0 6 $ - IECMSA 2021

web site : http://www.iecmsa.org/E-mail : [email protected]

Book of Abstracts 27-30 August 2019, Baku, Azerbaijan

8th INTERNATIONAL EURASIAN CONFERENCEON MATHEMATICAL SCIENCES AND APPLICATIONS

Dedicated to the 100th Anniversary of Baku State University

IECMSA - 2019

8TH INTERNATIONAL

EURASIAN CONFERENCE

ON

MATHEMATICAL SCIENCES

AND

APPLICATIONS

i

Foreword 1

Dear participants,

Azerbaijan is honored to host the 8th International Eurasian Conference on Mathematical Sciences

and Applications (IECMSA-2019). On behalf of the Baku State University (BSU), I welcome the

representatives of different countries and organizations as well as distinguished members of the inter-

national community, academy and universities. I would like to express my sincere appreciation to the

organizing committee who made this meeting possible in Baku.

The IECMSA is one of the flagship conference of the mathematical society. Organized every year, it

covers all theoretical, computational, and practical aspects of mathematics. Moreover, it is a great

honor for us that such a conference devoted to 100th anniversary of BSU, which is celebrated in 2019.

On September 1, 1919, the Parliament of the Azerbaijan Democratic Republic decided to establish a

university in Baku and approved its Charter. In addition, Mathematics at Baku State University has

a rich history, which originates from 1920. The faculty has since developed into a defining and leading

place in mathematical studies. Today faculty’s education program gives to students a chance to grow

into high level researchers. The mathematical atmosphere in BSU is a uniquely attractive environment

to learn and work, which every year enriched further by the special lecture series on actual topics on

state-of-the-art advances in science and the close collaboration with other faculties, such as physics,

computer science, and engineering, biology, chemistry and with different research centers in the region

and world.

I have no doubt that this conference will unite researchers of all scientific organizations of partici-

pating countries to work together and advance mathematics in a secular and inclusive atmosphere for

the betterment of our collaboration. More than 240 participants from about 43 countries all over theii

world will learn about the most recent developments and results and discuss new challenges from the-

oretical mathematics and its applications. With about seven invited state-of-the-art lectures, totaling

more than 200 presentations, IECMSA-2019 is by far one of the largest conferences so far. Besides

the IECMSA-2019 includes a poster session with 17 posters. These numbers are a clear indication

of the importance of the topics of this conference as a scientific discipline and a key basis for future

developments in numerous application areas.

World-renowned cultural and research centers, a thriving creative scene and rich history while with

modern architectures make Baku a popular place to live, work and travel. During a social program,

participants of the conference are invited to enjoy Baku’s historic city center - Icharishahar and main

sights with colleagues from all over the world. I hope that you will also find the time to take a look

around Baku on your own, to obtain a feeling for the vibrant lifestyle, and to explore the many at-

tractions of this wonderful city.

Finally, I wish to acknowledge, in particular, the members of the program committee, and the session’s

chairs, for setting up the scientific program. My sincere thanks go to the members of the organizing

committee and everyone involved in the local organization - for the many days, weeks and even months

of work. On behalf of our University, I would like to thank each one of the conference speakers and

attendees, as well as academic sponsors, for a successful IECMSA-2019.

I wish you all a pleasant and memorable IECMSA-2019 and a lot of exciting mathematics in the

open-minded and international atmosphere of Baku.

Sincerely

Dr. Elchin BABAYEV

Rector of Baku State University

August 2019

iii

Foreword 2

I welcome you to the 8th International Eurasian Conference on Mathematical Sciences and Applica-

tions (IECMSA-2019) on August 27-30, 2019 in Baku, Azerbaijan. It is an honor for me to inform you

that this conference is dedicated to the 100th Anniversary of the first university of Azerbaijan-Baku

State University which is a leader of educational institutions, has a rich history and today it is known

as one of the most famous scientific and educational centers of Azerbaijan Republic.

IECMSA-2019 is supported by Sakarya University, Baku State University, International Balkan Uni-

versity, Firat University, Tekirdag Namik Kemal University, Kocaeli University, Amasya University,

Gazi University, and Turkic World Mathematical Society.

The series of IECMSA provides a highly productive forum for reporting the latest developments in

the researches and applications of Mathematics. The previous seven conferences held annually since

2012 such that IECMSA-2012, Prishtine, Kosovo, IECMSA-2013, Sarajevo, Bosnia and Herzegovina,

IECMSA-2014, Vienna, Austria, IECMSA-2015, Athens, Greece, IECMSA-2016, Belgrade, Serbia,

IECMSA-2017, Budapest, Hungary, and IECMSA-2018, Kyiv, Ukraine.

The scientific committee members of IECMSA-2019 and the external reviewers invested significant

time in analyzing and assessing multiple papers, consequently, they hold and maintain a high stan-

dard of quality for this conference. The scientific program of the conference features invited talks,

followed by contributed oral and poster presentations in seven parallel sessions.

The conference program represents the efforts of many people. I would like to express my grati-

tude to all members of the scientific committee, external reviewers, sponsors and, honorary committee

for their continued support to the IECMSA. I also thank the invited speakers for presenting their talks

on current researches. Also, the success of IECMSA depends on the effort and talent of researchers in

mathematics and its applications that have written and submitted papers on a variety of topics. So, Iiv

would like to sincerely thank all participants of IECMSA-2019 for contributing to this great meeting

in many different ways. I believe and hope that each of you will get the maximum benefit from the

conference.

Welcome to Baku!

Prof. Dr. Murat TOSUN

Chairman

On behalf of the Organizing Committee

v

Honorary Committee

Prof. Dr. Fatih Savasan (Rector of Sakarya University)

Prof. Dr. Elcin Babayev (Rector of Baku State University)

Prof. Dr. Mehmet Dursun Erdem (Rector of International Balkan University)

Prof. Dr. Kutbeddin Demirdag (Rector of Fırat University)

Prof. Dr. Sadettin Hulagu (Rector of Kocaeli University)

Prof. Dr. Metin Orbay (Rector of Amasya University)

Prof. Dr. Ibrahim Uslan (Rector of Gazi University)

Prof. Dr. H. Hilmi Hacısalihoglu (Honorary President of TWMS)

vi

Scientific Committee

Prof. Dr. Abdeljalil Nachaoui (Universite de Nantes)

Prof. Dr. Ahmet Kucuk (Kocaeli Univeristy)

Prof. Dr. Alberto Cabada Fernandez (University of Santiago De Compostela)

Prof. Dr. Ali A. Ahmedov (Baku State University)

Prof. Dr. Anar Akhmedov (University of Minnesota)

Prof. Dr. Andrey A. Shkalikov (Moscow State University)

Prof. Dr. Araz R. Aliyev (Azerbaijan State Oil and Industry University)

Prof. Dr. Arsham Borumand Saeid (Shahid Bahonar University of Kerman)

Prof. Dr. Asaf Hajiyev (Baku State University)

Prof. Dr. Attila Gilanyi (University of Debrecen)

Prof. Dr. Azamat Akhtyamov (Bashkir State University)

Prof. Dr. Bayram Sahin (Ege University)

Prof. Dr. Bilal T. Bilalov (Baku State University, Azerbaijan)

Prof. Dr. Cengizhan Murathan (Uludag University)

Prof. Dr. Chang Chang Xi (Capital Normal University)

Prof. Dr. Cihan Ozgur (Balıkesir University)

Prof. Dr. Efim Zelmanov (University of California)

Prof. Dr. Emine Mısırlı (Ege University)

Prof. Dr. Fantuzzi Nicholas (University of Bologna)

Prof. Dr. F. Nejat Ekmekci (Ankara University)

Prof. Dr. Ferhan Atici (Western Kentucky University)

Prof. Dr. Grozio Stanilov (University of Sofia)

vii

Prof. Dr. Halis Aygun (Center of Assessment, Selection, and Place-

ment)

Prof. Dr. Hamzaaga Orucov (Baku State University)

Prof. Dr. Hari Mohan Srivastava (University of Victoria)

Prof. Dr. Hellmuth Stachel (Vienna Technical University)

Prof. Dr. Hidayet Huseyinov (Baku State University)

Prof. Dr. Idzhad Sabitov (Lomonosov Moscow State University)

Prof. Dr. Ismihan Bairamov (Izmir University of Economics)

Prof. Dr. Jinde Cao (Southeast University)

Prof. Dr. Josef Mikes (Palacky University Olomouc)

Prof. Dr. Kadri Arslan (Uludag University)

Prof. Dr. Kamil Aydazade (Baku State University)

Prof. Dr. Kazim Ilarslan (Kırıkkale University)

Prof. Dr. Leonid Bokut (Sobolev Institute of Mathematics)

Prof. Dr. Levent Kula (Kırsehir Ahi Evran University)

Prof. Dr. Lyudmila N. Romakina (Saratov State University)

Prof. Dr. Mahmut Ergut (Tekirdag Namık Kemal University)

Prof. Dr. Memmed Yaqubov (Baku State University)

Prof. Dr. Messoud Efendiyev (Helmholtz Zentrum Munchen )

Prof. Dr. Mikail Et (Fırat University)

Prof. Dr. Misir Mardanov (Azerbaijan National Academy of Sciences)

Prof. Dr. Mustafayev Heybetkulu (Van Yuzuncu Yıl University)

Prof. Dr. Mustafa Calıskan (Gazi University)

Prof. Dr. Nazim Kerimov (Khazar University)

Prof. Dr. Nuri Kuruoglu (Istanbul Gelisim University)

Prof. Dr. Oliver Schutze (Cinvestas)

Prof. Dr. Qalina Mehdiyeva (Baku State University)

Prof. Dr. Qeylani Penahov (Azerbaijan National Academy of Sciences)

Prof. Dr. Rauf Amirov (Cumhuriyet University)

Prof. Dr. Sabir Mirzayev (Baku State University)

viii

Prof. Dr. Sadık Keles (Inonu University)

Prof. Dr. Sadi Bayramov (Baku State University)

Prof. Dr. Senol Dost (Hacettepe University)

Prof. Dr. Sergiy Plaksa (National Academy of Science of Ukraine)

Prof. Dr. Sidney A. Morris (Federation University Australia)

Prof. Dr. Surkay Akbarov (Yildiz Technical University)

Prof. Dr. Urfat Nuriyev (Ege University)

Prof. Dr. Vaqif Ibrahimov (Baku State University)

Prof. Dr. Veli Kurt (Akdeniz University)

Prof. Dr. Vaqif S. Quliyev (Azerbaijan National Academy of Sciences)

Prof. Dr. Vijay Gupta (Netaji Subhas University of Technology)

Prof. Dr. Vitalii Shpakivskyi (National Academy of Sciences of Ukraine)

Prof. Dr. Vladimir V. Kisil (University of Leed)

Prof. Dr. Vuqar Mehrabov (Baku State University)

Prof. Dr. Yusif A. Mammadov (Baku State University)

Prof. Dr. Yusuf Yaylı (Ankara University)

ix

Organizing Committee

Prof. Dr. Murat Tosun (General Coordi-

nator)

Sakarya University

Prof. Dr. Arif Salimov Baku State University

Prof. Dr. Aydın H. Kazımzade Baku State University

Prof. Dr. Cristina Flaut Ovidius University

Prof. Dr. Dumitru Baleanu Cankaya University

Prof. Dr. Etibar S. Panahov Baku State University

Prof. Dr. Fikret Aliyev Baku State University

Prof. Dr. Irada N. Aliyeva Baku State University

Prof. Dr. Jasbir Singh Manhas Sultan Qaboos University

Prof. Dr. Ljubisa Kocinac Nis University

Prof. Dr. Mehemmed F. Mehdiyev Baku State University

Prof. Dr. Mehmet Ali Gungor Sakarya University

Prof. Dr. Mirsaid Aripov National University Of Uzbekistan

Prof. Dr. Mohammad Saeed Khan Sultan Qaboos University

Prof. Dr. Nizameddin Sh Iskenderov Baku State University

Prof. Dr. Pranesh Kumar University of Northern British Columbia

Prof. Dr. Soley Ersoy Sakarya University

Prof. Dr. Victor Martinez-Luaces Universidad de Montevideo

Prof. Dr. Yasar Mehraliyev Baku State University

Prof. Dr. Yusif Gasimov Azerbaijan University

Assoc. Prof. Dr. Elvin Azizbayov Baku State University

x

Assoc. Prof. Dr. Emrah Evren Kara Duzce University

Assoc. Prof. Dr. Fuat Usta Duzce University

Assoc. Prof. Dr. Mahmut Akyigit Sakarya University

Assist. Prof. Dr. Hidayet Huda Kosal Sakarya University

Lecturer Furkan Aydın Kahramanmaras Sutcu Imam University

xi

Contents

Foreword 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Foreword 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Honorary Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Scientific Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

INVITED SPEAKERS 1

Recent Results on Absorbing Ideals of Commutative Rings

(A. Badawi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Pure Tensor Fields and Their Applications

(A. Salimov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Differential Operators, Markov Semigroups and Positive Approximation Processes

(F. Altomare) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Are There any Genuine Continuous Multivariate Real-Valued Functions?

(S. A. Morris) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Relativization, Absolutization, and Latticization in Ring And Module Theory

(T. Albu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Blow Up of Solutions of Nonlinear Strongly Damped Wave Equations and Pseudoparabolic

Equations

(V. K. Kalantarov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A Generalized Π-Operator and its Application to the Hypercomplex Beltrami Equation

(W. Sproessig) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

ALGEBRA 11

Fuzzy Semi Maximal Filters in BL-algebras

(A. Paad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Tense Operators on BL-algebras

(A. Paad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Decomposition of Fuzzy Neutrosophic Soft Matrix

(A. Yalcıner) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Amply E-Radical Supplemented Modules

(C. Nebiyev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Cofinitely G-Radical Supplemented Modules

(C. Nebiyev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

H-basis Strata and Lifting Problem for Homogeneous Ideals

(E. Yılmaz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

On Tensor Fields of Type (0,2) in The Semi-Tangent Bundle

(F. Yıldırım) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Some Remarks Regarding Difference Equations of Degree n

(G-M. Zaharia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Some Properties of Local Cohomology Modules

(J. Azami) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Introduction to Fuzzy Topology on Soft Sets

(K. Veliyeva, C. Gunduz Aras, S. Bayramov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

An Extended Study of I-Functors and D-Rich Functors

(M. R. K. Ansari) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

On central Boolean rings and nearrings

(N. Hamsa, K. B. Srinivas, K. S. Prasad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Weakly 2-absorbing Ideals in Non-Commutative Rings

(N. Groenewald) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Abelianity Axiom is not Necessary to Define a Module

(N. Eroglu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A Note on d-Normal Modules

(N. Eroglu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Special Classes of Algebras and some of Their Applications

(R. Vasile) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Introduction on Neutrosophic Soft Lie Algebras

(S. Abdullayev, K. Veliyeva, S. Bayramov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

On Some Identities with Dual K− Pell Bicomplex Numbers

(S. Halıcı, S. Curuk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Zero Divisors of Split Octonion Algebra

(S. Halıcı, A. Karatas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

On Generalization of Fibonacci Dual Octonions

(S. Halıcı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

On Quaternion-Gaussian Lucas Numbers

(S. Halıcı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Essential Ideals and Dimension in Module over Nearrings

(S. P. Kuncham, S. Bhavanari, V. R. Paruchuri) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Quasi-Primary Spectrum and Some Sheaf-Theoretic Properties

(Z. Bilgin, N. A. Ozkirisci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ANALYSIS 47

On the Lambert W Function

(A. Witkowski) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A note on Modified Picard Integral Operators

(B. Yılmaz, D. Aydın Arı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Some New Fixed Point Theorems for Nonlinear Inclusions

(C. Temel, S. Polat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Approximation in Variable Exponent Spaces

(D. Israfilov, E. Kirhan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Approximation Properties of Kantorovich Type Bernstein-Chlodovsky Operators which

Preserve Exponential Function

(D. Aydın Arı, B. Yılmaz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A Study on Certain Sequence Spaces Using Jordan Totient Function

(E. E. Kara, M. Ilkhan, N. Simsek) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Some Order Properties of the Quotients of L-weakly Compact Operators

(E. Bayram) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

On the Matrix Representations of some Compact-like Operators

(E. Bayram) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Refined Some Inequalities for Frames with Specht’s Ratio

(F. Sultanzadeh, M. Hassani, M. E. Omidvar, R. A. Kamyabi Gol) . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A Note on Approximating Finite Hilbert Transform and Quadrature Formula

(F. Usta) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Characterization of Certain Matrix Classes Involving the Space |Cα|p(G. C. H. Gulec) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

The Space bvθk and Matrix Transformations

(G. C. H. Gulec, M. A. Sarıgol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

New Class of Probabilistic Normed Spaces and its Normability

(H. Panackal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Roducts of Weighted Composition Operators and Differentiation Operators between Weighted

Bergman Spacs and Weighted Banach Spaces of Analytic Functions

(J. S. Manhas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Extension of Order Bounded Operators

(K. H. Azar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Reconstruction of Signals from Short Time Fourier Transform

(K. T. Poumai) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Applications of Frames in Quantum Measurement

(K. T. Poumai, S. K. Kaushik) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Compact Operators in the Class(bvθk, bv

)(M. A. Sarıgol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A New Regular Matrix Defined by Jordan Totient Function and its Matrix Domain in `p

(M. Ilkhan, N. Simsek, E. Evren Kara) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

On The Finite Element Approximation of Quasi-variational Inequalities with vanishing zero

order term

(M. Boulbrachene) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Deferred Statistical Convergence in Metric Spaces

(M. Et, M. Cınar, H. Sengul) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A New Type of Generalized Difference Sequence Space m (φ, p, α) (∆nm)

(M. Et, R. Colak) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

On Generalized Deferred Cesaro Mean

(M. Et) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A survey of Neutrosophic Type Baire Spaces

(M. Kirisci, N. Simsek) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Representation of a Solution and Stability for a Sequential Fractional Impulsive Time-Delay

Linear Systems

(N. I. Mahmudov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Fixed Point Theorems on Neutrosophic Metric Spaces

(N. Simsek, M. Kirisci, M. Akyigit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

On Convolution of Boas Transform of Wavelets

(N. Khanna) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

The C∗-Algebra of Toeplitz Operators Associated with Discrete Heisenberg Group

(N. Buyukliev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A New Class of Operator Ideals Defined via s-Numbers and Lp(Φ) Sequence Space

(P. Zengin Alp, E. E. Kara) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A New Paranormed Sequence Space Defined by Catalan Conservative Matrix

(P. Zengin Alp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Statistical Convergence and Operator Valued Series

(R. Kama) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Perturbations of Frames in Quaternionic Hilbert Spaces

(S. K. Sharma, G. Singh, S. Sahu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Some Recent Results on Approximation by Linear Positive Operators

(T. Acar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Representation Theory for Finite Hankel-Clifford Transforms Using Complex Inversion

Operator

(V. R. Lakshmi Gorty) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

On an Existence of the Optimal Shape for One Functional Related with the Eigenvalues of

Pauli Operator

(Y. Gasimov, A. Aliyeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

APPLIED MATHEMATICS 97

A Highly Accurate Difference Method for Solving the Laplace Equation on a Rectangular

Parallelepiped with Boundary Values in Ck,λ

(A. A. Dosiyev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Effective Error Estimate for the Hexagonal Grid Solution of Laplace’s Equation on a Rectangle

(A. A. Dosiyev, S. C. Buranay) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

On Exact Controllability of Semilinear Systems

(A. E. Bashirov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Attractor for Nonlinear Transmission Acoustic Problem

(A. B. Aliev, S. E. Isayeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Differential Type Hysteresis Operators Describing Irreversible Processes in Ferroelectrics

(A. Skaliukh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Returned Sequences and Their Applications

(A. M. Akhmedov, Eldost U. Ismailov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Linear Stability of a Convective Flow in a Vertical Channel Generated by Internal Heat Sources

(A. Kolyshkin, V. Koliskina, I. Volodko) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Optimizing Wiener and Randic Indices of Graphs

(A. Mahasinghe, H. Erandi, S. Perera) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Curvature Stabilization and Thermally Driven Flows

(A. Cıbık) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Stability Analysis of a TB Epidemic Model in a Patchy Environment

(A. Jabbari, S. Fazeli) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Mathematical Analysis of a Fractional-Order Model of Tuberculosis Epidemic with Exogenous

Re-Infection

(A. Jabbari, H. Kheiri, F. Iranzad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Explicit Solutions and Conservation Laws of a Generalized Extended (3+1)-Dimensional

Jimbo-Miwa Equation

(C. M. Khalique) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A Very Efficient Approach for Pricing Geometric Asian Rainbow Options Described by the

Mixed Fractional Brownian Motion

(D. Ahmadian, L. V. Ballestra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Scattering Theory of Dirac Operator with the Impulsive Condition on Whole Axis

(E. Bairamov, S. Solmaz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Infimal Convolution Method for Duality in Second Order Discrete and Differential Inclusions

with Delay

(E. N. Mahmudov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Inverse Sturm-Liouville Problem in the Case Finite-Zoned Periodic Potentials

(E. S. Panakhov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Optimal Successive Complementary Expansion for Singular Differential Equations

(F. Say) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

On the Stability of Bodewadt Flow over a Rough Rotating Disk

(F. Say, B. Alveroglu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

The Method for Defining the Coefficient of Hydraulic Resistance on Different Areas of

Pump-Compressor Pipes in Gas Lift Process

(F. A. Aliev, N.S. Hajiyeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

An Identification Problem for Determining the Parameters of Discrete Dynamic System in

Gas-Lift Process

(F. Aliev, N.S. Hajiyeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Calculation Algorithm Defining the Coefficient of Hydraulic Resistance on Different Areas of

Pump-Compressor Pipes in Gas Lift Process

(F. Aliev, N. S. Hajiyeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

On a Way For Calculation Of The Double Definite Integrals

(G. Mehdiyeva, M. Boyukzade, M. Imanova, V. Ibarhimov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Recent Methods for the Numerical Solution of Hamiltonian Systems

(G. R. Hojjati, A. Abdi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A New Generalization of Dunkl Analogue of Szasz Operators

(G. Icoz, B. Cekim) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

On a New Generalization of Dunkl Analogue of Szasz-Mirakyan Operators

(G. Icoz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Beta Generalization of Stancu-Durrmeyer Operators Involving Analytic Functions

(G. Icoz, H. Eryigit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Continuous Dependence of An Invariant Measure on the Jump Rate of a Piecewise-Deterministic

Markov Process

(H. Wojewodka-Sciazko, D. Czapla, K. Horbacz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A Numerical Solution for Fractional Order Optimal Control in Infectious Disease Models

(H. Kheiri, M. Jafari, F. Iranzad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Mathematical Model for Eddy Current Testing of Cylindrical Structures

(I. Volodko, A. Kolyshkin, V. Koliskina) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Local Properties of Solutions of Trivial Monge-Amper Equation

(I. Kh. Sabitov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Existence of Positive Solution for Caputo Difference Equation and Applications

(K. Ghanbari, T. Haghi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Local and Nonlocal Boundary Value Problems for Hyperbolic Equations with a Caputo

Fractional Derivative

(M. H. Yagubov, S. Sh. Yusubov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

On Dynamical One-Dimensional Models of Thermoelastic Piezoelectric Bars

(M. Avalishvili, G. Avalishvili) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Higher Order Exponential Fuzzy Transform and its Application in Fluid Mechanics

(M. Zeinali, G. Eslami) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Analytical Solutions of Some Nonlinear Fractional-Order Differential Equations by Different

Methods

(M. Odabası, Z. Pınar, H. Kocak) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Investigation of Exact Solutions of Some Nonlinear Evolution Equations via an Analytical

Approach

(M. Odabası) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Uniformly Convergent Difference Schemes for Solving Singularly Perturbed Semilinear Problem

with Integral Boundary Condition

(M. Cakır) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

The Best Constant of Lyapunov-Type Inequality for Fourth-Order Linear Differential Equations

with Anti-Periodic Boundary Conditions

(M. F. Aktas, D. Cakmak, A. Ahmetoglu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

On Lyapunov-Type Inequalities for Various Types of Boundary Value Problems

(M. F. Aktas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

The Boundary-Value Problem for Two-Dimensional Laplace Equation with the Non-Local

Boundary Conditions on Rectangle

(N. A. Aliyev, M. B. Mursalova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

On The Solution of The Optimal Control Problem of Inventory of a Discrete Product In

Stochastic Model Of Regeneration

(N. A. Vakhtanov, P. V. Shnurkov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Modeling Deformation, Buckling and Post-Buckling of Thin Plates and Shells with Defects

under Tension

(N. Morozov, B. Semenov, P. Tovstik) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

On Two-Dimensional Boundary Layer Flows of a Psuedoplastic Fluid — Two Flow

Configurations

(N. C. Sacheti, P. Chandran, T. El-Bashir) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

The Scattering Problem for Hyperbolic System of Equations on Semi-Axis with Three Incident

Waves

(N.Sh. Iskenderov, K.A. Alimardanova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Numerical Method to Solve Fuzzy Boundary Value Problems

(N. Parandin, A. Hosseinpour) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

The Algorithm Solution of the Problem of Optimal Control in a Dynamic One-Sector Economic

Model with a Discrete Time Based on Dynamic Programming Method

(P. V. Shnurkov, A. O. Rudak) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Effects of Temperature Modulation on Natural Convection in a Non-Rectangular Permeable

Cavity

(P. Chandran, N. C. Sacheti, B. S. Bhadauria, A. K. Singh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Representation of Solutions of Neutral Time Delay Equations and Ulam-Hyers Stability

(P. Sabancigil, M. Kara, N. I. Mahmudov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Computational Modeling of the NO+CO Reaction over Composite Catalysts

(P. Katauskis, V. Skakauskas, R. Ciegis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Fractional Solutions of a k−Hypergeometric Differential Equation

(R. Yılmazer, K. K. Ali) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Solutions of Singular Differential Equations by means of Discrete Fractional Analysis

(R. Yılmazer, G. Oztas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

An Effective Computational Approach for Nonlinear Analysis of Imperfect Perforated

Compressed Laminates

(S. A. M. Ghannadpour, M. Mehrparvar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

The Study of One-Dimensional Mixed Problem for One Class of Fourth Order Differential

Equations

(S. J. Aliyev, F. M. Namazov, A. Q. Aliyeva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

On the Spectral Distribution of Symmetrized Toeplitz Sequences

(S. Hon, M. A. Mursaleen, S. Serra-Capizzano) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Quantum Correlation, Coherence & Uncertainty

(S.-M. Fei) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A Hybridized Discontinuous Galerkin Method for Solving Generalized Regularized Long Wave

Equations

(S. Baharlouei, R. Mokhtari) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Non-Instantaneous Impulsive Differential Equations with State Dependent Delay and Practical

Stability

(S. Hristova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

One-Dimensional Finite Element Simulations for Chemically Reactive Hypersonic Flows

(S. Cengizci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Some Numerical Experiments on Singularly Perturbed Problems with Multi-Parameters

(S. Cengizci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Incomplete Block-Matrix Factorization of M -Matrices Using Two Step Iterative Method for

Matrix Inversion and Preconditioning

(S. C. Buranay, O. C. Iyikal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Asymptotic Solutions of the Problem with Properties for Integro-Differential Equations with

Singular Perturbation

(T. H. Huseynov, A. T. Huseynova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Mixing Problems Modeled with Directed Graphs and Multigraphs: Results and Conjectures

(V. Martinez-Luaces) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Solvability and Long-Time Behaviour of Classical Solutions to a Model of Surface Reactions

over Composite Catalysts

(V. Skakauskas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Stability of the Transmission Plate Equation with a Delay Term in the Moment Feedback

Control

(W. Ghecham, S.-E. Rebiai, F. Z. Sidiali) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Lennard-Jones Potentials for Non-Metal Atoms Embedded in Tiv

(X. Yang, J. Hu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

A Time Nonlocal Inverse Problem for the Longitudinal Wave Propagation Equation with

Integral Conditions

(Y. T. Mehraliyev, E. I. Azizbayov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Mathematical Analysis for a Condition of the Hydrodynamic Characteristics

(Y. M. Sevdimaliyev, G. M. Salmanova, R. S. Akbarly) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Mathematical Modeling of the Dynamics of a Hydroelastic System - a Hollow Cylinder with

Inhomogeneous Initial Stresses and Incompressible Fluid

(Y. M. Sevdimaliyev, G. J. Valiyev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Optimal Symmetries of Option Pricing

(Z. Pınar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

On the Solutions of the Population Balance Model for Crystallization Problem

(Z. Pınar, H. Gulec, H. Kocak) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

DISCRETE MATHEMATICS 195

Disjunctive Total Domination Stability in Graphs

(C. Ciftci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Disjunctive Total Domination Number of Central and Middle Graphs of Certain Snake Graphs

(C. Ciftci, A. Aytac) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

On the Spectrum of Threshold Graphs

(E. Ghorbani) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Vectoral Angle Distance for DNA k-mers

(E. S. Oztas, F. Gursoy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Binary Linear Programming on Ramsey Graphs

(S. M. Ayat, A. Akrami, S. M. Ayat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

The Independence Number of Circulant Triangle-free Graphs

(S. M. Ayat, S. M. Ayat, M. Ghahramani) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

GEOMETRY 202

A General Notion of Coherent Systems

(A. H. W. Schmitt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Bezier-like Curves Based on Exponential Functions

(A. Yılmaz Ceylan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Geometry of Elastic Submanifolds in Trans-Sasakian Manifolds

(A. Cetinkaya) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

A New Version of Q-Surface Pencil in Euclidean 3-Space

(A. Yazla, M. T. Sarıaydın) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

New Methods to Construct Slant Helices from Hyperspherical Curves

(B. Altunkaya) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Helix Preserving Mappings

(B. Altunkaya, L. Kula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Constraint Manifolds for Some Spatial Mechanisms in Lorentz Space

(B. Aktas, O. Durmaz, H. Gundogan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

S-Manifolds and Their Slant Curves of Certain Types

(C. Ozgur, S. Guvenc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

On The Directional Evolutions of the Ruled Surfaces depend on A Timelike Space Curve

(C. Ekici, M. Dede, G. U. Kaymanlı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

On The Directional Associated Curves of Timelike Space Curve

(G. U. Kaymanlı, C. Ekici, M. Dede) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

On Some Curvature Conditions of Nearly α−Cosymplectic Manifolds

(G. Ayar, D. Demirhan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

The Homogeneous Lift of A Riemannian Metric in The Linear Coframe Bundle

(H. Fattayev) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

On The Generalized Taxicab Trigonometry

(H. B. Colakoglu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

The Transformation of the Involute Curves using by Lifts on R3 to Tangent Space TR3

(H. Cayır) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Euler-Lagrangian Dynamical Systems with Respect to Horizontal and Vertical Lifts on Tangent

Bundle

(H. Cayır) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Some Results on Null W -curves in E42

(H. Altın Erdem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Algorithm for Solving the Sylvester s-Conjugate Elliptic Quaternion Matrix Equations

(M. Tosun, H. H. Kosal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Rotary Mappings and Transformations

(J. Mikes, L. Ryparova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Some Results on Bertrand and Mannheim Curves

(K. Ilarslan, F. Gokcek) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Geometric Interpretation of Curvature Circles in Minkowski Plane

(K. Eren, S. Ersoy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

New Representation of Hasimoto Surfaces According to the Modified Orthogonal Frame

(K. Eren) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Geometry of Complex Coupled Dispersionless and Complex Short Pulse Equations by Using

Bishop Frames

(K. Eren) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

New Associated Curves and Their Some Geometric Properties in Euclidean 3-Space

(M. T. Sarıaydın) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

On The Directional Spherical Indicatrices of Timelike Space Curve

(M. Dede, G. U. Kaymanlı, C. Ekici) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

On Generalized Partially Null Mannheim Curves

(N. Kılıc Aslan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

On the Trajectory Ruled Surface of Framed Base Curves in E3

(O. G. Yıldız, M. Akyigit, M. Tosun) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

The gh-Gifts of Affine Connections on the Cotangent Bundle

(R. Cakan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

On the Involute of the Cubic Bezier Curves in E3

(S. . Kılıcoglu, S. Senyurt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Spinor Formulation of Bertrand Curves in E3

(T. Erisir, N. C. Kardag) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

MATHEMATICS EDUCATION 244

Kazan University and Development of Geometry in Azerbaijan

(R. M. Aslanov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Measurement of Achievement Distribution by Gini Coefficient Approach: An Application for

Statistics Course

(S. Guray) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Inverse Modeling Problems and Tasks Enrichment: Analysis of two Experiences with Spanish

Prospective Teachers

(V. Martinez-Luaces, J. A. Fernandez-Plaza, L. Rico) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

OTHER AREAS 250

Some New Results on Path Integration Methods

(A. Naess, L. Chen, E. R. Jakobsen) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Wrinkling of Annular Plates and Spherical Caps With Material Inhomogeneity

(E. Voronkova, S. Bauer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Some Properties of s-reducibility

(I. Chitaia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Omar Khayyam: Calendric Calculations, Cosmic poetry and Paintings Reflecting His Poetry

(V. Nikulina, M. R. K. Ansari) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

STATISTICS 257

Transmuted Lower Record Type Frechet Distribution

(C. Tanıs, B. Saracoglu, C. Kus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Goodness of Fit Test For Weibull Distribution Based on Kullback Leibler Divergence under

Progressive Hybrid Censoring

(I. Kınacı, G. Gencer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

A New Unit-Weibull Distribution

(K. Karakaya, I. Kınacı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Optimal Logistic Regression Estimator

(N. N. Urgan, D. Gungormez) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Bootstrap Confidence Intervals of Capability Index CPM Based on Progressively Censored

Data

(Y. Akdogan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

TOPOLOGY 264

On a∗-I -Open Sets and a Decomposition of Continuity

(A. Keskin Kaymakcı) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Common Fixed Point Results on Modular F-Metric Spaces and an Application

(D. Turkoglu, N. Manav) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

The Sheaves Representation of Hausdorff Spectra of Locally Convex Spaces

(E. I. Smirnov, S. A. Tikhomirov, E. A. Zubova) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Some Fixed Point Theorems for Multivalued Mappings on Complete Metric Spaces

(H. Aslan Hancer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

A Natural Way to Construct an Almost Hermitian B-Metric Structure

(M. Solgun, Y. Karababa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Some Fixed Point Theorems in Extended b−Metric Spaces with Applications

(M. S. Khan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

(Anti) Symmetrically Connected Extensions

(N. Javanshir, F. Yıldız) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

Some Solutions to the Recent Open Problems with Pata and Zamfirescu’s Techniques

(N. Yılmaz Ozgur, Nihal Tas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Some Results for Ψ− F−Geraghty Contraction on Metric-Like Sopace

(O. Acar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

On Virtual Braids and Virtual Links

(V. Bardakov) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

POSTERS 276

Evolution of Quaternionic Curve in the Semi-Euclidean Space E42

(A. Kızılay, O. G. Yıldız, O. Z. Okuyucu) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Balance and Symmetry in Abiyev Squares E42

(A. A. Abiyev, Y. Alizada) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Some Properties of Wajsberg Algebras

(C. Flaut) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Some Characterizations of Vi Helices in 4-dimensional Semi Euclidean Space with Index 2

(H. Altınbas, B. Altunkaya, L. Kula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Some Results on GBS Operators

(H. G. Ince Ilarslan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

On Generalized Expansive Mappings in the Setting of Elliptic Valued Metric Spaces

(I. Arda Kosal, M. Ozturk, H. H. Kosal ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

The Shannon Entropy as an Edge Detector in Grayscale Images

(J. Martınez-Aroza, J.F. Gomez-Lopera, D. Blanco-Navarro,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Investigation of The Sleep Quality of Cerebrovascular Patients

(K. Sanli Kula, A. Yetis, E. Gurlek) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

A Soft Set Approach for IFS

(K. Taskopru) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

On Lorentzian Ruled Surfaces in 4-Dimensional Semi Euclidean Space with Index 2

(K. Karakas, H. Altınbas, B. Altunkaya, L. Kula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Some New Associated Curves in Minkowski 3-Space

(M. Ergut, A. Kelleci ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

De-Moivre and Euler Formulae for Dual-Hyperbolic Numbers

(M. A. Gungor, E. Kahramani) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

On the Jerk in Motion Along a Space Curve

(M. Guner) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

On Asymptotic Aspect of Some Functional Equations in Metric Abelian Groups

(M. B. Moghimi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

A Unified Approach to Fractal Hilbert-type Inequalities

(T. Batbold, M. Krnic, P. Vukovic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

List of Participants of IECMSA-2019 297

INVITEDSPEAKERS

Recent Results on Absorbing Ideals of Commutative Rings

Ayman Badawi 1

Abstract. Let R be a commutative ring with 1 6= 0. Recall that a proper ideal I of R is called

a 2-absorbing ideal of R if a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I . A more

general concept than 2-absorbing ideals is the concept of n-absorbing ideals. Let n ≥ 1 be a positive

integer. A proper ideal I of R is called an n-absorbing ideal of R if a1, a2, ..., an+1 ∈ R and

a1a2 · · · an+1 ∈ I, then there are n of the ai’s whose product is in I. The concept of n-absorbing ideals

is a generalization of the concept of prime ideals (note that a prime ideal of R is a 1-absorbing ideal of

R). In this talk, we will state recent developments on the study of absorbing ideals of commutative rings.

Keyword: Prime, primary, weakly prime, weakly primary, 2-absorbing, n-absorbing, weakly 2-absorbing,

weakly n-absorbing, 2-absorbing primary, weakly 2-absorbing primary.

References

[1] D. F. Anderson and A. Badawi, On ((m,n)-closed ideals of commutative rings, To appear in Journal of Algebra and

Its Applications. DOI: 10.1142/S021949881750013X.

[2] D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39, 1646–1672, 2011.

[3] D. F. Anderson and A. Badawi, On (m,n)-closed ideals of commutative rings, J. Algebra Appl. 16 , no. 1, 1750013,

21, 2017.

[4] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75, 417–429, 2007.

[5] A. Badawi, n-absorbing ideals of commutative rings and recent progress on three conjectures: a survey, Rings,

polynomials, and modules, 33-52, Springer, Cham, 2017.

[6] A. Badawi, M. Issoual and N. Mahdou, On n-absorbing ideals and (m,n)-closed ideals in trivial ring extensions of

commutative rings, (Available on Line), to appear in Journal of Algebra and Its Applications.

[7] D. Bennis and A·B. Fahid, Rings in which every 2-absorbing ideal is prime, Beitr Algebra Geom. 59, 391–396, 2018.

1Department of Mathematics & Statistics, The American University of Sharjah, P.O. Box 26666, Sharjah, United

Arab Emirates, [email protected]

1

[8] P. J. Cahen, M. Fontana, S. Frisch, and S. Glaz, Open problems in commutative ring theory, Commutative Algebra.

Springer, 353–375, 2014.

[9] H. Seung Choi and A. Walker, The radical of an n-absorbing ideal, arXiv:1610.10077 [math.AC] (2016) (to appear

in Journal of Commutative Algebra).

[10] A. Yousefian Darani and E.R. Puczyowski, On 2-absorbing commutative semigroups and their applications to rings,

Semigroup Forum, 86, 83–91, 2013.

[11] . Issoual and N. Mahdou, Najib Trivial extensions defined by 2-absorbing-like conditions, J. Algebra Appl. 17, no.

11, 1850208, 10 pp, 2018.

[12] H. Fazaeli Moghimi and S. Rahimi Naghani, On n-absorbing ideals and the n-Krull dimension of a commutative

ring, J. Korean Math. Soc. 53, 1225-1236, 2016.

[13] H. Mostafanasab and A. Yousefian Darani, On n-absorbing ideals and two generalizations of semiprime ideals. (on

line), to appear in Thai Journal of Mathematics.

[14] P. Nasehpour, On the Anderson-Badawi ωR[X] (I [X]) = ωR (I) conjecture. Archivum Mathematicum (BRNO, 52,

71–78, 2016.

[15] M. Mukhtar, M. Tusif Ahmad and T. Dumitrescu, Commutative rings with two-absorbing factorization. Commun.

Algebra, 46, 970–978, 2018.

[16] A. Laradji, On n-absorbing rings and ideals, Colloq. Math., 147, 265–273, 2017.

2

Pure Tensor Fields and Their Applications

Arif Salimov 1

Abstract. In the first part of our presentation we give the fundamental results and some concepts

concerning geometry of hypercomplex manifolds which will be needed for the later treatment of some

types of hypercomplex manifolds. We introduce a pure tensor fields on hypercomplex manifolds and

show that such tensors are a real models of algebraic tensors. In the second part we show that if a

torsion tensor of anti-Hermitian metric connection [1] is pure, then the given anti-Hermitian manifold

is anti-Kahler. We prove that if an anti-Hermitian manifold is a conformally flat anti-Kahler-Codazzi

manifold, then the scalar curvature vanishes if and only if the given manifold is isotropic anti-Kahler.

We also consider anti-Hermitian metrics of Hessian type defined by holomorphic Hamiltonian func-

tions. Finally, we consider an example of anti-Kahler metrics on Walker 4-manifold. Let now Mn

be a differentiable manifold and T (Mn) its tangent bundle. Two types of lift problems have been

studied in the previous works: a) The lift of structures (functions, vector fields, forms, tensor fields,

linear connections, etc.) from the base manifold to the tangent bundle; b) The definition of geometric

structures on the total manifold T (Mn), by means of a specific geometric structure on Mn or on the

fibre bundle T (Mn). In the third part of present working we continue such a study by considering the

structure given by the dual numbers on the tangent bundle and defining new lifts of functions, vector

fields, forms, tensor fields and linear connections. Finally, we investigate the complete lift CϕT∗M of

almost complex structure ϕ to cotangent bundle and prove that it is a transform by symplectic-musical

isomorphism ωe′

of complete lift CϕT∗M to tangent bundle if the triple (M,ω, ϕ) with pure symplectic

2-form ω is an almost holomorphic Norden A-manifold [2].

Keyword: Hypercomplex algebra; anti-Hermitian (Norden) metrics; lift.

AMS 2010: 53C15; 53C12.

References

[1] A. Salimov, On anti-Hermitian metric connections, C. R. Math. Acad. Sci. Paris, 352, 731-735, 2014.

1Department of Algebra and Geometry, Baku State University, AZ1148, Baku, Azerbaijan, [email protected]

3

[2] A. Salimov, M. B. Asl and S. Kazimova, Problems of lifts in symplectic geometry, Chin. Ann. Math. Ser. B,40,

321-330, 2019.

4

Differential Operators, Markov Semigroups and Positive Approximation Processes

Francesco Altomare 1

Abstract. The talk will be centered about a topic concerning three interrelated subjects: positive

approximating operators, positive C0−semigroups of operators and initial-boundary value evolution

problems.

The main aim is to study those sequences (Ln)n≥1 of bounded linear operators on a Banach space E

which give rise to a C0−semigroup (T (t))t≥0 of operators on E such that for every t ≥ 0 and u ∈ E

T (t)u = limn→∞

Lk(n)n u in E, (1)

where (k(n))n≥1 is an arbitrary sequence of positive integers satisfying k(n)/n → t as n → ∞, and

each Lk(n)n denotes the iterate of order k(n) of Ln.

To such a semigroup there are naturally associated its infinitesimal generator A : D(A)→ E, which is

defined on a dense subspace D(A) of E, and the relevant abstract Cauchy problem, namely

du(t)

dt= Au(t) t ≥ 0,

u(0) = u0 u0 ∈ D(A).

(2)

When E is a continuous function space on a domain K of Rd, d ≥ 1, the operator A is, in fact, a

differential operator and problem (2) turns into an initial-boundary value evolution problem

∂u

∂t(x, t) = A(u(·, t))(x) x ∈ K, t ≥ 0,

u(x, 0) = u0(x) u0 ∈ D(A), x ∈ K,

(3)

the boundary conditions being incorporated in the domain D(A).

1University of Bari, Italy, [email protected]

5

Abstract. (Continuation) Moreover, problem (2) (resp. problem (3)) has a unique solution if and

only if u0 ∈ D(A). In such a case, the solution is given by

u(t) = T (t)u0 (t ≥ 0) (resp. u(x, t) = T (t)u0(x) (x ∈ K, t ≥ 0).

and hence, by using (1),

u(x, t) = T (t)(u0)(x) = limn→∞

Lk(n)n (u0)(x), (4)

where the limit is uniform with respect to x ∈ K.

Thus, if it is possible to determine the operator A and its domain D(A), the initial sequence (Ln)n≥1

become the key tool to approximate and to study (especially, from a qualitative point of view) the

solutions of problem (2) or (3).

The principal ideas and some of the more recent results on such functional analytic approach to

study problems like (2) or (3), will be discussed in the context of continuous function spaces by also

assuming that the operators Ln, n ≥ 1, are positive. Moreover, particular attention will be devoted

to the important case when the approximating operators are constructively generated by a given

positive linear operator T : C(K)→ C(K) which, in turn, allows to determine the differential operator

(A,D(A)) as well, K being a compact subset of Rd, d ≥ 1, having non-empty interior.

Initial-boundary value evolution problems corresponding to this particular setting, occur in the study

of diffusion problems arising from different areas such as biology, mathematical finance and physics.

For more details and for several other aspects related to the above outlined theory, the reader is referred

to [1] - [3].

References

[1] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa, Differential operators, Markov semigroups and

positive approximation processes associated with Markov operators, de Gruyter Series Studies in Mathematics, Vol.

61, De Gruyter, Berlin, 2014.

[2] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa, A generalization of Kantorovich operators for convex

compact subsets, Banach J. of Math. Anal., 11(3), 591 - 614, 2017.

[3] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa, Elliptic differential operators and positive semigroups

associated with generalized Kantorovich operators,J. of Math. Anal. and Appl., 458, 153 - 173, 2018.

6

Are There any Genuine Continuous Multivariate Real-Valued Functions?

Sidney A. Morris 1

Abstract. David Hilbert asked this question as his 13th question at the International Congress of

Mathematicians in Paris in 1900 and expected a positive answer. As even a beginning calculus student

might note every continuous real-valued function of two variables they meet is simply a composition

and addition of continuous functions of one variable. Hilbert conjectured that there are continuous

functions of 3 variables which cannot be expressed as composition and addition of continuous functions

of two variables. It took over 50 years to prove that Hilbert’s conjecture is false. The solution was pro-

vided by Kolmogorov and Arnol’d. In this talk the result, its proof and applications will be discussed.

The speaker’s contribution is only in terms of applications.

1Emeritus Professor, Federation University, Australia

Adjunct Professor, La Trobe University, Australia, [email protected]

7

Relativization, Absolutization, and Latticization in Ring And Module Theory

Toma Albu 1

Abstract. The aim of this talk is to illustrate a general strategy which consists on putting a ring/module-

theoretical result into a latticial frame (we call it latticization), in order to translate that result to

Grothendieck categories (we call it absolutization) and module categories equipped with hereditary

torsion theories (we call it relativization). The renowned Hopkins-Levitzki Theorem and Osofsky-Smith

Theorem from Ring and Module Theory are among the most relevant illustrations of this strategy.

An effort will be made to keep the exposition as self-contained as possible.

1Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania, [email protected]

8

Blow Up of Solutions of Nonlinear Strongly Damped Wave Equations and

Pseudoparabolic Equations

Varga K. Kalantarov 1

Abstract. The talk will be devoted to the problems of blow up in a finite time of solutions of the

Cauchy problem and initial boundary value problems for nonlinear strongly damped wave equations

and nonlinear pseudoparabolic equations. Recent results on blow up of solutions of quasilinear strongly

damped wave equations and quasilinear pseudoparabolic equations with arbitrary positive initial en-

ergy will be also discussed.

1Department of Mathematics, Koc University, Istanbul,

Department of General and Applied Mathematics, ASOIU, Baku, [email protected]

9

A Generalized Π-Operator and its Application to the Hypercomplex Beltrami

Equation

Wolfgang Sproessig 1

Abstract. The lecture is based on results obtained by K. Guerlebeck, U. Kaehler, H. Malonek, J.

Morais, M. Shapiro, N. Vasilevski and myself. We will introduce the complex Π-operator and study sev-

eral hypercomplex generalizations. In perticular we will obtain mapping properties of the correspond-

ing multidimensional generalizations in classes of Sobolev spaces. Norm estimates will deduced. Con-

nections to the operator calculus in hypercomplex function theory will be studied. All these results are

used to considered for the treatment of boundary value problems of the higher-dimensional Beltrami

equation. Further applications to so-called M-conformal mappings (M= monogenic) are presented.

1TU Bergakademie Freiberg - University, Freiberg, Germany, [email protected]

10

ALGEBRA

Fuzzy Semi Maximal Filters in BL-algebras

Akbar Paad 1

Abstract. In this paper, the concept of fuzzy semi maximal filter in BL-algebras is introduced and

several property of fuzzy semi maximal filters are proved. Using a level subset of a fuzzy set in a

BL-algebra, we give characterization of fuzzy semi maximal filters. Moreover, the homomorphic image

and preimage of fuzzy semi maximal filters are also fuzzy semi maximal filters are proved. Finally, we

study relationship between fuzzy semi maximal filters and semi simple BL-algebras.

Keyword: BL-algebra, fuzzy semi maximal filter, semi simple BL-algebra.

AMS 2010: 06D33, 06E99.

References

[1] R.A. Borzooei and A. Paad, Integral filters and integral bl-algebras, Italian Journal of Pure and Applied Mathematics,

30, 303-316, 2013.

[2] A. Paad and R.A. Borzooei, Generalization of integral filters in bl-algebras and n-fold integral bl-algebras, Afr. Mat.,

26,7-8, 1299-1311, 2015.

[3] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88, 467-490, 1958.

[4] A. Di Nola, G. Georgescu and A. Iorgulescu, Pseduo bl-algebras. part i., Mult Val Logic, 8,5-6, 673-714, 2002.

[5] S. Motamed, L. Torkzadeh, A.B. Saeid and N. Mohtashamnia, Radical of filters in bl-algebras, Math. Log. Quart.,

57(2), 166-179, 2011.

[6] L. Lianzhena and L. Kaitaia, Fuzzy boolean and positive implicative filter of bl-algebras, Fuzzy Sets and Systems,

152, 141-154, 2005.

[7] P. Hajek, Metamathematics of fuzzy logic, Klower Academic Publishers, Dordrecht 1999.

[8] E. Turunen, Bl-algebras and basic fuzzy logic, Mathware. Soft. Comput., 6, 49-61, 1999.

1University of Bojnord, Bojnord, Iran, [email protected], [email protected]

11

Tense Operators on BL-algebras

Akbar Paad 1

Abstract. In this paper, the notions of tense operators and tense filters in BL-algebras are studied

and several characterizations of them are obtained, for example tense filter generated by a nonempty

subset is characterized. Also, it is shown that the set of all tense filters of a BL-algebra is complete

sublattice of F (L) of all filters of BL-algebra L. Moreover, maximal tense filters and simple tense

BL-algebras are introduced and relation between them are studied.

Keyword: BL-algebra, tense operators, tense filter.

AMS 2010: 06D33, 06E99.

References

[1] J. Burges, Basic Tense Logic, in: D.M. Gabbay, F. Gunther (Eds.), Handbook of philosophical logic, vol. II, D. Reidel

Publ. Comp., 89-139, 1984.

[2] D. Diaconescu and G. Georgescu, Tense operators on MV-algebras and Lukasiewicz-Moisil algebras, Fundamenta

Informaticae, 81, 379-408, 2007.

[3] C. Lele and J. B. Nganou, MV -algebras derived from ideals in BL-algebra, Fuzzy Sets and Systems, 218, 103-113,

2013.

[4] P. Hajek, Metamathematics of fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1988.

[5] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic, 40, 467-473, 2001.

[6] E. Turunen and S. Sessa, BL-algebra and Basic Fuzzy Logic, Math Ware and Soft Compute, 49-61, 1999.

1University of Bojnord, Bojnord, Iran, [email protected], [email protected]

12

Decomposition of Fuzzy Neutrosophic Soft Matrix

Aynur Yalcıner 1

Abstract. The concept of neutrosophic set was introduced by Smarandache [4] which is a general-

ization of fuzzy logic and related systems. In [1,2], the authors defined fuzzy neutrosophic soft matrix

of the fuzzy neutrosophic soft set. Arokiarani and Sumathi introduced some operations on fuzzy

neutrosophic soft matrix in [3].

In this talk, we present new operators on fuzzy neutrosophic soft matrix. Then we obtain a decompo-

sition of fuzzy neutrosophic soft matrix by using this operators.

Keyword: Fuzzy neutrosophic soft matrix, fuzzy neutrosophic soft set

AMS 2010: 03E72, 15B15.

References

[1] I. Arockiarani and I.R.Sumathi, A fuzzy neutrosophic soft matrix approach in decision making, JGRMA, 2, 14-23,

2014.

[2] M. Dhar, S. Broumi and F. Smarandache, A note on square neutrosophic fuzzy matrices, Neutrosophic Sets and

Systems 3, 37-41, 2014.

[3] I.R. Sumathi and I. Arockiarani, New operations on fuzzy neutrosophic soft matrices, International Journal of Inno-

vative Research and Studies 13, 110-124, 2014.

[4] F. Smarandache, Neutrosophic set, a generialization of the intuituionistics fuzzy sets, Inter. J. Pure Appl. Math. 24,

287-297, 2005.

1Selcuk University, Konya, Turkey, [email protected]

13

Amply E-Radical Supplemented Modules

Celil Nebiyev 1

Abstract. In this work, all modules are unital left modules. Let M be an R−module. If every

essential submodule of M has a ample Rad-supplements in M , then M is called an amply e-radical

supplemented (or amply e-Rad-supplemented) module. Clearly we can see that every amply e-radical

supplemented module is amply e-radical supplemented. In this work, some properties of amply e-

radical supplemented modules are investigated.

Keyword: Radical, supplemented modules, radical (generalized) supplemented modules, E-radical sup-

plemented modules.

Some Results

Proposition 1. Let M be an amply e-radical supplemented module. Then M/RadM have no proper

essential submodules.

Lemma 1. Let M be an amply e-radical supplemented R−module. Then every factor module of M is

amply e-radical supplemented.

Corollary 1. Let M be an amply e-radical supplemented R−module. Then every homomorphic image

of M is amply e-radical supplemented.

Lemma 2. Let M be an R−module. If every submodule of M is e-Rad-supplemented, then M is amply

e-Rad-supplemented.

Proposition 2. Let R be any ring. Then every R−module is e-Rad-supplemented if and only if every

R−module is amply e-Rad-supplemented.

1Department of Mathematics, Ondokuz May University, Samsun, Turkey, [email protected]

14

References

[1] E. Buyukasık and C. Lomp, On a Recent Generalization of semiperfect rings, Bulletin of the Australian Mathematical

Society, 78, 317-325, 2008.

[2] Celil Nebiyev, E-radical supplemented modules, Presented in International Conference on Mathematics and Mathe-

matics Education’ , Ordu-Turkey, 2018.

[3] Y. Wang and N. Ding, Generalized supplemented modules, Taiwanese Journal of Mathematics, 10 No.6, 1589-1601,

2006.

[4] R. Wisbauer, Foundations of module and ring theory, Gordon and Breach, Philadelphia, 1991.

15

Cofinitely G-Radical Supplemented Modules

Celil Nebiyev 1

Abstract. In this work, all rings have unity and all modules are unital left modules. Let M be an

R−module. If every cofinite submodule of M has a g-radical supplement in M , then M is called a

cofinitely g-radical supplemented module. In this work some properties of cofinitely g-radical supple-

mented modules are investigated. It is clear that every cofinitely g-supplemented module is cofinitely g-

radical supplemented. Hence cofinitely g-radical supplemented modules are more general than cofinitely

g-supplemented modules.

Keyword: Small submodules, supplemented modules, G-supplemented modules, cofinitely G-supplemented

modules.

Results

Lemma 1. Let M be an R-module, K ≤ M and U be a cofinite submodule of M . If U + K has a

g-radical supplement in M and K is cofinitely g-radical supplemented, then U has a g-radical supple-

ment in M .

Corollary 1. Let M be an R-module, M1,M2, ...,Mk ≤ M and U be a cofinite submodule of M . If

U +M1 +M2 + ...+Mk has a g-radical supplement in M and Mi is cofinitely g-radical supplemented

for every i = 1, 2, ..., n, then U has a g-radical supplement in M .

