Conf. Proc., 1611, 78-83, 2014.
[14] Sahoo S., Ray, S.S., Improved fractional sub-equation method for (3+1)-dimensional
generalized fractional KdV–Zakharov–Kuznetsov equations, Computers and Mathematics
with Applications, 70, 158–166, 2015.
- 144 -
On Solutions of Some Second-Order Ordinary Differential Equations by Jacobi Last
Multiplier Method
Özlem Orhan1 and Teoman Özer2
Abstract. In this study, we present a new classification of a nonlinear equation and the
approach is used to obtain the general solutions by first integral and an effective technique for
analyzing first integral of nonlinear equation based on λ-jacobian approach is considered. The
effectiveness of this approach is demonstrated to find the general solution for the differential
equations defining geodesics on surfaces of revolution of constant curvature in a unified
manner.
Keywords. First Integral, Jacobi Last Multiplier Method, Ordinary Differential
Equations, Lagrangian.
References
[1] G.W. Bluman and S. Kumei, Symmetries and Differential Equations, 1989.
[2] M.C. Nucci, P.G.L. Leach, Jacobi’s last multiplier and symmetries for the Kepler problem
plus a lineal story, J. Phys. A 37, 7743–7753, 2004.
[3] Nucci MC, Jacobi last multiplier and Lie symmetries: a novel application of an old
Relationship, Journal of Nonlinear Mathematical Physics, 12, 284-304, 2005.
[4] Mustafa T., Al-Dweik Y., and Mara'beh A., On the Linearization of Second-Order
Ordinary Differential Equations to the Laguerre Form via Generalized Sundman
Transformations, Symmetry Integrability and Geometry: Methods and Applications, 9, 041,
2013.
1 Istanbul Technical University, Istanbul, Turkey, [email protected] 2 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 145 -
A Gravitational model with Y(R)F2-type Coupling
Özcan Sert1
Abstract. We investigate a gravitational model which involves a non-minimal
coupling between a function of curvature scalar and electromagnetic fields in Y (R)F2 form.
The models with RF2-type couplings [1–4] give more information about gravity and
electromagnetism. The natural extensions of the models to the Y (R)F2 gravity [5] were
studied in order to explain the late-time acceleration and inflation of the universe. Moreover,
the rotational curves of test particles around galaxies may be realized by considering a non-
minimal Y (R)F2 gravity [6,7]. We give the gravitational field equations obtained from a first-
order variation principle using the method of Lagrange multipliers and the algebra of exterior
differential forms. We discuss various spherically symmetric, electromagnetic solutions
consistent with observations.
Keywords. Gravitation, Einstein-Maxwell, Non-Minimal Coupling,
AMS 2010. 83C15, 83C22
References
[1] Prasanna, A.R., A new invariant for electromagnetic fields in curved space-time, Phys.
Lett. A, 37, 337, 1971.
[2] Horndeski, G.W., Conservation of Charge and the Einstein-Maxwell Field Equations, J.
Math. Phys., 17, 1980, 1976.
[3] Drummond, I.T., Hathrell, S.J., QED vacuum polarization in a background gravitational
field and its effect on the velocity of photons, Phys. Rev. D 22, 343, 1980.
[4] Dereli, T., Üçoluk, G., Kaluza-Klein reduction of generalised theories of gravity and non-
minimal gauge couplings, Class. Quantum Grav. 7, 1109, 1990.
[5] Bamba, K., Nojiri, S., Odintsov, S.D., The future of the universe in modified gravitational
theories: approaching a finite-time future singularity, JCAP 10, 045, 2008.
[6] Dereli, T., Sert, Ö., Non-minimal ln(R)F2 couplings of electromagnetic fields to gravity:
static, spherically symmetric solutions, Eur. Phys. J. C 71, 1589, 2011.
[7] Sert, Ö., Electromagnetic duality and new solutions of the non-minimally coupled Y(R)-
Maxwell gravity, Mod. Phys. Lett. A, 28, 12, 1350049, 2013.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 146 -
Some Aspects of Natural Convection in a Hydrodynamically and Thermally Anisotropic
Porous Non-Rectangular Cavity
Pallath Chandran1, Nirmal C. Sacheti2, Ashok K. Sing3 and Beer S. Bhadauria4
Abstract. Natural convection in a porous non-rectangular vertical cavity with a
sloping upper boundary with or without heat sources/sinks has been considered. It is assumed
that the cavity is filled with porous material subject to hydrodynamical and thermal
anisotropy. Assuming Darcy law to hold, together with Boussinesq approximation, the
governing partial differential equations have been solved numerically. To facilitate the
computation, the non-rectangular physical domain has been transformed to a square
computational domain using an algebraic grid generation method. The effect of a range of
parameters of interest such as internal heating parameter, slope of the upper boundary, Darcy-
Rayleigh number and aspect ratio, has been illustrated through plots of streamlines and
isotherms. Furthermore, the variation of the average Nusselt number has also been discussed
in relation to the anisotropic as well as heat source/sink parameters.
Keywords. Porous Media, Natural Convection, Anisotropy, Heat Source/Sink.
AMS 2010. 76S05, 76R10, 80A20.
1 Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected] 2 Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected] 3 Banaras Hindu University, Varanasi, India, [email protected] 4 Banaras Hindu University, Varanasi, India, [email protected]
- 147 -
Classic Metaheuristics and Evolutionary Optimization Algorithms for Routing
Problems: A Computational Study
Pantelis Z. Lappas1, Manolis N. Kritikos2 and George D. Ioannou3
Abstract. This paper focuses on the development of effective metaheuristic algorithms
for hard combinatorial optimization problems, such as routing problems. Routing problem is
the generic name given to a whole class of problems in which the transportation is necessary.
The Travelling Salesman Problem (TSP), the Vehicle Routing Problem (VRP) and the
Inventory Routing Problem (IRP) are representative examples of this category. Routing
problems have received much attention throughout the literature, because they are easily
described, widely applied to many transportation and distribution problems, and difficult to be
solved. The TSP is the most basic routing problem, as well as, a typical model of the
combinatorial optimization problems whose computation complexity is of non-polynomial
time (NP-Hard problem). The statement of the TSP is rather simple: a vehicle is to visit a
number of customers and the distance connecting two customers is known; the problem is to
find the shortest route that starts from a depot, visits all customers exactly once, and returns to
the depot. However, in transportation problems, customers have usually a demand, whereas
the depot consists of a fleet of vehicles, with limited and known capacity. This situation
reflects the VRP which generalizes the Multiple Travelling Salesman Problem (m-TSP), that
is the TSP with m vehicles. The VRP is even more complicated, since for small fleet sizes and
a moderate number of transportation requests, the planning task is highly complex. The
Vehicle Routing Problem with Time Windows (VRPTW) is a generalization of the VRP
involving the added complexity that every customer should be served within a given time
window. Furthermore, the IRP is an extension of the VRP, which integrates routing decisions
with inventory control. The problem arises in environments where Vendor Managed
Inventory (VMI) resupply policies are employed. Whereas VRPs typically deal with a single
day, IRPs have to deal with a longer horizon (a sequence of days). Actually, contrary to the
VRP, the vendor (supplier), not the customers, decides how much to deliver to which
customer each day. Moreover, the Inventory Routing Problem with Time Windows (IRPTW),
which is not addressed in the literature so far, is a generalization of the standard IRP involving
the added complexity that every customer should be served within a given time window. The
IRP is obviously NP-Hard, being a generalization of the single vehicle IRP, which reduces to
1 Athens University of Economics and Business, Athens, Greece, [email protected] 2 Athens University of Economics and Business, Athens, Greece, [email protected] 3 Athens University of Economics and Business, Athens, Greece, [email protected]
- 148 -
the TSP when time horizon is composed of one day, the inventory costs are zero, all the
customers need to be served and the transportation capacity is infinite. Due to the NP-hard
nature of the routing problems, it is very challenging to develop exact algorithms that can
solve large scale problems in a reasonable computational time. The TSP has been a primary
driving force for the development of novel optimization concepts and solution algorithms.
Several of the most important classic metaheuristics and evolutionary optimization algorithms
for the routing problems are summarized, implemented and compared, in the context of the
TSP. Classic metaheuristics that have been developed are related to algorithms such as: (1)
Iterated Local Search (ILS), (2) Guided Local Search (GLS), (3) Variable Neighborhood
Search (VNS), (4) Greedy Randomized Adaptive Search Procedure (GRASP), (5) Tabu
Search (TS) and (6) Simulated Annealing (SA). In addition, evolutionary optimization
algorithms that have been developed and compared to classic metaheuristics encompass
population-based approaches such as: (1) Genetic Algorithm (GA) and (2) Ant Colony
Optimization Algorithm (ACO). Some testing instances with different properties are
established to investigate the algorithmic performance, and the computational results are then
reported. Experimental results show that the GAs produce an optimal solution and reflect
superior performance compared to the classic algorithms. As a consequence, GAs have been
chosen and implemented to solve several variants of TSP, namely the m-TSP and the
VRPTW. Finally, an hybrid GA, combining Monte Carlo Simulation and GA, is used for
solving the IRPTW.
Keywords. Routing Problems, Metaheuristics, Evolutionary Optimization Algorithms.
- 149 -
The Lp Hardy Inequality with Two Weight Functions and Its Improved Versions
Semra Ahmetolan1 and İsmail Kombe2
Abstract. In this work, the Lp Hardy Inequality with two weight functions is obtained
for generalized Greiner vector fields:
∫ 𝜌𝛼|𝑧|𝑡|∇k∅|𝑝𝑑𝑑 Ω ≥ (𝑄+𝛼+𝑡−𝑝𝑝
)𝑝 ∫ 𝜌𝛼−𝑝|𝑧|𝑡|∇k𝜌|𝑝|∅|𝑝𝑑𝑑 Ω .
A generic point in ℝ2𝑛+1, 𝑛 ≥ 1, is defined by 𝑑 = (𝑧, 𝑙) = (𝑥,𝑦, 𝑙) ∈ ℝ2𝑛+1 where 𝑥,𝑦 ∈
ℝ𝑛 , 𝑧 = 𝑥 + √−1𝑦 and |𝑧| = (|𝑥|2 + |𝑦|2)1/2 . Here 𝑑𝑑 = 𝑑𝑧𝑑𝑙 denotes the Lebesgue
measure on ℝ2𝑛+1 and 𝜌(𝑑) = (|𝑧|4𝑘 + 𝑙2)1/4𝑘 . The sub elliptic gradient is the 2n
dimensional vector field given by ∇k= (𝑋1,𝑋2, … ,𝑋𝑛,𝑌,𝑌2, … ,𝑌𝑛) where 𝑋𝑗 = 𝜕𝜕𝑥𝑗
+
2𝑘𝑦𝑗|𝑧|2𝑘−2 𝜕𝜕𝜕
, and 𝑌𝑗 = 𝜕𝜕𝑦𝑗
− 2𝑘𝑥𝑗|𝑧|2𝑘−2 𝜕𝜕𝜕
, 𝑗 = 1,2, … ,𝑛, 𝑘 ≥ 1. Furthermore, we
also present its improved versions with remainder term by introducing a distance function. It
is well known that the Hardy inequality and its improved versions play important role in
partial differential equations with singular potentials [1-4]. Hence, there is a vast amount of
work in the literature related to the Hardy inequality [5-6].
Keywords. Hardy Inequality, Greiner Vector Fields, Sharp Constants.
AMS 2010. 22E30, 43A80, 26D10.
References
[1] Baras, P., Goldstein, J.A., The heat equation with a singular potential, Trasn. AMS, 284,
121-139, 1984.
[3] Cabré, X, Martel,Y. Existence versus explosion instantanée pour-des équations de la
chaleur linéaires avec potentiel singular, C.R. Acad. Sci. Paris, 329, 973-978, 1999.
[4] Goldstein, J.A., Kombe,I, Nonlinear parabolic differential equations with the singular
lower order term, Adv. Differential Equations, 10, 1153-1192, 2003.
[5] Niu, P., Ou, Y., Han, J., Several Hardy type inequalities with weights related to
Generalized Greiner operator, Canadian Mathematical Bulletin, 53, 153-162, 2010.
[6] Ahmetolan, S. , Kombe, I., A sharp uncertainty principle and Hardy-Poincaré inequalities
on sub Riemannian manifolds, Mathematical Inequalities & Applications, 15, 457-467, 2012.
1 Istanbul Technical University, Istanbul, Turkey, [email protected] 2 Istanbul Commerce University, Istanbul, Turkey, [email protected]
- 150 -
Static Analysis of Euler-Bernouilli Beams Resting on Foundation of Pasternak and
Winkler using Differential Transformation Method
Sema Bodur1 , Kanat Burak Bozdoğan2 and Galip Oturanç3
Abstract. In this study, differential equations of Euler – Bernouilli beams which play
an important role in engineering problems of Winkler and Pasternak foundation have been
examined [1,3.8]. Problems of beams, plates and shells rested on top of elastic foundation are
usually extreme due to the wideness of the area of their application. Some examples of these
applications are used in railway engineering, transportation pipes for fluids and gas, coastal
and seaport structures, missilse launchers, airports, aerospace, petro-chemical industries, bio-
mechanic and dentistry. Models like Winkler and Pasternak, Vlasov, Kerr and other types
have been developed for the resting foundation of structural elements such as beams and
plates[5,7]. Until today, problem of resting beam on elastic foundation, static, strain and
vibration, has been solved by calculating finite variations, finite elements, boundary elements,
Ritz, Galerkin using differential quadrature methods [2,7]. In this study, foundation models
that belong to Winkler and Pasternak foundations have been defined by Euler – Bernouilli
beam theory and equations of these problems calculated with the help of differential
transformation method [6]. Applicability between two foundation models has been analysed.
The differential transformation method used in analysis, is a technique of transformation
which is based on expansion of Taylor Series and analytic solutions of differential equations
[4]. In this method, specific transformation rules are applied to differential equations
belonging to a certain problem and boundary conditions to transform them into simple
analytic expressions. Characteristics of the described method have been defined and their
outcome compared to analytic results with the help of Maple 13 program. Subsidence values
have been presented in the table form to show the compliance of both outcome and analytic
results.
Keywords. Differential Transform Method, Winkler’s Model, Pasternak’s
Model, Euler – Bernouilli Beams
AMS 2010. 53A40, 20M15
References
[1] İnan M., Cisimlerin mukavemeti, İstanbul Teknik Üniversitesi Vakfı, 315-364, 2001.
1 Ege University, İzmir, Turkey, [email protected] 2 Kırklareli University, Kırklareli, Turkey, [email protected] 3 Selçuk University, Konya, Turkey, [email protected]
- 151 -
[2] Kılıç, V., Elastik zemine oturan plakların titreşimleri, Yüksek Lisans Tezi, İstanbul
Teknik Üniversitesi Fen Bilimleri Enstitüsü, İstanbul, 25-58, 2006.
[3] Özgan, K., Değiştirilmiş Vlasov modelini kullanarak elastik zemine oturan kirişlerin
serbest titreşim analizi, Yüksek Lisans Tezi, Karadeniz Teknik Üniversitesi Fen Bilimleri
Enstitüsü , Trabzon, 5-28, 2000.
[4] Oturanç, G., Keskin, Y., Diferansiyel Dönüşüm Yöntemiyle Diferansiyel Denklemlerin
Çözülmesi, Aybil Yayınları, Konya, 97-126, 2011.
[5] Bodur, S., Peker, H. A., Oturanç, G., Analysis of Winkler's Model of elastic Foundation
Using Differential Transform Method, III International Conference of the Georgian
Mathematical Union, Batumi, 2012.
[6] Bodur, S., Elastik Zemine Oturan Kiriş Modellerinin Diferansiyel Dönüşüm Methodu ve
Bilgisayar Destekli Analizi, Yüksek Lisans Tezi, Selçuk Üniversitesi Fen Bilimleri Enstitüsü,
2014.
[7] Civalek, Ö., Demir, Ç., Elastik zemine oturan kirişlerin ayrı tekil konvolüsyon ve
harmonik diferansiyel quadrature yöntemleriyle analizi, BAÜ FBE Dergisi, 11, 56-71, 2009.
[8] Kayan, İ., Cisimlerin Mukavemeti, İstanbul Teknik Üniversitesi İnşaat Fakültesi Matbaası,
İstanbul, 405-469, 1992.
- 152 -
Octonic Formulation of Dyons for Gravi-Electromagnetism
Süleyman Demir1, Murat Tanışlı2 and Mustafa Emre Kansu3
Abstract. Mathematically, octons are the eight-deimensional values composed of
scalars, vectors, pseudoscalars and pseudovectors. Mironov and Mironov [1] have proposed
this spatial object generating a closed noncommutative associative algebra and having a clear
well defined geometric interpretation. In relevant literature, octons have many physical
applications specially in electromagnetic theory [1,2], relativistic quantum mechanics [3,4]
and linear gravity [5]. In this work, a octonic model [6] is presented for the unification of
linear gravity and electromagnetism. Similarly, the generalized field equations are expressed
for the particles(dyons) carrying both electromagnetic and gravitational charges
simultaneously.
Keywords. Octon, Electromagnetic Theory, Linear Gravity, Dyon
AMS 2010. 00A06, 00A79, 12E12.
References
[1] Mironov V. L., Mironov S., Octonic representation of electromagnetic field equations, J.
Math. Phys. 50, 012901, 2009.
[2] Tolan, T., Tanışlı, M., Demir, S., Octonic form of Proca-Maxwell's equations and
relativistic derivation of electromagnetism, Int. J. Theor. Phys. 52, 4488–4506, 2013.
[3] Mironov V. L., Mironov S., Octonic first-order equations of relativistic quantum
mechanics, Int. J. Mod. Phys. A, 24, 4157, 2009.
[4] Mironov V. L., Mironov S., Octonic second-order equations of relativistic quantum
mechanics, J. Math. Phys. 50, 012302, 2009.
[5] Demir, S., Tanışlı, M., Tolan; T., Octonic gravitational field equations, Int. J. Mod. Phys.
A, 28, 1350112, 2013.
[6] Demir, S., Tanışlı, Kansu, M. E, Octonic massless field equations, Int. J. Mod. Phys. A,
accepted for publication, 2015.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected] 3 Dumlupinar University, Kutahya, Turkey, [email protected].
- 153 -
Evaluation of Soybean Hydration Model with Volume Variation
Seda Gülen1,2 and Turgut Öziş1
Abstract. Measurement of moisture during soybean hydration one of the most
important analyses on food product. During soaking, moisture profiles behave as distributed
parameter systems which predict moisture of grain over time by solution of partial differential
equations. One of the distributed parameter model in literature is known as Hsu model. Hsu
model has the advantage that it has effective diffusivity of water as its most important
parameter which depens on moisture content. Model parameters ( 0D , 1k , β ) from Hsu’s
(1983) which influence the behavior of the model and diffusivity are important.
In this study, Hsu model considered with volume variation during soybean hydration
was solved numerically. Different model parameters were obtained from literature and
optimal parameters were found at each temperature.
Keywords. Hsu Model, Finite Difference Method, Optimal Parameters.
AMS 2010. 65M06, 35K55.
References
[1] Nicolin, D.J., Coutinho, M., Andrade, C. M. D., Jorge, L. M. M., Hsu model analysis
considering grain volume variation during soybean hydration, Journal of Food
Engineering,111, 496-504, 2012.
[2] Hsu, K. H., A Diffusion Model with a Concentration- Dependent Diffusion Coefficient for
Describing Water Movement in Legumes During Soaking, Journal of Food Science, 48, 618-
623, 1983.
[3] M. R. Coutinho, P.R., Paraiso, R. M. M. Jorge, Application of the Hsu Model to Soybean
Grain Hydration, Cienc. Tecnol. Aliment., Vol.30, No.1, 2010.
[4] M. R. Coutinho, Modelagem, Simulaçeo Analise Da Hıdrataçao De Graos deSoja, Tese
(Doutorado submetida a Universidade Estadual de Maringa ), Brazil,.2006,
1 Ege University, Izmir, Turkey, [email protected] 2 Namik Kemal University, Tekirdag, Turkey
- 154 -
Stabilization of Solutions of Linear Volterra Implicit Integro-Differential
Equation of the First Order
S Iskandarov1
Abstract. All presented functions are continuous and relations is true 0 0, ;t t t tt≥ ≥ ≥
IDE - integro-differential equation; a stabilization of the solutions of linear first order Volterra
type IDE refers to the pursuit of finite limits for t →∞ all its solutions.
PROBLEM. To establish sufficient conditions for stabilization of solutions of the first
order IDE:
( ) ( ) ( )x t a t x t′ +0
[ ( , ) ( , )] ( ) ( ) ( ),t
t
K t Q t x d f t q tt t t t′+ + = +∫ 0.t t≥ (1)
This problem previously considered by the author (2002), author and D.N. Shabdanov
(2004). In this paper, a more general conditions imposed on the kernel ( , ) ( , )K t Q tt t+ and
the free term ( ) ( )f t q t+ than in the cited papers. Here presented one is of the results.
Let: 0 ( )tϕ< - some weighting function,
0( , ) ( , ),
n
ii
K t K tt t=
=∑0
( , ) ( , ),m
jj
Q t Q tt t=
=∑0
( ) ( ),n
ii
f t f t=
=∑0
( ) ( ),m
jj
q t q t=
=∑ ( )K , ( )Q , ( )f , ( )q
( ) ( 1.. ), ( ) ( 1.. )i jt i n t j mψ ϕ= = - some cutting functions,
1( , ) ( ) ( , )( ( ) ( ))i i i iR t t K t tt ϕ t ψ ψ t −≡ , 1( ) ( ) ( )( ( )) ,i i iE t t f t tϕ ψ −≡
)..1()()(),( 0 nitBtAttR iii =+= , ( )R
2( ) ( ) ( , )( ( ))j j iP t t Q t t tϕ ϕ −≡ , 1( , ) ( , )( ( )) ,j i jT t Q tt t ϕ t −≡ 1( ) ( ) ( )( ( )) ,i i jF t t q t tϕ ϕ −≡
( ) ( ) ( ) ( 1.. ),j j jP t M t N t j m= + = ( )P
1 2( ), ( ) ( 1.. ; 1.. )i jc t c t i n j m= = - some functions.
Note that the kernels ( , ) ( 1.. )iR t i nt = called cut; kernels ( , ) ( 1.. )iT t j mt = - partially cut.
THEOREM. Let 1) ( ) 0,tϕ > the conditions ( )K , ( )Q , ( )f , ( )q , ( )R , ( )P are true;
2) ( ) 0, ( ( ) ( )) 0;a t a t tϕ ′≥ ≤ 3) ( ) 0, ( ) 0, ( ) 0, ( , ) 0i i i iA t B t B t R tt t′ ′≥ ≥ ≤ ≥ , there are functions
1( ) ( , ),iA t L J R∗+∈ 1( ),ic t 1( ) ( , )iR t L J R∗
+∈ such that ( ) ( ) ( ),i i iA t A t A t∗′ ≤
( )2( ) ( ) ( )1( ) ( ) ( ),k k k
i i iE t B t c t≤ ( , ) ( ) ( , )it i iR t R t R tt tt t∗ ∗ ∗′≤ ( 1.. ; 0,1)j m k= = ; 4) ( ) 0,jM t >
1 Institute of Theoretical and Applied Mathematics of NAS of Kyrgyz Republic
- 155 -
( ) 0,jN t ≥ ( ) 0jN t′ ≤ , there are functions 1( ) ( , ),jM t L J R∗+∈ 2 ( )jc t such that
( ) ( ) ( ),j j iM t M t M t∗′ ≤ ( )2( ) ( ) ( )1( ) ( ) ( );k k k
j j jF t N t c t≤ 5) ( )0
12 20 0( )( ( )) ( ) ( ( , )) ( )
t
t
t f t t K t dϕ ϕ t ϕ t t−+ +∫
0
2 1 1( ) ( ( , )) ( ( )) ( , )t
j jt
t T t M d L J Rtϕ t t t−+′ ∈∫ ( 1.. )j m= . 6) ( ) 1 1( ) ( , \0)t L J Rϕ −
+∈ . Then for any
solution ( )x t of IDE (1): 1( ) ( , )x t L J R′ ∈ , i.e. any solution ( )x t of IDE (1) has a finite limit:
( ) ( )x t x→ ∞ under t →∞ .
Keywords. Integro-ordinary Differential Equations, Stabilization, Sufficient
Conditions.
AMS 2010. 53A40, 20M15.
- 156 -
Existence of a Solution for a General Class of Fredholm Integral Equations via F-
Contractive Non-Self-Mappings
Soomeyeh Khaleghizadeh Shahkhali
Abstract. This research concerns introducing new concepts of F-contractive non-self-
mapping and establishing the existence of PPF dependent fixed point theorems for such kinds of
contractive non-self-mappings in the Razumikhin class. This study also comprises an example
showing the validity of the researcher’s main result as well as an application in which an
existence and uniqueness of a solution for a general class of Fredholm integral equations of the
second kind has been proved.
- 157 -
Interval Oscillation Criteria for Functional Differential Equations of Fractional Order
Süleyman Öğrekçi1
Abstract. In this paper, we are concerned with the oscillatory behavior of a class of
fractional differential equations with functional terms. The fractional derivative defined in the
sense of the modified Riemann-Liouville derivative. By using a variable transformation,
generalized Riccati transformation, Philos type kernels, and averagaging technique we
establish new interval oscillation criteria. Illustrative examples also given.
Keywords. Differential Equations, Functional Term, Oscillation.
AMS 2010. 34C10, 34C15, 34K11.
1 Amasya University, Amasya, Turkey, [email protected]
- 158 -
Mathematical Aspects of Molecular Replacement: The Structure of Chiral Space
Groups Preferred by Proteins
Sajdeh Sajjadi1 and Gregory S. Chirikjian2
Abstract. The main goal of molecular replacement in macromolecular
crystallography is to find the appropriate rigid-body transformations that situate identical
copies of model proteins in the crystallographic unit cell. The search for such transformations
can be thought of as taking place in the coset space Γ/G where Γ is the chiral space group of
the macromolecular crystal, and G is the continuous group of rigid-body motions in Euclidean
space. Though the symmorphic case was addressed earlier in [1], [2] and [3], this paper is
concerned with decomposing nonsymmorphic Γ, which then gives freedom in how to choose
fundamental domains FΓ/G. Decompositions of Γ involve the concepts of Bieberbach
subgroups and maximal symmorphic subgroups. A number of new theorems are proven, and
it is shown how these concepts are related to the preferences that proteins have for
crystallizing in different space groups.
Keywords. Molecular Replacement, Bieberbach Group, Coset Space, Rigid-body
Motion.
AMS 2010. 53A40, 20M15.
References
[1] Chirikjian, G. S., Mathematical aspects of molecular replacement. I. Algebraic properties
of motion spaces, Acta Cryst. A: Foundations of Crystallography, 67(5), 435-446, 2011.
[2] Chirikjian, G. S., Yan, Y., Mathematical aspects of molecular replacement. II. Geometry
of motion space, Acta Cryst. A: Foundations of Crystallography, 68(2), 208-221, 2012.
[3] Chirikjian, G. S., Sajjadi, S., Toptygin, D., Yan, Y., Mathematical aspects of molecular
replacement. III. Properties of space groups preferred by proteins in the Protein Data Bank,
Acta Crystallographica Section A: Foundations and Advances, 71(2), 186-194, 2015.
[4] Janssen, T., Crystallographic Groups, Elsevier, 1973.
1 Johns Hopkins University, Baltimore, USA, [email protected] 2 Johns Hopkins University, Baltimore, USA, [email protected]
- 159 -
A New Extended Method For Solving Some Fractional Order Evolution Equations
Şerife Müge Ege1 and Emine Mısırlı2
Abstract. In this work, a new extended method is constructed to obtain the traveling
wave solutions of fractional order nonlinear equations. This method is effective, powerful and
can be used as an alternative to establish new solutions of different type of fractional
differential equations.
In this study, traveling wave solutions of the space-time fractional foam drainage
equation and the space-time fractional potential Kadomtsev-Petviashvili equation are derived
and compared with the solutions which obtained by means of modified Kudryashov method.
Keywords. Space-Time Fractional Foam Drainage Equation, Space-Time Fractional
Potential Kadomtsev-Petviashvili Equation, Modified Kudryashov Method.
AMS 2010. 35G20, 35G50.
References
[1] Kudryashov, N. A., One method for finding exact solutions of nonlinear differential
equations, Commun. Nonlinear Sci., 17, 2248–2253, 2012.
[2] Kudryashov, N. A., Exact solutions of the generalized Kuramoto–Sivashinsky equation,
Phys. Lett. A, 147, 287–91, 1990.
[3] Podlubny, I., Fractional differential equations, Math. Sci. Eng. Academic Press, New
York, USA, 1999.
[4] Jumarie, G., Fractional partial differential equations and modified Riemann-Liouville
derivative new methods for solution, J. Appl. Math. Compt., 24: 31-48, 2007.
[5] Ege S. M., Misirli E., The Modified Kudryashov Method for Solving Some Evolution
Equations, American Institute of Physics, 1470, 244, 2012.
[6] Ege S. M., Misirli E., Solutions of the Space-Time Fractional Foam Drainage Equation
and the Fractional Klein-Gordon Equation by Use of Modified Kudryashov Method,
International Journal of Research in Advent Technology, Vol.2, 384-388, 2014.
[7] Ege S. M., Misirli E., The Modified Kudryashov Method for solving some fractional-order
nonlinear equations, Advances in Difference Equations, 135, 1-13, 2014.
1 Ege University, Izmir, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
- 160 -
A Nonclassical and Nonautonomous Diffusion Equation with Infinite Delay
T. Caraballo 1 and A.M. Márquez Durán 2
Abstract. In this paper we consider the following nonclassical and nonautonomous
diffusion problem:
( ]
Ω∈∞−∈−=Ω×+∞=
Ω×+∞+=∆−
∆−
,,,),,(),(),(0
),(),()()(
xtxtxtuonu
inutfugutut
tu
t
ttf∂t
t∂∂g
∂∂
where τ ∈ is the initial time, nΩ ⊂ is a smooth bounded domain, and the time-dependent
delay term f(t, ut ) represents, for instance, the influence of an external force with some kind
of delay, memory or hereditary characteristic. Here, ut denotes a segment of solution, in other
words, given a function :u →Ω× , for each ∈t
we can define the mapping ut
(s)=u(t+s), s∈ (-∞,0].
In this way, this abstract formulation allows to consider several types of delay terms in a
unifed way.
When (t) is constant, this type of nonclassical parabolic equations has been very much
studied and is often used to model physical phenomena, such as non-Newtonian flows, soil
mechanics, heat conduction, etc (see, e.g., [1], [4], [5]). However, any physical model might
experiment some kind of natural or artificial changes, therefore it needs consistence under
perturbations (see, e.g., [3]). Moreover, in this paper we are interested in the case in which
some kind of delay is taken into account in the forcing term. This is an important variant of
the nondelay case because there are many situations in which the evolution of the model is
determined not only by the present state of the system but for its past history (see, e.g., [2]).
The existence and uniqueness of solution of our problem is first proved. We also
analyze the stationary problem and, under suitable conditions, we obtain exponential decay of
the solutions of the evolutionary problem to the stationary solution.
Keywords. Nonautonomous diffusion problem, infinite delay, stationary solution,
exponential decay.
AMS 2010. 35B20,35B50,35K55,37B55,37C70.
1 Universidad de Sevilla, Sevilla, España, [email protected] 2 Universidad Pablo de Olavide, Sevilla, España, [email protected]
- 161 -
References
[1] C.T. Anh and T.Q. Bao, Pullback attractors for a class of non-autonomous nonclassical
diffusion equations, Nonlinear Analysis, 73, 399-412, 2010.
[2] T. Caraballo and A.M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of
solutions for a nonclassical difusion equation with delay, Dynamics of Partial Differential
Equations,10, 3, 267-281, 2013.
[3] F. Rivero, Pullback attractor for non-autonomous non-classical parabolic equation,
Discrete Contin. Dyn. Syst. Ser. B, 18, 1, 209-221, 2013.
[4] C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic
Analysis, 59, 51-81, 2008.
[5] C. Sun, S. Wang y C. Zhong, Global attractors for a nonclassical diffusion equation,
Acta Math. Appl. Sin. Engl. Ser, 23, 1271-1280, 2007.
- 162 -
On a Spectral Expansion for Non-selfadjoint Boundary Value Problem
Volkan Ala1 and Khanlar R. Mamedov2
Abstract. Let us consider the boundary value problem generated by the differential
expression
[ )2( ) , 0,y q x y y xλ′′− + = ∈ ∞ (1)
with the boundary condition
2(0) (0) 0,y yλ′ − = (2)
where ( )q x is a complex-valued function, λ is a complex parameter.
In this work, the classification of spectrum for (1)-(2) is studied and the resolvent
operator is constructed. Eigenvalues and spectral singularities are investigated and under
certain conditions it is shown that spectral singularities have finite multiplicity. The principal
functions are defined and in terms of them the expansion formula is obtained.
Keywords. Sturm-Liouville Equation; Resolvent Operator; Spectral Singularities
AMS 2010. 53A40, 20M15.
References
[1] Naimark, M. A., Investigation on the spectrum and expansion in eigenfunctions of a non-
selfadjoint operator of the second order on a semi axis. AMS Translations, (2) 16, 103-193,
1960.
[2] Naimark, M. A., Linear differential operators I,II. Ungar, New York, 1968.
[3] Lyantse, V. E., A differential operator with spectral singularities I,II. AMS Translations,
(2) 60, 185-225 and 227-283, 1967.
1 Mersin University, Mersin, Turkey, [email protected] 2 Mersin University, Mersin, Turkey, [email protected]
- 163 -
Matrices with Null Columns in First Order Chemical Kinetics Mechanisms
Victor Martinez-Luaces1
Abstract. When modelling First Order Chemical Kinetics Mechanisms (F.O.C.K.M.),
the corresponding O.D.E. system associated matrix has a particular structure [1], [2]. As a
consequence of this fact, the Gershgorin Circles Theorem [3] can be applied to show that all
the eigenvalues are negative or zero. In several examples, like the conversion of grape juice
into wine and its final transformation into vinegar [1], the F.O.C.K.M. matrix has a null
column. Using an inverse modelling approach [4], [5], it can be proposed F.O.C.K.M. with
two or more final products and asociated matrices with two or more null columns. As a
consequence of this fact, the algebraic multiplicity of the null eigenvalue is greater than one
and therefore, the stability of the solutions must be studied. In this paper, F.O.C.K.M.
matrices with null columns are analyzed and the corresponding stability properties of the
solutions are studied. As an application of these results, the propagation of concentration
and/or surface concentration measurement errors are analyzed.
Keywords. Chemical Reactions, O.D.E. Systems, Null Column Matrices.
AMS 2010. 15A18, 34A30, 34D20.
References
[1] Martinez-Luaces,V., First Order Chemical Kinetics Matrices and Stability of O.D.E.
Systems, in Advances in Linear Algebra Research, Nova Science Publishers, New York,
U.S.A., 2015.
[2] Martinez-Luaces, V., Stability of O.D.E. solutions corresponding to chemical mechanisms
based-on unimolecular first order reactions, III - I.E.C.M.S.A., Vienna, Austria, 2014.
[3] Varga, R.S., Gershgorin and his circles, Springer-Verlag, Berlin, 2004.
[4] Martinez-Luaces, V., Modelling and Inverse Modelling: Experiences with O.D.E. linear
systems in engineering courses, International Journal of Mathematical Education in Science
and Technology, 40, 2, 259-268, 2009.
[5] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical
Kinetics, In Tech Open Science, Rijeka, Croatia, 2012.
1UdelaR, Montevideo, Uruguay, [email protected]
- 164 -
Stability of ODE Systems Associated to First order Chemical Kinetics Mechanisms
without Final Products
Victor Martinez-Luaces1
Abstract. First Order Chemical Kinetics Mechanisms (FOCKM) and their
corresponding O.D.E. system associated matrices were deeply studied in a book recently
published in U.S.A. [1]. In that publication was proved that all the eigenvalues have non-
positive real parts. This article can be considered as a further research on FOCKM matrices,
by analysing possible chemical mechanisms without final products. In this case the behaviour
of the solutions is absolutely different from other cases previously studied [2-3] since the
curves show oscilations which are not common in other cases where one or more final
products are present in the mechanism.
Some examples of this kind of mechanisms will be considered and the corresponding
ODE systems will be analysed from a qualitative viewpoint.
Finally, the algebraic and geometric multiplicity of the null eigenvalue in general
FOCKM matrices will be studied. As an application of algebraic theorems and chemical laws,
the stability of the solutions for these systems will be stated in all cases.
Keywords. Chemical Reactions, O.D.E. Systems, Stability of Solutions.
AMS 2010. 15A18, 34A30, 34D20.
References
[1] Martinez-Luaces,V., First Order Chemical Kinetics Matrices and Stability of O.D.E.
Systems, in Advances in Linear Algebra Research, Nova Science Publishers, New York,
U.S.A., 2015.
[2] Martinez-Luaces, V., Stability of O.D.E. solutions corresponding to chemical mechanisms
based-on unimolecular first order reactions, III - I.E.C.M.S.A., Vienna, Austria, 2014.
[3] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical
Kinetics, In Tech Open Science, Rijeka, Croatia, 2012.
1 UdelaR, Montevideo, Uruguay, [email protected]
- 165 -
A Two-Level Method for Emulating Parameterized Dynamic Partial Differential
Equation Models
Wei Xing1, V. Triantafyllidis1, Akeel Shah1, P.B. Nair2 and N. Zabaras1
Abstract. Model order reduction (MOR) techniques (e.g., proper orthogonal
decomposition (POD) and Krylov subspaces [1]) are often employed for numerical solutions
to PDE models when the computational expense of standard schemes is prohibitive. POD uses
a reduced-basis (RB) representation of the output space based on data from selected
simulations. The numerical formulation is restricted to the truncated basis and the coefficients
in the new basis become the targets for the numerical scheme. State-of-the-art methods are
global basis POD [2] and greedy RB [3]. We extend POD approaches for applications to
nonlinear, time-dependent PDEs with multiple parameters using a novel statistical emulator
that leverages nonlinear dimensionality reduction to learn the snapshots for a new parameter
value (standard methods are either unfeasible). The method is not restricted to POD and can
be used with other methods such as balanced truncation. Examples include 2D heat
conduction-convection (via finite volume) and a 1D burgers equation (using Galerkin FEM),
demonstrating the power of the method.
AMS 2010. 53A40, 20M15.
References
[1] Chen, P., Quarteroni, A., Rozza, G., A weighted empirical interpolation method: a priori
convergence analysis and applications, Math. Model. Numer. Anal., 48, 943-953, 2014.
[2] Jarvis, C., Reduced order model study of Burger's equation using proper orthogonal
decomposition, PhD thesis, Virginia Polytechnic Institute, 2012.
[3] Nouy, A., A generalized spectral decomposition technique to solve a class of linear
stochastic partial differential equations, Comput. Meth. Appl. Mech. Eng., 196, 4521-4537,
2007.
1 University of Warwick, Coventry, CV4 7AL, UK, [email protected] 2 University of Toronto Institute for Aerospace Studies, Toronto M3H 5T6, Canada
- 166 -
The Numerical Solution of the Symmetric RLW Equation by Using the Meshless Kernel
Based Method of Lines
Yılmaz Dereli1
Abstract. In this study, the Symmetric regularized long wave equation is solved
numerically by using meshless kernel method based of lines. The accuracy and efficiency of
the used method are tested by computing the invariants and error norms 𝐿2 and 𝐿∞ . The
numerical results are compared with numerical solutions of some earlier papers in the
literature.
Keywords. Meshless Method, Method of Lines, Symmetric RLW Equation
AMS 2010. 35C07, 35C08, 65M20
References
[1] Albert, J.; On the decay of solutions of generalized BBM equation, J. Math. Anal. Appl. 141, 2, 527-537, 1989.
[2] Amick, C. J., Bona, J. L., Schonbek, M. E.; Decay of solutions of nonlinear wave equations, J. Diff. Equa. 81, 1, 1-49, 1989.
[3] Seyler, C. E., Fenstermacher, D. L.; A symmetric regularized long wave equation, Physics of Fluids, 27, 1, 4-7., 1984.
[4] Peregrine, D. H.; Calculations of the development of an undular bore, J. Fluid Mech., 25, 321-330, 1966.
[5] Dong, X.; Numerical solutions of the Symmetric Regularized-Long-Wave equation by trigonometric integrator pseudospectral discretization., arXiv:1109.0764v1, 2011.
[6] Kaplan, A.G., Dereli, Y.; Numerical solutions of the Symmetric Regularized Long Wave Equation using radial basis functions, CMES, 84, 5, 423-437, 2012.
[7] Chen, L.; Stability and instability of solitary waves for generalized symmetric regularized long wave equation, Physica D, 118, 1-2, 53-68, 1998.