Proposition 1. Let M be an R−module and M =∑i∈IMi for Mi ≤M . If Mi is cofinitely g-radical

supplemented for every i ∈ I, then M is cofinitely g-radical supplemented.

Lemma 2. Every factor module of a cofinitely g-radical supplemented module is cofinitely g-radical

supplemented.

Corollary 2. Every homomorphic image of a cofinitely g-radical supplemented module is cofinitely

g-radical supplemented.

1Department of Mathematics, Ondokuz May University, Samsun, Turkey, [email protected]

16

References

[1] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting modules supplements and projectivity in module theory, Frontiers

in Mathematics, Birkhauser, Basel, 2006.

[2] B. Kosar, Cofinitely G-supplemented modules, British Journal of Mathematics and Computer Science, 17 No.4, 1-6,

2016.

[3] B. Kosar, C. Nebiyev and A. Pekin, A generalization of G-supplemented modules, Miskolc Mathematical

Notes(Accepted).

[4] B. Kosar, C. Nebiyev and N. Sokmez, G-supplemented modules, Ukrainian Mathematical Journal, 67 No.6, 975-980,

2015.

[5] C. Nebiyev and H. H. Okten, Weakly G-supplemented modules, European Journal of Pure and Applied Mathematics,

10 No.3, 521-528, 2017.

[6] Y. Wang and N. Ding, Generalized supplemented modules, Taiwanese Journal of Mathematics, 10 No.6, 1589-1601,

2006.

[7] R. Wisbauer, Foundations of module and ring theory, Gordon and Breach, Philadelphia, 1991.

17

H-basis Strata and Lifting Problem for Homogeneous Ideals

Erol Yılmaz 1

Abstract. Let K be a homogeneous ideal of K[x1, . . . , xn−1] where K is a field. An homogeneous

ideal I of K[x1, . . . , xn−1, xn] is called a lifting of J if

(a) xn is not a zero divisor in K[x1, . . . , xn−1, xn]/I;

(b) J = 〈f(x1, . . . , xn−1, 0)|f ∈ I〉.

Finding all liftings of a given ideal J is called the lifting problem for homogeneous ideals. This problem is

originally stated and studied in [1]. Since then many authors are investigated this interesting problem

(see [2],[3],[4]]. The lifting problem has been recently tried to solve via Grobner strata in [5]. The

Grobner strata approach however is contrary to the spirit of the problem. So their method involves

many unnecessary computations and does not find all the liftings of a homogeneous ideals. The lifting

problem and H-bases are associated for the first time in [6]. Using this relation they gave a method for

the solution of the lifting problem in case of monomial ideals. Later, the authors used syzygy modules

and H-bases and obtained the liftings of some homogeneous ideals in [4].

In this study, H-basis strata is defined. It is shown that H-basis strata is equivalent to the family of

lifting of a homogeneous ideal. Using the ideas given in [4], a method for finding H-basis strata is

obtained. This method is free of Grobner basis computation and finds all the liftings of a homogeneous

ideal. The further properties of H-basis strata are also investigated. The results are demonstrated with

examples.

Keyword: H-basis strata, lifting problem, syzygy modules.

AMS 2010: 13P10, 13D02.

References

[1] M. Roitman, On the lifting problem for homogeneous ideals in polynomial rings, J. Pure Appl. Algebra 51(1-2),

205-215, 1988.

[2] Leslie G. Roberts, On the lifting problem over an algebraically closed field, C. R. Math. Rep. Acad. Sci. Canada

11(1), 35-38, 1989.

[3] J. Migliore, U. Nagel, Lifting monomial ideals, Commun. Algebra 28(12), 5679-5701, 2000.

1Bolu Abant Izzet Baysal University, Bolu, Turkey, yilmaz [email protected]

18

[4] T. Luo and E. Yılmaz, On the lifting problem for homogeneous ideals, J. Pure Appl. Algebra 162(2-3), 327-335, 2001.

[5] C. Bertonea, F. Cioffi, M. Guida and M. Roggero, The scheme of liftings and applications, J. Pure Appl. Algebra

220, 34–54, 2016.

[6] G. Carra Ferro and L. Robbiano, On super G-bases, J. Pure Appl. Algebra 68(3), 279-292, 1990.

19

On Tensor Fields of Type (0,2) in The Semi-Tangent Bundle

Furkan Yıldırım 1

Abstract. In this paper the some lifts of tensor fields of type (0, 2) to semi-tangent bundle and their

lift problems are investigated.

Keyword: Complete lift, degenerate metric, horizontal lift, pull-back bundle, semi-tangent bundle.

AMS 2010: 53A45, 53B05, 53B30, 55R10, 55R65, 57R25.

References

[1] K. Yano, S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York, 1973.

[2] D. Husemoller, Fibre bundles, springer, New York, 1994.

[3] H. B. Lawson, M. L. Michelsohn, Spin geometry, Princeton University Press., Princeton, 1989.

[4] A. A. Salimov, E. Kadıoglu, Lifts of derivations to the semitangent bundle, Turk J. Math. 24, 259-266, 2000.

[5] N. Steenrod, The topology of fibre bundles, Princeton University Press., Princeton, 1951.

[6] L. S. Pontryagin, Characteristic cycles on differentiable manifolds, Amer. Math. Soc. Translation, no. 32, 72 pp.,

1950.

[7] W. A. Poor, Differential geometric structures, New York, McGraw-Hill 1981.

[8] N. M. Ostianu, Step-fibred spaces, Tr. Geom. Sem. 5, Moscow. (VINITI), 259-309, 1974.

[9] V. V. Vishnevskii, Integrable affinor structures and their plural interpretations, Geometry, 7.J. Math. Sci. (New York)

108, no. 2, 151-187, 2002.

[10] V. Vishnevskii, A.P. Shirokov and V.V. Shurygin, Spaces over algebras, Kazan. Kazan Gos. Univ. 1985 (in Russian).

1Ataturk University, Erzurum, Turkey, [email protected]

20

Some Remarks Regarding Difference Equations of Degree n

Geanina-Mariana Zaharia 1

Abstract. Difference equations have many applications in various domains. In the following, we

present some of these applications and directions in which these equations have found applicability, as

for example in Cryptography. For this purpose a new method for encrypting and decrypting messages

is presented. This new method provides multiple ways of finding the encryption and decryption keys,

with the added advantage that each natural number n has a unique representation by using the terms

of such a sequence.

Another application are related to connections between quaternions and some special number sequences,

as for example Fibonacci sequence. Are obtained Fibonacci quaternions or generalized Fibonacci

quaternions which are very helpful in the study of the generalized quaternion algebras, providing a

class of invertible elements in these algebras.

Keyword: Difference Equations , fibonacci quaternions, generalized quaternion algebras.

AMS 2010: 15A24, 15A06, 16G30, 1R52, 11R37,11B39.

References

[1] A. C. Atkinson, Tests of pseudo-random numbers, Applied Statistics, 29, 164-171, 1980.

[2] R. P. Agarwal, J. Y. Wong, Patricia, Advanced topics in difference equations, Springer Netherlands, 510 p., 1997.

[3] Jr. J. L. Brown, Note on complete sequences of integers, The American Mathematical Monthly, 68(6), 557-560, 1961.

[4] E. Cho, De-Moivre’s formula for quaternions, Appl. Math. Lett., 11 (6), 33-35, 1998.

[5] T.V. Didkivska, M. V. St’opochkina, Properties of Fibonacci-Narayana numbers, In the World of Mathematics, 9

(1), 29–36, 2003. [in Ukrainian]

[6] S. Eilenberg, I. Niven, The fundamental theorem of algebra for quaternions, Bull. Amer. Math. Soc., 50, 246-248,

1944.

[7] C. Flaut, Some application of difference equations inCryptography and Coding Theory, accepted in Journal of Dif-

ference Equations and Applications.

[8] C. Flaut, D. Savin, Quaternion algebras and generalized Fibonacci-Lucas quaternions, Adv. Appl. Clifford Algebras,

25(4), 853-862, 2015.

1Doctoral School of Mathematics, Ovidius University, Constanta, Romania, [email protected]

21

[9] C. Flaut, D. Savin, Some remarks regarding (a, b, x0, x1)-numbers and

(a, b, x0, x1)-quaternions, https://arxiv.org/pdf/1705.00361.pdf.

[10] C. Flaut, Savin, Some special number sequences obtained from a difference equation of degree three, Chaos, Solitons

& Fractals, 106, 67-71, 2018.

[11] C. Flaut, Shpakivskyi, V., An efficient method for solving equations in generalized quaternion and octonion algebras,

Adv. Appl. Clifford Algebras, 25(2), 337-350, 2015.

[12] A. F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68(1961), 455-459, 1961.

[13] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(1963), 289-291,

1963.

[14] M. E. Koroglu, I. Ozbek, I. Siap, Optimal codes from Fibonacci polynomials and secret saring schemes, Arab. J.

Math, 2017, 1-12, 2017.

[15] A. K. Lenstra, E. R. Verheul, Selecting cryptographic key sizes, J. Cryptology, 14(2001), 255–293, 2001.

[16] D. A. Mierzejewski, V. S. Szpakowski, On solutions of some types of quaternionic quadratic equations, Bull. Soc.

Sci. Lett. Lo dz 58, Ser. Rech. Deform., 55 (2008), 49-58, 2008.

[17] A. Pogoruy, R. M. Rodrigues-Dagnino, Some algebraic and analytical properties of coquaternion algebra, Adv. Appl.

Clifford Alg., 20 (2010), 79-84, 2010.

[18] R. D. Schafer, An Introduction to nonassociative algebras, Academic Press, New-York, 1966.

[19] W. D. Smith, Quaternions, octonions, and now, 16-ons, and 2n-ons; New kinds of numbers, www. math. temple.edu/

2dc wds/homepage/nce2.ps, 2004.

[20] http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

[21] http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/lucasNbs.html

[22] http://mathworld.wolfram.com/PellNumber.html

[23] E. Zeckendorf, Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de

Lucas, Bull. Soc. R. Sci. Licge, 41, 79-182, 1972.

22

Some Properties of Local Cohomology Modules

Jafar Azami 1

Abstract. Let (R,m) be a commutative Noetherian local ring and I be an ideal of R. For an R-

module M , the ith local cohomology module of M with respect to I is defined as

HiI(M) = lim−→

n≥1

ExtiR(R/In,M).

In this paper we consider some new properties of local cohomology modules. In particular, we obtain

some results about the finiteness of Bass-numbers, cofiniteness and cominimaxness of this modules.

Keyword: Local cohomology modules, cofiniteness, comonimaxness.

AMS 2010: 13D45, 13E05.

References

[1] M.P. Brodmann and R.Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge

University Press, Cambridge, 1998.

[2] W. Bruns, and J. Herzog, Cohen Macualay rings, Cambridge studies in advanced mathematics, 1997.

[3] A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lecture Notes in Math., 862 (Springer, New York,

1966).

[4] T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math.,

104, 2001, 519-525.

[5] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285, 649-668, 2005.

[6] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285, 649-668, 2005.

1University of Mohaghegh Ardabili, Faculty of Sciences, Ardabil, Iran, [email protected]

23

Introduction to Fuzzy Topology on Soft Sets

Kemale Veliyeva 1, Cigdem Gunduz Aras 2 and Sadi Bayramov 3

Abstract. Let X be an initial universe set, E be a set of parameters and SS(X,E) be a family of all

soft sets over X .

Definition 1. A fuzzy topology on a set X is defined to be a mapping τ : SS(X,E)→ [0, 1] satisfying:

1) τ(Φ) = τ(X) = 1,

2) τ((F,E)∩(G,E)) ≥ τ(F,E) ∧ τ(G,E), ∀(F,E), (G,E) ∈ SS(X,E),

3) τ

(∪i(Fi, E)

)≥ ∧

iτ(F,E), ∀(F,E) ∈ SS(X,E).

Then we denote (SS(X,E), τ) as (X,E, τ) and we call the triple (X,E, τ) as fuzzy soft topological

space.

Theorem 1. Let (X,E, τ) be a fuzzy soft topological space. Then for each r ∈ (0, 1], τr = (F,E) : τ(F,E) ≥ r

defines a soft topology on X . Also for r1 < r2, τr1 < τr2 is satisfied.

Theorem 2. Let τrr∈(0,1] be a descending family of soft topologies on X . Define τ : SS(X,E)→ I

by

τ(F,E) = ∨r : (F,E) ∈ τr

Then τ is a fuzzy soft topology on X .

Definition 2. (f, ϕ) : (X,E, τ)→ (Y,E′, τ ′) be a mapping. Then (f, ϕ) is called a continuous mapping

on the soft point xe if for arbitrary soft set (f(x))ϕ(e)∈ (G,E′) ∈ SS(Y,E′) and τ ′(G,E′) = r, there

exists soft set (F,E) ∈ SS(X,E) such that

1Department of Algebra and Geometry of Baku State University, Baku, Azerbaijan, [email protected]

2Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, Turkey, [email protected],

3Department of Algebra and Geometry of Baku State University, Baku, Azerbaijan, [email protected]

24

xe ∈ (F,E) ∈ SS(X,E), τ(F,E) ≥ r and (f, ϕ)((F,E)) ⊂ (G,E′)

If (f, ϕ) is a continuous mapping for all xe ∈ (F,E) ∈ SS(X,E), then (f, ϕ) is a continuous

mapping.

Theorem 3. (f, ϕ) is a continuous mapping if and only if τ ′(G,E′) ≤ τ((f, ϕ)−1(G,E′)) is satisfied,

for each (G,E′) ∈ SS(Y,E′)

Theorem 4. (f, ϕ) is a continuous mapping if and only if fr : (X,E, τr) → (Y,E′, τ ′r) is a soft

continuous mapping for each r ∈ (0, 1] .

Keyword: Soft set, fuzzy soft topological spaces, continuous mapping

AMS 2010: 54A40,03E72, 06F35.

References

[1] S. Bayramov and C. Gunduz, A new approach to separability and compactness in soft topological spaces, TWMS J.

Pure and Appl. Math., 9(1), 82-93, 2018.

[2] A.P.Sostak, On a fuzzy topological structure, Rendiconti Ciecolo Matematico Palermo (Suppl.Se. II), 89-103, 1985.

25

An Extended Study of I-Functors and D-Rich Functors

Muhammad Rashid Kamal Ansari 1

Abstract. Cotorsion completion functor of Matlis is studied in case of noncomutative rings. We

show that the category of cotorsion modules, which is a category with kernels and cokernels in case

of integral domains (commutative) is a category with kernels and cokernels in case of two sided Ore

domains also. Matlis [2] defines an I-functor and a rich functor in case of an integral domain. In our

study we carry over these concepts to a ring R which is a subring of a ring D. In section 2 we define

a D-rich functor which is a modification of the rich functor of Matlis [2] and provide examples in this

regard. Given an integral domain I there exists a cotorsion completion functor c(A) associated with

each reduced module A as given in Matlis [2]. In section 3 we establish the existence of such a functor

in case of two sided Ore domains. In this regard we also generalise other results of Matlis [2]. In case

of integral domains Matlis [2] proves that the category of cotorsion modules is a category with kernels

and cokernels. We show that this holds in case of two sided Ore domains also. Though, we consider R

to be a ring embedded in a ring D, however, in case of a left Ore domain, D will be replaced by Q the

minial left skew field of R. Note that in this situation, for modules in R-mod, the concepts of torsion,

reduced and cotorsion modules coincide with the concepts of Q-torsion, Q-reduced and Q-cotorsion

modules respectively. For basic homological concepts [3] can be referred. The results obtained will be

applied to study generalized module approximations as discussed in [1].

References

[1] A. Zaffar, M. R. K. Ansari, Some generalizations of module approximations, International Journal of Algebra, 7 (14),

661 - 666, 2013.

[2] E. Matlis, Cotorsion modules, Memoirs of Amer. Math. Soc., 49, 1964.

[3] J. Rotman, Homological algebra, 2nd Edition, Springer, 2018.

1Sir Syed University of Engineering and Technology, Karachi

26

On central Boolean Rings and Nearrings

Nayak Hamsa 1, Kedukodi Babushri Srinivas 2 and Kuncham Syam Prasad 3

Abstract. We present the concepts of central Boolean rings and nearrings and to show that these

structures are not commutative in general. We analyze conditions under which central Boolean near-

rings are commutative.

Keyword: Boolean ring, nearring

AMS 2010: 16Y30.

References

[1] H. E. Bell and G. Mason, On derivations in near-rings, Proceeding of Near-rings and Near-Fields, Tubigen, 1985,

North-Holland Mathematical Studies, 31–35, 1987.

[2] S. Bhavanari, S. P. Kuncham and B. S. Kedukodi, Graph of a nearring with respect to an ideal, Comm. Algebra, 38,

1957–1962, 2010.

[3] A. A. M Kamal and K. H. Al-Shaalan, Commutativity of near-rings with derivations by using algebraic substructures,

Indian J. Pure Appl. Math., 43(3), 211–225, 2012.

[4] H. Nayak, S.P. Kuncham and B.S. Kedukodi, Extensions of boolean rings and nearrings, Journal of Siberian Federal

University. Math. & Phys., 12(1), 58-67, 2019.

[5] G. Pilz , Near-rings and Near-fields (Handbook of Algebra), Edited by M.Hazewinkel, Elsevier Science B.V, 1996.

[6] Y. V. Reddy , Recent developments in boolean nearrings, Editors: S.P. Kuncham, B.S. Kedukodi, H. Panackal and

S. Bhavanari, Nearrings, Nearfields and Related Topics, World Scientific (Singapore), 2017.

1Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India, [email protected] Institute of Technology, Manipal Academy of Higher Education, Manipal, India,

[email protected]

3Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India, [email protected]

27

Weakly 2-absorbing Ideals in Non-Commutative Rings

Nico Groenewald 1

Abstract. Let R be a commutative ring with identity element. The concept of a 2-absorbing ideal

was introduced by Badawi in [5] as a generalization of a prime ideal. A proper ideal I of R is a

2-absorbing ideal of R if whenever a, b, c ∈ R with abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. Weakly

prime ideals introduced by Anderson [1] are also generalizations of prime ideals. A proper ideal Iof R

is a weakly prime ideal if whenever 0 6= ab ∈ I, then a ∈ I or b ∈ I. The concept of weakly prime ideal

was generalized to the concept of weakly 2-absorbing ideal in [2]. A proper ideal I of R is said to be

a weakly 2-absorbing ideal of R if whenever 0 6= abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. Up till now

research on these topics mainly concentrated on commutative rings. The concept of a 2-absorbing ideal

in a non-commutative ring was introduced by Groenewald in [3] as a generalization of a prime ideal.

A proper ideal I of a non-commutative ring R with identity is a 2-absorbing ideal if for a, b, c ∈ R such

that aRbRc ⊆ P, then ab ∈ P or ac ∈ P or bc ∈ P . A weakly prime ideal for non-commutative rings

was introduced by Hirano et. al. in [4]. They defined a proper ideal Iof a non-commutative ring R

with identity to be weakly prime if 0 6= JK ⊆ I implies either J ⊆ I or K ⊆ I for any ideals J,K of R.

They then showed that this is equivalent to: If a, b ∈ R such that 0 6= aRb ⊆ P , then a ∈ P or b ∈ P .

In this talk we introduce the notion of a weakly 2-absorbing ideal as a generalization of a weakly prime

ideal in a non-commutative ring with identity and show that many of the results in commutative rings

also hold in non-commutative rings with identity. For example we show that if I is an ideal of the

ring R such that I3 6= 0 then I is a weakly 2-absorbing ideal of R if and only if it is a 2-absorbing ideal.

Keywords: Prime, weakly prime, 2-absorbing, weakly 2-absorbing.

2000 Mathematics Subject Classification. 16N60, 16W99.

References

[1] D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29, 831-840, 2003.

[2] Ayman Badawi and Ahmad Darani, On weakly 2-Absorbing Ideals of Commutative Rings, Houston J. Math. 39, no.

2, 441-452, 2013.

1Nelson Mandela University, Port Elizabeth, South Africa, [email protected]

28

[3] N. J. Groenewald, On 2-absorbing ideals in non-commutative rings, JP Journal of Algebra, Number Theory and

Applications, 40, 855-867, 2018.

[4] Yasuyuki Hirano, Edward Poon and Hisaya Tsutsui, On rings in which every ideal is weakly prime, Bull. Korean

Math. Soc. 47, 1077-1087, 2010.

[5] Ayman Badawi, On 2-absorbing ideals in commutative rings, Bulletin of the Australian Mathematical Society 75,

417 - 429, 2007.

29

Abelianity Axiom is not Necessary to Define a Module

Nuray Eroglu 1

Abstract. We prove that the commutativity axiom with respect to the addition of some algebraic

systems follows from the other ones.

Keyword: Commutativity axiom, ring, group.

AMS 2010: 13A99, 12E99.

References

[1] V. Bryant, Reducing classical axioms, Math. Gaz. 55, 38-40, 1971.

1Tekirdag Namık Kemal University, Tekirdag, Turkey, [email protected]

30

A Note on d-Normal Modules

Nuray Eroglu 1

Abstract. In this study, some new necessary and sufficient conditions are given for a module M be

d-Normal.

Keyword: d-closure, d-normal module, d-closed submodule.

AMS 2010: 16D10, 16D25.

References

[1] M.S. Li and J.M. Zelmanowitz, On dominance, Comm. Algebra 22, 2703-2747, 1994.

[2] P.F. Smith, Modules for which every submodule has a unique closure, Proceedings of the Biennial Ohio State-Denison

Conference, 302-313, 1992.


31

Special Classes of Algebras and some of Their Applications

Radu Vasile 1

Abstract. The BCK-algebras were first introduced in mathematics in 1966 by Y. Imai and K. Iseki,

through the paper [3]. These algebras were presented as a generalization of the concept of set-theoretic

difference and propositional calculi. The class of BCK-algebras is a proper subclass of the class of

BCI-algebras. These algebras form an important class of logical algebras and have many applications

to various domains of mathematics. One of the recent applications of BCK-algebras was given in the

Coding Theory. (see [6] and [10]).

Since it is well known that each BCK-algebras of degree n+1 contains a subalgebra of degree n, in this

talk, we shortly present the properties of a BCK algebra of degree n+ 1 obtained from a BCK algebra

of degree n, using Iseki’s extension and other extensions like it. We emphasize what properties are

preserved and in what circumstances some properties are lost, by obtaining the so called BCK-trees, a

structure which give us a new and a good perspective regarding the above mentioned aspect.

Keyword: BCK algebra, Iseki extension, BCK-trees.

AMS 2010: 06F35, 94B60.

References

[1] H. A. S. Abujabal, M. Aslam, A.B. Thaheem, A representation of bounded commutative BCK-algebras, Internat. J.

Math. Math. Sci., 19(4), 733-736, 1996.

[2] R. L. O. Cignoli, I. M. L. D. Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Trends in

Logic, Studia Logica Library, Dordrecht, Kluwer Academic Publishers, 7, 2000.

[3] Y. Imai, K. Iseki, On axiom systems of propositional calculi, Proc. Japan Academic, 42, 19-22, 1966.

[4] A. Iorgulescu, Algebras of Logic as BCK Algebras, Editura ASE, Bucuresti, 2008.

[5] I. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Jpn. 23, 1–26, 1978.

[6] Y. B. Jun, S. Z. Song, Codes based on BCK-algebras, Inform. Sciences., 181, 5102-5109, 2011.

[7] Y. B. Jun, Satisfactory filters of BCK-algebras, Scientiae Mathematicae Japonicae Online, 9, 1–7, 2003.

[8] C. Flaut, BCK-algebras arising from block codes, Journal of Intelligent and Fuzzy Systems 28(4), 1829–1833, 2015.

[9] J. Meng, Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. Seoul, Korea, 1994.

1Doctoral School of Mathematics, Ovidius University, Constanta, Romania, [email protected]

32

[10] Z. Samaei, M. A. Azadani, L. Ranjbar, A Class of BCK-Algebras, Int. J. Algebra, 5,28, 1379 - 1385, 2011.

33

Introduction on Neutrosophic Soft Lie Algebras

Sebuhi Abdullayev 1, Kemale Veliyeva 2 and Sadi Bayramov 3

Abstract. We introduce the concept of neutrosophic soft Lie subalgebras of a Lie algebra and inves-

tigate some of their properties are investigated.

Let be a set of all parameters, L be Lie algebra and P (L) denotes all neutrosophic sets over L .

Then a pair (F , E) is called a neutrosophic soft Lie algebra over L, where, F is a mapping given by

F : E → P (L), if for ∀e ∈ E, F (e) = (TF (e), IF (e), FF )) is a neutrosophic Lie algebra over L.

We have following theorms:

Theorem 1. If (F 1, E1) and (F 2, E2) be two neutrosophic soft Lie subalgebra over L, then (F 1, E1)∩

(F 2, E2) = (F 3, E1 ∩ E2) is a neutrosophic soft Lie subalgebra over L.

Theorem 2. Let (F 1, E1) and (F 2, E2) be two neutrosophic soft Lie subalgebra over L. If E1∩E2 = ∅

, then (F 1, E1) ∪ (F 2, E2) = (F 3, E1 ∪ E2) is a neutrosophic soft Lie subalgebra over L.

Theorem 3. Let (F 1, E1) and (F 2, E2) be two neutrosophic soft Lie algebras over L1 and L2 respec-

tively. Then (F 1, E1) ∧ (F 2, E2) = (F 3, E1 × E2) is a neutrosophic soft Lie algebra over L.

Theorem 4. Let (F 1, E1) and (F 2, E2) be two neutrosophic soft Lie subalgebras of L, then (F 1, E1)×

(F 2, E2) neutrosophic soft Lie subalgebra of L× L .

Theorem 5. Let f : L1 → L2 epimorfizm of Lie algebras and (F , E) neutrosophic soft Lie subalgebra

of L1 ,then the homomorphic image of (F , E) is neutrosophic soft Lie subalgebra of L2.

Keyword: Lie algebra, subalgebra, neutrosophic soft set, neutrosophic soft Lie Algebras

AMS 2010: 03E72, 54A40.

References

[1] Akram, K.P.Shum : Intuitionistic Fuzzy Lie Algebras, Southerst Asian Bulletin of Mathematics, 31: 843-855, 2007.

[2] F.Smarandache Neutrosophic set, a generalization of the intuitionistic fuzzy sets, Inter. J. Pure Appl. Math. 24287-

297, 2005.

1Department of Algebra and Geometry of Baku State University, Baku, Azerbaijan, sebuhi [email protected]

[email protected],

[email protected]

34

[3] K.Veliyeva, .S.Abdullayev, and S.A. Bayramov : Derivative functor of inverse limit functor in the category of neu-

trosophic soft modules, Proceedings of the Institute of Mathematics and Mechanics 44, 2, (267-284), 2018.

35

On Some Identities with Dual K− Pell Bicomplex Numbers

Serpil Halıcı 1, Sule Curuk 2

Abstract. Abstract. In this study, we have considered the real and dual bicomplex numbers sep-

arately. Firstly, we examine the dual numbers and give them the characteristics of these numbers.

Then, we give the definition, feature and related concepts about bicomplex numbers. And then, we

define the number of dual k-Pell bicomplex numbers which are not found for the first time in the

literature and we examined the norm and conjugate properties of these numbers. We gave equations

about conjugates. In addition, we have given some important characteristics of these newly defined

numbers, and we have written the recursive correlations of these numbers. Using these relations we

have given some important identities such as Vajda’, Honsberger’s and d’Ocagne identities.

Keyword: Pell sequence, Dual numbers, Bicomplex numbers.

AMS 2010: 17A20, 11B39, 11B37.

References

[1] F. T. Aydin, On bicomplex Pell and Pell-Lucas numbers., arXiv preprint arXiv:1712.09595, 2017.

[2] F. Babadag, Fibonacci, Lucas numbers with Dual bicomplex numbers., Journal of Informatics and Mathematical

Sciences 10.1-2, 161-172, 2018.

[3] A. T. Benjamin, S. S. Plott, j. A. Sellers, Tiling proofs of recent sum identities involving Pell numbers. Annals of

Combinatorics, 12(3), 271-278, 2008.

[4] P. Catarino, On some identities and generating functions for k-Pell numbers, International Journal of Mathematical

Analysis, 7(38), 1877-1884, 2013.

[5] P. Catarino, Bicomplex k-Pell quaternions., Computational Methods and Function Theory : 1-12, 2018.

[6] S. Halıcı, On some Pell polynomials, Acta Universitatis Apulensis, (29), 105-112, 2012.

[7] T. Koshy, Fibonacci and Lucas numbers with applications (Vol. 1). John Wiley and Sons, 2017.

[8] Luna-Elizarraras, M. Elena, Michael Shapiro, Daniele C. Struppa, and Adrian Vajiac. Bicomplex holomorphic func-

tions: The algebra, geometry and analysis of bicomplex numbers. Birkhauser, 2015.

1Pamukkale University, Denizli, Turkey, [email protected]


36

[9] R. Melham, Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica, 56(3), 309-318, 1999.

[10] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Mathematische Annalen, 40(3),

413-467, 1892.

37

Zero Divisors of Split Octonion Algebra

Serpil Halıcı 1, Adnan Karatas 2

Abstract. Abstract. Split octonion algebra is an octonion algebra with isotropic norm form. Because

of the definition of multiplication operation, the split octonion algebra has non trivial zero divisors. In

this study, we give some equalities and identities for zero divisors in this algebra.

Keyword: Octonions , quaternions, recurrence relations

AMS 2010: 17A20, 11B39, 11B37.

References

[1] M. Aristidou, Idempotent elements in quaternion rings over Zp. International journal of Algebra, 6.27: 249-254, 2012.

[2] M. Aristidou, A note on nilpotent elements in quaternion rings over Zp. International Journal of Algebra, 6.14:

663-666, 2012.

[3] J. Baez, The octonions. Bulletin of the American Mathematical Society, 39.2: 145-205, 2002.

[4] R. D. Schafer, An introduction to nonassociative algebras. Courier Dover Publications, 2017.

[5] J. P. Ward, Quaternions and Cayley numbers: Algebra and applications. Springer Science and Business Media, 2012.

[6] C. J. Miguel, R. On the structure of quaternion rings over Zp. International Journal of Algebra, 5.27: 1313-1325,

2011.

[7] J. H. Conway, Derek A. On quaternions and octonions. AK Peters/CRC Press, 2003.

[8] S. Okubo, Introduction to octonion and other non-associative algebras in physics. Cambridge University Press, 1995.

[9] J. DH. Smith, An introduction to quasigroups and their representations. CRC Press, 2006.

[10] L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem. Annals of

Mathematics, 155-171, 1919.



38

On Generalization of Dual Fibonacci Octonions

Serpil Halıcı 1

Abstract. In this study, we examine all the second order linear recurrence relations over dual oc-

tonions. Actually, the study is a continuation of our work we have done for Horadam octonions in

references 14. Hence, we generalize Fibonacci-like relations over quaternions and octonions. For this

purpose we use the well-known Horadam sequence and obtain some fundamental and new identities

involving elements of this generalized sequence.

Keyword: Fibonacci numbers and generalization, Horadam sequence, octonions.

AMS 2010: 11B39, 11B37, 17A20.

References

[1] P. Catarino, On D-dual k-Pell quaternions and octonions, Mediterranean Journal of Mathematics, 14(2), 75, 2017.

[2] Cimen, Cennet Bolat and Ipek, Ahmet, On jacobsthal and jacobsthal–lucas octonions, Mediterranean Journal of

Mathematics, 14(2),37,2017.

[3] W. K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math. Soc., 4,381-395, 1873.

[4] A. F. Horadam, A generalized fibonacci sequence,The American Mathematical Monthly, 68(5), 455-459, 1961.

[5] A. F. Horadam, Special properties of the sequence Wn (a, b; p, q), Fibonacci Quart.,5(5), 424-434, 1967.

[6] M. W. Walker and L. Shao and R. A. Volz, Estimating 3-D location parameters using dual number quaternions,

CVGIP: Image Understanding, 54(3),358-367, 1991.

[7] K. Daniilidis, Hand-eye calibration using dual quaternions, The International Journal of Robotics Research, 18(3),

286-298, 1999.

[8] X. Wang and D. Han and C. Yu and Z. Zheng, The geometric structure of unit dual quaternion with application in

kinematic control, Journal of Mathematical Analysis and Applications, 398(2), 1352-1364, 2012.

[9] Y. Wu and X. Hu and D. Hu and T. Li and J. Lian, Strapdown inertial navigation system algorithms based on dual

quaternions, IEEE transactions on aerospace and electronic systems, 41(1), 110-132, 2005.

[10] C. Bolat Cimen and A. Ipek, On pell quaternions and pell-lucas quaternions,Advances in Applied Clifford Algebras,

26, 39-51, 2016.


39

[11] C. Flaut and D. Savin, Quaternion algebras and generalized fibonacci lucas quaternions, Advances in Applied

Clifford Algebras, 25(4), 853-862, 2015.

[12] S. Halıcı, On fibonacci quaternions, Advances in Applied Clifford Algebras, 22(2), 321-327, 2012.

[13] S. Halıcı, On complex fibonacci quaternions, Advances in Applied Clifford Algebras, 23(1), 105-112, 2013.

[14] A. Karatas, S. Halici, Horadam octonions, Analele Universitatii” Ovidius” Constanta-Seria Matematica, 25(3),

97-106, 2017.

40

On Quaternion-Gaussian Lucas Numbers

Serpil Halıcı 1

Abstract. In this study, we first discussed Gaussian Lucas numbers and we’ve given the properties

of these numbers. Then we have defined the quaternions that accept these numbers as a coefficient.

We have examined whether the numbers defined provide the existing some equations for quaternions in

the literature. We have also given some important properties of these numbers with the help of matrices.

Keyword: Recurrence Relations, Gaussian Numbers.

AMS 2010: 11B39, 11B83.

(This work was supported by Pau Bap with project 2019KKP051.)

References

[1] A. F. Horadam,. Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly,

70(3), 289-291, 1963.

[2] J. H. Jordan, Gaussian Fibonacci and Lucas Numbers. The Fibonacci Quarterly, 3(4), 315-318, 1965.

[3] T. Koshy, Fibonacci and Lucas numbers with applications. Wiley, 2019.

[4] W. R. Hamilton, Elements of quaternions. Longmans, Green and Company, 1866.

[5] S. Halici, On fibonacci quaternions. Adv. in appl.Clifford algebras, 22(2), 321-327, 2012.

[6] S. Halici, On complex Fibonacci quaternions. Adv. in appl. Clifford algebras, 23(1), 105-112, 2013.

[7] C. B. Cimen, A. Ipek, On Pell quaternions and Pell-Lucas quaternions. Advances in Applied Clifford Algebras, 26(1),

39-51, 2016.

[8] E. Polatli, S. Kesim, On quaternions with generalized Fibonacci and Lucas number components, Advances in Differ-

ence Equations, 2015(1), 169, 2015.

[9] A. Szynal-Liana, I. Wloch, A note on Jacobsthal quaternions. Advances in Applied Clifford Algebras, 26(1), 441-447,

2016.

[10] E. Tan, S. Yilmaz, M. Sahin, On a new generalization of Fibonacci quaternions. Chaos, Solitons, Fractals, 82, 1-4,

2016.


41

[11] S. Halici, S. Oz, On Gaussian Pell Polynomials and Their Some Properties. Palastine Journal of Mathematics, 7(1),

251-256, 2018.

[12] I. Okumus, Y. Soykan, E. Tasdemir, M. Gocen, Gaussian Generalized Tribonacci Numbers. Journal of Progressive

Research in Mathematics, 14(2), 2373-2387, 2018.

[13] S. Pethe, A. F. Horadam, Generalised Gaussian Fibonacci numbers. Bulletin of the Australian Mathematical Society,

33(1), 37-48, 1986.

[14] A. Sloin, A. Wiesel, Proper quaternion Gaussian graphical models. IEEE Transactions on Signal Processing, 62(20),

5487-5496, 2014.

[15] D. Tasci, F. Yalcin, Complex Fibonacci p− Numbers. Communications in Mathematics and Applications, 4(3),

213-218, 2013.

42

Essential Ideals and Dimension in Module over Nearrings

Syam Prasad Kuncham 1, Satyanarayana Bhavanari 2 and Venugopala Rao Paruchuri 3

Abstract. The concepts essential ideals and finite dimension play important role in development of

dimension theory of modules over associative rings. Finite dimension, essential, strictly essential, and

related concepts were studied in nearrings and N -groups by Reddy-Satyanarayana [3], Satyanarayana

- Syam Prasad [5, 6, 7, 8]. In this paper, the authors introduced the concept finite 1-dimension and

considered the relationship between finite dimension and finite 1-dimension. The notions H-essential

and strictly essential ideals of an N -group G are known. We determine the elementary properties of

essential ideals and strictly essential ideals, such as closed under finite intersections, transitive closures.

Consequently, we present i-uniform (i = 0, 1) ideals of an N -group and examine the cases wherein

these two concepts coincide. Some related examples were also presented.

Keyword: N -group, Essential ideal, Finite dimension

AMS 2010: 16Y30

References

[1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer Verlag, New York, 1974.

[2] G. Pilz, Nearrings, North Holland, 1983.

[3] Y. V. Reddy, Bh. Satyanarayana, A note on n-groups, Indian J. Pure-Appl. Math. 19 (1988) 842- 845, 1988.

[4] Bh. Satyanarayana, On modules with finite goldie dimension, J. Ramanujan Math. Society. 5 61-75, 1990.

[5] Bh. Satyanarayana, K. Syam Prasad, A result on e-direct systems in n-groups, Indian J. Pure-Appl. Math. 29 285 –

287, 1998.

[6] Bh. Satyanarayana, K. Syam Prasad, On direct and inverse systems in n-groups, Indian J. Math. (BN Prasad Birth

Commemoration Volume) 42 183-192, 2000.

1Manipal Institute of Technology, MAHE Manipal-576104, India, e-mail: [email protected] of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522510, India Email:

[email protected] of Mathematics, Andhra Loyola College (Autonomous), Vijayawada-520008, India Email:

[email protected]

43

[7] Bh. Satyanarayana, K. Syam Prasad, Linearly independent elements in n-groups with finite goldie dimension, Bulletin

of the Korean Mathematical Society, 42, No. 3, pp 433-441, 2005.

[8] Bh. Satyanarayana, K. Syam Prasad, D. Nagaraju, A theorem on modules with finite goldie dimension soochow

journal of mathematics, 32 (2) 311-315, 2006.

[9] K. Syam Prasad, Bh. Satyanarayana, Finite dimension in n-groups and fuzzy ideals of gamma nearrings, VDM Verlag,

Germany, ISBN: 978-3-639-36838-3, 2011.

[10] Bh. Satyanarayana, K. Syam Prasad, Near rings, fuzzy ideals, and graph theory, chapman and hall, taylor and

francis group (london, new york), isbn 13: 9781439873106, 2013.

[11] k. syam prasad, k. b. srinivas, p. k. harikrishnan, bh. satyanarayana, nearrings, nearfields and related topics, World

Scientific (Singapore), ISBN: 978-981-3207-35-6, 2017.

44

Quasi-Primary Spectrum and Some Sheaf-Theoretic Properties

Zehra Bilgin 1, Neslihan Aysen Ozkirisci 2

Abstract. In this work, we aim to investigate the set of quasi-primary ideals of a commutative ring

R equipped with a topology called quasi-primary spectrum. We define several properties and examine

some topological features of this notion. Moreover, we build a sheaf of rings on the quasi-primary

spectrum and we show that this sheaf is the direct image sheaf with respect to the inclusion map from

the prime spectrum of a ring to the quasi-primary spectrum of the same ring.

Keyword: Primary spectrum, quasi-primary ideal, quasi-primary spectrum, sheaf of rings.

AMS 2010: 13A15, 13A99, 14A99, 54F65.

References

[1] I. R. Shafarevich, Basic Algebraic Geometry 2: Schemes and complex manifolds, Third Edition, Springer-Verlag,

Berlin, 2013.

[2] K. Ueno, Algebraic geometry 1: from algebraic varieties to schemes, Translations of Mathematical Monographs, Vol.

185, American Mathematical Society, 1999.

[3] L. Fuchs, On quasi-primary ideals, Acta Sci. Math.(Szeged), 11, no.3, 174-183, 1947.

[4] M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, Addison-Wesley Publishing Company,

Inc., London, 1969.

[5] N. A. Ozkirisci, Z. Kılıc, S. Koc, A note on primary spectrum over commutative rings, An. Stiint. Univ. Al. I. Cuza

Iasi. Mat. (N.S.), 64(1), 111-119.

[6] R. Hartshorne, Algebraic geometry, Springer Science+Business Media, LLC, New York, 2000.

[7] R. Y. Sharp, Steps in commutative algebra, Second Edition, Cambridge University Press, 2000.

1Istanbul Medeniyet University, Istanbul, Turkey, [email protected]

2Yıldız Technical University, Istanbul, Turkey, [email protected]

45

ANALYSIS

On the Lambert W Function

Alfred Witkowski 1

Abstract. The Lambert W functions is defined as a solution of the equation

z = W (z)exp(W (z))

or as an inverse relation of the function f(z) = zez. Since f is not injective, the relation W is

multivalued in the complex domain. Restricting attention to the reals the relation is defined for

x ≥ −e−1 and is double-valued in (−e−1, 0). The constraint W ≥ −1 defines a single-valued function

W0, which is strictly increasing in [−e−1,∞), while the condition W ≤ −1 produces the branch called

W−1.

In this note we construct two operators whose iterations converge to the Lambert function W0 and

investigate the nature of convergence.

Keyword: Lambert W function, approximation

AMS 2010: 33B99, 26D20.

References

[1] A. Hoorfar and M.Hassani, Inequalities on the Lambert W function and hyperpower function, J. Ineq. Pure and

Appl. Math., 9(2), Article 51, 2008.

[2] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey AND D.E. Knuth, On the Lambert W function, Adv. Comput.

Math., 5(4), 329–359, 1996.

1UTP University of Science and Technology, Bydgoszcz, Poland, [email protected]

47

A note on Modified Picard Integral Operators

Basar Yılmaz 1, Didem Aydın Arı 2

Abstract. This study is a natural continuation of [1] where modified Picard operators preserving

exponential function are described. Firstly, we show these operators approximation processes in the

setting of large classes of weighted spaces. Then, we obtain weighted uniform convergence of the oper-

ators via exponential weighted modulus of smoothness. Finally, we get the shape preserving properties

by considering the exponential convexity.

Keyword: Picard operators, weighted modulus of continuity.

AMS 2010: 41A36, 41A25.

References

[1] O. Agratini, A. Aral, E. Deniz, On two classes of approximation processes of integral type, Positivity 21, 1189–1199,

2017.

[2] A. Aral, On Generalized Picard Integral Operators, Advances in Summability and Approximation Theory, 157–168,

Springer, Singapore, 2018.

[3] A. Aral, D. Cardenas-Morales, P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math.

Inequal, accepted.

[4] T. Coskun, Weighted approximation of unbounded continuous functions by sequences of linear positive operators,

Indian J. Pure Appl. Math., 34(3), 477–485, 2003.

[5] L. Rempulska, Z. Walczak, On modified Picard and Gauss - Weierstrass singular integrals, Ukr. Math. Zhurnal,

57(11), 1577–1584, 2005.

[6] T. Acar, A. Aral, D. Cardenas-Morales, P. Garrancho, Szasz-Mirakyan type operators which fix exponentials, Results

Math., 72(3), 1393–1404, 2017.

1Kırıkkale University, Kırıkkale, Turkey, [email protected]


48

Some New Fixed Point Theorems for Nonlinear Inclusions

Cesim Temel1, Suleyman Polat 2

Abstract. In this study, we present some new fixed point theorems for nonlinear operator inclusions

in WC-Banach algebras under weak topology. In particular, we introduce the existence of fixed points

of Krasnoselskii-mixed type operator inclusions with respect to weak topology in WC-Banach algebras,

without using the condition (P ).

Keywords: Krasnoselskii fixed point theorem, WC-Banach algebra, multivalued operator, nonlinear

operator inclusion.

AMS 2010: 34K13, 47H04, 47H10.

References

[1] R.P. Agarwal, D. O’Regan, Fixed-point theory for weakly sequentially upper-semicontinuous maps with applications

to differential inclusions, Nonlinear Oscillations, 3 (5), 277-286, 2002.

[2] C. Avramescu, A fixed points theorem for multivalued mapping, Electronic J. Qualitative Theory of Differential

Equations, 17, 1-10, 2004.

[3] A. Ben Amar, M. Boumaiza, D. O’Regan, Hybrid fixed point theorems for multivalued mappings in Banach algebras

under a weak topology setting, J. Fixed Point Theory Apple. 18, 327-350, 2016.

[4] B.C. Dage, Multivalued operators and fixed point theorems in Banach algebras, I. Taiwanese J. Math. 10, 1025-1045,

2006.

[5] J. R. Graef, J. Henderson, A. Ouahab, Multivalued versions of a Krasnosel’skii-type fixed point theorem, J. Fixed

Point Theory Appl. 19, 1059-1082, 2017.

[6] A. Jeribi, B. Krichen, B. Mefteh, Fixed point theory in WC-Banach algebras, Turk. J. Math. 40, 283-291, 2016.

[7] M.A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Trans. 10 (2), 345-409, 1958.

1Van Yuzuncu Yıl University, Faculty of Sciences, Department of Mathematics, 65080, Van, Turkey,

[email protected]

2Yasar University, Institute of Science, Department of Mathematics, Izmir, Turkey, [email protected]

49

Approximation in Variable Exponent Spaces

Daniyal Israfilov 1, Emine Kirhan 2

Abstract. In this talk we consider approximation problems in the variable exponent Smirnov classes

of analytic functions defined on domains of the complex plane. We study direct problem of approxi-

mation theory and prove one direct theorem in Smirnov classes defined on the finite simple connected

domains with regular boundary. Earlier, this type theorems were proved when the boundary of domain

is Dini smooth. This class of curves forms a subclass of class of regular curves. For the construction

of approximation polynomials we use one generalization of Faber series which is commonly used for

investigations of approximation problems in the complex plane.

This work was supported by Balikesir University grant 2018/071 D[16]: ”Inequalities in Variable

Exponent Spaces”.

Keyword: variable exponent, Smirnov class, regular curves, direct theorem.

AMS 2010: 30E10, 41A10, 41A30.

References

[1] D. M. Israfilov and A. Testici, Approximation by Matrix Transforms in Weighted Lebesgue Spaces with Variable

Exponent. Results Math 73:8, 1-25, 2018.

[2] D. M. Israfilov and A. Testici, Approximation problems in Lebesgue space with variable exponent. J Math Anal Appl,

459, 112-123, 2018.

[3] I. I. Sharapudinov, Approximation of functions by De Vallee Poussin means in the Lebesgue and Sobolev spaces with

variable exponent, Matem. Sb., Vol. 207:7, 131-158,2016.

1Balikesir University Balikesir Turkey [email protected]

2Balikesir University Balikesir Turkey [email protected]

50

Approximation Properties of Kantorovich Type Bernstein-Chlodovsky Operators

which Preserve Exponential Function

Didem Aydın Arı 1, Basar Yılmaz 2

Abstract. Inspire of the Bernstein-Chlodovsky operators which preserve exponential function, we

define integral extension of these operators by using different technique. We give weighted approxi-

mation properties including weighted uniform convergence and weighted quantitative theorem using

exponential weighted modulus of continuity. Then, we give Voronovskaya type theorem.

Keyword: Chlodovsky, Kantorovich, weighted modulus of continuity.

AMS 2010: 41A25, 41A36.

References

[1] J.P.King, Positive Linear Operators which preserve x2, Acta Math. Hungarica, 99 (3),203-208,2003.

[2] A.Holhos, The Rate of Approximation of functions in an infinite interval by positive linear operators,

Stud.Univ.Babes-Bolyai Math. 2, 133-142, 2010.

[3] T.Acar, A.Aral, H.Gonska, On Szasz Mirakyan Operators Preserving e2ax, a > 0, Mediterr. J.Math. 14:6, 2017.

[4] V.Gupta, A.Aral, A Note on Szasz-Mirakyan-Kantorovich type operators preserving e−x, Positivity, 22:415-423, 2018.

[5] I.Chlodovsky, Sur le developpement des fonctions defines dand un intervalle infini en series de polynomes de

M.S.Bernstein, Compositio Math., 4, 380-393, 1937.

[6] Radu Paltanea, A Note On Generalized Bernstein-Kantorovich Operators, Bulletin of The Transilvania University

of Brasov, Vol 6(55), No. 2, Series III: Mathematics, Informatics, Physics, 27-32, 2013.



51

A Study on Certain Sequence Spaces Using Jordan Totient Function

Emrah Evren Kara 1, Merve Ilkhan 2 and Necip Simsek 3

Abstract. In this presentation, we define some new Banach sequence spaces as the matrix domain of

a newly introduced regular matrix in the classical sequence spaces c0, c, `∞. Also, we compute α, β, γ-

duals of these spaces.

Keyword: Jordan totient function, sequence spaces, α, β, γ-duals.

AMS 2010: 11A25, 40C05, 46B45.

References

[1] F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012.

[2] M. Kirisci and F. Basar, Some new sequence spaces derived by the domain of generalized difference matrix, Computers

& Mathematics with Applications. 60, 1299-1309, 2010.

[3] M. Ilkhan and E.E. Kara, A new Banach space defined by Euler totient matrix operator, Operators and Matrices.

(in press).

[4] E. E. Kara and M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra.

64(11), 2208-2223, 2016.

[5] I. Schoenberg, The integrability of certain functions and related summability methods, The American Mathematical

Monthly. 66, 361-375, 1959.

1Duzce University, Duzce, Turkey, [email protected]


3Istanbul Commerce University, Istanbul, Turkey, [email protected]

52

Some Order Properties of the Quotients of L-weakly Compact Operators

Erdal Bayram 1

Abstract. The quotient spaces represent of the constructing new spaces from old ones. For this

reason, in this study we present certain order properties of quotients of the regular operators generated

by L-weakly compact operators which are Banach lattice. Moreover, we also give a representation of

the quotient space created by our operators.

Keyword: L-weakly compact operator, Quotient space, Regular operators.

AMS 2010: 46B42, 47B60.

References

[1] E. Bayram and A.W. Wickstead, Banach lattices of L-weakly and M -weakly compact operators, Arch. Math.(Basel)

108, 293-299, 2017.

[2] C. D. Aliprantis and O. Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press,

Inc., Orlando, FL, 1985.

[3] P. Meyer-Nieberg, Banach lattices, Springer-Verlag, Berlin, 1991.

[4] W.A.J. Luxemburg and A.C. Zaanen, Riesz spaces I,North-Holland Publ., Amsterdam, 1971.

[5] Z.L.Chen and A.W.Wickstead, L-weakly and M -weakly compact operators, Indag. Math. (N.S.), 10(3), 321-336,

1999.

[6] E. Bayram and W. Wnuk, Some algebra ideals of regular operators, Commentationes Mathematicae, 53-2, 127-133,

2013.

[7] Z.L. Chen, Y. Feng, J.X. Chen, The order continuity of the regular norm on regular operator spaces, Abstract and

Appl. Anal., Article ID 183786, 2013.

[8] M. Wojtowicz, Copies of `∞ in quotients of locally solid riesz spaces, Arch. Math. 80, 294-301, 2003.

[9] M. Gonzalez, E. Saksman, H.O. Tylli, Representing non-weakly compact operators, Studia Math., 113, 265-282, 1995.


53

On the Matrix Representations of some Compact-like Operators

Erdal Bayram 1

Abstract. A bounded linear operator which is defined between classical sequence spaces has an infi-

nite matrix representation. It is hence important to find necessary and sufficient conditions of entries

of this matrix representation. Therefore, we study on the necessary and sufficient conditions for the

matrix characterizations of L- and M-weakly compact operators which are defined on certain classical

sequence spaces as Banach lattices. It is known that these operators may coincide with both weakly

compact and compact operators on Banach lattices. Consequently, our study offers a different alterna-

tive to some known results for the matrix characterizations of compact and weakly compact operators

which are presented in terms of L- and M-weakly compactness.

Keyword: Matrix transformation, Weakly compact operator, Compact operator.

AMS 2010: 46A45, 46B15, 47B65.

References

[1] C. D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordecht, 2006.

[2] Y. Altın and M. Et, Generalized difference sequence spaces defined by a modulus function in a locally convex space,

Soochow J.Math., 31(2), 233-243, 2005.

[3] B. Aqzzouz, A. Elbour, A.W. Wickstead, Compactness of l-weakly and m-weakly compact operators on banach

lattices, Rend.Circ. Mat. Palermo, 60, 43-50, 2011.

[4] M. Basarır, E.E. Kara, On some difference sequence spaces of weighted means and compact operators,

Ann.Func.Anal., 2(2), 114-129, 2011.

[5] Z.L.Chen and A.W.Wickstead, L-weakly and M -weakly compact operators, Indag. Math. (N.S.), 10(3), 321-336,

1999.

[6] R.C. Cooke, Infinite matrices and sequence spaces, MacMillan and Co. Ltd, London, 1950

[7] I. Djolovic, Two ways to compactness, Filomat, 17, 15-21, 2003.

[8] I. Djolovic and E. Malkowsky, A note on compact operators on matrix domains, J.Math.Anal.Appl., 340, 291-303,

2008.


54

[9] A.M. Jarrah and E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat,

17, 59-78, 2003.

[10] I.J. Maddox, Infinite matrices of operators, Lecture Notes in Mathematics 780, Springer-Verlag, 1980.

[11] P. Meyer-Nieberg, Banach lattices, Springer-Verlag, Berlin Heidelberg New York, 1991.

[12] W.L.C. Sargent, On compact matrix transformations between sectionally bounded bk-spaces, Journal London

Math.Soc., 41, 79-87, 1966.

[13] M. Stieglitz, H. Tietz, Matrixtransformationen von folgenraumen eine ergebnisubersicht, Math. Zeitschrift, 154,

1-16, 1977.

[14] A. Wilansky, Summability through functional analysis, North-Holland Mathematical Studies 85, Elsevier Science

Publishers, 1984.

[15] A. Wilansky, Summability through functional analysis, North-Holland Mathematical Studies 85, Elsevier Science

Publishers, 1984.

55

Refined Some Inequalities for Frames with Specht’s Ratio

Fahimeh Sultanzadeh 1, Mahmood Hassani 2, Mohsen Erfanian Omidvar 3

and Rajab Ali kamyabi Gol 4

Abstract. We give a new lower bound in some inequalities for Frames in a Hilbert space. If fii∈I

be a Parseval frame for Hilbert space H with frame operator Sf =∑i∈I〈f, fi〉fi, , then for every J ⊂ I

and f ∈ H, we have

(1 + 2α

2 + 2α)‖f‖2 ≤

∑i∈J

|〈f, fi〉|2 + ‖∑i∈Jc

〈f, fi〉fi‖2,

where α = infR(‖SJcf‖‖SJf‖

) : f ∈ H, J ⊂ I, and R is Specht’s ratio. Several variations of this result

are given. Our results refine the remarkable results obtained by Balan et al. and Gavruta.

Keyword: Specht’s ratio, Parseval Frame, inequality.

References

[1] R. Balan, P.G. Casazza, D. Edidin, G. Kutyniok, A new identity for Parseval frames , Proc. Amer. Math. Soc. 135,

1007-1015, 2007.

[2] P.G. Casazza, The art of frame theory ,Taiwanese J. Math. 4, 129-201, 2000.

[3] Sh. Furuichi, Refined Young inequalities with Specht’s ratio , J. Egyptian. Math. Soc. 20, 46-49, 2012.

[4] L. Gavruta, Frames for operators , Appl Comput Harmon Anal. 32, 139-144, 2012.

[5] P. Gavruta, On some identities and inequalities for frames in Hilbert spaces , J. Math. Anal. Appl. 321, 469-478,

2006.

[6] Q.P. Guo, J.S. Leng, H.B. Li, Some equalities and inequalities for fusion frames , Springer Plus. 5, Article ID 121,

10 pages, 2016.

1Islamic Azad University, Mashhad, Iran, [email protected]



4Islamic Azad University, Mashhad, Iran

56

A Note on Approximating Finite Hilbert Transform and Quadrature Formula

Fuat Usta 1

Abstract. In this presentation, approximations for the finite Hilbert transform are given utilizing the

fundamental integral identity for absolutely continuous mappings. Then, a numerical integrations for

this transform is obtained. Finally some numerical experiments have been presented.