[8] Wendland, H.; Piecewise polynomial positive de.nite and compactly supported radial
functions of minimal degree, Adv. Comput. Math. 4, 389-396, 1995.
1 Anadolu University, Eskisehir, Turkey, [email protected]
- 167 -
One-Parameter Apostol-Genocchi Polynomials
Burak Kurt1
Abstract. In this work, we introduce and investigate a generalization of the one-
parameter Apostol-type polynomials by means of suitable generating functions. We establish
several interesting properties and relation between these polynomials. Furthermore, we give
explicit series representations for these polynomials. We prove the series expansion for the
product of the one-parameter Apostol-Genocchi polynomials G n(x,α,λ) and one-parameter
Apostol-Euler polynomials E_n(x,α,λ). Also, we give different form of the analogue of the
Srivastava-Pintér addition theorem.
Keywords. Bernoulli Numbers and Polynomials, Euler Polynomials and Numbers,
Genocchi Polynomials and Numbers, Apostol-Bernoulli Polynomials, Apostol-Euler
Polynomials, Apostol-Genocchi Polynomials, Stirling Numbers of Second Kind, One-
Parameter Apostol-Bernoulli Polynomials, Apostol-Genocchi Polynomials.
AMS 2010. 11B68, 11B73.
References
[1] Abromowitz M. and Stegun I. A., Handbook of Mathematical Functions with formulas,
Graps and Math. Tables, Dover Pub. Inc. New York, 1965.
[2] Apostol T. M., On the Lerch zeta function, Pasific J. Math., 1, 161-167, 1951.
[3] Brychkov Yu. A., On some properties of the generalized Bernoulli and Euler polynomials,
Int. Trans. and Special Functions, vol. 23, 10, 723-735, 2012.
[4] Luo Q.-M and Srivastava H. M., Some relationships between the Apostol-Bernoulli and
Apostol-Euler polynomials, Computers and Math. with Appl., 51, 631-642, 2006.
1 Akdeniz University, Antalya, Turkey, [email protected]
- 168 -
The Computational Complexity of Some Domination Parameters
Nader Jafari Rad 1
Abstract. A subset S ⊆ V is a dominating set of G if every vertex not in S is adjacent to
a vertex in S. The domination number of G, denoted by γ(G), is the minimum cardinality of a
dominating set of G. A dominating set S in a graph with no isolated vertex is a total
dominating set if the induced subgraph G[S] has no isolated vertex. The total domination
number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. The
bondage number of G, denoted by b(G), is the minimum number of edges whose removal
from G results in a graph with larger domination number. Let p ≥ 2 be a positive integer. The
p-bondage number of a graph G, is the minimum number of edges whose removal from G
results in a graph with larger p-domination number. The problem of NP-completeness has
received a huge amount of attention. The question of whether or not NP-complete problems
are intractable is one of the foremost problems, and determining whether or not a problem is
NP-complete plays an important role. Several parameters in the theory of domination have
been proved to be NP-complete, see for example [3, 4]. In this talk we study the complexity
issue of several parameters related to domination number and show that the decision problems
for these problems are NP-complete.
Keywords. Domination; p-domination; Bondage; Complexity.
AMS. 05C69
References
[1] J. E. Dunbar, T.W. Haynes, U. Teschner and L. Volkmann, Bondage, insensitivity and
rein-forcement, In: T. W. Haynes, S. T. Hedetniemi, P. J. Slater (eds.), Domination in Graphs:
Advanced Topics, Marcel Dekker, New York, 471-489, 1998.
[2] J. F. Fink, M. S. Jacobson, L. F. Kinch and J. Roberts, The bondage number of a
graph, Discrete Mathematics, 86 (1990), 47–57.
[3] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory
of NP-Completeness, Freeman, San Francisco, 1979.
[4]
1 Department of Mathematics, Shahrood University, Shahrood, Iran, Email: [email protected]
- 169 -
[5] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in
Graphs, Marcel Dekker, Inc. New York, 1998.
[6] F.-T. Hu, Y. Lu, J.-M. Xu, The total bondage number of grid graphs, Discrete Applied
Math-ematics 160, 2408–2418, 2012.
- 170 -
Construction of 4-Connected Graphic Matroids with Essential Elements
P. P. Malavadkar 1, M. P. Gadiya 2, S. B. Dhotrey 3 and M. M. Shikare 4
Abstract. An element e of an n-connected matroid M is called essential element if neither
M n e nor M=e is n-connected. Tutte proved that in a 3-connected matroid M every element is
essential if and only if M is wheel or whirl. We give construction of some families of 4-
connected graphic matroids in which every element is essential.
1 MIT College of Engineering, Savitribai Phule Pune University, Pune, India, _ [email protected] 2 MIT College of Engineering, Savitribai Phule Pune University, Pune, India, __ [email protected] 3 MIT College of Engineering, Savitribai Phule Pune University, Pune, India, y [email protected] 4 MIT College of Engineering, Savitribai Phule Pune University, Pune, India, [email protected]
- 171 -
Unified One-Parameter Apostol-Bernoulli, Euler and Genocchi Polynomials
Veli Kurt 1
Abstract. The aim of this paper is to introduce and investigate one-parameter Apostol-
Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between
these polynomials and the generalized power sums. We give explicit relations for these
polynomials and recurrence relations related to power sums.
Keywords. Bernoulli Polynomials and Numbers, Euler Polynomials and Numbers,
Apostol-Bernoulli Polynomials and Numbers, The Stirling Numbers of Second Kind, Unified
Apostol-Bernoulli, Euler and Genocchi Polynomials, One-Parameter Unified Apostol-Type
Polynomials, Modified Apostol-Type Polynomials.
AMS 2010. 05A10, 11B65, 11B68.
Reference
[1] B. S. El-Desouky and R. S. Gamma, A new unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials, App. Math. and Computer, 247, 695-702, 2014.
[2] M. El-Mikkaway and Faiz Altan, Derivation of identities involving some special polynomials and numbers via generating functions with applications, App. Math. and Computer, 230, 518-535, 2013.
[3] B. K. Karande and N. K. Thakare, On the unification of the Bernoulli and Euler polynomials, Indian J. of Pure App. Math., 6, 98-107, 1975.
[4] V. Kurt, Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums, Adv. in Diff. Equ., 32, 2013.
[5] M. A. Ozarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Computer and Math. with App., 62, 2452-2462, 2011.
1 Akdeniz University, Antalya, Turkey, [email protected]
- 172 -
Anti-Kahler-Codazzi Structures on Walker Manifolds
Arif Salimov1 and Sibel Turanlı2
Abstract. Theory of structures on manifolds is the one of the most interesting subjects
of modern differential geometry. One of these subjects is the complex structures. The
differential geometric aspects of manifolds which have such structures are very large and very
efficient areas for pseudo-Riemannian geometry.
Walker manifold is the triple in the form of (𝑀,𝑔,𝐷) , where M represents 4-
dimensional manifold, g represents indefinite metric and D the 2-dimensional parallel null
distribution. Canonical forms of such metrics were achieved by Walker (1950). It’s well
known that g has components
𝑔 = 0 0 1 00 0 0 11 0 𝑎 𝑐0 1 𝑐 𝑏
with respect to the adapted Walker coordinates ( 𝑥1, 𝑥2, 𝑥3, 𝑥4 ), where 𝑎, 𝑏 and 𝑐 are
functions of adapted coordinates.
The most intensive studies on Walker manifold were begun after 2004. Matsushita
(2004) built almost complex structures appropriate with Walker 4-manifolds. Chaichi (2005)
researched curvature features of 4-dimensional Walker metrics. Davidov (2006) examined
Almost Kahler Walker 4-manifolds and (2007) Hermitian Walker 4- manifolds, Salimov
(2010) analysed some features of Norden Walker metrics.
In this study, the purpose is introducing anti-Kahler-Codazzi manifolds which are new
class of the integrable almost anti-Hermitian manifolds and to investigate these structures on
Walker 4-manifolds.
Keywords. Walker Manifold, Anti-Kahler-Codazzi and Anti-Kahler Manifolds.
References
[1] Walker A.G., Canonical form for a Rimannian space with a paralel field of null planes, Quart. J. Math. Oxford 1 (2), 69-79, 1950.
1 Ataturk University, Erzurum, Turkey, [email protected] 2 Erzurum Technıcal University, Erzurum, Turkey, [email protected]
- 173 -
[2] Bonome A., Castro R., Hervella L. M. and MatsushitaY., Construction of Norden structures on neutral 4-manifolds, JP J. Geom. Topol.,5 (2) , 121-140, 2005.
[3] Chaichi M., García-Río E. and Matsushita Y. Curvature properties of four-dimensional Walker metrics, Classical and Quantum Gravity 22, 3, 559, 2005.
[4] Matsushita Y., Walker 4-manifolds with proper almost complex structure, J. Geom. Phys.,55 , 385-398, 2005.
[5] Davidov, J., Diaz-Ramos J.C., Garcia-Rio E., Matsushita Y., Muskarov O. and Vazquez-Lorenzo R., Almost Kähler Walker 4-manifolds, Journal of Geometry and Physics 57, 3, 1075-1088, 2007.
[6] Davidov, J., Diaz-Ramos J.C., Garcia-Rio E., Matsushita Y., Muskarov O. and Vazquez-Lorenzo R., Hermitian Walker 4-Manifolds, Journal of Geometry and Physics 58, 3, 307-323, 2008.
[7] Salimov A.A. and Iscan M., Some properties of Norden-Walker metrics, Kodai Mathematical Journal., 33, 2, 283-293, 2010.
[8] Salimov A.A., On operators associated with tensor fields, J. Geom., 99 (1-2), 107-145, 2010.
[9] Salimov A.A. and Turanli S., Curvature Properties of anti-Kahler-Codazzi Manifolds., C. R. Math. Acad. Sci. Paris., 351 (5-6), 225-227, 2013.
- 174 -
Classification of Geodesics on Sierpinski Gasket with the Intrinsic Metric
Bünyamin Demir 1, Yunus Özdemir 2 and Mustafa Saltan 3
Abstract. In this work, we explicitly define the intrinsic metric on Sierpinski Gasket
(SG) by which we determine the geodesics on SG. Moreover we give a classification of
geodesics on SG and we also show that there are at most five different geodesics between
two points on SG.
Key words: Sierpinski Gasket, Intrinsic Metric, Geodesic.
AMS 2010. 28A80, 51K99
References
[1] Burago, D., Burago, Y., Ivanov, S., A course in metric geometry, AMS, 2001.
[2] Barnsley, M. F., Fractals everywhere, Dover Publications, 2012.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected] 3 Anadolu University, Eskisehir, Turkey, [email protected]
- 175 -
On Ricci Solitons in Kenmotsu Manifolds with the Semi-Symmetric Non-Metric
Connection
Cumali Ekici1 and Hilal Betül Çetin2
Abstract. In this paper, we study 3-dimensional Kenmotsu manifolds with the semi-
symmetric non-metric connection. We obtain some results on Ricci solitons in Kenmotsu
manifolds with the semi-symmetric non-metric connection satisfying the conditions
0~).,(~=SXC ξ , 0~).,(~ =SXH ξ and 0~).,(~ =CXP ξ where C~ is quasi-conformal curvature
tensor, S~ is Ricci tensor, P~ is projective curvature tensor and H~ is conharmonic curvature
tensor. We also show that Ricci solitons are shrinking and expanding.
Keywords. Semi-Symmetric Non-Metrik Connection, Ricci Soliton, Kenmotsu
Manifold.
AMS 2010. 53C07, 53C25, 53D15.
References
[1] Bagewadi, C.S., Gurupadavva Ingalahalli and Ashoka, S.R., A study on Ricci solitons in Kenmotsu manifolds, International Scholarly Research Notices, Volume 2013, Article D 412593, 6 p., 2013.
[2] De, U. C. and Tripathi, M. M., Ricci tensor in 3-dimensional Trans-Sasakian manifolds, Kyungpook Mathematical Journal, 43, 247--255, 2003.
[3] Nagaraja, H. G. and Premalatha, C. R., Ricci solitons in Kenmotsu manifolds, Journal of Mathematical Analysis, 3, 2, 18—24, 2012.
[4] Tripathi, M. M., Ricci solitons in contact metric manifolds, http://arxiv. org / abs / 0801.4222, 9 p., 2008.
[5] Yıldız, A. and Çetinkaya, A., Kenmotsu manifolds with the semi-symmetric non-metric connection, preprint., 2013.
[6] Yıldız, A., De, U. C. and Turan, M., On 3-dimensional f-Kenmotsu manifolds and Ricci solitons, Ukrainian Mathematical Journal, 65, 5, 620-628, 2013.
∗ This study was supported by Scientific Research Projects Commission of Eskişehir Osmangazi University. Project code number: ESOGU-BAP 2013-282 1 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected]
- 176 -
Metallic Shaped Hypersurfaces in Lorentzian Space Forms
Cihan Özgür1 and Nihal Yılmaz Özgür2
Abstract. In [4], the present authors define the notion of a metallic shaped
hypersurface and give the classification of these kinds of hypersurfaces in real space forms. In
[5], Yang and Fu classified golden shaped hypersurfaces in Lorentzian space forms. In the
present study as a generalization of the results given in [5], we classify metallic shaped
hypersurfaces in Lorentzian space forms.
Keywords. Lorentzian Space Form, Metallic Shaped Hypersurface.
AMS 2010. 53C40, 53C42, 53A07.
References
[1] Abe N., Koike N., Yamaguchi S., Congruence theorems for proper semi-Riemannian
hypersurfaces in a real space form, Yokohama Math. J. 35, no. 1-2, 123—136, 1987.
[2] Crasmareanu M, Hretcanu C-E, Munteanu M-I., Golden and product-shaped
hypersurfaces in real space forms, Int J Geom Methods Mod Phys 2013; 10: 1320006, 9 pp.
[3] Magid, Martin A., Lorentzian isoparametric hypersurfaces. Pacific J. Math. 118, no. 1,
165—197, 1985.
[4] Özgür C., Yılmaz Özgür N., Classification of metallic shaped hypersurfaces in real space
forms, Turk J. Math. in press, DOI: 10.3906/mat-1408-17
[5] Yang D., Fu Yu., The classification of golden shaped hypersurfaces in Lorentz space
forms. J. Math. Anal. Appl. 412, no. 2, 1135—1139, 2014.
1 Balikesir University, Balikesir, Turkey, [email protected] 2 Balikesir University, Balikesir, Turkey, [email protected]
- 177 -
Homothetic Cayley Formula for Homothetic Motions around a Timelike Axis
and its Applications in Lorentzian Space
Doğan Ünal1, Mehmet Ali Güngör2 and Murat Tosun3
Abstract. In this study, homothetic Cayley mapping for a skew-symmetric matrix S
is defined. Some relations between skew-symmetric matrices and timelike vectors which are
corresponding to these matrices in 31 are given. Then, for homothetic motions around a
timelike axis in 31 , Rodrigues and Euler Parameters are obtained. Moreover, homothetic
rotation matrix which is obtained from a timelike vector is given and with the help of these,
some results, definitions, theorems and applications are obtained.
Keywords. Homothetic Cayley mapping, Cayley formula, homothetic motions.
AMS 2010. 53A17, 15A30.
References
[1] Bükçü, B., Cayley Formula and its Applications in . Ankara University Graduate School of Naturel and Applied Sciences Deparment of Mathematics, Ph.D. Thesis, (2003).
[2] Birman, G. S. and Nomizu, K., Trigonometry In Lorentzian Geometry. The American Mathematical Monthly, 91, 9, 543-549, 1984.
[3] Bottema, O. and Roth, B., Theoretical Kinematics. North-Holland Press, New York, 1979.
[4] Ergin, A. A., The Kinematic Geometry On The Lorentzian Plane. Ankara University Graduate School of Naturel and Applied Sciences Deparment of Mathematics, Ph.D. Thesis, (1989).
[5] Güngör, M. A., and TOSUN, M., One Parameter Lorentzian Motions In Lorentz 3-Space, Kragujevac Journal of Mathemathics , 31, 95 - 109, 2008.
[6] Hacısalioğlu, H. H. and ARSLAN, İ., The Sets of Homothetic Motions. Commun. Fac. Sci. Univ. Ank. Series A1, 39, 9-14 (11990), 1990.
[7] Keçilioğlu, O., Özkaldı, S. and Gündoğan, H., Rotations and Screw Motion with Timelike Vector in 3-Dimensional Lorentzian Space. Adv. App. Clifford Alg., 22, 1081-1091, 2012.
[8] Tosun, M., Küçük, A. and Güngör, M. A., The Homothetic Motions In The Lorentz 3-Space. Acta Mathematica Scientia, 26, 711-719, 2006.
1, 2, 3 Sakarya University, Sakarya, Turkey, [email protected], [email protected], [email protected]
31E
- 178 -
Screen Scalar Curvature in Screen Locally Conformal Coisotropic Lightlike
Submanifolds of a Semi-Euclidean Space
Erol Kılıç1, Sadık Keleş2 and Mehmet Gülbahar3
Abstract. The ideal screen distribution for screen locally conformal coisotropic
lightlike submanifolds is introduced in a semi-Euclidean space of index 2. Some inequalities
involving the screen scalar curvature on screen locally conformal coisotropic lightlike
submanifolds of a semi-Riemannian space form of index 2 are established.
Keywords. Ideal Screen Distribution, Lightlike Submanifold, Semi-Euclidean Space.
AMS 2010. 46C20, 53C40, 53C42.
Acknowledgements: The first author of this work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). (113F388 coded project.)
References
[1] Chen, B.-Y., Riemannian DNA, inequalities and their applications, Tamkang J. of Sci.
and Eng. 3(3), 123-130, 2000.
[2] Chen, B.-Y., Pseudo-Riemannian geometry, δ -invariants and applications. World
Scientific Publishing, Hackensack, NJ, 2011.
[3] Özgür, C., Tripathi, M. M., On submanifolds satisfying Chen’s equality in a real space
form, The Arab. J. for Sci. and Eng. 33, Number 2A, 320-330, 2008.
[4] Sahin, B., Screen conformal submersions between lightlike manifolds and semi
Riemannian manifolds and their harmonicity. Int. J. Geom. Methods Mod. Phys. 4(6), 987-
1003, 2007.
1Inonu University, Malatya, Turkey, [email protected] 2Inonu University, Malatya, Turkey, [email protected] 3Siirt University, Siirt, Turkey, [email protected]
- 179 -
The Fermi-Walker Derivative on the Spherical Indicatrix
Fatma Karakuş1 and Yusuf Yaylı2
Abstract. In this study Fermi-Walker derivative and Fermi-Walker parallelism and
non-rotating frame concepts are given along the spherical indicatrix of a timelike curve in
Minkowski 3-space. We consider a timelike curve in Minkowski space and investigate the
Fermi-Walker derivative along the principal normal indicatrix. The concepts which Fermi-
Walker derivative are analyzed along its principal normal indicatrix.
Keywords. Fermi-Walker Derivative, Fermi-Walker Parallelism, Non-Rotating
Frame, Principal Normal Indicatrix , Helix, Slant Helix.
AMS 2010. 53B20, 53B21, 53B50, 53Z05, 53Z99.
References
[1] Karakuş, F. and Yaylı, Y., On the Fermi-Walker derivative and Non-rotating frame, Int.
Journal of Geometric Methods in Modern Physics, Vol.9. No.8, 1250066 (11pages), 2012.
[2] Karakuş, F. and Yaylı, Y., The Fermi-Walker derivative on the Spherical Indicatrix of
Timelike Curve in Minkowski 3-Space, Advances in Applied Clifford Algebras,
DOI.:10.1007/s00006-015-0573-6.
[3] Balakrishnan, Radha., Space curves, anholonomy and nonlinearity, Pramana Journal of
Physics, 64(4), 607-615, 2005.
[4] Benn, I. M. and Tucker R. W. I, Wave mechanics and inertial guidance, Phys. Rev. D
39(6), 1594-1601, 1989.
[5] Fermi, E., Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Nat., 31, 184-306, 1922.
[6] Hawking, S.W.and Ellis, G.F.R., The large scale structure of spacetime, Cambridge Univ.
Press, 1973.
1 Sinop University, Sinop, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected]
- 180 -
On Hypersurfaces of Indefinite Quaternionic Kaehler Manifolds
Gülşah Aydın Şekerci1, Sibel Sevinç2 and A. Ceylan Çöken3
Abstract. In this study, some properties of hypersurfaces of indefinite quaternionic
Kaehler manifolds are analyzed. We research whether these hypersurfaces is indefinite
quaternionic Kaehler manifolds and some conclusions are obtained. Also, the differential
geometry of these manifolds is examined by using sectional curvature and Riemannian
curvature.
Keywords. Kaehler Manifold, Indefinite Kaehler Manifold, Hermitian Manifold,
Quaternionic Manifold.
AMS 2010. 53B35, 53C25, 53C26, 53C15.
References
[1] Mangione, V., QR-hypersurfaces of quaternionic Kaehler manifolds, Balkan Journal of
Geometry and Its Applications, vol. 8, no. 1, 63-70, 2003.
[2] Alekseevsky, D. V., Marchiafava, S., Hermitian and Kaehler submanifolds of a
quaternionic Kaehler manifold, Osaka J. Math., 38, 869-904, 2001.
[3] Ianuş, S., Mazzocco, R., Vilcu, G. E., Riemannian submersions from quaternionic
manifolds, Acta Appl. Math., 104, 83-89, 2008.
[4] Vilcu, G. E., A Schur-type theorem on indefinite quaternionic Kaehler manifolds, Int. J.
Contemp. Math. Sci., 2, 11, 529-536, 2007.
1 Süleyman Demirel University, Isparta, Turkey, [email protected] 2 Cumhuriyet University, Sivas, Turkey, [email protected] 3 Akdeniz University, Antalya, Turkey, [email protected]
- 181 -
On Holomorphically Projective Curvature Tensor in a Kahler-Weyl Space
Gülçin Çivi 1
Abstract. In this work, the tensor P, which is an invariant under the holomorphically
projective mapping between two Kahler -Weyl spaces is obtained and analogously to the
definition in a Kahler space, the tensor P is defined as the holomorphically projective
curvature tensor of Kahler-Weyl spaces. Then, the some basic identities of the
holomorphically projective curvature tensor by using the pure and hybrid tensors.
Keywords. Holomorphically Projective Mapping, Kahler-Weyl Space,
Holomorphically Projective Curvature Tensor.
MSC 2010. 53B35, 53B15; 53B20
References
[1] Bacso, S., Ilosvay, F.: On holomorphically projective mappings of special Kahler spaces. Acta Mathematica Academia Paedagogicae Nyiregyhaziensis, 15 , 41-44, 1999.
[2] Çivi , G., Arsan Gürpınar , G.: On holomorphically projective mapping of Kahler-Weyl spaces. Journal of Applied Mathematics, 5, number III, 79-83, 2012.
[3] Hlavaty, V.: Theorie d'immersion d'une $W_m$ dans $W_n$ , Ann. Soc.Polon. Math., 21, 196-206, 1949.
[4] Mikes, J.: On holomorphically projective mappings of Kahler spaces. Ukr, Geom. Sb., Kharkov, 23, 90-98, 1980.
[5] Mikes, J., Shiha, M., Vanzurova, A.: Invariant Objects by Holomorphically Projective Mappings of Parabolically Kahler Spaces, Journal of Applied Mathematics, Aplimat, Volume II, number I, 2009.
[6] Norden, A.: Affinely connected spaces. GRMFML, Moscow, (in Russian), 1976.
[7] Özdemir, F., Çivi Yıldırım, G.: On Conformally Recurrent Kahlerian Weyl Spaces, Topology and its Application, 153, 477-484, 2005.
[8] Pravanovic, M.: Holomorphically projective transformations of the Kahler spaces. Tensor, New ser.35, 99-104, 1981.
[9] Radulovic, Z., Mikes, J.: Geodesic and holomorphically projective mappings of conformally-Kahlerian spaces. Differential Geometry and its Applications, Proc. Conf. Opava (Czechoslavakia), silesian univ. 151-156, 1993.
1 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 182 -
Characterization of Null Scrolls in ℝ𝑣𝑚+𝑛
Gül Güner1 and F. Nejat Ekmekci2
Abstract. In this work, we expand the theory of null scrolls to the semi Riemannian manifolds with higher index. We construct a null scroll by
𝑌𝑀(𝑢,𝜃, 𝑡) = 𝑋(𝑢) + 𝑡(𝑛𝑇 + 𝑛𝑆)(𝑢), 𝑡 ∈ ℝ.
In our construction method, we assume the lightlike submanifold 𝑋(𝑈) = 𝑀 in ℝ𝑣𝑚+𝑛
as the base curve and a lightlike normal vector field in its transversal bundle as the directrix of
the null scroll. Note that this definition is the most general form of null scrolls so far. We also
give some examples.
Keywords. Null Scroll, Lightlike Submanifold, Sectional Curvature, Second
Fundamental Form.
AMS 2010. 53A35, 53C50.
References
[1] J. S. Pak, D. W. Yoon, On Null Scrolls Satisfying The Condition ∆H =100 AH, Comm. Korean Math. Soc. 15, No. 3, pp. 533–540, 2000.
[2] K. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Math. and Its App., Volume 364.105, 1996.
[3] K. Duggal, B. S¸ahin, Differential Geometry of Lightlike Submanifolds, Birk. Ver. AG. 2010.
[4] S. Izumiya, T. Sato, Lightlike Hypersurfaces along Spacelike Submanifolds in Minowski space-time, Journ. of Geom. and Phys. , 2013.
[5] B. O’Neill, Semi Riemannian Geometry with Applications to Relativity, 110 Academic Press, 1983.
[6] Y. H. Kim, D. W. Yoon, Classification of ruled surfaces in Minkowski 3 spaces, Journal of Geometry and Physics, No.49, pp. 89–100, 2004.
[7] L. K. Graves, Codimension One Isometric Immersions Between Lorentz Spaces,Tran. of The Amer. Math. Soc., Volume 252, pp. 115, 1979.
[8] H. Balgetir, M. Ergüt, H. Balgetir, M. Ergüt, Generalized Null Scrolls In The n-Dimensional Lorentzian space, Act. Math. Vietn., Vol. 29, No. 2, pp. 205-216, 2004.
1 Karadeniz Tech. University, Trabzon, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected]
- 183 -
On Affine Translation Surfaces
Hülya Gün Bozok1 and Mahmut Ergüt2
Abstract. In this paper we study the polynomial affine translation surfaces in 3E with
constant curvature. We derive some non-existence results for such surfaces. Several examples
are also given by figures.
Keywords. Affine Translation Surface, Polynomial Translation Surface, Gaussian
Curvature, Mean Curvature.
AMS 2010. 53A05, 53B25.
References
[1] Ali, A.T., Abdel Aziz, H.S. and Sorour A.H., On Curvatures and Points of the Translation Surfaces in Euclidean 3-space, Journal of the Egyp. Math. Soc., 23, 167-172, 2015.
[2] Liu, H. and Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad., 89, 9, 111-113, 2013.
[3] Munteanu, I.M and Nistor, A.I., On the geometry of the second fundamental form of translation surfaces in 3E , Houston J.Math. 37, 1087–1102, 2011.
[4] Yoon, D.W., Polynomial translation surfaces of Weingarten types in Euclidean 3-space, Central Eur. J. Math. 8, 3, 430–436, 2010.
1 Osmaniye Korkut Ata University, Osmaniye, Turkey, [email protected] 2 Namik Kemal University, Tekirdag, Turkey, [email protected]
- 184 -
On Quasi-Einstein Weyl Manifolds
İlhan Gül1 and Elif Özkara Canfes2
Abstract. In this work, first, we define the quasi-Einstein Weyl manifolds. Then, we
prove the existence of quasi-Einstein Weyl manifolds. Moreover, we examine quasi-Einstein
Weyl manifolds having semi-symmetric and Ricci-quarter symmetric connections and obtain
some results about them.
Keywords. Quasi-Einstein Weyl Manifolds, Semi-Symmetric Connection, Ricci-
Quarter Symmetric Connection.
AMS 2010. 53A30, 53A40, 53C25
References
[1] Chaki M. C. ; Maity R. K., On quasi Einstein manifolds. Publ. Math. Debrecen, 57, 297-
306, 2000.
[2] De U.C. and Ghosh G. C., On quasi Einstein manifolds. Periodica Mathematica
Hungarica, 48(1-2), 223-231, 2004.
[3] Hlavaty V., Theorie d'immersion d'une 𝑊𝑚 dans 𝑊𝑛. Ann Soc Polon. Math, 21, 196-206,
1949.
[4] Canfes, E.Ö.; Özdeger A., Some applications of prolonged covariant differentiation in
Weyl spaces. Journal of Geometry, 60(1/2), 7 -16, 1997.
[5] Canfes, E.Ö., Isotropic Weyl manifold with a semi-symmetric connection. Acta
Mathematica Scientia, 29B(1), 176 -180, 2009.
1 Istanbul Technical University, Istanbul, Turkey, [email protected] 2 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 185 -
On Constant Ratio Curves According to Bishop Frame in Minkowski 3-Space 31E
İlim Kişi1 and Günay Öztürk2
Abstract. In the present paper, we consider a curve in Minkowski 3-space 31E as a
curve whose position vector can be written as linear combination of its Bishop frame vectors.
In particular, we study the non-null curves in 31E and characterize such curves in terms of
their Bishop curvatures. Further, we obtain some results of T-constant and N-constant type
non-null curves in Minkowski 3-space 31E .
Keywords. Position Vector, Bishop Frame, Constant Ratio Curve.
AMS 2010. 53A04, 53A05.
References
[1] Bozkurt Z., Gök I., Okuyucu O. Z. and Ekmekci F. N., Characterization of rectifying,
normal and osculating curves in three dimensional compact Lie groups, Life Sci. 10, 819-
823, 2013.
[2] Bükçü B. and Karacan M.K., Bishop Frame of the Spacelike Curve with a Spacelike
Principal Normal in Minkowski 3-Space, Comm. de la Facul. des Sci. de l'Université
d'Ankara. Séries A1 57,1, 13-22, 2008.
[3] Chen B. Y., Constant ratio Hypersurfaces, Soochow J. Math. 28, 353-362, 2001.
[4] Chen B. Y., Convolution of Riemannian manifolds and its applications, Bull. Aust. Math.
Soc. 66, 177-191, 2002.
[5] Chen B.Y., Constant-ratio spacelike submanifolds in pseudo-Euclidean space. Houston J.
Math. 29, 281-294, 2003.
[6] Chen B. Y., More on convolution of Riemannian manifolds, Beitrage Algebra Geom. 44,
9-24, 2003.
1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected]
- 186 -
[7] Dugal K.L. and Bejancu A., Lightlike submanifolds of semi-Riemannian manifolds and
applications. Kluwer Academic, Dordrecht, 1996.
[8] Gürpınar S., Arslan K., Öztürk G., A Characterization of Constant-ratio Curves in
Euclidean 3-space E³, arXiv:1410.5577 ,2014.
[9] Karacan M.K. and Bükçü B., Bishop Frame of the Timelike Curve in Minkowski 3-Space,
S. Demirel Uni., Fac. of Sci. and Art, Jour. of Sci. 3,1, 80-90, 2008.
[10] O'Neill B., Semi-Riemannian Geometry with Application to Relativity. Academic Press,
1983.
[11] Walrave J., Curves and surfaces in Minkowski space, Doctoral thesis, K. U. Leuven, Fac.
Of Science, Leuven, 1995.
- 187 -
On Canal Surfaces According to Parallel Transport Frame in Euclidean Space E4
İlim Kişi1, Günay Öztürk2 and Kadri Arslan3
Abstract. In this study, we consider canal surfaces according to parallel transport
frame in Euclidean space E⁴. The curvature properties of these surfaces are investigated with
respect to k₁, k₂ and k₃ which are principal curvature functions according to parallel transport
frame. Finally, we point out that if spine curve γ is a straight line, then M is a Weingarten
canal surface and also M is linear Weingarten pipe surface.
Keywords. Parallel Transport Frame, Gaussian Curvature, Mean Curvature.
AMS 2010. 53C40, 53C42.
References
[1] Bishop L.R., There is more than one way to frame a curve, Amer. Math. Monthly, 82(3), 246-251, 1975.
[2] Bulca B., A Characterization of Surfaces in E⁴, PhD Thessis, Uludağ University, Graduate School of Natural on Applied Sciences, Bursa, 2012.
[3] Bulca B., Arslan K., Bayram B., Öztürk G., Canal Surfaces in 4-dimensional Euclidean Space, Preprint.
[4] Do Carmo, P. M., Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ.
[5] Farouki, R.T. and Neff, C. A. Algebraic Properties of Plane Offset Curves, Computer Aided Geometric Design, 7, 101-127, 1990.
[6] Gökçelik F., Bozkurt Z., Gök I., Ekmekci F. N. and Yaylı Y., Parallel Transport Frame in 4-dimensional Euclidean Space E4, Caspian J. of Math. Sci., 3, 1, 91-102, 2014.
[7] Karacan M. K., Bükcü B., On Natural Curvatures of Bishop Frame, Journal of Vectorial Relativity, 5, 34-41, 2010.
[8] Ro, S. J., Yoon, D.W., Tubes of Weingarten Types in a Euclidean 3-Space. Journal of the Chungcheong Mathematical Society, 22, 3, 359-366, 2009.
[9] Shani, U. and Ballard, D.H. Splines as Embeddings for Generalized Cylinders, Computer Vision, Graphics and Image Processing, 27,129-156, 1984.
1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected] 3 Uludag University, Bursa, Turkey, [email protected]
- 188 -
Complete System of Polynomial Invariants of Vectors in the Pseudo-Euclidean
Geometry of Index 1
İdris Ören1
Abstract. Let 𝐸1𝑛 be (n+1)-dimensional pseudo-Euclidean geometry of index 1, G be
the group M(n,1) of all motions of 𝐸1𝑛 or G is the subgroup of M(n,1) generated by rotations
and translations of 𝐸1𝑛.
Let 𝑈 be a subspace of 𝐸1𝑛 . For a subspace ⊂ 𝐸1𝑛 , denote the number of linearly
independent null vectors in U by 𝜘(𝑈).
This paper presents the system of generators of the set of all G –invariant polynomial
functions of vectors 𝑥1, 𝑥2, … , 𝑥𝑚 ∈ 𝐸1𝑛 . Correlations between the Gram matrix of vectors in
U and 𝜘(𝑈) are investigated.
Keywords. Pseudo-Euclidean Space, Invariant, Null Vector.
AMS 2010. 15A03, 15A63, 51M10, 83A05.
References
[1] Höfer , R., m-point invariants of real geometries, Beitrage Algebra Geom.,40, 261-266,
1999.
[2] Stasiak, E., Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of
index 1, Publ.Math.Debrecen, 57, 55-69, 2000.
[3] Misiak , A., Stasiak, E., Equivariant maps between certain G-spaces with G=O(n-1,1),
Math Bohem., 126, 555-560, 2001.
[4] Weyl, H., The classical groups. Their invariants and representations, Princeton
University Press, Princeton, NJ, 1997.
[5] Ören, İ., Complete system of invariants of subspaces of Lorentzian space, Iranian Journal
of Science and Technology: Transaction A-Science, 2015.(in press).
1 Karadeniz Technical University, Trabzon, Turkey, [email protected]
- 189 -
On Classification of Canal Surfaces in Minkowski 3-space
Kazım İlarslan1 and Ali Uçum2
Abstract. In this paper, we reconsider the canal surfaces for all Lorentz spheres which
are pseudo sphere, pseudo-hyperbolic sphere and lightlike cone. We find the parametrizations
of the surfaces. Moreover, we found the parametrization of the tubular surfaces with respect
to all Lorentz spheres.
Keywords. Canal Surfaces, Minkowski 3-Space, Space Curves.
AMS 2010. 53B30, 53C50, 53A35.
References
[1] Dillen, F., K¨uhnel, W., Ruled Weingarten surfaces in Minkowski 3-space. Manuscripte
Math. 98, 307–320, 1999.
[2] Karacan, M.K., Bukcu, B., An alternative moving frame for a tubular surface around a
spacelike curve with a spacelike normal in Minkowski 3-space. Rendiconti del Circolo
Matematico di Palermo 57, 193–201, 2008.
[3] Karacan, M.K., Tuncer, Y., Tubular surfaces of Weingarten types in Galilean and Pseudo-
Galilean. Bull. Math. Anal. Appl. 5, 87–100, 2013.
[4] Uçum A. And İlarslan K., New Types of Canal Surfaces in Minkowski 3-Space, Adv.
Appl. Clifford Algebras, DOI 10.1007/s00006-015-0556-7, 2015.
1 Kirikkale University, Kirikkale, Turkey, [email protected] 2 Kirikkale University, Kirikkale, Turkey, [email protected]
- 190 -
A Note on Matrix Representations of Split Quaternions
Mahmut Akyiğit1, Hidayet Hüda Kösal2 and Murat Tosun3
Abstract. In this study, we establish that there are a total of sixteen distinct ordered
sets of three 4 x 4 signed permutation matrices which will serve as the basis of an algebra of
split quaternions. After, we investigate properties of the fundamental matrices obtained from
the sixteen distinct triplets.
Keywords. Split Quaternion, Fundamental Matrices, Eigenvalues, Eigenvectors.
AMS 2010. 15R52, 15A99.
References
[1] Aragon G., Aragon J. L. and Rodriguez M. A. Clifford algebras and geometric algebra,
Adv. Appl. Clif. Algebras 7(2) 91-102, 1997.
[2] Farebrother R. W., Gro J. & Troschke S. Matrix representation of quaternions, Lin. Alg.
Appl. 362 251-255, 2003.
[3] J. Gro J., Trenkler G. & Troschke S. O., Quaternions: further contributions to a matrix
oriented approach, Linear Algebra Appl., 326, 205-213, 2001.
[4] Kula L. & Yayli Y., Split quaternions and rotations in semi Euclidean space, J. Korean
Math. Soc. 44 (6) 1313-1327, 2007.
[5] Ozdemir M. & Ergin A. A., Rotations with timelike quaternions in Minkowski 3- space, J.
Geom. Phys. 56 322-336, 2006.
[6] Rosenfeld B. A. Geometry of Lie Groups, Kluwer Academic Publishers, Dordrecht,
1997.Bruner, J. Acts of meaning. Cambridge, MA: Harvard University Press, 1990.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 191 -
On Integral Invariants of Parallel Ruled Surfaces with Darboux Frame∗
Muradiye Çimdiker1, Yasin Ünlütürk2 and Cumali Ekici3
Abstract. In this work, first, we obtain Steiner rotation vector of parallel ruled
surfaces with Darboux frame. Then, by using this rotation vector, we compute pitch length
and pitch angle of parallel ruled surfaces with Darboux frame. Also, we have some relations
among integral invariants of the surface in 𝐸3.
Keywords. Parallel Ruled Surface, Darboux Frame, Pitch Length, Pitch Angle.
AMS 2010. 53A05, 53A15.
References
[1] Blaschke, W., Diferensiyel Geometri Dersleri, İstanbul Üniversitesi Yay. No. 433, 1949.
[2] Brand, L., Vector and tensor analysis, John Wiley & Sons Inc., 1948.
[3] Craig, T., Note on parallel surfaces, Journal Für Die Reine und Angewandte Mathematik
(Crelle's Journal), Vol 94, 162--170, 1883.
[4] Darboux, G., Leçons sur la theorie generale des surfaces I-II-III-IV, Gauthier- Villars,
Paris, 1896.
[5] Hlavaty, V., Differentielle linien geometrie, Uitg P. Noorfhoff, Groningen, 1945.
[6] Hoschek, J., Integral invarianten von regelflachen, Arch, Math, XXIV, 1973.
[7] Müller, H.R., Verallgemeinerung einer formelvon Steiner, Abh. Braunschweig Wiss. Ges.,
31, 107-113, 1978.
[8] Saçlı, G.Y., Yüce, S., Characteristic properties of the ruled surface with Darboux frame
in 𝐸3, Kuwait Journal of Science, 2014 (in press).
[9] Sarıoğlugil, A., Tutar, A., On ruled surface in Euclidean space 𝐸3, Int. J. Contemp. Math.
Sci. 2, 1, 1-11, 2007.
∗ This work was supported by Scientific Research Projects Coordination Unit of Kırklareli University. Project code number: KLUBAP/054 1 Kirklareli University, Kirklareli, Turkey, [email protected] 2 Kirklareli University, Kirklareli, Turkey, [email protected] 3 Eskisehir Osmangazi University, Eskisehir, Turkey, e-mail: ç[email protected]
- 192 -
Directional Bertrand Curves
Mustafa Dede1 and Cumali Ekici2
Abstract. It is well known that a characteristic property of the Bertrand curve which
asserts the existence of a linear relation between curvature and torsion. In this paper, we
propose a new method for generating Bertrand curves, which avoids the basic restrictions.