Keywords: Finite Hilbert Transform, CPV (Cauchy Principal Value), Absolutely Continuous Mappings

AMS 2010: 41A15, 41A55.

References

[1] F. W. King, Hilbert Transforms, Volume 1, Encyclopedia of Mathematics and Its Applications 124, Cambridge

University Press, New York, 2009.

[2] N. M. Dragomir, S. S. Dragomir, P. M. Farrell and G. W. Baxter, A quadrature rule for the finite Hilbert transform

via trapezoid type inequalities, J. Appl. Math. Comput. 13, no. 1-2, 67-84, 2003.

[3] N. M. Dragomir, S.S. Dragomir & P. Farrell, Approximating the finite Hilbert transform via trapezoid type inequal-

ities, Comput. Math. Appl. 43, 10-11, 1359-1369, 2002.

[4] W. J. Liu and N. Lu, Approximating the finite Hilbert Transform via Simpson type inequalities and applications,

Politehnica University of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 77, no. 3, 107-122,

2015.

[5] S.S. Dragomir, Approximating the finite Hilbert transform via Ostrowski type inequalities for absolutely continuous

functions. Bull. Korean Math. Soc. 39(4), 543-559, 2002.

[6] W. Liu, X. Gao and Y. Wen, Approximating the finite Hilbert transform via some companions of Ostrowskis inequal-

ities. Bull. Malays. Math. Sci. Soc. 39, no. 4, 1499-1513, 2016.

[7] F. Usta, Approximating the finite hilbert transform for absolutely continuous mappings and applications in numerical

integration, Advances in Applied Clifford Algebras, 28: 78, 2018. https://doi.org/10.1007/s00006-018-0898-z

[8] F. Usta, On approximating the finite hilbert transform and applications in quadrature, Mathematical Methods in

the Applied Sciences, 2018; 10. https://doi.org/10.1002/mma.5252.


57

[9] S. Wang, X. Gao, N. Lu, A quadrature formula in approximating the finite Hilbert transform via perturbed trapezoid

type inequalities. J. Comput. Anal. Appl. 22, no. 2, 239-246, 2017.

[10] S. Wang, N. Lu, X. Gao, A quadrature rule for the finite Hilbert transform via Simpson type inequalities and

applications. J. Comput. Anal. Appl. 22, no. 2, 229-238, 2017.

[11] W. Liu, X. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function

of bounded variation and applications. Appl. Math. Comput. 247, 373-385, 2014.

58

Characterization of Certain Matrix Classes Involving the Space |Cα|p

G. Canan Hazar Gulec 1

Abstract. In this study we give the characterization of the matrix classes from the classical spaces

`∞, c, c0 and `1 to the space |Cα|p (p ≥ 1) which has been defined by Sarıgol in [5] for α > −1.

Furthermore, by using the Hausdorff measure of noncompactness, we characterize certain classes of

compact operators on this space.

Keyword: Absolute Cesaro spaces, Matrix operators, Compact operators.

AMS 2010: 40C05, 40F05, 46A45, 46B50.

References

[1] T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London

Math. Soc. 7, 113-141, 1957.

[2] G.C. Hazar Gulec, Compact Matrix Operators on Absolute Cesaro Spaces, Numer. Funct. Anal. Optim., 2019. DOI:

10.1080/01630563.2019.1633665

[3] G.C. Hazar, and M.A. Sarıgol, On absolute Norlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), 34

(5), 812-826, 2018.

[4] G.C. Hazar Gulec, and M.A. Sarıgol, Hausdorff measure of noncompactness of matrix mappings on Cesaro spaces,

Bol. Soc. Paran. Mat. (in press).

[5] M.A. Sarıgol, Spaces of Series Summable by Absolute Cesaro and Matrix Operators, Comm. Math Appl., 7 (1),

11-22, 2016.

[6] M.A. Sarıgol, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci., 42 (3), 28-35, 2015.

[7] M. Stieglitz and H. Tietz, Matrixtransformationen von folgenraumen eine ergebnisuberischt, Math Z., 154, 1-16,

1977.


59

The Space bvθk and Matrix Transformations

G. Canan Hazar Gulec 1, M.Ali Sarıgol 2

Abstract. In the present paper, we introduce the space bvθk, give its some algebraic and topological

properties, and also characterize some matrix operators defined on that space, which extend some well

known results.

Keyword: Sequence spaces, matrix transformations, BK spaces.

AMS 2010: 40C05, 40D25, 40F05, 46A45.

References

[1] F. Basar, B. Altay, and M. Mursaleen, Some generalizations of the space bvp of p-bounded variation sequences,

Nonlinear Analysis, 68 (2), 273-287, 2008.

[2] G.C. Hazar and M.A Sarıgol, On Absolute Norlund Spaces and Matrix Operators, Acta Math. Sinica, English Series,

34 (5), 812-826, 2018.

[3] E. E. Kara and M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra,

64 (11), 2208-2223, 2016.

[4] E. Malkowsky, V. Rakocevic and S. Zivkovic, Matrix transformations between the sequence space bvk and certain

BK spaces, Bull. Cl. Sci. Math. Nat. Sci. Math., 123 (27), 33-46, 2002.

[5] M. A. Sarıgol, Absolute Cesaro summability spaces and matrix operators on them, Comm. Math Appl., 7 (1), 11-22,

2016.

[6] M. A. Sarıgol, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci., 42 (3), 28-35, 2015.

[7] M. Stieglitz and H. Tietz, Matrixtransformationen von folgenraumen eine ergebnisuberischt, Math Z., 154, 1-16,

1977.



60

New Class of Probabilistic Normed Spaces and its Normability

Harikrishnan Panackal 1, Bernardo Lafuerza Guillen, Yeol Je Cho and K. T. Ravindran

Abstract. In this paper, we establish some properties of invertible operators, convex, balanced, ab-

sorbing sets and D−boundedness in Serstnev spaces. We have prove that some PN spaces (V, ν, τ, τ∗) ,

which are not Serstnev spaces, in which the triangle function τ∗ is not Archimedean can be endowed

with a structure of a topological vector space. Also, we have proved that the topological spaces ob-

tained in such a manner are normable under certain given conditions.

References

[1] C. Alsina, B. Schweizer and A. Sklar.: On the definition of a probabilistic normed space, Aequationes Math., 46,

91-98, 1993.

[2] C. Alsina, B. Schweizer and A. Sklar.: Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208,

446-452, 1997

[3] B. Lafuerza-Guillen, Panackal Harikrishnan.: Probabilistic Normed Spaces, Imperial College Press, World Scientific,

UK, London, 2014.

[4] B. Lafuerza-Guillen.: Finite products of probabilistic normed spaces, Radovi Matematicki, 13 , 111-117, 2004.

[5] B. Lafuerza-Guillen, A. Rodrıguez Lallena and C. Sempi.: A study of boundedness in probabilistic normed spaces,

J. Math. Anal. Appl., 232, 183-196, 1999.

[6] B. Lafuerza-Guillen.: D-bounded sets in probabilistic normed spaces and their products, Rend. Mat., Serie VII, 21,

17-28, 2001.

[7] B. Lafuerza-Guillen, Carlo Sempi, Gaoxun Zhang.: A Study of Boundedness in Probabilistic Normed Spaces, Non-

linear Analysis, 73 , 1127-1135, 2010.

[8] B. Lafuerza-Guillen, J.A. Rodrıguez Lallena, C. Sempi.: Normability of probabilistic normed spaces, Note di Matem-

atica, 29(1), 99-111, 2008.

[9] B. Lafuerza-Guillen, J.A. Rodrıguez Lallena, Carlo Sempi.: Probabilistic norms for linear operators,J. Math. Anal.

Appl., 220, 462-476, 1998.

1Department of Mathematics, Manipal Institute of Technology, MAHE, India, [email protected],

[email protected]

61

[10] B. Lafuerza-Guillen, J.A. Rodrıguez Lallena, C. Sempi.: Some classes of Probabilistic Normed Spaces. Rend. Mat.,

17, 237-252, 1997.

[11] B. Lafuerza-Guillen.: Primeros Resultados en el estudio de los espacios normados probabilisticos con nuevos con-

ceptos de acotacin, Ph.D Thesis, Universidad de Almeria, Spain, 1996.

[12] M. J. Frank, B.Schweizer.: On the duality of generalized infimal and supremal convolutions, Rend. Mat. (6) 12 no.

1, 1-23, 1979.

[13] Gaoxun Zhang, Minxian Zhang.: On the normability of generalized Serstnev PN spaces, J. Math. Anal. Appl. 340,

1000-1011, 2008.

[14] P.K. Harikrishnan, B. Lafuerza-Guillen, K.T. Ravindran.: Compactness and D− boundedness in Menger’s 2-

Probabilistic Normed Spaces, FILOMAT, 30(5), 1263-1272, 2016.

[15] P. K. Harikrishnan K. T. Ravindran.: Some Results Of Accretive Operators and Convex Sets in 2-Probabilistic

Normed Space, Journal of Prime Research in Mathematics, 8, 76-84, 2012.

[16] B. Jagadeesha, B. S. Kedukodi, S. P. Kuncham.: Interval valued L-fuzzy ideals based on t-norms and t-conorms, J.

Intell. Fuzzy Systems, 28 (6), 2631-2641, 2015.

[17] A.N. Kolmogoroff.: Zur Normierbarkeit eines allgemeinen topologischen linearen Raumes, Studia Math., 5 , 29-33

(1934); English translation in V. M. Tiklomirov (Ed.), Selected Works of A. N. Kolmogorov, Vol. I: Mathematics and

Mechanics, Kluwer, Dordrecht-Boston-London, 183-186.

[18] K. Menger.: Statistical Metrics. Proc Nat Acad Sci USA , 28,535-537, 1942.

[19] B. Schweizer, A. Sklar.: Probabilistic metric spaces, Springer, North Holland; 2nd ed., Dover, Mineola NY

(1983,2005)

[20] A. N. Serstnev.: On the motion of a random normed space, Dokl. Akad. Nauk SSSR, 149 , 280-283 (1963) (English

translation in Soviet Math. Dokl., 4, 388-390 (1963))

[21] H. Sherwood.: Complete probabilistic metric spaces, Z. Wahrsch. Verw. Gebiete, 20,pp.117-128, 1971.

[22] M. D.Taylor.: Introduction to Functional Analysis, Wiley, New York-London-Sydney, 1985.

[23] M. Zhang.: Representation theorem in finite dimensional probabilistic normed spaces. Sci. Math. Jpn., 2004.

62

Roducts of Weighted Composition Operators and Differentiation Operators

between Weighted Bergman Spacs and Weighted Banach Spaces of Analytic

Functions

Jasbir S. Manhas

Abstract. Let v and w be weights on the unit disc D. Let Av,p(D) be the weighted Bergman space of

analytic functions and H∞v (D) be the weighted Banach space of analytic functions. In this paper, we

investigate the analytic mappings φ : D → D and ψ : D → 20b5 which characterize the boundedness

and compactness of products of weighted composition operators and differentiation operators DWψ,φ

and Wψ,φ D between the weighted Bergman spaces and weighted Banach spaces of analytic functions.

Keyword:Weighted Bergman Spaces, Weighted Banach Spaces, Weighted Composition Operators, Dif-

ferentiation Operators, Bounded and Compact Operators. .

AMS 2010: 47B38, 47B33 .

63

Extension of Order Bounded Operators

Kazem Haghnejad Azar 1

Abstract. Assume that a normed lattice E is order dense majorizing of a vector lattice Et. There is

an monotone extension of the norm from E to Et, and so we can extend some lattice and topological

properties from normed lattice (E, ‖.‖) to new normed lattice (Et, ‖.‖t). For a Dedekind complete

Banach lattice F , T t is an extension of T from Et into F whenever T is an order bounded operator

from E into F . For each positive operator T , we have ‖T‖ = ‖T t‖ and we show that T t is a lattice

homomorphism from Et into F and moreover T t ∈ Ln(Et, F ) whenever 0 ≤ T ∈ Ln(E,F ) and

T (x ∧ y) = Tx ∧ Ty for each 0 ≤ x, y ∈ E. We also extend some lattice and topological properties of

T ∈ Lb(E,F ) to the extension operator T t ∈ Lb(Et, F ).

Assume that E is a normed lattice and sublattice of G, and E is order dense majorizing of a vector

lattice Et ⊆ G. The aim of this manuscript are in the following:

(1) We extend the norm from E to Et.

(2) Assume that T is an order bounded operator from E into Dedekind complete normed lattice F .

T t is a linear extension of T , from Et into F , in the sense that if S : Et → F is any operator

that extends T by same way, then T t = S.

(3) We also extend some lattice and topological properties from E and T for Et and T t, respectively.

Keyword: Order dense majorizing, universal completion, Vector lattice, Order bounded operator,

Positive extension operator

AMS 2010: 47B65, 46B40, 46B42.

References

[1] Y. A. Abramovich, C. D. Aliprantis, Locally Solid Riesz Spaces with Application to Economics, Mathematical

Surveys, vol. 105, American Mathematical Society, Providence, RI, 2003.

[2] C. D. Aliprantis, and O. Burkinshaw, Positive Operators, Springer, Berlin, 2006.

[3] P. Meyer-Nieberg, Banach lattices, Universitex. Springer, Berlin. MR1128093, 1991.

1Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.

Email: [email protected]

64

[4] O. van Gaans, Seminorms on ordered vector space that extend to Riesz seminorms on large spaces, Indag Mathem,

N8, 14(1), 15-30, 2003.

65

Reconstruction of Signals from Short Time Fourier Transform

Khole Timothy Poumai 1

Abstract. Short time Fourier transform (STFT) is studied based on frame theory analysis. The no-

tion of block discrete Fourier transform (BDFT) is introduced and shown that it is more advantageous

than discrete Fourier transform on sparsity and convolution of long signals. Using finite Zak transform

(FZT), we formulate a method to reconstruct a signal from STFT. Another method is designed to syn-

thesize a signal from STFT by taking another additional filter and this method is based on finite Zak

transform and the concept of frames. Also, a pair of biorthogonal discrete Gabor systems is defined to

get the reconstruction formula of a signal from STFT. Finally, uncertainty principle in term of sparsity

for BDFT and FZT is given.

Keyword: Frames,Short time Fourier Transform, Uncertainty Principle, Finite Zak transform.

AMS 2010: 42C15, 42C30, 42C05, 46B15.

References

[1] J. B. Allen and L. R. Rabiner, A unified approach to short-time fourier analysis and synthesis, Proceedings of the

IEEE, 65 (11), 1558-1564, 1977.

[2] H. Bolcskei and F. Hlawatsch, Discrete zak transforms, Polyphase Transforms, and Applications, IEEE Trans. Signal

Process. 45(4), 851-866, 1997.

[3] O. Christensen, An introduction to frames and riesz bases, Birkhaurer, Boston, 2016.

[4] D. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. APPL. MATH., 49(3), 906-931,

1989.

[5] A. J. E. M. Janssen, Duality and biorthogonality for discrete-time weyl-heisenberg frames, Technical report, Philips

Electronics, 1994.

[6] P. P. Vaidyanathan, Multirate systems and filter banks, prentice hall p t r, Englewood Cliffs, New Jersey o7632,

1993.

[7] Y. Y. Zeevi and I. Gertner, The finite zak transform: an efficient tool for image representation and analysis, Journal

of Visual Communication and Image Representation, 3(1), 13-23, 1992.

1Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi-110021, India,[email protected]

66

[8] M. Zibulski and Y. Y. Zeevi, Frame analysis of the discrete gabor-scheme, IEEE Transactions on Signal Processing,

24(4), 942-945, 1994.

67

Applications of Frames in Quantum Measurement

Khole Timothy Poumai 1, Shiv K. Kaushik 2

Abstract. In quantum theory, the outcome of a measurement is inherently probabilistic, with the

probabilities of the outcomes of any conceivable measurement determined by the state vector ψ ∈ H. In

this talk, we will discuss how the notion of frame can be used in quantum measurement. We show that,

in Hilbert spaces, Parseval block frames can represent Positive Operator Value Measure (POVM) and

give the existence of Parseval block frame from a given POVM. Also, we give the average probability of

an incorrect measurement by using a block frame. Further, we show that an orthonormal block frame

represents projection valued measure (PVM) in Hilbert spaces. Finally, we show Parseval block frame

can also PVM through dilation theorem of Parseval block frames.

Keyword: Frames, block frames, Riesz frames.

AMS 2010: 42C15, 42C30, 42C05, 46B15.

References

[1] O. Christensen, An introduction to frames and riesz bases, Birkhauser, Boston, 2016.

[2] C. Heil, A basis theory primer, Birkhauser, Boston, 2011.

[3] E. Desurvire, Classical and quantum information theory, Cambridge University Press, New York, 2009.

[4] Y. C. Eldar and G. D. Forney, Optimal tight frames and quantum measurement, IEEE Transactions on Information

Theory, Vol. 48(3), 599-610.

[5] M. Hayashi, Quantum information, Springer Berlin Heidelberg, New York, 2006.

[6] K.T.Poumai, S.K.Kaushik, and S. V. Djordjevic, Operator valued frames and applications to quantum channels,

SampTA 2017, IEEE Xplore 2017, https://doi.org/10.1109/SAMPTA.2017.8024361 .

1Department of Mathematics, Motilal Nehru College, University of Delhi, Delhi-110021, India,[email protected]

2Department of Mathematics, Kirori mal college, University of Delhi, Delhi-110007, India, [email protected]

68

Compact Operators in the Class(bvθk, bv

)Mehmet Ali Sarıgol 1

Abstract. The space bv of bounded variation sequence plays an important role in the summability.

More recently this space has been generalized to the space bvθk and the class(bvθk, bv

)of infinite matrices

has been characterized by Hazar Sarıgol [2]. In the present paper, for 1 < k < ∞, we give necessary

and sufficient conditions for a matrix in the same class to be compact, where θ is a sequence of positive

numbers.

Keyword: Sequence spaces; matrix transformations; bvθk spaces.

AMS 2010: 40C05, 40D25, 40F05, 46A45.

References

[1] E. Malkowsky, E., V.Rakocevic, V. and S. Zivkovic, Matrix transformations between the sequence space bvk and

certain BK spaces, Bull. Cl. Sci. Math. Nat. Sci. Math.123 (27), 33–46. 2002.

[2] G.C. Hazar and M.A. Sarıgol, The space bvθk and matrix transformations, IECMSA-2019, Baku, Azerbaijan.

[3] G.C. Hazar and M.A. Sarıgol, On absolute Norlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), 34

(5), 812-826, 2018.

[4] E. Malkowsky and V. Rakocevic, An introduction into the theory of sequence space and measures of noncompactness,

Zb. Rad. (Beogr) 9 (17), 143-234, 2000.

[5] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12, 87-120, 1998.

[6] M.A. Sarıgol, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci., 42 (2), 28-35, 2015.

[7] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnisuberischt, Math Z., 154, 1-16,

1977.

1Pamukkale University, Denizli, TURKEY, [email protected]

69

A New Regular Matrix Defined by Jordan Totient Function and its Matrix Domain

in `p

Merve Ilkhan 1, Necip Simsek 2 and Emrah Evren Kara 3

Abstract. In this presentation, we define a new regular matrix by the aid of Jordan totient function

and study the matrix domain of this newly introduced matrix in the classical sequence space `p.

Keyword: Jordan totient function, sequence spaces, matrix operators.

AMS 2010: 11A25, 40C05, 46B45.

References

[1] M. Ilkhan and E.E. Kara, A new Banach space defined by Euler totient matrix operator, Operators and Matrices.

(in press).

[2] E. E. Kara and M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra.

64(11), 2208-2223, 2016.

[3] I. Schoenberg, The integrability of certain functions and related summability methods, The American Mathematical

Monthly. 66, 361-375, 1959.

[4] M. Stieglitz and H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Mathematische

Zeitschrift. 154, 1-16, 1977.




70

On The Finite Element Approximation of Quasi-variational Inequalities with

vanishing zero order term

Messaoud Boulbrachene 1

Abstract. We are interested in the quasi-variational inequality (QVI) a(uα, v − uα) + α (uα, v − uα) ≥ (f, v − uα)∀v ∈ H1 (Ω)

v ≤Muα,uα ≤Muα%

as the zero order term α tends to 0.

Here Ω is a bounded open set in Rn, n ≥ 1, with smooth boundary, f is a given smooth function, (., .)

is the inner product in L2(Ω), a(., .) is the bilinear form defined by a(u, v) = (∇u,∇v), and Muα(x) =

k + inf uα(x+ ξ), ξ ≥ 0, x+ ξ ∈ Ω; k > 0, is the obstacle of impulse control [1]

Denoting by ωα = uα−uα with uα = (meas (Ω))−1∫

Ωuαdx, and λα = αuα, P.L. Lions and B.Perthame

[2] proved that the sequence ωα, λα converges (as α goes to 0) uniformly in Ω and strongly in H1 (Ω)

to ω0, λ0, the unique solution of the asymptotic problem a(ω0, v − ω0) ≥ (f − λ0, v − ω0)%∀v ∈ H1 (Ω)

v ≤Mω0,ω0 ≤Mω0%

In this paper, we are concerned with the standard finite element approximation in the L∞-norm. More

precisely, we establish the error estimates

‖ωαh − ω0‖L∞(Ω) ≤ Ch2 |lnh|3

|λαh − λ0| ≤ Ch2 |lnh|3

where ωαh and λαh are the finite element counterparts of ωα and λα, respectively, and C is a constant

independent of both α and h, the mesh size of the finite element triangulation.

Keyword: Quasi-variational inequality, Finite element, Error estimates.

AMS 2010: 65N30, 65N15.

1Department of Mathematics, Sultan Qaboos University, P.O. Box, 36, Muscat, Oman

71

References

[1] Bensoussan, A., Lions, J.L., Impulse control and quasi-variational inequalities. Gauthiers-Villars, Paris, 1984.

[2] P.L. Lions and B.Perthame, SIAM J. Control And Optimization, Vol 24, 4, 1986.

72

Deferred Statistical Convergence in Metric Spaces

Mikail Et 1, Muhammed Cınar 2 and Hacer Sengul 3

Abstract. A real valued sequence x = (xk) is said to be deferred statistically convergent to L, if for

each ε > 0

limn→∞

1

(qn − pn)|pn < k ≤ qn : |xk − L| ≥ ε| = 0,

where p = (pn) and q = (qn) are the sequences of non-negative integers satisfying

pn < qn and limn→∞

qn =∞.

In this study we introduce the concepts of deferred statistical convergence and deferred strong Cesaro

summability in general metric spaces. Also some relations between deferred strong Cesaro summability

and deferred statistical convergence are given in general metric spaces

Keyword: Statistical convergence, Deferred statistical convergence, Cesaro Summability, Deferred Ce-

saro Mean.

AMS 2010: 40A05, 40C05, 46A45.

References

[1] R. P. Agnew, On deferred Cesaro Mean, Comm. Ann. Math., 33, 413-421, 1932.

[2] B. Bilalov and T. Nazarova, On statistical convergence in metric space, Journal of Mathematics Research, 7(1), 37-43,

2015.

[3] E. Kayan, R. Colak and Y. Altın, d−statistical convergence of order α and d−statistical boundedness of order α in

metric spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 80(4), 229–238, 2018.

[4] M. Kucukaslan and M. Yılmazturk On deferred statistical convergence of sequences, Kyungpook Math. J. 56, 357-366,

2016.

1Firat University, Elazig, Turkey, [email protected]

2Mus Alparslan University, Mus, Turkey, [email protected]

3Harran University, Sanliurfa, Turkey, [email protected]

73

[5] M. Kucukaslan, U. Deger and O. Dovgoshey, On the statistical convergence of metric-valued sequences, Ukrainian

Math. J. 66(5), 796–805, 2014.

74

A New Type of Generalized Difference Sequence Space m (φ, p, α) (∆nm)

Mikail Et 1, Rifat Colak 2

Abstract. Let (φn) be a non-decreasing sequence of positive real numbers such that nφn+1 ≤

(n+ 1)φn for all n ∈ N. The class of all sequences (φn) is denoted by Φ. The sequence space

m (φ) was introduced by Sargent [2] and he studied some of its properties and obtained some relations

with the space `p. Later on this sequence space was investigated by Tripathy and Sen [3] and Tripa-

thy and Mahanta [4]. In this work using the generalized difference operator ∆nm, we generalize the

sequence space m (φ) to the sequence space m (φ, p, α) (∆nm) = x = (xk) : ∆n

mx ∈ m (φ, p, α), where

n,m ∈ N,∆0mx = x, ∆mx = (xk − xk+m) , ∆n

mx = (∆nmxk) =

(∆n−1m xk −∆n−1

m xk+m

), and so that

∆nmxk =

∑nv=0 (−1)n

(nv

)xk+mv, give some topological properties about this space and show that the

space m (φ, p, α) (∆nm) is a BK−space by a suitable norm.

Keyword: Difference Sequence, Cesaro Summability, Symmetric space, Normal space.

AMS 2010: 40A05, 40C05, 46A45.

References

[1] M. Et and R. Colak, On generalized difference sequence spaces, Soochow J. Math., 21(4), 377-386, 1995.

[2] W. L. C. Sargent, Some sequence spaces related to `p spaces, J. Lond. Math. Soc. 35, 161-171, 1960.

[3] B. C. Tripathy, M. Sen, On a new class of sequences related to the space `p, Tamkang J. Math. 33(2), 167-171, 2002.

[4] B. C. Tripathy and S. Mahanta, On a class of sequences related to the `p space defined by Orlicz functions, Soochow

J. Math. 29(4), 379–391, 2003.

[5] B . C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized difference Cesaro Sequence spaces, Soochow

J. Math. 31(3), 333-340, 2005.



75

On Generalized Deferred Cesaro Mean

Mikail Et 1

Abstract. In this work using the genealized difference operator ∆m, we geleralize the concept of

deferred Cesaro mean, give some topological properties about this concept and show that the sequence

spaces Cd1,0(∆m), Cd1 (∆m) and Cd∞(∆m) are Banach spaces by suitable norms.

Keyword: Difference Sequence, Cesaro Summability, Deferred Cesaro Mean.

AMS 2010: 40A05, 40C05, 46A45.

References

[1] R. P. Agnew, On deferred Cesaro Mean, Comm. Ann. Math., 33, 413-421, 1932.

[2] V. K. Bhardwaj and S. Gupta, Cesaro summable difference sequence space, J. Inequal. Appl. 2013:315, 9 pp, 2013.

[3] V. K. Bhardwaj, S. Gupta and R. Karan, R. Kothe-Toeplitz duals and matrix transformations of Cesaro difference

sequence spaces of second order, J. Math. Anal. 5(2), 1-11, 2014.

[4] M. Et, On some generalized Cesaro difference sequence spaces, Istanbul Univ. Fen Fak. Mat. Derg. 55/56, 221-229,

1996/97.

[5] M. Et and R. Colak, On generalized difference sequence spaces, Soochow J. Math., 21(4), 377-386, 1995.

[6] P. N. Ng and P.Y. Lee, Cesaro sequence spaces of non-absolute type, Comment Math. 20, 429-433, 1978.

[7] B .C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized difference Cesaro Sequence spaces, Soochow

J. Math. 31:3, 333-340, 2005.


76

A survey of Neutrosophic Type Baire Spaces

Murat Kirisci 1, Necip Simsek 2

Abstract. The fuzzy concept has invaded almost all branches of mathematics ever since the intro-

duction of fuzzy sets by L.A.Zadeh [10]. The theory of fuzzy topological space was introduced and

developed by C.L.Chang [2] and since then various notions in classical topology have been extended to

fuzzy topological space. The idea of intuitionistic fuzzy set was first published by Atanassov [1]. The

concept of fuzzy nowhere dense set in fuzzy topological space by G.Thangaraj and S.Anjalmose in [9].

Park given intuitionistic metric spaces related to the t-norm and the t-conorm. Coker [3] defined the

intuitionistic fuzzy topological spaces. Intuitionistic fuzzy Baire spaces was defined by Dhavaseelan [4].

The concept of neutrosophic sets was first introduced by Smarandache [7], [8] as a generalization of

intuitionistic fuzzy sets [?] where we have the degree of membership, the degree of indeterminacy and

the degree of non-membership of each element in X. Karatas and Kuru [3] redefined the neutrosophic

set operations in accordance with neutrophic topological structures. Kirisci and Simsek [4] defined the

neutrosophic metric spaces with respect to the t-norm and t-conorm.

In this presentation, a general survey of Neutrosophic Baire spaces has been conducted.

Keyword: Neutrosophic first category, neutrosophic second category, neutrosophic residual set, neu-

trosophic Baire space..

AMS 2010: 54A40, 03E72.

References

[1] K. Atanassov, lntuitionistic fuzzy sets, in: V. Sgurev, Ed., VII ITKR’s Session, So

a June 1983 Central Sci. and Techn. Library, Bulg. Academy of Sciences 1984.

[2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182–190, 1968.

[3] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88, 81–89, 1997.

[4] R. Dhavaseelan, Intuitionistic fuzzy Baire spaces, Annals of Fuzzy Mathematics and Informatics, 10, 215–219, 2015.

1Istanbul University-Cerrahpasa, Istanbul, Turkey, [email protected]


77

[5] S. Karatas and C. Kuru, Neutrosophic topology, Neutrosophic Sets and Systems, 13, 90–95, 2016.

[6] M. Kirisci and N. Simsek, Neutrosophic metric spaces, arxiv.org, arXiv:1907.00798.

[7] F. Smarandache, Neutrosophic set - a generalization of the intuitionistic fuzzy set, International Journal of Pure and

Applied Mathematics, 24, 287-297, 2005.

[8] F. Smarandache, Neutrosophy and neutrosophic logic, first international conference on neutrosophy, neutrosophic

logic, set, probability, and statistics, University of New Mexico, Gallup, NM 87301, USA(2002).

[9] G. Thangaraj and S. Anjalmose, On Fuzzy Baire Space, The Journal of Fuzzy Mathematics, 21, 667–676, 2013.

[10] L.A. Zadeh, Fuzzy sets, Inf. Cont. 8, 338–353, 1965.

78

Representation of a Solution and Stability for a Sequential Fractional Impulsive

Time-Delay Linear Systems

Nazim I. Mahmudov 1

Abstract. This note gives a representation of a solution to the initial value problem for a sequential

fractional impulsive time-delay linear system. We introduce the impulsive fractional delayed matri-

ces cosine/sine and establish some properties. Then, we use the method of variation of constants to

obtain the solution. Our results extend those for second order time-delay linear system. Moreover,

the representation of a solution is used to investigate a finite-time stability of the fractional impulsive

time-delay linear system.

Keyword: fractional calculus, time-delay equation, Caputo fractional derivative.

AMS 2010: 34K37, 26A33, 34A05, 34K06.

References

[1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier

Science B.V., 2006.

[2] D. Ya. Khusainov, J. Diblık, M. Ruzickova, J. Lukacova, Representation of a solution of the cauchy problem for an

oscillating system with pure delay, Nonlinear Oscil. (N. Y.) 11, No. 2, 276–285, 2008.

[3] J. Diblık, M. Feckan, and M. Pospısil, Representation of a solution of the Cauchy problem for an oscillating system

with two delays and permutable matrices, Ukrainian Mathematical Journal, vol. 65, pp. 58–69, 2013.

[4] B. Bonilla , M. Rivero, J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients

Applied Mathematics and Computation 187, 68–78, 2007.

[5] C. Liang, J. R. Wang, D. O’Regan, Representation of a solution for a fractional linear system with pure delay, Applied

Mathematics Letters 77, 72–78, 2018.

[6] C. Liang, J. R. Wang, D. O’Regan, Controllability of nonlinear delay oscillating systems, Electronic Journal of

Qualitative Theory of Differential Equations 2017, No. 47, 1–18.

1Eastern Mediterranean University, Famagusta, T.R. North Cyprus, [email protected]

79

[7] Cao X, Wang J. Finite-time stability of a class of oscillating systems with two delays. Math Meth Appl Sci., 41:4943–

4954, 2018.

[8] C.Liang, W. Wei, J. R. Wang, Stability of delay differential equations via delayed matrix sine and cosine of polynomial

degrees, Advances in Difference Equations, 2017:131, 2017.

[9] N.I. Mahmudov, A novel fractional delayed matrix cosine and sine. Appl. Math. Lett. 92, 41-48, 2019.

[10] A. Boichuk, J. Diblık, D. Khusainov, M. Ruzickova, Fredholms boundary-value problems for differential systems

with a single delay. Nonlinear Analysis 72, 2251-258, 2010.

[11] N.I. Mahmudov, Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay

differential equations. Math Meth Appl Sci. 2018;1–9. https://doi.org/10.1002/mma.5446

[12] N.I. Mahmudov, Representation of solutions of discrete linear delay systems with non permutable matrices, Applied

Mathematics Letters, Volume 85, November, 8-14, 2018.

80

Fixed Point Theorems on Neutrosophic Metric Spaces

Necip Simsek 1, Murat Kirisci 2 and Mahmut Akyigit 3

Abstract. The concept of neutrosophic sets was first introduced by Smarandache [7], [8] as a gener-

alization of intuitionistic fuzzy sets [1] where we have the degree of membership, the degree of indeter-

minacy and the degree of non-membership of each element in X. Karatas and Kuru [3] redefined the

neutrosophic set operations in accordance with neutrophic topological structures. Kirisci and Simsek

[4], using the idea of intuitionistic fuzzy sets, define the notion of neutrosophic metric spaces with the

help of continuous t-norms and continuous t-conorms as a generalization of intuitionistic fuzzy metric

space due to Park [6]. Grebiec [2] proved the contraction principle in the setting of fuzzy metric spaces

introduced by Kramosil and Michalek [5].

In this paper, we state some definitions and the uniform structure of neutrosophic metric spaces. This

is followed by concept of an neutrosophic contractive mapping. We conclude with extension of Banach

fixed point theorem and classical Banach contraction theorem on complete metric spaces to complete

neutrosophic metric spaces as our main results.

Keyword: complete neutrosophic metric space, neutrosophic contraction, neutrosophic fixed point.

AMS 2010: 54H25, 54A40.

References

[1] K. Atanassov, lntuitionistic fuzzy sets, in: V. Sgurev, Ed., VII ITKR’s Session, Sola June 1983 Central Sci. and

Techn. Library, Bulg. Academy of Sciences 1984.

[2] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst, 27, 385-389, 1988.

[3] S. Karatas and C. Kuru, Neutrosophic topology, Neutrosophic Sets and Systems, 13, 90–95, 2016.

[4] M. Kirisci and N. Simsek, Neutrosophic metric spaces, arxiv.org, arXiv:1907.00798.

[5] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334, 1975.

[6] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals 22, 1039–1046, 2004.


2Istanbul University-Cerrahpasa, Istanbul, Turkey, [email protected]

3Sakarya University, Sakarya, Turkey, [email protected]

81

[7] F. Smarandache, Neutrosophic set - a generalization of the intuitionistic fuzzy set, International Journal of Pure and

Applied Mathematics, 24, 287-297, 2005.

[8] F. Smarandache, Neutrosophy and neutrosophic logic, first international conference on neutrosophy, neutrosophic

logic, set, probability, and statistics, University of New Mexico, Gallup, NM 87301, USA(2002).

82

On Convolution of Boas Transform of Wavelets

Nikhil Khanna 1

Abstract. Boas [1] introduced an integral transform associated to the Hilbert transform which

emerged due to the study of the class of functions having Fourier transform which vanishes on a

finite interval. Later, in 1960, Goldberg [2] studied this transform in detail and gave some significant

results and properties. This transform was known by Boas transform. In this talk, we study Boas

transform of wavelets and give Boas transform wavelet convolution and cross-correlation theorems to

analyze Boas transform of convolved (cross-correlated) signals. Analogously to Bedrosian theorem,

Boas transform product theorem is also given.

Keyword: Boas transform, wavelets, Hilbert transform, Fourier transform; vanishing moments.

AMS 2010: 42A38, 42C40, 44A15, 44A60.

References

[1] R. P. Boas, Some theorems on Fourier transforms and conjugate trigonometric integrals, Transactions of the American

Mathematical Society, 40, no. 2, 287-308, 1936.

[2] R. R. Goldberg, An integral transform related to the Hilbert transform, J. London Math. Soc., 35, 200-204, 1960.

[3] P. Heywood, On a transform discussed by Goldberg, J. London Math. Soc., 38, 162-168, 1963.

[4] A.M. Jarrah, N. Khanna, Some results on vanishing moments of wavelet packets, convolution and cross correlation

of wavelets, Arab Journal of Mathematical Sciences (2018), https://doi.org/10.1016/j.ajmsc.2018.07.001.

[5] A. I. Zayed, Handbook of Function and Generalized Function Transformations, CRC Press, Boca Raton, FL.

1University of Delhi, Delhi, India, [email protected]

83

The C∗-Algebra of Toeplitz Operators Associated with Discrete Heisenberg Group

Nikolay Buyukliev 1

Abstract. We consider Toeplitz operators, associated with the discrete Heisenberg group H3 and its

positive semigroup P , where

H3 =

s =

1 a c

0 1 b

0 0 1

: a, b, c are integer

, P = s ∈ H3 : a, b, c ≥ 0

For f ∈ l1(H3) we define Toeplitz operator on l2(P ) by the formula:

(T (f)ξ)(t) =∑s∈H3

f(s)ξ(ts)1P (ts), ξ ∈ l2(P ).

We define the C∗-algebra of the Toeplitz operators T to be the C∗-algebra, generated by T (f) : f ∈

l1(H3).

We present T as a groupoid C∗-algebra: T ∼= C∗(G), where the groupoid G is a reduction of a

transformation group G = (Y ×H3)|X, and Y and X are explicitly described topological spaces.

This enables us to obtain the ideal structure of T .

Theorem 5. There is an increasing sequence of ideals:

0 ⊂ I0 ⊂ I1 ⊂ I1d ⊂ I2 ⊂ I3 = T ,

such that I0 ∼= K, I3/I2 ∼= C∗(H3), I2/I1d ∼= (C(T 2)×K)2, I1d/I1 ∼= C(T )×K and I1/I0 ∼= (C(T )×K)2.

We use cyclic cohomology to give an index formula of Fredholm operators in T .

Keyword: Toeplitz operator, discrete Heisenberg group, groupoid C∗-algebra.

AMS 2010: 47B35, 22A22.

1Sofia University, Sofia, Bulgaria, [email protected]

84

References

[1] A. Nika, Wiener-Hopf operators on the positive semigroup of a Haisenberg group, Preprint Series in Mathematics,

Bukuresti, N62/1988.

[2] P. Muhly, J. Renault, C∗-algebras of multivariable Wiener-Hopf operators, TAMS, 274, 1-44, 1982.

[3] J. Renault, A groupoid approach to C∗-algebras, Lect. notes in Math., 793, Springer Verlag, New York, 1980.

[4] Albert Jeu-Liang-Shew, On the Type of Wiener-Hopf algebras, Pros. Amer. Math Soc., 109, 4, 1990.

85

A New Class of Operator Ideals Defined via s-Numbers and Lp(Φ) Sequence Space

Pınar Zengin Alp 1, Emrah Evren Kara 2

Abstract. In this study, we define a new class of operator ideals via s-numbers and `p(Φ) sequence

space which is denoted by Lp,Φ (E,F ). Also it is proved that this class is a quasi-Banach operator ideal

by a quasi-norm defined on this class. Then some other classes defined by using different examples

of s-number sequences. Afterwards, these classes are examined if they are injective, surjective and

symmetric.

Keyword: Operator ideal, s-numbers, Euler totient matrix.

AMS 2010: 47B06,47B37,47L20.

References

[1] M. Ilkhan, E. E. Kara, A new Banach space defined by euler totient matrix operator, Operators and matrices, In

press.

[2] A. Pietsch, s-Numbers of operators in Banach spaces, Studia Mathematica, 51(3), 201-223,1974.

[3] A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

[4] A. Maji, P.D. Srivastava, Some class of operator ideals, Int. J. Pure Appl. Math., 83 (5), 731-740, 2013.



86

A New Paranormed Sequence Space Defined by Catalan Conservative Matrix

Pınar Zengin Alp 1

Abstract. In this study, by using Ilkhan Catalan conservative matrix, we give a new paranormed

sequence space `(C, p). Also we prove that the spaces `(C, p) and `(p) are linearly isomorphic. Then

we compute α−, β−, γ− duals and Schauder basis of this space.

Keyword: Catalan numbers, α−, β−, γ− duals, paranormed sequence space.

AMS 2010: 46A45, 46B45 .

References

[1] KG. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox. J. Math. Anal. Appl., 180(1),

223-238,1993.

[2] C. Aydın, F. Basar, Some new paranormed sequence spaces, Inform. sci.,160, 27-40, 2006.

[3] M. Ilkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces c and c0 ,

Linear and Multilinear Algebra,2019, DOI: 10.1080/03081087.2019.1635071

[4] R.P Stanley, Catalan Numbers, Cambridge University Press, New York,2015.


87

Statistical Convergence and Operator Valued Series

Ramazan Kama 1

Abstract. The concept of statistical convergence was firstly introduced by Fast [1] and later reintro-

duced by Schoenberg [4]. Then, this notion has been studied by many authors in various spaces. Fridy

[3] proved that a number sequence is statistical convergent if and only if it is statistical Cauchy. The

statistical convergence in Banach spaces was studied by Kolk [5]. Connor et al. [2] given important

results that relate the statistical convergence to classical properties of Banach spaces.

By ω, we denote the space of all real valued sequences. Any vector subspace of ω is called as a sequence

space. Let `∞, c and c0 be the spaces of all real valued bounded, convergent and null sequences

x = (xk), respectively, normed by ‖x‖∞ = supk∈N |xk|, where N denotes the set of positive integers.

Let X,Y be normed spaces, L(X,Y ) be also the space of continuous linear operators from X into Y

and∑i Ti be a series in L(X,Y ). λ be a vector space of X-valued sequences which contains c00(X), the

space of all sequences which are eventually 0. By `∞(X) and c0(X), we denote the X-valued sequence

spaces of bounded and convergence to zero, respectively. The series∑i Ti is λ-multiplier convergent

if the series∑i Tixi converges in Y for every sequence xi ∈ λ. The series

∑i Ti is λ-multiplier

Cauchy if the series∑i Tixi is Cauchy in Y for every sequence xi ∈ λ. If λ = `∞(X), a series

∑i Ti

is said to be `∞(X)−multiplier (Cauchy) convergent, and if λ = c0(X), a series∑i Ti is said to be

c0(X)−multiplier (Cauchy) convergent. For more information about operator valued series, see [6].

In this paper, we introduce and study some spaces of operator valued series by means of statistical

convergence, and also give some relationship between these spaces.

Keyword: Operator valued series, statistical convergence, completeness.

AMS 2010: 46B15, 40A05.

References

[1] H. Fast, Sur la convergence statistique, Colloq Math. 2, 241-244, 1951.

[2] J. Connor, M. Ganichev and V. Kadets, A characterization of Banach spaces with separable duals via weak statistical

convergence, J Math Anal Appl. 244, 251-261, 2000.

1Siirt University, Siirt, Turkey, [email protected]

88

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

[4] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer Math Monthly 66,

361-375, 1959.

[5] E. Kolk, The statistical convergence in Banach spaces, Acta et Comment Univ Tartu 928, 41-52, 1991.

[6] C. Swartz, Multiplier Convergent Series, World Scientific, Singapore, 2009.

89

Perturbations of Frames in Quaternionic Hilbert Spaces

S. K. Sharma 1, Ghanshyam Singh 2 and Soniya Sahu 3

Abstract. In this article, we study various types of perturbations of frames in a quaternionic Hilbert

space and obtain stability conditions. We perturb frames in a quaternionic Hilbert space by a non-zero

element and show with the help of examples that the perturbed sequence in quaternionic Hilbert space

may not be a frame. Also, we consider perturbation of a frame by a sequence of scalars and obtain a

sufficient condition for the stability of a frame in a quaternionic Hilbert space. Further, we consider

perturbation of a frame by a finite linearly independent set and obtain a necessary condition for the

stability of the same. Finally, we obtain sufficient conditions for the stability of finite sum of frames in

quaternionic Hilbert spaces.

Keyword: Frames, Quaternionic Hilbert spaces.

AMS 2010: 42C15, 42A38.

References

[1] S.L. Adler, Quaternionic quantum mechanics and quantum fields, Oxford University Press, New York, 1995.

[2] P.G. Casazza, The art of frame theory, Taiwanese J. of Math., 4 (2), 129-201, 2000.

[3] Q. Chen, P. Dang and T. Qian, A frame theory of hardy spaces with the quaternionic and the clifford algebra setting,

Adv. Appl. Clifford Algebras, 27, 1073-1101, 2017.

[4] O. Christensen, An introduction to Frames and Riesz Bases, Birkhauser, 2003.

[5] R.J. Duffin and A.C. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72, 341-366, 1952.

[6] R. Ghiloni, , V. Moretti and A. Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev.

Math. Phys. 25, 1350006, 2013.

[7] S.K. Kaushik, G. Singh and Virender, On perturbation of frames in Hilbert spaces, International Journal of Pure

and Applied Mathematics, 37 (1), 65-72, 2007.

1Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110 007, India,

[email protected]

2Department of Mathematics and Statistics, M.L.S University, Udaipur, India, [email protected]

3Department of Mathematics and Statistics, M.L.S University, Udaipur, India, [email protected]

90

[8] M. Khokulan, K. Thirulogasanthar and S. Srisatkunarajah, Discrete frames on finite dimensional quaternion Hilbert

spaces, Proceedings of Jaffna University International Research Conference (JUICE 2014).

[9] S.K. Sharma and Shashank Goel, Frames in quaternionic Hilbert space, J. Math. Phy., Anal., Geo., Accepted.

[10] S.K. Sharma and Virender, Dual frames on finite dimensioal quaternionic Hilbert space, Poincare J. Anal. Appl.,

(2), 79-88, 2016.

91

Some Recent Results on Approximation by Linear Positive Operators

Tuncer Acar 1

Abstract. Recently, linear positive operators preserving some exponential functions are the interest

of researchers since they are effective and useful in better rate of convergence in some certain senses. In

this talk, we present some sequence of linear positive operators preserving some certain functions and

we investigate its uniform convergence, rate of convergencve, pointwise convergence by Voronovskaya

type theorem.

Keyword: Exponential operators, Szasz-Mirakyan operators, Rate of convergence, Modulus of conti-

nuity.

AMS 2010: 41A25, 41A36.

References

[1] T. Acar, Asymptotic formulas for generalized Szasz-Mirakyan operators, Applied Mathematics and Computation,

263, 223-239, 2015.

[2] T. Acar, A. Aral, I. Rasa, Modified Bernstein-Durrmeyer operators, General Mathematics, 22(1), 27–41, 2014.

[3] T. Acar, A. Aral, D. Cardenas-Morales, P. Garrancho, Szasz-Mirakyan type operators which fix exponentials, Results

Math., 72, 1393–1404, 2017.

[4] T. Acar, P.N. Agrawal, A. Sathish Kumar, On a modification of (p, q)-Szasz-Mirakyan operators, Complex Anal.

Oper. Theory, 12(1), 155–167, 2018.

[5] T. Acar, M. Cappelletti Montano, P. Garrancho, V. Leonessa, On Bernstein-Chlodovsky operators preserving e−2x,

submitted.


92

Representation Theory for Finite Hankel-Clifford Transforms Using Complex

Inversion Operator

V. R. Lakshmi Gorty 1

Abstract. In this paper the author obtains a representation theory to complex inversion operator for

finite Hankel-Clifford transforms. Section is concluded by establishing representation theorem for the

determined function to be non-decreasing in every finite interval.

Keyword: Finite Hankel-Clifford transform, weak compactness, closed rectifiable curve.

AMS 2010: 33-XX, 44-XX.

References

[1] J. A. Dorta Diaz and J. M. R. Mendez-Perez, Dini’s series expansions and the Finite Hankel-Clifford transformations,

Jour. Inst. Math.and Comp.Sci., (Math. Ser.) Vol. 5, No. 1, 1-17, 1992.

[2] L. S. Dube, On finite Hankel transformation of generalized functions, Pacific J. Math. 62, 365-378, 1976.

[3] S. P. Malgonde, V. R. Lakshmi Gorty, Orthonormal series expansions of generalized functions and the finite gener-

alized Hankel-Clifford transformation of distributions, Rev. Acad. Canaria. Cienc. XX, Nums.1-2, 2009.

[4] I. N. Sneddon, The Use of integral transforms, Tata Mc Graw Hill Publishing Co. Ltd., New Delhi, 1974.

[5] G. N. Watson, A Treatise on the Theory of Bessel Functions, ( 2nd.ed.) Cambridge University Press, 1958.

[6] A. H. Zemanian, Generalized integral transformations, Inter science, New York, Republished by Dover, New York,

1987.

[7] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.

[8] A. Erdelyi, Higher Transcendental Functions, apps.nrbook.com/bateman/Vol1.pdf, 1953.

[9] D. T. Haimo, Integral Equations Associated with Hankel Convolutions, Transactions of the American Mathematical

Society, Vol. 116, pp. 330-375, 1965.

[10] E. T. Whittaker and G. N Watson, A Course Of Modern Analysis, Cambridge at the University Press, 1935.

1SVKM’s NMIMS University, MPSTME, Mumbai, India, [email protected]

93

On an Existence of the Optimal Shape for One Functional Related with the

Eigenvalues of Pauli Operator

Yusif Gasimov 1, Aynure Aliyeva 2

Abstract. In the work a shape optimization problem for the functional involving the eigenvalues of

Pauli operator is formulated. A theorem is given on the existence of the solution for the considered

problem.

It is known that eigenvalues of the different operators describe various physical or mechanical parame-

ters of the corresponding natural processes. For instance, eigenvalues of Schrodinger operator describe

full energy of the particle, biquadratic operator with corresponding boundary conditions - eigenfre-

quency of the free, clamped and pressed plates. Therefore investigation of the problems related with

the eigenvalues are of importance both from practical and theoretical points of view. A large class of

problems are reduced to the solution of the minimization problems for the eigenvalues. Traditionally

minimization in these problems are carried out over the physical parameters related to the considered

process such as (parameters of the materials of the plates, environment, boundary conditions etc.).

In mathematical formulation these parameters are described by the corresponding functions. But in

some problems require minimization with respect to the geometrical characteristics (for example form

of the plate). So we come to the shape optimization problems. In spite of different methods have been

developed to the investigation of the existence of the solutions in such problems, their analytical or

numerical construction, there is not any general method for their solution [2-4].

In this work we formulate shape optimization problem for the functional involving first two eigenvalues

of Pauli operator. It is known that this operator describes a motion of the particle in the external force

field and is a generalization of the Schrodinger operator. Its eigenvalues are indeed full energy of the

quantum particle [1].

1Azerbaijan University, Baku, Azerbaijan; Institute of Mathematics and Mechanics ANAS, Baku, Azerbaijan; Institute

for Problems of Physics, Baku State University, Baku, Azerbaijan, e-mail: [email protected]

2Sumgayit State University, Sumgayit, Azerbaijan, e-mail: [email protected]

94

Abstract. (Continuation) Consider the following eigenvalue problem

Pϕ = λϕ , x ∈ D , (1)

ϕ = 0 , x ∈ SD , (2)

where P is the Pauli operator defined as below

P = P (a, ν) · J + σB. (3)

Here

J =

1 0

0 1

, σ =

1 0

0 −1

, P = (a, v) = (−i∇− a)2 + V,

i is an imaginary unit; V is a smooth enough function; ∇ =∂∂x, ∂∂y

; a = (a1, a2) ∈ R2 is a vector

potential; B is a magnet field generated by the vector potential a

B =∂

∂xa2 −

∂

∂ya1.

If to consider all these definitions one can write two dimensional Pauli in the form

P =

(−i∇− a)2 + a2∂∂x − a1

∂∂y + V 0

0 (−i∇− a)2 − a2∂∂x + a1

∂∂y + V

=

=

−∆ + (2ia1 + a2) ∂∂x + (2ia2 − a1) ∂

∂y + a2 + V 0

0 −∆ + (2ia1 − a2) ∂∂x + (2ia2 + a1) ∂

∂y + a2 + V

.

(4)

By K we denote the set of all bounded, convex domains Ω with smooth boundary. Consider the

problem

min

λ2

1(Ω) + λ22(Ω)

λ1(Ω) + λ2(Ω): Ω ∈ K, mesΩ ≤ m

. (5)

Here λ1(Ω) and λ2(Ω) are the first and second Dirichlet eigenvalues of operator (4) in the domain

Ω ∈ K, m is a given number.

The following theorem is proved for this problem.

Theorem. Let m > 0 and K is large enough (generally, K should be as large as to contain the ball of

value 2N−1m). Then problem (5) has a solution on the set K.

Keyword: Pauli operator, eigenvalues, shape optimization, existence theorem.

AMS 2010: 49J45, 49Q10, 49R50.

References

[1] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrodinger Operators: With Application to Quantum Mechanics

and Global Geometry, Springer, 2009.

[2] Y.S. Gasimov, Some shape optimization problems for the eigenvalues. J. Phys. A: Math. Theor., 41, 5, 2008.

95

[3] Y.S. Gasimov, A. Nashaoui, A.A. Niftiyev, Nonlinear eigenvalue problems for p-Laplacian, Optimization Letters, 4,

2010.

[4] I. Elishakoff, Eigenvalues of Inhomogeneous Structures, CRC Press, Boca Raton, 2004.

96

APPLIEDMATHEMATICS

A Highly Accurate Difference Method for Solving the Laplace Equation on a

Rectangular Parallelepiped with Boundary Values in Ck,λ

Adiguzel A. Dosiyev 1

Abstract. A three-stage difference method for solving Dirichlet problem of Laplace’s equation on a

rectangular parallelepiped is proposed and justified. At the first stage, approximate values of the sum

of the pure fourth derivatives of the solution are defined on a cubic grid by the 14-point difference

operator. At the second stage approximate values of the sum of the pure sixth derivatives of the so-

lution are defined on a cubic grid by the simplest 6-point difference operator. At the third stage, the

system of difference equations the for the sought solution is constructed again by using the 6-point

difference operator with the corrections by the quantities determined of the first and the second stages.

It is proved that the proposed difference method converges uniformly with order of O(h6(|lnh| + 1)),

when the boundary functions on the faces are from C7,1, and on the edges their second and fourth

order derivatives satisfy the compatibility conditions which result from the Laplace equation. The

convergence of the method is also analyzed for other cases in Ck,λ, 0 < λ ≤ 1. Numerical experiment

is illustrated to support the theoretical results.

Keyword: Finite difference method, error estimations, numerical solution to the Laplace equation.

AMS 2010: 65M06, 65M12.

1Near East University, Nicosia,TRNC, Mersin 10, Turkey, [email protected]

97

Effective Error Estimate for the Hexagonal Grid Solution of Laplace’s Equation

on a Rectangle

Adiguzel A. Dosiyev 1, Suzan C. Buranay 2

Abstract. The error estimates of the approximate solution of PDEs obtained by the finite-difference

method as usual contain the maximum of absolute values of derivatives of the desired solution, i.e.,

these are not effective. In literature, some effective estimates, which depend only on the given data of

problems for the Laplace and Poisson equations for the square and special triangular grids were ob-

tained (see [1], [2]). In spite of the fact that the hexagonal grids are used in different applied problems

there is no effective error estimates given. In this paper, for the approximate solution obtained by

the hexagonal grid of Laplace’s equation on a rectangular domain, the effective error estimate of order

O(h4), in the maximum norm is given. This estimation is obtained by using the discrete and analytic

form of the Fourier representation of the difference and analytic solutions respectively, and depends on

the fifth derivatives of the boundary functions only. Numerical experiments are illustrated to support

the analysis made.

Keyword: Laplace’s equation, Dirichlet boundary value problem, hexagonal grids, effective error esti-

mate.

AMS 2010: 65N06, 65N15, 65N22.

References

[1] W. Wasow, On the truncation error in the solution of laplace’s equation by finite differences, Journal of Research of

the National Bureau of Standards, 48, 345-348, 1952.

[2] E.A. Volkov, Effective error estimates for solutions by the method of nets, of boundary value problems on a rectangle

and on certain triangles for the laplace and poisson equations, Proc.Steklov Inst. Math. 74, 57-90, 1966.