Our main result is that every space curve is a directional Bertrand curve with infinite
directional Bertrand mates.
Keywords. Bertrand Curves, Offset, Frenet Frame.
AMS 2010. 53A04, 68U05
References
[1] Ravani, B., Ku, T. S., Bertrand Offsets of ruled and developable surfaces, Comp. Aided
Geom. Design, 23, 2, 145-152, 1991.
[2] Tunçer, Y. and Ünal, S. New representations of Bertrand pairs in Euclidean 3-space,
Applied Mathematics and Computation 219, 1833-1842, 2012.
[3] Matsuda, H. Yorozu, S., Notes on bertrand curves, Yokohama Mathematical Journal, 50,
41-58, 2003.
[4] Ekmekçi, N. and Ilarslan, K., On Bertrand curves and their characterization, Differential
Geometry Dynamical System, 3, 17-24, 2001.
[5] Lucas, P. and Ortega-Yagües, J. A., Bertrand curves in the three-dimensional sphere,
Journal of Geometry and Physics, 62, 1903-1914, 2012.
[6] Öztekin, H. B. and Bektaş, M., Representation Formulae for Bertrand Curves in the
Minkowski 3-space, Scientia Magna, 6, 89-96, 2010.
[7] Papaioannou, S. G., Kiritsis, D., An application of Bertrand curves and surface to
CAD/CAM, Computer-Aided Design, 8, 17, 348-352, 1985.
[8] Coquillart, S., Computing offsets of B-spline curves, Computer-Aided Design, 19, 6, 305-
309, 1987.
1 Kilis 7 Aralik University, Kilis, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected]
- 193 -
On the Parallel Ruled Surfaces in Galilean Space
Mustafa Dede1 and Cumali Ekici2
Abstract. In this paper, we investigated the parallel surfaces of the ruled surfaces in
Galilean space. There are three types of ruled surfaces in Galilean space. We derived the
necessary conditions for each types of the ruled surfaces of the parallel surfaces to be ruled.
Consequently, we constructed some examples.
Keywords. Parallel Curves, Galilean Space, Curvatures.
AMS 2010. 53A35; 53Z05
References
[1] Çöken A. C. Çiftçi Ü. and Ekici C. On parallel timelike ruled surfaces with timelike
rulings. Kuwait Journal of Science & Engineering, 35:21-31, 2008.
[2] Divjak B. Curves in pseudo-Galilean geometry. Annales Universitatis Scientiarum
Budapest, 41:117-128, 1998.
[3] Dede M. Ekici C. and Çöken A. C. On the parallel surfaces in Galilean space. Hacettepe
Journal of Mathematics and Statistics, 42(6):605-615, 2013.
[4] Cekici C. & Dede M. On the Darboux vector of ruled surfaces in pseudo-Galilean space.
Math. and Comp. App., 16:830-838, 2011.
[5] Guggenheimer H. W. Differential geometry. New York: McGraw-Hill, 1963.
[6] Sağel M. K. and Hacısalihoğlu H. H. On the parallel hypersurfaces with constant
curvature. Commun. Fac. Sci. Univ. Ank. Series A, 40:1-5, 1991.
[7] Kamenarovic I. Existence theorems for ruled surfaces in the Galilean Space G₃. Rad
HAZU Math, 456:183-196, 1991.
[8] Milin-Sipus Z. Ruled Weingarten surfaces in Galilean space. Periodica Mathematica
Hungarica, 56:213—225, 2008.
[9] Milin-Sipus Z. and Divjak B. Special curves on ruled surfaces in Galilean and pseudo-
Galilean spaces. Acta Math. Hungar., 98:203-215, 2003.
[10] Ekici C. and Çöken A. C. The integral invariants of parallel timelike ruled surfaces.
Journal of Mathematical Analysis and Applications, 393:97-107, 2012.
[11] Ünlütürk Y. & Cekici C. Parallel surfaces of spacelike ruled Weingarten surfaces in
Minkowski 3-space. New Trends in Math. Sci., 1:85-92, 2013.
1 Kilis 7 Aralik University, Kilis, Turkey, [email protected] 2 Eskişehir Osmangazi University, Eskisehir, Turkey, [email protected]
- 194 -
Directional Tubular Surfaces
Mustafa Dede1, Cumali Ekici2 and Hatice Tozak2
Abstract. In this paper, we introduce a new version of tubular surfaces. We first
define a new adapted frame along a space curve, and denote this the q-frame. We then reveal
the relationship between the Frenet frame and the q-frame. We give a parametric
representation of a directional tubular surface using the q-frame. Finally, some comparative
examples are shown to confirm the effectiveness of the proposed method.
Keywords. Frenet Frame, Pipe Surface, Tube, Adapted Frame.
AMS 2010. 53A04; 53A05;
References
[1] Bishop, R.L., There is more than one way to frame a curve. Am. Math. Mon. 82, 246-251,
1975.
[2] Bloomenthal, J., Calculation of reference frames along a space curve, Graphics gems,
Academic Press Professional, Inc., San Diego, CA, 1990.
[3] Choi, B.K. and Lee, C. S., Sweep surfaces modelling via coordinate transformations and
blending, Computer Aided Design, 22 (2), 87-96, 1990.
[4] Dede, M., Tubular surfaces in Galilean space, Math. Commun., 18, 209-217, 2013.
[5] Dogan, F. and Yaylı, Y., Tubes with Darboux Frame, Int. J. Contemp. Math. Sciences, 7,
751-758, 2012.
[6] Farouki, R.T. and Han, C. Y., Rational approximation schemes for rotation-minimizing
frames on Pythagorean-hodograph curves, Computer Aided Geometric Design, 20(7), 435-
454, 2003.
[7] Guggenheimer, H., Computing frames along a trajectory. Comput. Aided Geom. Des. 6,
77-78, 1989.
1 Kilis 7 Aralik University, Kilis, Turkey, [email protected] 2 Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected] and [email protected]
- 195 -
[8] LÄu, W. and Pottmann, H., Pipe surfaces with rational spine curve are rational,
Computer Aided Geometric Design, 13, 621-628, 1996.
[9] Klok, F., Two moving coordinate frames for sweeping along a 3D trajectory. Comput.
Aided Geom. Des. 3, 217-229, 1986.
[10] Maekawa, T., Patrikalakis, N.M., Sakkalis, T., Yu, G., Analysis and applications of pipe
surfaces, Comput. Aided Geom. Design, 15, 437-458, 1988.
[11] Shin, H., Yoo, S. K., Cho, S. K., Chung, W. H., Directional Offset of a Spatial Curve for
Practical Engineering Design, ICCSA, 3, 711-720, 2003.
[12] Wang, W., JÄuttler, B., Zheng, D., Liu, Y., Computation of rotation minimizing frames.
ACM Trans. Graph. 27 (1), 1-18, 2008.
[13] Yilmaz S. and Turgut, M., A new version of Bishop frame and an application to
spherical images, J. Math. Anal. Appl., 371, 764-776, 2010.
- 196 -
Special Proper Pointwise Slant Surfaces of an Almost Constant Curvature Manifold
Mehmet Gülbahar1, Erol Kılıç2 and Semra Saraçoğlu Çelik3
Abstract. The structure of the pointwise slant submanifolds in an almost product
Riemannian manifold is investigated and the special proper pointwise slant
product surfaces of a locally product manifold are introduced. Two examples of proper
pointwise slant surfaces of a locally product manifold which one is special and the
other one is not special are given.
Keywords. Almost Product Riemannian Manifold, Special Slant Surface, Curvature.
AMS 2010. 53C15, 53C40, 53C42.
Acknowledgements: The second author of this work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). (113F388 coded project.)
References
[1] Chen B.-Y., Special slant surfaces and a basic inequality, Results Math. 33, 65-78, 1998.
[2] Dursun U., Turgay N. C., Minimal and Pseudo-umbilical rotational rurfaces in Euclidean
Space, 4E , Mediterr. J. Math. 10(1), 497-506, 2012.
[3] Sahin B., Slant submanifolds of an almost product Riemannian manifold, J. Korean Math.
Soc. 43(4), 717-732, 2006.
[4] Yano K., Kon M., Structures on manifolds, Series in Pure Mathematics, World Scientific
Publishing Co., Singapore, 1984.
1Siirt University, Siirt, Turkey, [email protected] 2Inonu University, Malatya, Turkey, [email protected] 3Bartin University, Bartin, Turkey, [email protected]
- 197 -
Self-Duality Operator on 2-Forms
Nedim Değirmenci1 and Hatice Zeybek2
Abstract. It is well known that hodge-duality operator and self-duality concept of a 2-
form have great importance in both mathematics and physics [1], [2]. In this work we define a
duality operator 𝑇𝜑 on the space of 2-forms Ω2(𝑅𝑛) for 𝑛 > 4, where 𝜑 is a special (𝑛 − 4) −
form on 𝑅𝑛. We prove that 𝑇𝜑 is symmetric, hence have real eigenvalues. We define self-dual
and anti-self-dual 2-forms over 𝑅𝑛 by using the eigenvalues of 𝑇𝜑. We give some explicit
constructions in dimensions 𝑛 = 5, 6, 7 and 8.
Keywords. Hodge-* operator, 2-form, Self-duality.
AMS 2010. 53Bxx, 58Cxx.
References
[1] Corrigan E, Devchand C., Fairlie D. and Nuyts J., First order equations for gauge fields
in spaces of dimensions greater than four, Nucl. Phys. B., 214, 452-464, 1983.
[2] Naber, G., Topology, Geometry and Gauge Fields (Foundations), Springer-Verlag, 2011.
[3] Bige, A.H., Dereli, T., Koçak, Ş. The Geometry of Self-dual 2-forms, Journal of Math.
Phys. 38, 8, 4804-4814, 1997.
[4] Morita, S., Geometry of differential Forms , AMS, Providence-Rhode Island, 2001.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Hacettepe University, Ankara, Turkey, [email protected]
- 198 -
Pointwise Slant Submersions from almost Contact Metric Manifolds
Sezin Aykurt Sepet1 and Mahmut Ergüt2
Abstract. In this paper, we characterize the pointwise slant submersions from almost
contact metric manifolds onto Riemannian manifolds and give several examples.
Keywords. Riemannian Submersion, Almost Contact Metric Manifold, Pointwise
Slant Submersion.
AMS 2010. 53C15, 53D15, 53C43.
References
[1] Baditoiu, G., Ianus, S., Semi-Riemannian submersions from real and complex pseudo- hyperbolic
spaces, Differ. Geom. Appl., 16(1), 79-94, 2002.
[2] Baird, P., Wood, J.C., Harmonic morphisms between Riemannian manifolds, London
Mathematical Society Monographs, No.29, Oxford University Press, The Clarendon Press,
Oxford, 2003.
[3] Blair, D.E., Riemannian geometry of contact and symplectic manifolds, Springer, 2010.
[4] Chinea, D., Almost contact metric submersions, Rend. Circ. Mat. Palrmo, 34(1), 89-104,
1985.
[5] Falcitelli, M., Ianus, S., Pastore, A.M., Riemannian Submersions and Related Topics,
World Scientific, River Edge, NJ, 2004.
[6] Gray, A., Psudo-Riemannian almost product manifolds and submersions, J. Math. Mech.,
16, 715-737, 1967.
[7] Gunduzalp, Y., Slant submersions from almost product Riemannian manifolds, Turk. J.
Math., 37, 863-873, 2013.
[8] Ianus, S., Mazocco, R., Vilcu, G.E., Riemannian submersions from quaternionic manifolds, Acta Appl. Math., 104(1), 83-89, 2008.
1 Ahi Evran University, Kirsehir, Turkey, [email protected] 1 Namik Kemal University, Tekirdag, Turkey, [email protected]
- 199 -
[9] Kupeli, I., Murathan, C., On Slant Riemannian submersions for cosymplectic manifolds,
arXiv:1311.3657v1 [Math.DG], 2013.
[10] Lee, J.W., Sahin, B., Pointwise slant submersions, Bull. Korean Math. Soc., 51(4), 1115-
1126, 2014.
[11] Marrero, J.C., Rocha, J., Locally conformal Kahler submersions, Geom. Dedicata, 52(3),
271-289, 1994.
[12] Olzsak, Z., On almost cosymplectic manifolds, Kodai Math J. 4, 239-250, 1981.
[13] O'Neill, B., The fundamental equations of a submersions, Mich. Math. J., 13, 459-469,
1966.
[14] Park, K.S., H-Semi-Slant Submersions from almost quaternionic hermitian manifolds,
Taiwan. J. Math., 18(6), 1909-1926, 2014.
[15] Park, K.S., H-Slant submersions, Bull. Korean Math. Soc. 49(2), 329-338, 2012.
[16] Park, K.S., Pointwise slant and pointwise semi-slant submanifolds in almost contact
metric manifolds, arXiv:1410.5587v2 [math.DG], 2014.
[17] Sahin, B., Anti-invariant Riemannian submerisons from almost Hermitian manifolds,
Cent. Eur. J. Math., 8(3), 437-447, 2010.
[18] Sahin, B., Riemannian Submersions from almost Hermitian manifolds, Taiwanese J.
Math., 17(2), 629-659, 2013.
[19] Sahin, B., Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci.
Math. Roumanie, 54(102), no.1, 93-105, 2011.
[20] Vilcu, G.E., 3-submersions from QR-hypersurfaces of quaternionic Kahler manifolds,
Ann. Pol. Math., 98(3), 301-309, 2010.
[21] Watson, B., Almost Hermitian submersions, J. Differential Geom., 11(1), 147-165, 1976.
- 200 -
On Constant Ratio Curves in Galilean Spaces
Sezgin Büyükkütük1, İlim Kişi2 and Günay Öztürk3
Abstract. In this study we consider a unit speed curve in Galilean spaces as a curve
whose position vector can be written as linear combination of its Serret-Frenet vectors. We
show that there is no T-constant curve in Galilen spaces and we obtain some results of N-
constant type of curves in Galilean spaces 3G and 4G .
Keywords. Position Vectors, Frenet Equations, Galilean Space.
AMS 2010. 53A35, 53B30.
References
[1] Ahmad T.A., Positin Vectors of Curves in Galiean Space 3G , Matematicki Vesnic, 64, 3, 200-210, 2012.
[2] Azak A.Z., Akyiğit M. and Ersoy S. Involute-Evolute Curves in Galiean Space 3G , Scientia Manga, 4, 6, 75-80, 2010.
[3] Balgetir Öztekin H. And Tatlıpınar S., On Some Curves in Galiean Plane and 3-Dimensional Galilean Space, Journal of Dynamical Systems and Geometric Theories, 10, 189-196, 2012.
[4] Chen B.Y., Constant ratio Hypersurfaces , Soochow J. Math., 28, 353-362 2001.
[5] Chen B.Y., More on convolution of Riemannian manifolds , Beitrage Algebra Geom, 44, 9-24, 2003.
[6] Gün Bozok H., Öztekin H., Inextensible Flows of Curves In The Equiform Geometry of 4-Dimensional Galilean Space 4G ., Facta Universtatis Ser. Math. Uniform, 30, 2, 209-216, 2015.
[7] Kamenarovic I., Existence Theorems for Ruled Surfaces In The Galilean Space 3G , Rad Hazu Math, 456, 10, 183-196, 1991.
[8] Kızıltuğ S., Inelastic Flows of Curves In 4D Galilean Space, J. Math. Comput. Sci. 36 ,1520-1532, 2013.
[9] Öğrenmiş A.O., Ergüt M. and Bektaş M. , On the Helices in the Galilean Space 3G , Iranian Journal of Science & Technology, Transaction A, 31, 177-181, 2007.
[10] Pavkovic B.J. and Kamenarovic I, The equiform differential geometry of curves in the Galilean space 3G , Glasnic Matematicki, 22, 42, 449-457, 1987.
1 Kocaeli University, Kocaeli, Turkey, [email protected] 2 Kocaeli University, Kocaeli, Turkey, [email protected] 3 Kocaeli University, Kocaeli, Turkey, [email protected]
- 201 -
On Z-Projective Change of Kropina Spaces
Salim Ceyhan1 and Gülçin Çivi2
Abstract. In this paper, we consider the projective change of metrics of
the Kropina space and the Finsler space , respectively. It is known that The Douglas
and the Weyl Curvature tensors remain invariant under the projective change of the Finsler
metrics. Moreover, h-curvature tensor in the Berwald connection is invariant under the a
special projective change named as Z-projective change. In [1] M. Fukui and T. Yamada
studied in the projective change between two Finsler spaces. Then, in [2], B.D. Kim and H.Y.
Park proved that if a symmetric space remains to be symmetric one under the Z-projective
change then the space is of zero curvature.
In present paper, we first investigated in the quantities which are invariant under the Z-
projective change between two Finsler spaces. Then, we obtained the necessary and sufficient
conditions for a projective change between a Kropina space and a
Finsler space to be a Z-projective change.
Keywords. Finsler Spaces, R-Curvature Tensor, Riemann Curvature, Z-Projective Transformations.
AMS 2010. 53C20, 53C60.
References
[1] M. Fukui and T. Yamada, On projective mapping in Finsler geometry, Tensor, N.S., 35, 216-222, 2000.
[2] B-D. Kim and Ha-Y. Park, On special Finsler spaces with common geodesics , Korean Math. Soc., 15, 331-338, 2000.
[3] E. Peyghan and A. Tayebi, Generalized Berwald Metrics, Turkish Journal Math., 35, 1-10, 2011.
[4] S.S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific, Singapore, New Jersey, London, Hong Kong, 2005.
[5] Z. Shen, Lectures on Finsler geometry, World Scientific, Singapore, New Jersey, London, Hong Kong, 2001.
[6] Z. Shen, Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001.
1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Istanbul Technical University, Istanbul, Turkey, [email protected]
- 202 -
Seiberg-Witten Equations on 8-Dimensional Manifolds
Serhan Eker1 and Nedim Değirmenci2
Abstract. Seiberg – Witten equations, which are used to analyze the structure of the
4 −dimensional manifolds, consist of two equations [2], [4], [7]. The first of these equations
is called Dirac equation and the latter is called Curvature equation. In higher dimensions, to
decribe the Curvature equation, generalized self-duality concept of a 2 form is needed.
Seiberg – Witten equations on 8 −dimensional manifolds, with Spin(7) structure were defined
in [1], [3] and also with SU(4) structure were defined in [5], [6]. In these works the self –
dual part is the Λ7 which is the first part of the decomposition of 2 − form spaces Λ2(𝑀) =
Λ7 ⊕ Λ21. In this work, the Curvature equation on 8 −dimensional manifolds is defined by
using the second part Λ21. We write down explicit form of these equations on 𝑀 = ℝ8 and
give some non− trivial solution.
Keywords. Seiberg-Witten Equations, Dirac Operator, Spin(7)-Structure
AMS 2010. 53A40, 20M15.
References
[1] Bilge, A.H, Dereli, T., Koçak, S., Monopole equations on 8-manifolds with Spin(7)
holonomy, Commun Math Phys; 203: 21–30, 1999.
[2] Salamon, D., Spin Geometry and Seiberg-Witten Invariants,1996 (preprint).
[3] Değirmenci, N., Özdemir N., Seiberg-Witten Like Equations On 8-Manifold With
Structure Group Spin(7), Journal of Dynamical Systems and Geometric Theories, Vol.7,
No.1, May., 2009.
[4] Friedrich, T., Dirac Operators in Riemannian Geometry, Providence, RI, USA: AMS,
2000.
[5] Gao, Y.H., Tian G., Instantons and the monopole-like equations in eight dimensions. J
High Energy Phys; 5: 036, 2000.
[6] Karapazar, Ş., Seiberg-Witten equations on 8-dimensional SU(4)-structure, International
Journal of Geometric Methods in Modern Physics, Vol. 10, No. 3, 2013.
[7] Witten, E., Monopoles and four manifolds, Math Res Lett; 1: 769–796, 1994.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected]
- 203 -
Bicomplex Fibonacci Numbers
Semra Kaya Nurkan1 and İlkay Arslan Güven2
Abstract. In this study, we define bicomplex Fibonacci and bicomplex Lucas numbers
and give some algebraic properties of them.
Keywords. Fibonacci Numbers, Bicomplex Fibonacci Number, Bicomplex Lucas
Number.
AMS 2010. 11B39, 11E88, 13A18.
References
[1] Akyiğit, M., Kösal, H.H and Tosun, M., Split Fibonacci quaternions, Adv. in Appl.
Clifford Algebras, 23, 535-545. 2013.
[2] Clifford, W. K., Preliminary sketch of bi-quaternions, Proc. London Math. Soc., 4, 381-
395, 1873.
[3] Dunlap, R. A., The golden ratio and Fibonacci numbers. World Scientific, 1997.
[4] Horadam, A. F., A generalized Fibonacci sequence. American Math. Monthly, 68, 455-
459, 1961.
[5] Koshy, T., Fibonacci and Lucas numbers with applications. A Wiley-Intersience
Publication, USA, 2001.
[6] Nurkan, S. K. and Güven, İ. A., Dual Fibonacci quaternions, Advances in Applied
Clifford Algebras , Volume 25, Issue 2, pp 403-414, June 2015.
[7] Rochon, D. And Shapiro, M., On algebraic properties of bicomplex and hiperbolic
numbers, Anal. Univ. Oreda Fascicola Matematica 11, 1-28, 2004.
[8] Vajda, S., Fibonacci and Lucas numbers and the golden section. Ellis Horwood Limited
Publ., England, 1989.
1 Usak University, Usak, Turkey, [email protected] 2 Gaziantep University, Gaziantep, Turkey, [email protected]
- 204 -
A New Version of Bishop Frame and Position Vector of a Timelike Curve in Minkowski
3-Space
Süha Yılmaz1, Yasin Ünlütürk2 and Abdullah Mağden 3
Abstract. In this work, using common vector field as the binormal vector of Serret-
Frenet frame, we introduce position vector of a timelike curve according to the Bishop frame
in . We call it "Type-2 Bishop frame" of timelike curves. Moreover, we obtain that the first
vector field of the type- Bishop frame of a regular curve satisfies a vector differential
equations of the third order in .
Keywords. Timelike Curve, Position Vector, Minkowski Space, Type-2 Bishop
Frame.
AMS 2010. 53A05, 53B25, 53B30.
References
[1] Ali, A.T., Turgut, M., Position vector of a time-like slant helix in Minkowski 3- Space, J.
Math. Anal. Appl. 365, 559-569, 2010.
[2] Bishop, L.R., There is more than one way to frame a curve, Amer. Math. Monthly, 82, 3,
246-251, 1975.
[3] Chen, B.Y., When does the position vector of a space curve always lie in its rectifying
plane?, Amer. Math. Mounthly, 110, 147-152, 2003.
[4] Ilarslan, K., Boyacioglu, O., Position vectors of a timelike and a null helix in Minkowski
3-space, Chaos, Solitons and Fractals, 38, 1383-1389, 2008.
[5] Turgut, M., On the invariants of time-like dual curves, Hacettepe J. Math. Stat., 37, 129-
133, 2008.
[6] Turgut, M., Yılmaz, S., Contributions to classical differential geometry of the curves in
E3, Sci. Magna, 4, 5-9, 2008.
[7] Yılmaz, S., Turgut, M., A new version of Bishop frame and an application to spherical
images, J. Math. Anal. Appl., 371, 764-776, 2010.
1 Dokuz Eylul University, Izmir, Turkey, [email protected] 2 Kirklareli University, Kirklareli, Turkey, [email protected] 3 Ataturk University, Erzurum, Turkey, [email protected]
- 205 -
[8] Yılmaz, S., Bishop spherical images of a spacelike curve in Minkowski 3-space,
International Journal of the Physical Sciences, 5, 898-905, 2010.
[9] Yılmaz, S., Contributions to differential geometry of isotropic curves in the complex
space, J. Math. Anal. Appl., 374, 673-680, 2011.
[10] Özyılmaz, E., Classical Differential Geometry of Curves According to Type-2 Bishop
Frame, Math. and Comp. Appl., 16, 4, 858-867, 2011.
- 206 -
On f-Biharmonic Submanifolds in Space Forms
Selcen Yüksel Perktaş1, Erol Kılıç2 and Sadık Keleş3
Abstract. As a generalization of biharmonic maps, f-biharmonic maps are the extrama
of f- bienergy functional. In this paper we study f-biharmonic submanifolds whose defining
isometric immersions are f-biharmonic.
Keywords. f-Biharmonic Maps, f-Biharmonic Submanifolds.
AMS 2010. 58E20, 58C43.
References
[1] Oniciuc, C., Biharmonic maps between Riemannian Manifolds, Ann. Stiint. Al. I. Cuza
Iaşi, Tomul XLVIII, s. I a, Matematica, 2002.
[2] Ouakkas, S., Nasri, R. and Djaa, M., On the f-harmonic and f-biharmonic maps, JP J.
Geom. Topol. 10 (1), 11-27, 2010.
[3] Wang, Z. -P. and Ou, Y. -L., Biharmonic Riemannian submersions from 3-manifolds,
Math. Z., 269, 917-925, 2011.
[4] Ou, Y. -L., and Lu, S., Biharmonic maps in two dimensions, Annali di Matematica Pura ed
Applicata, 192, 127-144, 2013.
1 Adiyaman University, Adiyaman, Turkey, [email protected] 2 Inonu University, Malatya, Turkey, [email protected] 3 Inonu University, Malatya, Turkey, [email protected]
- 207 -
An Examination on the Frenet Ruled Surfaces along the Bertrand Pairs Α and Α∗,
according to Their Normal Vector Fields in Euclidean 3-Space
Şeyda Kılıçoğlu1, Süleyman Şenyurt2 and H. Hilmi Hacısalihoğlu3
Abstract. In this paper we consider eight special Frenet ruled surfaces along to the
Bertrand pairs α∗, α respectively. First we find the excplit equations of Frenet ruled
surfaces along the Bertrand curve α, later we find the excplit equations of Frenet ruled
surfaces along its Bertrand mate α∗ in terms of the Frenet apparatus of its Bertrand curve α.
Further, normal vector fields of these Frenet ruled surfaces have been calculated in terms of
the Frenet apparatus of its Bertrand curve α too.
Finally we find a matrix which give us all sixteen positions of Normal vector fields of
eight Frenet ruled surfaces. Using that matrix we can examine positions, one by one, each pair
of Frenet ruled surfaces, in terms of Frenet apparatus of Bertrand curve α. For example one of
corallaries gives us which Frenet ruled surface are perpendicular.
Keywords. General Helix, Ruled Surface, Nil Space.
AMS 2010. 53A04, 53A05.
References
[1] Boyer C., A History of Mathematics, Wiley, New York 1968.
[2] do Carmo, M. P., Differential Geometry of Curves and Surfaces. Prentice-Hall, ISBN 0-
13-212589-7, 1976.
[3] Eisenhart, Luther P., A Treatise on the Differential Geometry of Curves and Surfaces,
Dover, ISBN 0-486-43820-1, 2004.
[4] Graves, L.K., Codimension one isometric immersions between Lorentz spaces. Trans.
Amer. Math. Soc. 252, 367—392, 1979.
[5] Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd
ed. Boca Raton, FL: CRC Press, p. 205, 1997.
1 Başkent Üniversitesi, Ankara, Turkey, [email protected] 2 Ordu Üniversitesi, Ordu, Turkey, [email protected] 3 [email protected]
- 208 -
[6] Hacisalihoğlu, H.H., Diferensiyel Geometri, Cilt 1, Ínönü Üniversitesi Yayinlari, Malatya
1994.
[7] Kiliçoğlu Ş., n-Boyutlu Lorentz uzayinda B-scrollar. Doktora tezi, Ankara Üniversitesi
Fen Bilimleri Enstitüsü, Ankara 2006.
[8] Kılıcoglu Ş.; On the b-scrolls with time-like directrix in 3-dimensional Minkowski Space.
Beykent University Journal of Science and Technology, 2(2):206-215, 2008.
[9] Kılıcoglu Ş.; On the generalized B-scrolls with p th degree in n- dimensional Minkowski
space and striction ( central spaces). Sakarya Üniversitesi Fen Edebiyat Dergisi, 10(2) , 15-
29 (ISSN: 1301-3769 )., 10(2):15-29, 2008.
[10] Kılıçoğlu, Ş., On the Involutive B-scrolls in the Euclidean Three-space E 3. XIIIth.
Geometry Integrability and Quantization, Varna, Bulgaria: Sofia , pp 205-214, 2012.
[11] Kılıçoğlu Ş., Hacısalihoğlu H.H., and Senyurt, S., On the fundamental forms of the B-
scroll with null directrix and Cartan frame in the Minkowskian 3-space. Applied
Mathematical sciences (accepted).
[12] L.J. Alias, Angel Ferrandez, Pascual Lucas and Miguel Angel Merono, On the Gauss
map of B-scrolls, Tsukuba J. Math. 22, 371-377, 1998.
[13] Hoschek, J.: Liniengeometrie. BI-Hochschulskripte, Z¨urich 1971.
[14] Izumiya, S.; Takeuchi, N.: Special curves and Ruled surfaces . Beitr¨age zur Algebra und
Geometrie Contributions to Algebra and Geometry, Volume 44, No. 1, 203-212, 2003.
[15] Lipschutz, M.M., Differential Geometry, Schaum's Outlines.
[16] Senyurt, S., and Kılıcoglu S¸ On the differential geometric elements of the involute D
scroll, Adv. Appl. Cliff ord Algebras, Springer Basel, doi:10.1007/s00006-015-0535-z, 2015.
- 209 -
The Steiner Formula and the Holditch Theorem for Homothetic Motions in the
Generalized Complex Plane
Tülay Erişir1, Mehmet Ali Güngör2 and Murat Tosun3
Abstract. In this study, we first obtained the Steiner area formula and the Holditch
theorem giving the relationship between the areas formed by points for homothetic motions in
the generalized complex plane (or p − complex plane). In this way, for p∈ we generalized
the Steiner Formula and Holditch theorem consisting the Euclidean ( 1)p = , Galilean ( 0)p =
and Lorentzian ( 1)p = − cases for homothetic motions.
Keywords. Generalized Complex Plane, Homothetic Motion, Holditch Theorem
AMS 2010. 53A17, 53B50, 11E88.
References
[1] H. Holditch, Geometrical Theorem, Q. J. Pure Appl. Math, 2, 1858.
[2] W. Blaschke, H. R. Müller, Ebene Kinematik, Verlag Oldenbourg, München, 1956.
[3] H. Potmann, Zum Satz von Holditch in der euklidischen Ebene, Elem. Math., 41, 1-6,
1986.
[4] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,
New-York, 1979.
[5] T. Erişir, M. A. Güngör and M. Tosun, A New Generalization of the Steiner Formula and
the Holditch Theorem, Adv. Appl. Clifford Algebr., DOI 10.1007/s00006-015-0559-4, 2015.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 210 -
Relaxed Elastic Line of Second Kind on an Oriented Surface in the Galilean Space
Tevfik Şahin1
Abstract. In this paper, we derive the Euler-Lagrange equations for a relaxed elastic
line of second kind on an oriented surface in the Galilean 3-dimensional space G3. These
equations will give direct and more geometric approach to questions concerning about relaxed
elastic lines of second kind on an oriented surface in G3.
Keywords. Galilean Space, Relaxed Elastic Line of Second Kind, Variational Method,
Geodesic.
AMS 2010. 53A35, 53A55, 49Q20.
References
[1] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, New
Jerse, 1976.
[2] Manning, G.S., Relaxed elastic line on a curved surface, Quart. Appl. Math. 45(3), 515-
527, 1987.
[3] Ünan, Z., Yılmaz, M., Elastic lines of second kind on an oriented surface, Ondokuz Mayıs
Üniv. Fen dergisi 8(1), 1-10, 1997.
[4] Röschel, O., Die Geometrie des Galileischen Raumes, Habilitationssch., Inst. für Mat und
Angew., Geometrie. 1984.
[5] Şahin, T., Intrinsic Equations for a Generalized Relaxed Elastic Line on an Oriented
Surface in The Galilean Space, Acta Mathematica Scientia, 33(3), 701- 711, 2013.
[6] Yaglom, I.M., A Simple Non-Euclidean Geometry and Physical Basis, Sprınger-Verlag,
Newyork, 1979.
[7] Yılmaz, M., Some relaxed elastic line on a curved hypersurface, Pure Appl. Math. Sci. 39,
59-67, 1994.
1 Amasya University, Amasya, Turkey, [email protected]
- 211 -
Centro-equiaffine Differential Invariants of Curve Families
Yasemin Sağıroğlu1
Abstract. The generator set of all centro-equiaffine differential invariant rational
functions field for arbitrary curves is obtained. By using these generators, the conditions of
equivalence for two curve families are found. Then the relations between elements of
generator set are investigated.
Keywords. Differential Invariant, Parametric Curve, Equivalence.
AMS 2010. 53A35, 53A55.
References
[1] Aripov, R.G., Khadzhiev, D., The Complete System of Global Differential and Integral
Invarians of a Curve in Euclidean Geometry, 51, 7, 1-14, 2007.
[2] Gardner, R.B., Wilkens, G.R., The Fundamental Theorems of Curves and Hypersurfaces
in Centro-affine Geometry, Bull. Belg. Math. Soc., 4, 379-401,1997.
[3] Pekşen, Ö., Khadjiev, D., On invariants of curves in centro-affine geometry, J. Math.
Kyoto Univ., 44, 3, 603-613, 2004.
[4] Sağıroğlu, Y., Global differential invariants of affine curves in R2, Far East Journal of
Mathematical Sciences, 96, 4, 497-515, 2015.
[5] Ünel, M., Wolovich, W.A., On the Construction of Complete Sets of Geometric Invariants
for Algebraic Curves, Advances in Applied Mathematics, 24, 65-87, 2000.
1 Karadeniz Technical University, Trabzon, Turkey, [email protected]
- 212 -
On Parallel Ruled Surfaces with Darboux Frame∗
Yasin Ünlütürk1, Muradiye Çimdiker2, Cumali Ekici3
Abstract. In this paper, parallel ruled surfaces with Darboux frame are introduced in
𝐸3. Some characteristic properties of parallel ruled surfaces with Darboux frame are given
such as developability and striction point. And then, integral invariants are computed for
parallel ruled surfaces with Darboux frame in 𝐸3.
Keywords. Parallel Ruled Surface, Darboux Frame, Integral Invariants.
AMS 2010. 53A05, 53A15.
References
[1] Blaschke, W., Diferensiyel Geometri Dersleri, İstanbul Üniversitesi Yay. No. 433, 1949.
[2] Brand, L., Vector and tensor analysis, John Wiley & Sons Inc., 1948.
[3] Craig, T., Note on parallel surfaces, Journal Für Die Reine und Angewandte Mathematik
(Crelle's Journal), Vol 94, 162--170, 1883.
[4] Darboux, G., Leçons sur la theorie generale des surfaces I-II-III-IV, Gauthier- Villars,
Paris, 1896.
[5] Hlavaty, V., Differentielle linien geometrie, Uitg P. Noorfhoff, Groningen, 1945.
[6] Hoschek, J., Integral invarianten von regelflachen, Arch, Math, XXIV, 1973.
[7] Müller, H.R., Verallgemeinerung einer formelvon Steiner, Abh. Braunschweig Wiss. Ges.,
31, 107-113, 1978.
[8] Saçlı, G.Y., Yüce, S., Characteristic properties of the ruled surface with Darboux frame
in 𝐸3, Kuwait Journal of Science, 2014 (in press).
[9] Sarıoğlugil, A., Tutar, A., On ruled surface in Euclidean space 𝐸3, Int. J. Contemp. Math.
Sci. 2, 1, 1-11, 2007.
∗ This work was supported by Scientific Research Projects Coordination Unit of Kırklareli University. Project code number: KLUBAP/054 1 Kirklareli University, Kirklareli, Turkey, [email protected] 2 Kirklareli University, Kirklareli, Turkey, [email protected] 3 Eskisehir Osmangazi University, Eskişehir, Turkey, ç[email protected]
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On the Metrics of Some Archimedean And Catalan Solids
Zeynep Can1 and Zeynep Çolak2
Abstract. Polyhedrons are exteremely important solids in 3 – dimensional analytical
space when they are convex. Some mathematicians have been studied on polyhedra and
improved Minkowski Geometry releated with them.
If a polyhedron has congruent faces which are regular polygons and same number of
faces intersect at each vertices then this polyhedron is a regular polyhedron, and they are only
five (Platonic Solids). If a polyhedron has congruent vertices and faces are regular polygons
then it is called semi–regular polyhedron, and they are thirteen (Archimedean Solids). Duals
of Archimedean solids are called Catalan Solids. Faces of a catalan solid are one kind of
polygon which are not regular.
Unit spheres of Minkowski geometries are some general, symmetric convex sets and
there is an important relation between these sets and metrics ([6]). In the previous studies it
has been seen that there are some metrics which unit spheres are some of the mentioned solids
([3],[4],[8]). Finding the unit sphere of a known metric depends on some basic calculations.
So, naturally a question on the contrary can be asked, “Is it possible to find the metric which
unit sphere is a known polyhedron?”.
In this study we introduce two new metrics which unit spheres are a Catalan solid
Disdyakis Triacontahedron and an Archimedean solid Truncated Octahedron. Also we will
give a family of metrics and show that the spheres of the 3 – dimensional analytical space
covered by these metrics are some well – known polyhedra.
Keywords. Archimedean Solids, Catalan Solids, Metric, Disdyakis Triacontahedron,
Truncated Octahedron, Chinese Checkers Metric.
AMS 2010. 51B20, 51F99, 51K05, 51K99, 51N25.
References
[1] Cromwell, P., Polyhedra, Cambridge University Press, 1999.
1 Akasaray University, Aksaray, Turkey, [email protected] 2 Canakkale Onsekiz Mart University, Canakkale, Turkey, [email protected]
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[2] Chen, G., Lines and Circles in Taxicab Geometry, Master Thesis, Department of
Mathematics and Computer Science, University of Central Missouri, 1992.
[3] Ermiş, T., Kaya, R., On the Isometries the of 3- Dimensional Maximum Space, Konuralp
Journal of Mathematics, Vol.3, No. 1., 103-114, 2015.
[4] Gelişgen, O., Kaya, R., Ozcan, M., Distance Formulae in The Chinese Checker Space, Int.
J. Pure Appl. Math. 26 , no.1, 35-44, 2006.
[5] Krause, E.F., Taxicab Geometry, Addison-Wesley Publishing Company, Menlo Park, CA,
88p., 1975.
[6] Millmann, R.S. and Parker, G.D., Geometry a Metric Approach with Models,
Springer,370p., 1991.
[7] Thompson, A.C. Minkowski Geometry, Cambridge University Press, Cambridge, 1996.
[8] Field, J.V., Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca
Pacioli, Leonardo da Vinci, Albrecht DÄurer, Daniele Barbaro, and Johannes Kepler, Archive
for History of Exact Sciences, 50(3-4), 241-289, 1997.
[9] Gelişgen O. and Kaya R., The Taxicab Space Group, Acta Mathematica Hungarica,
DOI:10.1007/s10474-008-8006-9, 122(1-2), 187-200, 2009.
[10] Can Z., Çolak Z., Gelişgen O., A Note On The Metrics Induced By Triakis Icosahedron
And Disdyakis Triacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences
Journal, Vol. 1, 1-11, 2015.
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Tetrakis Hexahedron Space Isometry Group
Zeynep Çolak1 and Zeynep Can2
Abstract. There are many thinkers that worked on polyhedra among the ancient
Greeks. Early civilizations worked out mathematics as problems and their solutions.
Polyhedra have interesting symmetries. Therefore they have attracted the attention of
scientists and artists from past to present. Thus polyhedra are discussed in a lot of scientific
and artistic works. There are only five regular convex polyhedra known as the platonic solids.
Semi-regular convex polyhedron which are composed of two or more types of regular
polygons meeting in identical vertices are called Archimedean solids. The duals of the
Archimedean solids are known as the Catalan solids.
Three essential methods for geometric investigations; synthetic, metric and group
approach. The group approach takes isometry groups of a geometry and convex sets plays an
substantial role in indication of the group of isometries of geometries. Those properties are
invariant under the group of motions and geometry studies those properties.
Some mathematicians have studied isometry groups of Taxicab space, Maximum
space and Chinese Checker space in three dimensional space ([2], [1], [4]). We worked on
isometry groups of Tetrakis Hexahedron which is a Catalan solid. In this study we show that
the group of isometries of the 3-dimensional space with respect to Tetrakis Hexahedron
metric is the semi-direct product of octahedral group Oh and T(3), where Oh is the
(Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the
3-dimensional space.
Keywords. Isometry Group, Catalan Solid, Tetrakis Hexahedron, Polyhedra.
AMS 2010. 51B20, 51F99, 51K05, 51K99, 51N25.
References
[1] Ermiş, T., Kaya, R., On the Isometries the of 3- Dimensional Maximum Space, Konuralp
Journal of Mathematics,Vol.3, No.1, 103-114, 2015.
[2] Gelişgen, Ö., Kaya, R., The Taxicab Space Group, Acta Mathematica Hungarica,
DOI:10.1007/s10474-008-8006-9, 122, No.1-2, 187-200, 2009.
1 Çanakkale Onsekiz Mart University, Çanakkale, Turkey, [email protected] 2 Akasaray University, Aksaray, Turkey, [email protected]
- 216 -
[3] Gelişgen, Ö., Kaya, R., Ozcan, M., Distance Formulae in The Chinese Checker Space, Int.
J. Pure Appl. Math. 26, no.1,35-44, 2006.
[4] Gelişgen, Ö., Kaya, R., The Isometry Group Of Chınese Checker Space, International
Electronic Journal of Geometry, (to appear 2015).
[5] Can Z., Çolak Z., Gelişgen O., A Note On The Metrics Induced By Triakis Icosahedron
And Disdyakis Triacontahedron, Eurasian Academy of Sciences Eurasian Life Sciences
Journal, Vol. 1, 1-11, 2015.
- 217 -
Conformal Weyl-Euler-Lagrange Equations on Lorentzian Trans
and Para Sasakian Manifolds
Zeki Kasap1
Abstract. Contact geometry deals with certain manifolds of odd dimension. It is close
to symplectic geometry and like the latter it is originated in questions of classical mechanics.
Contact geometry can be considered as symplectic geometry such that it has broad
applications in physics, geometrical optics, classical mechanics, analytical mechanics,
mechanical systems, thermodynamics, geometric quantization and applied mathematics such
as control theory. Locally, every symplectic manifold admits a Kähler structure. Every Kähler
manifold has a symplectic manifold. Kähler manifolds are of tremendous importance in
modern physics. Sasakian manifolds are an analog one dimensional of Kähler manifolds. It is
well known that one way of solving problems in classical mechanics use the Euler-Lagrange
equations. Weyl o¤ered a conformal structure and this structure was transferred to the
mechanical systems. In this study, Weyl-Euler-Lagrange equations as representing the motion
of the body were found on Sasakian manifolds. Also, these solution of differential equations
are solved by symbolic computation program.
Keywords. Weyl Manifold, Conformal Geometry, Sasakian Manifold, Lorentzian,
Mechanical System, Dynamic Equation, Lagrangian Formalism.
AMS 2010. 34B20, 34N05, 53A30, 53D10, 70S05, 81Q05, 82C21.
References
[1] B. Laha, B.Das and A.Bhattacharyya, Submanifolds of Some Indefinite Contact and
Paracontact Manifolds, Gulf Journal of Mathematics, Vol.1, Issue 2, 180-188, 2013.
[2] S. Keles, E. Kilic, M.M. Tripathi and S.Y. Perktas, Einstein Like (ε)-Para Sasakian
Manifolds, 1-6, 2012.
[3] R. Miron, D. Hrimiuc, H. Shimada, S.V. Sabau, The Geometry of Hamilton and Lagrange
Spaces, Kluwer Academic Publishers, 2002.
[4] S.S. Shukla and A. Yadav, Lightlike Submanifolds of Indefinite Para-Sasakian Manifolds,
Matematıqkı Vesnik, 1-15, 2013.
1 Pamukkale University, Denizli, Turkey, [email protected]
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[5] G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory, World
Scientific, ISBN 981-02-2045-6, 1995.
[6] Z. Kasap, Weyl-Mechanical Systems on Tangent Manifolds of Constant W-sectional
Curvature, IJGMMP, Vol.10, No.10, 1-13, 2013.
[7] M. Tekkoyun, On Para-Euler-Lagrange and Para-Hamiltonian Equations, Phys. Lett. A,
340, 7-11, 2005.
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Euler-Lagrange Equations on Almost Paracontact Metric Manifolds
Zeki Kasap1
Abstract. Almost paracontact metric manifolds are the famous examples of almost
para-CR manifolds. Symplectic geometry has its origins in the Lagrangian formulation of
classical mechanics such that the phase space of certain classical systems takes on the
structure of a symplectic manifold. It is well known that a preferred method to solve the
problems of classical mechanics is to with the Euler-Lagrange equations. Classical field
theory uses a simple solution method of Euler-Lagrangian dynamics. This theory was
extended to time-dependent classical mechanics. In this study, Euler-Lagrange equations, as
representing the orbits of moving objects in this space that its are geodesic modeling, found
on Lorentzian trans and para Sasakian manifolds. Also, these solutions of differential
equations solved by symbolic computation program.
Keywords. Sasakian Manifold, Lorentzian, Mechanical System, Dynamic Equation, Lagrangian Formalism.
AMS 2010. 34B20, 34N05, 53A30, 53D10, 70S05, 81Q05, 82C21.
References
[1] J. Wełyczko, Para-CR Structures on almost Paracontact Metric Manifolds, 1-18, 2012.
[2] E. Peyghan, A. Tayebi, Almost Paracontact Structure on Finslerian Indicatrix, An. S t.
Univ. Ovidius Constanta, Vol. 19 (3), 151-162, 2011.
[3] M.M. Tripathi, E. Kilic, S.Y. Perktas and S. Keles, Indefinite Almost Paracontact Metric
Manifolds, International Journal of Mathematics and Mathematical Sciences, 1-19, 2010.
[4] T. Takahashi, Sasakian Manifold with Pseudo-Riemannian Metric, Tohoku Math. Journ.,
21, 271-290, 1969.
[5] B. Pandey, On Indefinite Almost Paracontact Metric Manifold, International Mathematical
Forum, Vol.6, No.22, 1071-1078, 2011.
[6] Z. Kasap, M. Tekkoyun, Mechanical Systems on Almost Para/Pseudo-Kähler--Weyl
Manifolds, IJGMMP, Vol.10, No.5, 1-8, 2013.
[7] M. Manev and M. Staikova, On almost Paracontact Riemannian Manifolds of Type (n,n),
J. Geom., 72, 108-114, 2001.
1 Pamukkale University, Denizli, Turkey, [email protected]
- 220 -
Examining Pre-Service Mathematics Teachers’ Math Literacy And Their Attitudes
Towards Mathematics Education Courses
Dilek Çağırgan Gülten1
Abstract. Teachers’ attitudes towards mathematics and mathematics teaching
influence students’ attitudes. One needs to have knowledge of mathematics and pedagogical
knowledge in mathematics in order to teach mathematics. Teachers need to have not only
accurate knowledge of concepts and operations but also they need to understand the
connections between mathematical thoughts. In this context, it is of essence for mathematics
teachers to become math literates.
An individual needs to study in the faculty of education so as to be a mathematics
teacher. Therefore, pre-service teachers should be taught mathematics at the faculties of
education in the way they are asked to teach it. Starting from this point, the aim of this study
is to examine pre-service mathematics teachers’ mathematical literacy and their attitudes
towards mathematics education courses. The research study was conducted with 181 pre-
service teachers enrolled in the program of Primary Mathematics Education at İstanbul
University.
The research data were collected using the “Math Literacy Self-Efficacy Scale”
developed by Özgen and Bindak (2008) and the “Attitude Scale for Mathematics Education
Courses” developed by Turanlı, Türker, and Keçeli (2008). The study was carried out
considering the variables of grade, order of preference, and reason for preference and the
relationship between mathematical literacy and the attitude towards mathematics education
courses was determined. Some suggestions were made for further research and researchers in
line with the findings.
Keywords. Mathematics Education Courses, Attitude, Mathematical Literacy.
References
[1] Turanlı, N., Türker, N.K ve Keçeli, V., Matematik alan derslerine yönelik tutum ölçeği
geliştirilmesi. Hacettepe Üniversitesi Eğitim Fakültesi 34, 254-262, 2008.
[2] Özgen, K., Bindak, R. Matematik okuryazarlığı öz-yeterlik ölçeğinin geliştirilmesi,
Kastamonu Eğitim Dergisi, 16 (2), 517-528, 2008.
1 Istanbul University, Istanbul, Turkey, [email protected]
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The Inclusion of Geometric Thinking in Elementary School Mathematics Textbooks
Deniz Özen1 and Nilüfer Yavuzsoy Köse2
Abstract. Geometric thinking includes geometry-specific ideas which individuals may
encounter in their daily lives as well as in their learning process and components that create
these ideas. The Model of Geometric Habits of Mind suggested by Driscoll, Wing DiMatteo,
Nikula and Egan (2007) describes geometric thinking with four basic geometric habits of
mind, which are “reasoning with relationships”, "generalizing geometric ideas", "investigating
invariants" and "balancing exploration and reflection" [1].
In elementary school mathematics curriculum it is aimed to foster geometric thinking,
nevertheless to enhance geometric thinking becomes difficult if the problems and activities in
the textbooks do not support these geometric habits of mind. Textbooks are of great
importance in the teaching process considering that geometric thinking is supported with
geometric habits of mind and these habits are acquired through problems and activities. In this
study it is aimed to determine to what extent mathematics textbooks foster geometric thinking
within the framework of Geometric Habits of Mind. Accordingly, four mathematics textbooks
being taught in 5, 6, 7 and 8th grades of elementary school were examined and geometry
problems and activities in these textbooks were analyzed based on the theoretical framework
of Geometric Habits of Mind.
Document analysis which is a qualitative research method has been carried out in data
analysis process. In this process, subjects related to geometry learning were determined
primarily, and then geometry activities and problems were categorized with reference to the
geometric habits of mind for each grade level. For the reliability of the study, activities and
problems were coded separately by two researchers based on geometric habits of mind
framework. The differences between coders have been resolved through discussions and of
compromises of the researchers.
Considering the necessity of supporting of geometric thinking, importance of gaining
geometric habits of mind for the students and the inclusion of geometric thinking in the
textbooks, it is thought that this study will bring light for further studies.
Keywords. Geometric Thinking, Geometric Habits of Mind, Mathematics Textbooks
References [1] Driscoll, M., Wing DiMatteo, R., Nikula, J. ve Egan, M., Fostering geometric thinking: A guide for teachers grades 5-10. Portsmouth, NH: Heineman, 2007.
1 Adnan Menderes University, Aydin, Turkey, [email protected] 2 Anadolu University, Eskisehir, Turkey, [email protected]
- 222 -
Examination of 6th
Students’ Quantitative Reasoning Skills and Developments
in Their Problem Solving Process
Dilek Tanışlı1 and Mehmet Dur2
Abstract. The purpose of this research is to examine middle school grade 6 students’
quantitative reasoning skills and developments in their problem solving process. In this
research, qualitative research methods were adopted for data collection, analysis and
interpretation. In this context, teaching experiment model was used. The implementations of
the research were carried out with 4 sixth grade students, two girls and two boys in public
school. During the entire teaching experiment in which it was aimed to identify students’
quantitative reasoning skills and observe their developments, clinical interviews, worksheets,
clinical interviews with videotapes of teaching process, student journals, and researcher
journals were used as data collection tools, and a thematic analysis was used for data analysis.
As results of the research, it was observed that students who had low quantitative
reasoning skills in pre-interviews improved their quantitative reasoning skills notably in post-
interviews as a result of the teaching experiment. It was identified that students had higher
successes in understanding the problems, selecting appropriate strategies to the problem
situation, carrying out the selected strategy and evaluating stages after the teaching
experiment. Also, it was observed that using diagrams, tables, and visual representations
which take efficient roles during quantitative reasoning became as a habit after the teaching
experiment.. Besides, it was observed that the development of students' quantitative reasoning
contributed to their algebraic development and achievement levels in mathematics.
Keywords. Quantitative Reasoning, Algebra, Problem Solving
References
[1] Thompson P. W., Quantitative concepts as a foundation for algebraic reasoning:
sufficiency, necessity, and cognitive obstacles. M. Behr, C. Lacampagne and M. Wheeler
(Ed.), Proceedings of the Annual Conference of the International Group for the Psychology of
Mathematics Education, 163-170, 1988.
1 Anadolu University, Eskisehir, Turkey, [email protected] 2 Ministry of National Education, Eskisehir, Turkey, [email protected]
- 223 -
[2] Kieran, C., The learning and teaching of school algebra. In D. A. Grouws (Ed.),
Handbook of research on mathematics teaching and learning (pp. 390-419). New York:
Macmillan, 1992.
[3] Thompson P. W., Quantitative reasoning, complexity, and additive structures. Educational
Studies in Mathematics, 25 (3), 165-208, 1993.
- 224 -
The Investigation of Using Mathematical Language of 7th Graders
When Identifying Circle and It’s Elements
Esra Akarsu1 and Süha Yılmaz2
Abstract: Mathematical language is the combination of the whole rules which are
used together with mathematical concepts, operations and symbols that have the property of
expressing scientific thoughts easily [1]. Usage of correct branch language is really important
to eliminate the misconceptions which occur in students’ minds [2].
Through this research it is aimed to observe 7th grade students’ ability of using
mathematical language when identifying circle and it’s elements. The using of mathematical
language skills of students identifying circle and it’s elements when given in scenario and
when their names were given were compared. The sample of this study consists of 138
seventh graders selected from several primary schools located in Manisa. A scenario which
consists of 7 questions about identifying circle and it’s elements (center, radius, diameter,
secant, chord and tangent ) was prepared by the researchers and students were asked to
answer these questions. Furthermore, students were asked to identify circle and it’s elements
when their names were given. The mathematical language skills of the students were
examined in terms of their ability to identify visually and verbally. It was concluded that the
mathematical language skills of students identifying circle and it’s elements when given in
scenario need to be developed. Also, students’ mathematical language skills identifying them
when their names were given are sufficient. It was observed that students have most difficulty
in defining tangent.
Keywords. Circle, The Elements of Circle, Mathematical Language.
References
[1] Çalıkoğlu Bali, G.,Opinions Of Prospective Mathematics Teachers About Language In
Mathematics Teaching. Hacettepe University Journal of Faculty of Education.25: 19-25,
2003.
[2] Yeşildere, S., The Competencies of Prospective Primary School Mathematics Teachers in
Using Mathematical Language. Boğaziçi University Journal of Education. 24(2), 61-70, 2007.
1 Dokuz Eylul University, Izmir, Turkey, [email protected] 2 Dokuz Eylul University, Izmir, Turkey, [email protected]
- 225 -
Linear Algebra with SAGE
Erdal Özüsağlam 1
Abstract. Usage of Mathematical software has been increasing constantly since
computers are used in education. Sage, Open source Software, is a powerful system for
studying and exploring many different areas of mathematics.
Teachers talk about linear algebra using many linear text books in classroom.
However, the usage of indispensable teaching method for some issues in linear algebra that is
difficult to teach. Therefore, I introduce the principles of working process of Sage and give
some examples about basic linear algebra including matrices, determinants.
Keywords. Linear Algebra, Open Source Software, Sage, Mathematics Education
AMS 2010. 97B40, 97C80.
References
[1] Özüsağlam, E. Teknoloji Destekli Matematik Öğretiminin Öğretimi, Matematik
Etkinlikleri 2004, Ankara, 2004.
[2] Kleinfeld, E. and Kleinfeld, M., Understanding Linear Algebra Using MATLAB, Prentice
Hall, 2001.
[3] Shiskowski, K. and Frinkle, K., Principles of Linear Algebra with Maple, Wiley, 2010.
[4] E. Özüsağlam, Mathematica Destekli On-Line Matematik Dersi Sunumu Üzerine Bir
Çalışma, Bilişim Teknolojileri Işığında Teknoloji Eğitimi Konferansı, ODTU, Ankara, 2001.
1 Aksaray University, Aksaray, Turkey, [email protected]
- 226 -
Views of High School Mathematics Teachers and Students on Computer-Assisted
Mathematics Instruction: Mathematica Case
Mehmet Alper Ardıç1 and Tevfik İşleyen2
Abstract. In this study, high school mathematics teachers, who did not know about
computer-assisted mathematics instruction before, were trained on it in the first place, and
they were enabled to use Mathematica in their classes for the graphics of quadratic functions
(parabola). Obtained through the semi-structured interview, this study then focuses on the
views of both teachers and students about computer-assisted mathematics instruction and
about Mathematica, a computer algebra system (CAS) benefited during the process. The
interview data were examined with the descriptive analysis and content analysis methods to
get some codes and themes about the subject. The results show that all the teachers found
computer-assisted mathematics instruction interesting on the students’ part just as the students
did. While all the students wanted to benefit from computer-assisted mathematics instruction
in mathematics and geometry classes, most of the students (67%) wanted to benefit from
computer-assisted instruction in different classes as well. It is seen that students did not have
any problem with Mathematica, used during the activities of computer-assisted mathematics
instruction. However, it is also seen that one of the teachers and of the students believed that
the constant application of computer-assisted mathematics instruction in classes would hinder
their studies on university entrance exam.
Keywords. Computer-Assisted Mathematics Instruction, Computer Algebra System,
Mathematica, Teacher’s View, Student’s View.
AMS 2010. 97A99, 97D40, 97U50
References
[1] Aktümen, M. and Kaçar, A., Effects of computer algebra systems on attıtutes towards
mathematıcs, Hacettepe University Journal of Education, 35(2008), 13-26, 2008.
[2] Baki, A., Güven, B. ve Karataş, İ., Dinamik Geometri Yazılımı Cabri İle Keşfederek
Öğrenme, V. Ulusal Fen Bilimleri ve Matematik Eğitimi Kongresi Bildiriler Kitabı, Cilt: II,
884-891, ODTÜ, Ankara, 2002.
1 Adiyaman University, Adiyaman, Turkey, [email protected] 2 Ataturk University, Erzurum, Turkey, [email protected]
- 227 -
[3] Ersoy, M. and Akbulut, Y., Cognitive and affective implications of persuasive technology
use on mathematics instruction, Computers & Education, 75, 253–262, 2014, 2014.
[4] Hohenwarter, M. and Fuchs, K., Combination of Dynamic Geometry, Algebra and
Calculus in the Software System GeoGebra, in Computer Algebra Systems and Dynamic
Geometry Systems in Mathematics Teaching Conference. P´ecs, Hungary, 2004.
[5] Taşlıbeyaz, E., Ortaöğretim öğrencilerinin bilgisayar destekli matematik öğretiminde
matematik algılarına yönelik durum çalışması: lise 3.sınıf uygulaması. Yayınlanmamış
yüksek lisans tezi, Atatürk Üniversitesi, Fen Bilimleri Enstitüsü, Erzurum, 2010.
[6] Tatar, E., Kağızmanlı, B.T. ve Akkaya, A., Türkiye’deki teknoloji destekli matematik
eğitimi araştırmalarının içerik analizi, Buca Eğitim Fakültesi Dergisi, 35, 33-50, 2013.
[7] Vlachos, P. & Kehagias, A., A computer algebra system and a new approach for teaching
business calculus, The International Journal of Computer Algebra in Mathematics Education,
7(2), 2000.
- 228 -
Mental Components of Mathematical Literacy Success of Secondary School Students
Murat Altun1 and Işıl Bozkurt2
Abstract. Mathematical Literacy (ML) concept and to what extent mathematics
curriculum develops it have been frequently argued. With the participations of students [1],
teacher candidates [2] and teachers to examine the levels of ML, various studies have been
conducted from different viewpoints such as mental processes, subject fields, mathematical
skill levels etc., and some comparisons through PISA results related to ML took place[3].
Differently from those studies, in this study, a Mathematical Literacy Test (MLT) which
includes a total of 15 questions selected from different topic fields has been applied to 221
secondary students most of whom are from eight grade. Data tables were formed by
specifying the grades that students got for each question and factor analysis was applied to the
data. The factors that account for ML success were named as (i) skill to make algorithmic
operations, (ii) mathematical proposal development, (iii) justification of mathematical result
(iv)understanding and interpretation of the mathematical content and (v) establishing
relationships between different data sets and interpretation. It is expected that the results of
this study will contribute to curriculum development and organization of the instructional
activities.
Keywords. Problem Solving, Mathematical Literacy, PISA
References
[1] Okur, S., Students’ Strategies, Episodes and Metacognitions in The Context of PISA 2003
Mathematical Literacy Items. (Yüksek Lisans Tezi, ODTÜ, Fen Bilimleri Enstitüsü,
Ankara). http://tez2.yok.gov.tradresinden alınmıştır, 2008.
[2] Saenz, C., The role of contextual, conceptual and procedural knowledge in activating
mathematical competencies (PISA). Educational Studies in Mathematics, 71, 123-143, 2008.
[3] Rautalin, M. ve Alasuutari, P., The uses of the national PISA results by Finnish officials in
central goverment. Journal of Education Policy, 24 (5), 539-556, 2009.
1 Uludag University, Bursa, Turkey, [email protected] 2 Uludag University, Bursa, Turkey, [email protected]
- 229 -
The Competence of Students in Understanding the Properties of a Function
from Its Graph
Nevin Mahir1
Abstract. Visualization is very important in helping students to understand functions
and their properties. The graph of a function, as the picture of a function, can be considered a
visual aid while examining the function and its properties. Most of the properties of a function
can be determined with the help of its graph. Therefore, it is important for students of
mathematics to extract some of the properties of a function such as limit, continuity,
derivative, growth and concavity from its graph. In this study, I investigated the competence
of students who have taken Calculus I-II courses in understanding the properties of a function
from its graph. To this purpose, an examination was given to science and engineering students
at a Turkish university. The results showed that the students were unable to extract the
properties related to the function from its graph.
Keywords. Function, Graph, Limit
References
[1] Asiala, M., et al., The Development of Students’ Graphical Understanding of The
Derivative, Journal of Mathematical Behavior, 16, 399-431, 1997.
[2] Aspinwal et al., Uncontrollable Mental Imagery: Graphical Connections between a Function
and its Derivative, Educational Studies in Mathematics, 33, 301-317, 1997.
[3] Vinner, S., Advanced Mathematical Thinking, edited by D. Tall, Dordrecht: Kluwer, 1991.
[4] Stylianou, D. A. and Silver, E. A., The Role of Visual Representations in Advanced
Mathematical Problem Solving: An Examination of Expert-Novice Similarities and
Differences, Mathematical Thinking and Learning, 6(4), 353-387, 2004.
1 Anadolu University, Eskisehir, Turkey, [email protected]
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Reasoning in Mathematics Education
Nilüfer Yavuzsoy Köse1
Abstract. One of the most important objectives of mathematics teaching is to have
students believe they can do mathematics. It will be quite important for the development of
students’ mathematical thinking if they can defend their thoughts with appropriate arguments,
make logical deductions, and share and discuss using an appropriate language. The most basic
component for this development is mathematical reasoning.
Mathematical reasoning is defined as the process of gaining new information by using
thinking techniques (induction, deduction, comparison, generalization and so on) and tools
specific to mathematics (symbols, definitions, relationships and so on) [1]. In other words,
mathematical reasoning is a process in which individuals convince themselves and others to
confirm a certain assumption and in which they develop such ways of thinking or argument as
solving a problem or gathering various thoughts [2]. In national and international literature,
there are several terms used in reasoning studies in relation to mathematical reasoning such as
developing assumptions, mathematical explanations, justification, verification, mathematical
arguments, mathematical exploration and generalization, and these terms are associated with
types of reasoning. Then, what associations can be established between these terms and
approaches to reasoning? The answers to this question are likely to vary depending on
learners’ level of education and their age, their individual characteristics and attitudes, the
learning environment, the context/subject, the mathematics instructor’s theoretical
perspective, and on the educational policies according to which curricula and course books
are developed. In line with this, the present study discusses the place of mathematical
reasoning in elementary school mathematics teaching and provides a synthesis by examining
the current theoretical structures.
Keywords. Mathematics Education, Mathematical Reasoning
References
[1] MEB. (2013) Ortaokul matematik dersi (5, 6, 7 ve 8 sınıflar) öğretim
programı, http://ttkb.meb.gov.tr/www/guncellenen-ogretim-programlari-ve-kurul-
kararlari/icerik/150.
[2] Brodie, K. Teaching mathematical reasoning in secondary school classrooms. London:
Springer, 2010.
1 Anadolu University, Eskisehir, Turkey, [email protected]
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Concept Association of Prospective Elementary and Secondary Mathematics Teachers
Şüheda Güray1, Şeyda Kılıçoğlu 2 and Merve Koştur 3
Abstract. Concept knowledge competency of prospective teachers is in the scope of
reasearchers in mathematics education (Ball, 1991; Even, 1993; Wilson, Shulman & Richert,
1987). While investigating concept knowledge, it is important to inquire into the relationship
of the concepts with each other and defining the concepts interrelatedly (Adams, T. L., 2012;
Ball, 1990). In the present study, concept associations of prospective elementary and
secondary mathematics teachers were investigated. In the study, the participants were 60
prospective elementary mathematics teachers in 1st and 4th grades and 47 prospective
secondary mathematics teachers in 1st, 3rd and 4th grades at a university in Turkey. The
definitions for 9 concepts gathered from the prospective teachers were grouped in two as “full
definition” and “incomplete definition”. Afterwards, concepts were grouped in two and
concept associations of every prospective teacher were analyzed via chi square analysis. As a
result, regardless of grade differences, the dependence and independence of concepts with
each other were investigated and reported in the study.
Keywords: Mathematical Concept Knowledge, Concept Associations, Mathematics
Educations, Prospective Teacher.
References
[1] Adams, T. L., Prospective elementary teachers' mathematics subject matter knowledge:
The real number system. Action in Teacher Education, 20(2), 35-48, 1998.
[2] Ball, D. L., The mathematical understandings that prospective teachers bring to teacher
education. The Elementary School Journal, 90(4), 449-466, 1990.
[3] Ball, D. Research on teaching mathematics: Making subject matter knowledge part of the
equation. In J. Brophy (Ed.), Advances in research on teaching: Teachers’ subject matter
knowledge and classroom instruction, Vol. 2, (pp. 1-48). Greenwich: CT JAI Press., 1991.
1 Baskent University, Ankara, Turkey, [email protected] 2 Baskent University, Ankara, Turkey, [email protected] 3 Baskent University, Ankara, Turkey, [email protected]
- 232 -
[4] Even, R., Subject-matter knowledge and pedagogical content knowledge: Prospective
secondary teachers and the function concept. Journal for Research in Math. Education, 24(2),
94-1 16. 1993.
[5] Wilson, S. M., Shulman, L. S., & Richert, A. E., 150 different ways of knowing:
Representations of knowledge in teaching, 1987.
- 233 -
Investigating the Concept Knowledge of Prospective Elementary and Secondary
Mathematics Teachers
Şeyda Kılıçoğlu1, Şüheda Güray2 and Merve Koştur3
Abstract. Concept knowledge competency of prospective teachers is in the scope of
researchers in mathematics education (Ball, 1991; Even, 1993; Wilson, Shulman & Richert,
1987). In recent years, concept knowledge has been analyzed qualitatively and investigated
deeply by focusing on knowledge and definition (Ball, 1988, 1991; Even, 1990; Lienhardt &
Smith; 1985; Shulman, 1986; Wilson et al., 1987). In defining the concepts, the correct use of
mathematical language and supporting the definition by multiple representations are of
importance. In this regard, definitions of the total of 107 prospective elementary and
secondary mathematics teachers for selected mathematics concepts were investigated. In the
study, prospective teachers were asked to define 9 concepts selected among fundamental
mathematics concepts in the literature. By using content analysis, the definitions were coded
with 5 themes as “complete definition”, “exampling”, “making a drawing”, and “empty”. The
results were reported by comparing concept knowledge of elementary and secondary
mathematics education program students according to grade and program differences. In
addition, the data gathered from the participants via semi structured interviews were analyzed
and probable reasons of common misconceptions and incomplete definitions were explained.
Key words: Mathematical Concept Knowledge, Concept Associations, Mathematics
Educations, Prospective Teachers.
References
[1] Ball, D. L. The Subject Matter Preparation of Prospective Mathematics Teachers:
Challenging the Myths., 1988.
[2] Ball, D. Research on teaching mathematics: Making subject matter knowledge part of the
equation. In J. Brophy (Ed.), Advances in research on teaching: Teachers ’ subject matter
knowledge and classroom instruction, Vol. 2, (pp. 1-48). Greenwich: CT JAI Press., 1991.
1 Baskent University, Ankara, Turkey, [email protected] 2 Baskent University, Ankara, Turkey, [email protected] 3 Baskent University, Ankara, Turkey, [email protected]
- 234 -
[3] Even, R. Subject-matter knowledge and pedagogical content knowledge: Prospective
secondary teachers and the function concept. Journal for Research in Mathematics Education,
24(2), 94-1 16, 1993.
[4] Leinhardt, G., & Smith, D. A., Expertise in mathematics instruction: Subject matter
knowledge. Journal of educational psychology, 77(3), 247, 1985.
[5] Shulman, L. S. Those who understand: Knowledge growth in teaching. Educational
researcher, 4-14, 1986.
[6] Wilson, S. M., Shulman, L. S., & Richert, A. E. 150 different ways" of knowing:
Representations of knowledge in teaching, 1987.
- 235 -
Asymptotic Results for a Semi-Markovian Random Walk with Generalized Beta
Interferences
Cihan Aksop1 and Tahir Khaniyev2,3
Abstract. In this study, a semi-Markovian random walk with interferences distributed
on the interval [𝑠, 𝑆] is investigated. Asymptotic expansions for the moments of this process
are obtained as a function of moments of ladder heights and moments of the interferences,
when the interferences have a generalized beta distribution and 𝛾 ≡ 𝑆 − 𝑠 → ∞. Moreover,
accuracy of the proposed asymptotic expansions is examined with a Monte Carlo simulation
method.
Keywords. Ladder Heights, Asymptotic Expansions, Ergodic Moments, Semi-
Markovian Random Walk, Generalized Beta Distribution
AMS 2010. 60G50.
References
[1] Feller, W., Introduction to Probability Theory and Its Applications II, John Wiley, 1971.
[2] Gihman, I.I., Skorohod A.V., Theory of Stochastic Processes II, Springer, Berlin, 1975.
[3] Aliyev, R., Kucuk, Z., Khaniyev, T. Three-term Asymptotic Expansions for the Moments
of the Random Walk with Triangular Distributed Inteference of Chance. Applied
Mathematical Modelling, 34: 3599-3607, 2010.
[4] Khaniyev, T., Kucuk, Z. Asymptotic Expansions for the Moments of the Gaussian
Random Walk with Two Barriers, Statistics & Probability Letters, 69: 91-103, 2004.
[5] Khaniyev, T., Mammadova, Z. On the Stationary Characteristics of the Extended Model
of Type (s,S) with Gaussian Distribution of Summands. Journal of Statistical Computation
and Simulation. 76: 861-874, 2006.
1 The Scientific and Technological Research Council of Turkey, Ankara, Turkey, [email protected] 2 TOBB University of Economics and Technology, Ankara, Turkey, [email protected]. 3 Azerbaijan National Academy of Sciences, Institute of Cybernetics, Baku, Azerbaijan, [email protected]
- 236 -
A Queueing System Equipped with Two Components Subject to Random Failures and
Heterogeneous General Service with Fluctuating Rates Depending on State of the
Components
Kailash C. Madan1
Abstract. We study a batch arrival queueing system with a single server equipped
with two components 1C and 2C both subject to random failures with different failure rates.
There are two repair facilities, one each for the two components. The system works in full
efficiency as long as both components are in working state. However, the system works in
reduced efficiency if 2C is in the failed state, it works in low efficiency if 1C is in the failed
state and the system is completely down if both 1C and 2C are in the failed state. The
system provides service to customers with different general service times with different
service rates depending on the fluctuations in the system efficiency. Steady state
probability generating functions are obtained for various states of the system and results for
some particular cases have been derived.
Keywords. Batch Arrivals, General Service, Random Breakdowns, Repairs,
Fluctuating Efficiency, Steady
AMS 2010: 60K25 References [1] Avi-Itzhak, B. and Naor, P., Some Queueing Problems with the Service Station Subject to Breakdowns, Operations Research, 11, pp. 303–320, 1963.
[2] Choudhury, G. and Tadj, L., An M/G/1 Queue with Two Phases of Service Subject to the Server Breakdown and Delayed Repair, Applied Mathematical Modeling, 33, pp. 2699–2709, 2009.
[3] Dorda, M., On Two Modifications of E2/E2/1/m Queueing System with a Server Subject to
Breakdowns, Applied Mathematical Sciences, 7-11, pp. 539–550,2013.
[4] Gaver, D.P., A Waiting Line with Interrupted Service Including Priorities, Journal of Royal
Statistical Society, Ser.- B, 24, pp. 73–90, 1962.
1 Ahlia University, Manama, Kingdom of Bahrain, [email protected]
- 237 -
Higher-Order Adjustments of the Signed Scoring Rule Root Statistic
Laura Ventura1, V. Mameli and M. Musio2
Abstract. Proper scoring rules (see e.g. Parry et al., 2012), different from the log-
score, can be used as an alternative to the full likelihood, when the aim is to increase the
robustness or to simplify computations. Proper scoring rule inference is usually based on the
first-order approximations to the distribution of the scoring rule estimator or of the scoring
rule ratio test statistic. However, several examples (see Dawid et al., 2015, Mameli and
Ventura, 2015) illustrate the inaccuracy of first-order methods, even in models with a scalar
parameter, when the sample size is small or moderate. Analytical higher-order asymptotic
expansions for proper scoring rules, generalizing results for likelihood quantities but allowing
for the failure of the information identity, have been discussed in Mameli and Ventura (2015).
However, the calculation of the quantities involved in the analytical adjustments of the signed
and signed profile scoring rule root statistic is cumbersome, even for simple models. The aim
of this work is to discuss the alternative approach to higher-order adjustments, based on a
parametric bootstrap. In particular, focus is on the signed profile scoring rule root statistic.
Keywords. Bootstrap, Higher-Order Asymptotics, Robustness, Scoring Rules.
AMS 2010. 62F05, 62F35.
References
[1] Dawid A.P., Musio M. Theory and Applications of Proper Scoring Rules, Metron, 72, 169-
183, 2014.
[2] Dawid, A. P., Musio, M., Ventura, L., Minimum scoring rule inference. Scandinavian
Journal of Statistics, to appear, 2015.
[3] Mameli, V., Ventura, L., Higher-order asymptotics for scoring rules. Journal of Statistical
Planning and Inference, 165, 13-26, 2015.
1 University of Padova, Italy, [email protected] 2 University of Cagliari, Italy, [email protected], [email protected]
- 238 -
In the Ridge Regression Method, ‘k Point Estimation Method’ (Approach) in the
Estimation of the Parameter ‘‘k’’
Mücahit Kurtuluş1
Abstract. In this study, multiple linear regression analysis, Ridge Regression method
that is one of the methods to overcome the problems arising among the independent variables
in the case of multicolinearity was studied. Ridge Regression method was compared with
Least squares method. In ridge Regression method, ‘k Point Estimation Method’ as an
alternative to the methods that was used to determine the parameters ‘‘k’’ has been proposed.
(1)
Keywords. Ridge Trace, Multicolinearity, ‘‘k Point Estimation Method’’, Mean
Squared Error.
References
[1] Kurtuluş, Mücahit. Ridge Regresyon üzerine bir çalışma, (Danışman: Prof.Dr.Soner Gönen), Gazi Üniversitesi, Fen Bilimleri Enstitüsü, İstatistik AB.D., Yüksek Lisans Tezi, 2001.
[2] A.E.,Hoerl, Application of Ridge Analysis to Regression Problems, Chemical Engineering Progress., 58, 77-88, 1962.
[3] A.E.,Hoerl and R.W., Kennard., Baldwin, Communication in Statistics, 4, 105-123, 1975.
[4] A.E.,Hoerl and R.W., Kennard., Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics. 12, 55-67, 1970a.
[5] A.E.,Hoerl and R.W., Kennard., Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics. 12, 69-82, 1970b.
[6] A.E., Hoerl, R.W., Kennard. and K.F, Baldwin, Ridge Regression: Iterative Estimation of the Baising Parameter, Communication in Statistics, 5, 77-88, 1970b.
1 Sinop University, Sinop, Turkey, [email protected]
𝒌 = 𝝀𝝀𝝀𝝀 − 𝟏𝟏𝟏𝝀𝝀𝟏𝟏𝟗𝟗
- 239 -
A Comparison of Information Criteria in Clustering based on Mixture of Multivariate
Normal Distributions
Serkan Akoğul1 and Murat Erişoğlu2
Abstract. Clustering analysis based on mixture of multivariate normal distributions is
commonly used in the clustering of multidimensional data sets which are unknown natural
cluster structure. The most important problems are to choose the number of components and
to identify the structure of variance-covariance matrix in the clustering based on modelling
with multivariate normal distributions of each cluster (components). In this study, the
efficiency of information criterion which is commonly used in the model selection is
examined. The effectiveness of information criteria has been determined according to the
success in the estimation of the number of components and in the selection of appropriate
variance-covariance matrix.
Keywords. Cluster Analysis, Mixture Models, Information Criteria.
AMS 2010. 62H30, 62-07.
References
[1] Biernacki, C. and Govaert, G., Using the Classification Likelihood to Choose the Number
of Clusters, Computing Science and Statistics, 29 (2), 451-457, 1997.
[2] Everitt, B. S., Cluster analysis, 3rd ed. London: Edward Arnold, 1993.
[3] Oliveira, B, A., and Martins, F., V., Assessing The Number Of Components In Mixture
Models: A Review, Fep Working Papers, 197, 2005.
[4] Schwarz, G., Estimating the dimension of a model,Annals of Statistics 6(2):461-464, 1978.
[5] Symons, M.J., Clustering Criteria and Multivariate Normal Mixtures, Biometrics, 37, 35-
43, 1981.
1 Yildiz Technical University, Istanbul, Turkey, [email protected] 2 Necmettin Erbakan University, Konya, Turkey, [email protected]
- 240 -
The Performance of the Optimal Extended Balanced Loss Function Estimators
and Predictors
Selahattin Kaçıranlar 1 and Issam Dawoud 2
Abstract. The optimal heterogeneous, homogenous and homogenous unbiased
estimators of the mean square error (MSE) are introduced and discussed by many authors but
the goodness of fitted model criterion is quite often ignored which is used to investigate the
performance of estimators. Therefore, Shalabh et al. (2009) proposed the extended balanced
loss function (EBLF) in which the MSE, the Zellner’s balanced loss function and the
predictive loss function are just special cases of it. So, we derive the optimal heterogeneous,
homogenous and homogenous unbiased estimators of the EBLF and discuss the performances
of these estimators and their predictors.
Keywords: Linear Model; Estimation; Extended Balanced Loss Function; Optimal
Heterogeneous Estimator; Optimal Homogenous Estimator, Prediction Mean Square Error.
AMS Subject Classifications: 62J05, 62J07.
References [1] Chaturvedi, A., Shalabh, Bayesian Estimation of Regression Coefficients Under Extended Balanced Loss Function. Comm. Stat. Theory Methods 43, 4253-4264, 2014.
[2] Dwivedi, T.D., Srivastava, V. K., On the minimum mean squared error estimators in a regression model. Comm. Stat. Theory Methods 7, 487-494, 1978.
[3] Giles, J. A., Giles, D. E. A., Ohtani, K., The exact risks of some pre-test and Stein-type regression estimates under balanced loss. Comm. Stat. Theory Methods 25, 2901–2924, 1996.
[4] Goldberger, A. S., Best linear unbiased prediction in the generalized regression model. J. American Stat. Assoc., 57, 369–375, 1962.
[5] Gruber, M. H. J., The efficiency of shrinkage estimators with respect to Zellner’s balanced loss function. Comm. Stat. Theory Methods 33, 235–249, 2004.
1 Cukurova University, Adana, Turkey., , [email protected] 2 Cukurova University, Adana, Turkey., [email protected]
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Stein-Rule Restricted Ridge Regression Estimator
Sadullah Sakallıoğlu1 and Selahattin Kaçıranlar 2
Abstract. Stein-rule and ridge estimators have been extensively used for estimating
the coefficient vector in a regression model. These estimators lead to an improvement in the
risk properties of the ordinary least squares (OLS) estimator. Instead of using one or the other
estimator, both of them may be appropriately combined. We introduce an alternative
estimator that combines the approaches followed in obtaining the restricted Stein-rule
estimation and the ridge regression estimation. A Monte Carlo simulation is performed to
compare the behavior of the proposed estimator.
Keywords. Stein-Rule Estimator; Ridge Regression Estimator; Restricted
Estimator; Quadratic Loss Function.
AMS Subject Classifications: 62J05, 62J07.
References
[1] Rao, C. R., Linear Statistical Inference and Its Applications. 2nd edition, Wiley, New
York, 1973.
[2] Vinod H. D., Ullah A., Recent Advances in Regression Methods. Marcel Dekker, Inc,
361p., 1981.
[3] Chaturvedi A., Wan A.T.K., Singh S.P., Stein-Rule Restricted Rehression Estimator In A
Linear Regression Model With Nonspherical Disturbances, Comm. Stat. Theory Methods
30(1), 55-68, 2001.