1Department of Mathematics, Faculty of Arts and Sciences, Near East University, Nicosia, North Cyprus, Via Mersin

10, Turkey, [email protected] of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North

Cyprus, Via Mersin 10, Turkey, [email protected]

98

On Exact Controllability of Semilinear Systems

Agamirza E. Bashirov 1

Abstract. The exact controllability is a property of control systems consisting in attaining every

point in the state space from every initial state point. It was defined by R. E. Kalman in [1]. A weaker

concept is the approximate controllability which is an ability to approximate every point in the state

space instead of attaining. The controllable systems are very important in engineering applications.

A simple example is a robot which has to move any object in some area to some another location in

the same area. Such a robot is functional if the control system describing the movement of its arm is

controllable. Another example is a network which is able to communicate every input to every output.

There are different papers proving sufficient conditions of exact controllability for deterministic semilin-

ear control systems. These papers use different fixed-point theorems. The method by fixed-point theo-

rems is not an efficient method. It requires validity of some extra inappropriate conditions. Moreover,

in stochastic case, this method requires the coerciveness of the stochastic analogue of controllability

operator, while it is proved that it is never coercive [2]. Therefore, there is a need in an alternative

method for investigation of the exact controllability avoiding fixed-point theorems.

We suggest such an alternative method and prove the exact controllability of a semilinear systems

under essentially two conditions besides the conditions for the existence and uniqueness of its solution.

The first condition is the coerciveness of the controllability operator at all non-initial time moments. In

turn, this means that the linear part of semilinear system must be exactly controllable on all intervals.

The second one is the boundedness of the nonlinear terms disturbing the linear part. In the stochastic

case, this is two functions representing nonlinear drift and diffusion terms. But in the deterministic

case, this reduces to one function.

Keyword: Exact controllability, semilinear system, deterministic system, stochastic system.

AMS 2010: 93B05.

1Eastern Mediterranean University, Gazimagusa, North Cyprus ([email protected]) and Institute of

Control Systems, Baku, Azerbaijan.

99

References

[1] R.E. Kalman, Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 5,

102-119, 1960.

[2] A.E. Bashirov and N. Ghahramanlou, On partial s-controllability of semilinear partially observable systems, Inter-

national Journal of Control, 88, 969-892, 2015.

100

Attractor for Nonlinear Transmission Acoustic Problem

Akbar B. Aliev 1, Sevda E. Isayeva 2

Abstract. Let Ω be a bounded domain in R3 with smooth boundary Γ1, Ω2 ⊂ Ω1 is a subdomain with

smooth boundary Γ2 and Ω1 = Ω\(Ω2 ∪ Γ2) is a subdomain with boundary Γ = Γ1 ∪ Γ2, Γ1 ∩ Γ2 = ∅.

The nonlinear transmission acoustic problem considered here is

utt −∆u+ αut + u+ f1(u) = 0 in Ω1 × (0,∞),

ϑtt −∆ϑ+ αϑt + ϑ+ f2(ϑ) = 0 in Ω2 × (0,∞),

δtt + βδt + δ = −ut on Γ2 × (0,∞),

u = 0 on Γ1 × (0,∞),

u = ϑ, δt =∂u

∂ν− ∂ϑ

∂νon Γ2 × (0,∞),

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω1,

ϑ(x, 0) = ϑ0(x), ϑt(x, 0) = ϑ1(x), x ∈ Ω2,

δ(x, 0) = δ0(x), δt(x, 0) =∂u0

∂ν− ∂ϑ0

∂ν≡ δ1 x ∈ Γ2

where ν is the unit outward normal vector to Γ; α > 0 and β > 0; fi : R → R (i = 1, 2), u0, u1 :

Ω1 → R, ϑ0, ϑ1 : Ω2 → R, δ0 : Γ2 → R.

Assume that fi ∈ C1(R), i = 1, 2 and there exist constants ci ≥ 0, i = 1, 2, such that

|f ′i(s)| ≤ ci(1 + s2), lim inf|s|→∞

fi(s)

s> −1. (6)

The problem (??)-(??) can be formulated in the energy space

V =w = (w1, w2, w3, w4, w5, w6) : w1 ∈ H1

Γ1

(Ω1),

w2 ∈ L2(Ω1), w3 ∈ H1(Ω2), w4 ∈ L2(Ω2), w5 ∈ L2(Γ2), w6 ∈ L2(Γ2), w1|Γ2

= w3|Γ2

.

1Institute of Mathematics and Mechanics of Azerbaijan National Academy of Sciences, Baku, Azerbaijan,

[email protected]

2Baku State University, Baku, Azerbaijan, [email protected]

101

Abstract. (Continuation) We introduce the linear unbounded operator A : D(A) ⊂ V → V :

Aw = (w2,∆w1 − w1 − α1w2, w4,∆w3 − w3 − α2w4, w6,−w2 − w5 − βw6) .

Furthermore, we introduce the nonlinear function Φ : V → V :

Φ(w) = (0,−f1(w1), 0,−f2(w3), 0, 0)

for every w ∈ V .

Then the problem can be put in the form wt = Aw + Φ(w),

w(0) = w0,

(7)

where w = (u, ut, ϑ, ϑt, δ, δt) and w0 = (u0, u1, ϑ0, ϑ1, δ0, δ1) ∈ V .

Theorem. Let (6) holds and w0 ∈ V . Then the problem (7) possesses a unique global attractor A in

the energy phase space V .

Keyword: Transmission acoustic condition, attractor, nonlinear hyperbolic equation.

AMS 2010: 35L05, 35L70.

102

Differential Type Hysteresis Operators Describing Irreversible Processes in

Ferroelectrics

Alexander Skaliukh 1

Abstract. For quasistatic polarization and deformation processes of polycrystalline ferroelectric con-

tinuum, nonlinear irreversible dependencies between the sought-for and determining parameters are

studied. For irreversible parts of polarization and deformation a rate independent dielectric and de-

formation hysteresis operators are constructed, which are a system of equations in differentials [1].

For the reversible parts the linear algebraic operators are derived, connecting them with the electric

field vector and mechanical stress tensor [2]. The results are used in the finite element analysis for

polycrystalline ferroelectric materials [3].

This work was supported by the Russian Foundation for Basic Research, grant 17-08-00860-a.

Keyword: Hysteresis, ferroelectrics, polarization, deformation.

AMS 2010: 74F15.

References

[1] A.V. Belokon and A.S. Skaliukh, Mathematical modeling of irreversible polarization processes, FIZMATLIT, Moscow,

2010.

[2] A.S. Skaliukh, Functional dependence of physical characteristics on irreversible parameters under electromechanical

effects on ferroelectric ceramics, Tomsk State University Bulletin, Mathematics and Mechanics, 58, 128-141, 2019.

[3] A.S. Skaliukh, Finite-element modeling irreversible polarization process of ferroelectric ceramics, Mathematics and

Mathematical Modeling, 5, 13-16, 2019.

1Institute mathematics, mechanics and computer sciences named after Vorovich I.I. of Southern Federal University,

Rostov on Don, Russia, [email protected]

103

Returned Sequences and Their Applications

Ali M. Akhmedov 1, Eldost U. Ismailov 2

Abstract. In this work, we study the behavior of the sequence an of complex numbers satisfying the

relation an+k = q1an+q2an+1 + . . .+qkan+k−1; where qn is a fixed sequence of complex numbers. Such

kind of sequences arise in problems of analysis, fixed point theory, dynamical systems, theory of chaos,

etc. [1]-[4]. Investigating the spectra of triple and more than triple band triangle operator-matrices,

the behavior of such sequence required [5,6]. From the point of application, the proved results and

formulas in the literature for the spectra of the operator-matrices look like very complicated. In this

work, we eliminate the indicated flaws and apply new approach where the formulas for the spectra

describe circular domains. Also we apply receiving results to some natural processes.

Keyword: Spectrum, difference operator-matrices, the sequence space, returned sequence, circular

domain.

AMS 2010: 42B20, 42B25, 42B35.

References

[1] A.M. Akhmedov and F. Basar, The fine spectra of the difference operator ∆ over the sequence space

bvp (1 ≤ p <∞), Acta Math. Sin. (Engl. Ser.), 23, 1757-1768, 2007.

[2] D. Popa, Hyers - Ulam, Stability of the linear recurrence with constant coefficients, Advances in Difference Equations,

2005(2), 101-102, 2005.

[3] B. Slavisa, Presic surune classe inequations aux differences finies et sur la convergence de certaines suites, Publ.de

I’nstitut Mathematique Nouvelle serie fome, 5(19), 75-78, 1965.

[4] M. S. Khan, M. Berzig and B. Samet, Some convergence results for iterative sequences of presic type and applications,

Advances in Difference Equations, 2012, 38: 19 p., 2012.

[5] H. Bilgic, H. Furkan and B. Altay, On the fine spectrum of the operator b(r; s; t) over c0 and c, Computers and

Mathematics with Applications, 63, 6989-998, 2007.

[6] H. Furkan, H. Bilgic and F. Basar, On the fine spectrum of the operator b(r; s; t) over the sequence spaces lp and

bvp, (1 < p < 1), Comp., Math., Appl. 60, 2141-2152, 2010.



104

Linear Stability of a Convective Flow in a Vertical Channel Generated by

Internal Heat Sources

Andrei Kolyshkin 1, Valentina Koliskina 2 and Inta Volodko 3

Abstract. Consider a steady flow of a viscous incompressible fluid in a vertical channel. The flow is

generated by internal heat sources of density Q distributed within the fluid. Fluid motion is described

by the system of the Navier-Stokes equations under the Boussinesq approximation. Linear stability

of the steady flow is investigated using the method of normal modes. The corresponding system of

ordinary differential equations is solved by the Chebyshev collocation method.

Channels of different shapes and different forms of the function Q are considered in the paper. The

analysis of the stability problem for the flow in a pipe caused by internal heat generation with constant

density Q is performed in [1]. Reasonable agreement is found between linear stability calculations

in [1] and available experimental data. We analyze also the case where the density of heat sources

Q = Q0(1 + αr) is a linear function of the radial coordinate [2].

Problems related to biomass thermal conversion [3] require extensive mathematical modeling in order to

determine optimal parameters that ensure effective energy conversion process. Hydrodynamic stability

analysis is used in the paper to analyze linear stability problem caused by internal heat generation

in accordance with the Arrhenius Law: Q = Q0 exp[−E/(RT )], where T is the temperature of the

fluid. The linear stability problem is solved for case of a flow in an annulus. The solution of a

nonlinear steady boundary value problem is obtained numerically. Linear stability analysis shows that

either axisymmetric or the first asymmetric mode is the most unstable depending of the radius ratio

between the cylinders. For large Prandtl numbers instability is associated with thermal waves that are

propagating downstream with sufficiently large phase velocity.

1Riga Technical University, Riga, Latvia, [email protected]



105

Abstract. The effect of a transverse magnetic field on the stability characteristics is also investigated.

In particular, the flow in a vertical channel caused by internal heat sources with density Q = Q0(1+αx)

and Q = Q0 exp(−αx) in the tranverse magnetic field is analyzed.

Keyword: Thermal instability, heat transfer, collocation method.

AMS 2010: 76E05, 80A20.

References

[1] A. Kolyshkin and V. Koliskina, Stability of a convective flow in a pipe caused by internal heat generation, JP Journal

of Heat and Mass Transfer, 15, 515-530, 2018.

[2] V. Koliskina and A. Kolyshkin, Linear stability analysis of a convective flow in a pipe due to radially distributed

heat sources in engineering for rural development, 1289-1294, 2018.

[3] M. Abricka, I. Barmina, R. Valdmanis, M. Zake and H. Kalis, Experimental and numerical studies on integrated

gasification and combustion of biomass, Chemical Engineering Transactions, 50, 127 - 132, 2016.

106

Optimizing Wiener and Randic Indices of Graphs

Anuradha Mahasinghe 1, Hasitha Erandi 2 and Sanjeewa Perera 3

Abstract. Wiener and Randic indices have long been studied in chemical graph theory as measures

of connection strength of graphs [1, 2]. Later on, these indices were used in different fields such as

network analysis [3]. We consider two optimization problems related to these indices, with potential

applications to network theory, in particular to epidemiological networks. Given a connected graph

and a fixed total edge weight, we investigate how the individual weights must be assigned to each edge,

minimizing the connection strength of the graph. In order to capture the connection strength, we

consider weighted Wiener index and a modified version of ordinary Randic index. The corresponding

optimization problems turn out to be both non–linear and non–convex, hence we adopt the technique

of separable programming and reduce them down to mixed integer linear programs [4]. We present our

experimental results by applying the relevant algorithms to several graphs.

Keyword: Chemical graph theory, separable programming,

AMS 2010: 94C15, 90C90.

References

[1] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69, 17-20, 1947.

[2] C. Delorme, O. Favaron and D. Rautenbach, On the randic index, Discrete Math. 257, 29-38, 2002.

[3] S.C. Basak, V.R. Magnuson, G.J. Niemi, R.R. Regal and G.D. Veith, Topological indices: their nature, mutual

relatedness, and applications, Math. Model. 8, 300-305, 1987.

[4] H.M. Markowitz and A.S. Manne, On the solution of discrete programming problems, Econometrica, 84-110, 1957.

1University of Colombo, Colombo, Sri Lanka, [email protected]



107

Curvature Stabilization and Thermally Driven Flows

Aytekin Cıbık 1

Abstract. We study the curvature stabilization of thermally driven flows, so called Boussinesq sys-

tem, in this study. The method we propose is accurate, effective and unconditionally stable. Accuracy

in time is proven and the convergence results for the fully discrete solutions of problem variables are

given. Numerical examples including the famous Marsigli’s flow are given to support the obtained

theoretical results and demonstrate the efficiency and the accuracy of the method.

Keyword: Boussinesq equations, finite element method, stabilization, curvature.

AMS 2010: 76R10, 65K15.

References

[1] C. Trenchea, Stability of partitioned imex methods for systems of evolution equations with skew-symmetric coupling,

ROMAI J. 10, 175-189, 2014.

[2] J. G. Liu, C. Wang and H. Johnston, A fourth order scheme for incompressible boussinesq equations, J. Sci. Comput.

18, 253-285, 2003.

[3] C. Trenchea, Second order implicit for local effects and explicit for nonlocal effects is unconditionally stable, ROMAI

J. 1, 163-178, 2016.

[4] N. Jiang, M. Mohebujjaman, L. G. Rebholz and C. Trenchea, An optimally accurate discrete regularization for second

order time-stepping methods for navier-stokes equations, Comput. Methods Appl. Mech. Engrg. 310, 388-405, 2016.

1Gazi University, Ankara, Turkey, [email protected]

108

Stability Analysis of a TB Epidemic Model in a Patchy Environment

Azizeh Jabbari 1, Somayyeh Fazeli 2

Abstract. In this paper, a two-patch model, is used to analyze the spread of tuberculosis, in which

only susceptible individuals can travel freely between the patches. The existence and uniqueness of the

associated equilibria are discussed. The model supports a globally-asymptotically stable disease-free

equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be

locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one.

Keyword: Patches, tuberculosis, stability.

AMS 2010: 93A30, 37B25.

References

[1] A. Jabbari, C. Castillo-Chavez, F. Nazari, B. Song and H. Kheiri, A two-strain tb model with multiple stagee,

Mathematical Biosciences and Engineering, 13(4), 741-785, 2016.

[2] H. Kheiri and M. Jafari, Stability analysis of a fractional order model for the hiv/aids epidemic in a patchy environ-

ment, Journal of Computational and Applied Mathematics, 346, 323339, 2019.

[3] J. Tewa, S. Bowon and B. Mewoli, Mathematical analysis of two-patch model for the dynamical transmission of

tuberculosis, Applied Mathematical Modelling, 36, 2466-2485, 2012.

1Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran, a [email protected]

2Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran, [email protected]

109

Mathematical Analysis of a Fractional-Order Model of Tuberculosis Epidemic with

Exogenous Re-Infection

A. Jabbari 1, H. Kheiri 2 and F. Iranzad 3

Abstract. In this paper, we present a fractional-order model of tuberculosis (TB) epidemic with

exogenous re-infection among the latently infected individuals. We have mainly found that the model

exhibits the phenomenon of backward bifurcation, where the stable disease-free equilibrium coexists

with a stable endemic equilibrium, when the basic reproduction ratio is less than unity. In this case,

the model have multiple boundary equilibria. It is shown that backward bifurcation dynamics feature

is caused by the re-infection of latently infected individuals.

Keyword: Fractional order derivatives, tuberculosis, backward bifurcation.

AMS 2010: 26A33, 93A30.

References




ment, Journal of Computational and Applied Mathematics, 346, 323-339, 2019.

[3] H. Kheiri and M. Jafari, Optimal control of a fractional-order model for the hiv/aids epidemic, International Journal

of Biomathematics, 11(7), DOI: 10.1142/S1793524518500869, 2018.

1Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran, a [email protected]

2Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, [email protected]


110

Explicit Solutions and Conservation Laws of a Generalized Extended

(3+1)-Dimensional Jimbo-Miwa Equation

Chaudry Masood Khalique 1

Abstract. In this talk we study a generalized extended (3+1)- dimensional Jimbo-Miwa equation.

Using symmetry methods we obtain its explicit solutions in terms of the incomplete elliptic integral

function. Moreover we present conservation laws of the underlying equation.

Keyword: Jimbo-Miwa equation, exact solutions, conservation laws.

AMS 2010: 35B06, 35L65.

References

[1] L.V. Ovsiannikov, Group Analysis of Differential Equations (English Translation by W.F. Ames), Academy Press,

New York, 1982.

[2] G.W. Bluman and S. Kumei, Symmetries of differential equations, Springer-Verlag, New York, 1989.

[3] P.J. Olver, Applications of lie groups to differential equations, graduate texts in mathematics, 107, 2nd edition,

Springer-Verlag, Berlin, 1993.

[4] I. Simbanefayi and C.M. Khalique, Travelling wave solutions and conservation laws for the korteweg-de vries-bejamin-

bona-mahony equation, Results in Physics, 8, 57-63, 2018.

[5] M. Abramowitz and I. Stegun, Handbook of mathematical functions, New York, Dover, 1972.

[6] N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333, 311–328, 2007.

1International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathemati-

cal Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa,

[email protected]

111

A Very Efficient Approach for Pricing Geometric Asian Rainbow Options Described

by the Mixed Fractional Brownian Motion

D. Ahmadian 1, L. V. Ballestra 2

Abstract. We deal with the problem of pricing geometric Asian rainbow options on assets described

the mixed fractional Brownian motion. Based on standard no-arbitrage arguments, we derive a partial

differential problem in several independent variables, which we solve by applying suitable changes of

variables and theoretical results established in [1] and [2]. Numerical simulations reveal that the pro-

posed method is extremely accurate and fast, and performs significantly better than the finite difference

method.

Keyword: Mixed fractional Brownian motion, rainbow options, Asian option, multi-asset option, non-

Markov process.

AMS 2010: 35R99, 91B25, 91G60.

References

[1] H. Johnson, Options on the maximum or the minimum of the several assets, Bull. J. FINANC. QUANT. ANAL. 287,

273-288, 1987.

[2] P. Cheridito, Arbitrage in fractional brownian motion models, FINANC. STOCH. 7, 533-553, 2003.

1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran,[email protected] of Statistical Sciences, Alma Mater Studiorum University of Bologna, Via delle Belle Arti 41, 40126

Bologna, Italy, [email protected]

112

Scattering Theory of Dirac Operator with the Impulsive Condition on Whole Axis

Elgiz Bairamov 1, Seyda Solmaz 2

Abstract. In this paper, we investigate the Jost solutions of the impulsive Dirac systems (IDS) on

entire axis and study analytic and asymptotic properties of these solutions. Furthermore, characteristic

properties of the scattering matrix of the IDS have been examined. Finally, we also compare the similar

properties for the IDS with the mass m on entire axis and give an example.

Keyword: Differential equations, Dirac systems, Jost solutions, scattering matrix.

AMS 2010: 34B37, 35P25, 47A75.

References

[1] L.D. Faddaev, The inverse problem of quantum theory of scattering, J. Math. Phys., Vol 4/1, 72-104, 1963.

[2] M.G. Gasymov, The inverse scattering problem for a system of dirac equations of order 2n, Trans. Moscow Math.

Sot. 19, 41, 1968.

[3] Fam Loa Vu, The inverse scattering problem for a system of dirac equations on the whole axis, (Russian) Ukrain.

Mat. Zh. 24, 666-674, 716. (Reviewer: J. Wiesner) 47E05 (81.47), 1972.

[4] I.S. Frolov, An inverse scattering problem for the dirac system on the entire axis, (Russian) Dokl. Akad. Nauk SSSR

207, 44-47. (Reviewer: R. C. Gilbert) 34B25 (47E05), 1972.

[5] I.P.P. Syroid, Conditions for the absence of spectral singularities for a non-self-adjoint dirac operator in terms of the

potential, Ukrain. Math. Zh. Vol. 38/3, 359-364, 1986.

[6] E. Ugurlu, Dirac systems with regular and singular transmission effects, Turk J. Math. 41, 193-210, 2017.

[7] E. Bairamov and S. Solmaz, Spectrum and scattering function of the impulsive discrete dirac systems, Turk. J. Math.

42, 3182-3194, 2018.

1Ankara University, Ankara, Turkey, [email protected]

2Ankara University, Ankara, Turkey, [email protected]

113

Infimal Convolution Method for Duality in Second Order Discrete and

Differential Inclusions with Delay

Elimhan N. Mahmudov 1, 2

Abstract. The paper deals with the optimal control problem described by second order evolution

differential inclusions (DFIs) with delay; to this end first we use an auxiliary problem with second or-

der discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex

functions, step by step we construct the dual problems for discrete, discrete-approximate and DFIs

and prove duality results. It seems that the Euler-Lagrange type inclusions are ”duality relations” for

both primary and dual problems and that the dual problem for discrete-approximate problem make

a bridge between them. Finally, relying to the method described within the framework of the idea of

this paper a dual problem can be obtained for any higher order DFIs. In this way for computation of

the conjugate and support functions of discrete-approximate problems a Pascal triangle with binomial

coefficients, can be successfully used for any ”higher order” calculations.

As is well known, many extremal problems, important from a practical point of view, are described

in terms of set-valued mappings and form a component part of the modern mathematical theory of

controlled dynamical systems and mathematical economics [1]-[6]. Along with these duality theory

occupies a central place in classical convex optimization. Thus, the present paper is dedicated to one

of the difficult and interesting fields construction of duality of optimization problems with second order

ordinary DFIs with delay: infimum ϕ(x(1), x′(1)

)subject to x′′(t) ∈ F

(x(t), x′(t), x(t − h), t

)a.e.

t ∈ [0, 1], x(t) ∈ Ω(t), t ∈ [0, 1];x(t) = ξ(t), t ∈ [−1, 0), x(0) = θ, h > 0, where F (·, t) : R3n→→ Rn and ϕ

are time dependent set-valued mapping and continuous proper function, respectively, ξ(t), t ∈ [−h, 0) is

an absolutely continuous initial function, θ is a fixed vector, Ω : [0, 1]→→Rn is a set-valued mapping. It

is required to find an absolutely continuous function x(t), t ∈ [−h, 1] minimizing the Mayer functional

over a set of feasible trajectories. Keyword: Infimal convolution, duality, conjugate, Euler-Lagrange,

approximation.

AMS 2010: 34A60, 49N15, 49M25, 90C46.

1Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

2Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan. [email protected]

114

References

[1] E.N. Mahmudov, Approximation and optimization of discrete and differential inclusions, Elsevier, Boston, USA,

2011.

[2] E.N. Mahmudov, Approximation and optimization of higher order discrete and differential inclusions, Nonlinear Diff.

Equ. Appl., NoDEA 21, 1-26, 2014.

[3] E.N. Mahmudov, Convex optimization of second order discrete and differential inclusions with inequality constraints,

J. Convex Anal. 25, 1-26, 2018.

[4] E.N. Mahmudov, Optimization of mayer problem with sturm-liouville type differential inclusions, J. Optim.Theory

Appl. 177, 345-375, 2018.

[5] E.N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: Control,

Optimisation and Calculus of Variations, DOI: https://doi.org/10.1051/cocv/2019018

[6] B.S. Mordukhovich, N.M. Nam, R.B. Rector and T. Tran, Variational geometric approach to generalized differential

and conjugate calculi in convex analysis, Set-Valued Var. Anal. 25, 731–755, 2017.

115

Inverse Sturm-Liouville Problem in the Case Finite-Zoned Periodic Potentials

E. S. Panakhov 1

Abstract. In this speech we assume that the potential q(x) is a smooth periodic function of period

Ie. We consider the two Sturm-Liouville problems

−y′′ + q (x) y = λy,

A) y (0) = y (π) ,

y′ (0) = y′ (π) ,

B) y (0) = −y (π) ,

y′ (0) = −y′ (π) ,

We call the first problem a periodic Sturm-Liouville problem, and the second an antiperiodic problem,

We denote the spectrum of the first problem by λ0 < λ3 ≤ λ4 < λ7 ≤ λ8 < ...and and the spectrum of

the second by λ1 ≤ λ2 < λ5 ≤ λ6 ≤ ...Both spectra can be arranged in a single chain of inequalities,

namely, λ0 < λ1 ≤ λ2 < λ3 ≤ λ4...

Along with the problems (A) and (B), we consider yet another problem

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

We denote the spectrum of the problem (C) by v1 < v2 < ...It is well known that λ1 ≤ v1 < λ2 < λ3 ≤

v2 ≤ λ6 < ...

We say the potential q(x) is N-zoned if

λ2N+1 = λ2N+2, λ2N+3 = λ2N+4, ...

In the case of an N-zoned potential we have, for all n > N

λ2N+1 = vn = λ2N+2 (1.2)

1Institute of Applied Mathematics, Bakun State University, Baku, Azerbaijan, [email protected]

116

Abstract. (Continuation) We denote by c (x, λ) and s (x, λ) solutions of (1.1) satisfying the initial

conditions c′ (0, λ) = s (0, λ) = 0.Further, we put

f (λ) =1

2

[c (π, λ) + s′ (π, λ)

](1.3)

We call f (λ) a Lyapunov function. We consider integral equation

K (x, s) + F (x, s) +

∫ x

0

K (x, s)F (x, s) = 0, (0 ≤ s ≤ x ≤ π) (1.4)

Based on this representation, we investigate the generalized degeneration of the kernel of the integral

equation (1.4). And we can also prove the Hochstadt’s theorem in this case

117

Optimal Successive Complementary Expansion for Singular Differential Equations

Fatih Say 1

Abstract. Singular differential equations with small parameters and their applications are a highly

topical field in applied mathematics and physics. Recently the method of successive complementary

expansion (SCEM), which works efficiently, is introduced in [1]. However, since the asymptotic rep-

resentation of the singular differential equations is mostly divergent, the method misses some useful

features for these types of equations. For instance, truncation of the series at random may cause to in-

crease the truncation error and hence may decrease the accuracy of the asymptotic solution. Moreover,

crucial information about these equations is hidden by the algebraic order solutions, and they can be

extracted by exponential asymptotics [2, 3, 4]. In this work, we study a singular ordinary differential

equation of a two-point boundary value problem asymptotically. We particularly consider the SCEM

with asymptotics beyond all orders [5] along with its benefits and drawbacks. In doing this, we use the

well-known WKB approach for the instructive differential equation of [6, 7]. We extend the SCEM for

singular differential equations by implementing techniques in exponential asymptotics and therefore

we did not eliminate the growing exponentials and their asymptotic behaviour across certain lines as

they have a plethora applications in many areas.

Keyword: Asymptotic expansion, singular perturbation problems, asymptotics beyond all orders, ex-

ponential smallness, Stokes lines.

AMS 2010: 34E20, 34M35, 34M40, 34M60.

References

[1] J. Cousteix and J. Mauss, Asymptotic analysis and boundary layers, Springer Science & Business Media, Berlin,

Heidelberg, 2007.

[2] M.V. Berry, Stokes’s phenomenon; smoothing a victorian discontinuity, Publ. Math-Paris, 68(1), 211-221, 1988.

[3] G.G. Stokes, Mathematical and physical papers by the late sir george gabriel stokes v, Cambridge University Press,

Cambridge, 1905.

[4] J.P. Boyd, Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge, SIAM Rev.,

47(3), 553-575, 2005.

1Department of Mathematics, Ordu University, Ordu, Turkey, [email protected]

118

[5] M.V. Berry, Asymptotics, superasymptotics, hyperasymptotics, in: H. Segur, S. Tanveer and H. Levine (eds), Asymp-

totics Beyond all Orders, Springer, pp. 1-14, Boston, 1991.

[6] F.W. Dorr and S.V. Parter, Singular perturbations of nonlinear boundary value problems with turning points, J.

Math. Anal. Appl. 29(2), 273-293, 1970.

[7] D. Kamowitz, Multigrid applied to singular perturbation problems, Appl. Math. Comput. 25(2), 145-174, 1988.

119

On the Stability of Bodewadt Flow over a Rough Rotating Disk

Fatih Say 1†, Burhan Alveroglu 2†

Abstract. The rotating disk boundary layer flows [1] have received increased attention of the re-

searches as they are relevant to many industrial devices such as rotor-stator systems and turbine

engines. Some of the studies could be found in [2, 3]. One of the important ones of those types of

the flows is the Bodewadt boundary layer flow [4]. It arises due to the rotation of the steady flow at

a larger distance from a stationary disk. This boundary layer flow is extended to the case where the

disk surface admits partial slip and investigated here analytically for large Reynolds numbers using an

asymptotic approach. Surface roughness is modelled imposing a partial-slip boundary condition at the

wall. We are particularly interested in the effects of the roughness on the inviscid Type I instabilities

that are shown as stationary crossflow vortices. The basic flow solution is obtained as an exact solution

of the Navier-Stokes equation.

In this study, we follow the approaches of Stephen [5, 6] and Culverhouse [7] using the techniques

of perturbation theory to investigate the linear stability of the inviscid Type I instabilities over the

appropriate asymptotic regions. The results will be compared with the those of the recent studies that

have been performed using numerical methods [8]. Since the pressure difference is large, we need to

extend the velocity and pressure in the wall layer. To do this, we redefine the perturbation parameter

and analyse the equation with a fast scale which enables us to study the equation in and near the

inviscid zone.

Keyword: Boundary layer, rotating-disc, asymptotics, roughness.

AMS 2010: 76D05, 76U05, 65L10, 37E35, 76M45.

References

[1] P. Hall, An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc,

P. Roy. Soc. Lond. A Mat., 406 (1830), 93-106, 1986.

[2] J.S.B. Gajjar, Nonlinear critical layers in the boundary layer on a rotating disk, J. Eng Math, 57(3), 205-217, 2007.

1Department of Mathematics, Ordu University, Ordu, Turkey, [email protected] of Mathematics, Bursa Technical University, Bursa, Turkey, [email protected]

†Both authors contributed equally to this work.

120

[3] P.T. Griffiths, Hydrodynamic stability of non-newtonian rotating boundary-layer flows, PhD Thesis, University of

Birmingham, 2016.

[4] S.O. Mackerrell, Stability of bodewadt flow, Phil. Trans. R. Soc. A, 363, 1181-1187, 2005.

[5] S.O. Stephen, Effects of partial slip on rotating-disc boundary-layer flows, 20th Australasian Fluid Mechanics Con-

ference, 2016.

[6] S.O. Stephen, Effects of partial slip on viscous instabilities in rotating-disc boundary-layer flows, 8th AIAA Theoretical

Fluid Mechanics Conference, 2017.

[7] N.A. Culverhouse, The hydrodynamic stability of crossflow vortices in the bodewadt boundary layer, PhD Thesis,

University of Birmingham, 2009.

[8] B. Alveroglu, A. Segalini and S.J. Garrett, The effect of surface roughness on the convective instability of the bek

family of boundary-layer flows, Eur. J. Mech. B/Fluids, 56, 178-187, 2016.

121

The Method for Defining the Coefficient of Hydraulic Resistance on Different

Areas of Pump-Compressor Pipes in Gas Lift Process

Fikret Aliev 1, N. S. Hajiyeva 2

Abstract. In this paper the process of gas-lift in the oil production is considered. In this process

the motions of gas and gas-liquid mixture (GLM) are described by the system of partial differential

equations of hyperbolic type. Applying lines method [1] the system of partial differential equations of

hyperbolic type is reduced to the system of ordinary differential equations with respect to the volumes

of gas, GLM and their pressures. Applying least-squares method, the coefficient of hydraulic resistance

(CHR) is obtained on different areas of pump-compressor pipes. On the concrete example [2] the ade-

quacy of the mathematical model is shown.

Keyword: Identification, the coefficient of hydraulic resistance, the least-squares method.

AMS 2010: 49J15, 49J35.

References

[1] N.S. Hajiyeva, N.A. Safarova and N.A. Ismailov, Algorithm defining the hydraulic resistance coefficient by lines

method in gas-lift process, 18, 771-777, 2017.

[2] 2. F.A. Aliev, M.Kh. Ilyasov and M.A. Dzhamalbekov, The modeling gas lift wells operation, 4, 107-115, 2008 (in

Russian).

1Baku State University, Institute of Applied Mathematics, Baku, Azerbaijan, f [email protected]

2Baku State University, Institute of Applied Mathematics, Baku, Azerbaijan, [email protected]

122

An Identification Problem for Determining the Parameters of Discrete Dynamic

System in Gas-Lift Process

F. Aliev 1, N. S. Hajiyeva 2

Abstract. In the paper the identification problem [1] to determine the parameters of dynamic sys-

tem in discrete case is considered. Firstly, nonlinear discrete equation is linearized using the method

of quasilinearization [2]. Then using the statistical data the quadratic functional and its gradient are

derived. At the end the calculation algorithm is proposed.The results are illustrated on the example

[3] from oil industry which shows adequacy of the mathematical model.

Keyword: Discrete equation, quasilinearization, identification.

AMS 2010: 49J15, 49J35.

References

[1] N.S. Hajiyeva, N.A. Safarova and N.A. Ismailov, Algorithm defining the hydraulic resistance coefficient by lines

method in gas-lift process, Miskolc Mathematical Notes, 18, 771-777, 2017.

[2] F.A. Aliev, N.A. Ismailov, E.V. Mamedova and N.S. Mukhtarova, Computational algorithm for solving problem of

optimal boundary-control with nonseparated boundary conditions, J. Comput. Syst. Sci. Int. 55, 700-711, 2016.

[3] N.S. Mukhtarova and N.A. Ismailov, Algorithm to solution of the optimization problem with periodic condition and

boundary control, TWMS J. Pure Appl. Math. 5, 130-137, 2014.

1Baku State University, Institute of Applied Mathematics Baku, Azerbaijan, f [email protected]


123

Calculation Algorithm Defining the Coefficient of Hydraulic Resistance on

Different Areas of Pump-Compressor Pipes in Gas Lift Process

F. Aliev 1, N. S. Hajiyeva 2

Abstract. In this paper the process of gas-lift in the oil production is considered. In this process

the motions of gas and gas-liquid mixture (GLM) are described by the system of partial differential

equations of hyperbolic type. Applying lines method [1] the system of partial differential equations of

hyperbolic type is reduced to the system of ordinary differential equations with respect to the volumes

of gas, GLM and their pressures. Applying least-squares method, the coefficient of hydraulic resistance

(CHR) is obtained on different areas of pump-compressor pipes. On the concrete example [2] the ade-

quacy of the mathematical model is shown.

Keyword: Gas lift, identification, the coefficient of hydraulic resistance.

1Baku State University,Institute of Applied Mathematics,Baku, Azerbaijan


124

On a Way For Calculation Of The Double Definite Integrals

Galina Mehdiyeva 1, Maryam Boyukzade 2, Mehriban Imanova 3 and Vagif Ibarhimov 4

Abstract. Let us consider to calculation of double definite integral which can be written as the

following:

I =

b∫a

d∫c

f(s, t)dsdt. (8)

Usually, with the calculation of these integrals encounter then finding the volume of a geometric figure.

For the calculation of the integral (1) proposed, here to use the following function:

u(x, y) =

x∫a

y∫c

f(s, t)dsdt, a ≤ x ≤ b, c ≤ y ≤ d. (9)

It follows from here that u(b, d) = I.

As was noted above the aim of our investigation contained in the calculation of the integral (1), which

is reduces to solve the initial-value problem for the ODEs.

It is not difficult to understand that the integral (2) can be written as:

∂2u(x, y)

∂x∂y= f(x, t); a ≤ x ≤ b, c ≤ y ≤ d. (10)

As is known the problem (3) can be solved by using the finite-difference method. In simple form one

can use the following formula:

(∂2u

∂x∂y

)∣∣∣∣x=xi

y=yj

=1

hτ(ui,j − ui−1,j − ui,j−1 + ui−1,j−1). (11)

Here ui,j = u(xi, yj).

1Baku State University, Baku, Azerbaijan, imn [email protected]




125

Abstract. (Continuation) It is known that the double definite integral can be calculated by the

following formula:

b∫a

d∫c

f(s, t)dsdt = u(b, d)− u(a, d)− u(b, c) + u(a, c). (12)

If here replace a, b, c and d by the xi−1, xi, yj−1 and yj , then receive the method of (5).

For calculation the value of u(b, d), here proposed to use the solution of the following problems:

F ′y(x, y) = f(xi, y), F (xi, c) = 0, i = 0, 1, 2, ..., n, F (x, y) =

y∫c

f(x, t)dt,

u′x(xi, d) = F (xi, d), u(a, y) = o, i = 0, 1, 2, ..., n; x0 = a.

By solving these system we can find the value u(b, d) .

Keyword: Double definite integral, initial-value problem, finite-difference method.

AMS 2010: 65M06, 35E15.

126

Recent Methods for the Numerical Solution of Hamiltonian Systems

Gholam Reza Hojjati 1, Ali Abdi 2

Abstract. We introduce a class of methods for the numerical solution of initial value problems of ordi-

nary differential equations with special structures. These methods which are explored within the class

of a large family of second derivative general linear methods should be equipped to some properties to

preserve qualitative geometrical properties of the problem along the long-time integration. Numerical

experiments of the proposed methods on the well-known Hamiltonian problems confirm capability of

the methods in solving such problems.

Keyword: Initial value problems, general linear methods, second derivative methods, Hamiltonian

systems.

AMS 2010: 65L05.

References

[1] J.C. Butcher and G. Hojjati, Second derivative methods with rk stability, Numer. Algorithms, 40, 415-429, 2005.

[2] A. Abdi and G. Hojjati, An extension of general linear methods, Numer. Algorithms, 57, 149–167, 2011.

[3] M. Hosseini Nasab, G. Hojjati and A. Abdi, G-symplectic second derivative general linear methods for hamiltonian

problems, J. Comput. Appl. Math. 313, 486-498, 2017.

1University of Tabriz, Tabriz, Iran, [email protected]

2University of Tabriz, Tabriz, Iran, a [email protected]

127

A New Generalization of Dunkl Analogue of Szasz Operators

Gurhan Icoz 1, Bayram Cekim 2

Abstract. The aim of this article is to construct Stancu-type linear positive operators generated by

Dunkl generalization. We first give approximation properties with the help of well-known Korovkin-

type theorem and weighted Korovkin-type theorem. Then we obtain the rate of convergence in terms of

classical modulus of continuity, the class of Lipschitz functions, Peetre’s K-functional and second-order

modulus of continuity.

Keyword: Dunkl exponential, Szasz operators, modulus of continuity.

AMS 2010: 41A25, 41A36.

References

[1] S. Sucu, G. Icoz and S. Varma, On some extensions of szasz operators including boas-buck type polynomials, Abstract

and Applied Analysis, 2012, 1-15, 2012.

[2] S. Sucu, Dunkl Analogue of szasz operators, Applied Mathematics and Computation, 244, 42-48, 2014.

[3] P. P. Korovkin, Convergence of positive linear operators in the space of continuous functions, (Russian) Doklady

Akademii Nauk SSSR, 90, 961-964, 1953.

[4] A. D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems

analogues to that of P. P. Korovkin. Soviet Mathematics Doklady, 15(5), 1453-1436, 1974.

[5] P. L. Butzer and H. Berens, Semi-groups of operators and approximation, Springer, Berlin-Heidelberg-New York,

1967.



128

On a New Generalization of Dunkl Analogue of Szasz-Mirakyan Operators

Gurhan Icoz 1

Abstract. The aim of this article is to construct sequences of Dunkl analogue that are base on func-

tion Ψ verifying some features of Szasz-Mirakyan operators. Firstly, we have defined the operators and

obtained test values and central moments for these operators. We have given classical and weighted

Korovkin theorem for the operators and then, investigated approximation properties of these operators

by means of some inequality on the function spaces CB [0,∞), C1B [0,∞) and C2

B [0,∞).

Keyword: Dunkl exponential, Szasz operators, modulus of continuity, Voronovskaja type asymptotic

formula.

AMS 2010: 41A25, 41A36.

References

[1] M. Rosenblum, Generalized hermite polynomials and the bose-like oscillator calculus, Oper. Theory: Adv. Appl. 73,

369-396, 1994.

[2] D. Cardenas-Morales, P. Garrancho and I. Rasa, Asymptotic formulae via a korovkin-type result, Abstr. Appl. Anal.

Article ID 217464, 2012.

[3] S. Sucu, Dunkl Analogue of szasz operators, Applied Mathematics and Computation, 244, 42-48, 2014.

[4] P. P. Korovkin, Convergence of positive linear operators in the space of continuous functions, (Russian) Doklady

Akademii Nauk SSSR, 90, 961-964, 1953.

[5] A. D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems

analogues to that of P. P. Korovkin Soviet Mathematics Doklady, 15(5), 1453-1436, 1974.

[6] P. L. Butzer and H. Berens, Semi-groups of operators and approximation, Springer, Berlin-Heidelberg-New York,

1967.


129

Beta Generalization of Stancu-Durrmeyer Operators Involving Analytic Functions

Gurhan Icoz 1, Hatice Eryigit 2

Abstract. We deal with beta generalization of Stancu-Durrmeyer operators with the help of analytic

functions. Approximation properties are investigating and convergence results are estimating for the

operators.

Keyword: Generating functions, Stancu type generalization, Durrmeyer type integral.

AMS 2010: 41A25, 41A36.

References

[1] S. Varma, S. Sucu and G. Icoz, Generalization of szasz operators involving brenke type polynomials, Comput. Math.

Appl. 64, 121-127, 2012.

[2] S. Sucu, G. Icoz and S. Varma, On some extensions of szasz operators including boas-buck-type polynomials, Abst.

Appl. Anal., 2012.

[3] S. Sucu, G. Icoz and S. Varma, Approximation by operators including generalized appell polynomials, Filomat, 30,

429-440, 2016.

[4] G. Icoz and B. Cekim, Stancu-type generalizations of the chan-chyan-srivastava operators, Filomat, 30, 3733-3742,

2016.

[5] S. Sucu and S. Varma, Approximation by sequence of operators involving analytic functions, Mathematics, 7(2), 188,

2019.

[6] S. Sucu and S. Varma, Generalization of jakimovski- leviatan type szasz operators, Appl. Math. and Comp. 270,

977-983, 2015.



130

Continuous Dependence of An Invariant Measure on the Jump Rate of a

Piecewise-Deterministic Markov Process

Hanna Wojewodka-Sciazko 1, Dawid Czapla 2 and Katarzyna Horbacz 3

Abstract. We investigate a piecewise-deterministic Markov process, whose deterministic behaviour

between random jumps is governed by some semi-flow, and any state right after the jump is attained

by a randomly selected continuous transformation. It is assumed that the jumps appear at random

moments, which coincide with the jump times of a Poisson process with intensity λ > 0. The model

of this type, although in a more general version, was examined in [1], where we have shown, among

others, that the Markov process under consideration possesses a unique invariant probability measure,

say µ∗λ.

The aim now is to prove that the map λ 7→ µ∗λ is continuous (in the topology of weak convergence of

measures). To do this, we refer to the results already established in [1], as well as to some well-known

facts from mathematical or functional analysis.

The studied dynamical system is inspired by certain models of gene expression (cf. [1, 2, 3]), and hence

our results may be interesting not only from the mathematical, but also biological point of view.

Keyword: Markov process, random dynamical system, invariant measure.

AMS 2010: Firstly 60J05, 60J25, Secondly 37A30, 37A25.

Acknowledgements: The work of Hanna Wojewodka-Sciazko has been supported by the National

Science Centre of Poland, grant number 2018/02/X/ST1/01518.

References

[1] D. Czapla, K. Horbacz and H. Wojewodka, Ergodic properties of some piecewise-deterministic markov process with

application to gene expression modelling, arXiv:1707.06489v3, 2018.

[2] A. Lasota and M. Mackey, Cell division and the stability of cellular populations, J. Math. Biol. 38, 241-261, 1999.

1University of Silesia in Katowice, Katowice, Poland, [email protected]



131

[3] S. Hille, K. Horbacz and T. Szarek, Existence of a unique invariant measure for a class of equicontinuous markov

operators with application to a stochastic model for an autoregulated gene, Ann. Math. Blaise Pascal, 23, 171-217,

2016.

132

A Numerical Solution for Fractional Order Optimal Control in Infectious Disease

Models

H. Kheiri 1, M. Jafari 2, A. Jabbari3 and F. Iranzad 4

Abstract. In most texts, fractional optimal control problems (FOCPs) have formulated in terms of

the left and right fractional derivatives. In this paper, we give an appropriate technique that convert

the right derivative to the left derivative. This technique provides the employment of one fractional

numerical scheme in Forward-Backward sweep method (FBSM) for solving FOCPs. We apply the

FBSM together with the Adams-type predictor-corrector method. We apply it for Optimal Control

in infectious disease Models. The simulation of the models is done. The numerical results show that

implementing the control efforts decreases significantly the number of infected people.

Keyword: Fractional optimal control, infectious disease, fractional calculus, epidemic model.

AMS 2010: 49J15, 93A30.

References


ment, Journal of Computational and Applied Mathematics, 346, 323-339, 2019.



[3] H. Kheiri and M. Jafari, Optimal control of a fractional-order model for the hiv/aids epidemic, International Journal

of Biomathematics, 11(7), DOI: 10.1142/S1793524518500869, 2018.



3Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran, [email protected]


133

Mathematical Model for Eddy Current Testing of Cylindrical Structures

Inta Volodko 1, Andrei Kolyshkin 2 and Valentina Koliskina 3

Abstract. Quasi-analytical solutions of direct eddy current testing problems are obtained in the

paper. The physical model is described as follows. An air-core coil carrying alternating current is

located above an electically conducting multilayered cylindrical structure which has flaws (or inclusions)

of cylindrical shape. Different types of cylindrical flaws are considered: (a) surface cylindrical flaws

located at the top or bottom of a two-layered plate of a cylindrical shape, (b) volumetric cylindrical

flaws located inside the conducting medium, (c) two surface cylindrical flaws (one at the top and the

other at the bottom of a conducting cylinder). It is assumed that in all cases considered the coil is

symmetric with respect to cylindrical flaws.

The method of truncated eigenfunction expansions is used to solve all the above mentioned problems [1].

Assuming that the electromagnetic field is zero at some radial distance b from the center of the coil we

solve the corresponding boundary-value problem for the system of Maxwell’s equations by separation

of variables. Using the interface conditions we obtain a nonlinear equation of the form f(λ) = 0,

where λ is a complex eigenvalue. The method proposed in [2] is used to calculate complex eigenvalues.

The system of linear algebraic equations is solved numerically in order to find the coefficients of the

eigenfunction expansions in each domain.

Quasi-analytical method for the solution of direct eddy current testing problems is compared with finite

element method implemented in Comsol [3]. Good agreement is found between the proposed model

and finite element method.

Keyword: Maxwell’s equations, eddy currents, eigenfunction expansions.

AMS 2010: 78A55, 35Q61.




134

References

[1] T.P. Theodoulidis and E. E. Kriezis, Eddy current canonical problems (with applications to nondestructive testing),

Tech Science, Duluth, 2006.

[2] L.M. Delves and J.N. Lyness, A numerical method for locating the zeros of an analytic function, Math. Comp. 21,

543-560, 1967.

[3] V. Koliskina, A. Kolyshkin, R. Gordon and O. Martens, Direct eddy current method for volumetric flaws of cylindrical

shape, in Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering,

vol.4, 7659-7665, 2016.

135

Local Properties of Solutions of Trivial Monge-Amper Equation

I. Kh. Sabitov 1,

Abstract. We call the equation

zxxzyy − z2xy = 0 (1)

trivial Monge-Amper equation for a function z(x, y) ∈ C2. Geometrically the surface S : z = z(x, y) is

a surface with locally Euclidean metric and as such it composed by straight generatrices along which

the tangent plane is the same. The solutions of (1) possess some properties slightly seeming to ones for

solutions of elliptic types equations. Namely their regularity can be better than it is supposed initially

and their singularity at isolated points can be removed as for harmonic functions. These properties are

given by following theorems

Theorem 1.([1],[2]) Let for a solution z(x, y) ∈ Cn, n ≥ 2, of the equation (1) the condition zxx 6= 0

be satisfied. Thenzxyzxx∈ Cn−1,

zyyzxx∈ Cn−1.

Theorem 2. Let D and (o

D) note the domains x2 + y2 < r2 and 0 < x2 + y2 < r2 and let a solution

z(x, y) of the equation (1) belong to the class Cn(o

D), n ≥ 2. Then z(x, y) can be continuously prolonged

to the point (0, 0) such that it will become of class Cn(D).

As to C1-smooth ruled developable surfaces z = z(x, y) with an isolated singular point for them also

there is a theorem making more precise the behavior of surface in this point:

Theorem 3. Let a ruled developable surface S : z = z(x, y) ∈ C1(o

D). Then the function z(x, y) can

be continuously prolonged to the point (0, 0) such that it becomes of class C1(o

D) ∩ C(D).

But in general we can’t affirm that for such a surface an analogue of theorem 2 for n = 1 will be valid,

an example is given by the cone z =√x2 + y2.

The work is supported by a grant of Scientific School-6222.2018.1.

Keywords: zero curvature surfaces, Monge-Amper equation, propeties of solutions.

AMS 2010: 53A05+53C45

1Lomonosov Moscow State University, Moscow, Russia, [email protected]

136

References

[1] V. Ushakov, The explicit general solution of trivial Monge–Ampere equation, Comment. Math. Helvetici, 75, p.

125-133, 2000.

[2] I. Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics.- Series ¡¡Reviews in Mathematics

and Mathematical Physics¿¿, vol. 13, Part 1, edited by A.T. Fomenko. Cambridge Scientific Publishers, 2009, 276p.

137

Existence of Positive Solution for Caputo Difference Equation and Applications

Kazem Ghanbari 1, Tahereh Haghi 2

Abstract. In this presentation we consider a typical Caputo fractional difference equation depen-

dent to a parameter that appear in important applications such as modelling of medicine distributing

throughout the body via injection. Using fixed point theory of linear operators we find a parameter

interval for which this boundary value problem has a unique positive solution. Moreover, the exact

solutions are computed in some cases.

Keyword: Positive solution, fixed point theory, fractional diffrence equation.

AMS 2010: 39A12, 44A25.

References

[1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Review, 18,

602-709, 1976.

[2] A. Almusharrf, Development of fractional trigonometry and an application of fractional calculus to pharmacokinetic

model, Master Thesis, Western Kentucky University, 2011.

[3] Introduction to Pharmacokinetics and Pharmacodynamics, American Society of Health-System Pharmacists, Re-

trieved March 13, 2014.

[4] C. Zhai and L. Xu, Properties of positive solutions to a class of four-point boundary value problem of caputo

fractional differential equations with a parameter, Communications in Nonlinear Science and Numerical Simulation

19, 2820-2827, 2014.

[5] Y. Zhao, S. Sun and Y. Zhang, Existence and uniqueness of solutions to a fractional difference equation with p-

laplacian operator, Appl. Math. Comput. 54, 183-197, 2017.

1Sahand University of Technology, Tabriz, Iran, [email protected]

2Sahand University of Technology, Tabriz, Iran, [email protected]

138

Local and Nonlocal Boundary Value Problems for Hyperbolic Equations with a

Caputo Fractional Derivative

Mammad H. Yagubov 1, Shakir Sh. Yusubov 2

Abstract. The fractional calculus deals with extensions of derivatives and integrals to noninteger

orders. The field of fractional differential equations has been subjected to an intensive development of

the theory and applications in mathematical physics, finance, hydrology, biophysics, thermodynamics,

control theory, statistical mechanics, astrophysics, cosmology and bioengineering. In recent years,

several qualitative results for ordinary and partial fractional differential equations have been obtained.

Nonlocal boundary value problems are usually called problems with given conditions that connect the

values of the desired solution and/or its derivatives either at different points of the boundary or at

boundary points and some interior points. Note that nonlocal problems for hyperbolic differential

equations and the corresponding optimal control problems are being actively studied at present time.

But, nonlocal problems for the hyperbolic equations of fractional order are less investigated.

In this paper we study local and nonlocal boundary value problems for the hyperbolic equations of the

general form with variable coefficients and with a Caputo fractional derivative. For the investigation

the posed problem, one functional space of fractional order is introduced. The posed problem is re-

duced to the integral equation and the existence of its solution is proved by the help of a priori estimate.

Keyword: Nonlocal problem, hyperbolic differential equation, fractional derivative, Riemann -Liouville

integral, Caputo derivative.

AMS 2010: 26A33, 35R11, 34K37.

1Baku State University, Baku, Azerbaijan, yaqubov [email protected]

2Baku State University, Baku, Azerbaijan, yusubov [email protected]

139

On Dynamical One-Dimensional Models of Thermoelastic Piezoelectric Bars

Mariam Avalishvili 1, Gia Avalishvili 2

Abstract. Piezoelectrics are one of the most popular smart materials that are incorporated in smart

structures, which has the capability to respond to changing external and internal environments. Appli-

cations of smart structures range from aerospace and submarine systems to civil structures, and medical

systems. Various parts of smart structures are plates, shells and bars, and consequently piezoelectric

structures of these shapes need to be well modeled.

In the present paper, we consider thermoelastic piezoelectric bar with variable cross-section, which may

vanish on the butt ends, consisting of inhomogeneous anisotropic material. We obtain variational for-

mulation of the initial-boundary value problem corresponding to the linear dynamical three-dimensional

model [1, 2] of the thermoelastic piezoelectric solid with regard to magnetic field, when along one of the

butt ends of the bar with positive area electric and magnetic potentials vanish, and on the remaining

parts of the boundary normal components of electric displacement, magnetic induction and heat flux,

and density of surface force are given. On the basis of the variational formulation applying general-

ization of the dimensional reduction method suggested by I. Vekua in the classical theory of elasticity

for plates with variable thickness [3] we construct a sequence of subspaces with special structure of the

spaces corresponding to the original three-dimensional problem and by projecting the three-dimensional

problem on these subspaces we obtain a hierarchy of dynamical one-dimensional models.

We investigate the constructed one-dimensional initial-boundary value problems in suitable spaces of

vector-valued distributions with respect to the time variable with values in corresponding weighted

Sobolev spaces and prove the existence and uniqueness results. Moreover, we prove that the sequence

of vector-functions of three space variables restored from the solutions of the one-dimensional problems

converges in the corresponding function spaces to the exact solution of the three-dimensional initial-

boundary value problem and under additional conditions we estimate the rate of convergence.

1University of Georgia, Tbilisi, Georgia, [email protected]

2I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, [email protected]

140

Abstract. (Continuation) This work was supported by Shota Rustaveli National Science Foundation

(SRNSF) [217596, Construction and investigation of hierarchical models for thermoelastic piezoelectric

structures].

Keyword: Thermo-electro-magnetoelasticity, initial-boundary value problems, hierarchical one-dimensional

models, modeling error estimate.

AMS 2010: 35Q74, 74F15, 74H15, 74K10.

References

[1] J.Y. Li, Uniqueness and Reciprocity Theorems for Linear Thermo-electro-magnetoelasticity, Quart. J. Mech. Appl.

Math. 56, 1, 35-43, 2003.

[2] D. Natroshvili, Mathematical Problems of Thermo-electro-magneto-elasticity, Lecture Notes of TICMI 12, Tbilisi

State University Press, Tbilisi, 2011.

[3] I.N. Vekua, On a Way of Calculating of Prismatic Shells, Proc. of A. Razmadze Inst. Math. Georgian Acad. Sci. 21,

191-259, 1955 (in Russian).