1 Cukurova University, Adana, Turkey. [email protected] 2 Cukurova University, Adana, Turkey., [email protected]
- 242 -
On the Stationary Characteristics of a Renewal Reward Processes
with Generalized Reflecting Barrier
Tahir Khaniyev 1, Basak Gever 1 and Zulfiyya Mammadova 2
Abstract. In this study, a semi–Markovian renewal–reward process with generalized
reflecting barrier is investigated and the ergodicity of the process is proved under some weak
conditions. Next, in general case, the explicit form of the first four moments for the ergodic
distribution is found and using these expressions, the asymptotic expansions for the ergodic
moments are obtained. To give these results, construct the considered process mathematically.
The independent and identically distributed initial random pairs (ξn,ηn), n = 1,2,3, …
defined on a same probability space (Ω,ℱ,𝑃) . Moreover, the positive–valued random
variables 𝜉𝑛 and 𝜂𝑛 are mutually independent. Their distribution functions are notated as
follows:
Φ(𝑡) = 𝑃ξn ≤ 𝑡; F(𝑥) = 𝑃ηn ≤ 𝑥; 𝑛 = 1,2, … ; 𝑥, 𝑡 ≥ 0.
Using the initial random pairs, construct the following renewal sequences 𝑇𝑛 and 𝑆𝑛:
T0 = S0; Tn = ξi
n
i=1
; Sn = ηi
n
i=1
; 𝑛 = 1,2, …
With the help of the renewal sequence 𝑆𝑛, the following integer – valued random variables
are defined:
𝑁0 = 0; 𝜁0 = 𝑧 ≥ 0 ; 𝑁1 ≡ 𝑁1(𝜆𝑧) = inf𝑘 ≥ 1: 𝜆𝑧 − 𝑆𝑘 < 0 ;
𝑁𝑛 ≡ 𝑁𝑛(𝜆𝜁𝑛−1) = inf𝑘 ≥ 𝑁𝑛−1 + 1: 𝜆𝜁𝑛−1 − 𝑆𝑘 − 𝑆𝑁𝑛−1 < 0 ;
𝜁𝑛 ≡ 𝜁𝑛(𝜆𝜁𝑛−1) = 𝜆𝜁𝑛−1 − 𝑆𝑁𝑛 − 𝑆𝑁𝑛−1; 𝑛 = 1,2, …
Here, 𝜆 ≥ 1 is an arbitrary positive constant. By means of Nn , define the sequence
𝜏𝑛, 𝑛 = 1,2, … as follows:
𝜏0 ≡ 0; 𝜏1 ≡ 𝜏1(𝜆𝑧) = ξi
N1
i=1
; 𝜏2 = ξi
N2
i=1
; … ; 𝜏𝑛 = ξi
Nn
i=1
;𝑛 = 1,2, …
Moreover, define 𝜈(𝑡) = min𝑛 ≥ 1: Tn > 𝑡 , 𝑡 > 0 . Now, the considered process can be
constructed mathematically as follows:
X(𝑡) = 𝜆ζn − 𝑆𝜈(𝑡)−1 − 𝑆Nn; ∀ 𝜏𝑛 ≤ 𝑡 < 𝜏𝑛+1; 𝑛 = 1,2, …
1 TOBB University of Economics and Technology, Ankara, Turkey, [email protected]; [email protected] 2 Karabuk University, Karabuk, Turkey, [email protected]
- 243 -
The defined process X(𝑡) is called as “Renewal reward process with generalized reflecting
barrier”. The aim of this study is to investigate the ergodic moments of the considered
process. Define the following notations to give the main result of this study.
mn ≡ E(η1n); E(Xn) = limt→∞
E(X(t)n) ; n = 1,2, …
Now, present the main result of the study as follows:
Theorem. Assume that the initial random variables ξn and ηn, n = 1,2, … are satisfied the
following supplementary conditions:
i) 0 < E(ξ1) < ∞; ii) E(η1) > 0; iii) E(η1n+2) < ∞;
iv) η1, is a non – arithmetic random variable.
Then, the following asymptotic expansions can be written for the nth order moment for the
ergodic moments of the process 𝑋(𝑡), when 𝜆 → ∞:
E(Xn) =2mn+2
(n + 1)(n + 2)m2λn + Bnλn−1 + Cnλn−2 + o(λn−2);𝑛 = 1,2, …
Here, the coefficient Bn and Cn are indicated in the exclusive version of the study.
Keywords. Renewal Reward Process, Reflecting Barrier, Ergodic Distribution,
Ergodic Moments, Asymptotic Expansion.
AMS 2010. 60K05, 60K15.
References
[1] Aliyev R., Khaniyev T. and Kesemen T., Asymptotic expansions for the moments of a
semi-Markovian random walk with Gamma distributed interference of chance,
Communications in Statistics - Theory and Methods, 39.1, 130-143, 2010.
[2] Aliyev R., Kucuk Z. and Khaniyev T., Tree-term asymptotic expansions for the
moments of the random walk with triangular distributed interference of chance, Applied
Mathematical Modelling, 34.11, 3599-3607, 2010.
[3] Janssen A.J.E.M. and van Leeuwaarden J.S.H., On Lerch's transcendent and the
Gaussian random walk, Annals of Applied Probability, 17.2, 421-439, 2007.
[4] Khaniyev T.A. and Mammadova Z.I., On the stationary characteristics of the extended
model of type (s,S) with Gaussian distribution of summands, Journal of Statistical
Computation and Simulation, 76.10, 861-874, 2006.
[5] Brown M. and Solomon H., A second - order approximation for the variance of a
renewal - reward process, Stochastic Processes and Applications, 3, 301-314, 1975.
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Ideal Rothberger Spaces
Aslı Güldürdek1
Abstract. In this work we introduce the notion of ideal Rothberger space, and
examine some basic properties. Also, we give some characterizations of almost Rothberger
and weakly Rothberger spaces by using ideal Rothberger property.
Keywords. Ideal Topological Space, Rothberger, Almost Rothberger, Weakly
Rothberger.
AMS 2010. 54D20, 54D30.
References
[1] Hamlett, T.R., Lindelöf with respect to an ideal, New Zealand Journal of Mathematics, 42,
115-120, 2012.
[2] Hamlett, T.R, Rose, D., Jankovic, D., Paracompactness with respect to an ideal, Internat.
J. Math. & Math. Sci., 20, 3, 433-442, 1997.
[3] Song, Y.K., Remarks on almost Rothberger spaces and weakly Rothberger spaces,
Quaestiones Mathematicae, 38, 3, 317-325, 2015.
1 Izmir University of Economics, Izmir, Turkey, [email protected]
- 245 -
A New Almost Continuity for δ-b-Open Sets
Aynur Keskin Kaymakci1
Abstract. In this talk, first of all we introduce a weaker form of R-maps which is
called almost δ-b-continuity. Then, we investigate and obtain its some properties and
characterizations. Finally, we show that ƒ: (X,τ) → (Y,ϕ) is almost δ-b-continuous if and
only if ƒ: (X, τs) → (Y, ϕs ) is b-continuous, where τs and ϕs are the semi-regularizations of
τ and ϕ, respectively.
Keywords. R-Maps, Almost δ-b-Continuity, b-Continuity, Semi-Regularization.
AMS 2010. 54C08
References
[1] Andrijević, D., On b-open sets, Mat. Vesnik, 48, 59-64, 1996.
[2] Ekici, E., On δ-semiopen sets and a generalization of functions, Bol. Soc. Paran. Mat.
(35)23, 1-2, 73-84, 2005.
[3] Ekici, E., Generalization of perfectly continuous, regular set connected and clopen
functions, Acta Math. Hungar., 107(3), 193-206, 2005.
[4] A-El-Atik, A., A study on some types of mappings on topological spaces, M. Sc. Thesis,
Tanta University, Egypt, 1997.
[5] Herrington, L. L., Properties of nearly-compact spaces, Proc. Amer. Math. Soc., 45(3),
431-436, 1974.
[6] Keskin, A., Noiri, T., Almost b-continuous functions, Chaos Solitons and Fractals, 41(1),
72-81, 2009.
[7] Keskin Kaymakci, A., On δ-b-open sets, (submitted).
[8] Levine, N., Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly, 70, 36-41, 1963.
1 Selcuk University, Konya, Turkey, [email protected]
- 246 -
[9] I. EL-Magharabi, A. Mubarki, A. M., Z-open sets and Z-continuity in topological space,
International Journal of Mathematical Archive, 2(10), 1819-1827, 2011.
[10] Munshi, B. M., Bassan, D. S., Super continuous mappings, Indian J. Pure Appl. Math.,
13, 229-236, 1982.
[11] Noiri, T., Remarks on δ-semi-open sets and δ-preopen sets, Demonstratio Math., 36,
1007-1020, 2003.
[12] Noiri, T., Hyperconnectedness and preopen sets, Rev. Roum. Math. Pure Appl., 29, 329-
334, 1984.
[13] Park, J.H., Lee B.Y., Son, M.J., On δ-semiopen sets in topological spaces, J. Indian
Acad. Math., 19(1), 59-67, 1997.
[14] Raychaudhuri, S., Mukherje, M.N., On δ-almost continuity and δ-preopen sets, Bull.
Inst. Math. Acad. Sinica, 21, 357-366. 1993.
[15] Stone, M.H., Applications of the theory of Boolean rings to general topology, Trans.
Amer. Math. Soc., 41, 375-481. 1937.
[16] Velićko, N.V., H-closed topological spaces, Amer. Math. Soc. Transl. (2), 78, 103-118,
1968.
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Cardinal invariants of the Vietoris Topology on the Space of Minimal CUSCO Maps
Branislav Novotný1 and Ľubica Holá2
Abstract. Let X be a topological space. Among upper semicontinuous multifunctions
with values in real numbers, there is an important class, namely CUSCO maps; that is upper
semi-continuous and non-empty, compact and convex valued maps. Identifying
multifunctions with their graphs, we can equip the space of CUSCO maps with the Vietoris
topology inherited from the space of subsets of X × R.
Let MC(X) be the space of all minimal CUSCO maps on X; minimal with respect to
the ordering of their graphs by ⊆. W e investigate cardinal invariants and related properties on
this space equipped with the Vietoris topology depending on the properties of X.
Keywords. Vietoris Topology, Minimal CUSCO Map, Cardinal Invariant.
AMS 2010. 54A25, 54C35, 54C60.
References
[1] Ľ. Holá, R.A. McCoy. Relations approximated by continuous functions. Proc. Amer.
Math. Soc. 133:2173–2182, 2005.
[2] Ľ. Holá, T. Jain, R.A. McCoy. Topological properties of the multifunction space L(X) of
cusco maps. Math. Slovaca 58:763–780, 2008.
[3] R. Engelking. General topology. PWN, Warszawa, 1977.
[4] I. Juhász. Cardinal Functions in Topology - Ten Years Later. Matematisch Centrum,
Amsterdam, 1980.
1 Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia, [email protected]. 2 Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia, [email protected].
- 248 -
Fg Morphisms and Their Some Properties
Ceren Sultan Elmalı1 and Tamer Uğur2
Abstract. We defined Fan-Gottesman morphism (FG-morphism, for short). We mean
a continuous map 𝑞:𝑋 → 𝑌 such that 𝑞(𝑋) is openly dense in 𝑌 and the topology of 𝑋 is the
inverse image of 𝑌 by 𝑞. We investigated the properties of FG-Morphism.
Keywords: Fan-Gottesman Compactification, Categories
AMS 2010. 54D35, 18A40.
References
[1] M. Baran, Separation Properties, Indian J. Pure and Appl. Math. 23, 333-342 1992.
[2] K. Belaid and O. Echi, 𝑇(𝛼,𝛽) Space and the Wallman Compactification, International
Journal of Mathematics and Mathematical Sciences Vol.2004, Issue 68, 3717-3735, 2004.
[4] C. Elmalı and T. Ugur, Fan-Gottesman Compactification of some specific space is
Wallman type compactification, Chaos,Soliton and Fractals, Vol.42, no.1, 17-19 2009.
[5] C. Elmalı and T. Ugur, Fg-Morphism and Fg-Extension, Hacettepe Journal of
Mathematics and Statistic, Volume 43 (6), 915 – 922, 2014.
[6] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott,
Continuous Lattices and Domains, Cambridge University Press, Cambridge, England, 2003.
[7] A. Grothendieck and J. Dieudonné, Éléments de Géometrie Algébrique, Die Grundlehren
der mathematischen Wissenchaften, vol.166, Springer-Verlag, New York, 1971.
[8] D. Harris, The Wallman compactification is an epireflection, Proc. Amer. Math. Soc.
31,265-267, 1972.
1 Erzurum Technical University, Erzurum, Turkey, [email protected] 2 Atatürk University, Erzurum, Turkey, [email protected]
- 249 -
Simplicial Cohomology Rings of a Connected Sum of Minimal Simple Closed Surfaces
Gülseli Burak1 and İsmet Karaca2
Abstract. In this study, we research digital versions of some concepts of algebraic
topology [1]. We present some general notions of digital images and compute simplicial
cohomology groups of a connected sum of minimal simple closed surfaces [2]. We define a
simplicial cup product for digital images and use it to establish ring structure of digital
cohomology [3]. We give an example about computing the cohomology ring of a connected
sum of a minimal simple closed surface. Then we present algebra structures of digital
cohomology with the cup product. Finally, we show that ),(, GXH k∗ is a graded G-algebra
with the cup product.
Keywords. Digital Simplicial Cohomology Group, Cup Product, Cohomology Ring.
AMS 2010. 55U20, 55N99, 68U05, 68U10.
References
[1] J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley Publishing Company,
1984.
[2] I. Karaca, G. Burak, Simplicial relative cohomology rings of digital images, Applied
Mathematics and Information Sciences 8, No. 5, 2375-2387, 2014.
[3] V.V. Prasolov, Elements of Homology Theory, American Mathematical Society, 2007.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
- 250 -
Introduction to Disoriented Knot Theory
İsmet Altıntaş1
Abstract. This paper is an introduction to disoriented knot theory, which is a
generalization of the oriented knot and link diagrams and an exposition of new ideas and
constructions, including the basic definitions, some numerical invariants such as the
disoriented diagram number, the linking number and the complete writhe and the polynomial
invariants such as the extended bracket polynomial, the Jones polynomial, the complete Jones
polynomial for the disoriented knot and link diagrams.
Keywords. Disoriented Diagrams, Disoriented Diagram Number, Complete Jones
Polynomial.
AMS 2010. 57M25.
References
[1] V.F.R. Jones, A new knot polynomial and Von Neuman algebra, Bull. Amer. Math. Soc.
12, pp. 103-112, 1985.
[2] V.F.R. Jones, Hecke algebra representations of braid groups and link polynomial, Ann.
Math. 126, pp. 335-388, 1987.
[3] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lİckoricsh and A. Ocneau, A new polynomial
invariant of knots and links, Bul. Amer. Math. Soc. 12: pp. 239-246, 1985.
[4] L.H. Kauffman, State models and the Jones polynomial, Topology 26, pp. 395-407, 1987.
[5] L.H. Kauffman, Knot and physics, Word Scientific, Singapore, (third edition), 2001.
[6] L.H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly vol. 95, pp.
195-242, 1988.
[7] L.H. Kauffman, An extended bracket polynomial for virtual knots and links, JKTR, Vol.
18, No 10, pp. 1369-1422, arXiv:0712.2546, 2009.
[8] L.H. Kauffman, Introduction to virtual knot theory, arXiv:1101.0665v2, 2012.
1Sakarya University, Sakarya, Turkey, [email protected]
- 251 -
[9] H.A. Dye and L.H. Kauffman, Virtual crossing number and the arrow polynomial, JKTR,
Vol. 18, No 10, pp. 1335-1356, arXiv:0810.3858, 2009.
[10] H. A. Dye, L. H. Kauffman and V. O. Manturov, On two categorifications of the arrow
polynomial for virtual knots, arXiv:0906.3408v3 [math.GT], 2010.
[11] K. Murasugi (Translated by Bohdam Kurpido), Knot theory and its applications,
Boston,1996.
[12] C.C. Adams, The knot book An elementary introduction to the mathematical theory of
knot, W.H. Freeman Company, New York, (Third printing), 1999.
- 252 -
Digital Homotopy Fixed Point Theory
İsmet Karaca1 and Özgür Ege2
Abstract. Fixed point theory is in relationship with several areas of mathematics such
as mathematical analysis, general topology and functional analysis. There are various
applications of fixed point theory in mathematics, game theory, computer science,
engineering, image processing. Fixed point theorems are used to solve some problems in
mathematics and engineering. In this study, we construct the Notion of the digital homotopy
fixed point property. Some results regarding digital retractions and the fixed point property
are given. We prove that the digital homotopy fixed point property is a topological invariant.
Finally we give a nice application of the digital homotopy fixed point theory to digital images.
Keywords. Digital Image, Digital Homotopy, Fixed Point.
AMS 2010. 47H10, 54E35, 68U10.
References
[1] L. Boxer, Digitally continuous functions, Pattern Recognition Letters, 15, 833-839, 1994.
[2] L. Boxer, Homotopy properties of sphere-like digital images, Journal of Mathematical
Imaging and Vision, 24, 167-175, 2006.
[3] O. Ege and I. Karaca, The Lefschetz fixed point theorem for digital images, Fixed Point
Theory and Applications doi:10.1186/10.1186/1687-1812-2013-253, 2013.
[4] O. Ege and I. Karaca, Applications of the Lefschetz number to digital images, The Bulletin
of the Belgian Mathematical Society-Simon Stevin 21, no. 5, 823-839, 2014.
[5] G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image
Processing, 55, 381-396, 1993.
[6] A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letters, Vol.
4, 177-184, 1986.
[7] M. Szymik, Homotopies and the Universal Fixed Point Property, Order,
doi:10.1007/s11083-014-9332-x, 2014.
1 Ege University, Izmir, Turkey, [email protected] 2 Celal Bayar University, Manisa, Turkey, [email protected]
- 253 -
Some Classes of Functions Between Continuous and Quasicontinuous Functions
Ján Borsík1
Abstract. A real function f is quasicontinuous at a point x if for every positive ε
and for every neighbourhood U of x there is an open nonempty subset G of U such that |f(y)-
f(x)|<ε for each y from G. A function f is quasicontinuous if it is such at each point.
Quasicontinuous functions need not be measurable. Further, the set of all points of
discontinuity of a quasicontinuous function is of first category, however it need not be of
mneasure zero. In the talk, we will investigate classes of functions between continuous and
quasicontinuous functions for which the set of discontinuity points is of measure zero or even
σ-porous.
Keywords. Quasicontinuous Functions, Points of Discontinuity, Porosity.
AMS 2010. 54C08, 54C30.
References
[1] Borsík, J., Holos, J., Some properties of porouscontinuous functions, Math. Slovaca 64,
741-750, 2014.
[2] Borsík, J., Some classes of strongly quasicontinuous functions, Real Anal. Exchange 30,
689-702, 2004-05.
1 Mathematical Institute of Slovak Academy of Sciences, Kosice, Slovakia, [email protected]
- 254 -
HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials
Kemal Taşköprü1 and İsmet Altıntaş2
Abstract. The focus of this paper is to study the HOMFLY polynomial of (2,n)-torus
link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of
generalized Fibonacci and Lucas polynomials and provide their some fundamental properties.
We define the HOMFLY polynomials of (2,n)-torus link with a way similar to our generalized
Fibonacci polynomials and provide its fundamental properties. We also show that the
HOMFLY polynomials of (2,n)-torus link can be obtained from its Alexander-Conway
polynomial or the classical Fibonacci polynomial. We finally give the matrix representations
and prove important identities, which are similar to the Fibonacci identities, for the our
generalized Fibonacci polynomial and the HOMFLY polynomial of (2,n)-torus link.
Keywords. HOMFLY Polynomial, Alexander-Conway Polynomial, Torus Link,
Fibonacci Polynomial, Binet's Formula, Fibonacci Identities.
AMS 2010. 57M25, 11B39, 11C08.
References
[1] Alexander, J. W., Topological invariants of knots and links, Trans. Amer. Math. Soc., 30,
275-308, 1928.
[2] Brandt, R. D, Lickorish, W. B. R., Millett, K. C., A polynomial invariant for unoriented
knots and links, Invent. Math. 84, 563-573, 1986.
[3] Conway, J. H., An enumeration of knots and links, and some of their algebraic properties,
in: Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon,
Oxford, pp. 329-358, 1970.
[4] Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K., Ocneau, A., A new
polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12, 239-246, 1985.
[5] Jones, V. F. R., A polynomial invariant for knots via von Neumann algebras, Bull. Amer.
Math. Soc., 12, 103-111, 1985.
[6] Kauffman, L. H., On Knots, Vol. 115 of Annals of Mathematics Study, Princeton
University Press, 1987.
1 Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 255 -
[7] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, (3), 395-407,
1987.
[8] Kauffman, L. H., New invariants in the theory of knots, Amer. Math. Monthly, 95, (3),
195-242, 1988.
[9] Kauffman, L. H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 417-471,
1990.
[10] Altıntaş, İ., An oriented state model for the Jones polynomial and its applications
alternating links, Applied Math. and Comp., 194, (1), 168-178, 2007.
[11] Murasugi, K., Knot Theory and Its Applications, Birkhuser, Basel, 1996.
[12] Fox, R. H., Free differential calculus I: Derivation in the free group ring, Annals of
Math., 57, (3), 547-560, 1953.
[13] Crowell, R. H., Fox, R. H., Introduction to Knot Theory, Vol. 57 of Graduate Texts in
Mathematics, Springer, 1963.
[14] Koseleff, P.-V., Pecker, D., On Fibonacci knots, Fibonacci Quart., 48, (2), 137-143,
2010.
[15] Amdeberhan, T., Moll, V., Chen, X., Sagan, B., Generalized Fibonacci polynomials and
Fibonomial coefficients, Annals of Combin., 18, (4), 541-562, 2014.
[16] Catalani, M,. Generalized bivariate Fibonacci polynomials, arXiv:math/0211366v2
[math.CO].
[17] Koshy, T., Fibonacci and Lucas Numbers with Applications, Pure and Applied
Mathematics: A Wiley Series of Texts, Monographs and Tracts, Wiley, 2011.
[18] Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos,
Solitons and Fractals, 42, (5), 3179-3186, 2009.
[19] Panwar, Y. K., Singh, M., Generalized bivariate Fibonacci-like polynomials, Inter.
Journal of Pure Math., 1, 8-13, 2014.
- 256 -
Minimal Usco and Minimal Cusco Maps and Compactness
Ľubica Holá1 and Dušan Holý2
Abstract. We prove Ascoli-type theorems for the space of rninimal usco and minimal
cusco maps. Let X be a locally cornpact space, (Y, d) be a metric space and MU (X, Y) be the
space of minimal usco maps frorn X to Y. The family Ɛ, subset of MU(X,Y) is compact in
MU(X,Y) equipped with tlie topology ΤUC of uniforrn convergence on cornpact sets if and only
if Ɛ is closed, compactly bounded and densely equicontinuous. If (Y,d) is a complete rnetric
space the farnily Ɛ, subset of (MU(X,Y), ΤUC) is compact if and only if Ɛ is closed, pointwise
bounded and densely equicontinuous. The same result holds also for compact subsets of
(MC(X,Y), ΤUC), the space of minimal cusco maps frorn X to a Banach space Y.
Keywords. Densely Equicontinuous, Minirnal Usco Map, Minirnal Cusco Map.
AMS 2010. Primáry 54C60, 46B25; Secondary 54B20.
References
[1] L. Holá, D. Holý, Minirnal usco rnaps, densely continuous forrns and upper
sernicontinuous functions, Rocky Mountain J. Math. 39, 545—562, 2009.
[2] L. Holá, D. Holý, Relation betwen rninirnal usco and rninirnal cusco rnaps, Portugaliae
Mathematica 70, Fracs 3, 211—224, 2013.
[3] L. Holá, D. Holý, New characterization of rninirnal cusco maps, Rocky Mountain J.
Math. 44, 1851-1866, 2014.
1 Academy of Sciences, Institute of Mathematics, Bratislava, Slovakia, [email protected] 2 Trnava University, Trnava, Slovakia, [email protected]
- 257 -
On 𝜸-P-Open Sets in Bitopological Spaces
Merve İlkhan 1, Mahmut Akyiğit2 and Emrah Evren Kara3
Abstract. In this study, we introduce the concept of 𝛾-P-open sets in bitopological
spaces. Also, we investigate some properties related to this concept.
Keywords. Bitopological Spaces, 𝛾-Open Sets, (𝑖, 𝑗)-𝛾-P-Open Sets.
AMS 2010. 54A05, 54D10.
References
[1] Ogata, H., Operation on topological spaces and associated topology, Math. Japonica., 36,
175-184, 1991.
[2] Kelly, J. C., Bitopological Spaces, Proc. London Math. Soc. 13, 3, 71-89, 1962.
[3] Khalaf, A. B., Ibrahim, H. Z., Some applications of 𝛾-P-open sets in topological spaces,
Int. J. Pure Appl. Math. Sci., 5, 81-96, 2011.
[4] Al-Areefi, S. M., Operation-separation axioms in bitopological spaces, An. Şt. Univ.
Ovidius Constanta, 17, 2, 5-18, 2009.
1 Duzce University, Duzce, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Duzce University, Duzce, Turkey, [email protected]
- 258 -
Invariant Measures and Controllability of Affine Control Systems
Memet Kule1
Abstract. In this paper, which is a continuation of the paper [6], we show that affine
control systems on general Lie groups to be controllable is used the concept of measure of
invariant vector fields and is established the desired implication for analytic systems under a
Poisson stability condition.
Keywords. Affine Control Systems, Lie Groups, Lie Algebras, Invariant Measures,
Poisson Stability
AMS 2010. 37A05, 37N35, 93B05, 93C10.
References
[1] Agrachev, A.A., Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Springer Verlag, 2004.
[2] Helgason, H., Differential Geometry, Lie Groups and Symmetric Spaces. AMS, Providence, RI, 2001.
[3] Jouan Ph., Invariant measures and controllability of finite systems on compact manifolds. ESAIM: Control Optim Calc Var. 18, 643-655, 2012.
[4] Jouan, Ph., Equivalence of control systems with linear systems on Lie groups and homogeneous spaces, ESAIM: Control Optim Calc Var. , 16, 956-973, 2010.
[5] Jurdjevic, V., Sallet, G., Controllability Properties of Affine Systems, SIAM J.Control Opt., 22, 501-518, 1984.
[6] Kule, M., Controllability of Affine Control Systems on Lie Groups, Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0522-6.
[7] Kara, A., Kule, M., Controllability of Affine Control Systems on Carnot Groups, Int. J. of Contemp. Math. Sci., 5, 2167-2172, 2010.
[8] Kara, A., San Martin. L., Controllability of Affine Control System for the Generalized
Heisenberg Lie Groups, Int. J. Pure Appl. Math., 29, 1-6, 2006.
1 Kilis 7 Aralık University, Kilis, Turkey, [email protected]
- 259 -
Soft Topological Questions and Answers
Milan Matejdes1
Abstract. The paper deals with a few questions stated for a soft topological space in
[5]. The main goal is to point out that any soft topological space is homeomorphic to a
topological space (A × X, τA×X) where τA×X is a topology on the product A × X, consequently
many soft topological notions and results can be derived from general topology. Examples
answering the questions of [5] are given.
Keywords. Soft Set, Topological Soft Space, Soft Open (closed) Set, Soft Interior
(closure), Soft Separation Axioms, Soft e-continuity.
AMS 2010. 54C60, 54C65, 26E25.
References
[1] A. Aygünoğlu and H. Aygün, Some notes on soft topological spaces, Neural. Comput.
Appl., 21, 113-119, 2011.
[2] N. Cağman, S. Karataş and S. Enginoglu, Soft topology, Comput. Math. Appl., 62, 351-
358, 2011.
[3] M. Caldas, S. Jafari and M. M. Kovár, Some properties of θ-open sets, Divulgaciones Ma-
temáticas, 12, No. 2, 161-169, 2004.
[4] D. N. Georgiou and A. C. Megaritis, Soft set theory and topology, Appl. Gen. Topol. 15,
no. 1, 93-109. 2014.
[5] D. N. Georgiou, A. C. Megaritis and V. I. Petropoulos, On Soft Topological Spaces, Appl.
Math. Inf. Sci. 7 no. 5, 1889-1901, 2013.
[6] S. Hussain and B. Ahmad, Some properties of soft topological spaces, Comput. Math.
Appl. 62, 4058-4067, 2011.
[7] W. K. Min, A note on soft topological spaces, Comput. Math. Appl. 62, 3524-3528. 2011.
1 Technical University in Zvolen, Zvolen, Slovakia, [email protected]
- 260 -
[8] Nirmala Rebecca Paul, Remarks on soft ω-closed sets in soft topological spaces, Bol. Soc.
Paran. Mat., 33, 181-190. 2015.
[9] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl. 61, 1786-1799.
2011.
[10] S. Yüksel, N. Tozlu and Z. Güzel Ergül, Soft regular generalized closed sets in soft topo-
logical spaces, Int. Journal of Math. Analysis, 8, 355-367. 2014.
[11] I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces,
Annals of Fuzzy Mathematics and Informatics 3, no. 2, 171-185. 2012.
- 261 -
A Fixed Point Criterion for 𝒑-adic Actions
Mehmet Onat1
Abstract. In this study, it is given that the criterion for the existence of fixed point for
effective actions of some compact, connected, abelian non-Lie groups. This criterion is
implemented for effective actions of 𝑝-adic solenoid.
Keywords. Classifying Space, 𝑝-Adic Solenoid, Compact Connected Abelian Groups.
AMS 2010. 57S25, 55N91.
References
[1] Hofmann K.H. and Mostert P.S., The cohomology of compact abelian groups, Bull. Amer.
Math. Soc. 74, 975-978, 1968.
[2] Hofmann K.H. and Mostert P.S., The Structure of compact groups. Berlin, Germany: de
Gruyter, 1998.
[3] Lee J.S., Totally disconnected groups, 𝑝- adic groups and the Hilbert-Smith conjecture.
Comm Korean Math Soc; 12: 691-699, 1997.
[4] Montgomery D, Zippin L., Topological transformation groups. New York, NY, USA:
Interscience, 1955.
[5] Yamabe H., A generalization of a theorem of Gleason. Ann Math. 58: 351-365,1953.
1 Cukurova University, Adana, Turkey, [email protected]
- 262 -
Complex Valued Dislocated Metric Spaces and an Application to Differential Equations
Özgur Ege1 and İsmet Karaca2
Abstract. In this paper, we introduce complex valued dislocated metric spaces. We
prove Banach contraction principle in this new space. Moreover, we give an application of the
theory to differential equations.
Keywords. Fixed Point, Dislocated Metric Space, Banach Contraction Principle.
AMS 2010. 47H10, 54E35, 54H25.
References
[1] Aage, C.T., Salunke, J.N., The results on fixed point theorems in dislocated and dislocated
quasi-metric space, Applied Math. Sci., 2, 2941-2948, 2008.
[2] Azam, A., Fisher, B., Khan, M., Common fixed point theorems in complex valued metric
spaces, Number. Funct. Anal. Optim., 32, 243-253, 2011.
[3] Hitzler, P., Seda, A.K., Dislocated topologies, Journal of Electrical Engineering, 51, 3-7,
2000.
[4] Jha, K., Panthi, D., A common fixed point theorem in dislocated metric space, Applied
Mathematical Sciences, 6, 4497-4503, 2012.
[5] Matthews, S.G., Metric domains for completeness, Ph.D. Thesis, University of Warwick,
1-127, 1986.
[6] Pasicki, L., Dislocated metric and fixed point theorems, Fixed Point Theory and
Applications, 2015:82, 2015.
[7] Rahman, M.U., Sarwar, M., Fixed point theorems for expanding mappings in dislocated
metric space, Mathematical Sciences Letters, 4, 69-73, 2015.
[8] Rao, K.P.R., Rangaswamy, P., A coincidence point theorem for four mappings in
dislocated metric spaces, Int. J. Contemp. Math. Sciences, 6, 1675-1680, 2011.
1 Celal Bayar University, Manisa, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
- 263 -
Principal P-adic Bundles Over Circle Groups
Seda Eren1
Abstract. In this work it is shown that p-adic solenoid is a principal p-adic bundle
over circle groups.
Keywords. p-adic Solenoid, Principal G Bundle
AMS 2010. 54B25, 57S10.
References
[1] M. C. McCord, Inverse limit sequences with covering maps, theorem 5-6, 1963.
[2] V. Matijevic, Classifying finited-sheeted coverings of paracompact spaces, Revista Mat
Comput 16, 311-327, 2003.
[3] Edwin H. Spanier, Algebraic Topology, 1966.
[4] R. Engelking, General Topology, 1989.
1 Cukurova University, Adana, Turkey, [email protected]
- 264 -
Topological Group-2-Groupoids and Topological 2G-Crossed Modules
Sedat Temel1 and Nazmiye Alemdar 2
Abstract. The main purpose of this paper is to construct the group structure on a
topological 2-groupoid which we call topological group-2-groupoid and to obtain an algebraic
structure which we call topological 2G-crossed module by using topological crossed modules
corresponding to topological group-2-groupoids. Moreover we prove that the category of
topological group-2-groupoids and of topological 2G-crossed modules are equivalent.
Keywords. Topological 2-Groupoid, Topological Group-2-Groupoid, Topological
2G-Crossed Module.
AMS 2010. 18B30, 18D05, 22A05, 22A22, 54H11.
References
[1] Baez J. C., Stevenson D., The classifying space of a topological 2-group. Algebraic
Topology Abel Symposia, 4: pp 1-31, 2009.
[2] Baez J. C., Baratin A., Freidel L., Wise D.K. Infinite-dimensional representations of 2-
groups. Memoirs of the American Mathematical Society, 219: Number 1032, 2012.
[3] Brown R., Spencer C.B., G-groupoids, crossed modules and the fundamental groupoid of
a topological group. Proc. Konn. Ned. Akad., 79: 296-302, 1976.
[4] Noohi B., Notes on 2-groupoids, 2-groups and crossed modules. Homology Homotopy
Appl., 9: Number 1, 75-106, 2007.
[5] Mucuk O., Sahan T., Alemdar N. Normality and quotients in crossed modules and group-
groupoids. Appl. Categor. Struct., DOI 10.1007/s10485-013-9335-6, 2013.
1 Erciyes University, Erciyes, Turkey, [email protected] 2 Erciyes University, Erciyes, Turkey, [email protected]
- 265 -
Textures and Approximation Spaces
Şenol Dost 1
Abstract. By a texturing [1] of a set S we mean a subset δ of the power set P(S)
which is a point separating complete, completely distributive lattice with respect to inclusion
which contains S and ∅ , and for which arbitrary meets coincide with intersections and finite
joins coincide with unions. The pair ( ),S δ is then called a texture space, or simply texture.
Difunctions [2] arise often in the study of textures and ditopological texture spaces. A
difunction is a direlation ( ),f F satisfying certain additional conditions.
The theory of rough set is firstly introduced by Pawlak [5]. It should be noted that the basic
tools for this theory are lower and upper approximation operators. A discussion is presented
on rough set theory from the textural point of view [3]. Here is observed that the presections
which are defined in terms of direlations are generalizations for rough sets. The dual operators
of a textural rough set algebra which are defined on a complemented texture spaces are called
approximation operators. Furthermore, if (r;R) is a complemented direlation then the inverse
of the relation r(the corelation R) of is a lower (an upper) approximation operator.
The aim of this study is to investigate the link between rough set approximation operators and
textural approximation operators in view of categorical approach. We consider generalized
interior-closure spaces (gic-space) [4] which their operators satisfy some conditions, and they
are called i-c* spaces. Further, we observe that a textural rough set algebra is a complemented
gic-space. Bicontinuous difunction between gic-spaces are used as morphisms, and some
categories of the above mentioned approximation operators are formed.
Keywords. Rough Set, Interior-Closure Space, Texture, Textural Approximation
Space
AMS 2010. 54A40, 06A15, 18B30.
References
[1] Brown, L. M., Ertürk, R., Fuzzy sets as texture spaces, I. Representation theorems, Fuzzy
Sets and Systems, 110 (2), 227236, 2000.
1 Hacettepe University, Ankara, Turkey, [email protected]
- 266 -
[2] Brown L. M., Ertürk R., Dost Ş. Ditopological texture spaces and fuzzy topology, I. Basic
Concepts, Fuzzy Sets and Systems, 147 (2), 171-199, 2004.
[3] Diker M. Textural approach to generalized rough sets based on relations, Information
Sciences, 180, 1418-1433, 2010.
[4] Diker, M., Dost, Ş., Altay Ugur, A. Interior and closure operators on texture spaces-I:
Basic concepts and Cech closure operators, Fuzzy Sets and Systems, 161935-953, 2009.
[5] Pawlak Z. Rough sets, International jornal of computer and information sciences, 341-
356, 1982.
- 267 -
A Numerical Analysis of Non Newtonian flow through Microchannels
Arunabha Chanda1
Abstract. Understanding non-Newtonian flow in microchannels is required for various
micro fluidic devices to explore both fundamental and practical significance of the flow with a
special reference to the effect of viscosity, slip velocity and different power law indices. Flow
channel cross sections in the order of micro scale gives rise to very high heat and mass transfer in
non-Newtonian flow which has a connection to slip flow at microchannel walls and electro
viscous effect. The occurrence of slip near solid boundary has created a challenging problem in
fluid mechanics. Flow through microchannels introduces an electric field which applies an
electric resistance on charged fluid in the opposite direction of the fluid flow. This is known as
electro viscosity effect which retards the flow and on contrary, wall slip phenomena increases the
flow velocity. The role of different indices of power law to the combined effect of wall slip and
electro viscosity has not been well studied. Different models along with dimensionless numbers
has been proposed to describe non Newtonian flow through microchannels but still there exists
discrepancies in results.
A finite difference analysis has been carried out in the present work in an attempt to
investigate numerically the combined effect of wall slip and electro viscosity for the different
indices of power law. Different combinations of steady state, laminar and constant fluid
properties in power law model can yield various results which can be used to justify the
numerical analysis.
Keywords: Microchannels, Electro Viscosity, Wall Slip
Reference
[1] G.H. Tang et al, “Electroviscous effect on non-Newtonian fluid flow in microchannels”, J
Non-Newtonian Fluid Mech, 165, 435-440
1-Jadavpur University, Kolkata, India
- 268 -
Beautiful number 6174
Asra Rezafadaei
Abstract. In this article we try to solve "Mysterious number 6174" or "Kaprekar's
constant". We solve this problem in two step or two theorem. This problem is this:
Consider a 4-digit number (which is not a multiple for 1111). Sort the digits in bigger-
smaller, and reverse order. Subtract the smaller number from the bigger one. Perform the
same operation with the remainder (it is called Kaprekar operation). After a number of steps
we reach 6174. For example:
3452 => 5432 – 2345 = 3087
3087 => 8730 – 378 = 8352
8352 => 8532 – 2358 = 6174
The question is why these numbers reach 6174?
This question was told by an Indian scientist Kaprekar who had lived from 1905 till
1986. A question that after about 50 years no one is able to solve it with mathematical rules
and reasons.
As what was talked about, lots of people have tried to solve this abstruse question but
they couldn’t yet.
However in all of their detection, s.th is collective and that is:
If set all the numbers, after a level we have a particular group of digit numbers that
the magnitude of them is 91. And again if set them from the smallest till the biggest, the
magnitude will change to 30. And they are shown in series named “S”.
S=9810 , 9771 , 9621 , 9531 , 9441 , 8820 , 8721 , 8622 , 8532 , 8442 , 8730 , 7731 , 7632 ,
7533 , 7443 , 8640 , 7641 , 6642 , 6543 , 6444 , 8550 , 7551 , 6552 , 5553 , 5544 , 9990
,9981 ,9972 ,9963 ,9954
In first step or theorem, I proof that why these numbers arrive to the series "S" with
parameter and rules of mathematic. After that I proof that why the numbers of series "S"
arrive to 6174 with
- 269 -
Density Functional Theory in The Solid State Science
Engin Deligöz1 and Haci Özışık2
Abstract. Computational material science uses computers to model, understand and
predict material properties. One of the most innovative and challenging areas of materials
theory has centered on predicting some physical properties. This predictive ability in some
areas competes with experimental measurements. Density functional theory (DFT) is a topic
of interest in mathematics, physics and in chemistry. Modern DFT simulation codes can
calculate a vast range of structural, elastic, electronic, vibrational and thermodynamic
phenomena. We present an overview of the capabilities of solid-state DFT simulations in all
of these topics, illustrated with recent example for TiAsTe compound using the VASP
computer program. The obtained results are in the agreement with the available experimental
results.