141

Higher Order Exponential Fuzzy Transform and its Application in Fluid Mechanics

Masoumeh Zeinali 1, Ghiyam Eslami 2

Abstract. In this work, Fm-transform with the exponential basis functions constructed and discussed

in details. In order to have more consistency, the generating function and consequently basic functions

supposed to be exponential as well. In this type of fuzzy transform, there is free parameter α. Hat

shaped and sinusoidal basic functions can be obtained by changing α. Furthermore, the behavior of the

approximation changes with changing α. This shows the reflexibility of this kind of fuzzy transform. In

order to show the efficiency of this approximation method, we apply exponential Fm-transform method

for numerical solution of some problems in the field of fluid mechanics.

Keyword: Fuzzy transform, Numerical soluion, Fluid mechanics.

References

[1] R. Alikhani, M. Zeinali, F. Bahrami, S. Shahmorad, I. Perfilieva, Trigonometric Fm-transform and its approximative

properties, Soft Computing 21, 3567-3577, 2017.

[2] I. Perfilieva, Fuzzy transforms: theory and applications, Fuzzy Sets and Syst. 157, 993-1023, 2006.

[3] I. Perfilieva, M. Dankova, B. Bede, Toward a higher degree F-trasform, Fuzzy Sets and Syt. 180, 3-109, 2011.

[4] Stefanini, Luciano. F-transform with parametric generalized fuzzy partitions. Fuzzy Sets and Systems 180, 98-120,

2011.

[5] S. Tomasiello, An alternative use of fuzzy transform with application to a class of delay differential equations, Int J

Comput Math. 1-8, 2016.

[6] M. Zeinali, R. Alikhani, S. Shahmorad, F. Bahrami, I. Perfilieva, On the structural properties of Fm-transform with

applications, Fuzzy Sets and Systems 342, 32-52, 2018.


2Department of Mechanical Engineering, Ahar branch, Islamic Azad University, Ahar, Iran

142

Analytical Solutions of Some Nonlinear Fractional-Order Differential Equations

by Different Methods

Meryem Odabası 1, Zehra Pınar 2 and Huseyin Kocak 3

Abstract. In this work, we investigate exact analytical solutions of some fractional-order differential

equations arising in mathematical physics. We consider the space-time fractional Kaup-Kupershmidt,

the space-time fractional Fokas, and the space-time fractional breaking soliton equations which have im-

portant applications in science and engineering. Exact traveling wave solutions of these equations have

been established by different methods. Finding analytical methods and solutions for fractional-order

differential equations may help to understand the nature of the nonlinear phenomena they characterize.

Keyword: Fractional-order differential equations, exact solutions, analytical methods.

AMS 2010: 35R11, 35C07.

References

[1] I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999.

[2] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-

Holland Mathematical Studies, Elsevier, Amsterdam, 2006.

[3] K. Mille and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New

York, 1993.

[4] R. Khalil, M.A. Horani and A.M.Y. Sababheh, A New Defnition of Fractional Derivative, J. Comput. Appl. Math.

264, 65-70, 2014.

[5] J.H. Choi and H. Kim, Soliton Solutions for the Space-Time Nonlinear Partial Differential Equations with Fractional-

Orders, Chinese J. Phys. 55, 556-565, 2017.

1Ege University, Izmir, Turkey, [email protected]

2Namık Kemal University, Tekirdag, Turkey, [email protected]


143

Investigation of Exact Solutions of Some Nonlinear Evolution Equations via an

Analytical Approach

Meryem Odabası 1

Abstract. This study investigates exact analytical solutions of some nonlinear partial differential

equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation,

the Gardner equation, the ZK-BBM, and the KP-BBM equations have been considered. With the

implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling

wave solutions of these equations have been attained. The solutions have been checked and graphs

have been given via package programs to see the behavior of the waves.

Keyword: Nonlinear evolution equations, traveling wave solutions, analytical methods.

AMS 2010: 35C07, 35Q99.

References

[1] J. Lu, New Exact Solutions for Kudryashov-Sinelshchikov Equation, Adv. Difference Equ. 374, 2018.

[2] D. Daghan and O. Donmez, Exact Solutions of the Gardner Equation and their Applications to the Different Physical

Plasmas, Braz. J. Phys. 46, 321-333, 2016.

[3] A.M. Wazwaz, The Extended Tanh Method for New Compact and Noncompact Solutions for the KP-BBM and the

ZK-BBM Equations, Chaos Soliton Fract. 38, 1505-1516, 2008.

[4] J. Akter and M.A. Akbar, Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation via the

Modified Simple Equation Method, J. Part. Diff. Eq. 29(2), 143-160, 2016.

[5] C.S. Liu, Trial Equation Method and Its Applications to Nonlinear Evolution Equations, Acta. Phys. Sin. 54, 2505-

2510, 2005.

[6] M. Odabasi and E. Misirli, A Note on the Traveling Wave Solutions of some Nonlinear Evolution Equations,Optik.

142, 394-400, 2017.

This research is supported by Ege University, Scientific Research Project (BAP), Project Number:

2017-TKMYO-002.

1Ege University, Izmir, Turkey, [email protected], [email protected]

144

Uniformly Convergent Difference Schemes for Solving Singularly Perturbed

Semilinear Problem with Integral Boundary Condition

Musa Cakır 1

Abstract. This work deals with a singularly perturbed semilinear boundary value problem with Neu-

mann and integral boundary conditions. The main aim of the paper is to give a uniform convergence

numerical method. First, parameter explicit theoretical bounds on the continuous solution and its first

derivative are derived. Then, finite difference scheme on piecewise uniform mesh (Shishkin type mesh)

is constructed. The scheme is based on the method of integral identities with the use of exponential

basis functions and interpolating quadrature rules with wight and remainder term in integral forms.

It is shown that the scheme is of almost first order convergent in the discrete maximum norm with

respect to the perturbation parameter. At the end of the paper some numerical experiments are given

to demonstrate our theoretical estimates.

Keyword: Singularly perturbed problems, finite difference schemes, Shishkin type mesh.

AMS 2010: 65L05, 65L12, 65L20, 65L70.

References

[1] E.P. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Method for Problems with Initial and Boundary

Layers, Boole Press, Dublin, 1980.

[2] P.A. Farell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Robust Computational Techniques for

Boundary Layers, Chapman Hall/CRC, New York, 2000.

[3] M. Cakir, A Numerical Study on the Difference Solution of Singularly Perturbed Semilinear Problem with Integral

Boundary Condition, Math. Modell. and Anal. 21(5), 644-658, 2016.

[4] M. Cakir and G.M. Amiraliyev, A Finite Difference Method for the Singularly Perturbed Problem with Nonlocal

Boundary Condition, Appl. Math. and Comput. 160, 539-549, 2005.

[5] R. Ciegis, The Difference Scheme for Problems with Nonlocal Conditions, Informatica (Lietuva), 2, 155-170, 1991.

[6] M. Kuda and G.M. Amiraliyev, Finite Difference Method for a Singularly Perturbed Differential Equations with

Integral Boundary Condition, Inter. J. Math. Comput. 26(3), 72-79, 2015.

1Van Yuzuncu Yil University, Van, Turkey, [email protected]

145

The Best Constant of Lyapunov-Type Inequality for Fourth-Order Linear

Differential Equations with Anti-Periodic Boundary Conditions

Mustafa Fahri Aktas 1, Devrim Cakmak 2 and Abdullah Ahmetoglu 3

Abstract. This paper is concerned with Lyupanov-type inequalities for fourth-order linear differen-

tial equations with anti-periodic boundary conditions. Our study is based on the absolute maximum

of Green’s function corresponding to anti-periodic boundary value problem.

Keyword: Green’s functions, Lyapunov-type inequalities.

AMS 2010: 34C10, 26D10, 34B05.

References

[1] A. Abdurrahman, F. Anton and J. Bordes, Half-String Oscillator Approach to String Field Theory, Nuclear Physics

B 397, 260-282, 1993.

[2] M.F. Aktas, D. Cakmak and A. Tiryaki, Lyapunov-Type Inequality for Quasilinear Systems with Anti-Periodic

Boundary Conditions, J. Math. Inequal. 8, 313-320, 2014.

[3] M.F. Aktas, Lyapunov-Type Inequalities for n-Dimensional Quasilinear Systems, Elect. J. of Diff. Eq. 67, 1-8, 2013.

[4] G. Borg, On a Liapounoff Criterion of Stability, Amer. J. of Math. 71, 67–70, 1949.

[5] A. Cabada, J.A. Cid and B. Maquez-Villamarin, Computation of Green’s Functions for Boundary Value Problems

with Mathematica, Appl. Math. Comput. 219, 1919-1936, 2012.

[6] E.L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1926.

[7] W.G. Kelley, A.C. Peterson, The Theory of Differential Equations, Classical and Qualitative, Universitext 278,

Springer Science+Business Media, LLC 2010.

[8] M.K. Kwong, On Lyapunov’s Inequality for Disfocality, J. Math. Anal. Appl. 83, 486–494, 1981.

[9] A.M. Liapunov, Probleme General de la Stabilite du Mouvement, Ann. Fac. Sci. Univ. Toulouse 2, 203–407, 1907.

[10] D. Cakmak and A. Tiryaki, On Lyapunov-Type Inequality for Quasilinear Systems, Appl. Math. Comput. 216,

3584–3591, 2010.

1University, City, Country, e-mail

2University, City, Country, e-mail

3Gazi University, Ankara, Turkey, [email protected], [email protected]

146

On Lyapunov-Type Inequalities for Various Types of Boundary Value Problems

Mustafa Fahri Aktas 1

Abstract. In this paper, we establish new Lyapunov-type inequalities via the absolute maximums of

Green’s functions corresponding to the various types of boundary value problems. In addition, some

applications of the obtained inequalities are given.

Keyword: Green’s functions, Lyapunov-type inequalities.

AMS 2010: 34C10, 34B15, 34L15.

References

[1] M.F. Aktas and D. Cakmak, Lyapunov-Type Inequalities for Third-Order Linear Differential Equations Under the

Non-Conjugate Boundary Conditions, Differ. Equ. Appl. 10, 219-226, 2018.

[2] M.F. Aktas and D. Cakmak, Lyapunov-Type Inequalities for Third-Order Linear Differential Equations, Elect. J. of

Diff. Eq. 139, 1-14, 2017.

[3] M.F. Aktas, D. Cakmak and A. Tiryaki, On the Lyapunov-Type Inequalities of A Three-Point Boundary Value

Problem for Third Order Linear Differential Equations, Appl. Math. Lett. 45, 1-6, 2015.

[4] P.R. Beesack, On the Green’s Function of An N -Point Boundary Value Problem, Pacific J. Math. 12, 801-812, 1962.

[5] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[6] E.L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1926.

[7] W.G. Kelley and A.C. Peterson, The Theory of Differential Equations, Classical and Qualitative, Universitext 278,

Springer Science+Business Media, LLC 2010.

[8] M.K. Kwong, On Lyapunov’s Inequality for Disfocality, J. Math. Anal. Appl. 83, 486-494, 1981.

[9] A.M. Liapunov, Probleme General de la Stabilite du Mouvement, Ann. Fac. Sci. Univ. Toulouse 2, 203-407, 1907.

[10] G.F. Roach, Green’s Functions, Cambridge University Press, Cambridge, 1982.

1Gazi University, Ankara, Turkey, [email protected], [email protected]

147

The Boundary-Value Problem for Two-Dimensional Laplace Equation with the

Non-Local Boundary Conditions on Rectangle

Nihan A. Aliyev 1, Metanet B. Mursalova 2

Abstract. The presented article deals with investigation of solutions of the boundary-value problem

for the two-dimensional Laplace equation in case when the simultaneous motion of the four points

along the boundary satisfies the Carleman condition:

∆u(x1, x2) = 0, (13)[2∑j=1

α(1)ij (x1)

∂u(x)

∂xj+ α

(1)io (x1)u(x)

] ∣∣x1

=at

x2=0

+

[2∑j=1

α(2)ij (x2)

∂u(x)

∂xj+ α

(2)io (x2)u(x)

] ∣∣x1

=a

x2=b(1−t)

+

+

[2∑j=1

α(3)ij (x1)

∂u(x)

∂xj+ α

(3)io (x1)u(x)

] ∣∣x1

=a(1−t)

x2=b

+

+

[2∑j=1

α(4)ij (x2)

∂u(x)

∂xj+ α

(4)io (x2)u(x)

] ∣∣x1

=0

x2=bt

= ϕi(t), t ∈ [0; 1] , i = 1, 4. (14)

Here D = (0; 1)× (0; b) is rectangle that stays in the first quarter, all dates in (2) are real continuous

functions and boundary conditions are linearly independent and satisfies the Carleman condition when

t varies on [0, 1].

The Fredholm property of considered problem was proved.

Keyword: Non-local boundary problem, fundamental solution, singularity, regularization, Fredholm

property.

AMS 2010: 34B10, 35J25.



148

On The Solution of The Optimal Control Problem of Inventory of a Discrete

Product In Stochastic Model Of Regeneration

N. A. Vakhtanov 1, P. V. Shnurkov 2

Abstract. The work considers a new complete model of discrete product inventory control in regen-

eration scheme with a Poisson flow of customer requirements and random delivery delay. In the system

deferred demand is allowed, the volume of which is limited by a given value N0. The control parameter

r is the level of the stock, at which achievement it is necessary to make an order for replenishment,

and this parameter is determined in accordance with a discrete probability distribution, which plays

the role of a control strategy.

As an indicator of control efficiency, we consider the average specific profit obtained during the re-

generation period. In order to obtain an explicit representation for this indicator, a special version of

the classical ergodic theorem [1] was proved for the additive cost functional, which has an additional

component associated with regeneration moments. The optimal control problem is solved on the basis

of the statement about the extremum of a fractional-linear integral functional on the set of discrete

probability distributions [2]. It is established that the optimal control strategy is deterministic and

is determined by the point of global extremum of the function, which is a stationary cost efficiency

indicator and depends on the control parameter.

1National Research University Higher School of Economics, Moscow, Russian Federation, [email protected]


149

Abstract. In the work, explicit representations are derived for the mathematical expectations of the

increments of the profit functional on the regeneration period under all possible conditions on the

control parameter. These analytical representations enable us to explicitly obtain the stationary cost

indicator of control efficiency as a function of the control parameter and, for given model characteris-

tics, numerically determine the optimal value of the control, which contributes to solving one of the

important applied problems of the modern economy.

Keyword: Inventory management, controlled regenerative process, stationary cost indicator of control

efficiency.

References

[1] H. Mine and S. Osaki, Markovian decision processes, New York, NY: Elsevier. 142 p., 1970.

[2] P.V. Shnurkov, Solution of the unconditional extremum problem for a linear-fractional integral functional on a set of

probability measures, Dokl. Math. 94(2), 550-554, 2016.

150

Modeling Deformation, Buckling and Post-Buckling of Thin Plates and Shells with

Defects under Tension

Nikita Morozov 1, Boris Semenov 2 and Petr Tovstik 3

Abstract. Thin-walled elements are widely used in various designs. When analyzing their bearing

capacity, it is necessary to take into account not only the loads leading to their destruction, but also

the loads under which occurs the loss of stability. It should be noted that the stability loss can occur

both during compression of these elements, and during stretching in the presence of defects such as

cuts and inclusions in them, since in the vicinity of these defects there are areas of compressive stresses,

which can lead to local buckling .

The problem of the loss of the plane form of deformation of the plate, weakened by a crack, under

uniaxial tension was studied in a number of papers ([1] - [5]).

However, the question of post-buckling deformation and its effect on fracture did not receive a final

answer. In this regard, we can point to the work [5], in which the experimental results for stretching

paper sheets with a central crack were presented and it was stated that after the plate buckling the

stress intensity decreases in the vicinity of the crack tip. At the same time, experiments on stretching

of metal sheets with a central crack show that local buckling in the vicinity of the crack leads to an

increase in the stress concentration in the vicinity of the crack tips, i.e. to reduce the fracture load [4].

In the framework of this article the post-buckling deformation of a plates and cylindrical shells with

defects (cracks, holes) is analyzed and the effect of buckling on stress concentration near these defects

is estimated. For plates with cracks the stress state in the initial postcritical stage is investigated and

the approximate analytical solution is suggested.

Keyword: Finite elements method, plate, shell, buckling, crack, hole, stress concentration.

AMS 2010: 74K20, 74K25, 74G60.

Acknowledgements: This work is supported by Russian Science Foundation with project number 15-

19-00182.

1St. Petersburg State University, St.Peterburg, Russia, [email protected]



151

References

[1] G.P. Cherepanov, On the buckling under tension of a membrane containing holes, J. Appl. Math. Mech., 27(2),

405-420, 1963.

[2] J.R. Dixon and J.S. Stranningan, Stress distribution and buckling in thin sheets with central slits, Proc. 2nd Int.

Conf. Fracture. Brighton, 1969.

[3] K. Markstrom and B. Storakers, Buckling of cracked members under tension, Int. J. Sol. Struct. 16, 217-229, 1980.

[4] M.S. Dyshel, Stability and fracture of plates with a central and an edge crack under tension, Int J. Appl Mech, 38,

472-476, 2002.

[5] C. Li, R. Espinosa and P. Stahle, Fracture mechanics for membranes, Proc. XVth European Conf. on Fracture

(ECF15), Stockholm, 2004.

152

On Two-Dimensional Boundary Layer Flows of a Psuedoplastic Fluid — Two Flow

Configurations

Nirmal C. Sacheti 1, Pallath Chandran 2 and Tayfour El-Bashir 3

Abstract. The flow of an inelastic fluid subject to shear thinning phenomenon is considered. Using

the Williamson constitutive equation to model the pseudoplastic effects, the governing boundary layer

equations for steady laminar flow near a horizontal flat rigid surface, have been subjected to a similarity

analysis. Two specific flow configurations corresponding to (i) Blasius flow and (ii) the Sakiadis flow,

respectively, have been investigated. The resulting nonlinear boundary value problem for each flow has

been solved using a perturbation expansion followed by numerical integration. The focus of this work is

on bringing out the effect of the rheological parameter, and also the relative higher order effects on the

flows. It is concluded that higher order effects arising due to the non-Newtonian effects, do influence

the flows to varying degrees.

1Department of Mathematics, College of Science, Sultan Qaboos University, PC 123, Al Khod, Muscat, Sultanate of

Oman, [email protected] of Mathematics, College of Science, Sultan Qaboos University, PC 123, Al Khod, Muscat, Sultanate of

Oman, [email protected] of Mathematics, College of Science, Sultan Qaboos University, PC 123, Al Khod, Muscat, Sultanate of

Oman, [email protected]

153

The Scattering Problem for Hyperbolic System of Equations on Semi-Axis with

Three Incident Waves

N. Sh. Iskenderov 1, K. A. Alimardanova 2

Abstract. Let us consider the hyperbolic system of six equations on the semi-axis x > 0

ξi∂ψi(x, t)

∂t− ∂ψi(x, t)

∂x=

6∑j=1

Cij(x, t)uj (x, t) , i = 1, 6

ξ1 > ξ2 > ξ3 > 0 > ξ4 > ξ5 > ξ6,

where ψ1(x, t), ...ψ6(x, t, −∞ < t < +∞, is unknown function, Cij(x, t) are complex-valued measurable

by x and t functions satisfying the conditions

|Cij(x, t)| ≤ C[(1 + |x|)(1 + |t|)]−1−ε

and Cii(x, t) = 0, I, J = 1, 6 c > 0, ε > 0 are constants.

We consider two problems for the system (1) on a semi-axis: to find the solution of the system satisfying

one of the boundary conditions:

ϕ12(0, t) = H1ϕ

11(0, t), (3)

ϕ22(0, t) = H2ϕ

21(0, t), (4)

where ϕ1(x, t) = ψ1(x, t), ψ2(x, t), ψ3(x, t) , ϕ2(x, t) = ψ4(x, t), ψ5(x, t), ψ6(x, t) , H1 = diag 1, 1, 1 , H2 =0 0 1

1 0 0

0 1 0

by the given incident waves a1(t+ξ1x), a2(t+ξ2x), a3(t+ξ3x), determining as x→ +∞

asymptotic of the solutions:

1Institute of mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan, nizameddin [email protected]

2Institute of mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan, [email protected]

154

Abstract. (Continuation)

ψkj (x, t) = aj(t+ ξjx) + o(1), x→ +∞, k = 1, 2; j = 1, 2, 3 (5)

Theorem 1. Let the coefficients of the system (1) satisfy the conditions (2). Then there exists the

unique solution of the scattering problem on the semi-axis for the system (1) with arbitrary incident

waves Ai(S) ∈ L∞(R), R = (−∞,+∞), i = 1, 3.

Each solution assumes in space L∞(R) the asymptotic representation for ψkj (x, t), j = 4, 5, 6.

ψkj (x, t) = bkj ((t+ ξjx) + o(1), k = 1, 2, j = 4, 5, 6, x→ +∞ (6)

Based on Theorem 1, according to (6), to each vector-function a(t) = (a1(t), a2(t), a3(t)) ∈ L∞(R)

giving the incident waves there correspond two solutions of the system (1)-the solutions of 1st and 2nd

problems with the boundary conditions (3) and (4). These solutions define according to (6) the profiles

of the scattering waves bk(t) = (bk4(t), bk5(t), bk6(t)), k = 1, 2. Thereby in the space L∞(R) we determine

the operator S = (S1, S2) that takes a(t) to b(t)

b(t) = S · a(t) (7)

This operator is called the scattering operator for the system (1) on the semi-axis x ≥ 0.

Theorem 2.Under conditions of Theorem 1 and C6j(x, t) = 0, j = 1, 5 the coefficients of the system

are uniquely determined by the operator S.

Keyword: Scarttering problem, asymptotic, incident waves, scattering operator.

AMS 2010: 35L02, 35L05.

References

[1] L.P. Nizhnik, Inverse scattering problems for hyperbolic equations, 232 pp., 1991.

[2] n.Sh. Inkenderov, M.I. Ismailov, Inverse non-stationary scattering problem for hyperbolic system of four first order

equations on semi-axis, Proc. of IMM of NASA, IV (XII), 161-168, 1996.

[3] M.I. Ismailov, Inverse scattering problem for hyperbolic systems on a semi-axis in the case of equal number of incident

and scattered waves, Inverse Problems, 22, 955-974, 2006.

155

Numerical Method to Solve Fuzzy Boundary Value Problems

N. Parandin 1, A. Hosseinpour 2

Abstract. The present paper is concerned with a numerical solution of fuzzy heat equation with

nonlocal boundary conditions. We first express the necessary materials and definitions, then consider

a difference scheme for one dimensional heat equation. However, the integrals in the boundary con-

ditions are approximated by the composite trapezoid rule. We also express the necessary conditions

for existence of answer. In final part, we give an example for checking the numerical results. In this

example, we obtain the Hausdorff distance between exact solution and approximate solution.

Keyword: Fuzzy numbers, Fuzzy heat equation, finite difference scheme, stability.

AMS 2010: Firstly, Secondly.

References

[1] D. Dubois and H. Prade, Towards fuzzy differential calculus: Part 3, Differentiation, Fuzzy Sets and Systems. 8,

225-233, 1987.

[2] G.D. Smith, Numerical solution of partial differential equations, 1993.

[3] H. Kima and R. Sakthivel, Numerical solution of hybrid fuzzy differential equations using improved predic-

toræ¼¿orrector method, Communications in Nonlinear Science and Numerical Simulation. 17, 3788-3794, 2012.

[4] K. Kanagarajan and M. Sambath, Numerical solution of fuzzy differential equations by third order runge-kutta

method, International Journal of Applied Mathematics and Computation. 2, 1-8, 2010.

[5] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, FSS. 96, 201-209, 1998.

[6] M.L. Puri and D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91, 321-325, 1983.

[7] M. Puri and D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 409-422, 1986.

1Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran,

n−[email protected] Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran,

a−[email protected]

156

The Algorithm Solution of the Problem of Optimal Control in a Dynamic

One-Sector Economic Model with a Discrete Time Based on Dynamic Programming

Method

P. V. Shnurkov 1, A. O. Rudak 2

Abstract. In this paper we study a new formulation of the optimal control problem in a dynamic

single-sector eco-nomic model with discrete time. In the task, the states are the values of the specific

capital, that is, the total amount of capital related to the unit of labor resources. The role of manage-

ment is played by a parameter rep-resenting the proportion of the specific product produced that is

directed to investment. The target functionali-ty is the sum of two components. The first one expresses

the specific consumption accumulated during the evolution of the system. The second is expressed as

a given function of the value of the specific capital at the final point in time and describes the level of

technological development in the system formed at that moment. The main limitation is the dynamic

ratio for the specific capital, describing its change under the influence of management. The initial state

in the system is assumed to be fixed. The study is based on the dynamic pro-gramming method. The

Bellman equations for the problem are obtained. Based on the well-known theoretical assertions, it is

established that the sequence of controls satisfying the Bellman equations is optimal. An algo-rithm

has been created and described in detail that allows one to solve the Bellman functional equations

nu-merically and find a sequence of optimal controls for the problem posed.

Keyword: Dynamic programming, optimal control problem, discrete time, Bellman equations, one-

sector model of an economic system.

References

[1] R. Bellman, Dynamic programming, 6th ed. Princeton, NJ: Princeton University Press, 1972.

[2] R. Bellman and S. Dreyfus, Applied dynamic programming, London: Oxford University Press, 1962.

[3] M. Intriligator, Mathematical methods of optimization and economic theory, Philadelphia: SIAM, 2002.



157

[4] S.A. Ashmanov, Matematicheskie modeli i metody v ekonomike, M.: Izdatel’stvo Moskovskogo Universiteta, 1980.

[5] M. Kamien and N. Schwartz, Dynamic optimization, New York: Elsevier North Holland, 1981.

[6] R. Barro and X. Sala-i-Martin, Economic growth, second ed. London: The MIT Press, 2004.

[7] A.D. Ioffe and V.M. Tihomirov, Teoriya ekstremal’nyh zadach, M.: Nauka, 1974.

158

Effects of Temperature Modulation on Natural Convection in a Non-Rectangular

Permeable Cavity

Pallath Chandran 1, Nirmal C. Sacheti 2, B.S. Bhadauria 3 and Ashok K. Singh 4

Abstract. Natural convection in vertical trapezoidal enclosures with or without embedded permeable

material finds applications in a number of engineering and geophysical fields. In such convective flows,

the nature of heating mechanisms at the bounding surfaces play important roles in the ensuing flow.

In this work, we have considered a special type of thermal condition on one of the vertical walls of

an isotropic porous cavity. This thermal condition relates to introducing the cosinusoidal modulation

effects on the left vertical wall. However, the non-vertical walls are subjected to adiabatic conditions

while the right wall is maintained at uniform temperature. It is assumed that the Darcy law governs

the flow in the cavity. The governing equations describing such a convective flow, subject to Boussinesq

approximation, have been solved numerically. A number of parameters arise in the study describing

physical and geometrical aspects — Darcy-Rayleigh number, aspect ratio, inclination angle, modu-

lation amplitude and frequency. The influence of these parameters on the convective flow and heat

transfer have been analyzed through a range of plots for streamlines and isotherms. Some additional

heat transfer related features have also been analyzed.

Keyword: Natural convection, trapezoidal cavity, porous medium, temperature modulation.

AMS 2010: 76R10, 76S05.

1Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected]

2Sultan Qaboos University, Al Khod, Muscat, Sultanate of Oman, [email protected]

3B.B. Ambedkar University, Lucknow, India, [email protected]

4Banaras Hindu University, Varanasi, India, [email protected]

159

Representation of Solutions of Neutral Time Delay Equations and Ulam-Hyers

Stability

Pembe Sabancigil 1, Mustafa Kara 2 and Nazim I. Mahmudov 3

Abstract. In this paper we studied representation of solutions of the initial value problem for neutral

differential equation with one delay with square matrices.

x(t) = Ax(t) +Bx(t− τ) + Cx(t− τ) for t ≥ 0

x(t) = ϕ(t) −τ ≤ t ≤ 0

(1)

where ϕ ∈ C1 ([−τ, 0] ,Rn) , A,B,C are n× n matrices. We prove the Ulam-Hyers stability for (1) on

the compact interval I = [a, b] and Ulam-Hyers-Rassias stability on I = [a,∞].

Keyword: Neutral differential equations, delayed equations,Ulam-Hyers stability, Ulam-Hyers-Rassias

stability.

AMS 2010: 34K37, 26A33, 34A05, 34K06.

References

[1] M. Pospısil and L. Skripkova, Representation of neutral differential equations with delay and linear parts defined by

pairwise permutable matrices, Miskolc Mathematical Notes, Vol. 16, No 1, pp. 423-438, 2015.

[2] D. Otrocal and V. Ilea, Ulam stability for a delay differential equation, Central European Journal of Mathematics,

11(7), 1296-1303, 2013.

[3] M. P. Lazarevic, D. Lj. Debeljkovic and Lj. Nenadic, Finite-time stability of delayed systems, IMA Journal of

Mathematical Control & Information 17, 101-109, 2000.




160

Computational Modeling of the NO+CO Reaction over Composite Catalysts

Pranas Katauskis 1, Vladas Skakauskas 2 and Raimondas Ciegis 3

Abstract. A mathematical model for effective computational simulation of the carbon monoxide

(CO) oxidation with nitrogen monoxide (NO) reaction occurring on composite catalysts [1, 2] is pro-

posed. The model is described by a coupled system of partial differential equations. Some PDEs are

considered in the domain, and the other part is solved on the boundary of the domain subject to non-

classical conjugation conditions. The model is based on the Langmuir–Hinshelwood surface reaction

mechanism and includes the bulk diffusion of both reactants and reaction products, adsorption and

desorption of particles of both reactants, and surface diffusion of adsorbed molecules. The readsorption

of the reaction product N2O is also investigated. The bulk diffusion is described by the Fick law while

the surface diffusion of the adsorbed particles is based on the particle jumping mechanism [3]. The

spillover phenomenon [4, 5] is taking into account.

The PDE model is approximated by using the finite volume method in space and the alternating

direction implicit (ADI) finite difference technique for integration in time [6]. The influence of the

rate constants of the adsorbed particle jumping via the catalyst–support interface and reaction rate

constants on the surface reactivity is investigated. The turnover rates of the CO and NO into products

N2O, CO2, and N2 may possess one or two maxima. Conditions for arising of the second maximum

are studied. The dependence of the turnover rates on the N2O readsorption is analysed.

Keywords: Heterogeneous reactions, spillover, surface diffusion.

AMS 2010: 00A69, 35K61.

References

[1] V.P. Zhdanov and B. Kasemo, Simulations of the reaction kinetics on nanometer supported catalyst particles, Surf.

Sci. Rep. 39, 25-104, 2000.

1Vilnius University, Vilnius, Lithuania, [email protected]


3Vilnius Gediminas technical university, Vilnius, Lithuania, [email protected]

161

[2] L. Cwiklik, B. Jagoda-Cwiklik and M. Frankowicz, Influence of spatial distribution of active centers on the kinetics

model heterogeneous catalytic processes, Surf. Sci. 572, 318-328, 2004.

[3] A.N. Gorban, H.P. Sargsyan and H.A. Wahab, Quasichemical models of multicomponent nonlinear diffusion, Math.

Model. Nat. Phenom. 6, 184-262, 2011.

[4] V. Skakauskas and P. Katauskis, Spillover in monomer-monomer reactions on supported catalysts – dynamic mean-

field study, J. Math. Chem. 52, 1350-1363, 2014.

[5] V. Skakauskas and P. Katauskis, Computational study of the dimer–trimer and trimer–trimer reactions on the

supported catalysts, Comput. Theor. Chem. 1070, 102-107, 2015.

[6] R. Ciegis, P. Katauskis and V. Skakauskas, The robust finite volume schemes for modeling non-classical surface

reactions, Nonlinear Anal. Model. Control, 23, 234-250, 2018.

162

Fractional Solutions of a k−Hypergeometric Differential Equation

Resat Yılmazer 1, Karmina K. Ali 2

Abstract. One of the most popular research interests of science and engineering is the fractional

calculus theory in recent times. Discrete fractional calculus has also an important position in fractional

calculus. It is well known that many phenomena in physical and technical applications are governed

by a variety of ordinary and partial differential equations.

In this article, we obtained discrete fractional solutions for the second-order non-homogeneous k−hypergeometric

differential equation.

Keyword: Nabla operator, discrete fractional calculus, k−hypergeometric differential equation.

AMS 2010: 26A33, 34A08.

References

[1] I. Podlubny, Fractional differential equations, mathematics in science and engineering, vol. 198, Academic Press, San

Diego, 1999.

[2] D. Baleanu, Z.B. Guven and J.A.T. Machado, New trends in nanotechnology and fractional calculus applications,

Springer, Berlin/Heidelberg, Germany, 2010.

[3] S. Li and Y. Dong, k−hypergeometric series solutions to one type of non-homogeneous k−hypergeometric equations,

Symmetry, 262, 1-11, 2019.

[4] F.M. Atıcı, P.W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ.,

3, 1-12, 2009.

[5] R. Yilmazer, et al., Particular solutions of the confluent hypergeometric differential equation by using the nabla

fractional calculus operator, Entropy, 18, 49, 1-6, 2016.

[6] R. Yilmazer, Discrete fractional solutıon of a non-homogeneous non-fuchsian differential equations, Thermal Science,

23, Suppl. 1, S121-S127, 2019.



163

Solutions of Singular Differential Equations by means of Discrete Fractional

Analysis

Resat Yılmazer 1, Gonul Oztas 2

Abstract. One of the most popular research interests of science and engineering is the fractional

calculus theory in recent times. Discrete fractional calculus has also an important position in fractional

calculus.

In this study, we will consider a general class of linear differential equations with singular points. The

particular solutions of this equation will be obtained with the nabla-discrete fractional calculus opera-

tor out of the known methods.

Keyword: Nabla operator, discrete fractional calculus, ordinary differential equation.

AMS 2010: 26A33, 34A08.

References

[1] I. Podlubny, Fractional differential equations, mathematics in science and engineering, vol. 198, Academic Press, San

Diego, 1999.

[2] D. Baleanu, Z.B. Guven and J.A.T. Machado, New trends in nanotechnology and fractional calculus applications,

Springer, Berlin/Heidelberg, Germany, 2010.

[3] F.M. Atıcı, P.W. Eloe, A transform method in discrete fractional calculus, International Journal of Difference Equa-

tions 2, 165-176, 2007.

[4] F.M. Atıcı, P.W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ.,

3, 1–12, 2009.

[5] J.J. Mohan, Analysis of nonlinear fractional nabla difference equations, Int. J. Analysis Applications 7, 79-95, 2015.

[6] R. Yilmazer, et al., Particular solutions of the confluent hypergeometric differential equation by using the nabla

fractional calculus operator, Entropy, 18 , 49, 1-6, 2016.



164

An Effective Computational Approach for Nonlinear Analysis of Imperfect

Perforated Compressed Laminates

S. A. M. Ghannadpour 1, M. Mehrparvar 2

Abstract. Penalty methods are a certain category of procedures for solving constrained optimization

problems. A penalty method substitutes a constrained optimization problem by a series of uncon-

strained problems whose answers preferably converge to the solution of the original constrained one.

The unconstrained problems are formed by adding a term, called a penalty function, to the impartial

function that consists of a penalty parameter multiplied by a measure of violation of the constraints.

The measure of violation is nonzero when the constraints are violated and is zero in the region where

constraints are not violated [1]. Perforated plates are extensively used as structural members for

weight optimization purposes, openings for hardware and wiring to pass through and in case of fuse-

lage windows and doors. Nevertheless, the presence of holes may cause the plates stability decreases

significantly and alter the membrane stress in the plates. Therefore, inspecting the buckling of such

structures is inevitable. Ghannadpour et al. [2] analyzed the buckling behavior of cross-ply laminated

plates with circular and elliptical cutouts by FEM, in which the effects of cutout shape, plate aspect

ratio and boundary conditions had been studied. On the other hand, composite laminates may endure

further loads even after buckling takes place and hence the post-buckling behavior of such structures

has been of significant research interest. Ovesy, Ghannadpour and Nassirnia [3] carried out a study

on post-buckling behavior of rectangular FGPs in thermal environments using a semi-analytical finite

strip method.

1Shahid Beheshti University, G.C, Tehran, Iran, a [email protected]

2Shahid Beheshti University, G.C, Tehran, Iran, [email protected]

165

Abstract. In this study, post-buckling analysis of rectangular composite plates, which contain holes,

is investigated. The laminates are assumed to have an initial geometric imperfection shape. The first-

order shear deformation plate theory is employed to account for the transverse shear strains, and the

Von Karman-type nonlinear strain-displacement relationship is adopted. The displacement fields are

selected such that to satisfy the boundary conditions and the principle of minimum potential energy is

applied to obtain a nonlinear equilibrium equations system. It is also noted that the Legendre polyno-

mials are used as basis functions for displacement fields. The whole plate potential energy form, which

is the summation of the element’s potential energy, obtained by the above assumptions can be written

as quadratic, cubic and quartic energy terms and the related integrals are taken numerically by using

Gauss quadrature and Double exponential integration methods. The obtained nonlinear equations can

be solved using an iterative procedure and here it is the quadratic extrapolation procedure. The effects

of different values of initial imperfection and also the cutout shape, size and location on post-buckling

behavior of rectangular laminates are examined.

Keyword: Penalty methods, double exponential integration, quadratic extrapolation, post-buckling

behavior, composite laminates, imperfection.


References

[1] J. Nocedal and J. Wright, Numerical optimization, Springer-Verlag, New York, 1999.

[2] S.A.M. Ghannadpour, A. Najafi and B. Mohammadi, On the buckling behavior of cross-ply laminated composite

plates due to circular/elliptical cutouts, Composite Structures, 75, 3-6, 2006.

[3] S.A.M. Ghannadpour, H.R. Ovesy and M. Nassirnia, An investigation on buckling behavior of functionally graded

plates using finite strip method, Applied Mechanics and Materials, 152-154, 1470-1476, 2012.

166

The Study of One-Dimensional Mixed Problem for One Class of Fourth Order

Differential Equations

Samed J. Aliyev 1, Faig M. Namazov 2 and Arzu Q. Aliyeva 3

Abstract. This work is dedicated to the study of existence in small of an almost everywhere solution

for the following one-dimensional mixed problem:

utt(t, x)− uxx(t, x)− αuttxx(t, x) =

= F (t, x, u(t, x), ux(t, x), uxx(t, x), ut(t, x), utx(t, x), utxx(t, x)) (0 ≤ t ≤ T, 0 ≤ x ≤ π), (1)

u(0, x) = ϕ(x) (0 ≤ x ≤ π), ut(0, x) = ψ(x) (0 ≤ x ≤ π), (2)

u(t, 0) = u(t, π) = 0 (0 ≤ t ≤ T ), (3)

where α > 0 is a given number; 0 < T < +∞; F, ϕ, ψ are the given functions, and u(x, t) is a sought

function.

In this work, using contracted mappings principle and Shauder’ fixed point principle the following

existence in small (i.e. for sufficiently small values of T ) theorem for the almost everywhere solution

of problem (1)-(3) is proved.

Theorem. Let

1) ϕ(x) ∈ C(1) ([0, π]) , ϕ′′(x) ∈ L2(0, 1) and ϕ(0) = ϕ(π) = 0;

ψ(x) ∈ C(1) ([0, π]) , ψ′′(x) ∈ L2(0, 1) and ψ(0) = ψ(π) = 0.

2) F (t, x, u1, ..., u6) ∈ C([0, T ]× [0, π]× (−∞,∞)6

).

3) ∀R > 0 in [0, T ]× [0, π]× [−R,R]4 × (−∞,∞)2

|F (t, x, u1, u2, u3, u4, u5, u6)− F (t, x, u1, u2, u3, u4, u5, u6)| ≤ CR · (|u5 − u5|+ |u6 − u6|) ,

where CR > 0 is a constant.

Then problem (1)-(3) has an almost everywhere solution.

Keyword: Mixed problem, almost everywhere solution, fourth order differential equation.

AMS 2010: 35L76, 35L82.



3Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, [email protected]

167

On the Spectral Distribution of Symmetrized Toeplitz Sequences

Sean Hon 1, Mohammad Ayman Mursaleen 2 and Stefano Serra-Capizzano 3

Abstract. Let Tn [f ] denote the Toeplitz matrix generated by f ∈ L−1 [−π, π] and Yn be the anti-

identity matrix. In this work, we furnish the singular value distribution of YnTn [f ]n and the asymp-

totic inertia of YnTn [f ] that is an evaluation of the number of positive, negative, and zero eigenvalues.

Considering the symmetrized Toeplitz matrix sequences YnTn [f ]n, we show that its singular value

distribution can be obtained analytically. In particular, we provide a detailed proof that relates the

asymptotic distribution of |Cn [f ]| and that of YnTn. That way we can clearly see that the symmetric

positive definite (SPD) |Cn [f ]| is naturally a good preconditioner for YnTn [f ]. Moreover, while its

eigenvalues are of course real, their modulus coincides with the singular value and precise informa-

tion on the distribution of their sign can be provided. Specifically, we show that roughly half of the

eigenvalues of YnTn [f ] are negative/positive, when the dimension is sufficiently large and f is sparsely

vanishing, i.e. its set of zeros is of (Lebesgue) measure zero.

Results of this presentation are recently published in [Linear Algebra and its Applications, 579 (2019)

32-50] by the same authors.

1Mathematical Institute, University of Oxford, Radcli e Observatory Quarter, Oxford, OX2 6GG, United Kingdom

2Department of Science and High Technology, University of Insubria, Via Valleggio 11, Como, 22100, Italy

3Department of Science and high Technology, University of Insubria, Via Valleggio 11, Como, 22100, Italy

168

Quantum Correlation, Coherence & Uncertainty

Shao-Ming Fei 1

Abstract. Quantum correlations, quantum coherence and quantum uncertainty relations play sig-

nificant roles in quantum information processing such as quantum communication and computation.

The operational characterization of quantum correlations and quantum coherence are also the impor-

tant aspects of the corresponding resource theory. We introduce some recent results on the theory

of quantum entanglement; coherence quantifier based on max-relative entropy and its implications to

subchannel discriminations; as well as the related quantum uncertainty relations such as error and

disturbance based trade-off relation, quantum information masking including both deterministic and

probabilistic masking machines, quantum coherence and energy flow.

References

[1] W. Ma, B. Chen, Y. Liu, M. Wang, X. Ye, F. Kong, F. Shi, S.M Fei, and J. Du, Experimental demonstration of

uncertainty relations for the triple components of angular momentum, Phys. Rev. Lett. 118, 180402, 2017.

[2] K. Bu, U. Singh, S.M. Fei, A.K. Pati and J. Wu, Maximum relative entropy of coherence: an operational coherence

measure, Phys. Rev. Lett. 119, 150405, 2017.

[3] W. Zheng, Z. Ma, H. Wang, S.M. Fei and X. Peng, Experimental demonstration of observability and operability of

robustness of coherence, Phys. Rev. Lett. 120, 230504, 2018.

[4] Y.L. Mao, Z. Ma, R.B. Jin, Q.C. Sun, S.M. Fei, Q. Zhang, J. Fan and J.W. Pan, Error and disturbance trade-off

relation based on statistical distance, Phys. Rev. Lett. 122, 090404, 2019.

[5] H.H. Qin, T.G. Zhang, L. Jost, C.P. Sun, X. Li-Jost and S.M. Fei, Uncertainties of genuinely incompatible triple

measurements based on statistical distance, Phys. Rev. A, 99, 032107, 2019.

[6] Z.X. Jin and S.M. Fei, Super-activation of monogamy relations for non-additive quantum correlation measures, Phys.

Rev. A, 99, 032343, 2019.

[7] Z.X. Xiong, M.S. Li, Z.J. Zheng, C.J. Zhu and S.M. Fei, Positive-partial-transpose distinguishability for lattice-type

maximally entangled states, Phys. Rev. A, 99, 032346, 2019.

[8] H. Wang, Z. Ma, S. Wu, W. Zheng, Z. Cao, Z. Chen, Z. Li, S.M. Fei, X. Peng, J. Du and V. Vedral, Uncertainty

equality with quantum memory and its experimental verification, NPJ Quant. Inform. 5, 39, 2019.

1Capital Normal University, Beijing 100048, China, [email protected]

169

[9] B. Li, S. Jiang, X.B. Liang, X. Li-Jost, H. Fan and S.M. Fei, Quantum information masking: deterministic versus

probabilistic, Phys. Rev. A, 99, 052343, 2019.

[10] T. Ma, M.J. Zhao, S.M. Fei and M.H. Yung, Necessity for Quantum Coherence of Nondegeneracy in Energy Flow,

Phys. Rev. A, 99, 062303, 2019.

[11] Y. Xi, T.G. Zhang, Z. Zheng, X. Li-Jost and S.M. Fei, Converting Coherence to Genuine Multipartite Entanglement

and Nonlocality, to appear in Phys. Rev. A, 2019.

170

A Hybridized Discontinuous Galerkin Method for Solving Generalized Regularized

Long Wave Equations

Shima Baharlouei1, Reza Mokhtari2

Abstract. The regularized long wave (RLW) or Benjamin-Bona-Mahony equation has been becoming

attractive ever since Peregrine introduced it as an alternative to the KdV equation for investigating

soliton phenomena and as a model for small amplitude long waves on the surface of water [3, 4, 3].

On the other hand, the hybridized discontinuous Galerkin (HDG) method is one of the outstanding

and successful methods for solving evolution equations [1, 2]. In this paper, we aim to construct and

present an HDG method for solving the following generalized regularized long wave (GRLW) equation

ut + f(u)x − α2uxxt = r(x, t), x ∈ Ω = [xL

, xR

] ⊂ R, t ∈ (0, T ], (15)

where f(u) = u + α1

m+1um+1, m is a positive integer, α1 and α2 are positive constants and r is a given

function. Actually, we employ here an HDG scheme for the spatial discretization and a backward Euler

method for the temporal discretization. For obtaining weak formulation, we need to define a numerical

flux which is in term of the numerical trace u of u. Unlike the approximate values corresponding to u

and its first derivative, u is a global unknown. By enforcing conservation of the numerical flux on the

element faces, one extra global equation is obtained which helps us to find the global unknown.

1Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran,

[email protected] of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran,

[email protected]

171

Abstract. Finally, weak equations as well as the global equation lead to a nonlinear system of equa-

tions which is solved by using a Newton-Raphson method. Moreover, we proved that if equation (15) is

equipped with periodic or homogeneous Dirichlet boundary conditions then the proposed HDG method

is stable under the proper choice of stabilization parameters. By testing some different examples, we

observe that for a mesh with k-th order elements, approximate solution and its derivative show optimal

convergence at order k+1. Some physical attributes of the model problem such as the motion of single

solitons, interaction of solitary waves and soliton generation using the Maxwellian initial condition are

simulated very well and also some invariant quantities are conserved numerically by the method.

Keyword: HDG method, GRLW equation, stability analysis.

AMS 2010: 65M60, 65M12.

References

[1] R.M. Kirby, S.J. Sherwin and B. Cockburn, To cg or to hdg: a comparative study, J. Sci. Comput. 51, 183-212, 2012.

[2] D.M. Luo, W.Z. Huang and J.X. Qiu, An hybrid ldg-hweno scheme for kdv-type equations, J. Comput. Phys. 313,

754-774, 2016.

[3] M. Mohammadi and R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing

kernel space, J. Comput. Appl. Math. 235, 4003-4014, 2011.

[4] R. Mokhtari and M. Mohammadi, Numerical solution of grlw equation using sinc-collocation method, Comput. Phys.

Commun. 181, 1266-1274, 2010.

[5] D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid. Mech. 25, 321-330, 1966.

172

Non-Instantaneous Impulsive Differential Equations with State Dependent Delay

and Practical Stability

Snezhana Hristova 1

Abstract. Many evolution processes are characterized by the states abrupt changes. Adequate models

in this case are impulsive differential equations. In the literature there are two popular types of impulses:

instantaneous impulses ( whose duration is relatively short compared to the overall duration of the

whole process) and non-instantaneous impulses (which start their action at some points and remain

active on a finite time interval). Recently some results about various types of differential equations

with non-instantaneous impulses are obtained, for example, in [2], [3], [4], [5]. An overview of the

main properties of the presence of non-instantaneous impulses to ordinary differential equations and to

fractional differential equations is given in the book [1]. Note non-instantaneous impulsive differential

equations are natural generalizations of impulsive differential equations.

The study of the case of differential equations with delays and non-instantaneous impulses is developing

rather slowly due to a number of technical and theoretical difficulties related to the phenomena of

”beating” of the solutions, bifurcation, loss of the property of autonomy, etc.. The great possibilities

for application to mathematical simulations require the obtaining of criteria for various types of stability

of their solutions.

In this paper nonlinear differential equations with non - instantaneous impulses and variable delays

are presented and practical stability of the solutions is studied. The delay depends on both the time

and the state variable which is a generalization of the time variable delay. Comparison principle and

Razumikhin method are applied. Nonlinear non-instantaneous impulsive differential equations without

any delay are used as comparison equations. It makes the practical application of the obtained results

easier. Some sufficient conditions for practical stability and strong practical stability are obtained.

Examples are given to illustrate the results.

Keyword: Non-instantaneous impulses, differential equations, practical stability.

AMS 2010: 34K45, 34K20.

Acknowledgments. Research was partially supported by the Fund NPD, University of Plovdiv

”Paisii Hilendarski”, No. FP19-FMI-002.

1Plovdiv University ”Paisii Hilendarski”’, Plovdiv, Bulgaria, e-mail: [email protected]

173

References

[1] R. Agarwal, S. Hristova, D. O’Regan, Non-instantaneous impulses in differential equations, Springer, 2017.

[2] S. Liu, J.R. Wang, D. Shen and D. O’Regan, Iterative learning control for noninstantaneous impulsive fractional-order

systems with varying trial lengths, Int. J. Robust Nonlinear Control, 28, 6202-6238, 2018.

[3] M. Pierri , H. R. Henriquez, A. Prokopczyk, Global solutions for abstract differential equations with non-instantaneous

impulses, Mediterr. J. Math., 13, 1685-1708, 2016.

[4] M. Pierri, D. O’Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not

instantaneous impulses, Appl. Math. Comput. 219, 6743–6749, 2013.

[5] A. Sood and S. K. Srivastava, On stability of differential systems with non-instantaneous impulses, Math. Probl.

Eng., Article ID 691687, 5 p., 2015.

174

One-Dimensional Finite Element Simulations for Chemically Reactive Hypersonic

Flows

Suleyman Cengizci 1

Abstract. Hypersonic vehicles are utilized in recent years massively for both military and civilian

purposes. As the vehicles fly through an atmosphere at hypersonic speeds (generally considered as

Mach > 5), they experience critical physical and chemical interactions due to extremely high (several

thousands of Kelvin) temperatures generated around the vehicles. This high-temperature effects may

cause vibrational excitation, dissociation and ionization of atoms and molecules. Therefore, the perfect

gas hypothesis is no longer valid for air and high temperature effects also should be included in the

mathematical model.

In this study, one-dimensional compressible multi-species Navier-Stokes Equations for thermo-chemical

non-equilibrium are simulated utilizing a Galerkin Finite Element Method. Air mixture is considered

as a combination of atoms Oxygen (O), Nitrogen (N) and molecules Nitric oxide (NO), Dioxygen (O2),

Dinitrogen (N2).

Keyword: Hypersonic, non-equilibrium flow, chemically reactive, finite element, Vibration-dissociation

coupling

AMS 2010: 76K05, 76N15.

References

[1] J. Hao, W. Jingying and L. Chunhian, Assessment of vibration dissociation coupling models for hypersonic nonequi-

librium simulations, Aerospace Science and Technology 67, 433-442, 2017.

[2] A. Logg, M. Kent-Andre, W. Garth, eds. Automated solution of differential equations by the finite element method:

The FEniCS book. Vol. 84. Springer Science & Business Media, 2012.

[3] B. Kirk, B. Steven and B. Ryan, A streamline-upwind petrov-galerkin finite element scheme for non-ionized hypersonic

flows in thermochemical nonequilibrium, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum

and Aerospace Exposition. 2011.

1Antalya Bilim University, Antalya, Turkey, [email protected]

175

[4] T:J:R: Hughes, S. Guglielmo and T.E. Tezduyar, Stabilized methods for compressible flows, Journal of Scientific

Computing 43.3, 343-368, 2010.

[5] T.E. Tezduyar and Y.J. Park, Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-

reaction equations, Computer Methods in Applied Mechanics and Engineering 59.3, 307-325, 1986.

[6] P.A. Gnoffo, R.N. Gupta and J.L. Shinn, Conservation equations and physical models for hypersonic air flows in ther-

mal and chemical nonequilibrium, No. N-89-16115; NASA-TP-2867; L-16477; NAS-1.60: 2867. National Aeronautics

and Space Administration, Hampton, VA (USA). Langley Research Center, 1989.

176

Some Numerical Experiments on Singularly Perturbed Problems with

Multi-Parameters

Suleyman Cengizci 1

Abstract. In this study, numerical behavior of singular perturbed ordinary differential equations that

depend on positive small parameters is investigated. An efficient method that combines the well-known

Finite Element Method (FEM) and an asymptotic approach so-called Successice Complementary Ex-

pansion Method (SCEM) is employed for numerical simulations of the multi-parameter problems.

Keyword: Asymptotic approximation, singular perturbation, finite element method, multi- parameter

problem, SCEM.

AMS 2010: 34E15, 65L11, 65L60.

References

[1] A. Logg, K.A. Mardal and G. Wells, eds. Automated solution of differential equations by the finite element method,

The FEniCS book. Vol. 84. Springer Science and Business Media, 2012.

[2] J. Cousteix and J. Mauss, Asymptotic analysis and boundary layers, Springer Science and Business Media, 2007.

[3] M.G. Larson and F. Bengzon, The finite element method: theory, implementation, and applications, Vol. 10. Springer

Science and Business Media, 2013.

[4] T. Lin and H.G. Roos, Analysis of a finite-difference scheme for a singularly perturbed problem with two small

parameters, Journal of Mathematical Analysis and Applications 289.2, 355-366, 2004.

[5] S. Natesan, J.L. Gracia and C. Clavero, Singularly perturbed boundary-value problems with two small parameters-a

defect correction approach.” proceedings of the international conference on boundary and interior layerscomputational

and asymptotic methods, BAIL. 2004.

1Antalya Bilim University, Antalya, Turkey, [email protected]

177

Incomplete Block-Matrix Factorization of M-Matrices Using Two Step Iterative

Method for Matrix Inversion and Preconditioning

Suzan C. Buranay 1, Ovgu C. Iyikal 2

Abstract. It is given that the block versions of incomplete factorization are more efficient with re-

spect to computer time than pointwise versions and do not require more storage [1]-[4]. Hence in this

study recursive approach to construct incomplete block factorization of M -matrices using two step

iterative method to approximate the inverse of diagonal pivoting block matrices at each stage of the

recursion by which the matrix multiplications and additions for calculating matrix polynomials in the

inverse finding algorithm are reduced through factorizations and nested loops are proposed. The given

incomplete block factorization of M -matrices are used to precondition some iterative methods as one

step stationary iterative method. Certain applications are conducted on M -matrices occurring from

the discretization of boundary value problems of partial differential equations using finite difference

methods. Numerical results justify that the proposed incomplete block factorization of M -matrices

using the two step iterative method to approximate the inverse of diagonal pivoting block matrices at

each stage give robust preconditioners and the numerical results are presented via tables and figures.

Keyword: M -matrices, incomplete block matrix factorization, approximate inverse of matrix, one step

stationary iterative method, preconditioning.

AMS 2010: 65F08, 65F10, 65M06.

References

[1] O. Axelsson, A general incomplete block-matrix factorization method, Linear algebra and Its Applications, 74, 179-

190, 1986.

[2] O. Axelsson, Iterative solution methods, Cambrige University Press, New York, 1994.

1Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North

Cyprus, Via Mersin 10, Turkey, [email protected] of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North

Cyprus, Via Mersin 10, Turkey, [email protected]

178

[3] O. Axelsson, S. Brinkkemper and V.P. Ilin, On some versions of incomplete block-matrix factorization iterative

methods, Linear Algebra and Its Application, 58, 3-15, 1984.

[4] P. Concus, G.H. Golub and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM Journal,

6(1), 220-252, 1985.

179

Asymptotic Solutions of the Problem with Properties for Integro-Differential

Equations with Singular Perturbation

Tofig H. Huseynov 1, Aygun T. Huseynova 2

Abstract. The work is devoted to the construction of asymptotic solution for initial problem of some

singular perturbed integro-differential equations, analysis of its properties and shown important role

of integral part for boundedness of solution.

In [1-4] the asymptotic of the following system of equations is constructed and investigated:

dxdt

= L1[x, y] + f1(t),

ε dydt

= L2[x, y] + f2(t),

(16)

with boundary conditions

x(0, ε) = x0, y(T, ε) = y0, (17)

where

Li[x, y] = Ai1

(t)x+Ai2

(t)y +

α∫0

[Ki1

(t, s)x(s) +Ki2

(t, s)y(s)]ds, i = 1, 2,

and ε > 0 small parameter, α = t and α = T, 0 ≤ t ≤ T, x, f1 is n dimensional; y, f2 is m dimensional

vectors, A11

,K11

− (n×n);A12

,K12

− (n×m);A22

,K22

− (m×m);A21

,K21

− (m×n) dimensional enough

smooth matrices K22

(t, s) 6= 0.

Let the characteristic values λi(t) of the matrix A22

(t) satisfy the condition

Reλi(t) < 0, i = 1,m, 0 ≤ t ≤ T. (18)

Note that, by fulfillment of the condition (3) and when the integral part is absent problem (1), (2)

generally has no bounded by ε → 0 solution, at the same time (1), (2) has bounded ε → 0 solution

x(t, ε), x(t, ε).

Therefore is shown that appearing of the integral part leads to quantitative changes of the behavior

of the solution of the boundary problem.


2Baku State University, Baku, Azerbaijan

180

References

[1] A.B. Vasilyeva and M.G. Dmitriyev, Singular perturbation in the optimal control problems, Resume of the Scie. And

Tech., Math. Analysis, 20, 3-77, 1982.

[2] A.T. Huseynova, On an asymptotical solution of some integro-differential equations, Referat of the Ph.D. Thesis,

2005.

[3] T.H. Huseynov, On an asymptotics and some its features for the solution of the optimal control problem for singular

perturbed system of integro-differential equations, Referat of the Ph.D.Thesis, 1981.

[4] N.N. Nefyodov and A.G. Nikitin, The cauchy problem for the singular perturbed integro-differential fredholm equa-

tions, Jour. of Vich. Math. And Math. Phys. 47(4), 655-664, 2007.

181

Mixing Problems Modeled with Directed Graphs and Multigraphs: Results and

Conjectures

Victor Martinez-Luaces 1

Abstract. In this paper, open and closed mixing problems are modeled with direct graphs and multi-

graphs, depending whether there is recirculation or not [1-2]. Classical examples like cycles, wheels,

trees, cubes, complete graphs and bipartite graphs and a few directed multigraphs are analyzed [3-4].

This analysis is focused on the qualitative behavior of the ODE system solutions [5-6] associated to

mixing problems which structure can be modeled using Graph Theory tools. Finally, several theorems

and corollaries as well as some conjectures and open questions, which are relevant for further research,

are discussed.

Keyword: Mixing problems, linear ODE systems, directed graphs and multigraphs.

AMS 2010: 34A30, 05C20, 37C75.

References

[1] V. Martinez-Luaces, Matrices in chemical problems: characterization, properties and consequences about the stability

of ode systems, in: Advances in Mathematics Research. Chapter 1, pp. 1-33. New York: Nova Science Publishers,

2017.

[2] V. Martinez-Luaces, Square matrices associated to mixing problems, in: matrix theory: applications and theorems,

Chapter 3, pp. 41-58. London, UK: In Tech Open Science, 2018.

[3] J.L. Gross, J. Yellen, eds., Handbook of graph theory, CRC press, 2004.

[4] N. Trinajstic, Chemical graph theory, Routledge, 2018.

[5] M. Braun, Differential equations and their applications, 3rd Edition. New York: Springer, 2013.

[6] R. Bellman, Stability theory of differential equations, Courier Corporation, 2008.

1UdelaR, Montevideo, Uruguay, [email protected]

182

Solvability and Long-Time Behaviour of Classical Solutions to a Model of Surface

Reactions over Composite Catalysts

Vladas Skakauskas 1

Abstract. Coupled systems of nonlinear parabolic differential equations with nonlinear boundary

conditions usually arise from applications and have recently been extensively studied in literature.

We consider a coupled system of nonlinear parabolic equations that arise in modelling of reactions

proceeding between two polyatomic reactants over surfaces of composite catalysts. According to Lang-

muir molecules of reactants adsorb on the active sites of the catalyst surface, diffuse on it, and react

to produce a product. Two types of the reactants adsorption are known: (i) molecules of different

reactants adsorb on sites of different type, (ii) particles of both reactants compete for adsorption sites.

We consider a mean-field PDEs model for reactions of the first reactants adsorption type. Data of the

composite catalyst model (kinetic coefficients) possess a jump discontinuity on a surface which divides

a given domain into two subdomains. We formulate non-classic conjugate conditions on the interface

of subdomains. Under some restrictions on the boundary and model data we prove the existence and

uniqueness theorem of a nonnegative classic solution in domains of continuity of the model data and

study. The existence is based on the lower and upper solutions technique and potential theory.

We also study the long-time behaviour of classical solutions of time-dependent model and prove that

solutions of parabolic system tend to solutions of corresponding elliptic equations.

Keywords: Parabolic systems, surface reactions, surface diffusion.

AMS 2010: 00A69, 35K61.

References

[1] V. Skakauskas and P. Katauskis, Spillover in monomer-monomer reactions on supported catalysts – dynamic mean-

field study, J. Math. Chem. 52, 1350-1363, 2014.

[2] V. Skakauskas and P. Katauskis, Computational study of the dimer–trimer and trimer–trimer reactions on the

supported catalysts, Comput. Theor. Chem. 1070, 102-107, 2015.


183

[3] A. Ambrazevicius and V. Skakauskas, Solvability of a model for monomer-monomer surface reactions, Nonlinear

Anal.: Real World Appl. 35, 211-228, 2017.

[4] C.V. Pao and W.H. Ruan, Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions, J.

Math. Anal Appl. 333, 472-499, 2007.

[5] A. Garcia Cantu Ros, J.S. McEven and P. Gaspard, Effect of ultrafast diffusion on adsorption, desorption, and

reaction processes over heterogeneous surfaces, Phys. Rev. E, 83, 021604, 2011.

[6] C.V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.

184

Stability of the Transmission Plate Equation with a Delay Term in the Moment

Feedback Control

Wassila Ghecham 1, Salah-Eddine Rebiai 2 and Fatima Zohra Sidiali 3

Abstract. In this paper, we study the problem of stability for a system of transmission of plate

equation with moment feedback control that contains a delay term and that acts on the exterior

boundary.

Let Ω1 ⊂ Ω ⊂ Rn, n ≥ 2, be strictly convex, bounded domains with smooth boundaries Γ1 = ∂Ω1,Γ =

∂Ω,Γ1 ∩ Γ = ∅. Then O = Ω\Ω1 is a bounded, connected domain with boundary ∂O = Γ1 ∪ Γ. We

are going to study the following mixed boundary value problem

(∂2t + c2∆2)u1(x, t) = 0 in Ω1 × (0,+∞),

(∂2t + ∆2)u2(x, t) = 0 in O × (0,+∞),

u1(x, t) = u2(x, t), ∂νu1 = ∂νu2(x, t), on Γ1 × (0,+∞),

c∆u1(x, t) = ∆u2(x, t), c∂ν∆u1(x, t) = ∂ν∆u2(x, t) on Γ1 × (0,+∞),

u2(x, t) = 0,∆u2(x, t) = −α1∂ν∂tu2(x, t)− α2∂ν∂tu2(x, t− τ) on Γ× (0,+∞),

u1(x, 0) = u01(x), ∂tu(x, 0) = u1

1(x) in Ω1,

u2(x, 0) = u02(x), ∂tu(x, 0) = u1

2(x) in O,

∂tu2(x, t− τ) = f0(x, t− τ) on Γ× (0,+∞).

(19)

where ν(x) denotes the outer unit normal vector to the point x ∈ Γ and ∂νu is the normal derivative.

Moreover, c > 1 is a constant, τ > 0 is the time delay, α1 and α2 are positive real numbers, and the

initial data (u01, u

11, u

02, u

12, f0) belong to a suitable Hilbert space.

1University of Batna 2, Batna, Algeria, [email protected]



185

Abstract. If α1 > 0 and α2 = 0, it was proved in ([1]) that the solutions of (19) are exponentially

stable in the energy space. In this paper, we consider the case where both α1and α2 are strictly positive,

and under appropriate assumptions, we establish exponential stability of the solutions. This result is

obtained by introducing suitable energy functionnals and by proving an observability estimate.

Keyword: Transmission problem, boundary stabilization, Bernoulli-Euler plate equation, time delay. .

AMS 2010: 35B05, 93D15.

References

[1] K. Ammari and G. Vodev, Boundary stabilization of the transmission problem for the bernoulli-euler plate equation,

Cubo A Mathematical Journal, 11, 39-49, 2009.

186

Lennard-Jones Potentials for Non-Metal Atoms Embedded in Tiv

Xinhua Yang,

Jian Hu

Abstract. As a kind of clean energy, nuclear energy is a potential solution to the future energy crisis.

TiV alloys are the main candidate materials of the first wall in future fusion reactors. In service, they

will be inevitably invaded by impurity atoms in the environment, such as H, He, C, and O, so that their

properties will be changed. As the first step of properties variation investigation, it is very important

for design and improvement of materials to explore the interactions between the impurity atoms and

the ones in the base alloy. In the present work, Lennard-Jones (L-J) potential was used to characterize

the interactions of H, He, C, and O atoms with Ti and V atoms. Eight groups of computational models

were created for the impurity atoms embedded in TiV alloy. The relations between the binding energy

and the atomic distance were calculated with the first-principles approach so that the L-J potential

curves and their potential parameters were obtained. These curves were found to be well consistent

with some published results.

Keyword: Non-metal atom, TiV, First-principles, Lennard-Jones potential.

AMS 2010: 65E05, 81U10.

References

[1] J.Y. Yan, H. Ehteshami, P.A. Korzhavyi and A. Borgenstam, Sigma 3(111) grain boundary of body-centered cubic

ti-mo and ti-v alloys: first-principles and model calculations, Physical Review Materials, 1(2): 023602, 2017.

[2] C.X. Li, H.B. Luo, Q.M. Hu, R. Yang, F.X. Yin, O. Umezawa and L.Vitos, Lattice parameters and relative stability

of alpha phase in binary titanium alloys from first-principles calculations, Solid State Communications, 159: 70-75,

2013.

[3] V.M. Chernov, V.A. Romanov and A.O. Krutskikh, Atomic mechanisms and energetics of thermally activated pro-

cesses of helium redistribution in vanadium, Journal of Nuclear Materials, 271-272, 1999.

187

A Time Nonlocal Inverse Problem for the Longitudinal Wave Propagation Equation

with Integral Conditions

Yashar T. Mehraliyev 1, Elvin I. Azizbayov 2

Abstract. This paper investigates the inverse problem of finding the functions u(x, t) ∈ C(2,2)(DT )

and q(t) ∈ C[0, T ], connected in the rectangular domain DT := (x, t) : 0 ≤ x ≤ 1, 0 ≤ t ≤ T for the

equation [1]

utt(x, t)− uttxx(x, t)− uxx(x, t) = q(t)u(x, t) + f(x, t) (x, t) ∈ DT ,

with the conditions

u(x, 0) =

T∫0

P1(x, t)u(x, t)dx+ ϕ(x), ut(x, 0) =

T∫0

P2(x, t)u(x, t)dx+ ψ(x), 0 ≤ x ≤ 1,

ux(0, t) = u(1, t) = 0, u(0, t) = h(t), 0 ≤ t ≤ T,

where f(x, t), P1(x, t), P2(x, t), ϕ(x), ψ(x), h(t) are given functions and

C(2,2)(DT ) := u(x, t) : u(x, t) ∈ C2(DT ), uttx(x, t), utxx(x, t), uttxx(x, t) ∈ C(DT ).

The authors consider in the present paper a time nonlocal inverse boundary-value problem for the

equation of longitudinal wave propagation. First, the original problem is reduced to an equivalent

problem. Further, the existence and uniqueness of the solution of the equivalent problem are proved

using a contraction mapping. Finally, using the equivalency, the existence and uniqueness of a solution

of the considered problem is obtained.

Keyword: Inverse value problem, longitudinal wave propagation equation, nonlocal integral condition.

AMS 2010: 35R30, 35L82, 49K20.

References

[1] S.A. Gabov and B.B. Orazov, The equation ddt2

[uxx−u]+uxx = 0 and several problems associated with it (in Russian),

Computational Mathematics and Mathematical Physics, 26(1), 58-64, 1986.

1Baku State University, Baku, Azerbaijan, yashar [email protected]


188

Mathematical Analysis for a Condition of the Hydrodynamic Characteristics

Yusif M. Sevdimaliyev 1, Gulnar M. Salmanova 2 and Reyhan S. Akbarly 3

Abstract. In this article a thin-walled isotropic infinitely long-circular cylindrical tube and the pulsed

flow of an aerohydroelastic system within a concise bubble fluid have been presented in 3 D dimension.

The equation has been solved for a symmetrical case to the axis.The form and frequency of the specific

dances formed in the coating-fluid dynamic system have been determined. The obtained results have

been calculated for the cases where thin-walled pipe material is made of various modifications of high-

strength steel and non-classic polymer materials, and as being bi-phase fluid for cases where as the

carrier liquid that consists of spherical air bubbles taken together with water.

The fundamental iteration technique has been used to compute eigenvalues and corresponding eigen-

functions to represent field quantities with the help of MATLAB software. The numerical results have

been presented graphically.

Keyword: Oximetric waves propagation, liquid-gas environment, viscous liquid.

References

[1] R.I. Nigmatulin, Multiphase Dynamics, P.1, M., 464p, 1987.

[2] R.S. Akbarly, Waves propagation in the fluid flowing in an elastic tube, considering viscoelastic friction of surrounding

medium, International Journal on Technical and Physical Problems of Engineering 35, 39-42, 2018.




189

Mathematical Modeling of the Dynamics of a Hydroelastic System - a Hollow

Cylinder with Inhomogeneous Initial Stresses and Incompressible Fluid

Yusif M. Sevdimaliyev 1, Gurbanali J. Valiyev 2

Abstract. The work is devoted to the study of the dynamics of a deformable circular cylinder (thick-

walled pipe) and an incompressible fluid filling it under the assumption that they form a hydroelastic

system. Such multi-phase (two or more) systems are found in many areas of modern technology and

industry.

A boundary-value problem simulating a stress-deformable state (SDS) of a system with initial inho-

mogeneous stresses, a flow process and phase interaction in a 3D formulation is formed. Attention is

paid to analytical and numerical solution methods that allow one to get complete information about all

hydrodynamic characteristics of joint motion in the form of free oscillation and propagation of elastic

waves in deformable media. Initial-boundary and contact conditions are compiled for joint motion,

based on the physical properties and kinematic data of the liquid phase.

The effects of initial inhomogeneous stresses on the distribution of amplitude and frequency in time in

the continuum are established.

Keyword: Natural frequency, fluid-filled hollow tube, inhomogeneous initial stress.

AMS 2010: 37N10, 74B10.

References

[1] N.N. Moiseev, Introduction to the theory of oscllations of liquid containing bodies. advances in applied mechanics,

vol.YIII, Academik Press, New York and London, 1964.

[2] S.D. Akrbarov, H.H. Guliyev, Y.M. Sevdimaliyev and N. Yahnioglu, The discrete-analitical soliton method for inves-

tiqation dynamics of the sphere with inhomogeneous initial stresses, CMC, 55(2), 359-380, 2018.



190

Optimal Symmetries of Option Pricing

Zehra Pınar 1

Abstract. For the option pricing, the Black-Scholes model is considered. To obtain an analytical

solutions of the Black-Scholes equation, the combination of Lie group transformation and Chebyshev

approximation is considered. In this work, instead of the classical Black-Scholes equation, we consider

the parametric expansion of the Black-Scholes equation.

Keyword: Black-Scholes model, Chebyshev equation, Lie group transformation.

AMS 2010: 22E70, 20C35, 35Q91.

References

[1] L. Debnath, Nonlinear partial differential equations for scientists and engineers (2nd ed.) Birkhauser, Boston, 2005.

[2] G.B. Whitham,A general approach to linear and nonlinear waves using a lagrangian, J. Fluid Mech., 22, 273-283,

1965.

[3] E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge Univ. Press, Cambridge, 1927.

[4] J.P. Singh and S. Parabakaran, Group properties of the black-scholes equation and its solution, Electronic Journal

of Theoretical Physics EJTP 5 No.18, 51-60, 2008.

[5] G.Y. Xuan Liu, Z. Shun-Li, Q.C. Zheng, Symmetry breaking for black-scholes equations, Commun. Theor.Phys. 47,

995-1000, 2007.

[6] X. Zeng and X. Yong, A new mapping method and its applications to nonlinear partial differential equations, Phy.

Lett. A. 372, 66026607, 2008.

[7] G.I. Burde, Expand lie group transformations and similarity reductions of differential equations, Proc. Ins. Math.

NAS of Ukraine, Vol.43, Part I, 93-101, 2002.

[8] P.J. Olver, Applications of lie groups to differential equations, GTM, V.107,Second edn., Springer-Verlag, New York,

1986.

[9] A.H. Davison and S. Mamba, Symmetry methods for option pricing, Commun Nonlinear Sci Numer Simulat 47,

421425, 2017.

1Tekirdag Namik Kemal University, Tekirdag, Turkey,[email protected]

191

[10] N.H. Ibragimov and R.K. Gazizov, Lie symmetry analysis of differential equations in finance, Nonlinear Dynamics

17:387-407, 1998.

[11] Sirendaoreji, Auxiliary equation method and new solutions of kleingordon equations, Chaos, Solitons and Fractals

31, 943950, 2007.

[12] R. Polat and T. Ozis, Expanded lie group transformations and similarity reductions for the celebrity black- scholes

equation in finance, Applied and Computational Mathematics, 13(1), 71-77, 2014.

[13] X. Lv, S. Lai, Y.H. Wu, An auxiliary equation technique and exact solutions for a nonlinear kleingordon equation,

Chaos, Solitons and Fractals 41, 8290, 2009.

[14] E. Yomba, A generalized auxiliary equation method and its application to nonlinear klein-gordon and generalized

nonlinear camassa-holm equations, Physics Letters A, 372, 10481060, 2008.

192

On the Solutions of the Population Balance Model for Crystallization Problem

Zehra Pınar 1, Hasret Gulec 2 and Huseyin Kocak 3

Abstract. Crystallization problem one of the popular problems in wide area of science. It is modeled

by the population balance model, which is one of the important models of mathematical biology and

engineering, is a nonlinear partial integro-differential equation and examines the exchange of particles

and the production of new particles in a system of particles. For the crystallization problem, the

considered population balance equation includes aggregation, nucleation and growth particles. For

aggregation kernel, three different cases are considered. The semi-analytical solutions are obtained via

the well-known Adomian decomposition method.

Keyword: Crystallization, population balance equation, Adomian decomposition method.

AMS 2010: 35R09, 35R10 .

References

[1] P. Marchal, R. David, J.P. Klein and J. Villermaux, Crystallization and precipitation engineering, An Efficient

Method for Solving Population Balance in Crystallization with Agglomeration, Chemical Engineering Science, 43,

59-67, 1988.

[2] D.Ramkrishna and M.R. Singh, Population balance modeling: current status and future prospects, Annu. Rev. Chem.

Biomol. Eng, 5, 123-146, 2014.

[3] D. Ramkrishna, Population balances: theory and applications to particulate systems in engineering, Academic Press,

355, New York, 2000.

[4] A.D. Randolph and M.A. Larson, Theory of particulate processes: analysis and techniques of continuous crystalliza-

tion, New York: Academic Press, 1971.

[5] R. Zauner and A.G. Jones, Determination of nucleation, growth, aggregation and disruption kinetics from experi-

mental precipitation data: the calcium oxalate system, Chemical Eng. Sci., 55, 4219-4232, 2000.

1Tekirdag Namik Kemal University, Tekirdag, Turkey,[email protected]

2Tekirdag Namik Kemal University, Tekirdag, Turkey, [email protected]


193

[6] A. Hasseine, A. Bellagoun and H.-J. Bart, Analytical solution of the droplet breakup equation by the adomian

decomposition method, Applied Mathematics and Computation, 218, 2249-2258, 2011.

194

DISCREATE MATHEMATICS

Disjunctive Total Domination Stability in Graphs

Canan Ciftci 1

Abstract. The concept of domination stability in graphs was introduced by Bauer et al. [1] and has

been studied, for instance, in [2]. With the thought that the definition of domination stability, stability

of some variations of domination is studied by some authors, see [3, 4, 5]. One of variations of domina-

tion is disjunctive total domination [6]. A set S of vertices in a graph G is a disjunctive total dominating

set of G if every vertex has a neighbor in S or has at least two vertices in S at distance 2 from it. The

disjunctive total domination number is the minimum cardinality of such a set. In this work, we study

on disjunctive total domination stability which is the minimum size of a non-isolating set of vertices

in G whose removal changes the disjunctive total domination number. We determine exact values of

some special classes of graphs. Moreover, we give some results on disjunctive total domination stability.

Keyword: Domination, disjunctive total domination, stability.

AMS 2010: 05C12, 05C69.

References

[1] D. Bauer, F. Harary, J. Nieminen and C.L. Sujel, Domination alteration sets in graphs, Discrete Mathematics, 47,

153-161, 1983.

[2] N.J. Rad, E. Sharifi and M. Krzywkowski, Domination stability in graphs, discrete mathematics, 339(7), 1909-1914,

2016.

[3] M.A. Henning and M. Krzywkowski, Total domination stability in graphs, Discrete Applied Mathematics, 236, 246-

255, 2018.

[4] W.J. Desormeaux, Total domination in graphs and graph modifications, PhD, University of Johannesburg, 2012.

[5] Z. Li, Z. Shao and S.J. Xu, 2-rainbow domination stability of graphs, Journal of Combinatorial Optimization, 1-10,

2019.

[6] M.A. Henning and V. Naicker, Disjunctive total domination in graphs, Journal of Combinatorial Optimization, 31(3),

1090-1110, 2016.

1Ordu University, Department of Mathematics, Ordu, Turkey, [email protected]

195

Disjunctive Total Domination Number of Central and Middle Graphs of Certain

Snake Graphs

Canan Ciftci 1, Aysun Aytac 2

Abstract. Domination is well-studied topic in graph theory [1]. There are many variations of domi-

nation one of which is disjunctive total domination [2]. A set S of vertices in a graph G is a disjunctive

total dominating set (DTD-set) of G if every vertex is adjacent to a vertex of S or has at least two

vertices in S at distance 2 from it. The disjunctive total domination number is the minimum car-

dinality of a DTD-set in G. Disjunctive total domination is studied on some graphs such as trees

[2, 3], claw-free graphs [2], grids [4], permutation graphs [5] and Harary graphs [6]. In this study, we

focus on disjunctive total domination number of central [7] and middle graphs [8]. We determine exact

values of disjunctive total domination number of central and middle graphs of triangular snake, double

triangular snake and diamond snake graphs.

Keyword: Distance, disjunctive total domination, central graph, middle graph, triangular snake, double

triangular snake, diamond snake.

AMS 2010: 05C12, 05C69, 05C76.

References

[1] T.W. Haynes, S. T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker Inc., New

York, 1998.

[2] M. A. Henning and V. Naicker, Disjunctive total domination in graphs, journal of combinatorial optimization, 31(3),

1090-1110, 2016.

[3] M. A. Henning and V. Naicker, Bounds on the disjunctive total domination number of a tree, Discussiones Mathe-

maticae Graph Theory, 36(1), 153-171, 2016.

[4] C. F. Lin, S. L. Peng and H. D. Yang, Disjunctive total domination numbers of grid graphs, International Computer

Symposium (ICS), IEEE, 80-83, 2016.

1Department of Mathematics, Ordu University, Ordu, Turkey, [email protected]

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

196

[5] E. Yi, Disjunctive total domination in permutation graphs, discrete mathematics, Algorithms and Applications, 9

(1), 1750009 (20 pages), 2017.

[6] C. Ciftci and A. Aytac, Disjunctive total domination in harary graphs, submitted, 2018.

[7] J. Vernold Vivin, Harmonious coloring of total graphs, n-leaf, Central Graphs and Circumdetic Graphs, PhD Thesis,

Bharathiar University, Coimbatore, India, 2007.

[8] D. Michalak, On middle and total graphs with coarseness number equal 1, Springer Verlag Graph Theory, Lagow

Proceedings, Berlin Heidelberg, New York, Tokyo, 13–150, 1981.

197

On the Spectrum of Threshold Graphs

Ebrahim Ghorbani 1

Abstract. A threshold graph is a graph that can be constructed from a one-vertex graph by repeated

addition of a single isolated vertex to the graph, or addition of a single vertex that is adjacent to all

other vertices. An equivalent definition is the following: a graph is a threshold graph if there are a

real number S and for each vertex v a real vertex weight w(v) such that two vertices u, v are adjacent

if and only if w(u) + w(v) > S. This justifies the name “threshold graph” as S is the threshold for

being adjacent. This talk deals with the eigenvalues of the adjacency matrices of threshold graphs. In

particular, eigenvalue-free intervals for threshold graphs will be discussed.

Keyword: Threshold graph, Spectrum.

AMS 2010: 05C50.

References

[1] C.O. Aguilar, J. Lee, E. Piato, and B.J. Schweitzer, Spectral characterizations of anti-regular graphs, Linear Algebra

Appl., 557, 84–104, 2018.

[2] M. Andelic and S.K. Simic, Some notes on the threshold graphs, Discrete Math., 310, 2241–2248, 2010.

[3] A. Brandstadt, V.B. Le, and J.P. Spinrad, Graph Classes: A Survey, Society for Industrial and Applied Mathematics

(SIAM), Philadelphia, PA, 1999.

[4] E. Ghorbani, Spectral properties of cographs and P5-free graphs, Linear Multilinear Algebra, 67, 1701–1710, 2019.

[5] F. Harary, The structure of threshold graphs, Riv. Mat. Sci. Econom. Social., 2, 169–172, 1979.

[6] N.V.R. Mahadev and U.N. Peled, Threshold Graphs and Related Topics, Annals of Discrete Mathematics, North

Holland Publishing Co., Amsterdam, 1995.

[7] P. Manca, On a simple characterisation of threshold graphs, Riv. Mat. Sci. Econom. Social., 2 , 3–8, 1979.

1Department of Mathematics, K. N. Toosi University of Technology, P. O. Box 16315-1618, Tehran, Iran and

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran,

e [email protected]

198

Vectoral Angle Distance for DNA k-mers

Elif Segah Oztas 1, Fatmanur Gursoy 2

Abstract. In this study, a new distance form is introduced for DNA k-mers instead of Hamming

distance etc. The new distance form measures the angle between DNA strands according to a convert-

ing method from DNA strings to vectors, which is defined by authors. We apply this vectoral angle

distance to DNA codes which are created by algebraic error correcting codes.

Acknowledgements: Computing resources used in this work were provided by the National Cen-

ter for High Performance Computing of Turkey (UHeM) under grant number 1006492019.

Keyword: DNA k-mers, distance

AMS 2010: 92D20, 94B05.

1Karamanoglu Mehmetbey University, Karaman, Turkey, [email protected]

2Yildiz Technical University, Istanbul, Turkey, [email protected]

199

Binary Linear Programming on Ramsey Graphs

S. Masih Ayat 1, Abbas Akrami 2 and S. Majid Ayat 3

Abstract. (s, t, n)-Ramsey graphs are simple n-vertex graphs without any s -cliques and t-independent

sets. In this paper, some extensions of (s, t, n)-graphs to (s, t, n + 1)-graphs by use of binary linear

programming are presented.

Keyword: Ramsey Graphs, Binary Linear Programming. .

AMS 2010: 05D10, 05C69, 90C09

References

[1] J. M. AlJaam, Experiments of intelligent algorithms on ramsey graphs, The International Arab Journal of Information

Technology, 2,161-167, 2007.

[2] G. Exoo, Some new computer generated results in ramsey graph theory, Computers and Mathematics with Applica-

tions, 15, 255-257, 1988.

[3] G. Exoo, On two classical ramsey numbers of the form r(3,n), SIAM Journal on Discrete mathematics, 2, 488-490,

1989.

[4] G. Exoo, Some new ramsey colorings, the electronic journal of combinatorics, 5, 1-5, 1998.

[5] B. D. McKay and S. P. Radziszowski, Linear programming in some ramsey problems, J. Comb. Theory, Ser. B, 61,

125-132, 1994.

[6] K. Piwakowski, Applying tabu search to determine new ramsey graphs, the electronic journal of combinatorics, 3,

1-4, 1996.

1University of Zabol, Zabol, Iran, [email protected]


3Isfahan University Of Technology, Isfahan, Iran

200

The Independence Number of Circulant Triangle-free Graphs

S. Masih Ayat 1, S. Majid Ayat 2 and Meysam Ghahramani 3

Abstract. The independence number of circulant triangle-free graphs for 2, 3-regular graphs are in-

vestigated. It is shown that the independence ratio of circulant triangle-free graphs for 3-regular graphs

is at least 3/8. Some class of r-regular circulant triangle-free graphs with independence number equal

to r are determined.

Keyword: Triangular-free Graphs, Circulant Graphs, Independence Number. .

AMS 2010: 05C69, 05C75..

References

[1] Bauer, D., Regular Kn-free graphs, J. Combin. Theory Ser. B, Journal of Combinatorial Theory. Series B,35, 193-200,

1983.

[2] Brandt, S., Triangle-free graphs whose independence number equals the degree, Discrete Math., 310, 666-669, 2010.

[3] Heckman, C., Thomas, C., A new proof of the independence ratio of triangle-free cubic graphs, Discrete Mathematics,

233, 233-237, 2001.

[4] Punnim, N., The clique numbers of regular graphs, Graphs and Combinatorics, 18, 781-785, 2002.

[5] Ringeisen, R. D., Roberts, F. S., Applications of discrete mathematics, Proceedings of the Third SIAM Conference on

Discrete Mathematics held at Clemson University, Clemson, South Carolina, May 14–16, 1986.Society for Industrial

and Applied Mathematics (SIAM),Philadelphia, PA,1988.

[6] Sidorenko, A. F., Triangle-free regular graphs, Discrete Math., 91, 215-217, 1991.

[7] Staton, W., Some Ramsey-type numbers and the independence ratio, Transaction of the American Mathematical

Society, 256, 353-370, 1979.


2Isfahan University Of Technology, Isfahan, Iran

3Shiraz University of Technology, Shiraz, Iran, [email protected]

201

GEOMETRY

A General Notion of Coherent Systems

Alexander H. W. Schmitt1

Abstract. Bradlow, Brambila-Paz, Garcıa-Prada, and Gothen suggested to study coherent systems

for Higgs bundles in order to get a better understanding of the geometry of moduli spaces of Higgs

bundles.

We will look at a wider class of coherent systems for decorated vector bundles and propose a notion of

semistability. In the special case of tensor powers, we will study this notion more closely by doing some

non-trivial constructions and computations in geometric invariant theory. It is an interesting aspect

that ampleness of the linearization in the geometric invariant theory construction yields a bound on

the stability parameter for coherent systems.

Our work builds on the papers [1] and [2] by King/Newstead and Le Potier, respectively, which contain

the fundamentals of the theory of classical coherent systems, relevant, e.g., for Brill–Noether theory of

vector bundles, and general techniques for dealing with decorated vector bundles from [3].

Keywords: Coherent system, moduli space, geometric invariant theory, semistability, linearization,

ampleness.

AMS 2010: 14H60, 14D20, 14L24.

References

[1] A.D. King, P.E. Newstead, Moduli of Brill–Noether pairs on algebraic curves, Internat. J. Math. 6, 733-48, 1995.

[2] J. Le Potier, Systemes coherents et structures de niveau, Asterisque, vol. 214, Societe Mathematique de France, Paris,

143 pp, 1993.

[3] A.H.W. Schmitt, Geometric invariant theory and decorated principal bundles, Zurich Lectures in Advanced Mathe-

matics, European Mathematical Society, Zurich, vii+389pp, 2008.

1Freie Universitat Berlin, Germany, [email protected]

202

Bezier-like Curves Based on Exponential Functions

Ayse Yılmaz Ceylan 1

Abstract. In this work we construct a new Bezier-like basis in the space Γn = span1, t, t2, ...,

tn−2, et, e−t by an integral approach and define Bezier-like curves based on this basis. We then compare

these basis and curves with the Bernstein basis and the Bezier curves in polynomial spaces respectively.

Keyword: Bezier-like basis function, Bezier-like curve, Bernstein basis, Bezier curve.

AMS 2010: 65D07, 65D17.

References

[1] J.W. Zhang, C-curves: An extension of cubic curves, Computer Aided Geometric Design, 13, 199-217, 1996.

[2] J. Sanchez-Reyes, Harmonic rational Bezier curves, p-Bezier curves and trigonometric polynomials. Computer Aided

Geometric Design 15, 909-923, 1998.

[3] E. Mainer, J.M. Pena and J. Sanchez-Reyes, Shape preserving alternatives to the rational Bezier model, Computer

Aided Geometric Design, 18, 37-60, 2001.

[4] Q.Y. Chen and G.Z. Wang, A class of Bezier-like curves, Computer Aided Geometric Desing, 20, 29-39, 2003.

[5] Y. Wang, The Theory and Application of H-Bezier Curves, M.S. Thesis, Northwest University, 2006.

[6] H. Zhu and J. Tan, Bezier-like curves based on algebraic and exponential polynomials and their connection conditions,

J. Inf. Comput. Sci., 9, 1499-1510, 2012.

[7] Y. Zhu and X. Han, Curves and surfaces construction based on new basis with exponential fuctions, Acta Appl.

Math., 129, 183-203, 2014.

1Akdeniz University, Antalya, Turkey, [email protected]

203

Geometry of Elastic Submanifolds in Trans-Sasakian Manifolds

Azime Cetinkaya 1

Abstract. In this article, firstly we define elastic submanifolds in Trans-Sasakian manifolds. We

investigate some properties of this type of submanifold using a special quarter symmetric non-metric

connection, and finally we give some important examples.

Keyword: elastic submanifold, Laplace, quarter symmetric non-metric connection, Trans-Sasakian

manifold.

AMS 2010: 53C15, 53C40.

References

[1] A.Rosso, A. K. Hartmann, and W. Krauth, Depinning of elastic manifolds, Physical Review E 67, 021602,2003.

[2] D.A.Singer, Lectures on Elastic Curves and Rods, AIP Conference Proceedings 1002, 3, 2008.

[3] D.E. Blair,and J.A. Oubina, Conformal and related changes of metric on the product of

two almost contact metric manifolds, Publ. Mat.,34(1),1990.

[4] G. Ozkan, and A. Yucesan, Relaxed elastic line in a Riemannian manifold, Turk J. Math., 38, 746-752, 2014.

[5] K. Yano, and M. Kon, Structures on Manifolds,Series in Pure Mathematics, Volume 3, World Scientic

Publishing Corp., Singapore, 1984.

[6] M.P. Carmo, Differential geometry of curves and surfaces, Prentice Hall, Englewood Cliffs., 1976.

[7] S. Golab, On semi-symmetric and quarter-symmetric linear connections, Tensor, N. S., 29, 249-254, 1975.

1Piri Reis University, Istanbul, Turkey, [email protected]

204

A New Version of Q-Surface Pencil in Euclidean 3-Space

Aziz Yazla 1, Muhammed Talat Sarıaydın 2

Abstract. In this paper, the q-surface pencil is studied in Euclidean 3-space. By using q-Frame

in Euclidean space, we give the necessary and sufficient condition for a q-surface pencil. Finally, we

construct the corresponding surfaces which possessing some representative curves as lines of curvature.

Keyword: The line of curvature, q-frame, Surface Pencil.

AMS 2010: 53A05.

References

[1] F. Akbulut, Darboux vectors of a curve on the surface, Ege Universitesi Fen . Fak. Izmir, 1983.

[2] A. Akutagawa, S. Nishikawa, The gauss map and spacelike surface with prescribed mean curvature in minkowski

3-space, Tohoku Math. J., 42, 67-82, 1990.

[3] M.P. Do Carmo, Differential geometry of curves and surfaces, Englewood Cliffs, Prentice Hall, 1976.

[4] F. Chen, J. Zheng, T.W. Sederberg, The mu-basis of a rational ruled surface, Comp. Aid. Des., 18, 61-72, 2001.

[5] F.C. Park, J. Yu, C. Chun, B. Ravani, Design of developable surfaces using optimal control, ASME J. Mech. Design,

124(4), 602-608, 2002.

[6] M.K. Saad, H.S. Abdel-Aziz, G. Weiss, M. Solimman, Relations among darboux frames of null bertrand curves in

pseudo-euclidean space, 1st Int. WLGK11, Paphos, 25-30, 2011.

[7] G.J. Wanga, K. Tangb, C.L. Taic, Parametric representation of a surface pencil with a common spatial geodesic,

Computer-Aided Design, 5(36), 447-459, 2004.



205

New Methods to Construct Slant Helices from Hyperspherical Curves

Bulent Altunkaya 1

Abstract. In this study, we give methods to construct slant helices from arclength parameterized

hyperspherical curves. By means of these methods, we construct slant helices and Salkowski curves

that lie on 2n-dimensional hyperboloid. We also construct rectifying slant helices which are geodesics

of 2n-dimensional cone.

Keyword: Slant helix, Salkowski curve, rectifying curve, geodesic of a hypersurface.

AMS 2010: 53A04, 53C40.

References

[1] T. A. Ahmad, M. Turgut, Some characterizations of slant helices in the Euclidean space En, Hac. J. Math. Sta. 39,

327-336, 2010.

[2] K. Arslan, Y. Celik, C. Deszcz, C Ozgur, Submanifolds all of whose normal sections are W-curves, Far East J. Math.

Sci. 5, 537-544, 1997.

[3] B. Altunkaya, F.K. Aksoyak, L. Kula, C. Aytekin, On rectifying slant helices in Euclidean 3-space, Kon. J. Math. 4,

17-24, 2016.

[4] B. Altunkaya, L Kula, General helices that lie on the sphere S2n in Euclidean space E2n+1, Uni. J. Math. App. 1,

166-170, 2018.

[5] C. Camci, K. Ilarslan, L Kula, H.H. Hacisalihoglu, Harmonic cuvature and general helices, Chaos Solitons Fractals.

40, 2590-2596, 2009.

[6] S. Cambie, W. Goemans, I. Van Den Bussche, Rectifying curves in the n-dimensional Euclidean space, Turkish J.

Math. 40, 210-223, 2016.

[7] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly.

110, 147-152, 2003.

[8] B. Y. Chen, F. Dillen, Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Aca. Sinica. 33, 77-90,

2005.

[9] B. Y. Chen, Differential geometry of rectifying submanifolds, Int. Elec. J. Geo. 9, 1-8, 2016.

1Ahi Evran University, Kırsehir, Turkey, [email protected]

206

[10] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48, 209-214, 2017.

[11] S. Deshmukh, B.Y. Chen, S.H. Alshammari, On rectifying curves in Euclidean 3-space. Turkish J. Math. 42, 609-620,

2017.

[12] H. Gluck, Higher curvatures of curves in Euclidean space, Amer. Math. Monthly. 73, 699-704, 1966.

[13] K. Ilarslan, E. Nesovic, Some characterizations of rectifying curves in Euclidean space E4, Turkish J. Math. 32,

21-30, 2008.

[14] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish J. Math. 28, 153-163, 2004.

[15] S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom. 74, 97-109, 2002.

[16] L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, App. Math. Comp. 169, 600-607, 2005.

[17] L. Kula, N. Ekmekci, Y. Yaylı, K. Ilarslan, Characterizations of slant helices in Euclidean 3-space, Turkish J. Math.

34, 261-273, 2010.

[18] P. Lucas, J.A. Ortega-Yagues, Rectifying curves in the three-dimensional sphere, J. Math. Anal. Appl. 421, 1855-

1868, 2015.

[19] J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Com.

Aided Geo. Design. 26(3), 271-278, 2009.

[20] B. O’Neill, Elementary Differential Geometry, Academic Press, London, 2006.

[21] E. Salkowski, Zur transformation von raumkurven, Mathematische Annalen. 66(4), 517-557, 1909.

[22] Y. Yayli, E. Ziplar, On slant helices and general helices in Euclidean n-space, Mathematica Aeterna. 1, 599-610,

2010.

[23] Y. Yayli, I. Gok, H.H. Hacisalihoglu, Extended rectifying curves as new kind of modified Darboux vectors, TWMS.

J. Pure Appl. Math. 9, 18-31, 2018.

207

Helix Preserving Mappings

Bulent Altunkaya 1, Levent Kula 2

Abstract. In this work, we define mappings that preserve general helices in Euclidean spaces and

we give some characterizations about them. Furthermore, we find invariants of these mappings. In

addition, we generate polynomial, rational, conical, ellipsoidal, hyperboloidal helices by using these

mappings.

Keyword: Curvatures, helix, map, invariant, surface of revolutions.

AMS 2010: 53A04,53A05,58C25

References

[1] B. Altunkaya, L. Kula, On polynomial general helices in n-dimensional Euclidean space Rn, Adv. Appl. Clifford

Algebras, 28:4, 2018.

[2] C. Camci, K. Ilarslan, L. Kula, and H. H. Hacisalihoglu, Harmonic cuvature and general helices, Chaos Solitons &

Fractals, 40, 2590-2596, 2009.

[3] R. T. Farouk, T. Sakkalis, Pythagorean hodographs, IBM J. Res. D. 34, 736-752, 1990.

[4] H. W. Guggenheimer, Differential Geometry, Dover, New-York, 1977.

[5] G. Kim, S. Lee, Pythagorean-hodograph preserving mappings, J. Comput. Appl. Math., 216, 217-226, 2008.

[6] J. H. Kong, S. Lee, G. Kim, Minkowski Pythagorean-hodograph preserving mappings, J. Comput. Appl. Math., 308,

166-176, 2016.

[7] D. J. Struik, Lectures on classical differential geometry, Dover, New-York, 1988.



208

Constraint Manifolds for Some Spatial Mechanisms in Lorentz Space

Busra Aktas 1, Olgun Durmaz 2 and Halit Gundogan 3

Abstract. A parameterized curves, surface or hypersurface in the image space which is related to the

degree of freedom of the chain is called as the constraint manifold of the open chain. Geometrically, the

constraint that is being imposed on the positions of the last link by the rest of the chain is presented

by it. In this paper, we represent the constraint manifolds of 2C and 3C spatial open chains by using

the form of planar open chains in Lorentz plane.

Keyword: Spatial Open Chain, Constraint Manifolds, Split Quaternion.

AMS 2010: Firstly 70B15,76E07, Secondly 93B17.

References

[1] E.T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, Cambridge, Cambridge University

Press, 1994.

[2] R.C. Hibbeler, Kinematics and kinetics of a particle, Engineering Mechanics:Dynamics, Jurong, Sigapure: Prentice

Hall, 2009.

[3] A.C. Gozutok, S. OzkaldıKarakus and H. Gundogan, Conics and quadrics in lorentz space, Math Sci Apply E-Notes,6,

58-63, 2018.

[4] P.P. Teodorescu, Kinematics, Mechanical systems, classical models: particle mechanics, Dordrecht, Springer, 2007.

[5] J.M. McCharthy, An introduction to theoretical kinemastics, The MIT Pres, Cambridge, 1990.

[6] R.G. Ratcliffe, Foundations of hyperbolic manifolds, Springer-Verlag, New York, 1994.

[7] R. Lopez, Differential geometry of curves and surfaces in lorentz-minkowski space, arXiv: 0810.335lvl [Math.DG],

2008.

[8] S. Ozkaldı and H. Gundogan, Cayley Formula, Euler parameters and rotations in 3-dimensional lorentzian space,

Adv. Appl. Clifford Alg. 20, 367-377, 2010.


2Ataturk University, Erzurum, Turkey, [email protected]


209

[9] D. Knossow, R. Ronfard and R. Horaud, Human motion tracking with a kinematic parametrization of extremal

contours, Int J. Comput Vision, Springer-Verlag, 79, 247-269, 2008.

210

S-Manifolds and Their Slant Curves of Certain Types

Cihan Ozgur 1, Saban Guvenc 2

Abstract. We define and study C-parallel and C-proper slant curves of S-manifolds. We prove that

a curve γ in an S-manifold of order r ≥ 3, under certain conditions, is C-parallel or C-parallel in the

normal bundle if and only if it is a non-Legendre slant helix or Legendre helix, respectively. Moreover,

under certain conditions, we show that γ is C-proper or C-proper in the normal bundle if and only if

it is a non-Legendre slant curve or Legendre curve, respectively. We also give two examples of such

curves in R2m+s(−3s).

This talk is supported by Balıkesir University Scientific Research Project numbered 2018/016.

Keyword: C-parallel curve, C-proper curve, slant curve, S-manifold.

AMS 2010: 53C25, 53C40, 53A04.

References

[1] J. Arroyo, M. Barros, O. J. Garay, A characterization of helices and Cornu spirals in real space forms, Bull. Austral.

Math. Soc. 56, 37–49, 1997.

[2] C. Baikoussis, D. E. Blair, Integral surfaces of Sasakian space forms, J. Geom. 43, 30–40, 1992.

[3] B.-Y. Chen, Null 2-type surfaces in Euclidean space, Algebra, analysis and geometry (Taipei, 1988), 1–18, World Sci.

Publ., Teaneck, NJ, 1989.

[4] J. T. Cho, J. Inoguchi, J.-E. Lee, On slant curves in Sasakian 3-manifolds, Bull. Austral. Math. Soc. 74, 359–367,

2006.

[5] S. Guvenc, C. Ozgur, On slant curves in trans-Sasakian manifolds, Rev. Un. Mat. Argentina 55, 81–100, 2014.

[6] J. Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math. 100,

163–179, 2004.

[7] J.-E. Lee, Y. J. Suh, H. Lee, C-parallel mean curvature vector fields along slant curves in Sasakian 3-manifolds,

Kyungpook Math. J. 52, 49–59, 2012.

1Department of Mathematics, Balıkesir University, Balıkesir, Turkey, [email protected]

2Department of Mathematics, Balıkesir University, Balıkesir, Turkey, [email protected]

211

On The Directional Evolutions of the Ruled Surfaces depend on A Timelike Space

Curve

Cumali Ekici 1, Mustafa Dede 2 and Gul Ugur Kaymanlı 3

Abstract. In this paper, we work on the directional evolutions of the ruled surfaces generated by the

quasi normal and quasi binormal vector fields of timelike space curve in Minkowski 3-space by using

q-frame. Evolutions of both quasi normal and quasi binormal ruled surfaces are investigated by using

their directrices. Then some examples are constructed and plotted.

Keyword: Evolution surface, Minkowski space, q-frame, Ruled surface.

AMS 2010: 14J26, 51B20, 57R25.

References

[1] H. N. Abd-Ellah, Evolution of translation Surfaces in Euclidean 3 space E3, Applied Mathematics and Information

Science. 9(2), 661-668, 2015.

[2] K. Akutagawa and S. Nishikawa, The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski

3-space, Tohoku Math. J. 42(2): 67-82, 1990.

[3] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82: 246–251, 1975.

[4] J. Bloomenthal, Calculation of Reference Frames Along a Space Curve, Graphics gems, Academic Press Professional

Inc., San Diego, CA, 1990.

[5] S. Coquillart, Computing offsets of B-spline curves, Computer-Aided Design, 19(6): 305-09, 1987.

[6] M. Dede, C. Ekici and A. Gorgulu, Directional q-frame along a space curve, IJARCSSE. 5(12), 775-780, 2015.

[7] M. Dede, C. Ekici, H. Tozak, Directional tubular surfaces, International Journal of Algebra. 9(12), 527-535, 2015.

[8] M. Dede, G. Tarım and C. Ekici, Timelike Directional Bertrand Curves in Minkowski Space, 15th International

Geometry Symposium, Amasya, Turkey 2017.

[9] C. Ekici, M. Dede, H. Tozak, Timelike directional tubular surfaces, Int. J. Mathematical Anal., 8(5), 1-11, 2017.

[10] H. Guggenheimer, Computing frames along a trajectory, Comput. Aided Geom. Des., 6: 77-78, 1989.

1Eskisehir Osmangazi University, Eskisehir, Turkey, [email protected]

2Kilis 7 Aralık University, Kilis, Turkey, [email protected]

3Cankırı Karatekin University, Cankırı, Turkey, [email protected]

212

[11] R. A. Hussien and T. Youssef, Evolution of Special Ruled Surfaces via the Evolution of Their Directrices in Euclidean

3-Space E3, Applied Mathematics and Information Science. 10, 1949-1956, 2016.

[12] G. U. Kaymanlı, C. Ekici and M. Dede, Directional canal surfaces in E3, Current Academic Studies in Natural

Sciences and Mathematics Sciences, 63-80, 2018.

[13] B. O‘Neill, Semi-Riemannian geometry with applications to relativity. Academic Press, New York, 1983.

[14] M. A. Soliman, N. H. Abdel-All, R.A. Hussien and T. Youssef, Evolutions of the Ruled Surface via the Evolution of

Their Directrix Using Quasi Frame along a Space Curve, Journal of Applied Mathematics and Physics. 6, 1748-1756,

2018.

[15] W. Wang, B. Juttler, D. Zheng and Y. Liu, Computation of rotation minimizing frame, ACM Trans. Graph, 27 (1)

(2008), 18 pages.

213

On The Directional Associated Curves of Timelike Space Curve

Gul Ugur Kaymanlı1, Cumali Ekici 2 and Mustafa Dede 3

Abstract. In this work, the directional associated curves of timelike space curve in Minkowski 3-space

by using q-frame are studied. We inverstigate quasi normal and quasi binormal direction and donor

curves of the timelike curve with q-frame. Finally, some new associated curves are constructed and

plotted.

Keyword: Associated curves, Minkowski space, q-frame.

AMS 2010: 14H50, 51B20, 57R25.

References


3-space, Tohoku Math. J. 42(2): 67-82, 1990.

[2] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82: 246–251, 1975.

[3] J. Bloomenthal, Calculation of Reference Frames Along a Space Curve, Graphics gems, Academic Press Professional

Inc., San Diego, CA, 1990.

[4] J. H. Choi, Y. H. Kim, A. T. Ali, Some associated curves of Frenet non-lightlike curves in E31. J Math Anal Appl.,

394, 712-723, 2012.

[5] S. Coquillart, Computing offsets of B-spline curves, Computer-Aided Design, 19(6): 305-09, 1987.




[8] C. Ekici, M. Dede, H. Tozak, Timelike directional tubular surfaces, Int. J. Math. Anal., 8(5), 1-11, 2017.

[9] H. Guggenheimer, Computing frames along a trajectory, Comput. Aided Geom. Des., 6: 77-78, 1989.

[10] G. U. Kaymanlı, C. Ekici and M. Dede, Directional canal surfaces in E3, Current Academic Studies in Natural

Sciences and Mathematics Sciences, 63-80, 2018.




214

[11] T. Korpınar, M. T. Sarıaydın and E. Turhan, Associated Curves According to Bishop Frame in Euclidean 3-space,

AMO. 15, 713-717, 2013.

[12] R. Lopez, Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int Elect Journ Geom, 3(2),

67-101, 2010.

[13] N. Macit and M. Duldul, Some New Associated curves of a Frenet Curve in E3 and E4, Turkish Journal of Mathe-

matics. 38, 1023-1037, 2014.


[15] Y. Unluturk, S. Yılmaz, M. Cimdiker, S. Simsek, Associated curves of non-lightlike curves due to the Bishop frame

of type-1 in Minkowski 3-space, Advanced Modeling and Optimization, 20(1), 313-327, 2018.

[16] Y. Unluturk and S. Yılmaz, Associated Curves of the Spacelike Curve via the Bishop Frame of type-2 in E31, Journal

of Mahani Mathematical Research Center. 8(1-2), 1-12, 2019.

[17] S. Yılmaz, Characterizations of Some Associated and Special Curves to Type-2 Bishop Frame in E3, Kirklareli

University Journal of Engineering and Science. 1, 66-77, 2015.

215

On Some Curvature Conditions of Nearly α−Cosymplectic Manifolds

Gulhan Ayar 1, Dilek Demirhan 2

Abstract. In the present study, we have focused on nearly alpha cosymplectic manifolds. After

defining nearly α−cosymplectic manifolds, we have tried to show certain curvature conditions and ba-

sic properties of nearly α−cosymplectic manifolds.

Keyword: Nearly α−Cosymplectic Manifolds, Cosymplectic Manifolds.

AMS 2010: 53D10, 53D15, 53D25.

References

[1] A. D. Nicola, G. Dilo and I. Yudin, On nearly sasakian and nearly cosymplectic manifolds, arXiv:1603.09209v2

[math.DG] 26 May 2017.

[2] D. G. Prakasha and B.S. Hadimani, -ricci solitons on lorentzian para-sasakian manifolds, Journal of Geometry, DOI

10.1007/s00022-016-0345-z.

[3] V. A. Khan, M. A. Khan and S. Uddin, Totally umbilical semi-invariant submanifolds of a nearly kenmotsu manifold,

Soochow Journal Of Mathematics, Volume 33, No. 4, pp. 563-568, October 2007

[4] J. Kim , X. Liu and M. M.Tripathi, On semi-invariant submanifolds of nearly trans-sasakian manifolds, Int. J. Pure

& Appl. Math. Sci. Vol. 1(2004), pp. 15-34.

[5] I. K. Erken, P. Dacko and C. Murathan, On the existence of proper nearly kenmotsu manifolds, Mediterr. J. Math.

13 (2016), 4497-4507 ,DOI 10.1007/s00009-016-0758-9,1660-5446/16/064497-11,June 24, 2016

[6] H. Endo, On the curvature tensor of nearly cosymplectic manifolds of constant Φ-sectional curvature, Analele Stiin-

tifice ale Universitatii Ovidius Constanta, f2, 2005.



216

The Homogeneous Lift of A Riemannian Metric in The Linear Coframe Bundle

Habil Fattayev 1

Abstract. It is well - known that the research connected with Sasaki metric in the fiber bundles one

of the basic foundations of the modern differential geometry [1], [2], [3]. However, the Sasaki metric

is non homogeneous in the fibers of above mentioned bundles and is not used for study the global

properties of these bundles. In [4], Miron introduced a homogeneous lift G of a Riemannian metric

g to slit tangent bundle which together with the natural almost complex structure F give rise to a

conformal Kahlerian structure on slit tangent bundle. Similar homogeneous lift of the Riemannian

metric g and the associated almost complex structure in the slit cotangent bundle were investigated by

Stavre and Popescu [4]. The homogeneous lift of a Riemannian metric g to slit tensor bundle of type

(1, 1) and the curvature properties of the Levi-Civita connection of this metric are studied in [6].

This report is devoted to the investigation of homogeneous lift of a Riemannian metric in the coframe

bundle. Let Mn be an n−dimensional Riemannian manifold of class C∞, and

F ∗(Mn) = (x, u∗ ) |x ∈M n , u∗ : basis (coframe) for T ∗x (Mn)

be the linear coframe bundle over Mn. Firstly we define the homogeneous lift g of a Riemannian metric

g to the linear coframe bundle F ∗(Mn) as follows:

g = gijdxi ⊗ dxj + 1

hδαβg

ijδXαi ⊗ δXβ

j ,

where h is a function defined by

h =

n∑α=1

‖Xα‖2 =

n∑α=1

gijXαi X

αj =

n∑α=1

g−1(Xα, Xα).

Also we study the Levi-Civita connection of homogeneous lift g.

Keyword: Riemannian metric, coframe bundle, homogeneous lift, Levi-Civita connection.

AMS 2010: 55R10, 53C07, 53C15.


217

References

[1] O.Kowalski and M.Sekizawa, On curvatures of linear frame bundle with naturally lifted metrics, Rend. Sem. Mat.

Univ. Pol. Torino, (3) 63, 283-296, 2005.

[2] A.A.Salimov and A.Filiz, Some properties of Sasakian metrics in cotangent bundles, Mediterr. J. Math., 8, 243-255,

2011.

[3] H.D.Fattayev and A.A.Salimov, Diagonal lifts of metrics to coframe bundle, Proc. of IMM of NAS of Azerbaijan, (2)

44, 328-337, 2018.