Keywords: Density Functional Theory, ab initio Calculations
1Aksaray University, Aksaray, Turkey, [email protected] 2Aksaray University, Aksaray, Turkey, [email protected]
- 270 -
Mathematical Ratio in Painting
Elif Gürsoy1 and Semra Kaya Nurkan2
Abstract. Mathematical ratio or Golden ratio is one of the oldest rate system which
can be described as the relation of the part with the whole and the relation of the objects with
each other, in terms of symmetry and balance. And it brings to our lives balance, harmony and
coherence.
In this study we sought the mathematical ratio in painting. Works of art are dated back
to Medieval Europe and examples of religious painting.
Keywords. Mathematical Ratio, Mathematical Ratio in Painting, Painting.
AMS 2010. 00A67, 97M80.
References
[1] Akdeniz, F., Doğada, sanatta, mimaride altın oran ve Fibonacci sayıları, Nobel Kitabevi,
2007.
[2] Beksaç, E., Ortaçağ Avrupasında resim sanatı, Paradigma Yayınları, 2012.
[3] Bergil, M.S., Doğada bilimde sanatta altın oran, Arkeoloji ve Sanat Yayınları, 2009.
[4] Cömert, B., Mitoloji ve ikonografi, Ayraç Yayınları, 1999.
[5] Çağlarca, S., Altın oran, İnkılap Kitabevi, 1997.
[6] Çakmak, M.S., Evrenin geometrik şifresi altın oran kaos fraktal simetri, Grifin, 2011.
1Usak University, Usak, Turkey, [email protected] 2Usak University, Usak, Turkey, [email protected]
- 271 -
Non-linear Analysis of Rotating Cracked FGM Beams
G. Pohit1
Abstract. The dynamic characteristics of cracked structures are of considerable
importance in structural health monitoring. It is known that a crack in a structure introduces a
local flexibility, reduces the stiffness and may change the dynamic behavior of the structure
[1, 2]. Presence of crack in rotating structure may pose higher risk of failure. The present
analysis is carried out for a cantilever beam fixed with a rotating hub. The effect of centrifugal
stiffening on the nonlinear response of a cracked Timoshenko beam is determined. Material is
assumed to have a gradation relation along the depth of the beam. Crack is modeled as a mass
less rotational spring. Centrifugal force and axial displacement raised due to the rotating hub
is incorporated in the strain energy equations. Ritz approximation followed by an iterative
technique is employed to obtain the nonlinear vibration responses. Effects of rotational
speeds, hub radius, crack depths, crack locations and different gradation relations on the
nonlinear frequencies are obtained for different vibration modes. Effect of crack on the
system characteristics are obtained by varying crack size and position. Comparison study of
some of the results with the available solutions confirms the accuracy of the method.
Keywords. Rotating Beam, Crack, Nonlinear Vibration, Functionally Graded
Material, Timoshenko Beam
References
[1] Chondros, T. G. Dimarogonas A. D. and Yao, J. A., Continuous cracked beam vibration
theory, Journal of Sound and Vibration, 215, 17-34, 1998.
[2] Rezaee, M. and Hassannejad, R., A new approach to free vibration analysis of a beam
with a breathing crack based on mechanical energy balance method, Acta Mechanica Solida
Sinica, 24, 185-194, 2011.
1 Jadavpur University, Kolkata, India, [email protected]
- 272 -
Different Viewpoint for Puzzle Problems as Artificial Intelligence Toy Problems:
Heuristic Angular Metric Approach
İhsan Ömür Bucak1 and Murat Tatlılıoğlu2
Abstract. In this study, the behaviours of heuristic metrics for the solutions of the
puzzle problems with different sizes have been investigated. It has been expressed
systematically that what features a good metric should require have been expressed
systematically by comparing Manhattan, Euclidean, Chebyshev heuristic metrics up to 99-
puzzle. A novel angular metric approach has been developed and a different perspective to the
metric concept has been revealed. The developed angular metric approach has been tested and
proven to be correct. Heuristic metrics provide nearest-optimal solutions in the shortest time
possible in such problems having variables with astronomical values. As in toy problems such
as 8-Puzzle, heuristic metrics which do the search, selection and optimization in a very short
period of time can be seen very important. Comparison of the heuristic metrics on the puzzle
problems can easily be done and observed. Solving behaviour of different metrics are easily
observable when we increase the size of the puzzle. This allows one to examine the heuristic
metrics much better. Under the light of three features of heuristic functions such as
dominance, admissibility and consistency, the metrics can become a straight-line starting from
a half-circle. Having it a half-circle is due to the necessity of triangle rule. In this approach, in
the geometric representation of the metric, a heuristic acquisition is obtained in proportion to
the length of a circle slice. As part of this study, operational and experimental results have
been provided in detail in the original manuscript. As a result, the characters of heuristic
metrics regarded as solid foundations of Artificial Intelligence can be identified easily
according to the approach developed here.
Keywords. Angular Metric Approach, Heuristic Metrics, Heuristic Functions.
1 Melikşah University, Kayseri, Turkey, [email protected] 2 Meliksah University, Kayseri, Turkey, [email protected]
- 273 -
Development of Mathematical Model using Group Contribution Method to Predict
Exposure Limit Values in Air for Safeguarding Health
Mohanad El-Harbawi1 and Phung Thi Kieu Trang2
Abstract. Occupational Exposure Limits (OELs) are representing the amount of a
workplace health hazard that most workers can be exposed to without harming their health. In
this work, a new Quantitative Structure Property Relationship (QSPR) model to estimate
occupational exposure limits values has been developed. The model was developed based on
a set of 100 exposure limit values, which were published by the American Conference of
Governmental Industrial Hygienists (ACGIH). MATLAB software was employed to develop
the model based on a combination between Multiple Linear Regression (MLR) and
polynomial models. The results showed that the model is able to predict the exposure limits
with high accuracy, R2 = 0.9998. The model can be considered scientifically useful and
convenient alternative to experimental assessments.
Keywords: OELs, Group Contribution Method, QSAR, MATLAB.
1 King Saud University University, Riyadh, Saudi Arabia, [email protected] 2 Universiti Teknologi PETRONAS, Perak, Malaysia, [email protected]
- 274 -
The Way Ahead for Bug-fix time Prediction
Meera Sharma1
Abstract.The bug-fix time i.e. the time to fix a bug after the bug was introduced is an
important factor for bug related analysis, such as measuring software quality [1] or
coordinating the development effort during bug triaging to maintain the software systems
effectively [2]. In literature efforts have been made to construct many bug-fix time prediction
models, based on machine learning algorithms, on both open source and commercial projects.
[3-5]. Previous work has proposed many bug-fix time prediction models that use various bug
attributes (number of developers who participated in fixing the bug, bug severity, bug-
opener’s reputation, number of patches) for predicting the fix time of a newly reported bug. In
this paper we investigate the associations between bug attributes and the bug-fix time. Our
prediction method is based on the association rule mining method which was first explored by
[6].We have proposed an approach to apply association mining by using Apriori algorithm to
predict the fix time of a newly coming bug based on the bug’s severity, priority summary
terms and assignee. We demonstrate our approach on collection of 1,695 bug reports of
Thunderbird, AddOnSDK and Bugzilla products of Mozilla open source project.
Keywords. Bug-fix Time, Association Rule Mining, Association Rules.
AMS 2010. 53A40, 20M15.
References
[1] Kim, S., Whitehead, J. E., How long did it take to fix bugs? Int. Workshop Mining
Software Repositories, New York, NY, USA, ACM, 173–174, 2006.
[2] Hooimeijer, P., Weimer, W., Modeling bug report quality.ASE, 2007.
[3] Anbalagan, P., and Vouk, M., On predicting the time taken to correct bug reports in open
source projects. Int. Conf. Software Management, Edmonton, AB, IEEE, 523-526, 2009.
[4] Bhattacharya, P. and Neamtiu, I., Bug-fix Time Prediction Models: Can We Do Better? 8th
Working Conf. Mining Software Repositories (New York, NY, USA,2012). ACM, 207-210,
2010.
1University of Delhi, Delhi, India, [email protected]
- 275 -
[5] Giger, E., Pinzger, M., and Gall, H., Predicting the fix time of bugs. Int. Workshop
Recommendation Systems on Software Enginnering (New York, NY, USA), ACM, 52-56,
2010.
[6] Agrawal, R., Imielinski, T., Swami, A., Mining Association Rules between Sets of Items in
Large Databases. SIGMOD Conf. Management of Data, ACM. May 1993.
- 276 -
Construction of Lossless Broadband Matching Networks with Lumped Elements
Metin Şengül1
Abstract. For microwave and communication engineers, construction of broadband
matching networks has been regarded as a vital problem [1]. For this purpose, broadband
matching analytic theory [2], [3] and computer aided design (CAD) tools are available [4]-[6].
But it is well known that analytic theory is quite difficult to use and is not practical.
Therefore, it is always preferred to employ CAD tools to construct broadband matching
networks. All CAD tools optimize the performance of the system. At the end of optimization,
the element values of the broadband matching network are obtained. Here it should be
emphasized that performance optimization is highly nonlinear with respect to element values
and needs very good initials. So initial element value selection is important for successful
optimization. Therefore, in this work, a well established initialization process is introduced for
construction of broadband matching networks. It is expected that the proposed algorithm can
be used as a front-end for commercially available CAD tools to design practical broadband
matching networks for microwave communication systems.
Keywords. Broadband Matching, Real Frequency Techniques, Matching Networks,
Lossless Networks.
AMS 2010. 53A40, 20M15.
References
[1] Yarman, B. S., Broadband networks. Wiley Encyclopedia of Electrical and Electronics
Engineering, 2, 589-604, 1999.
[2] Youla, D. C., A new theory of broadband matching, IEEE Trans. Circuit Theory, 11, 30-
50, 1964.
[3] Fano, R. M., Theoretical limitations on the broadband matching of arbitrary impedances,
J. Franklin Inst., 249, 57-83, 1950.
[4] AWR: Microwave Office of Applied Wave Research Inc.: www.appwave.com
[5] EDL/Ansoft Designer of Ansoft Corp; www.ansoft.com/products.cfm
[6] ADS of Agilent Techologies; www.home.agilent.com
1 Kadir Has University, Istanbul, Turkey, [email protected]
- 277 -
American options with regime-switching uncertainty
Saul D. Jacka1 and Adriana Ocejo2
Abstract. We study the minimal payoff scenario for the holder of an American-style
option in the presence of regime-switching uncertainty, for a large class of payoff functions.
We assume that the transition rates are only known to lie within level-dependent compact sets,
and so the holder takes a worst-case scenario position. We show that the minimal payoff
identifies with the value function of an optimal stopping problem associated with a certain
extremal rate matrix and characterized as the probabilistic solution of a free-boundary
problem. The approach is via time-change and classical PDE techniques. We apply our results
to the context of American option pricing under the Markov-modulated constant elasticity of
variance and Markov-modulated geometric Brownian motion models.
Keywords. American Option, Regime-switching, Markov-modulated, Stochastic
Control, Time-change, Stochastic Volatility, Uncertainty.
AMS 2010. 93E20, 49J20, 60G40, 91G20, 91G80.
1 The University of Warwick, Coventry, UK, [email protected] 2 University of North Carolina at Charlotte, Charlotte, USA, [email protected]
- 278 -
Designed Filter with the New Generation Current Conveyor and Analysis Of Ecg
Şükrü Kitiş1, Etem Köklükaya2 and Rüştü Güntürkün3
Abstract. In this study, a model was designed and practiced for ECG measurements
which are tend to be used in detecting heart disorders [1]. This model was designed covering
CCII+ structure. Before designing the model, filter structures [2] and amplifier circuits was
simulated PSPICE program. In this research, an ECG circuit without CCII+ was examined
and practiced in order to observe the advantages of CCII+ structures. In the ongoing process,
filter and amplifier parts of that circuit was detected and redesigned as involving CCII+
structures. The difference between them and also designed ECG circuit and other ECG
circuits [3],[4],[5],[6] were determined.
Keywords. ECG, Next-Generation Current Conveyors, CCII+, Instrumentation,
Amplifier, Low-Pass Filter, High-Pass Filter, Inverting Amplifier.
References
[1] Kemaloğlu, S. Kara, S. EKG işaretleri ile kalp seslerinin eş zamanlı alınması ve ölçüm
düzeneği. Erciyes Üniversitesi Fen Bilimleri Enstitüsü dergisi, 18, (1-2), 28-33, 2002.
[2] Smith K. C. and Sedra A., A second generation current conveyor and its applications,
IEEE Trans. Circuit Theory, CT-17, pp. 132-134, 1970.
[3] Martins, R. Selberherr, S. Vaz, F.A., A CMOS IC for portable EEG acquisition systems,
IEEE transactions on instrument and measurement, Vol.47. No:5, October 1998.
[4] Harrison, R.R. and Charles, C., A low-power low-noise CMOS amplifier for neural
recording applications, IEEE Journal of Solid-State Circuits, Vol.38, No. 6. June 2003.
[5] Yazicioglu, R.F. Merken, P. Puers, R. Hoof, C.V., A 200 µW eight-channel acquisition
ASIC for ambulatory EEG systems, IEEE International Solid-State Circuits Conference 2008.
[6] Chen, C.H. Chang, C.L. Chang, C.W., A low-power bio-potential acquisition system with
flexible PMDS dry-electrodes for portable ubiquitous helatcare applications, Sensors
2013,13,3077 309 ;doi:10.3390/s130303077
1-3 Dumlupinar University, Kutahya, Turkey, [email protected] 2 Gazi University, Ankara, Turkey, [email protected]
- 279 -
A Study on Mathematical Modelling for Open Source Software
Optimal Release Planning
V.B.Singh 1
Abstract: The main characteristics of open source software are the release early and
release often [1]. The software engineering community has devoted little attention to release
engineering [2]. Regarding releases, volunteer-driven open source projects usually employ
one of two strategies. Many projects issue a new release after implementing a certain set of
features [3 and 4]. This involves numerous challenges. Alternatively, projects might adopt a
time-based strategy, in which releases are planned for a specific date as mentioned by [3 and
5]. The frequent changes in the source code make the source code complex. It is evident that
for fixing bugs, new features introduction and feature improvements, different files of the
software need to be changed. It is always desirable to meet the users need before releasing
the next version of the software. Release time problem for proprietary software has been
widely discussed by considering only one factor, the bugs which have been fixed in different
releases.In this paper, we have studied a quantified approach for open source release time
problem based on the implementations of bug fixing, new feature introduction and feature
enhancement.
Keywords. Release Planning, Code Changes, Open Source.
AMS 2010. 53A40, 20M15.
References
[1] Raymond, E. S., The Cathedral and the Bazaar: Musings on Linux and Open Source by an Accidental Revolutionary, ISBN 1-56592-724-9, 1997.
[2] Wright, H. K. and Perry, D. E., Release Engineering Practices and Pitfalls, 34th International Conference on Software Engineering, 1281–1284, 2012.
[3] Raymond, E. S. The Cathedral and the Bazaar, O’Reilly, 2001.
[4] Adams, B. et al., eds., Proc. 1st Int'l Workshop Release Engineering (RELENG 13), doi:10.1109/ICSE.2013.6606779. 2013.
[5] Kerzazi, N. and Khomh, F., Factors Impacting Software Release Engineering: A Longitudinal Study, Proceedings 2nd Workshop Release Engineering, 2014.
1 University of Delhi, Delhi, India, [email protected]
- 280 -
Equivariantly Formal Solenoidal Actions
Ali Arslan Özkurt1
Abstract. X is said to be equivariantly formal G -space if * * *
Gi : H ( X ,k ) H ( X ,k )→ is onto. It is well gnown fact that equivariantly formal compact
Lie group actions have fixed points. In this study, it is shown that the same statement is true
for equivariantly formal solenoidal actions.
Keywords. Equivariantly Formal Actions, Solenoidal Groups
AMS 2010. 57S25, 55N91.
References
[1] Allday C, Puppe V., Cohomological methods in transformation groups. Cambridge:
Cambridge University Press, 1993.
[2] Bredon GE., Sheaf theory, 2nd ed. Graduate Texts in Mathematics 170. New York:
Springer, 1997.
[3] Bredon GE, Raymond F, Williams RF., p-adic groups of transformations. Trans. Amer.
Math.Soc.. 99, 488-498, 1961.
[4] Hofmann KH, Morris SA., The structure of compact groups. Berlin: de Gruyter, 1998.
[5] Löwen R., Locally compact connected groups acting on euclidean space with Lie isotropy
groups are Lie. Geom. Dedicata 5, 171-174, 1976.
[6] Milnor J., Construction of Universal Bundles. I The Annals of Mathematics. 63, 272-284,
1956.
[7] Mitchell B., Theory of categories. New York: Academic Press, 1965.
[8] Spanier E., Algebraic topology. New York: Springer-Verlag, 1989.
1 Cukurova University, Adana, Turkey, [email protected]
- 281 -
An Extension of Srivastava's Triple Hypergeometric Function HC
A. Çetinkaya 1, M. B. Yağbasan2 and İ. O. Kıymaz3
Abstract. Recently, some extensions of the well-known special functions such as
hypergeometric, confluent hypergeometric and Mittag-Leffler have been studied by several
authors. The main object of this study is to introduce an extension of Srivastava's triple
hypergeometric function HC by using extended beta function. Some integral representations
are also obtained for this extension.
Keywords. Extended Beta Function, Srivastava's Triple Hypergeometric Function HC,
Integral Representations.
AMS 2010. 33C60, 33C65, 33C70
Acknowledgement: This work was supported by Ahi Evran University Scientific
Research Projects Coordination Unit. Project Number: PYO-FEN.4001.14.008
References
[1] Chaudhry M. A., Qadir A., Rafique M., Zubair S. M., Extension of Euler's beta function,
J. Comput. Appl. Math. 78, 19-32, 1997.
[2] Chaudhry M. A., Qadir A., Srivastava H. M., Paris R. B., Extended hypergeometric and
confluent hypergeometric functions, Appl. Math. Comput. 159, 589-602, 2004.
[3] Choi J., Hasanov A., Srivastava H. M., Turaev M., Integral representations for
Srivastava's triple hypergeometric functions. Taiwanese J. Math. 15, no:6, 2751-2762, 2011.
[4] Choi J., Hasanov A., Turaev M., Integral representations for Srivastava's hypergeometric
function HC., Honam Math. J. 34, no. 4, 473–482, 2012.
[5] Çetinkaya, A., Yağbasan, M. B., Kıymaz, İ. O., An Extension of Srivastava's Triple
Hypergeometric Function HA, 2nd International Eurasian Conference on Mathematical
Sciences and Applications, 26-29 August, Sarajevo, Bosnia Herzegovina, 2013.
[6] Srivastava H. M., Hypergeometric functions of three variables, Ganita 15, 97-108, 1964.
[7] Srivastava H. M., Some integrals representing triple hypergeometric functions, Rend.
Circ. Mat. Palermo 16, 99-115, 1967.
1 Ahi Evran University, Kirsehir, Turkey, [email protected] 2 Ahi Evran University, Kirsehir, Turkey, [email protected] 3 Ahi Evran University, Kirsehir, Turkey, [email protected]
- 282 -
Construction of Independent Spanning Trees for Bi-Rotator Graphs
Cheng-Jhe Lee1, Chiun-Chieh Hsu2, Yu-Ting Tsai3 and Yu-Chun Chu4
Abstract. The families of Cayley graphs, including star, hypercube, and rotator graphs,
have been extensively studied in recent years [1], [2], [3], [4]. All these graphs possess the
rich structural properties such as symmetry, recursive construction, small diameter, and small
degree. In addition, these properties have been proved to be very useful and outperform those
of others in terms of network transmission, computation, broadcasting, communication, fault
tolerance, and so on [2], [3], [4]. Rotator graphs were modified by adding generation
functions to make all edges bi-directional, which is called bi-rotator graphs. A bi-rotator graph
with size n (abbreviated to n-BR), a member of Cayley graphs, contains !n nodes, where each
node is labeled with a unique permutation of n distinct symbols of 1, 2, 3, …, n, and has
degree of 2n-3 [1]. (n-1)-BR subgraphs can be connected with each other via the highest order
of generation functions.
A spanning tree of a graph is composed of all nodes of the original graph and parts of
edges which connect all nodes and form no cycles. Spanning trees are said to be independent
if a directed edge can only be contained in one tree. If a number of independent spanning
trees rooted at a source node can be found, a message can then be sent to one node through
different paths via the different independent spanning trees. Moreover, fault-tolerant
broadcasting can also be achieved by utilizing these independent spanning trees.
The method proposed in this paper adopts recursive construction for finding all
independent spanning trees. With the increasing of the scale, two edges are added to each
node. Based on the 4-BR, we can construct all independent spanning trees of the 5-BR using
the rich structural properties and the added neighbors. In a similar way, we can construct
independent spanning trees of the n-BR based on the spanning tree construction of the (n-1)-
BR. Using the explored properties of symmetry and recursive construction, the paper
proposes an efficient algorithm of constructing independent spanning trees for a bi-rotator
graph. The proposed method constructs 2n-3 independent spanning trees rooted at a
designated node for an n-BR, where the number of the spanning tree is proved to be
maximum.
1 National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. [email protected] 2 National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. [email protected] 3 National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. [email protected] 4 National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. [email protected]
- 283 -
Keywords. Bi-Rotator Graph, Independent Spanning Trees Algorithm, Fault
Tolerance Broadcasting
References
[1] Corbett, P. F., Rotator graphs: An efficient topology for point-to-point multiprocessor
networks, IEEE Trans. on Parallel and Distributed Systems, 3, 5, 622-626, Sept. 1992
[2] Kuo, C. J., Hsu, C. C., Lin, H. R., Lin, K. K., An efficient algorithm for minimum feedback
vertex sets in rotator graphs, Information Processing Letters, 109, 9, 450–453, 2009.
[3] Stevens, B., Williams, A., Hamilton Cycles in Restricted and Incomplete Rotator Graphs,
Journal of Graph Algorithms and Applications, 16, 4, 785-810, 2012.
[4] Suzuki, Y., Kaneko, K., Nakamori, M., Node-Disjoint Paths Algorithm in a Transposition
Graph, IEICE Transactions on Information and Systems, E89-D, 10, 2600-2605, 2006.
- 284 -
Pre-service Mathematics Teachers’ Views of Language in Mathematics Teaching and
Their Math Literacy Self-Efficacies
Dilek Çağırgan Gülten1 and Yavuz Yaman2
Abstract. Learning and teaching mathematics requires good knowledge of language,
which is a tool used by individuals to communicate in a society. Using language in
mathematics is of great concern to the first grade primary students who are in the stage of
learning and developing a language. Mathematical literacy involves using mathematical
knowledge properly in everyday life, having an idea about the historical development of
mathematics, using the language of mathematics to communicate and problem solving skills.
The concept of self-efficacy is vital to the development of mathematical literacy.
In consideration of the above mentioned, this study aims to investigate pre-service
mathematics teachers’ views of language used in mathematics teaching and their self-
efficacies in terms of certain variables. The research data were obtained from the study
conducted with the first and fourth grade university students enrolled in the program of
Primary Mathematics Education at İstanbul University. The data were collected using the
“Language in Mathematics Teaching Scale” developed by Bali-Çalıkoğlu (2002), the “Math
Literacy Self-Efficacy Scale” developed by Özgen and Bindak (2008) and a demographical
form. While there is not a similar research study to compare with, the results of the data
analysis will pave the way for further research contributing both to the literature and the
training of mathematics teachers. The findings were discussed in the light of the literature and
some suggestions were made for further research and researchers.
Keywords. Language in Mathematics Teaching, Mathematical Literacy, Self-Efficacy.
References
[1] Bali-Çalıkoğlu, G. Matematik öğretmen adaylarının matematik öğretiminde dile ilişkin
görüşleri. Hacettepe Üniversitesi Eğitim Fakültesi dergisi, 25, 19-25, 2003.
[2] Özgen, K., Bindak, R. Matematik okuryazarlığı öz-yeterlik ölçeğinin geliştirilmesi,
Kastamonu Eğitim Dergisi, 16 (2), 517-528, 2008.
1 Istanbul University, Istanbul, Turkey, [email protected] 2 Istanbul University, Istanbul, Turkey, [email protected]
- 285 -
Fixed Points of Involutions in a Lie Algebra of the Form 𝑭/𝑹
Dilek Ersalan 1 and Naime Ekici 2
Abstract: Let 𝐹 be a free Lie algebra of rank two and 𝜑 be the automorphism of 𝐹
which is permuting the generators. If 𝑅 is a 𝜑-invaryant ideal of 𝐹 then 𝜑 induces an
automorphism 𝜑 of the Lie algebra 𝐿 = 𝐹/𝑅. We investigate for which cases the
automorphism 𝜑 has no non-trivial fixed points. Also we prove that if 𝐿 is a free metabelian
Lie algbera then 𝜑 can has non-trivial fixed points.
Keywords. Free Lie Algebra, Automorphism, Fixed Point
AMS 2010. 17B01, 17B40
References:
[1] Bryant R.M, On the fixed points of finite groups acting on a free Lie algbera, J. London
Math. Soc. 43(2), 215-214, 1991.
[2] Ekici N., Sönmez D., Fixed points of IA-endomorphisms of a free metabelizn lie algebra,
Proc. Indian Acad. Sci.(Math.Sci.)Vol.121, No.4, 405-416, 2011.
[3] Ekici N., Fixed points of certain automorphisms of free solvable lie algberas, IECMSA-
2015, Athens-Greece, 2015.
[4] Esmerligil Z., Ekici N., Fixed point of automorphisms permuting free generators of free
metabelian Lie algebras, IECMSA-2015, Athens-Greece, 2015.
[5] O.Macedonska, W.Tomoszewski, Fixed points induced by the permutation of generators
in a free group of rank 2, Infinite Groops 94, Eds.: de Giovanni/Newell, Walter de Gruyder
Co.Berlin, New York 1995.
1 Cukurova University, Adana, Turkey, [email protected] 2 Cukurova University, Adana, Turkey, [email protected]
- 286 -
Stability of vertical Cylindrical Cavity with Circular Gross –Section view of Elastic-Plastic Deformations at Small Homogenous Subcritical States
E.A.Hazar1, M.K.Cerrahoğlu2 and E. Cerrahoğlu 3
Abstract. Within a three dimensional linear stability theory, using the general solution
method, exact solutions for some problems in the case of homogenous initial tension have
been obtained. Approximate analytical solutions have also been built by employing a small
parameter method while very strong homogeneous initial plastic deformations are
present.(1),(2),(3).
Keywords. Stability, Elastic - Plasticity, Cavity, Critical Load.
References
[1] A.N. Guz. Stability of three-dimensional deformable bodies .//Edition – Nauka dumka.-К.-
1971.-C-276.
[2] A.N. Guz. Stability of elastic bodies under final deformations.// Edition – Nauka dumka,-
К.- С.270, 1973.
[3] Sporykhin A.N., Shashkin A.I. Stability of vertical mine workings in the hardening plastic
massifs.// Applied mechanics.- 10.-11.C-.76-80, 1974.
1 Kyrgyz-Turkish ManasUniversity, Manas, Kyrgyz, [email protected] 2 Sakarya University, Sakarya, Turkey. [email protected] 3 Kocaeli University, Kocaeli, [email protected]
- 287 -
The Mathematical Equations of a New Five Phases Segmental
Switched Reluctance Motor
Erdal Büyükbıçakcı1 and Ali Fuat Boz2
Abstract. In this study, the mathematical equations of five phase segmental switched
reluctance motor (SARM) was established by using basic electrical motor model. Phase
currents and magnetic flow variations with respect to prominent inductance of SARM which
has different rotor structure were calculated. In addition, the momentum equation of SARM
was obtained by determining the situations of different phases. It was understood that the
magnetic flow equations depend not only on the function of rotor position, but also the
changing current.
Keywords. Switched Reluctance Motor, Mathematical Model, State Equation
AMS 2010. 93A30, 97M50
References
[1] Uygun, D., Bal, G., Sefa, I., Linear model of a novel 5-phase segment type
switched reluctance motor, Elektronika Ir Elektrotechnika Journal, Vol.20, pp.3–7, 2014.
[2] Büyükbıçakcı, E., Design and applicaiıon of PI controlled drive system for a bipolar excıted segmental 5-phase switched reluctance motor, PhD Thesis, Sakarya University, 2013
[3] Zhen Zhong Ye, Martin, T.W., Balda, J.C. An An investigation of multiphase excitation
nodes of a 10/8 switched reluctance motor with short flux pathto maximize its average torque,
Industrial Electronics IEEE International Symp.,Vol.2, pp.390 – 395, 2000.
[4] Thachuk, V., Klytta, M. Switched reluctance motor and its mathematical model,
Compatibility in Power Electronics, CPE’07, pp.1 – 5, 2007.
[5] Edrington, C.S., Fahimi, B., Bipolar switched reluctance machines, IEEE Pow. Eng. Soc.
Gen. Meeting, vol.2, pp. 1351-1358, June 2004.
[6] Chen, Xiaoyuan, Zhiquan Deng, Xiaolin Wang, Peng, Jingjing, Li, Xiangsheng, New designs of switched reluctance motors withsegmental rotors, 5th IET International Conference Power Electronics, pp.1–6, 2010.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 288 -
Dynamic Simulation Results of a New Five Phases Segmental
Switched Reluctance Motor
Erdal Büyükbıçakcı1, Ali Fuat Boz2 and Zeynep Büyükbıçakcı1
Abstract. In this study, the dynamic simulation results of a new five phases segmental
switched reluctance motor were developed by using gained statement equations of them. In
computer media, the SARM was rotated from 0° to 90° by 1° increment in order to see the
change of function of current by using algorithmic code parts. In every rotation angle of
SARM, the adjacent two phases were energized to generate common inductance. In analyses,
the variation of obtained phase currents was explained by simulating and visualizing every
current.
Keywords. Switched Reluctance Motor, Mathematical Model, Simulation
AMS 2010. 93A30, 93B40
References
[1] Mecrow, B.C. ; Finch, J.W. ; El-Kharashi, E.A. ; Jack, A.G., Switched reluctance motors
with segmental rotors, Electric Power Application, pp. 245 – 254, 2002.
[2] Uygun, D., Bal, G., Sefa, I., Linear model of a novel 5-phase segment type
switched reluctance motor, Elektronika Ir Elektrotechnika Journal, Vol.20, pp.3 – 7, 2014.
[3] Büyükbıçakcı, E., Design and appiıcation of PI controlled drive system for a bipolar excited segmental 5-phase switched reluctance motor, PhD Thesis, Sakarya University, 2013.
[4] Edrington, C.S., Fahimi, B., Bipolar switched reluctance machines, IEEE Pow. Eng. Soc.
Gen. Meeting, vol.2, pp. 1351-1358, June 2004.
[5] Chen, Xiaoyuan, Zhiquan Deng, Xiaolin Wang, Peng, Jingjing, Li, Xiangsheng, New
designs of switched reluctance motors withsegmental rotors, 5th IET International Conference
Power Electronics, pp.1 – 6, 2010.
[6] J.M. Kokernak, D.A. Torrey, Magnetic circuit model for the mutually coupled switched
reluctance machine, IEEE Transactions on Magnetics, vol.36, pp. 500–507, 2000.
1 Sakarya University, Sakarya, Turkey, [email protected] , [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 289 -
Investigation of the Effect of Project – Based Learning Method on Academic Success for
Mathematic Course in Higher Education
Erdal Büyükbıçakcı1 and Zeynep Büyükbıçakcı1
Abstract. The aim of this study is to investigate the effect of project-based learning
method on academic success for mathematic course by comparing the traditional learning
methods and applying ideas of students the in higher education. A qualitative type research
was applied to first year students of Sakarya University Vocational School of Karasu
Department of Computer Technologies. In first semester, the Mathematic course was done by
traditional education, and in second semester, the project-based learning method was applied
to students. After that, the students were questioned to compare the two learning methods by
completing a questionnaire form. At the end of the study, the project-based learning method
was seen more favourable than traditional one.
Keywords. Project-Based Learning, Mathematic Education, Traditional Learning
AMS 2010. 97B40, 97B70.
References
[1] Han, S., Capraro, R., Capraro, M., How science, technology, engineering, and mathematics (stem) project-based learning (pbl) affects high, middle, and low achievers differently: the ımpact of student factors on achıevement, International Journal of Science and Mathematics Education, National Science Council, Taiwan 2014.
[2] Ay, Ş., Ttraınee teachers’ vıews on project-based learnıng and tradıtıonal educatıon, Hacettepe University Journal of Education, vol.28(1), pp. 53-67, 2013.
[3] Kalaycı, N., An application related to project based learning in higher education analysis in terms of students directing the project, Education and Science vol.33-147, pp.85-105, 2008.
[4] Natasha, M.S., John, P. S., Aladar, H., Collegiate mathematics teaching: An unexamined practice, The Journal of Mathematical Behavior, vol.29, pp. 99-114, 2010.
[5] Meredith, A.P.R., Dionne, I.C., Melissa, S.G., Amy, E.T., Gayle, A.B., First year implementaiıon of a project-based learning approach: the need for addressing teachers’ oiıentaiıons in the era of reform, International Journal of Science and Mathematics Education, National Science Council, Taiwan 2011.
1 Sakarya University, Sakarya, Turkey, [email protected], [email protected]
- 290 -
On the Existince of Bertrand Curves in Dual Space 𝑫𝟒
Emel Karaca1 and Mustafa Çalışkan2
Abstract. Bertrand curves were defined and some characterizations for these curves
were studied in 𝐼𝐼3 by J. Bertrand in 1850. According to definition, a Bertrand curve in 𝐼𝐼3
is a curve such that its principal normal vectors are the principal normal vectors of an other
curve. Aminov proved that a Bertrand curve does not exist in 𝐼𝐼𝑛 if n ≥ 4. In this study, we
defined Bertrand curves in 𝐼𝐼4 and proved that the curves whose principal normal vectors are
pure dual vectors are Bertrand curves on the unit dual sphere in 𝐼𝐼4 . We gave two
conclusions under special conditions in 𝐼𝐼4. One of them is that two dual curves are Bertrand
curves if 𝜆1 and 𝜆2 are obtained pure reel. The other is that there is no curve whose principal
normal vectors are linear dependent if the dual curves and 𝛽 are unique dual vectors.
Keywords. Dual Frenet Frames, Bertrand Curves, Dual Space.
AMS 2010. 53A04
References
[1] Kim, C.Y., Park, J., Yorozu, S., Curves On The Unit 3-Sphere In Euclidean 4-Space, Bull.
Korean Math. Soc. 50, No. 5, pp. 1599-1622, 2013.
[2] Karaca, E., Çalışkan, M., Some Characterizations For Geodesic Sprays In Dual Space, International Journal of Engineering and Applied sciences pp 59-63 Vol 04. No. 10, 2014.
[3] Hacısalihoğlu, H. H., Hareket geometrisi ve kuaterniyonlar teorisi, Gazi Üniversitesi, Fen-
Edebiyat Fakultesi Yayinlari 2, 1983.
[4] Hacısalihoğlu, H. H., Diferensiyel Geometri, Ankara Üniversitesi Fen Fakültesi, 2000.
[5] Güven, İ. A., Ağaoğlu, İ., The Properties of Bertrand Curves In Dual Space, Cornell
University Library, 2013.
[6] Aminov, Yu., Differential Geometry and Topology of Curves, Gordon and Breach Science
Publishers, Amsterdam, 2000.
1 Gazi University, Ankara, Turkey, [email protected]. 2 Gazi University, Ankara, Turkey, [email protected].
- 291 -
Surface Growth Kinematics in Minkowski 3 Space
Gül Güner1, Zehra Özdemir2 and F. Nejat Ekmekci3
Abstract. In [1], the authors developed the modelling of the surface growth by taking
a curve evolving in space. Such a curve generate some surfaces like seashells and horns. We
generalize this modelling to the curves in Minkowski 3 Space. Hence, we can analyze the
surface growth by means of space and time.
Keywords. Frenet Frame, Growth Velocity, Seashell, Mathematical Model.
AMS 2010. 53A35, 92B99.
References
[1] D. E. Moulton, A. Goriely, Surface growth kinematics via local curve evolution, J. Math.
Biol., 68:81-108, 2014.
[2] D. E. Moulton, A. Goriely, R. Chirat, Mechanical growth and morphogenesis of seashells,
OCCAM, Report Number 12/01.
[3] R. J Low, Framing Curves in Euclidean and Minkowski Space, Journal of Geometry &
Symmetry in Physics, 2012.
[4] A. Yücesan, A. C. Çöken, N. Ayyıldız, On the Darboux rotation axis of Lorentz space
curve, Applied Mathematics and Computation 155, pp. 345–351, 2004.
1 Karadeniz Tech. University, Trabzon, Turkey, [email protected] 2 Ankara University, Ankara, Turkey, [email protected] 3 Ankara University, Ankara, Turkey, [email protected]
- 292 -
On the Codimension-Two and -Three Bifurcations of a Food Web of Four Species
Hsiu-Chuan Wei1, Yuh-Yih Chen2, Shin-Feng Hwang3 and Jenn-Tsann Lin4
Abstract. This work is concerned with codimension-two and -three bifurcations of a
food web developed by Bockelman and Deng [1]. It contains a bottom prey X, two competing
predators Y and Z on X, and a super predator W only on Y. Parameter conditions for a part of
codimension-two bifurcations and a codimension-three bifurcation are derived. Three-
parameter bifurcation diagrams are computed using an adaptive grid method [2,3] to locate
the bifurcations determined by the eigenvalues of equilibria.
Keywords. Food Web, Bifurcation, Numerical Computation.
AMS 2010. 34D20, 37C35, 92D25, 92D40.
References
[1] Bockelman, B., Deng, B., Food web chaos without subchain oscillators, Int. J. Bifurcation
and Chaos, 15, 3481-3492, 2005.
[2] Wei, H. C., On the bifurcation analysis of a food web of four species, Appli. Math.
Comput., 215, 3280-3292, 2010.
[3] Wei, H. C., A modified numerical method for bifurcations of fixed points of ODE systems
with periodically pulsed inputs, Appli. Math. Comput., 236, 373-383, 2014.
1 Department of Applied Mathematics, Feng Chia University, TaiChung, Taiwan, hsiucwei@ fcu.edu.tw 2 Department of Applied Mathematics, Feng Chia University, TaiChung, Taiwan, [email protected] 3 Department of Applied Mathematics, Feng Chia University, TaiChung, Taiwan, [email protected] 4 Department of Applied Mathematics, Feng Chia University, TaiChung, Taiwan, [email protected]
- 293 -
Algorithms for Calculating the Limits of Convergent Infinite Series
Héctor Luna García1, Luz María García Cruz2 and A. E. García3
Abstract. We use algorithms to calculate the exact limits of a wide range of
convergent infinite series by means of special functions, this are polygamma functions.
However, in the case of alternating series, these algorithms do not allow the use of such
functions, but allow them to find the limits of this series. Finally, these methods are used as a
powerful and simple tool for calculating the limits of many infinite series as shown in the
examples included.You write the text of your abstract.
Keywords. Polygamma Functions, Convergent Infinite Series, Laplace Transform.
AMS 2010. 33E50, 40A05, 44A10.
1 Universidad Autónoma Metropolitana-Unidad Azcapotzalco, México City, México, [email protected] 2 Universidad Autónoma Metropolitana-Unidad Azcapotzalco, México City, México, [email protected] 2 Universidad Autónoma Metropolitana-Unidad Azcapotzalco, México City, México, [email protected]
- 294 -
An Extension of Srivastava's Triple Hypergeometric Function HB
İ. O. Kıymaz1, M. B. Yağbasan2 and A. Çetinkaya3
Abstract. Recently, some extensions of the well-known special functions such as
hypergeometric, confluent hypergeometric and Mittag-Leffler have been studied by several
authors. The main object of this study is to introduce an extension of Srivastava's triple
hypergeometric function HB by using extended beta function. Some integral representations
are also obtained for this extension.
Keywords. Extended Beta Function, Srivastava's Triple Hypergeometric Function HB,
Integral Representations.
AMS 2010. 33C60, 33C65, 33C70
Acknowledgement: This work was supported by Ahi Evran University Scientific
Research Projects Coordination Unit. Project Number: PYO-FEN.4001.14.014
References
[1] Chaudhry M. A., Qadir A., Rafique M., Zubair S. M., Extension of Euler's beta function,
J. Comput. Appl. Math. 78, 19-32, 1997.
[2] Chaudhry M. A., Qadir A., Srivastava H. M., Paris R. B., Extended hypergeometric and
confluent hypergeometric functions, Appl. Math. Comput. 159, 589-602, 2004.
[3] Choi J., Hasanov A., Srivastava H. M., Turaev M., Integral representations for
Srivastava's triple hypergeometric functions. Taiwanese J. Math. 15, no:6, 2751-2762, 2011.
[4] Choi J., Hasanov A., Turaev M., Integral representations for Srivastava's hypergeometric
function HB. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 19, no. 2, 137–145, 2012.
[5] Çetinkaya, A., Yağbasan, M. B., Kıymaz, İ. O., An Extension of Srivastava's Triple
Hypergeometric Function HA, 2nd International Eurasian Conference on Mathematical
Sciences and Applications, 26-29 August, Sarajevo, Bosnia Herzegovina, 2013.
[6] Srivastava H. M., Hypergeometric functions of three variables, Ganita 15, 97-108, 1964.
[7] Srivastava H. M., Some integrals representing triple hypergeometric functions, Rend.
Circ. Mat. Palermo 16, 99-115, 1967.
1 Ahi Evran University, Kırşehir, Turkey, [email protected] 2 Ahi Evran University, Kırşehir, Turkey, [email protected] 3 Ahi Evran University, Kırşehir, Turkey, [email protected]
- 295 -
Canonical Transformations to the Schrodinger Equation: Hypergeometric Solutions and
Exponential-Type Potentials
J. J. Peña1 , J. Morales2 and J. García-Ravelo3
Abstract. In this work, a direct point canonical transformation and a gauge
transformation is applied to the Schrödinger equation to convert it into a hypergeometric
differential equation. This method leads to exactly solvable multiparameter exponential-type
potentials with eigen-functions given in terms of the hypergeometric function. The proposal is
general because it permits to identify one-dimensional exponential potentials as well as radial
exponential potentials by means of a simple choice of the involved parameters. As useful
applications, the method shows the treatment of some specific exponential-type potentials for
the cases of singular and non-singular exponential potentials, such as Wood-Saxon [1] and
Schiöberg [2] potentials, among others, all of them with hypergeometric-type solutions.
Keywords. Canonical Transformation, Hypergeometric Solution, Exponential-Type
Potentials.
AMS 2010. 44A203, 81Q05.
References
[1] Berkdemir A., et al. Eigenvalues and eigenfunctins of Wood-Saxon potential in PT-
symmetric quantum mechanics. Mod. Phys Lett. A, Vol. 21, 20-87, 2006.
[2] Ortakaya S., Non relativistic l-state solutions for Schiöberg molecular potential in
hyperspherical coordinates, Few-Body Syst 54:1901-1909. DOI 10.1007/s00601-013-0712-3
1 Universidad Autónoma Metropolitana Azc. México. [email protected] 2 Universidad Autónoma Metropolitana Azc. México. [email protected] 3 Instituto Politécnico Nacional ESFM, México. [email protected]
- 296 -
Siacci’s Resolution of the Acceleration Vector for a Non-Null Space Curve in
Minkowski Space
Kahraman Esen Özen1, Murat Tosun2 and Mahmut Akyiğit3
Abstract. In [1], a resolution of the acceleration vector is well known by Siacci for
motion of a material point along a space curve, In this resolution, the acceleration vector is
expressed as the sum of two special oblique components in the osculating plane to the curve.
In this paper, we have studied the Siacci’s theorem for the curves defined in Minkowski
space. Also, an example is given for a helix lying on a cylinder.
Keywords. Siacci’s theorem, Minkowski space, kinematics.
AMS (2010). 51B20, 53A17.
References
[1] F. Siacci, Moto per una linea gobba. Atti R Accad Sci. Torino, 14, 946-951, 1879.
[2] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
[3] S. P. Radzevich, Geometry of Surfaces: A practical Guide for Mechanical Engineers , Wiley, 2013.
[4] J. Casey, Siacci’s Resolution of the Acceleration Vector for a Space Curve. Meccanica, 46, 471-476 DOI: 10.1007/s11012-010-9296-x, 2011.
[5] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn.Cambridge University Press, Cambridge. Dover, New York, 1944.
[6] N. Grossman, The Sheer Joy of Celestial Mechanics. Birkhauser, Basel, 1996.
[7] M.Yıldırım Yılmaz, M. Bektaş, Z. Küçükarslan, Siacci’s Theorem for Curves in Finsler Manifold , 3F Turkish Journial of Science and Technology, 7, No. 2, 181-185, 2012.
[8] J.Walrave, Curves and surfaces in Minkowski space. Dissertation, K. U. Leuven, Fac. of Science, Leuven,1995.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 297 -
Two Parameter Homothetic Motions on the Galilean Plane
Muhsin Çelik1 and Mehmet Ali Güngör2
Abstract. The one-parameter motions on the Galilean plane, the relations between
absolute, relative, sliding velocities and accelerations and pole curves were studied in ref [9].
In this study, sliding velocity, pole lines, Hodograph and acceleration poles of two parameter
homothetic motions on the Galilean plane at ( , )∀ λ µ positions are obtained. Some
characteristic properties about the velocity vectors, the acceleration vectors and the pole
curves are given.
Keywords. Two Parameter Motion, Shear Motion, Galilean Plane, Planar Kinematics,
Homothetic Motion.
AMS 2010. 53A17, 53A35, 53A40.
References [1] Müller H.R. Zur Bewegunssgeometrie in Räumen Höherer Dimension. Monotshefte für Math. 70 Band, 1 Heft. 47-57, 1966.
[2] Nomizu K., Kinematics and Differential Geometry of Submanifolds, Tohoku Math. Journ. 30(4), 623-637, 1978.
[3] Blaschke W., Müller H.R. Ebene Kinematik, Verlag Oldenbourg, München, 1956.
[4] Karacan M.K., Yaylı Y., Special Two Parameter Motion in Lorentzian Plane. Thai Journal of Mathematics 22, 239-246, 2004.
[5] Çelik M., Ünal D., Güngör M.A., On the Two Parameter Lorentzian Homothetic Motions, Advances in Difference Equations, Vol.42, pp. 1-20 - 20, 1, 2014.
[6] Ünal D., Çelik M., Güngör M.A., On the Two Parameter Homothetıc Motıons in Complex Plane, TWMS J. Pure Appl. Math., Vol. 4.2 ,pp. 204 - 214, 2013.
[7] Çelik M., Güngör M.A., Hiperbolik Düzlemde İki Parametreli Homotetik Hareketler Üzerine Bir Çalışma, Beykent University Journal of Science and Engineering, 7(2), 1-20, 2014.
[8] Yaglom I.M., A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York, 1979.
[9] Akar M., Yüce S., Kuruoğlu N., One-parameter planar motion on the Galilean plane, International Electronic Journal of Geometry, vol.6, no.1, 79-88, 2013.
[10] Tütüncü E. E., The geometry of motions in the Galilean Space, Ph.D.Thesis, Ankara University, Institute of Natural Sciences, 2009.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 298 -
Exact Traveling Wave Solutions of some Nonlinear Evolution Equations
Meryem Odabaşı1 and Emine Mısırlı2
Abstract. In nonlinear sciences, it is important to obtain traveling wave solutions of
nonlinear evolution equations to understand the phenomena they describe. In this study, we
obtained the exact traveling wave solutions of the two-dimensional Bratu equation,
generalized heat conduction equation, generalized Benjamin-Bona-Mahony equation and
coupled nonlinear Klein-Gordon equations by means of the trial equation method and the
complete discrimination system. This method is reliable, effective and enables to get soliton,
single-kink and compacton solutions of the generalized nonlinear evolution equations and
systems of equations.
Keywords. Generalized Heat Conduction Equation, Generalized Benjamin-Bona-
Mahony Equation, Coupled Nonlinear Klein-Gordon Equations, Trial Equation Method.
AMS 2010. 35Q79, 35Q35, 35Q40, 35A25.
References
[1] Boyd, J.P., An analytical and numerical study of the two-dimensional Bratu equation,
Journal of Scientific Computing, 1, 183-206, 1986.
[2] Kabir, M.M., Analytic solutions for generalized forms of the nonlinear heat conduction
equation, Nonlinear Analysis: Real World Applications, 12, 2681-2691, 2011.
[3] Li, W. and Zhao, Y.M., Exact solutions for a BBM (m,n) equation with generalized
evolution, Applied Mathematical Sciences, 6, 27, 1325-1334, 2012.
[4] Alagesan, T., Chung, Y. and Nakkeeran, K., Soliton solutions of coupled nonlinear Klein–
Gordon equations, Chaos, Solitons and Fractals, 21, 879–882, 2004.
[5] Liu, C.S, A new trial equation method and its applications, Communications in
Theoretical Physics, 45, 3, 395-397, 2006.
[6] Odabasi, M. and Misirli, E., On the solutions of the nonlinear fractional differential
equations via the modified trial equation method, Mathematical Methods in the Applied
Sciences, DOI: 10.1002/mma.3533, 2015.
1 Ege University, Tire Kutsan Vocational School, Izmir, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
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On Reflections and Cartan-Dieudonné Theorem in Minkowski 3-space
Mustafa Özdemir1 and Melek Erdoğdu2
Abstract. In this paper, we investigate the reflections in Minkowski 3-space by three
different appoach. Firstly, we define Lorentzian reflections with Lorentzian inner product.
Then, we examine Lorentzian reflections in view of Lorentzian Householder matrices.
Finally, we use pure split quaternions to derive Lorentzian reflections. For each case, we find
the matrix representation of Lorentzian reflections and characterize the plane of reflection by
using this matrix representation. Morever, we prove the Cartan-Dieudonne Theorem in the
Minkowski space.
Keywords. Minkowski Space, Reflections, Rotations, Cartan-Dieudonne Theorem.
AMS 2010. 5B10, 15A16, 53B30
References
[1] Özdemir M., Erdoğdu M., Şimşek H., On the Eigenvalues and Eigenvectors of a
Lorentzian Rotation Matrix by Using Split Quaternions. Adv. Appl. Clifford Algebras 24,
179-192, 2014.
[2] Özdemir, M., A.A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space,
Journal of Geometry and Physics. 56, 322-336, 2006.
[3] Özdemir, M., The Roots of a Split Quaternion, Applied Mathematics Letters. 22, 258-263,
2009.
[4] Gonzalez, G. A., Aragon, J. L., Rodriguez-Andrade, M. A., Verde, L., Reflections,
Rotations and Pythagorean Numbers, Adv. Appl. Clifford Algebras. 19, 1-14, 2009.
[5] Gonzalez, G. A., Aragon, J. L., Rodriguez-Andrade, M. A., Pythagorean Vectors and
Clifford Numbers, Adv. Appl. Clifford Algebras. 21, 247-258, 2011.
[6] Rodriguez-Andrade, M. A., Gonzalez, G. A., Aragon, J. L., Verde, L., An Algorithm for
Cartan-Dieudonne Theorem on Generalized Scalar Product Space, Linear Algebra and Its
Applications. 434, 1238-1254, 2011.
1 Akdeniz University, Antalya, Turkey, [email protected], [email protected] 2 Necmettin Erbakan University, Konya, Turkey, [email protected]
- 300 -
Analytical Problem Solution About Initial Step of Pressing Powder Material Tube
M.Ya. Flax1, A.V. Bochkov2, V.A. Goloveshkin3 and A.V.Ponomarev4
Abstract. Prediction of finite size in the process of hot isostatic pressing (HIP) of
powder material tubes is a difficult task which is important for practical purposes. In this
paper, we propose an analytical problem solution about initial step of the process.
Full problem statement is represented as follows [1-2]. Suppose that in the area of the
cylindrical coordinate system area 21 RrR ≤≤ , 43 RrR ≤≤ (capsule) filled by plastically
incompressible material, area 32 RrR ≤≤ filled by plastically compressible powder
material. It is assumed that the deformation rate zε is constant throughout the entire volume.
The initial moment of the pressing process is considered, so it is assumed that the density is
constant throughout the entire volume. Let ( )ru - radial displacement speed.
Steady-state equation: 0rrddr r
ϕσ σσ −+ = .
The equation of the yield surface for the powder material is taken in the form of
Green:
( ) ( ) ( ) ( )2
2 2 2 22 2
2 1
1 2 2 29 6
r zr z r z z r T
f fϕ
ϕ ϕ ϕ
σ σ σσ σ σ σ σ σ σ σ σ
+ + + − − + − − + − − = .
The equation of the yield surface to plastically incompressible material has the form:
( ) ( ) ( )2 2 2 21
1 2 2 26 r z r z z r Tϕ ϕ ϕσ σ σ σ σ σ σ σ σ − − + − − + − − =
.
Solution of the problem:
Using the flow law and equilibrium equation to determine the speeds of movement in
the powder material have the following equation:
( ) ( ) ( )2 2 22rz z z z r z z
dd C D C D D D C D D Ddr dr
ϕϕ ϕ ϕ ϕ
εεε ε ε ε ε ε ε ε ε ε ε + + + + + − + − − +
( ) ( )2 2 2 2 0rr z r r z zC D
rϕ
ϕ ϕ ϕ
ε εε ε ε ε ε ε ε ε ε
− + + + + + + = .
The parametric representation of its solution has the form:
1 Moscow State University of Information Technologies, Moscow, Russia, [email protected] 2 Moscow State University of Information Technologies, Moscow, Russia, [email protected] 3 Moscow State University of Information Technologies, Moscow, Russia, [email protected] 4 Moscow State University of Information Technologies, Moscow, Russia, [email protected]
- 301 -
( ) ( ) ψψψδδψψψδ cossin 00
0
−+−=
shchRr ,
( ) ( )( )
+−−−
−+−=
ααψψδγ
ψψψδδψψψδε
cos1cossh
cosshsinchRu 0
00
0z ,
( )
−−−−= 2
2
0z 21sh
δδψψδγεεϕ ,
( ) ( ) ( )( )
−+−
++
−−−−= 0022
2
0zr shtgch12
21sh ψψδδψψψδ
δγδ
δδψψδγεε , 0;
2πψ ≤
.
Investigation of this solution allows to reveal the various modes of deformation
process.
Keywords. Powder Material. Green's Condition. Hot Isostatic Pressing. Analytical
Solution. Predictive Analytics
AMS 2010. 74Cxx
References
[1] Anokhina A.V., Goloveshkin V.A., Pirumov A.R., Flax M.Ya., Research of the initial step
of pressing powder material tube with the vertical shrinkage, Composite Mechanics and
Design, All-Russian scientific journal, том 9, 2, 2003.
[2] Flax M.Ya., Axisymmetric Process of Plastic Compressible Materials in Inhomogeneous
Stress State Conditions, Ph.D. thesis in Technical Science. Moscow State Academy of
Instrument Engineering and Computer Science, 2004.
- 302 -
X-ray and Computational Studies of (2Z, 3E)-3-(((E)-3-Ethoxy-2-
Hydroxybenzylidene)Hydrazono)Butan-2-One Oxime
Nezihe Çalışkan1, Çiğdem Yüksektepe Ataol2, Hümeyra Batı3, Numan Kurban1, Pelin
Kurnaz3 and Güler Ekmekçi4
Abstract. (2Z, 3E)-3-(((E)-3-Ethoxy-2-Hydroxybenzylidene)Hydrazono)Butan-2-One
Oxime (1) has been synthesized and characterized by IR, UV/vis and X-ray diffraction. The
molecular structure of the title compound in the ground state (in vacuo) were optimized by
Density Functional Theory (DFT) to include correlation corrections with the 6–311G(d,
p) and B3LYP/6-31G basis sets. In DFT calculations, hybrid functionals are also used,
including the Becke’s three-parameter functional (B3) [1], which defines the exchange
functional as the linear combination of Hartree-Fock, local, and gradient-corrected exchange
terms. The B3 hybrid functional was used in combination with the correlation functionals of
Lee et al. [2]. In addition to the experimental studies, the optimized structure, vibrational
parameters, chemical shifts, molecular orbital energies, thermodynamic properties, ionization
energy, electron affinity, electronegativity, global chemical hardness and chemical softness of
the molecule have been investigated by using DFT. The HOMO and LUMO energies were
calculated by time-dependent TD-DFT approach. The experimental results of the compound
have been compared with theoretical results and it is found to show good agreement with
calculated values. Single crystal X-ray results show that 1 crystallizes in the monoclinic
system, space group P21/c.
Keywords. Oxime, Hydrazone, DFT.
References
[1] Becke, A. D., Density-functional thermochemistry. III. The role of exact Exchange, J.
Chem. Phys. 98, 5648-5652, 1993.
[2] Lee, C., Yang, W., Parr, R. G., Development of the Colle-Salvetti correlation-energy
formula into a functional of the electron density. Phys. Rev. B37, 785-789, 1988.
1 Gazi University, Ankara, Turkey, [email protected] 2 Cankiri Karatekin University, Cankiri, Turkey, [email protected] 3 Ondokuz Mayis University, Samsun, Turkey, [email protected] 4 Gazi University, Ankara, Turkey, [email protected]
- 303 -
Fixed Points of Certain Automorphisms of Free Solvable Lie Algebras
Naime Ekici1
Abstract: Let 𝐹𝑛 be a free Lie algebra of finite rank 𝑛 (𝑛 > 1) over a field 𝐾 and 𝜃 be
an automorphism of 𝐹𝑛 of finite order 𝑛 which has no nontrivial fixed points and permutes
the generators of 𝐹𝑛 cyclically. We describe form of the fixed points of an automorphism of
the free solvable Lie algebra 𝛿𝑚−1(𝐹𝑛)/𝛿𝑚(𝐹𝑛) which is induced by 𝜃, where 𝛿𝑚(𝐹𝑛) is the
m-th derived term and 𝜔 ∈ 𝛿𝑚−1(𝐹𝑛)/𝛿𝑚(𝐹𝑛). Form of the fixed point subalgebras of an
automorphism of a free metabelian Lie algebra were given by N. Ekici and Z. Esmerligil. We
generalize this result for solvable Lie algebras. This work is an extension of results of W.
Tomaszewski in group theory for Lie algebras.
Keywords. Free Solvable Lie Algebra, Automorphism, Fixed Point
AMS 2010. 17B01, 17B40
References
[1] Bryant R.M, On the fixed points of finite group acting on a free Lie algebra, J. London
Math. Soc. 43 (2), 215-224, 1991.
[2] Bryant,R.M., Papistas A.I, On the fixed points of a finite group acting on a relatively free
Lie algebra, Glasg.Math.J. 42,167-181, 2000.
[3] Drensky V., Fixed algebras of residually nilpotent Lie algebras, Proc. Amer. Math.
Soc.120, 1021-1028, 1994.
[4] Ekici N., Sönmez D., Fixed points of IA- endomorphisms of a free metabelian Lie algebra,
Proc. Indian Acad. Sci.(Math. Sci.) Vol. 121, No. 4, 405-416, 2011.
[5] Esmerligil Z., Ekici N., Fixed point of automorphisms permuting free generators of free
metabelian Lie algebras, IECMSA-2015, Athens-Greece.
[6] Witold T., Fixed points of automorphisms preserving the length of words in free solvable
groups, Arch.Math.,99, 425-432, 2012.
1 Cukurova University, Adana, Turkey, [email protected]
- 304 -
Generalizations of Sherman's Inequality by Lidstone's Interpolating Polynomial
Ravi P. Agarwal1, Slavica Ivelić Bradanović2 and Josip Pečarić3
Abstract. In majorization theory, the well known Majorization theorem plays a very
important role. More general result is obtained by S. Sherman. In this paper, concerning 2n-
convex functions, we get generalizations of these results applying Lidstone's interpolating
polynomials and Čebyšev functional. Using obtained results, we generate new familly of
exponentially convex functions. The outcome are some new classes of two-parameter
Cauchy-type means.
Keywords. Majorization, n-convexity, Schur-convexity, Sherman's Theorem,
Lidstone Interpolating Polynomial, Čebyšev Functional, Grüss Type Inequalities, Ostrowsky-
Type Inequalities, Exponentially convex Functions, Log-convex Functions, Means
AMS 2010. Primary 39B82; Secondary 44B20, 46C05.
References
[1] R. P. Agarwal, P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and their
Applications, Kluwer Academic Publisher, Dordrecht, 1993.
[2] P. Cerone, S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional
and applications, J. Math. Inequal. 8 (1), 159-170, 2014.
[3] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, 2nd
ed., Cambridge 1952.
[4] J. Jakšetić, J. Pečarić, Exponential Convexity Method, J. Convex Anal. 20, No. 1, 181-197,
2013.
[5] J. E. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings, and
Statistical Applications, Academic Press, Inc.
[6] S: S. Sherman, On a theorem of Hardy, Littlewood, Pólya and Blackwell, Proc. Nat. Acad.
Sci. USA, 37 (1), 826-831, 1957.
1 University-Kingsville, Texas, [email protected] 2 University of Split, Matice, Croatia, [email protected] 3 University of Zagreb, Zagreb, Croatia, [email protected]
- 305 -
A simple State Observer Design for Linear Dynamic Systems by Using Taylor Series
Approximation
Saadettin Aksoy 1 and Hakan Kizmaz 2
Abstract. State variables that determine a system’s dynamics should be known for
analysis and control of dynamical systems [1-2]. Specifically, dynamics feedback for pole
placement is required. Furthermore, estimation of state variables in real time is a very
important problem in adaptive control applications [3]. Unfortunately, all of the state variables
cannot be measured in practice. As a result, use of a suitable state observer or estimator is
unavoidable in order to obtain immeasurable state variables. There exit a variety of state
observers in the literature [4-5]. Implementation of state observers that use only input and
output measurements of the systems are carried out via solution of the observer state integral
equations pertinent to the observer. There are several numerical solution algorithms for a
solution of the observer state integral equations in the literature [6]. Even though the Runge-
Kutta numerical integration algorithm is frequently used for this purpose, it has several
drawbacks that depend on the step-size h. First, accuracy gets poorer as h increases. Second,
computation time becomes an issue if h is too small. Third, round-off errors may become
important for small values of h because the number of cycles required to cover the desired
time interval [0, t] increases. Note that equations are evaluated for each t in the interval [0, t]
in all of the above mentioned algorithms.
In this study, a simple general algorithm is proposed for state variables estimation of
linear, time-invariant multi-input multi output systems. The proposed algorithm is based on
Taylor series approximation and has an analog solution. The solution that results from the
proposed algorithm gets closer to the true solution when more and more terms are kept in the
Taylor series. Finally, the proposed method gives the approximate solution of the estimation
vector ˆ (t)x as a function of time in the interval [0,t]. Consequently, computation of the state
integral equations for each t is eliminated. The Taylor series are defined on the interval
t∈[0,1] and have the orthogonality property like the Walsh, Chebyshev and Legendre series
[7-8]. The proposed algorithm uses some important properties such as the operational matrix
of integration for Taylor vector [9-10]. The algorithm consists of four steps. In the first step,
the feedback gain matrix G, which will force the estimation error to go to zero in a short time,
is determined by using a suitable method [4]. In the second step, the observer state equation is
1 Siirt University, Siirt, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 306 -
converted into integral equation by integrating the terms on either side of the equation. After
some algebraic manipulations, the time dependent terms on either side of the integral equation
are removed. Hence, the problem is reduced to a set of nonlinear equations with constant
coefficients. System outputs are used by the observer equations. Therefore, we have to
calculate it’s as the function of time. They can obtained from plant output measurement by
using curve fitting methods such as Linear Least Squares, Lebenberg-Marquardt and Gauss-
Newton [11]. Finally, in the last step, nonlinear equations for unknown state vector are
converted into a recursive form whose solution can be obtained easily by a computer program.
The proposed estimation algorithm was implemented in MATLABTM and it was applied to
different cases. Results obtained by the proposed algorithm are in harmony with the real
results.
Keywords. State Estimation, Taylor Series, State Observers, Curve Fitting
References
[1] Daughlas B. Miron, Design of Feedback Control Systems, Harcourt Brace Jovanovic Inc.,
USA, 1989.
[2] Brasch, F.M., and Pearson, J. B., Pole Placement Using Dynamic Compensators, IEEE
Trans. Automatic Control, AC -15, pp. 34-43, 1970.
[3] Aström, K.J., and Wittermark, B., Adaptive Control, Addison Wesley Pub. Inc., USA,
1989.
[4] Kailath, T., Linear Systems, Prentice-Hall Inc., Tokyo, 1980.
[5] O. Reilly, J., Observers for Linear Systems, London Academic Press, London,1983.
[6] Hildebrand,F.B., Introduction to Numerical Analysis, 2d edition, McGraw-Hill, New
York, 1937.
[7] G. Sansone, Orthogonal Functions, Interscience Publishers, Inc., New York, 1991.
[8] Cheng-Chilan Liu and Y. P. Shih, Analysis and optimal control of time-varying systems
via Chebyshev polynomials, Int. J. Control, Vol.38, No.5, pp. 1003-012, 1983.
[9] Spares P.D., and Moutroutsas S. G., Analysis and Optimal Control of Time-Varying
Linear Systems via Taylor Series, Int. Journal Cont., Vol.41, pp. 831-842, 1985.
[10] Mouroutsos S. G. and Sparis P. D., Taylor Series Approach to System Identification
Analysis and Optimal Control, Journal of Frank. Inst., vol. 319, no. 3, pp. 359-371, 1985.
[11] Steven C. Chapra, Raymond P. Canale, Numerical Methods for Engineers, McGraw-Hill,
USA, 2009.
- 307 -
On the Classical Zariski Topology over Prime Spectrum of a Module
Seçil Çeken1 and Mustafa Alkan 2
Abstract. Let R be an associative ring with identity and Spec(M) denote the set of all
prime submodules of a right R-module M. In this talk, we deal with the classical Zariski
topology on Spec(M) which is denoted by τc. We prove that if (Spec(M), τc) is a Noetherian
topological space, then M has finitely many minimal prime submodules. We characterize all
the irreducible components of (Spec(M), τc ) and all the minimal prime submodules of M for a
non-zero flat module M over a commutative ring R. We obtain some results concerning
compactness and connectedness of (Spec(M), τc) by using algebraic properties of the module
M. We give some equivalent conditions for (Spec(M), τc) to be a Hausdorff space or T₁-space
when M is a right module over a left perfect ring R.
Keywords. Prime Submodule, Prime Spectrum, Classical Zariski Topology. AMS 2010. 16D10, 13C11, 54B99.
References
[1] A. Abbasi and D. Hassanzadeh-Lelekaami, Modules and spectral spaces, Comm. Algebra 40, no. 11, 4111-4129, 2012.
[2] Ansari-Toroghy, H., Ovlyaee-Sarmazdeh, R. On the prime spectrum of a module and zariski topologies. Comm. Algebra 38, 4461-4475, 2010.
[3] M. Behboodi and M. R. Haddadi, Classical Zariski Topology of Modules and Spectral Spaces I, Inter. Electronic J. Alg. 4, 104-130, 2008.
[4] M. Behboodi and M. R. Haddadi, Classical Zariski Topology of Modules and Spectral Spaces II, Inter. Electronic J. Alg. 4, 131-148, 2008.
[5] S. Çeken, M. Alkan and P. F. Smith, Second modules over noncommutative rings, Communications in Algebra, 41 (1), 83-98, 2013.
[6] T. Duraivel, Topology on spectrum of modules. J. Ramanujan Math. Soc. 9:25—34, 1994.
[7] C-P. Lu, Prime submodules of modules. Comm. Math. Univ. Sancti Pauli 33, 61-69, 1984.
[8] C.P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25 (3) 417-432, 1999.
[9] R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25, 79-103, 1997.
1 Akdeniz University , Antalya, Turkey, [email protected] 2 Akdeniz University , Antalya, Turkey, [email protected]
- 308 -
Strongly k-Spaces
Soley Ersoy 1, İbrahim İnce 2 and Merve Bilgin3
Abstract. In this paper, we introduce the notion of strongly k-spaces (with the weak
(=finest) pre-topology generated by their strongly compact subsets). We characterize the
strongly k-spaces and investigate the relationships between pre-closedness, locally strongly
compactness, pre-first countability and being strongly k-space.
Keywords. Strongly Compact Sets, Preopen Sets, k-Spaces.
AMS 2010. 54A05; 54C08, 54D50.
References
[1] R. F. Arens, A topology for spaces of transformations, Ann. of Math. (2), 47 480-495.
1946.
[2] R. H. Atia, S. N. El-Deeb, and I. A. Hasanein, A note on strong compactness and S-
closedness, Mat. Vesnik 6(19)(34), no. 1, 23-28, 1982.
[3] D. E. Cohen, Spaces with weak topology, Quart.J. Math. Oxford Ser. 5, 77-80, 1953.
[4] J. Dontchev, Survey on preopen sets, The Proceedings of the Yatsushiro Topological
Conference, 1-18, 1998.
[5] S. Jafari, T. Noiri, More on strongly compact spaces, Missouri J. Math. Sci. 19 (1), 52-61,
2007.
[6] Y. B. Jun, S. W. Jeong., H. J. Lee, and J. W. Lee, Applications of pre-open sets, Appl.
Gen. Topol. 9, no. 2 213-228, 2008.
[7] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak
precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53, 47-53, 1982.
[8] A. S. Mashhour, I. A. Hasanein, S. N. El-Deeb, A note on semicontinuity and pre-
continuity, Indian J. Pure Appl. Math. 13, no. 10 1119-1123, 1982.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 309 -
[9] A. S. Mashhour, M. E. Abd El-Monsef, I. A. Hasanein, and T. Noiri, Strongly compact
spaces, Delta J. Sci. 8, 30-46, 1984.
[10] M. C. McCord, Classifying spaces and in_nite symmetric products, Trans. Amer. Math.
Soc. 146, 273-298, 1969.
[11] I. Nasir , Strongly C-compactness, Journal of Al-Nahrain University 15(1), 140-144,
2012.
[12] N. L. Noble, k-spaces and some generalizations, Thesis (Ph.D.)-University of Rochester
113, 1967.
[13] L. Norman, K-spaces, their antispaces and related maps, Pacific J. Math. 59 2, 1975.
[14] T. M. Nour, Pre-unique sequential spaces, Indian J. Pure Appl. Math. 32, no. 6, 797-800,
2001.
[15] V. Popa, Characterization of H-almost continuous functions, Glasnik Math. Vol. 22(42)
157-161, 1987.
[16] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14, 133-
152, 1967.
[17] R. C. Steinlage, On Ascoli theorems and the product of k-spaces, Kyungpook Math. J.
12, 145-151, 1972.
[18] D. D. Weddington, On k-spaces, Proc. Amer. Math. Soc. 22, 635—638, 1969.
- 310 -
On the Study of the Matrix in a Model of Economic Dynamics
Sabir I. Hamidov1
Abstract. In the study of some reproduction models equilibrium mechanisms are often
used. In this paper we consider a diversified Leontief type model. In the paper [1] it is shown
that the equilibrium vector ( )nx~,...,x~x~ 1= is a solution of the equation 0),( =Ψ xf , where
( )nψψ=Ψ ,...,1 is a mapping of nR+ to itself.
It is also proved there that the function ),( xfΨ is differentiable with respect to x , and
x∂ψ∂ is a block matrix, the elements of which depends on the matrix )~(2 i
ixi xA ϕ∇= of the
second partial derivatives of the function iϕ calculated in point ix~ . Then we introduce the
matrix niAqC iii ,1, == .
In the present paper the case 3=n is considered in detail and Metzler matrix
,)( 1nijijcС == i.e. the matrix with )(0 jicij ≠≥ is investigated. The estimations for the norms
C and 1−iC are obtained.
Keywords. Discrete dynamical models, consumption, Cobb-Douglas production
function.
References
[1] Hamidov, S.I., Equilibrium mechanism in models of reproduction with a fixed budget,
American Journal of Applied Mathematics. Vol 3, N 3, 2015, pp 146-150.
1 Baku State University, Baku, Azerbaijan, [email protected]
- 311 -
Numerical Solutions of Steady Incompressible Dilatant Flow in an Enclosed
Cavity Region
Serpil Şahin1 and Hüseyin Demir2
Abstract. In this study, we consider flow properties of Dilatant fluid motion generated
by top wall motion for 2-D steady incompressible flows. Pseudo time derivative is used to
solve the continuity and momentum equations with suitable initial and boundary conditions.
Therefore, the governing equations of fluid of vorticity-stream function formulations are
solved numerically using finite difference and Gauss Elimination method. The stream
function and vorticity results are obtained for the steady two-dimensional Dilatant
incompressible flow. These results are presented both in tables and figures. The stream
function and vorticity equations are solved separately with the numerical solution method
used in this study. Each equation with pseudo time parameter on very fine grid mesh is solved
step by step with a pair of tridiagonal system. The advantage of this process is that it gives the
solution of the flow problems effectively and accurately.
Keywords. Finite Difference Method, Pseudo Time Parameter, Dilatant Fluid.
AMS 2010. 76D05, 76M20, 76M25.
References
[1] Barragy, E., Carey, G.F., Stream function-vorticity driven cavity solutions using p finite
elements, Computers and Fluids, 26, 453-468, 1997.
[2] Benjamin, A.S., Denny, V.E., On the convergence of numerical solutions for 2-D flows in
a cavity at large Re, J. Comp. Physics, 33, 340-358, 1979.
[3] Botella, O., Peyret, R., Benchmark spectral results on the lid-driven cavity flow,
Computers and Fluids, 27, 421-433, 1998.
[4] Ertürk, E., Corke, T.C., Gökçöl, C., Numerical solutions of 2-D steady incompressible
driven cavity flow at high Reynolds numbers, J. Numer. Meth. Fluids, 48, 747-774, 2005.
[5] Gupta, M.M., High accuracy solutions of incompressible N-S equations, J. Comp. Physics,
93, 343-359, 1991.
[6 ] Hou, S., Zou, Q., Chen, S., Doolen, G., Cogley, A.C., Simulation of cavity flow by the
Lattice Boltzmann method, J. Comp. Physics, 118, 329-347, 1995.
1 Amasya University, Amasya, Turkey, [email protected] 2 Ondokuz Mayis University, Samsun, Turkey, [email protected]
- 312 -
Smarandache Curves of Mannheim Curve Couple According to Frenet Frame
Süleyman Şenyurt 1 and Abdussamet Çalışkan 2
Abstract: In this paper, when the Frenet vectors of the partner curve of Mannheim
curve are taken as the position vectors, the curvature and the torsion of Smarandache curves
are calculated. These values are expressed depending upon the Mannheim curve. Besides,
special Smarandache curves belonging to *α Mannheim partner curve such as ** NT , **BN , **BT and **BNT drawn by Frenet frame are defined and some related results are given.
Keywords. Mannheim Curve, Mannheim Partner Curve, Smarandache Curves, Frenet
Invariants
AMS 2010. 53A04.
References
[1] Ali, A. T., Special Smarandache Curves in the Euclidean Space, Intenational Journal of
Mathematical Combinatorics, 2, 30-36, 2010.
[2] Bayrak, N., Bektaş, Ö. and Yüce, S., Special Smarandache Curves in 31E , International
Conference on Applied Analysis and Algebra, Yıldız Techinical University, 209, İstanbul, 20-
24 June 2012.
[3] Bektaş, Ö. and Yüce, S., Special Smarandache Curves According to Dardoux Frame in
Euclidean 3-Space, Romanian Journal of Mathematics and Computer science, 3, 1, 48-59,
2013.
[4] Çalışkan, A. and Şenyurt, S., Smarandache Curves In terms of Sabban Frame of Spherical
Indicatrix Curves, Gen. Math. Notes, impress, 2015.
[5] Çalışkan, A. and Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Fixed
Pole Curve, Boletim da Sociedade parananse de Mathemtica, 34, 2, 53-62, 2016.
[6] Çalışkan, A. and Şenyurt, S., * *
N C − Smarandache Curves of Mannheim Curve Couple
According to Frenet Frame, International J.Math. Combin., 1, 1--13, 2015.
1 Ordu University, Ordu, Turkey, [email protected] 2 Ordu University, Ordu, Turkey, [email protected]
- 313 -
[7] Hacısalihoğlu, H. H., Differential Geometry, İnönü University, Malatya, Mat. no.7, 1983.
[8] Liu, H. and Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, 88, no.
1-2, 120-126(7), 2008.
[9] Orbay, K. and Kasap, E., On Mannheim partner curves in, International Journal of
Physical Sciences, 4, 5, 261-264, 2009.
[10] Sabuncuoğlu, A., Differential Geometry, Nobel Publications, Ankara, 2006.
[11] Şenyurt, S. and Sivas, S.,. An Application of Smarandache Curve, Ordu Univ. J. Sci.
Tech., 3, 1, 46--60, 2013.
[12] Taşköprü, K. and Tosun, M., Smarandache Curves According to Sabban Frame on 2S
Boletim da Sociedade parananse de Mathemtica 3 srie., 32, 1, 51-59, ıssn-0037-8712, 2014.
[13] Turgut, M. and Yılmaz, S., Smarandache Curves in Minkowski space-time, International
Journal of Mathematical Combinatorics, 3, 51-55, 2008.
[14] Wang, F.,and Liu, H., Mannheim Partner Curves in 3-Space, Mathematics in Practice
and Theory, 37, 141-143, 2007.
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Smarandache Curves of Involute-Evolute Curve Couple According to Frenet Frame
Süleyman Şenyurt1, Selin Sivas2 and Abdussamet Çalışkan3
Abstract. In this paper, when the Frenet vectors of involute curve are taken as the
position vectors, the curvature and the torsion of Smarandache curves are calculated. These
values are expressed depending upon the evolute curve. Besides, special Smarandache curves
belonging to *α involute curve such as ** NT , **BN , **BT ve **BNT drawn by Frenet
frame are defined and some related results are given.
Keywords. Evolute Curve, Involute Curve, Smarandache Curves, Frenet Invariants.
AMS 2010. 53A04.
References
[1] Ali, A. T., Special Smarandache Curves in the Euclidean Space, Intenational Journal of
Mathematical Combinatorics, 2, 30-36, 2010.
[2] Bayrak, N., Bektaş, Ö. and Yüce, S., Special Smarandache Curves in 31E , International
Conference on Applied Analysis and Algebra, Yıldız Techinical University, 209, İstanbul, 20-
24 June 2012.
[3] Bektaş, Ö. and Yüce, S., Special Smarandache Curves According to Dardoux Frame in
Euclidean 3-Space, Romanian Journal of Mathematics and Computer science, 3, 1, 48-59,
2013.
[4] Bilici, M., The Curvatures and the natural lifts of the spherical indicator curves of the
involute-evolute curve, Master's thesis. Institute of science, Ondokuz Mayıs University,
Samsun, 49, 1999.
[5] Bilici, M. and Çalışkan, M., Some New Notes on the Involutes of the Timelike Curves in
Minkowski 3-Space, Int. J. Contemp. Math. Sciences, 6, no. 41, 2019 - 2030, 2011.
[6] Çalışkan, A. and Şenyurt, S., Smarandache Curves In terms of Sabban Frame of Spherical
Indicatrix Curves, Gen. Math. Notes, impress, 2015.
1 Ordu University, Ordu, Turkey [email protected] 2 Ordu University, Ordu, Turkey, [email protected] 3 Ordu University, Ordu, Turkey, [email protected]
- 315 -
[7] Çalışkan, A. and Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Fixed
Pole Curve, Boletim da Sociedade parananse de Mathemtica, 34, 2, 53-62, 2016.
[8] Çalışkan, A. and Şenyurt, S., * *
N C − Smarandache Curves of Mannheim Curve Couple
According to Frenet Frame, International J.Math. Combin., 1, 1--13, 2015.
[9] Çalışkan, M. and Bilici, M., Some Characterizations for The Pair of involute-evolute
Curves in Euclidean Space 3E , Bulletin of Pure and Applied Sciences, 21, 2, 289-294. 2002.
[10] Çalışkan, M. and Bilici, M., On The involutes of Timelike Curves in 31R , IV. International
Geometry Symposium, Zonguldak Karaelmas University, 17-21 July 2006.
[11] Hacısalihoğlu, H. H., Differential Geometry, İnönü University, Malatya, Mat. no.7, 1983.
[12] Millman, R.S. and Parker, G.D., Elements of Differential Geometry, Prentice-Hall Inc.,
Englewood Cliffs, New Jersey, 265, 1977.
[13] Sabuncuoğlu, A., Differential Geometry, Nobel Publications, Ankara, 2006
[14] Şenyurt, S. and Sivas, S.,. Şenyurt, S. and Sivas, S.,. An Application of Smarandache
Curve, Ordu Univ. J. Sci. Tech., 3, 1, 46--60, 2013.
[15] Taşköprü, K. and Tosun, M., Smarandache Curves According to Sabban Frame on 2S
Boletim da Sociedade parananse de Mathemtica 3 srie., 32, 1, 51-59, 2014, ıssn-0037-8712.
[16] Turgut, M. and Yılmaz, S., Smarandache Curves in Minkowski space-time, International
Journal of Mathematical Combinatorics, 3, 51-55, 2008.
[17] Turgut, A. and Esin, E., Involute-Evolute Curve Couples Of Higher Order in and Their
Hortizonal Lifts in , Commun. Fac. Sci. Univ. Ank. Series, 41, 125-132, 1992.