[4] R.Miron, The homogeneous lift of a Riemannian metric, An. St. Univ. ”Al. I. Cuza” Iasi, 46, 73-81, 2000.

[5] P.Stavre and L.Popescu, The homogeneous lift on the cotangent bundle, Novi Sad J. Math., (2)32, 1-7, 2002.

[6] E.Peyghan, H.Nasrabadi and A.Tayebi, The homogeneous lift to the tensor bundle of a Riemannian metric, Int. J.

of Geom. Meth. in Modern Physics, (4) 10, 1-19, 2013.

218

On The Generalized Taxicab Trigonometry

Harun Barıs Colakoglu 1

Abstract. In this study, we define generalized taxicab trigonometric functions as the natural gen-

eralized taxicab versions of the trigonometric functions of the Euclidean plane, using the generalized

taxicab radian notion which is also the natural generalized taxicab version of the radian notion of the

Euclidean plane. Then, we give two simple applications which determine changes in the Euclidean

and the generalized taxicab lengths of a line segment under the generalized taxicab and the Euclidean

rotations, respectively.

AMS 2010: 51K05, 51K99, 51N99.

References

[1] Z. Akca and R. Kaya, On the taxicab trigonometry, Jour. of Inst. of Math. & Comp. Sci. (Math. Ser.), Vol. 10(3),

151-159, 1997.

[2] R. Brisbin and P. Artola, Taxicab trigonometry, Pi Mu Epsilon Journal, Vol. 8, 89-95, 1984.

[3] H.B. Colakoglu and R. Kaya, A generalization of some well-known distances and related isometries, Math. Commun.,

Vol. 16, 21-35, 2011.

[4] H.B. Colakoglu, The generalized taxicab group, Int. Electron. J. Geom. Vol. 11, 83-89, 2018.

[5] E. Ekmekci, A. Bayar and A.K. Altıntas, On the group of isometries of the generalized taxicab plane, International

Journal of Contemporary Mathematical Sciences, Vol. 10(4), 159-166, 2015.

[6] E. Ekmekci, Z. Akca and A.K. Altıntas, On trigonometric functions and norm in the generalized taxicab metric,

Mathematical Sciences And Applications E-Notes, Vol. 3(2), 27-33, 2015.

[7] M. Ozcan, S. Ekmekci and A. Bayar, A note on the variation of the taxicab lengths under rotations, Pi Mu Epsilon

Journal, Vol. 11(7), 381-384, 2002.

[8] K.P. Thompson and T. Dray, Taxicab angles and trigonometry, The Pi Mu Epsilon Journal, Vol. 11(2), 87-96, 2000.

1Akdeniz University, Vocational School of Technical Sciences, Department of Computer Technologies, 07070,

Konyaaltı, Antalya, TURKIYE, [email protected].

219

The Transformation of the Involute Curves using by Lifts on R3 to Tangent Space

TR3

Hasim Cayır 1

Abstract. ”How we can speak about the features of involute curve on space TR3 by looking at the

characteristics of the first curve α?” In this paper, we investigate the answer of this question using by

lifts. In this direction firstly, we define the involute curve of any curve with respect to the vertical,

complete and horizontal lifts on space R3 to its tangent space TR3 = R6. Secondly, we examine the

Frenet-Serret aparatus T ∗(s), N∗(s), B∗(s), κ∗, τ∗ and the unit Darboux vector D∗ of the involute

curve α∗ according to the vertical, complete and horizontal lifts on TR3 depending on the lifting of

Frenet-Serret aparatus T (s), N(s), B(s), κ, τ of the first curve α on space R3. In addition, we include

all special cases the curvature κ∗(s) and torsion τ∗(s) of the Frenet-Serret aparatus of the involute

curve α∗ with respect to lifts on space R3 to its tangent space TR3. As a result of this transformation

on space R3 to its tangent space TR3, we could have some information about the features of involute

curve of any curve on space TR3 by looking at the characteristics of the first curve α. Moreover, we

get the transformation of the involute curves using by lifts on R3 to tangent space TR3. Finally, some

examples are given for each curve transformation to validated our theorical claims.

Keyword: Vector fields, involute curve, vertical lift, complete lift, horizontal lift, tangent space.

AMS 2010: 28A51,53A04,57R25

References

[1] M. Bilici and M. Calıskan, Some characterizations for the pair of involute-evolute curves is Euclidian E3, Bulletin of

Pure and Applied Sciences, vol.21E(2), pp.289-294, 2002.

[2] S. Gur, S. Senyurt, Frenet Vectors and Geodesic Curvatures of Spheric Indicators Of Salkowski Curves in E3, Hadronic

Journal, vol.33, no.5, pp. 485-512, 2010

[3] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC

Press, 205, 1997.

1Department of Mathematics, Faculty of Arts and Sciences,

Giresun University, 28100, Giresun, Turkey, e-mail: [email protected]

220

[4] H.H. Hacısalihoglu, Differential Geometry(in Turkish), vol.1, Inonu University Publications, 1994.

[5] Izumiya, S., Takeuchi, N., Special curves and Ruled surfaces, Beitrage zur Algebra und Geometrie Contributions to

Algebra and Geometry,44(1), 203-212, 2003.

[6] A. A. Salimov, Tensor Operators and Their applications, Nova Science Publ., New York, 2013.

[7] A. A. Salimov, H. Cayır, Some Notes On Almost Paracontact Structures, Comptes Rendus de 1’Acedemie Bulgare

Des Sciences, 66 (3),331-338, 2013.

[8] B. Senoussi, M. Bekkar, Characterization of General Helix in the 3− Dimensional Lorentz-Heisenberg Space, Inter-

national Electronic Journal of Geometry, 6(1), pp. 46-55, 2013

[9] S. Senyurt, O. F. Calıskan, The Natural Lift Curves and Geodesic Curvatures of the Spherical of the Timelike

Bertrand Curve Couple, International Electronic Journal of Geometry, 6(2), 88-99, 2013.

[10] M. Tekkoyun, Lifting Frenet Formulas, arXiv:0902.3567v1[math-ph] 20 Feb 2009.

[11] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker Inc., New York, 1973.

221

Euler-Lagrangian Dynamical Systems with Respect to Horizontal and Vertical

Lifts on Tangent Bundle

Hasim Cayır 1

Abstract. The differential geometry and mahthematical physics has lots of applications. The Euler-

Lagrangian mechanics are very important tools for differential geometry, classical and analytical ma-

chanics. There are many studies about Euler-Lagrangian dynamics, mechanics, formalisms, systems

and equations. The classic mechanics firstly introduced by J. L. Lagrange in 1788. Because of the

investigation of tensorial structures on manifolds and extension by using the lifts to the tangent or

cotangent bundle, it is possible to generalize to differentiable structures on any space (resp. manifold)

to extended spaces (resp. extended manifolds) [5, 6, 9]. In this study, the Euler-Lagrangian theories,

which are mathematical models of mechanical systems are structured on the horizontal and the vertical

lifts of an almost complex structure in tangent bundle TM. In the end, the geometrical and physical

results related to Euler-Lagrangian dynamical systems are concluded.

Keyword: Euler-Lagrangian equations, Dynamical Systems, Horizontal Lift, Vertical Lift, Tangent

Bundle.

AMS 2010: 15A72, 53A45, 53C15, 53D05, 70H03, 34N05

References

[1] R. Abraham, J.E. Marsden ant T. Ratiu, Manifolds tensor analysis and applications, Springer, 2001.

[2] M. de Leon and P.R. Rodrigues, Methods of differential geometry in analytical mechanics, Elsevier Sc. Pub. Com.

Inc., 1989.

[3] Z. Kasap, Weyl-mechanical systems on tangent manifolds of constant W-sectional curvature, IJGMMP, 10(10), 1-13,

2013.

[4] J. Klein, Escapes varialionnels et mecanique, Ann. Inst. Fourier, Gronoble, 12, 1962.

[5] A. A. Salimov, Tensor operators and their applications, Nova Science Publ., New York, 2013.

1Department of Mathematics, Faculty of Arts and Sciences,

Giresun University, 28100, Giresun, Turkey, e-mail: [email protected]

222

[6] S. Sasaki, On the diferential geometry of tangent bundles of Riemannian manifolds, Tohoku Math J, 10, 338-358,

1958.

[7] M. Tekkoyun, On para-Euler-Lagrange and para-Hamilton equations, Phys. Lett. A, 340, 7-11, 2005.

[8] M. Tekkoyun, Mechanical systems on manifolds, Geometry Balkan Press, Bucharest, Romania, 2014.

[9] K. Yano and S. Ishihara, Tangent and cotangent bundles, New York, NY, USA: Marcel Dekker, 1973.

223

Some Results on Null W -curves in E42

Hatice Altın Erdem1, Kazım Ilarslan2

Abstract. It is well known that a curve γ is called a W -curve or curve with constant curvature, if it

has constant Frenet curvatures (in R3, W-curves are circular helices). W -curves are the orbits of the

instantaneous space motions. In this study, we classify all null W -curves in E42, 4-dimensional semi-

Euclidean space with indeks 2. Since all three curvatures k1, k2 and k3 are constant, the classification

is reduced mainly to differential equations with constant coefficients in E42. We also give some examples.

Keyword: W -Curve, null curves, curvatures, Frenet equations, semi-Euclidean space.

AMS 2010: 53A04.

References

[1] H. Altın Erdem, C. Unal, K. Ilarslan and N. Kılıc Aslan, Non-null W -curves with non-null normals in E42 , Submited,

2018.

[2] K. Arslan, Y. Celik and H. H. Hacısalihoglu, On harmonic curvatures of a Frenet curve, Common. Fac. Sci. Univ.

Ank. Series AV 1, 49, 15-23, 2000.

[3] K. Ilarslan, O. Boyacıoglu, Position vectors of spacelike W -curves in Minkowski space E42 , Bull. Korean Math. Soc.,

44, 3, 429-438, 2007.

[4] F. Klein, S. Lie, Aber diejenigen ebenen Curven welche durch ein geschlossenes system von einfach unendlich vielen

vertauschbaren linearen Transformationen in sich Abergeben, Math. Ann., 4, 50-84, 1871.

[5] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, Braunschweig, Wiesbaden, 1999.

[6] M. Petrovic-Torgasev, E. Sucurovic, W -curves in Minkowski space-time, Novi Sad. J. Math., 2(32), 55-65, 2002.

1Kırıkkale University, Kırıkkale, Turkiye, hatice [email protected]

2Kırıkkale University, Kırıkkale, Turkiye, [email protected]

224

Algorithm for Solving the Sylvester s-Conjugate Elliptic Quaternion Matrix

Equations

Murat Tosun 1, Hidayet Huda Kosal 2

Abstract. In this study, the existence of the solution to Sylvester s-conjugate elliptic quaternion

matrix equations is characterized and the solution is obtained in an explicit form by means of real

representation of an elliptic quaternion matrix. Moreover, a pseudo-code for our method is presented.

Actually, Sylvester conjugate matrix equation over the complex field form a special class of Sylvester

s− conjugate elliptic quaternion matrix equations. Thus, the obtained results extend, generalize and

complement the scope of Sylvester conjugate matrix equations known in the literature.

Keyword: Elliptic quaternions and their matrices, Real matrix representations, Sylvester conjugate

matrix equation.

References

[1] M. Dehghan and M. Hajarian, Efficient iterative method for solving the second-order sylvester matrix equation

EV F 2 −AV F − CV = BW , IET Contr. Theory Appl., 3(10), 1401-1408, 2009.

[2] B. Zhou, Z. Li, G. Duan, and Y.Wang, Weighted least squares solutions to general coupled Sylvester matrix equations,

J. Comput. Appl. Math., 224(2), 759-776, 2009.

[3] C. Song and G. Chen, On solutions of matrix equations XF − AX = C and XF − A∼X = C over quaternion field,

J. Appl. Math. Comput., 37(1-2), 57-68, 2011.

[4] J.H. Bevis, F.J. Hall, R.E. Hartwing, Consimilarity and the matrix equation AX - XB = C, in: Current Trends in

Matrix Theory, Auburn, Ala., 1986, North-Holland, New York, 1987, pp. 51-64.

[5] J.H. Bevis, F.J. Hall, R.E. Hartwig, The matrix equation AX -XB = C and its special cases, SIAM Journal on Matrix

Analysis and Applications, 9 (3), 348-359, 1988.

[6] A.G. Wu, G.R. Duan, H.H. Yu, On solutions of XF − AX = C and XF − AX = C, Applied Mathematics and

Applications, 182 (2), 932-941, 2006.

[7] T. Jiang, S. Ling, On a solution of the quaternion matrix equation A∼X −XB = C and its applications, Adv. Appl.

Clifford Algebr., 23, 689-699, 2013.



225

[8] H. H. Kosal, M. Tosun, Some equivalence relations and results over the commutative quaternions and their matrices,

An. S.t. Univ. Ovidius Constanta., 25, 125-142, 2017.

[9] F. Catoni, R. Cannata, P. Zampetti, An introduction to commutative quaternions, Adv. Appl. Clifford Algebr., 16,

1-28, 2006.

[10] H. H. Kosal, An Algorithm for solutions to the elliptic quaternion matrix equation AX = B, Conference Proceedings

of Science and Technology, 1(1), 36-40, 2018.

226

Rotary Mappings and Transformations

Josef Mikes 1, Lenka Ryparova 2

Abstract. The term of isoperimetric extremal of rotation was defined Leiko in work [2]. These curves

are a solution of special variational problem on Riemannian spaces. Conditions of the existence of the

rotary mappings were significantly specified in paper [5], see [4, 127-131].

A diffeomorfism f between two-dimensional manifold An and (pseudo-) Riemannian manifold Vn is

called rotary mapping if any geodesic on An is mapped onto isoperimetric extremal of rotation on

Vn. Chuda, Mikes and Sochor [1] formulated necessary and sufficient condition for space An to admit

rotary mapping onto space Vn, and this was the existence of a special torse-forming vector field θ which

satisfies

∇Xθ = θ · (Θ(X) +∇XK/K) + ν ·X (1)

for any tangent vector X, where ∇ is the Levi-Civita connection on V2, K is the Gaussian curvature,

ν is a function, the form Θ is defined as Θ(X) = g(θ,X), and g is a metric of V2.

The above mentioned condition is also a necessary and sufficient condition of the existence of the rotary

mapping between Riemanian spaces, see [2].

In work [2] by Leiko, there is stated that from the condition (1) follows the isometry with surfaces of

revolution. This statement was proved to be invalid. A contra-example to this problem was constructed

in paper [3] by Ryparova, Mikes and Chuda.

For rotary vector field θ satisfying (1) we obtain [6]: Theorem A two-dimensional (pseudo-) Rie-

mannian manifold V2 admits rotary vector field θ if and only if the following closed Cauchy type

system of PDE’s in covariant derivatives has a solution with respect to functions θi(x) and ν(x):

θi,j = θi(θj + ∂jK/K) + νgij , ν,i = ν(θi − ∂iK/K) − Kθi − θαθβgαβ∂iK/K + θigαβθα∂βK/K. The

general solution depends on no more than 3 real parameters.

We obtained new simpler fundamental equations of rotary transformations.

Keyword: Rotary mapping, rotary transformation, isoperimetric extremal of rotation.

AMS 2010: 53B20, 53A05, 53B30, 53C22.

1Palacky University, Olomouc, Czech Republic, Josef.Mikes@upol,cz

2Palacky University, Olomouc, Czech Republic, Lenka.Ryparova01@upol,cz

227

References

[1] H. Chuda, J. Mikes, M. Sochor, Rotary diffeomorphism onto manifolds with affine connection, Geometry, integrability

and quantization XVIII, Bulgar. Acad. Sci., Sofia, 130-137, 2017.

[2] S.G. Leiko, Rotary diffeomorphisms on euclidean spaces, Math. Notes, 47(3-4), 261-264, 1990.

[3] J. Mikes, J., L. Ryparova, H. Chuda, On theory of rotary mappings, Math. Notes, 104(3-4), 617-620, 2018.

[4] J. Mikes et al., Differential geometry of special mappings, Palacky University Press, Olomouc, 2015.

[5] J. Mikes, M. Sochor, T. Stepanova, On the existence of isoperimetric extremals of rotation and the fundamental

equations of rotary diffeomorphisms, Filomat, 29(3), 517-523, 2015.

[6] L. Ryparova, J. Krızek, J. Mikes, On fundamental equations of rotary vector fields, Proc. 18th Conf. APLIMAT,

Bratislava 1030-1034, 2019.

228

Some Results on Bertrand and Mannheim Curves

Kazım Ilarslan1, Fatma Gokcek2

Abstract. In the theory of curves in Euclidean space, one of the important and interesting problem

is characterization of a regular curve. There are two ways widely used to solve these problems: to

figure out the relationship between the Frenet vectors of the curves and to determine the shape and

size of a regular curve by using its curvature functions. Bertrand and Mannheim curves are interesting

examples of the relationship between the Frenet vectors of the curves. In this talk we discuss necessary

and sufficient conditions for a rectifying curve, a normal curve and a osculating curve to be a Bertrand

curve or a Mannheim curve.

Keyword: Bertrand Curve, Mannheim curve, rectifying curve, normal curve, osculating curve.

AMS 2010: 53A04.

References

[1] J. M. Bertrand, Memoire sur la theorie des courbes a double courbure, Comptes Rendus, 36, 1850.

[2] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Montly,

110, 147-152, 2003.

[3] F. Kaymaz and F. Kahraman Aksoyak, Some special curves and Mannheim curves in three dimensional Euclidean

space. Math. Sci. Appl. E-Notes 5, no. 1, 34-39, 2017.


[5] H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 88 , 120-126, 2008.

[6] O. Tigano, Sulla determinazione delle curve di Mannheim, Matematiche Catania 3, 25-29, 1948.



229

Geometric Interpretation of Curvature Circles in Minkowski Plane

Kemal Eren 1, Soley Ersoy 2

Abstract. The main aim of this study is to present the geometric interpretations of curvature cir-

cles of motion at the initial position of the Minkowski plane. Preliminarily, the basic principles of

one-parameter motion and the instantaneous invariants in Minkowski plane are summarized. Then

the equations of the circular point curve and center point curve of the motion are given. In these

regards, the position of these curves with respect to each other and their special cases are interpreted

geometrically.

Keywords: Curvature circles, Instantaneous Invariants, Burmester Theory, Minkowski plane.

AMS 2010: 53A10, 53B30, 53C42.

References

[1] O. Bottema, On instantaneous invariants, Proceedings of the International Conference for Teachers of Mechanisms,

New Haven (CT): Yale University; 159-164, 1961.

[2] O. Bottema, On the determination of Burmester points for five distinct positions of a moving plane; and other topics,

Advanced Science Seminar on Mechanisms, Yale University, (1963), July 6-August 3.

[3] O. Bottema, B. Roth, Theoretical kinematics, New York (NY): Dover; 1990.

[4] B. Roth, On the advantages of instantaneous invariants and geometric kinematics, Mech. Mach. Theory,89, 5-13,

2015.

[5] F. Freudenstein, Higher path-curvature analysis in plane kinematics, ASME J. Eng. Ind. 87, 184-190, 1965.

[6] F. Freudenstein and G. N. Sandor, On the Burmester points of a plane, Journal of Applied Mechanics, Transactions

of the ASME, Series E, Vol. 83, March, 41-49, 1961.

[7] G.R. Veldkamp, Curvature theory in plane kinematics [Doctoral dissertation], Groningen: T.H. Delft. 1963.

[8] G.R. Veldkamp, Some remarks on higher curvature theory. J. Manuf. Sci. Eng. 89, 84-86, 1967.

[9] G.R. Veldkamp, Canonical systems and instantaneous invariants in spatial kinematics. J. Mech. 2, 329-388, 1967.

1Fatsa Science High School, Ordu, Turkey, [email protected]


230

[10] K. Eren and S. Ersoy, Circling-point curve in Lorentz plane, Conference Proceedings of Science and Technology,

Vol. 1, No. 1, 1-6, 2018.

231

New Representation of Hasimoto Surfaces According to the Modified Orthogonal

Frame

Kemal Eren 1

Abstract. In this study, we investigate Hasimoto surfaces with modified orthogonal frame. Firstly,

the relations between the Frenet frame and the modified orthogonal frame is given. We give the defini-

tions and some new theorems about Hasimoto surfaces. After that, the first and second fundamental

form, mean curvature and Gaussian of the Hasimoto surface according to the modified frame are calcu-

lated. Finally, we have expressed the properties of parameter curves of Hasimoto surfaces with modified

frame in Euclidean space.

Keywords: Hasimoto surfaces, modified orthogonal frame.

AMS 2010: 53A05, 37K25.

References

[1] N. H. Abdel-All, R. A. Hussien and T. Youssef, Hasimoto surfaces, Life Sci. J. 9, no. 3, 556–560, 2012.

[2] A. Cakmak, Parallel surfaces of Hasimoto surfaces in Euclidean 3-space, BEU J. of Sci. 7, no. 1, 125–132, 2018.

[3] M. Erdogdu and M. Ozdemir, Geometry of Hasimoto surfaces in Minkowski 3-space, Math. Phys. Anal. Geom. 17,

169–181, 2014.

[4] M. Erdogdu and M. Ozdemir, Hasimoto surfaces in Minkowski 3-space with parallel frame, 14th Int. Geom. Symp.,

Denizli, Turkey, 2016.

[5] M. Grbovic and E. Nesovic, On Backlund transformation and vortex filament equation for null Cartan curve in

Minkowski 3-space, Math. Phys. Anal. Geom. 23, 1–15, 2016.

[6] M. Grbovic and E. Nesovic, On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space, J.

Math. Anal. Appl. 461, no. 1, 219–233, 2018.

[7] N. Gurbuz, The motion of timelike surfaces in timelike geodesic coordinates, Int. J. Math. Anal. (Ruse) 4, no. 5–8,

349–356, 2010.

[8] N. Gurbuz, Hasimoto surfaces according to three classes of curve evolution with Darboux frame in Euclidean space,

Gece Publishing, 1, 49–62, 2018.


232

[9] H. Hasimoto, Motion of a vortex filament and its relation to elastica, J. Phys. Soc. Jpn. 31, 293–294, 1971.

[10] H. Hasimoto, A soliton on a vortex filament, J. Fluid. Mech. 51, 477-485, 1972.

[11] T.A. Ivey, Helices, Hasimoto surfaces and Backlund transformations, Canad. Math. Bull. 43, no. 4, 427–439, 2000.

[12] A. Kelleci, M. Bektas and M. Ergut, Parallel Hasimoto surfaces in Minkowski 3-spaces, International Conference on

Mathematics and Mathematics Education, 2016.

[13] A. Kelleci, M. Bektas and M. Ergut, The Hasimoto surface according to Bishop frame, Adiyaman Uni. J. of Sci. 9,

no. 1, 13–22, 2019.

[14] H.C. Pak, Motion of vortex filaments in 3-manifolds, Bull. Korean Math. Soc. 42, no. 1, 75–85, 2005.

233

Geometry of Complex Coupled Dispersionless and Complex Short Pulse Equations

by Using Bishop Frames

Kemal Eren 1

Abstract. In this study, we consider generalized coupled dispersionless, complex coupled dispersion-

less and complex short pulse equations from both geometric and algebraic point of views. Firstly,

generalized coupled dispersionless equations are given by Bishop-1 frame and complex coupled disper-

sionless equations are obtained from Bishop-2 frame. Then, using the Hodograph transformation, the

complex short pulse equations are obtained from generalized coupled dispersionless and complex cou-

pled dispersionless equations. Finally, the integrability conditions of the obtained differential equations

by finding the Lax pair is found.

Keywords: Complex coupled dispersionless equations, short pulse equations, Bishop frame, Lax pairs.

AMS 2010: 34A26, 37K40, 35A30.

References

[1] K. Konno, H. Oono, New coupled dispersionless equations, J. Phys. Soc. Jpn.63:377-378, 1994.

[2] K. Konno, H. Kakuhata, Interaction among growing, decaying and stationary solitons for coupled and stationary

solitons, J. Phys. Soc. Jpn. 64:2707-2709, 1995.

[3] H. Kakuhata, K. Konno, A generalization of coupled integrable, dispersionless system, J. Phys. Soc. Jpn. 65:340-341,

1996.

[4] T. Schafer, C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D 196:90-105,

2004.

[5] B.-F. Feng, Complex short pulse and coupled complex short pulse equations, Physica D 297:62-75, 2015.

[6] A. Sakovich, S. Sakovich, The short pulse equation is integrable, J. Phys. Soc. Jpn.74:239-241, 2005.

[7] J. C. Brunelli, The short pulse hierarchy, J. Math. Phys. 46:123507, 2005.

[8] Y. Matsuno, Periodic solutions of the short pulse model equation, J. Math. Phys.49:073508, 2008.

[9] B.-F. Feng, K. Maruno, Y. Ohta, Self-adaptive moving mesh schemes for short pulse type equations and their Lax

pairs, Pac. J. Math. Ind. 6:7-20, 2014.


234

[10] S. F. Shen, B. F. Feng, Y. Ohta, From the real and complex coupled dispersionless equations to the real and complex

short pulse equations, Stud. Appl. Math., 136, 64-88, 2016.

[11] L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, Volume 82, Issue 3, 246-251,

1975.

[12] S. Yılmaz, M. Yılmaz, A new version of Bishop frame and application to spherical images, J. Math. Analy.Appl.,

371: 764-776, 2010.

[13] E. Ozyılmaz, Classical differential geometry of curves according to type-2 Bishop trihedra, Mathematical and Com-

putational Applications, 16(4): 858- 867, 2011.

[14] C. Rogers And W. K. Schief, Backlund and Darboux transformations: Geometry and modern applications in soliton

theory, Cambridge University Press, Cambridge, 2002.

[15] B.-F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno, Y. Ohta, Discrete integrable systems and hodograph transforma-

tions arising from motions of discrete plane curves, J. Phys. A: Math. Theor. 44:395201, 2011.

235

New Associated Curves and Their Some Geometric Properties in Euclidean 3-Space

Muhammed Talat Sarıaydın 1

Abstract. In this paper, we define some new associated curves in Euclidean 3-space. We give some

relationships between curvatures of these curves. By using these associated curves, we give some geo-

metric properties and illustrative examples according to quasi frame in Euclidean 3-space.

Keyword: Associated Curves, Direction Curve, q-Frame.

AMS 2010: 53A04.

References

[1] I.J. Kolar and W.M. Peter, Natural operations in differential geometry, 1999.

[2] T.Korpinar, M.T. Sariaydin, and E. Turhan. Associated Curves According to Bishop Frame in Euclidean 3 Space,

Advanced Modeling and Optimization 15(3), 713-717, 2013.

[3] N. Macit, M. Duldul, Some new associated curves of a Frenet curve in Eˆ3 and Eˆ4, Turkish Journal of Mathematics

38(6), 1023-1037, 2014.

[4] B. O’neill, Elementary differential geometry. Elsevier, 2006.

[5] B. Sahiner, Quaternionic Direction Curves, Kyungpook Mathematical Journal 58(2), 377-388, 2018.

[6] B. Sahiner, Bir egrinin asli normaller gostergesinin dogrultu egrileri, Teknik Bilimler Dergisi 8(2), 46-54,2018.

[7] S. Yilmaz, Characterizations of some Associated and special curves to type -2 Bishop frame in Eˆ3, Kırklareli

Universitesi Muhendislik ve Fen Bilimleri Dergisi 1(1) 66-77.


236

On The Directional Spherical Indicatrices of Timelike Space Curve

Mustafa Dede 1, Gul Ugur Kaymanlı2 and Cumali Ekici 3

Abstract. In this study, the directional spherical indicatrices of a timelike space curve using tangent

and quasi normal vectors with q-frame is introduced. Then we work on the condition that a timelike

space curve to be slant helix by using the geodesic curvature of the spherical image of the directional

normal indicatrix. Finally, some applications of the results are given.

Keyword: Minkowski space, q-frame, Slant helix, Spherical Indicatrix.

AMS 2010: 51B20, 57R25.

References


3-space, Tohoku Math. J. 42(2), 67-82, 1990.

[2] A.T. Ali, New special curves and their spherical indicatrix, Glob. J Adv. Res. Class. Mod. Geom. 1(2), 28-38, 2012.

[3] S. Bas and T. Korpınar, Directional Inextensible Flows of Curves by Quasi Frame, Journal of Advanced Physics, 7,

1-3, 2018.

[4] B. Bukcu and M.K. Karacan, The slant helices according to Bishop frame, Int. J. Comput. Math. Sci. 3, 67-70, 2009.

[5] S. Coquillart, Computing offsets of B-spline curves, Computer-Aided Design, 19(6), 305-09, 1987.


[7] C. Ekici, M. Dede, H. Tozak, Timelike directional tubular surfaces, Int. J. Mathematical Anal., 8(5), 1-11, 2017.

[8] C. Ekici, G. U. Kaymanlı and M. Dede, Spherical Indicatrices of Directional Space Curve, 17th International Geometry

Symposium, Erzincan, Turkey, 2019.



[10] S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk. J. Math, 28, 153-163, 2004.




237

[11] T. Korpınar, E. Turhan and V. Asil, Tangent Bishop spherical images of a biharmonic B-slant helix in the Heisenberg

group Heis3, IJST, A4, 265-271, 2011.

[12] L. Kula, Y. Yaylı, On Slant Helix and Its Spherical Indicatrices, Appl. Math. Comput. 169, 600-607, 2005.


[14] W. Wang, B. Juttler, D. Zheng and Y. Liu, Computation of rotation minimizing frame, ACM Trans. Graph, 27(1),

18 pages, 2008.

[15] S. Yılmaz, E. Ozyılmaz and M. Turgut, New Spherical Indicatrices and Their Characterizations, Analele Stiint

Univ. 18, 337-354, 2010.

238

On Generalized Partially Null Mannheim Curves

Nihal Kılıc Aslan1, Kazım Ilarslan2

Abstract. It is well known that there are many associated curves whose Frenet frame’s vector fields

of which satisfy some extra conditions. For instance Mannheim curves which are discovered by A.

Mannheim in 1887. They are defined as the curves having a property that their principal normal lines

coincide with binormal lines of their mate curves at the corresponding points. In this talk, a char-

acterization method for generalized partially null Mannheim curves and their generalized Mannheim

mate curves are presented by considering the cases when the corresponding mate curve is a spacelike,

a timelike, null Cartan, partially null or pseudo null Frenet curve.

Keyword: Generalized Mannheim curve, semi-Euclidean space, partially null curve, curvature functions

AMS 2010: 53A04.

References

[1] K. Ilarslan, A. Ucum, E. Nesovic, On generalized spacelike Mannheim curves in Minkowski space-time, Proc. Nat.

Ac. Sci. 86 (2),2016.

[2] M. Grbovic, K. Ilarslan, E. Nesovic, On generalized null Mannheim curves in Minkowski space-time. Publ. De l’Inst.

Math, 99 : 77-98, 2016.

[3] M. Grbovic, K. Ilarslan, E. Nesovic, On null and pseudo null Mannheim curves in Minkowski 3-space. Jour. Geo.,

105 (1): 177-183, 2014.


[5] H. Liu and F. Wang, Mannheim partner curves in 3-space, Journal of Geometry, 88 , 120-126, 2008.

[6] O. Tigano, Sulla determinazione delle curve di Mannheim, Matematiche Catania 3, 25-29, 1948.



239

On the Trajectory Ruled Surface of Framed Base Curves in E3

Onder Gokmen Yıldız 1, Mahmut Akyigit 2 and Murat Tosun 3

Abstract. In this paper, we study trajectory ruled surface of a space curve with singular points. By

using theory of framed curve, we investigate the trajectory ruled surface and give some results about

invariants of these surface. Moreover, we determine local diffeomorphic image of these surface.

Keyword: Framed base curve, ruled surface, trajectory.

AMS 2010: Firstly 53A05, Secondly 53A04, 58K05.

References

[1] J. W. Bruce and P. J.Giblin, Curves and singularities. A geometrical introduction to singularity theory, Second

edition, University Press, Cambridge, 1992.

[2] T. Fukunaga and M. Takahashi, Framed surfaces in the Euclidean space, Bulletin of the Brazilian Mathematical

Society, New Series 50(1), 37-65, 2019.

[3] S. Honda, Rectifying developable surfaces of framed base curves and framed helices, Singularities in Generic Geometry,

Mathematical Society of Japan, 273-292, 2018.

[4] S. Honda and M. Takahashi, Framed curves in the Euclidean space, Advances in geometry 16(3), 265-276, 2016.

[5] T. Jiang and S. Ling, Algebraic Methods for Condiagonalization Under Consimilarity of Quaternion Matrices in

Quaternionic Quantum Mechanics, Adv. Appl. Clifford Alg., 23, 405-415, 2013.

[6] A. Kucuk, On the developable time-like trajectory ruled surfaces in a Lorentz 3-space R13, Applied mathematics and

computation, 157(2), 483-489, 2004.

1Bilecik Seyh Edebali University, Bilecik, Turkey, [email protected]



240

The gh-Gifts of Affine Connections on the Cotangent Bundle

Rabia Cakan 1

Abstract. In this paper, we determine the gh-lift of the affine connection via the musical isomor-

phism on the cotangent bundle. We obtain the torsion tensor and the curvature tensor of the gh-lift of

the affine connection. Finally, we investigate the properties of the geodesic curve of the gh-lift of the

Levi-Civita connection.

Keyword: gh-lift, horizontal lift, connection, curvature tensor, musical isomorphism, geodesic.

AMS 2010: Primary 55R10, Secondary 53C05.

References

[1] M. Altunbas and A. Gezer, On affine connnections induced on the (1, 1)−tensor bundle, Chin. Ann. Math. Ser. B

39(4), 683-694, 2018.

[2] R. Cakan, K. Akbulut and A. A. Salimov, Musical isomorphisms and problems of lifts, Chin. Ann. Math. Ser.B 37(3),

323-330, 2016.

[3] R. Cakan, On gh-lifts of some tensor fields, C. R. Acad. Bulgare Sci. 71(3), 317-324, 2018.

[4] M. Kures, Naturel lifts of classical linear connections to the cotangent bundle, Suppl. Rend. Circolo Mat. Palermo II

43, 181-187, 1996.

[5] Z. Pogoda, Horizontal lifts and foliation, Suppl. Rend. Circolo Mat. Palermo II, 21, 279-289, 1989.

[6] A. Salimov and H. Fattayev, Connections on the coframe bundle, Int. Electron. J. Geom. 12(1), 93-101, 2019.

[7] K. Yano and M. Patterson, Horizontal lifts from a manifold to its cotangent bundle, J. Math. Soc. Japan 19(2),

185–198, 1967.

[8] K. Yano and S. Ishihara, Tangent and cotangent bundles, Pure and Applied Mathematics, Marcel Dekker, New York,

1973.

1Kafkas University, Kars, Turkey, [email protected]

241

On the Involute of the Cubic Bezier Curves in E3

S. eyda Kılıcoglu 1, Suleyman Senyurt 2

Abstract. In 1962 Bezier curves were studied by the French engineer Pierre Bezier, who used them to

design automobile bodies. But the study of these curves was first developed in 1959 by mathematician

Paul de Casteljau. We have already examined the Frenet apparatus of any cubic Bezier curve in E3.

In this study we have examined, involute of the cubic Bezier curve in E3. Frenet vector fields and also

curvatures of involute of the cubic Bezier curve in E3 are examined based on the Frenet apparatus of

cubic Bezier curve of the first cubic Bezier curve in E3.

References

[1] H. Hagen, Bezier-curves with curvature and torsion continuity. Rocky Mountain J. Math. 16, no. 3, 629–638, 1986.

[2] H.H. Hacısalihoglu, Diferensiyel Geometri, Cilt 1, Inonu Universitesi Yayinlari, Malatya 1994.02/05/2010.

[3] S. Michael, Bezier Curves and Surfaces Lecture 8, Floater Oslo Oct., 2003.

[4] S. Kılıcoglu and S. Senyurt, An examination on Frenet apparatus of Cubic Bezier Curves in E3, (IECMSA–2018).

11Baskent University, Ankara, Turkey, [email protected]

2Ordu University, Ordu, Turkey, [email protected]

242

Spinor Formulation of Bertrand Curves in E3

Tulay Erisir 1, Neslihan Cansu Kardag

Abstract. In this paper, we have studied on spinors with two complex components and we have given

spinor representations of Bertrand curves in E3. Firstly, we have introduced spinor representations of

Frenet vectors of curve in three dimensional Euclidean space E3. Moreover we have chosen arbitrary

two curves corresponding two spinor with complex components. Then, we have considered that these

curves are Bertrand curves. So, we have investigated the answer of question ”How are the relations

between the spinors corresponding to the Bertrand curves. Finally, we have given an example which

crosscheck to theorems throughout this study.

Keyword: Spinors, Bertrand Curves.

AMS 2010: 11B39, 11R52.

References

[1] E. Cartan, The theory of spinors, The M.I.T. Press, Cambridge, MA, 1966.

[2] G. F. T. Del Castillo and G. S. Barrales, Spinor formulation of the differential geometry of curves, revista colombiana

de matematicas, 38, 27-34, 2004.

[3] T. Erisir, M. A. Gungor and M. Tosun, Geometry of the hyperbolic spinors corresponding to alternative frame, Adv.

in Appl. Cliff. Algebr., 25, 4, 799-810, 2015.

[4] Y. Balci, T. Erisir and M. A. Gungor, Hyperbolic spinor darboux equations of spacelike curves in minkowski 3-space,

Journal of the Chungcheong Mathematical Society, 28, 4, 525-535, 2015.

[5] Z. Ketenci, T. Erisir and M.A. Gungor, A construction of hyperbolic spinors according to frenet frame in minkowski

space, Journal of Dynamical Systems and Geometric Thedories, 13, 2, 179-193, 2015.

[6] D. Unal, I. Kisi and M. Tosun, Spinor bishop equation of curves in euclidean 3-space, Adv. in Appl. Cliff. Algebr.,

23, 3, 757-765, 2013.

[7] I. Kisi and M. Tosun, Spinor darboux equations of curves in euclidean 3-space, Math. Morav., 19, 1, 87-93, 2015.

1Erzincan Binali Yildirim University, Erzincan, Turkey, [email protected]

243

MATHEMATICSEDUCATION

Kazan University and Development of Geometry in Azerbaijan

R. M. Aslanov 1

Abstract. The book ”Kazan University and Geometry Development in Azerbaijan” prepared by us

(authors M.J.Mardanov, R.M. Aslanov) is dedicated to the 100th anniversary of Baku State University.

The manuscript is devoted to a brief history of the Kazan University and its role in the development of

geometry in Azerbaijan. The book consists of two chapters. The first chapter includes 28 paragraphs.

The book tells about the role of Lobachevsky in the development of mathematics and mathematics

education at Kazan University. The book also gives a brief history of the creation of the Faculty of

Mechanics and Mathematics, mathematical departments, including the creators of these departments.

The manuscript tells about the mathematicians of the Kazan University: N.I. Lobachevsky, A.V.

Vasiliev, N.G. Chebotarev, P.A. Shirokov, N.G. Chetaev, A.P. Norden, B.L. Laptev, F.D. Gakhov,

V.V. Morozov, M.P. Nuzhin, L.I. Chibrikova, A.P.Shirokov, V.V. Vishnevsky, M.M. Arslanov, A.V.

Suldin, B.M. Gagayev, S. R. Nasirov on their brief biography and scientific heritage, and much more. In

the book, a lot of attention is paid to geometers and the Department of Geometry at Kazan University.

The second chapter of the book is called Geometers of Azerbaijan. It is about the creation and develop-

ment of geometry in Azerbaijan. This chapter is devoted to the creators of the geometry of Azerbaijan,

including N.Tusi, B.A. Rosenfeld, MA Javadov, M.T. Abbasov, B.G.Salayeva, N.L. Nasrullaev and

others, as well as the role of professors Javadov M.A., Salimov A.A. and Kazan University in the cre-

ation of a geometric school in Azerbaijan. In the manuscript, a wide place is given to the departments

of geometry of higher educational institutions of Azerbaijan and their relations with Kazan University.

The book includes two applications of Mirza Mohammed Ali Kazembek and Kazan University and

the city of Baku and Kazan. The book will be published by the Kazan State University Publishing

House. The scientific editor of the book is a professor of Kazan University, follover of A.P. Norden

V.V. Shurygin.

1Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbajan,

e-mail r [email protected]

244

Abstract. (Continuation) The reviewers were Shakirova Liliana Rafikovna - Doctor of Pedagogical,

Professor, Head of the department of Theory and Technology Teaching of Mathematics and Informatics

Institute of Mathematics and Mechanics named after N.I. Lobachevsky Kazan (Privolzhsk) Federal

University (KSU), Doctor of Physical and Mathematical Sciences, Professor, Head of the chair of

Algebra and Geometry of Baku State University (BSU) Salimov Arif Agadzhan oglu and Candidate

of Physical Mathematical Sciences, Doctor of Pedagogical Sciences, Head of the chair of Applied

Mathematics and informatics Kostroma State Pedagogical University named after N. A. Nekrasov,

professor Sekovanov V.S.

It is necessary to remark the deep impression that the meetings and conversations made with the

academician of the Academy of Sciences of the Republic of Tatarstan, Professor M.M. Arslanov, Cor-

responding Member of the Academy of Sciences of the Republic of Tatarstan, Professor S.R. Nasirov,

Professor L.R. Shakirova, Professor L.L. Salekhova, Professor A.A. Salimov, Professor MS Dzhabrailov

and others. Without these meetings, the book would not have been written.

This book will be useful for students, postgraduate students, master students, teachers and those who

want to know the history of geometry and the development of the geometry of Azerbaijan. In the

report we will speak in detail about the structure and content of the book.

245

Measurement of Achievement Distribution by Gini Coefficient Approach: An

Application for Statistics Course

Suheda Guray 1

Abstract. Aim of the study is to analyze the distribution of 2014-2019 Faculty of Education stu-

dents’ academic achievements in three statistics courses (Statistics, Statistics of Education, Statistics

and Probability) by using Lorenz curve and Gini coefficient.

The empirical data used in the study may differ in academic periods. The sample of the study consists

of students between 2014-2019 academic periods. The sample size is academic achievements data of

561 students. Distribution of those data among courses is as follows; Statistics: 242 students, Statistics

of Education: 207 students, Statistics and Probability: 112 students.

Main questions of the study are whether the academic achievement is evenly distributed among different

courses in 2014-2019 and whether the academic achievement for the same course is evenly distributed

by years.

In the conclusion of the study, according to academic achievement distribution for the department,

distribution of academic achievement, obtained by the Gini coefficient, for three different statistics

courses is interpreted.

Keyword: Gini coefficient, academic achievement, Lorenz Diagram.

References

[1] E. Albay and S. Kılıcoglu, Lorenz egrisi ve Gini katsayısının egitimde olcme ve degerlendirme uygulaması, Baskent

Universitesi Bitirme Tezi, 2018.

[2] E. Erdem and S. Coban, Turkiye’de ıller bazında egitim esitsizliginin olculmesi ve ekonomik gelismislik farklılıklarıyla

ıliskisi; egitimin Gini katsayıları, 14. Istatistik Arastırma Sempozyumu 2005 bildirileri, 5-6 Mayıs 2005, Ankara 188-

2004.

[3] J. Crespo Cuaresma, K.C. Samir and P. Sauer, Age-specific education inequality, Education Mobility and Income

Growth, WWWforEurope Working Papers series 6, WWWforEurope, 2013.

1Baskent University Ankara, Turkey, [email protected]

246

[4] S. Coban, The Relationships among mortality rates, Income and Educational Inequality in Terms of Economic

Growth: A Comparison between Turkey and Euro Area. MPRA, 2008.

[5] F. Senses, Iktisada (farklı bir) giris, Iktisadı Ogrencileri ve Iktisada Ilgi Duyanlar Icin Yardımcı Kitap, Iletisim

Yayınları, Istanbul, 2017.

[6] V. Thomas, Y. Wang, and X. Fan, Measuring education inequality: Gini coefficients of education, World Bank Policy

Research Working Paper, 2001.

247

Inverse Modeling Problems and Tasks Enrichment: Analysis of two Experiences

with Spanish Prospective Teachers

Victor Martinez-Luaces 1, Jose Antonio Fernandez-Plaza 2 and Luis Rico 3

Abstract. Modeling and applications are widely considered as hot topics in mathematics education

research [1]. It should be noted that when searching for modeling situations, real-life problems are usu-

ally posed in an inverse form [2]. Thus, when both characteristics are combined, the so-called inverse

modeling problems [3] are obtained. One of the main reasons of the relevance is their potential for tasks

enrichment. Consequently, since 2016, a research project was carried out by using inverse modeling

problems to develop prospective teacher‘s tasks enrichment skills. Some results of this experience that

took place in 2017 have been analyzed and explained in a book chapter [4]. Then, during 2018 and after

the first experience, a new research design was proposed and implemented at the beginning of 2019.

The new results showed interesting differences and few similarities. In this paper, both experiences are

analyzed and lastly, findings and final conclusions are reported.

Keyword: Inverse problems, tasks enrichment, prospective teachers, solution sketches, mathematical

modeling.

AMS 2010: 97B50, 97D50, 97C70.

References

[1] W. Blum, ICMI Study 14: applications and modelling in mathematics education discussion document. Educational

Studies in Mathematics, 51 (1-2), 149-171, 2002.

[2] C. W. Groetsch, Inverse problems: activities for undergraduates, Washington, D.C.: Mathematical Association of

America, 1999.

[3] V. Martinez-Luaces, Inverse modeling problems and their potential in mathematics education. in m. vargas (ed.)

teaching and learning: principles, approaches and impact assessment, pp. 151-185. New York: Nova Science Publish-

ers, 2016.

1University of Granada, Granada, Spain, [email protected]



248

[4] V. Martinez-Luaces, L. Rico, J. F. Ruiz-Hidalgo and J. A. Fernandez-Plaza, Inverse modeling problems and task

enrichment in teacher training course,. In R.V. Nata (Ed.) Progress in Education, Vol. 53, pp. 185-214. New York:

Nova Science Publishers, 2018.

249

OTHERAREAS

Some New Results on Path Integration Methods

A. Naess 1, L. Chen 2 and E. R. Jakobsen 3

Abstract. We study a numerical method to compute probability density functions of solutions of

stochastic differential equations. The method is sometimes called the (numerical) path integration

(PI) method and has been shown to be fast and accurate in application oriented fields.

The PI/density tracking approach (i.e. simulating the PDFs) enjoys several favorable properties. First,

it introduces an extra perspective to the system, which enables deeper insights and invites broader

mathematical tools. Secondly, as an explicit method, one can formally implement the path integration

algorithm on a vast number of scenarios. Since the formulation is deterministic, it is also free from

perturbation by extreme outcomes during stochastic simulation. Finally, the result of the method is an

explicit density function rather than bundle of random paths. This means that many characteristics of

the system become more transparent, and can be captured and displayed by e.g. visualization methods.

In our proposed paper we discuss rigorous analyses of the method that covers systems of equations

with unbounded coefficients.

Working in a natural space for densities, L1, we obtain stability, consistency, and new convergence

results for the method, new well-posedness and semigroup generation results for the related Fokker-

Planck-Kolmogorov equation, and a new and rigorous connection to the corresponding probability

density functions for both the approximate and the exact problems.

To prove the results we combine semigroup and PDE arguments in a new way that is of independent

interest.

Keywords: Stochastic Differential Equations, Path Integration, Density Tracking, Probability Density,

Semigroup Generation, Convergence

AMS 2010: 60H35, 65M12, 47D07.

1NTNU, Trondheim, Norway, [email protected]



250

Wrinkling of Annular Plates and Spherical Caps With Material Inhomogeneity

Eva Voronkova 1, Svetlana Bauer 2

Abstract. This work is concerned with wrinkling of internally pressurised spherical caps with material

inhomogeneity. Unsymmetrical buckling can be observed in, for example, metal sheets or in biological

tissues such as human skin or blood vessels, and has been discussed by many authors [1], [2].

We consider a shallow spherical shell of uniform thickness with a small circular opening at the top.

The apex rise of the shell is much less than the curvature radius.

The shell is loaded with normal uniformly distributed inner pressure. It is assumed that the outer edge

of the shell is clamped but can move freely in the radial direction without rotation. The inner edge of

the shell is supposed to be supported by a roller which can slide along a vertical wall.

Meridional material inhomogeneity is assumed for the shell, i.e. Young’s modulus is spatially dependent.

We derived the Donnell-Mushtari-Vlasov equations are derived for a spherical cap with meridional

inhomogeneity and sought the asymmetric part of the solution in terms of multiples of the angular

harmonics.

Prebuckling stress-state in a narrow zone near the shells edge makes a major contribution to the un-

symmetrical buckling mode and the value of the critical load. It is shown that if the elasticity modulus

decreases away from the center of a plate, the critical pressure for unsymmetrical buckling is sufficiently

lower than for a plate with constant mechanical properties. Number of waves in the circumferential

direction increases with the degree of nonuniformity. The buckling load and corresponding mode

number increase as the shallowness parameter grows. For a truncated shallow shell the wrinkling pres-

sure increases as the radius of the opening increases, while the buckling mode decreases.

Keyword: wrinkling, spherical shell, annular plate

AMS 2010: 74K25, 74G60..

1St. Petersburg State University, St. Petersburg, Russia, [email protected]

2St. Petersburg State University, St. Petersburg, Russia, [email protected]

251

References

[1] C. D. Coman, A. P. Bassom, Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap.

International Journal of Non-Linear Mechanics. 81, 8-18, 2016.

[2] S. M. Bauer, E. B. Voronkova, Models of shells and plates in the problems of ophthalmology. Vestnik St. Petersburg

University: Mathematics, 47(3), 123-139, 2014.

252

Some Properties of s-reducibility

Irakli Chitaia 1

Abstract. The reducibility known as s-reducibility is a restricted version of enumeration reducibility.

We recall that any computably enumerable (or, c.e.) set W defines an enumeration operator (for short:

e-operator), i.e. a mapping ΦW from the power set of ω to the power set of ω such that, for A ⊆ ω,

ΦW (A) = x : (∃u) [〈x, u〉 ∈W and Du ⊆ A],

where Du is the finite set with canonical index u: We often identify finite sets with their canonical

indices, thus writing, for instance, 〈x,D〉 instead of 〈x, u〉 if D = Du. If A = Φ(B) for some e-operator

Φ then we say that A is enumeration reducible to B (or, more simply, A is e-reducible to B; in symbols:

A ≤e B) via Φ. An e-operator Φ is said to be an s-operator, if Φ is defined by a c.e. set W such that

(∀ finite D)(∀x)[〈x,D〉 ∈W ⇒ |D| ≤ 1]

(where the symbol |X| denotes the cardinality of a given set X). We say that A is s-reducible to B (in

symbols: A ≤s B) if A = Φ(B), for some s-operator Φ (see [2]).

During the research of strong enumeration reducibility, some interesting and important results have

been obtained by us, namely it has been shown: if A is any infinite set then A is hyperimmune

(respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has KssB (re-

spectively, K sB): here ≤s is the finite-branch version of s-reducibility, ≤ss is the computably bounded

version of ≤s, and K is the complement of the halting set (see [1]). We also show that degs(K) is

hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed (see[1]);

and etc.

Despite the mentioned results which shows that the research of s-reducibility gives very interesting

results, many issues are still unexplained and undetermined. In this talk we will present some recently

obtained results about structural properties of s-reducibility.

Keyword: s-reducibility, enumeration operator, computably enumerable set.

AMS 2010: 03D25, 03D30.

1Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia, [email protected]

253

References

[1] I. O. Chitaia, R. Sh. Omanadze, A. Sorbi, Immunity properties and strong positive reducibilities, Arch. Math. Logic.

50(3-4): 341–352, 2011.

[2] R. Sh. Omanadze and A. Sorbi, Strong enumeration reducibilities, Arch. Math. Logic. 45(7):869–912, 2006.

254

Omar Khayyam: Calendric Calculations, Cosmic poetry and Paintings Reflecting

His Poetry

Victoria Nikulina 1, Muhammad Rashid Kamal Ansari 2

Abstract. Omar Khayyam was a prominent mathematician, astronomer and cosmologist with a num-

ber of academic contributions in the form of papers and books. In view of recent calculations also, his

calendar appears to be more accurate than the calendars in use [5]. His poetry reflects his scientific

and philosophical ideas. His famous quatrains which were originally in Persian are translated in many

languages of the world. As usual the translations are influenced by the thoughts of the translators

themselves. Also, a large number of artists attempted to reflect the theme of the quatrains in their

paintings. Unfortunately, these paintings are also influenced by the thoughts of the artists. In a previ-

ous publication [9] the authors of this communication discussed the cosmic poetry of Omar Khayyam in

the perspective of his eight quatrains comparing seven translations. This study exhibits the paintings

of the first author based on a Russian translation of some cosmic and astronomical quatrains of Omar

Khayyam. These paintings illustrate the cosmic and philosophical point of view of Omar Khayyam.

The geometrical ideas of Omar Khayyam and his contemporaries are used in forming the patterns in

Islamic Art. In this regard Geometrical Patterns of Islamic art are also discussed. This presentation

reviews the important features of the calendar of Omar Khayyam. Finally, four paintings on Islamic

Art by Victoria Nikulina are included and their prominent feature are discussed.

References

[1] B. Rumer Osip, Umar Hayyam, Academia, 113, 1936.

[2] A. Dashti, Dame-Ba-Khayym, Tehran, 1967.

[3] A. Borel, Mathematics: Art and Science, The mathematical intelligencer, 5(4), Springer-Veflag New York, 1983.

[4] P. Bruter, Claude, ed. Mathematics and art: mathematical visualization in art and education, Heidelberg, Germany,

http : //www.springer.de/cgi/svcat/searchbook.pl?isbn = 3− 540− 43422− 4.

1Sir Syed University of Engineering and Technology, Karachi

2Sir Syed University of Engineering and Technology

255

[5] A. Sajjad, M. R. K. Ansari and J. Quamar, The problem of calendar reform, Arabian Journal for Science and

Engineering, Volume 30 (2A): 249-256, 2005.

[6] F. Abullalaeva, N. Chalisova, Ch. Melvill, The russian perception of omar khayyam,

http://www.academia.edu/25409662/, 2010.

[7] K. Kamran, Z. Asma, M. R. K.Ansari, Islamic Art, Mathematics and heritage of sindh, The S.U. Jour. of Ed. 40,

58-73, 2010-2011.

[8] V. Nikulina and M. R. K. Ansari, Islamic Art, Mathematics and eternity: heritage and traditions of sindh, The S.U.

Jour. of Ed. 41, 93-109, 2011-2012.

[9] V. Nikulina, M. R. K. Ansari, Cosmic poetry of omar khayyam and its artistic exposition using batik techniques,

Mystic Thoughts 1, 61-94, 2015.

256

STATISTICS

Transmuted Lower Record Type Frechet Distribution

Caner Tanıs 1, Bugra Saracoglu 2 and Coskun Kus 3

Abstract. In this paper, we have introduced a new lifetime distribution called “Transmuted lower

record type Frechet (TLRTF)”. Our introduced distribution is constracted by mixing two lower record

values. Several statistical properties such as moments, incomplete moments, Bonferroni and Lorenz

curves are studied. The maximum likelihood (ML) and Bayes estimates of parameters are derived. In

Bayesian analysis, the MCMC random walk algorith is used. The MLE based approximate confidence

intervals, credible and highest density posterior confidence intervals are obtained. The MLE based

parametric Bootstrap CI are also included. A Monte Carlo simulation study is performed to observe

the risk behaviour of the ML and Bayes estimates for different sample sizes. In addition, the coverage

probabilities and mean length of the all confidence intervals are investigated in the simulation study.

Finally, two real data applications are given to illustrate the modelling capability of TLRTF distribu-

tion.

Keyword: Bayes estimates, credible interval, Frechet distribution, highest density posterior, lower

records, maximum likelihood

AMS 2010: 62F10, 62H10, 62F15.

References

[1] M. R. Mahmoud, R. M. Mandouh, On the transmuted Frechet distribution. Journal of Applied Sciences Research. 9,

5553-5561, 2013.

[2] I. Elbatal, G. Asha, A. V. Raja, transmuted exponentiated Frechet distribution: properties and applications, Journal

of Statistics Applications and Probability. 3, 379, 2014.