- 316 -
The Analysis of the Mathematical Modelling Activities in the Ninth Grade
Mathematics Coursebook
Selin Urhan1 and Şenol Dost2
Abstract. Mathematical modelling is a process of translating the real life problems
into the language of mathematics using mathematical terms, or expressing a real life situation
mathematically (Cheng, 2001). Model eliciting activities, on the other hand, are problem-
solving activities which include nonroutine, open-ended, and complicated real life situations
with nontraditional problems and which have various possible solutions (Lesh & Doer, 2003).
According to Lesh et al. (2000), a model construction activity must involve the following
principles: the model construction principle, the reality principle, the self-assessment
principle, the model documentation (construct documentation) principle, the model
generalizability principle, and the effective prototype principle.
In the secondary mathematics education program in Turkey, the significance of creating
learning environments based on modelling activities that are appropriate for the level of
students and that can ensure active participation is highly emphasized (MoNE, 2013). Yet,
modelling activities are seldom used in courses (Tekin & Bukova Güzel, 2011; Urhan & Dost,
2015). One of the reasons behind this situation is that the coursebooks are qualitatively and
quantitatively insufficient as far as modelling activities are concerned (Vural et al., 2013;
Urhan & Dost, 2015).
The aim of the current study is to analyze the activities in the ninth grade mathematics
coursebook that was prepared in line with the new curriculum based on the model
construction activity principles. The data collection procedure involved document analysis,
which is one of the qualitative data collection methods. Frequency and percentage values
were calculated to reveal the congruence between the activities analyzed and the principles.
The results of the analysis indicate that model construction activities that employ all the
principles are seldom found in the coursebook. It was found that of the seventy three activities
in the coursebook, fourteen activities (19%) are model construction activities. It was revealed
that the model construction activities are mostly seen in “Equations and Inequalities” unit.
Modelling activities were not found in “Congruence and Similarity in Triangles” and “Right
Angled Triangle, Trigonometry, Area of the Triangle, Vectors” units. While five of the model
construction activities partially follow the self-assessment principle, they fully follow all the
1 Hacettepe University, Ankara, Turkey, [email protected] 2 Hacettepe University, Ankara, Turkey, [email protected]
- 317 -
other principles. Within the framework the obtained results, it is recommended that ninth
grade mathematics coursebooks be revised in terms of the modellling activities they involve,
and that similar analyses be conducted on other coursebooks as well.
Keywords. Mathematical Modelling, Modelling Activities, Modelling Principles,
Coursebooks
AMS 2010. 53A40, 20M15.
References
[1] Cheng, A. K., Teaching mathematical modelling in singapore schools. The Mathematics
Educator, 6(1), 63-75, 2001.
[2] Çiltaş, A., Deniz, D., Akgün, L., Işık, A., ve Bayrakdar, Z., İlköğretim İkinci Kademede
Görev Yapmakta Olan Matematik Öğretmenlerinin Matematiksel Modelleme ile İlgili
Görüşlerinin İncelenmesi. 10. Matematik Sempozyumu, Şile/İstanbul, 2011.
[3] Lesh, R., & Doerr, H. M., (Eds.). Beyond constructivism: Models and modeling
perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ:Lawrence
Erlbaum, 2003.
[4] Lesh, R., Hoover, M., Hole, B., Kelly, E., & Post, T., Principles for developing thought-
revealing activities for students and teachers. Mahaway, NJ: Lawrence Erlbaum, 2000.
[5] Milli Eğitim Bakanlığı [MEB], Ortaöğretim matematik dersi (9-12 sınıflar), öğretimi
programı, 2013.
[6] Tekin, A., Bukova Güzel, E., Ortaöğretim Matematik Öğretmenlerinin Matematiksel
Modellemeye İlişkin Görüşlerinin Belirlenmesi. 20. Eğitim Bilimleri Kurultayı, Mehmet Akif
Ersoy Üniversitesi, Burdur, 2011.
[7] Urhan, S., Dost, Ş., Matematik Öğretmenlerinin Matematiksel Modelleme Hakkındaki
Görüşleri. 5. Uluslararası Öğretmen Eğitiminde Politika ve Sorunlar Sempozyumu,
Bakü/Azerbaycan, 2015.
[8] Vural, D. Ö., Çetinkaya, B., Erbaş, A. K., Alacacı, C., ve Çakıroğlu, E., Lise matematik
öğretmenlerinin modelleme ve modellemenin matematik öğretiminde kullanılmasına yönelik
düşünceleri: Bir hizmet içi eğitim programının etkisi. 1. Türk Bilgisayar ve Matematik
Eğitimi Sempozyumu, Fatih Eğitim Fakültesi, Trabzon, 2013.
- 318 -
On Geodesic Paracontact CR-Lightlike Submanifolds
Selcen Yüksel Perktaş1 and Bilal Eftal Acet2
Abstract. In this paper we study geodesic paracontact CR-lightlike submanifolds of
para-Sasakian manifolds. We derive some necessary and sufficient conditions for totally
geodesic, D∧
− geodesic, 'D − geodesic and mixed geodesic paracontact CR-lightlike
submanifolds. Also we examine geodesic paracontact screen CR-lightlike submanifolds of
para-Sasakian manifolds.
Keywords. Para-Sasakian Manifolds, Lightlike Submanifolds.
AMS 2010. 53C15, 53C25.
References
[1] Duggal, K. L. and Bejancu, A., Lightlike submanifolds of semi-Riemannian manifolds and
applications, Mathematics and Its Applications, Kluwer Publisher, 1996.
[2] Duggal, K. L. and Sahin, B., Differential geometry of lightlike submanifolds, Frontiers in
Mathematics, 2010.
[3] Sahin, B., and Güneş, R., Geodesic CR-lightlike submanifolds , Contributions to Algebra
and Geometry, Vol. 42, No. 2, 583-594, 2001.
[4] Zamkovoy, S., Canonical connection on paracontact manifolds, Ann. Glob. Anal. Geo.,
36, 37-60, 2009.
[5] Acet, B. E., Yüksel Perktaş, S., Kılıç, E., On lightlike geometry of para-Sasakian
manifolds, Scientific World J., Article ID 696231, 2014.
1 Adiyaman University, Adiyaman, Turkey, [email protected] 2 Adiyaman University, Adiyaman, Turkey, [email protected]
- 319 -
A Generalized Method for Centres of Trajectories in Kinematics
Tülay Erişir1, Mehmet Ali Güngör2 and Murat Tosun3
Abstract. In this paper, we study on one-parameter planar motion in generalized
complex plane or p − complex plane 2: , ,p x iy x y i p= + ∈ = which is defined as system of
the generalized complex numbers. Firstly, we define a canonical relative system for one-
parameter planar motion in p − complex plane. From this, we obtain the generalized Euler-
Savary formula, which gives the relationship between the curvatures of trajectory curves in
the generalized complex plane.
Keywords. Generalized Complex Plane, Generalized Euler Savary Formula
AMS 2010. 53A17, 53B50, 11E88.
References
[1] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, 1968.
[2] W. Blaschke, H. R. Müller, Ebene Kinematik, Verlag Oldenbourg, München, 1956.
[3] A. A. Harkin and J. B. Harkin, Geometry of generalized complex numbers, Math. Mag.,
77, 2, 2004.
[4] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,
New-York, 1979.
[5] S. Ersoy and M. Akyiğit, One-parameter homothetic motion in the hyperbolic plane and
Euler-Savary Formula, Adv. Appl. Clifford Algebr., 21, 2, 297-313, 2011.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 320 -
A New Generalization of the Steiner Formula and the Holditch Theorem
Tülay Erişir1, Mehmet Ali Güngör2 and Murat Tosun3
Abstract. In this study, we first obtained the Steiner area formula in the generalized
complex plane. Then, with the aid of this formula, we determined a new approach for the
Holditch theorem giving the relationship between the areas formed by points in the
generalized complex plane (or p − complex plane). Finally, according to the special values
of p∈ we examined the cases of the Steiner Formula and Holditch Theorem. In this way,
for p∈R we generalized the Steiner Formula and Holditch theorem consisting the Euclidean
( 1)p = , Galilean ( 0)p = and Lorentzian ( 1)p = − cases.
Keywords. Generalized Complex Plane, The Steiner Formula, The Holditch Theorem
AMS 2010. 53A17, 53B50, 11E88.
This paper was funded by Sakarya University BAPK (No: 2014-50-02-023).
References
[1] H. Holditch, Geometrical Theorem, Q. J. Pure Appl. Math., 2, 1858.
[2] W. Blaschke, H. R. Müller, Ebene Kinematik, Verlag Oldenbourg, München, 1956.
[3] H. Potmann, Zum Satz von Holditch in der euklidischen Ebene, Elem. Math., 41, 1-6,
1986.
[4] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,
New-York, 1979.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 321 -
On the Construction of Generalized Bobillier Formula
Tülay Erişir1, Mehmet Ali Güngör2 and Soley Ersoy3
Abstract. In this paper, one-parameter planar motion in the generalized complex plane
(or p − complex plane) 2: , ,p x iy x y i p= + ∈ = which is defined as a system of
generalized complex numbers is studied. Firstly, generalized Bobillier formula is obtained by
using the geometric interpretation of generalized Euler-Savary formula in the p − complex
plane. Moreover, it is shown that the Bobillier formula may be obtained by an alternative
method without the use of Euler-Savary formula in the generalized complex plane. Thus, this
formula generalizes the Euclidean, Lorentzian and Galilean cases.
Keywords. Generalized Complex Plane, Generalized Bobillier Formula, Generalized
Euler Savary Formula
AMS 2010. 53A17, 53B50, 11E88.
References
[1] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, New York, 1968.
[2] W. Blaschke, H. R. Müller, Ebene Kinematik, Verlag Oldenbourg, München, 1956.
[3] A. A. Harkin and J. B. Harkin, Geometry of generalized complex numbers, Math. Mag.,
77, 2, 2004.
[4] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag,
New-York, 1979.
[5] S. Ersoy and N. Bayrak, Bobillier Formula for One Parameter Motions in the Complex
Plane, J. Mechanisms Robotics, 4, 2, 024501-1-024501-4, 2012.
[6] M. Fayet, Bobillier formula as a fundamental law in planar motion, Z. Angew. Math.
Mech. 82, 3, 207-210, 2002.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected] 3 Sakarya University, Sakarya, Turkey, [email protected]
- 322 -
Generalized 𝑭-Expansion Method and Application to Nonlinear Fractional
Differential Equation
Yusuf Ali Tandoğan 1 and Yusuf Pandır 2
Abstract. Many researches have been done to create the solutions of fractional
differential equations [1-4]. Firstly, Zhang and Xia offer generalized F-expansion method to
solve partial differential equations according to the power series [5]. In this study, we apply
generalized F-expansion method in order to achieve new jakobi elliptic function classes of the
fractional differential equations within modified Riemann-Liouville derivative. By using this
approach, we find some new results for the nonlinear differential equation. As a result, many
non-travelling wave solutions are acquired such as single and combined non-degenerate
Jacobi elliptic function solutions, soliton solutions and trigonometric function solutions.
Keywords. Generalized F-Expansion Method, Nonlinear Fractional Differential,
Equations, Single and Combined Non-Degenerate Jacobi Elliptic Function Solutions,
AMS 2010. 35R11, 26A33, 35C07, 35C08, 35C10.
References
[1] Pandir Y., Yildirim A., New exact solutions of the space-time fractional potential
Kadomtsev-Petviashvili (pKP) equation, AIP Conf. Proc., 1648, 370016, 2015.
[2] Bekir A., Application of the Subequation method to some differential equations of time
fractional order, J. Comput. Nonlinear Sci., 10(5), 054503, 2015.
[3] Lu B., The first integral method for some time fractional differential equations, J. Math.
Anal. Appl., 395, 684-693, 2012.
[4] Pandir Y., Gurefe Y., Misirli E., The extended trial equation method for some time-
fractional differential equations, Discrete Dyn. Nat. Soc., 2013, 491359, 2013.
[5] Zhang S., Xia T., A generalized F-expansion method with symbolic computation exactly
solving Broer-Kaup equations, Appl. Math. Comput., 189(1), 836-843, 2007.
1 Bozok University, Yozgat, Turkey, [email protected] 2 Bozok University, Yozgat, Turkey, [email protected]
- 323 -
Views of Pre-service Elementary Mathematics Teachers Toward The Reasons for
Student’s Mistakes About Fractions and Preventing These Mistakes
Yasemin Kıymaz1
Abstract. To evaluate the students’ works, to analyze the students’ mistakes and with
taking into account these situations to provide learning environments for students that
facilitate their understanding are some of the important tasks that pre-service teachers will
confront in the future. In this study, pre-service elementary mathematics teachers are asked
the reasons for student’s mistakes about fractions and how they prevent these mistakes. Data
were collected through a survey that contained open ended questions. 71 pre-service teachers
who attended a state university participated in this study. The collected data will be analyzed
through content analysis.
Keywords. Misconception, Fractions, Teacher Education
AMS 2010. 97B50, 97C70, 97D70.
Acknowledgement: This work is supported by Ahi Evran University PYO with
project number PYO-EGT.4001.15.005.
References
[1] Alacaci C., "Öğrencilerin Kesirler Konusundaki Kavram Yanılgıları", In Bingölbali, E. &
Özmantar M.F. (Eds.), İlköğretimde Karşılaşılan Matematiksel Zorluklar ve Çözüm Önerileri
(ss.63-95), PEGEM AKADEMİ, Ankara, 2009.
[2] Pesen, C. Öğrencilerin Kesirlerle İlgili Kavram Yanılgıları. Eğitim ve Bilim, 32(143), 79-
88, 2007.
[3] Soylu, Y. & Soylu, C., İlköğretim Beşinci Sınıf Öğrencilerinin Kesirler Konusundaki
Öğrenme Güçlükleri: Kesirlerde Sıralama, Toplama, Çıkarma, Çarpma ve Kesirlerle İlgili
Problemler. Erzincan Eğitim Fakültesi Dergisi. 7(2), 101-117, 2005.
[4] Baştürk, S., Mutlak Değer Kavramı Örneğinde Öğretmen Adaylarının Öğrenci Hatalarına
Yaklaşımları. Necatibey Eğitim Fakültesi Dergisi. Sayı 1. Cilt 3, 174-194, 2009.
1 Ahi Evran University, Kirsehir, Turkey, [email protected]
- 324 -
Interface Design for Comparative Solution of Mathematical Equations by Classical
Interpolation Methods and Artificial Nerve Network Approaches
Zeynep Batık1 and Erdal Büyükbıçakcı2
Abstract. As a result of the experiments, measurements, observations and calculations
realized in applied sciences, many data is obtained. Methods such as interpolation, smallest
squares, halving, tangent-beam, Newton Raphson and Regula–False. In recent years'
developments, artificial nerve networks, blur logic, intuitional and generic algorithms are used
in sciences designing new systems by replicating the human brain functions. Classical
Interpolation Methods (CIM) and Artificial Nerve Networks (ANN) are used in many fields
such as math, physics, electric, computer engineering and many applications such as
optimization, control, sample completion-matching, voice and vision detection, calculation
and classification.
In this study solution of mathematical equations were given comparatively by using
interpolation methods and artificial nerve network approaches. In simulations realized in
MATLAB GUI ambiance designed by graphical user interface program, results of equation
systems are shown as numerical and graphical interface. At the same time the program
ensures us to select the mathematical function on interface with the help of interpolation
method parameters and artificial nerve network parameters.
Keywords. Classical Interpolation, Nerve Network, Matlab Interface Design
AMS 2010. 65D05, 65D17, 68R10
References
[1] Caruso, C., Interpolation Methods Comparison, Computers, Math. Applic. Vol. 35 - 12,
pp. 109-126, 1998.
[2] Nabiyev, V.V., Yapay Zeka, Seçkin Yayincilik, Ankara, 2005.
[3] The MathWorks Inc. MATLAB, 2007.
[4] Sagiroglu, G., BeGdok, E., Erler, M., Mühendislikte Yapay Zeka Uygulamaları-I: Yapay
Sinir Ağları, Ufuk Kitap Kırtasiye Yayıncılık Tic. Ltd. Şti., Kayseri, 2003.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 325 -
Interface Design for Genetic Algorithm Based Solution of Polynomial Equations
Zeynep Batık1 and Erdal Büyükbıçakcı2
Abstract. Various analytical methods have been and are being developed fir solution
of mathematical equation systems (linear and nonlinear equations). In stead of using classical
methods in solution of mathematical equation systems using computer supported numerical
analysis methods that realizes multiple iterations more rapidly ease the solution. In recent
years computer supported Genetic Algorithms (GA) are prevalently used in solution of all
kinds of optimization problems. Generic algorithms are especially used in calculating roots of
high degree polynomials easily. In order to ensure the solution of equations mathematical
operations such as start up conditions, root interval and iterations are necessary.
In this study genetic algorithms which are used in optimization field in recent years are
adopted to polynomial equation solutions and accordingly graphical interface program was
designed and results were compared with other numerical methods. Bu using the graphical
user interface program designed in MATLAB GUI ambiance, roots of determined polynomial
equations are calculated by using generic algorithm and other numerical methods. This
program which can also be used for education purpose, polynomial coefficients and
sensitivity values are inputed and the roots can be calculated by generic algorithms and
classical equation solution methods on intervals determined by the user.
Keywords. Genetic Algorithm, Polynomial Root Calculation, Matlab Interface
AMS 2010. 65H04, 68U07, 68R10
References
[1] Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning,
Addison-Wesley Publishing Company Inc.,USA, 1989.
[2] ÖZTÜRK N., ÇELİK E., Polinom olmayan denklemlerin genetik algoritma Tabanlı
Çözümü, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(4): 322-328, 2012.
[3] The Genetic Algorithm and Direct Search Toolbox, MATLAB, The MathWorks, Inc.
[4] The MathWorks Inc. MATLAB, 2007.
1 Sakarya University, Sakarya, Turkey, [email protected] 2 Sakarya University, Sakarya, Turkey, [email protected]
- 326 -
Fixed Point of Automorphisms Permuting Free Generators of Free
Metabelian Lie Algebras
Zerrin EsmerligilP
1 and Naime EkiciP
2
Abstract: Let 𝐴𝑛 and 𝑀𝑛 be a free abelian Lie algebra and a free metabelian Lie
algebra of finite rank 𝑛 respectively.
We describe fixed point subalgebras of an automorphism of permuting cyclically free
generators of 𝐴𝑛 and free generators of 𝑀𝑛.
We prove that if 𝜃 is an automorphisms of order 𝑛 which permutes free generators of
𝑀𝑛 then the form of the fixed point subalgebra of 𝑀𝑛 is 𝑣 + 𝜃(𝑣) + ⋯+ 𝜃𝑛−1(𝑣):𝑣𝑣𝑀𝑛.
The motivation of this work is based on the results of C. Baginski and W. Tomaszewski.
Keywords. Free Solvable Lie Algebra, Automorphism, Fixed Point
AMS 2010. 17B01, 17B40
References:
[1] Baginski C., Tomaszewski W., Automorphisms of Prime Order of Free Metabelian
Groups, Commun. Alg., Vol. 30, No. 10,4985-4996, 2002.
[2] Bryant R.M, On the fixed points of finite group acting on a free Lie algebra, J. London
Math. Soc. 43 (2), 215-224, 1991.
[3] Bryant,R.M., Papistas A.I, On the fixed points of a finite group acting on a relatively free
Lie algebra, Glasg.Math.J. 42,167-181, 2000.
[4] Ekici N., Sönmez D., Fixed points of IA- endomorphisms of a free metabelian Lie algebra,
Proc. Indian Acad. Sci.(Math. Sci.) Vol. 121, No. 4, 405-416, 2011.
[5] Shpilrain, V., Fixed points of endomorphisms of a free metabelian group, Math. Proc.
Camb. Phil. Soc., 123, 75-83, 1998.
[6] Tomaszewski W., Fixed point of automorphisms preserving the length of words in free
solvable groups, Arch. Math. 99, 425-432, 2012.
1 Cukurova University, Adana, Turkey, [email protected] 2 Cukurova University, Adana, Turkey, [email protected]
- 327 -
Mechanics Equations of Frenet-Serret Frame on Minkowski Space
Zeki Kasap1 and Emin Özyılmaz2
Abstract. It is well known that Minkowski space is the mathematical space setting and
Einstein.s theory of special relativity is most appropriate formulated. Dynamical systems
theory is an area of mathematics used to describe the behavior of complex dynamical systems
in which usually by employing differential equations or difference equations. The Frenet-
Serret formulas describe the kinematic properties of a particle which moves along a
continuous, differentiable curve in Euclidean space three-dimensional real space or the
geometric properties of the curve itself in any case of any motion. The Frenet-Serret trihedron
plays a key role in the differential geometry of curves such that its shows ultimately leading to
a more or less complete classification of smooth curves in Euclidean space up to congruence.
In this paper, we established mechanics Equations of Frenet-Serret frame on Minkowski space
and we considered a relativistic for an electromagnetic field that it is moving under the
influence of its own Frenet-Serret curvatures. Also, we obtained the mechanical equations of
motion for several curvatures dependent actions of interest in physics.
Keywords. Frenet-Serret Curvature, Mechanical System, Minkowski Space,
Lagrangian Equation.
AMS 2010. 51B20, 70S05, 70Q05.
References
[1] G. Arreaga, R. Capovilla and J. Guven, Frenet-Serret Dynamics, Class Quantum Grav., 18, 5065-5083, 2001.
[2] S. Ali and A.H. Sarkar, Serret-Frenet Equations in Minkowski Space, Dhaka Univ. J. Sci., 61 (1), 87-92, 2013.
[3] S. Yilmaz, E. Ozyilmaz, Y.Yayli and M. Turgut, Tangent and Trinormal Spherical Images of A Time-Like Curve on the Pseudohyperbolic Space 3
0H , Proceedings of the Estonian Academy of Sciences, 59, 3, 216-224, 2010.
[4] G. Arreaga, R. Capovilla and J. Guven, Frenet--Serret Dynamics, Class Quantum Grav., 18, 5065-5083, 2001.
[5] M.G. Calkin, Lagrangian and Hamiltonian Mechanics: Solutions to The Exercises, World Scientific Pub. Co. Inc., 1999.
[6] Z. Kasap, M. Tekkoyun, Mechanical Systems on Almost Para/Pseudo-Kähler--Weyl Manifolds, IJGMMP, Vol.10, No.5, 1-8, 2013.
1 Pamukkale University, Denizli, Turkey, [email protected] 2 Ege University, Izmir, Turkey, [email protected]
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Test Elements of Free Metabelian Leibniz Algebras
Zeynep Özkurt1
Abstract. Let F(M) be a free metabelian Leibniz algebra in two variables over a field
K of characteristic zero. In this paper it is determined some test elements of F(M).
Keywords. Free Metabelian Leibniz Algebra, Test Element, Automorphism.
AMS 2010. 17A32,17A50.
References
[1] Abdykhalikov, A. T.; Mikhalev, A.A.; Umirbaev, U.U., Automorphism of Two-Generated
Free Leibniz Algebras. Commun. Algebra 29(7), 2953-2960, 2001.
[2] Loday, J.-L.; Prashvili, T., Universal Enveloping Algebras of Leibniz algebras and
(co)Homology. Math. Ann. 296,139-158, 1993.
[3] Mikhalev, A.A.;Umirbaev, U.U., Subalgebras of Free Leibniz Algebras. Commun.
Algebra 26, 435-446, 1998.
[4] Drensky, V.,Piacentini Cattaneo , G.M., Varieties of Metabelian Leibniz Algebras, Journal
of Algebra and its Applications, 1(1), 31-50, 2002.
1Cukurova Universty, Adana, Turkey, [email protected]
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Title Name Surname UniversityProf. Dr. Abderrahim Benhadda President of Mohammed V University
Prof. Dr. Abdullah Mağden Atatürk University
Prof. Dr. Ahmet Bekir Eskisehir Osmangazi University
Prof. Dr. Antonio Fernández-Carrión University of Sevilla
Prof. Dr. Anuradha Gupta Delhi College of Arts and Commerce
Prof. Dr. Arunabha Chanda Jadavpur University
Prof. Dr. Auzhan Sakabekov Kazakh-British Technical University
Prof. Dr. Azmi Özcan Bilecik Seyh Edebali University
Prof. Dr. Bünyamin Demir Anadolu University
Prof. Dr. Chiun-Chieh Hsu National Taiwan University of Science and Technology
Prof. Dr. Cihan Özgür Balikesir University
Prof. Dr. Debasis Giri Haldia Institue of Technology
Prof. Dr. Deepika Rani Indian Institute of Technology Roorkee
Prof. Dr. Edgar Martinez Moro Valladolid University
Prof. Dr. Ekrem Savaş Istanbul Commerce University
Prof. Dr. Erol Kılıç İnönü University
Prof. Dr. Ertan İbikli Ankara University
Prof. Dr. Etibar Penahlı (Panakhov) Firat University
Prof. Dr. F. Nejat Ekmekci Ankara University
Prof. Dr. Faraj.A.Abunabi Algabel Garbi University
Prof. Dr. Fawad Hussain Hazara University
Prof. Dr. Francisco J. Naranjo-Naranjo University of Sevilla
Prof. Dr. Goutam Pohit Jadavpur University
Prof. Dr. H. Hilmi Hacısalihoğlu Bilecik Seyh Edebali University
Prof. Dr. Héctor Martín Luna Garcia Universidad Autonoma Metropolitana
Prof. Dr. Hikmet Özarslan Erciyes University
Prof. Dr. Hsiu-Chuan Wei Feng Chia University
Prof. Dr. Idriss Mansouri Président de l’Université Hassan II
Prof. Dr. İlham A. Aliyev Akdeniz University
Prof. Dr. İsmail Kocayusufoğlu International Balkan University
Prof. Dr. İsmet Karaca Ege University
Prof. Dr. Jose Juan Peña Gil Universidad Autonoma Metropolitana
Prof. Dr. Kailash C. Madan College of Information Technology
Prof. Dr. Kazım İlarslan Kirikkale University
Prof. Dr. Laura Ventura University of Padova
Prof. Dr. Lubica Hola Slovak Academy of Sciences
Prof. Dr. Mahaveer Gadiya MIT College of Engineering
Prof. Dr. Mahir Rasulov Beykent University
Prof. Dr. Mahmut Ergüt Namik Kemal University
Prof. Dr. Maksat Kalimoldayev Institute of Mathematics and Mathematical Modeling
Prof. Dr. Manaf Manafov Adiyaman University
Prof. Dr. Meera Kaushik University of Delhi
Prof. Dr. Mehmet Ali Sarıgöl Pamukkale University
Prof. Dr. Monireh Sedghi Azarbijan Shahid Madani University
Prof. Dr. Montserrat Alsina Universitat Politècnica de Catalunya
Prof. Dr. Muhammad Shahzad Hazara Unviersity
Prof. Dr. Murat Altun Uludag University
Prof. Dr. Murat Tosun Sakarya University
Prof. Dr. Mustafa Çalışkan Gazi University
Prof. Dr. Muvasharkhan Jenaliyev Institute of Mathematics and Mathematical Modeling
Prof. Dr. Muzaffer Elmas Sakarya University
Prof. Dr. Naime Ekici Cukurova University
Prof. Dr. Nedim Değirmenci Anadolu University
Prof. Dr. Nezihe Çalışkan Gazi University
Prof. Dr. Nihal Yılmaz Özgür Balikesir University
Prof. Dr. Nikolaos G. Tzanakis Crete University
Prof. Dr. Nirmal C. Sacheti Sultan Qaboos University
Prof. Dr. Nuri Kuruoğlu Bahcesehir University
Prof. Dr. Pallath Chandran Sultan Qaboos University
Prof. Dr. Pragati Gautam University of Delhi
Prof. Dr. Prasanta Sahoo Jadavpur University
Prof. Dr. Ram N. Mohapatra Central Florida University
Prof. Dr. Renu Chugh Maharshi Dayanand University
Prof. Dr. Reza Naghipour University of Tabriz and IPM
Prof. Dr. Saadettin Aksoy Siirt University
Prof. Dr. Saaid Amzazi President of Mohammed V University
List of Participants
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Prof. Dr. Sadaoui Boualem Khemis Miliana University
Prof. Dr. Sadettin Hülagü Kocaeli University
Prof. Dr. Sadık Keleş İnönü University
Prof. Dr. Sadullah Sakallıoğlu Cukurova University
Prof. Dr. Samandar Iskandarov National Academy of Science of Kyrgyz Republic
Prof. Dr. Selahattin Kaçıranlar Cukurova University
Prof. Dr. Slavica Ivelić Bradanović University of Split
Prof. Dr. Snezhana Hristova Plovdiv University
Prof. Dr. Süleyman Demir Anadolu University
Prof. Dr. Teoman Özer Istanbul Technical University
Prof. Dr. V. B. Singh University of Delhi
Prof. Dr. Varga Kalantarov Koc University
Prof. Dr. Veli Kurt Akdeniz University
Prof. Dr. Victor Jimenez Lopez Universidad de Murcia
Prof. Dr. Victor Martinez-Luaces Electrochemistry Engineering Multidisciplinary Research Group
Prof. Dr. Yusuf Avcı Bahcesehir University
Assoc. Prof. Dr. A. Neşe Dernek Marmara University
Assoc. Prof. Dr. Adem Cengiz Çevikel Yildiz Technical University
Assoc. Prof. Dr. Ali Arslan Özkurt Cukurova University
Assoc. Prof. Dr. Alp Arslan Kıraç Pamukkale University
Assoc. Prof. Dr. Antonio Miguel Márquez Durán Pablo de Olavide University
Assoc. Prof. Dr. Aynur Keskin Kaymakçı Selcuk University
Assoc. Prof. Dr. Ayşegül Çetinkaya Ahi Evran University
Assoc. Prof. Dr. Bahaddin Sinsoysal Beykent University
Assoc. Prof. Dr. Bülent Yılmaz Marmara University
Assoc. Prof. Dr. Ceren Sultan Elmalı Erzurum Technical University
Assoc. Prof. Dr. Cesim Temel Yuzuncu Yil University
Assoc. Prof. Dr. Cumali Ekici Eskisehir Osmangazi University
Assoc. Prof. Dr. Dilek Çağırgan Gülten Istanbul University
Assoc. Prof. Dr. Dilek Ersalan Cukurova University
Assoc. Prof. Dr. Dilek Tanışlı Anadolu University
Assoc. Prof. Dr. Dušan Holy Trnava University
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Assoc. Prof. Dr. Engin Deligöz Aksaray University
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Assoc. Prof. Dr. Faruk Uçar Marmara University
Assoc. Prof. Dr. Figen Öke Trakya University
Assoc. Prof. Dr. Gülçin Çivi Istanbul Technical University
Assoc. Prof. Dr. Günay Öztürk Kocaeli University
Assoc. Prof. Dr. Handan Çerdik Yaslan Pamukkale University
Assoc. Prof. Dr. İ. Onur Kıymaz Ahi Evran University
Assoc. Prof. Dr. İhsan Ömür Bucak Meliksah University
Assoc. Prof. Dr. İlkay Yaslan Karaca Ege University
Assoc. Prof. Dr. İsa Yıldırım Atatürk University
Assoc. Prof. Dr. İsmail Yaslan Pamukkale University
Assoc. Prof. Dr. İsmet Altıntaş Sakarya University
Assoc. Prof. Dr. Jan Borsik Mathematical Institute of Slovak Academy of Science
Assoc. Prof. Dr. Kashi Nath Saha Jadavpur University
Assoc. Prof. Dr. Mehmet Ali Güngör Sakarya University
Assoc. Prof. Dr. Merce Olle Unıversıtat Polıtecnıca De Catalunya
Assoc. Prof. Dr. Metin Şengül Kadir Has University
Assoc. Prof. Dr. Milan Matejdes Technical University in Zvolen
Assoc. Prof. Dr. Musa Çakır Yuzuncu Yil University
Assoc. Prof. Dr. Mustafa Alkan Akdeniz University
Assoc. Prof. Dr. Mustafa Özdemir Akdeniz University
Assoc. Prof. Dr. Nevin Mahir Anadolu University
Assoc. Prof. Dr. Nilüfer Yavuzsoy Köse Anadolu University
Assoc. Prof. Dr. Özcan Sert Pamukkale University
Assoc. Prof. Dr. Özlem Girgin Atlıhan Pamukkale University
Assoc. Prof. Dr. Selcen Yüksel Perktaş Adiyaman University
Assoc. Prof. Dr. Semra Ahmetolan Istanbul Technical University
Assoc. Prof. Dr. Sergey Kostyrko St. Petersburg State University
Assoc. Prof. Dr. Soley Ersoy Sakarya University
Assoc. Prof. Dr. Süha Yılmaz Dokuz Eylul University
Assoc. Prof. Dr. Şehmuş Fındık Cukurova University
Assoc. Prof. Dr. Şenol Dost Hacettepe University
Assoc. Prof. Dr. Tamer Uğur Atatürk University
Assoc. Prof. Dr. Yasemin Sağıroğlu Karadeniz Technical University
Assoc. Prof. Dr. Yılmaz Dereli Anadolu University
Assoc. Prof. Dr. Zerrin Esmerligil Cukurova University
Assist. Prof. Dr. Adriana Ocejo University of North Carolina at Charlotte
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Assist. Prof. Dr. Ali Akgül Siirt University
Assist. Prof. Dr. Ali Hikmet Değer Karadeniz Technical University
Assist. Prof. Dr. Arzu Özkoç Duzce University
Assist. Prof. Dr. Aslı Güldürdek Izmir Economy University
Assist. Prof. Dr. Aynur Çöl Sinop University
Assist. Prof. Dr. Ayten Koç Istanbul Kultur University
Assist. Prof. Dr. Canan Celep Yücel Pamukkale University
Assist. Prof. Dr. Cennet Eskal Osmaniye Korkut Ata University
Assist. Prof. Dr. Derya Çelik Anadolu University
Assist. Prof. Dr. Elif Gürsoy Usak University
Assist. Prof. Dr. Emin Aygün Erciyes University
Assist. Prof. Dr. Fatma Karakuş Sinop University
Assist. Prof. Dr. Gonca İnceoğlu Anadolu University
Assist. Prof. Dr. Gülistan Kaya Gök Hakkari University
Assist. Prof. Dr. Gülseli Burak Pamukkale University
Assist. Prof. Dr. Gültekin Tınaztepe Akdeniz University
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Assist. Prof. Dr. İrma Hacinliyan Istanbul Technical University
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Assist. Prof. Dr. Mahpeyker Öztürk Sakarya University
Assist. Prof. Dr. Mehmet Gülbahar Siirt University
Assist. Prof. Dr. Memet Kule Kilis 7 Aralik University
Assist. Prof. Dr. Merve Avcı Ardıç Adiyaman University
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Assist. Prof. Dr. Mohanad El-Harbawi King Saud University
Assist. Prof. Dr. Murat Candan İnönü University
Assist. Prof. Dr. Mustafa Dede Kilis 7 Aralik University
Assist. Prof. Dr. Mustafa Kemal Cerrahoğlu Sakarya University
Assist. Prof. Dr. Nader Jafari Rad Shahrood University
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Assist. Prof. Dr. Nedret Elmas Celal Bayar University
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Assist. Prof. Dr. Ricardo Almedia University of Aveiro
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Assist. Prof. Dr. Semra Kaya Nurkan Usak University
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Assist. Prof. Dr. Süleyman Şenyurt Ordu University
Assist. Prof. Dr. Şahin Ceran Pamukkale University
Assist. Prof. Dr. Şeyda Kılıçoğlu Baskent University
Assist. Prof. Dr. Tevfik Şahin Amasya University
Assist. Prof. Dr. Yasemin Kıymaz Ahi Evran University
Assist. Prof. Dr. Yasin Ünlütürk Kırklareli University
Assist. Prof. Dr. Yusuf Ali Tandoğan Bozok University
Assist. Prof. Dr. Zeki Kasap Pamukkale University
Assist. Prof. Dr. Zeynep Özkurt Cukurova University
Lecturer Dr. Erdal Büyükbıçakcı Sakarya University
Lecturer Dr. Mahmoud Jafari Shah Belaghi Bahcesehir University
Lecturer Dr. Murat Beşenk Karadeniz Technical University
Lecturer Dr. Zeynep Büyükbıçakcı Sakarya University
Rsc. Assist. Dr. Canay Aykol Yüce Ankara University
Dr. Akeel Shah University of Warwick
Dr. Branislav Novotny Mathematical Institute of Slovak Academy of Science
Dr. Burak Kurt Akdeniz University
Dr. Cihan Aksop The Scientific and Technological Research Council of Turkey
Dr. Deniz Özen Adnan Menderes University
Dr. George Ciprian Modoi Babes-Bolyai University
Dr. Jasbir S. Manhas Sultan Qaboos University
Dr. Kok Fong See Universiti Sains Malaysia
Dr. Matvey Flax Moscow State University of Information Technologies
Dr. Sajdeh Sajjadi Johns Hopkins University
Dr. Sandeep Singh Thapar University
Lecturer Doğan Ünal Sakarya University
Lecturer Muhsin Çelik Sakarya University
Lecturer Mücahit Kurtuluş Sinop University
Lecturer Şüheda Güray Baskent University
- 332 -
Lecturer Şükrü Kitiş Dumlupınar University
Lecturer Zeynep Can Aksaray University
Rsc. Assist. Alper Ülker Ege University
Rsc. Assist. Aslı Bektaş Kamışlık Recep Tayyip Erdoğan University
Rsc. Assist. Bahar Korkmaz Anadolu University
Rsc. Assist. Başak Gever TOBB University of Economics and Technology
Rsc. Assist. Burhan Tiryakioğlu Marmara University
Rsc. Assist. Emre Kişi Sakarya University
Rsc. Assist. Erkam Lüy Erciyes University
Rsc. Assist. Fatih Aylıkçı Yildiz Technical University
Rsc. Assist. Fatma Aydın Akgün Yildiz Technical University
Rsc. Assist. Gül Güner Karadeniz Technical University
Rsc. Assist. Gülşah Aydın Şekerci Süleyman Demirel University
Rsc. Assist. Hidayet Hüda Kösal Sakarya University
Rsc. Assist. Hülya Gün Bozok Osmaniye Korkut Ata University
Rsc. Assist. İlhan Gül Istanbul Technical University
Rsc. Assist. İlim Kişi Kocaeli University
Rsc. Assist. Kemal Taşköprü Bilecik Seyh Edebali University
Rsc. Assist. Mehmet Alper Ardıç Adiyaman University
Rsc. Assist. Mehmet Onat Cukurova University
Rsc. Assist. Melike Kaplan Eskisehir Osmangazi University
Rsc. Assist. Merve İlkhan Duzce University
Rsc. Assist. Muhammed Abdussamed Maldar Aksaray University
Rsc. Assist. Muradiye Çimdiker Kirklareli University
Rsc. Assist. Mustafa Özkan Trakya University
Rsc. Assist. Nagehan Kılınç Geçer Ahi Evran University
Rsc. Assist. Pantelis Z. Lappas Athens University of Economics and Business
Rsc. Assist. Seda Gülen Ege University
Rsc. Assist. Sedat Temel Erciyes University
Rsc. Assist. Sefa Anıl Sezer Ege University
Rsc. Assist. Sema Bodur Ege University
Rsc. Assist. Serhan Eker Anadolu University
Rsc. Assist. Serkan Akoğul Yildiz Technical University
Rsc. Assist. Sezgin Büyükkütük Kocaeli University
Rsc. Assist. Sezin Aykurt Sepet Ahi Evran University
Rsc. Assist. Sibel Koparal Kocaeli University
Rsc. Assist. Sibel Sevinç Cumhuriyet University
Rsc. Assist. Sibel Turanlı Erzurum Technical University
Rsc. Assist. Şerife Müge Ege Ege University
Rsc. Assist. Tülay Erişir Sakarya University
Rsc. Assist. Volkan Ala Mersin University
Rsc. Assist. Zeynep Çolak Canakkale 18 Mart University
Ahmet Karakaş Erciyes University
Antoine Emil Zambelli Imperial College London
Asra Rezafadaei
Burcu Ayhan Eskisehir Osmangazi University
Elif Ardıyok Duzce University
Esra Akarsu Dokuz Eylul University
Ethem İlhan Şahin Istanbul Technical University
Gülesin Balaban Marmara University
Hans Jörg Ruppen Ecole polytechnique fédérale EPFL
Hatice Tozak Eskisehir Osmangazi University
Hilal Betül Çetin Eskisehir Osmangazi University
Işıl Arda Kösal Sakarya University
İsmail Gülten Yildiz Technical University
Leonid Stanislavovich Mikhlin Saint Petersburg State University
Melike Saruhan Sakarya University
Neslihan Gündoğdu Sakarya University
Nurcan Öztürk The Scientific and Technological Research Council of Turkey
Özge Güller Ankara University
Seda Eren Cukurova University
Sevil Kıvrak Marmara University
Vasileios Triantafyllidis University of Warwick
Wei Xing University of Warwick
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