[3] W. T. Shaw,I. R. Buckley, The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-

kurtotic-normal distribution from a rank transmutation map. arXiv preprint arXiv:0901.0434, 2009.




257

[4] D. C. T. Granzotto,F. Louzada, N. Balakrishnan, Cubic rank transmuted distributions: inferential issues and appli-

cations. Journal of Statistical Computation and Simulation. 87(14), 2760-2778, 2017.

258

Goodness of Fit Test For Weibull Distribution Based on Kullback Leibler

Divergence under Progressive Hybrid Censoring

Ismail Kınacı 1, Gulcan Gencer 2

Abstract. Goodness of fit test has an important role in data modelling. Various goodness of fit tests

are proposed by several authors. Kullback-Leiblier divergence can measure the difference between two

probability distribution. Because of time and cost constraints, censored samples are frequently used for

lifetime analysis. Recently, some goodness of fit tests for some distributions based on Kullback-Leiblier

divergence under progressive censoring. In this study, the goodness of fit test for Weibull distribution

based on Kullback-Leibler divergence is studied under progressive hybrid censoring. A simulation study

is performed to observe critical values and corresponding powers of test.

Keyword: Goodness of fit test, Kullback-Leibler divergence, progressive hybrid censoring, Weibull

distribution.

AMS 2010: 60E05, 62N01.



259

A New Unit-Weibull Distribution

Kadir Karakaya 1 Ismail Kınacı 2

Abstract. Although there is a lot of data between (0, 1) interval in various fields, there are a few

distributions such as Beta and Kumaraswamy in modeling these data. Therefore, a lot of new unit

distributions have recently been introduced by several authors. They used different transformation

methods on existing lifetime distributions. Some of them are Y = exp (−X), Y = X/ (X + 1) and

etc. In this study a new transformation is suggested to obtain a new probabilty distribution with

support (0,1). We propese two parameters unit-Weibull distribution by considering the transforma-

tion Y = tanh (X) ,where X ∼ Weibull (α, β). Some mathematical properties are studied such as

moments, skewness, kurtosis, Bonferroni and Lorenz curves. The method of maximum likelihood is

used to estimate the parameters, reliability and quantile functions. Monte Carlo simulation is carried

out to examine the bias and mean square errors of the maximum likelihood estimates of parameters.

The confidence intervals and the coverage probabilities are obtained for the parameters, reliability and

quantile function at given a point. Two numerical examples are provided to illustrate the methodology.

Keyword: Confidence intervals, Maximum likelihood estimators, Monte Carlo simulation, Weibull

distribution

AMS 2010: 60E05, 62N01.

1Selcuk University, Konya, TURKEY, [email protected]

2Selcuk University, Konya, TURKEY, [email protected]

260

Optimal Logistic Regression Estimator

Nurkut Nuray Urgan 1, Demet Gungormez 2

Abstract. Logistic regression model is used to describe the relationship between binary data set

and the predictors. Least Square Estimation (LSE) and Maximum Likelihood-estimation (MLE) are

widely used to estimate the parameters in logistic regression. But, if the predictors are highly corre-

lated, known as multicollinearity, the variance will be inflated, so the Mean Square Error (MSE) of

the estimator gets became bigger. Ridge Logistic Estimator and Liu Logistic Estimator are proposed

to combat the multicollinearity in the sense of mean square criterion(MSE) as an alternative to the

MLE. In this study, an optimal method is proposed for logistic regression(OLRE), when the data are

multicollinear. Furthermore, to investigate the theoretical results about the performance of this new

estimator, it is compared with MLE method under the relative mean squared error criterion (RMSE).

Keyword: Logistic Regression, Maximum Likelihood Estimation, Multicollinearity, Liu Logistic Esti-

mator.

AMS 2010: 62J07, 62J12.

References

[1] Hoerl, A.E. and Kennard, R.W.. Ridge regression: biased estimation for nonorthogonal problems. Techonomics 12,

55-67, 1970.

[2] Hoerl, A.E. and Kennard, R.W.. Ridge regression: iterative estimation of the biasing parameter. Communications in

Statistics: Theory and Methods 5, 77-88, 1976.

[3] P. McCullagh, J.A. Nelder. Generalized linear models (2nd ed.), Chapman and Hall, 1989.

[4] Liu, K.. A new class of biased estimate in linear regression. Communications in Statistics: Theory and Methods 22,

393-402, 1993.

[5] Urgan, N. N., Tez, M.. Liu estimator in logistic regression when the data are collinear. In: 20th EURO Mini Conference

Continuous Optimization and Knowledge Based Technologies Lithuania, Selected papers, Vilnius, 323-327, 2008.

1Namik Kemal University, Tekirdag, Turkey, [email protected]

2Tekirdag, Turkey, [email protected]

261

[6] Su, L. and Bondell, H.D.. Best linear estimation via minimization of relative mean squared error. Stat.Comput.

https://doi.org/10.1007/s11222-017-9792-0, 2017.

[7] Varathan, N. and Wijekoon, P.. Optimal generalized logistic estimator.Communications in Statistics - Theory and

Methods, 47,2, 463-474, 2018

262

Bootstrap Confidence Intervals of Capability Index CPM Based on Progressively

Censored Data

Yunus Akdogan 1

Abstract. One of the indicators to decide on the capability of a process is the process capability

index. There are several capability indexes such as Cp, Cpk, Cpm, and Cpmk. Hsiang and Taguchi

(1985) and Chan et al. (1988) independently introduced a Cpm as a process capability index. In this

paper, parametric bootstrap confidence intervals of the Cpm are studied under progressive censoring.

The maximum likelihood method is used to estimate the Cpm index. Several bootstrap confidence

intervals such as normal and percentile methods are conducted for obtaining confidence intervals of

Cpm. A Monte Carlo simulation is performed to observe the estimated coverage probabilities and aver-

age width of the bootstrap confidence intervals. An illustrative example is presented to close the paper.

Keyword: capability index, Monte Carlo simulation, progressive censoring, bootstrap, confidence in-

terval.

AMS 2010: 62F25, 62F40, 60E05, 62N01.

References

[1] T.C. Hsiang, G. Taguchi, A tutorial on quality control and assurance: the Taguchi methods. Joint Meetings of the

American Statistical Association, Las Vegas, NV, 188, 1985.

[2] L.K. Chan, S.W. Cheng, F.A. Spiring, A new measure of process capability Cpm. Journal of Quality Technology

20(3), 162–175, 1988.


263

TOPOLOGY

On a∗-I -Open Sets and a Decomposition of Continuity

Aynur Keskin Kaymakcı 1

Abstract. We introduce a new set namely a∗-I -open in ideal topological spaces. Besides, we give

some characterizations and properties of it. Then, we obtain that it is stronger than pre∗-I -open

and b-open and weaker than δβI -open. Finally, we give a decomposition of continuity by using a

notion of a∗-I -open as stated the following: f : (X, τ, I) −→ (Y, ϕ) is continuous if and only if it is

a∗-I-continuous and strongly AI -continuous.

Keywords: ideal topological spaces, a∗-I -open sets, decomposition of continuity.

AMS 2010: 54A05, 54A10, 54C08, 54C10.

References

[1] D. Andrijevic, On b-open sets, Mat. Vesnik, 48, 59-64, 1996.

[2] J. Dontchev, On pre-I-open sets and decompositon of I-continuity, Banyan Math. J., 2, 1996.

[3] E. Ekici and T. Noiri, On subsets and decompositions of continuity in ideal topological spaces, Arab. J. Sci. Eng.

Sect. A Sci. 34, 165-167, 2009.

[4] E. Hatir, A more on δα-I-open sets and semi∗-I-open sets, Math. Commun., 16, 433-445, 2011.

[5] E. Hatir, On decompositions of continuity and complete continuity in ideal topological spaces, European Journal of

Pure and Applied Math., 6(3), 352-362, 2013.

[6] S. Yuksel, A. Acikgoz and T. Noiri, On δ-I-continuous functions, Turkish Journal of Mathematics, 29, 39-51, 2005.

1University of Selcuk, Konya,Turkey,[email protected]

264

Common Fixed Point Results on Modular F-Metric Spaces and an Application

Duran Turkoglu 1, Nesrin Manav 2

Abstract. Jleli and Samet (2018) introduced a new concept, named an F-metric space, as a general-

ization of the notion of a metric space. In this paper, we prove certain common fixed point theorems in

F-metric spaces. As consequences of our results, we obtain results of modular F-metrics in any given

spaces. An application in dynamic programming is also given.

Keyword: Fatou property, fixed point, generalized modular metric space

AMS 2010: 54H25, 47H10.

References

[1] M. Jleli and B. Samet, On a new generalization of metric spaces J. Fixed Point Theory Appl., 2018, 20:128

https://doi.org/10.1007/s11784-018-0606-6.

[2] Turkoglu, D. and Manav, N. Fixed Point Theorems in New Type of Modular Metric Spaces, Fixed Point Theory and

Applications, 2018, https://doi.org/10.1186/s13663-018-0650-3.

[3] V. V. Chistyakov, Metric Modular Spaces Theory and Applications, SpringerBriefs in Mathematics, ISSN 2191-

8198(electronic) Library of Congress Control Number: 2015956774, 73, 2015, DOI 10.1007/978-3-319-25283-4.


2Erzincan BY University, Erzincan, Turkey, [email protected]

265

The Sheaves Representation of Hausdorff Spectra of Locally Convex Spaces

E. I. Smirnov 1, S. A. Tikhomirov 2 and E. A. Zubova 3

Abstract. We introduce here new concepts of functional analysis: Hausdorff spectrum and Hausdorff

limit or H-limit of Hausdorff spectrum of locally convex spaces of E.I.Smirnov with point of view using

sheaves theory. Particular cases of regular H-limit are projective and inductive limits of separated

locally convex spaces. The class of H-spaces contains Frechet spaces and is stable under the operations

of forming countable inductive and projective limits, closed subspaces and factor-spaces. Besides, for

H-space the strengthened variant of the closed graph theorem holds true. The space of germs of holo-

morphic functions on connected bounded subset will be provided with the topology (in general not

separated) of uniform convergence on the compact subsets and with the locally convex topology of the

H-limit. We also present an essentially new approach to the study of sheaves based on the notion of

Hausdorff spectra associated with the presheaf.

Keyword: spectrum, closed graph, sheaf.

AMS 2010: 46A13, 14F05.

References

[1] E.I. Smirnov, Homological spectra in functional analysis, Springer-Verlag, London, 2002.

[2] E.I. Smirnov, Weierstrass’s global division theorem and continuity of linear operators, British Journal of Mathematics

and Computer Science. 4 (3), 307–321, 2014.

[3] E.I. Smirnov, S.A. Tikhomirov, The limit object of Hausdorff spectrum in the category TLC, Journal of Mathematical

and Computational Science. 5 (2), 222–236, 2015.

[4] A.A. Kytmanov, A.S. Tikhomirov, S.A. Tikhomirov, Series of rational moduli components of stable rank two vector

bundles on P3, Selecta Mathematica, New Series. 25:29, 2019.

1Yaroslavl State Pedagogical University, Yaroslavl, Russia, [email protected]

2Yaroslavl State Pedagogical University, Yaroslavl, Russia, [email protected]

3Branch of Ural State University of Railway Transport in Tyumen, Tyumen, Russia, [email protected]

266

Some Fixed Point Theorems for Multivalued Mappings on Complete Metric Spaces

Hatice Aslan Hancer 1

Abstract. In this study, considering the recent techniques, which is used by Popescu for fixed points

of single valued mappings and by Klim-Wardowski for fixed points of multivalued mappings, we in-

troduce new contractive condition for multivalued mappings and present some fixed point results for

such mappings on complete metric space. Our results are proper generalizations of some earlier related

fixed point theorems.

Keyword: Multi valued mapping, complete metric space, fixed point.

AMS 2010: 54H25, 47H10.

References

[1] V. Berinde and M. Pacurar, The role of the pompeiu-hausdorff metric in fixed point theory, Creat. Math. Inform.,

22 (2), 35-42, 2013.

[2] V. I. Istratescu, Fixed point theory an introduction, Dordrecht D. Reidel Publishing Company 1981.

[3] S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30, 475-488, 1969.

[4] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 4 (5), 26-42, 1972.

[5] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math.

Anal. Appl., 141, 177-188, 1989.

[6] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math.

Anal. Appl., 334, 132-139, 2007.

[7] Y. Feng and S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type map-

pings, J. Math. Anal. Appl., 317, 103-112, 2006.

[8] O. Popescu, A new type of contractive mappings in complete metric spaces, submitted.

[9] I. Altun, G. Durmaz, M. Olgun, P -contractive mappings on metric spaces, Journal of Nonlinear Functional Analysis,

2018 (2018), Article ID 43, pp. 1-7, 2018.

[10] A. Fulga and A. Proca, A new generalization of Wardowski fixed point theorem in complete metric spaces, Advances

in the Theory of Nonlinear Analysis and its Applications, 1 (1), 57-63, 2017.


267

[11] A. Fulga and A. M. Proca, Fixed points for ϕE-Geraghty contractions, J. Nonlinear Sci. Appl., 10 (9), 5125-5131,

2017.

268

A Natural Way to Construct an Almost Hermitian B-Metric Structure

Mehmet Solgun 1, Yasemin Karababa 2

Abstract. In this work, we construct an almost Hermitian B-metric structure on a manifold M ×R,

where M is equipped with an almost contact B-metric structure. Further, we give some relations

between the classes of almost contact B-metric structure and the obtained almost Hermitian B-metric

structure.

Keyword: Almost Hermitian B-metric structure, almost contact B-metric structure, Norden metric.

AMS 2010: 53C10, 53C25, 53C27.

References

[1] G. T. Ganchev and A. V. Borisov, Note on the Complex Manifolds with a Norden Metric, Comptesrendus de

l’AcadA©mie bulgare des Sciences, 39(5), 31-34, 1986.

[2] M. Manev, K. Gribachev. Contactly conformal transformations of almost contact manifolds with B-metric, Serdica,

19, 1993, 287-299.

[3] Mancho Manev, Natural Connection with Totally Skew-Symmetric Torsion on Almost Contact Manifolds with B-

metric, arXiv: 1001.3800v3 [math.DG], Dec 2011.

[4] Hristo Manev, On the Structure Tensors of Almost Contact B-metric Manifolds, arXiv: 1405.3088v1 [math.DG],May

2014.

[5] Mancho Manev and Miroslava Ivanova, A Natural Connection on Some Classes of Almost Contact Manifolds with

B-metric, arXiv: 1110.3023v1 [math.DG], Oct 2011.

[6] Georgi Ganchev and Vesselka Mihova, Canonical Connection and the Canonical Conformal Group on an Almost Com-

plex Manifold with B-metric, Annuaire de L’Universite de Sofia ”St. Kliment Ohridski” Faculte de Mathematiques

et Informatique, 1(81), 195-206, 1987.



269

Some Fixed Point Theorems in Extended b−Metric Spaces with Applications

Mohammad Saeed Khan 1

Abstract. The purpose of this paper is to obtain fixed points of Ciric type operators in the framework

of extended b-metric space introduced by Kamran et al. [A generalization of b-metric space and some

fixed point theorems,Mathematics, 5(19) (2017), 7 pages]. Our results unify and improve the results of

Alqahtani et al. [Non-unique fixed point results in extended metric space,Mathematics, 6(68),(2018),

11 pages], Alsulami et al. [Ciric type nonunique fixed point theorems on b- metric spaces, Filomat,

3(11), (2017), 3147–3156] and others. As an application of our result, we establish the existence of

solution of a non-linear Fredholm integral equation. A numerical example is also presented to support

our result.

Keywords: Non unique fixed point, comparison function,Orbital admissible, orbitally continuous, ex-

tended b-metric space, Fredholm integral equation.

AMS 2010: 47H10; 54H25.

References

[1] J. Achari, On Ciri c’s non-unique fixed points, Mat. Vesnik, 13(28), 355-257, 1976.

[2] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially

ordered b-metric spaces, Math. Slovaca, 64(4), 941-960, 2014.

[3] B. Alqahtani, A. Fulga, E. Karapinar, Non-unique fixed point results in extended metric space, Mathematics, 6(68),

11 pages, 2018.

[4] H. H. Alsulami, E. Karapinar, V. Rako cevic, Ciric type nonunique fixed point theorems on b- metric spaces, Filomat,

3(11), 3147–3156, 2017.

[5] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Gos. Ped. Inst. Unianowsk,

30, 26–37, 1989.

[6] V. Berinde, Generalized contractions in quasimetric spaces, Seminar on fixed point theory, Babe s-Bolyai Unversity,

3, 3–9, 1993.

[7] V. Berinde, Sequences of operators and fixed points in quasimetric spaces, Studia Univ. Babe s-Bolyai Math., 41(4),

23–27, 1996.

1Sultan Qaboos University, Muscat, Oman, [email protected]

270

[8] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19(1), 7–22,

2003.

[9] M. Boriceanu, M. Bota, A. Petru susel, Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8(2), 367–377,

2010.

[10] Lj. Ciric, Generalized contraction and fixed point theorems, Publ. Inst. Math., 12(26), 19–26, 1971.

[11] Lj. Ciric, On contraction type mappings, Math.Balkanica, 1, 52–57, 1974.

[12] Lj. Ciric, On some maps with a nonunique fixed point, Publ. Inst. Math., 17(31), 52–58, 1974.

271

(Anti) Symmetrically Connected Extensions

Nezakat Javanshir 1, Filiz Yıldız 2

Abstract. In this talk, we will consider some approaches to the theories of symmetrically and anti-

symmetrically connected extensions for T0-quasi-metric spaces. Following that some various properties

of the corresponding T0-quasi-metric subspaces of symmetrically and antisymmetrically connected T0-

quasi-metric spaces are also discussed under the suitable condition of density with respect to the

symmetrization topology.

Keyword: T0-quasi-metric, symmetrically connected space, antisymmetric connected extension, sym-

metric pair

AMS 2010: 54D05, 54E35, 54D40

References

[1] M.J. Campion, E. Indurain, G. Ochoa and O. Valero, Functional equations related to weightable quasi-metrics,

Hacettepe J. Mat. Stat. 44 (4), 775–787, 2015.

[2] S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Springer, Basel, 2013.

[3] N. Demetriou and H.-P.A. Kunzi, A study of quasi-pseudometrics, Hacettepe J. Math. Stat. 46 (1), 33–52, 2017.

[4] A. Hellwig and L. Volkmann, The connectivity of a graph and its complement, Discrete Appl. Math. 156, 3325–3328,

2008.

[5] F. Yıldız and H.-P. A. Kunzi, Symmetric connectedness in T0-quasi-metric spaces, preprint.

1Hacettepe University, Ankara, Turkey, [email protected]

2Hacettepe University, Ankara, Turkey, [email protected]

272

Some Solutions to the Recent Open Problems with Pata and Zamfirescu’s Techniques

Nihal Yılmaz Ozgur 1, Nihal Tas 2

Abstract. In this talk, we present new solutions to the Rhoades’ open problem on the discontinuity

at fixed point and the fixed-circle problem on the geometry of Fix(T ), the set of fixed points of a

self-mapping T , on S-metric spaces. For this purpose, we modify some Pata and Zamfirescu’s results.

Also, we give some illustrative examples of our obtained theoretical results.

Keyword: Rhoades’ open problem, fixed-circle problem, discontinuity.

AMS 2010: 54H25, 47H10.

Acknowledgement: This work is financially supported by Balıkesir University under the Grant no.

BAP 2018 /019.

References

[1] V. Berinde, Comments on some fixed point theorems in metric spaces, Creat. Math. Inform., 27(1), 15-20, 2018.

[2] G. K. Jacob, M.S. Khan, C. Park and S. Yun, On generalized pata type contractions, mathematics, 6, 25, 2018.

[3] N. Y. Ozgur and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc., 2017.

https://doi.org/10.1007/s40840-017-0555-z

[4] N. Y. Ozgur and N. Tas, A new solution to the rhoades’ open problem with an application, submitted for publication.

[5] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl., 240, 284-289, 1999.

[6] V. Pata, A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10, 299-305, 2011.

[7] B. E. Rhoades, Contractive definitions and continuity, Contemp. Math., 72, 233-245, 1988.

[8] N. Tas and N. Y. Ozgur, On the geometry of fixed points of self-mappings on s-metric spaces, submitted for publi-

cation.

[9] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23, 292-298, 1972.

1Balıkesir University, Balıkesir, Turkey, [email protected]

2Balıkesir University, Balıkesir, Turkey, [email protected]

273

Some Results for Ψ− F−Geraghty Contraction on Metric-Like Space

Ozlem Acar 1

Abstract. In this talk, I introduced a new type of Geraghty type contractions and proved a fixed

point theorem in the class of metric-like spaces. In the end of talk I give an illustrative example.

Keyword: Fixed Point, ΨF -Geraghty Contraction, Metric-like.

AMS 2010: 54H25, 47H10.

References

[1] A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory and Applications,

2012(1),204, 2012.

[2] Aydi, Hassen, Abdelbasset Felhi, and Hojjat Afshari. ”New Geraghty type contractions on metric-like spaces, J.

Nonlinear Sci. Appl 10.2 (2017): 780-788.

[3] H. Aydi, F. Abdelbasset, and S. Slah, A Suzuki fixed point theorem for generalized multivalued mappings on metric-

like spaces, Glasnik matematicki, 52(1), 147-161, 2017.

[4] H. Aydi, and F. Abdelbasset, On best proximity points for various U3b1-proximal contractions on metric-like spaces,

J. Nonlinear Sci. Appl, 9(8), 5202-5218, 2016.


274

On Virtual Braids and Virtual Links

Valeriy Bardakov 1

Abstract. Virtual Knot Theory was defined by L. Kauffman [1] as a generalization of the Classical

Knot Theory. Also he defined the virtual braid group V Bn, which contains the classical braid group Bn

on n-strands. The group V Bn plays the fundamental role in the Virtual Knot Theory. In particular,

any virtual knot is the closure of a virtual braid. In [2] was introduced the group of virtual pure braid

group V Pn and was proved that V Bn is the semi-direct product of V Pn and the symmetric group Sn.

In my talk I will explain possible presentations of V Bn by automorphisms of some groups. Using these

representations, will be defined groups of virtual knots and links that are a strong invariants (see [3-4]).

Keyword: Braids; virtual braids; representations by automorphism.

AMS 2010: 57M25, 57M27.

References

[1] L. H. Kauffman, Virtual knot theory, Eur. J. Comb. 20(7), 663-690, 1999.

[2] V. G. Bardakov, The virtual and universal braids, Fund. Math. 181, 1-18, 2004.

[3] V. G. Bardakov, M. V. Neshchadim, On a representation of virtual braids by automorphisms. (Russian) Algebra

Logika 56, 539-547, 2017.

[4] V. G. Bardakov, Yu. A. Mikhalchishina, M. V. Neshchadim, Representations of virtual braids by automorphisms and

virtual knot group, Journal of Knot Theory and Its Ramifications, 26, 2017.

1Sobolev Institute of Mathematics, Novosibirsk, Russia, [email protected]

275

POSTER

Evolution of Quaternionic Curve in the Semi-Euclidean Space E42

Alperen Kızılay 1, Onder Gokmen Yıldız 2 and Osman Zeki Okuyucu 3

Abstract. In this paper, kinematics of quaternionic curve in semi-Euclidean space E42 is obtained in

terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic

curve are obtained. Also, examples of evolution equations of curvatures are given.

Keyword: Quaternionic curve, evolution, inextensible flow, semi-Euclidean space.

AMS 2010: 54C44, 22E15.

References

[1] N. Abdel-All, S. Mohamed, and M. Al-Dossary, Evolution of generalized space curve as a function of its local geometry,

Applied Mathematics. 5, 2381-2392, 2014 doi: 10.4236/am.2014.515230.

[2] T. Korpınar and S. Bas, Characterization of quaternionic curves by inextenaible flows, Prespacetime Journal, 7,

1680-1684, 2016.

[3] D. Y. Kwon, F. C. Park and D. P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18,

1156-1162, 2005.

[4] O. G. Yıldız and M. Tosun, A note on evolution of curves in the minkowski spaces, Adv. Appl. Clifford Algebras 27,

2873-2884, 2017.




276

Balance and Symmetry in Abiyev Squares

Asker Ali Abiyev 1, Yusif Alizada 2

Abstract. Magic squares are simple arrangements of numbers and symbols such that the figures in

each vertical, horizontal, and diagonal rows add up to the same values [1, 2]. The main object of

interest in mathematics is the properties they hold within.

One of the most important properties of the magic squares is that if the numbers in their respective lo-

cations in the magic squares are considered as point-masses, then the mass centre and geometric centre

of such a system will be the same [3, 4]. Calculations of the centre of mass of magic squares existing to

date show that all of them uphold this property, therefore enabling to call them ”balanced squares”.

Four arithmetic sequences (+b,+c,−b,−c) and symmetric cycles are used in the algorithm for Abiyev

Squares. In such squares of even order n, the sum of mass-position vectors (m−→r −mr vectors) of each

concentric frame of order k is expressed by the parameters of these arithmetic sequences.

When replacing even numbers - within any one frame or across different frames - that are symmetric

about orthogonal axes crossing the centre of the square, this distribution and the absolute values of

sum of mr vectors remain constant. On the other hand, since frames and cycles are different in Abiyev

Squares of odd order, the distribution of x, y coordinates by frame order k has a different pattern. The

outlined properties are unique to Abiyev Squares and not present in other magic squares.

Keyword: Keyword one, keyword two, keyword three.


1Azerbaijan National Academy of Sciences - Institute of Radiation Problems, Baku, Azerbaijan,

[email protected]

2Toronto, Canada, [email protected]

277

References

[1] A. Abiyev, Balanced (Magic) Squares http://askeraliabiyev.com/en/squares.html

[2] A. Abiyev, Dogal Sihirli Kareler (Natural Magic Squares), Bilim ve Teknik, Ekim (Science and Technology, October)

395, 87-89, 2000.

[3] A. Abiyev, A. Arslan, Azer Abiyev, A Comparison with Abiyev Balanced Square and Other Magic Squares,

IMS’2008 6th International Symposium on Intelligent and Manufacturing Systems ”Feature, Strategies and Inno-

vation”, Sakarya, Turkey. October 14-17, 2008.

[4] P. D. Loly, Franklin squares – a chapter in the scientific studies of magical squares, Complex Syst. 17, 143-161, 2007.

[5] N. S. Yanofsky, M. Zelcer, The Role of Symmetry in Mathematics, July 15, 2018.

278

Some Properties of Wajsberg Algebras

Cristina Flaut 1

Abstract. BCK-algebras were first introduced in mathematics by Y. Imai and K. Iseki, in 1966,

through the paper citeII; 66, as a generalization of the concept of set theoretic difference and proposi-

tional calculi. These algebras form an important class of logical algebras and have many applications

to various domains of mathematics (group theory, functional analyses, sets theory, etc.). Because of

the necessity to establish certain rational logic systems as a logical foundation for uncertain informa-

tion processing, various types of logical systems have been proposed. For this purpose, some logical

algebras appeared and have been researched ([20]). One of these algebras are MV-algebras, where MV

is referred to ”many valued”( [11]), which were originally introduced by Chang in [3]. He tried to pro-

vide a new proof for the completeness of the Lukasiewicz axioms for infinite valued propositional logic.

These algebras appeared in the specialty literature under equivalent names: bounded commutative

BCK-algebras or Wajsberg algebras, ([5]). Wajsberg algebras were introduced in 1984, by Font, Ro-

driguez and Torrens, through the paper [9] as an alternative model for the infinite valued Lukasiewicz

propositional logic.

In this presentation, we will give some exemple of finite bounded commutative BCK-algebras. In the

finite case, it is very useful to have many examples of such algebras. But, such examples, in general,

are not so easy to found. A method for this purpose can be Iseki’s extension. But, from the above,

we remark that the Iseki’s extension can’t be always used to obtain examples of finite commutative

bounded BCK-algebras with given initial properties, since the commutativity, or other properties, can

be lost. From this reason, we use other technique to provide examples of such algebras. For this

purpose, we use the connections between these algebras and Wajsberg algebras and the algoritm and

examples given in the papers [6] and [10].

Keywords: MV-algebras, Wajsberg algebras.

AMS 2010: 06F35, 06F99.

1Ovidius University of Constanta, Romania, [email protected]

279

References

[1] H. A. A. Abujabal, M. Aslam, A. B. Thaheem, A representation of bounded commutative BCK-algebras, Internat.

J. Math. & Math. Sci., 19(4), 733-736, 1996.

[2] D. Busneag, Categories of algebraic logic, Editura Academiei Romane, 2006.

[3] C. C. Chang, Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc. 88, 467-490, 1958.

[4] R. L. O. Cignoli, I. M. L .D. Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Trends in

Logic, Studia Logica Library, Dordrecht, Kluwer Academic Publishers, 7, 2000.

[5] R. Cignoli, A. T. Torell, Boolean Products of MV-Algebras: Hypernormal MV-Algebras, J Math Anal Appl (199),

637-653, 1996.

[6] C. Flaut, S. Hoskova-Mayerova, A. B. Saeid, R. Vasile, Wajsberg algebras of order, n, n ≤ 9,

https://arxiv.org/pdf/1905.05755.pdf

[7] C. Flaut, BCK-algebras arising from block codes, Journal of Intelligent and Fuzzy Systems 28(4), 1829–1833, 2015.

[8] C. Flaut, Some connections between binary block codes and hilbert algebras, in A. Maturo et all, Recent Trends in

Social Systems: Quantitative Theories and Quantitative Models, Springer 2017, p. 249-256.

[9] J. M. Font, A. J. Rodriguez, A. Torrens, Wajsberg Algebras, Stochastica, 8(1), 5-30, 1984.

[10] C. Flaut, R. Vasile, Wajsberg algebras arising from binary block codes, https://arxiv.org/pdf/1904.07169.pdf

[11] H. Gaitan, About quasivarieties of p-algebras and Wajsberg algebras, 1990, Retrospective Theses and Dissertations,

9440, https://lib.dr.iastate.edu/rtd/9440

[12] U. Hohle, S. E. Rodabaugh, Mathematics of fuzzy sets: logic, Topology and Measure Theory, Springer Science and

Business Media, LLC, 1999.

[13] Y. Imai, K. Iseki, On axiom systems of propositional calculi, Proc Japan Academic 42, 19–22, 1966.

[14] A. Iorgulescu, Algebras of Logic as BCK Algebras, Editura ASE, Bucuresti, 2008.

[15] Y. B. Jun, S. Z. Song, Codes based on BCK-algebras, Inform. Sciences., 181(2011), 5102-5109, 2011.

[16] Y. B. Jun, Satisfactory filters of BCK-algebras, Scientiae Mathematicae Japonicae Online, 9, 1–7, 2003.

[17] Meng, J., Jun, Y. B., BCK-algebras, Kyung Moon Sa Co. Seoul, Korea, 1994.

[18] D. Mundici, MV-algebras-a short tutorial, Department of Mathematics Ulisse Dini, University of Florence, 2007.

[19] D. Piciu, Algebras of Fuzzy Logic, Editura Universitaria, Craiova, 2007.

[20] J. T. Wang, B. Davvaz, P. F. He, On derivations of MV-algebras, https://arxiv.org/pdf/1709.04814.pdf

280

Some Characterizations of Vi Helices in 4-dimensional Semi Euclidean Space with

Index 2

Hasan Altınbas 1, Bulent Altunkaya 2 and Levent Kula 3

Abstract. In this study, we define harmonic curvatures of Vi helices in 4-dimensional semi Euclidean

space with index 2. Then, we give some characterizations of Vi helices that depend on harmonic cur-

vatures by using Serret-Frenet frame. Moreover, we give examples of Vi helices. Finally, we project

these curves on an arbitrary plane.

Keyword: Harmonic curvature, spacelike(timelike) curve, Vi helices.

AMS 2010: 53A35, 53C25.

References

[1] B. Altunkaya and L. Kula, On polynomial helices in n-dimensional Euclidean space Rn, Advances in Applied Clifford

Algebras, 28:4, 2018.

[2] B. O’Neil, Semi-Riemannian Geometry, Academic Press, New-York, 1983.

[3] T. A. Ahmad and M. Turgut, Some Characterizaton of Slant Helices in the Euclidean Space En, Hacettepe Journal

of Math. and Stat., 39(3), 327-336, 2010.

[4] E. Ozdamar and H. H. Hacisalihoglu, A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci.

Univ. Ankara, Ser A1, 24: 15-23, 1975.

[5] K. Ilarslan, N. Kilic and H. A. Erdem, Osculating curves in 4-dimensional semi-Euclidean space with index 2, Open

Math, 15: 562-567, 2017.

[6] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

[7] H. H. Hacisalihoglu, Diferensiyel Geometri 1, 3. Edition, 1998.

[8] L. Kula, N.Ekmekci and K. Ilarslan, Characterizations of slant helices in Euclidean 3-space, Turk J Math. Vol 34,

261-273, 2010.

[9] A. Sabuncuoglu, Diferensiyel Geometri (5. Edition), Nobel Press, 2014.

1Kirsehir Ahi Evran University, Kirsehir, Turkey, [email protected]



281

Some Results on GBS Operators

Hatice Gul Ince Ilarslan1

Abstract. We construct Generalized Boolean Sum operators associated with combinations of the

Szasz-Kantorovich operators based on Brenke-type polynomials. We obtain the rate of convergence for

the GBS operators with the help of the mixed modulus of continuity and the Lipschitz class of Bogel

continuous functions.

Keyword: SzA¡sz-Kantorovich operators, Brenke polynomials, GBS operators.

AMS 2010: 41A10, 41A25.

References

[1] C.Atakut, I.Buyukyazıcı, Approximation by Kantorovich-Szasz type operators based on Brenke type polynomials,

Numer. Funct. Anal. Optim vol.37, 12, pp.1488-1502, 2016.

[2] C. Badea, I. Badea, H. H. Gonska, A test function theorem and approximation by pseudo polynomials, Bull. Austral.

Math. Soc., 34, 53–64, 1986.

[3] C. Badea, I. Badea, C. Cottin, H. H. Gonska, Notes on the degree of approximation of B-continuous and B-

differentiable functions, J. Approx. Theory Appl., 4, 95–108, 1988.

[4] C. Badea, C.Cottin, Korovkin-type theorems for Generalised Boolean Sum operators, Colloquia Mathematica Soci-

etatis Janos Bolyai.Approximation Theory, Kecskemt (Hungary), 58, 51-67, 1990.

[5] I. Badea, Modulus of continuity in Bogel sense and some applications for approximation by a Bernstein-type operator,

Studia Univ. Babes-Bolyai, Ser. Math-Mech, 18, 2, 69–78 (Romanian), 1973.

1Gazi University, Ankara, Turkiye, [email protected]

282

On Generalized Expansive Mappings in the Setting of Elliptic Valued Metric Spaces

Isıl Arda Kosal 1, Mahpeyker Ozturk 2 and Hidayet Huda Kosal 3

Abstract. In this study, we first aim to define a new metric space which is a generalization of complex

valued metric spaces using the set of elliptic numbers

Ep =z = x+ iy : x, y ∈ R, i2 = p < 0

,

and called this space as an elliptic valued metric space. We investigate some topological properties of

this new space and give some comparisons between existing literature. Also we obtain some fixed point

results in the setting of elliptic valued metric spaces by introducing new classes of expansive mappings.

We see that our results are real generalizations of the consequences of several fixed point theorems.

Keyword: Elliptic valued metric space, expansive mappings, common fixed point.

AMS 2010: 54H25, 47H10, 54E50.

References

[1] A. Harkin, J. Harkin, Geometry of generalized complex numbers, Mathematics Magazine, 77(2), 118-129, 2004.

[2] A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional

Analysis and Optimization, 32(3), 243-253, 2011.

[3] M. Ozturk, Common fixed points theorems satisfying contractive type conditions in complex valued metric spaces,

Abstract and Applied Analysis, 2014, 7 pages, 2014.

[4] V. Popa, Some fixed point theorems of expansion mappings, Demonstratio Math., 19, 699-702, 1986.

[5] A. Constantin, On fixed points in noncomplete metric spaces, Publ. Math. Debrecen, 40, 297-301, 1992.

[6] S. Z. Wang, B. Y. Li, Z. M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Japonica, 29,

631-636, 1984.




283

The Shannon Entropy as an Edge Detector in Grayscale Images

J. Martınez-Aroza, J.F. Gomez-Lopera, D. Blanco-Navarro

and J. Rodrıguez Camacho1

Abstract. Shannon entropy H can be useful to evaluate the quantity of information in images. This

measure tends to become saturated, that is, to reach high values, when dealing with a large scale of gray

levels, as well as with textures or degradations such as noise or blurring. This is due to a large amount

of irrelevant information which makes this measure useless for measuring significant information. In

this paper we present a corrected information measure, the clustered entropy CH, based on clustering

local histograms. CH has a zero value for quasi-homogeneous regions and reaches high values for

regions containing edges. In this paper we use CH as an edge detector, by centering a sliding window

on every pixel of the image, and calculating the clustered entropy of the corresponding histogram.

A search for local maxima throughout the resulting matrix of CH provides the final image of edges.

The mathematical properties of CH are studied, a comparison between CH and H is done, and some

comparative experiments of edge detection are shown in this paper.

Keywords: image segmentation; histogram clustering; entropic edge detection; Shannon entropy; clus-

tered entropy; gray level quantization.

AMS 2010: 68U10.

References

[1] M.R. Anderberg, R.K. Blashfield, Cluster Analysis, Sage Publication Inc, 1984.

[2] Z. Atae-Allah, J.Martınez-Aroza, A filter to remove Gaussian noise by clustering the gray scale, J. of Mathematical

Imaging and Vision 17, 15-25, 2002.

[3] V. Barranco-Lopez, P. Luque-Escamilla, J. Martınez-Aroza, R. Roman-Roldan, Entropic texture-edge detection for

image segmentation, Electronic Letters 31 (11), 867-869, 1995.

[4] J. Canny, A computational approach to edge detection, IEEE Transactions on pattern Recognition and machine

Intelligence 8 (6), 679-698, 1986.

[5] R.C. Dubes, A.K. Jain, Clustering techniques: the user’s dilemma, Pattern Recognition 8, 247-260, 1976.

1University of Granada, Spain, respectively [email protected], [email protected], [email protected] and [email protected]

284

[6] J.F. Gomez-Lopera, J. Martınez-Aroza, M.A. Rodrıguez-Valverde, M.A. Cabrerizo-Vılchez, F.J. Montes-Ruiz-

Cabello, Entropic image segmentation of sessile drops over patterned acetate, Mathematics and Computers in Simu-

lation 118, 239-247, 2015.

[7] M. Gray, Entropy and Information Theory, Springer-Verlag, Nueva York, 1990.

[8] S. Guiasu, Information Theory with Applications, Library of Congress Cataloging in Publication Data, 1977.

[9] R.M. Haralick, L.G. Shapiro, Image Segmentation Techniques, Computer Vision, Graphics and Image Processing 29,

100-132, 1985.

[10] A.K. Jain, R.C. Dubes, Algorithms for Clustering Data, Englewood Cliffs, NJ: Prentice-Hall, 1988.

[11] Q.D. Katatbeh, J. Martınez-Aroza, J.F. Gomez-Lopera, D. Blanco-Navarro, An optimal segmentation method using

Jensen-Shannon divergence via a multi-size sliding window technique, Entropy 17, 7996-8006, 2015.

285

Investigation of The Sleep Quality of Cerebrovascular Patients

Kamile Sanli Kula 1, Aysu Yetis 2 and Emrah Gurlek 3

Abstract. In this study, it was aimed to investigate the sleep quality of cerebrovascular patients and

to examine them according to various variables. For this purpose, 158 cerebrovascular patients at

Kirsehir Ahi Evran University Training and Research Hospital between July 2017 and July 2018 were

applied to Pittsburg Sleep Quality Index. 44% of these patients were females, while 56% were males.

At the end of the study we obtain the result: 69% of patients had bad sleep quality of these patients.

Sleep quality of male patients were significantly better than female patients.

Keyword: Cerebrovascular disease, stroke, sleep.

AMS 2010: 62P10.

This work was supported by the Scientific Research Projects Council of Ahi Evran University, Kirsehir,

Turkey under Grant FEF.A4.17.016.

References

[1] Y. Agargun, H. Kara and O. Anlar, The validity and reliability of the pittsburgh sleep quality index, Turk Psikiyatri

Dergisi, 7, 107-115, 1996.

[2] A. G. Harvey, K. Stinson, K.L. Whitaker, D. Moskovitz and H. Virk, The subjective meaning of sleep quality: a

comparison of individuals with and without insomnia, Sleep, 31 (3), 383-393, 2008.

[3] A. M. Karadag, Classification of sleep disorders, Akciger Arsivi, 8, 88-91, 2007.

[4] A.D. Krystal, Edinger, J.D., Measuring sleep quality, Sleep Medicine, 9 (1), 10–17, 2008.

[5] S. Ozturk, Epidemiology of cerebrovascular diseases and risk factors-perspectives of the world and turkey, Turkish

Journal of Geriatrics, 13 (1), 51-58, 2009.

1Kırsehir Ahi Evran University, Kirsehir, Turkey, [email protected]

2Kırsehir Ahi Evran University Education and Research Hospital, Kirsehir, Turkey, [email protected]

3Kirsehir Ahi Evran University, Kirsehir, TURKEY,[email protected]

286

A Soft Set Approach for IFS

Kemal Taskopru 1

Abstract. Iterated function systems (IFS) is a common way to generate fractals that are complex,

irregular and unpredictable objects existing in nature. However, soft set theory is emerged to deal

with the complexity of uncertain data. Many works associated and applied the soft sets to various

mathematical structures. In this work, we combine such two phenomena and we define soft IFS which

present a soft set approach for the IFSs. We also give some properties of the proposed approach and

illustrate some examples.

Keyword: soft set, iterated function system, fractal

AMS 2010: 03E99, 28A80, 54C50.

References

[1] D. Molodtsov, Soft set theory–first results, Comput. Math. Appl. 37 (4), 19–31, 1999.

[2] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (4), 555–562, 2003.

[3] M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl. 61 (7), 1786–1799, 2011.

[4] H. Aktas, N. Cagman, Soft sets and soft groups, Inform. Sci. 177 (13), 2726–2735, 2007.

[5] Y. B. Jun, C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (11), 2466–

2475, 2008.

[6] P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl.

44 (8), 1077–1083, 2002.

[7] Y. Zou, Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowl.-Based Syst. 21 (8),

941–945, 2008.

[8] F. Feng, C. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft

Comput. 14 (9), 899–911, 2010.

[9] J. C. R. Alcantud, Some formal relationships among soft sets, fuzzy sets, and their extensions, Int. J. Approx.

Reason. 68, 45–53, 2016.

[10] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (5), 713–747, 1981.


287

[11] M. F. Barnsley, Fractals Everywhere, 2nd ed., Academic Press, Boston, MA, 1993.

[12] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., John Wiley & Sons Inc.,

Hoboken, NJ, 2003.

[13] A. Edalat, Power domains and iterated function systems, Inform. and Comput. 124 (2), 182–197, 1996.

[14] P. F. Duvall, L. S. Husch, Attractors of iterated function systems, Proc. Amer. Math. Soc. 116 (1), 279–284, 1992.

288

On Lorentzian Ruled Surfaces in 4-Dimensional Semi Euclidean Space with Index 2

Kıvanc Karakas 1, Hasan Altınbas 2, Bulent Altunkaya 3 and Levent Kula 4

Abstract. In this work, we investigate Lorentzian ruled surface couples which generated from the

split quaternion product of constant vectors and a space curve in 4-dimensional semi Euclidean space

with index 2. Furthermore, we obtain some special relations between the tangent spaces and normal

spaces of these surface couples. Moreover, we support our findings with examples.

Keyword: Lorentzian ruled surface, split quaternion.

AMS 2010: 53A35, 53C25.

This work is supported by Kırsehir Ahi Evran University Scientific Research Project Coor-

dination Unit. Project number: FEF.A3.17.002.

References

[1] X. Wang and R. Goldman, Quaternion rational surfaces: Rational surfaces generated from the quaternion product

of two rational space curves, Journal of Graphical Models, no.81, 18-32, 2015.

[2] F. Chen, J. Zheng and T. Sederberg, µ-basis of a rational ruled surface, Journal of Computer Aided Geometric

Design, 2001, no.18, 61-72.

[3] L. Kula, Split quaternions and the geometrical applications, Ph.D. Thesis, Ankara University, Institute of Science,

Ankara, 2003.

[4] L. Kula and Y. Yayli, Split quaternions and rotations in semi-Euclidean space E42 ,Journal of the Korean Mathematical

Society, 2007, 44-6, 1313-1327.

[5] B. Altunkaya and L. Kula, On polynomial helices in n-dimensional Euclidean space Rn, Advances in Applied Clifford

Algebras, 28:4, 2018.

[6] B. O’Neil, Semi-Riemannian Geometry, Academic Press, New-York, 1983.

1Kırsehir Ahi Evran University, Kırsehir, Turkey, [email protected]




289

[7] E. Ozdamar and H. H. Hacisalihoglu, A characterization of inclined curves in Euclidean n-space, Comm. Fac. Sci.

Univ. Ankara Ser A1, 24: 15-23, 1975.

[8] K. Ilarslan, N. Kilic and H. A. Erdem, Osculating curves in 4-dimensional semi-Euclidean space with index 2, Open

Math, 15: 562-567, 2017.

[9] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

[10] H. H. Hacisalihoglu, Diferensiyel Geometri 1, 3. Edition, 1998.

[11] A. Sabuncuoglu, Diferensiyel Geometri (5. Edition), Nobel Press, 2014.

290

Some New Associated Curves in Minkowski 3-Space

Mahmut Ergut 1, Alev Kelleci 2

Abstract. In this paper, we introduce the notion of some new associated curves of the non-null Frenet

curve in the Minkowski 3-space, by moving from the notion defined in [1]. The main aim of that paper

is to find some relationships between a non-null Frenet curve and its associated curve.

Keyword: Minkowski 3-space, Slant helix, Associated curves.

AMS 2010: 53B30, 53A35.

References

[1] S. Deshmukh, B.-Y. Chen and A. Alghanemi, Natural mates of frenet curves in euclidean 3-space, Turkish Journal

of Mathematics (5) 42, 2826–2840, 2018.

[2] J. H. Choi, Y. H. Kim and A. T. Ali, Some associated curves of frenet non-lightlike curves in e13, Journal of

Mathematical Analysis and Applications, (2) 394 , 712–723, 2012.

[3] B.-Y. Chen and F. Dillen, Rectifying curves as centrodes and extremal curves, Bulletin-Institute of Mathematics

Academia Sinica (2) 33, 77, 2005.

[4] A. T. Ali and R. Lopez, On slant helices in minkowski space 31, J. Korean Math. Soc., 48, 159–167, 2011.

[5] K. Ilarslan and E. Nesovic, On rectifying curves as centrodes and extremal curves in the minkowski 3-space, Novi

Sad J. Math (1) 37, 53–64, 2007.

[6] A. Kelleci, Conjugate mates for non-null Frenet curves, Sakarya University Journal of Science, 23(4), 600-604, 2019.

[7] A. Kelleci, Natural mates of non-null Frenet curves in Minkowski 3-space, arXiv:1804.04705, 2018.

[8] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, volume 16 (Weisbaden: Braunschweig, 1999).

[9] B. O’neill, Semi-Riemannian geometry with applications to relativity, volume 103 (Academic press, 1983).


2Fırat University, Elazıg, Turkey, [email protected]

291

De-Moivre and Euler Formulae for Dual-Hyperbolic Numbers

Mehmet Ali Gungor 1, Elma Kahramani 2

Abstract. In this study, we generalize the well-known formulae of De-Moivre and Euler of hyper-

bolic numbers to dual-hyperbolic numbers. Furthermore, we investigate the roots and powers of a

dual-hyperbolic number by using these formulae. Consequently, we give some examples to illustrate

the main results in this paper. .

Keyword: Hyperbolic number, Dual number, De Moivre’s formula and Euler’s formula.

AMS 2010: 51M10, 47L50.

References

[1] Cho, E., De-Moivres formula for quaternions. Appl. Math. Lett. 11(6), 33-35 ,1998.

[2] Yuce, S. ve Ercan, Z., On Properties of the Dual quaternions, European Journal of Pure and Applied Mathematics,

4 (2): 142-146, 2011.

[3] Kabadayi, H., Yayla, Y., De-Moivres formula for dual quaternions. Kuwait J. Sci. Technol. 38(1), 15-23, 2011.

[4] Isil Arda Kosal, A note on hyperbolic quaternions, Universal journal of mathematics and Applications. 1(3), 155-159,

2018.

[5] V. Majernik, Multicomponent number systems, Acta Physica Polonica A, 3(90), 491-498, 1996.

[6] Messelmi F., Dual-complex numbers and their holomorphic functions. https://hal.archives-ouvertes.fr/hal-01114178,

2015.

[7] Cihan A. and Gungor, M.A., On dual-hyperbolic numbers with generalized fibonacci and lucas numbers components.

submitted.

[8] Gungor, M.A. and Tetik A., De-Moivre and Euler formulae for dual-complex numbers. submitted.

[9] Ozdemir, M., The roots of a split quaternion, Appl. Math. Lett. 22, 258-263, 2009.

[10] MacFarlane, A., Hyperbolic quaternions, Proc. Roy. Soc. Edinburgh, 169-181, 1900.



292

On the Jerk in Motion Along a Space Curve

Mehmet Guner 1

Abstract. The jerk vector of a moving particle is the third time derivative of the position vector and

thus the time derivative of the acceleration vector. A useful resolution of the acceleration vector of a

particle traveling along a space curve is well known in the literature [1]. A similar resolution of the

jerk vector is given as a new contribution to field [2]. In the present article, we take into account of

a particle moving on a space curve which is equipped with the Bishop frame and study the aforesaid

resolution of the jerk vector for this particle. Furthermore, we have given an illustrative example to

explain how the our result works.

Keyword: Kinematics of a particle, plane and space curves, Siacci, jerk, Bishop frame.

AMS 2010: 70B05, 14H50.

References

[1] F. Siacci, Moto per una linea gobba, Atti R Accad Sci. Torino. 14, 946-951, 1879.

[2] K.E. Ozen, M. Tosun, F.S. Dundar, An alternative approach to jerk in motion along a space curve with applications,

Journal of Theoretical and Applied Mechanics. 57, 435-444, 2019.

[3] J. Casey, Siacci’s resolution of the acceleration vector for a space curve, Meccanica. 46, 471-476, 2011.

[4] N. Grossman, The sheer joy of celestial mechanics, Birkhauser, Basel, 1996.

[5] Z. Kucukarslan, M.Y. Yılmaz, M. Bektas, Siacci’s theorem for curves in finsler manifold f3, Turkish Journial of

Science and Technology. 7, 181-185, 2012.

[6] K.E. Ozen, M. Tosun, M. Akyigit, Siacci’s theorem according to darboux frame, An. St. Univ. Ovidius Constanta.

25, 155–165, 2017.

[7] S.H Schot, Jerk: the time rate of change of acceleration, American Journal of Physics. 46, 1090-1094, 1978.

[8] M. Tsirlin, Jerk by axes in motion along a space curve, Journal of Theoretical and Applied Mechanics. 55, 1437-1441,

2017.

[9] R.L. Bishop, There is more than one way to frame a curve, The American Mathematical Monthly. 82, 246-251, 1975.


293

[10] B. Bukcu, M.K. Karacan, The slant helices according to bishop frame, International Journal of Computational and

Mathematical Sciences. 3, 67-70, 2009.

294

On Asymptotic Aspect of Some Functional Equations in Metric Abelian Groups

M. B. Moghimi 1

Abstract. In this note we deal the following functional equations:

f(x + y)& = f(x) + f(y)

f(x + y) + f(x− y)& = 2f(x)

f(x + y) + f(x− y)& = 2f(x) + 2f(y)

f(x + y) + f(x− y)& = 2f(x) + f(y) + f(−y)

We investigate the asymptotic stability behavior of the above functional equations. Indeed, we show

that if these equations hold approximately for large arguments with an upper bound ε, then they are

also valid approximately everywhere with a new upper bound which is a constant multiple of ε. We

applied these results to the study of asymptotic properties of these functional equations. We also

obtain some results of hyperstability character for these functional equations..

Keywords: functional equation, stability, asymptotic stability, metric group.

AMS 2010: 39B82, 39B62.

References

[1] A. Bahyrycz, Zs. Pales and M. Piszczek, Asymptotic stability of the Cauchy and Jensen functional equations, Acta

Math. Hungar., 150, 131–141, 2016.

[2] B. Khosravi, M. B. Moghimi and A. Najati, Asymptotic aspect of Drygas, quadratic and Jensen functional equations

in metric abelian groups, Acta Math. Hungar. 155, no. 2, 248–265, 2018.

[3] D. Molaei and A. Najati, Hyperstability of the general linear equation on restricted domains, Acta Math. Hungar.,

149, 238–253, 2016.

[4] M. Piszczek and J. Szczawinska, Stability of the Drygas functional equation on restricted domain, Results. Math. 68,

11–24, 2015.

1University of Mohagheghe Ardabili, Ardabil, Iran, [email protected]

295

A Unified Approach to Fractal Hilbert-type Inequalities

Tserendorj Batbold 1, Mario Krnic 2 and Predrag Vukovic 3

Abstract. In the present study we provide a unified treatment of fractal Hilbert-type inequalities.

Our main result is a pair of equivalent fractal Hilbert-type inequalities including a general kernel and

weight functions. A particular emphasis is devoted to a class of homogeneous kernels. In addition,

we impose appropriate conditions for which the constants appearing on the right-hand sides of the

established inequalities are the best possible. As an application, our results are compared with some

previously known from the literature.

Keyword: Hilbert inequality, conjugate parameters, local fractional integral.

AMS 2010: 26D15.

References

[1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, second edition, Cambridge University Press, Cambridge, 1967.

[2] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong

Kong, 2011.

[3] M. Krnic, J. Pecaric, I. Peric, P. Vukovic, Recent advances in Hilbert-type inequalities, Element, Zagreb, 2012.

[4] G-S. Chen, Generalizations of Holder’s and some related integral inequalities on fractal space, Journal of Function

Spaces and Applications, 9 pp., 2013.

[5] V. Adiyasuren, Ts. Batbold, M. Krnic, Multiple Hilbert-type inequalities involving some differential operators, Banach

J. Math. Anal. 10(2), 320–337, 2016.

[6] Ts. Batbold, M. Krnic, J. Pecaric, P. Vukovic, Further development of Hilbert-type inequalities, Element, Zagreb,

2017.

1Department of Mathematics, National University of Mongolia, Ulaanbaatar 14201, Mongolia, [email protected]

2University of Zagreb, Faculty of Electrical Engineering and Computing, Croatia, [email protected]

3University of Zagreb, Faculty of Teacher Education, Croatia, [email protected]

296

List of Participants of IECMSA-2019

Invited Speakers

Prof. Dr. Arif Salimov (Baku State University, Azerbaijan)

Prof. Dr. Ayman Rateb Badawi (American University of Sharjah, UEA)

Prof. Dr. Francesco Altomare (University of Bari Aldo Moro, Italy)

Prof. Dr. Sidney A. Morris (Federation University Australia, Australia)

Prof. Dr. Toma Albu (IMAR, Romania)

Prof. Dr. Varga Kalantarov (Koc University, Turkey)

Prof. Dr. Wolfgang Sproessig (TU Bergakademie Freiberg, Germany)

Participants

Prof. Dr. Adıguzel Dosiyev (Near East University, Cyprus)

Prof. Dr. Agamirza Bashirov (Eastern Mediterranean University, Turkey)

Prof. Dr. Akbar A. Aliev (Azerbaijan National Acad. of Sci., Azerbaijan)

Prof. Dr. Alexander Schmitt (Freie Universitat Berlin, Germany)

Prof. Dr. Ali Akhmedov (Baku State University, Azerbaijan)

Prof. Dr. Arvid Naess (Norwegian University, Norway)

Prof. Dr. Asker Ali Abiyev (Azerbaijan National Acad. of Sci., Azerbaijan)

Prof. Dr. Aynur Kaymakcı (Selcuk University, Turkey)

Prof. Dr. Cesim Temel (Van Yuzuncu Yıl University, Turkey)

Prof. Dr. Chaudry Masood Khalique (North-West University, SouthAfrica)

Prof. Dr. Cihan Ozgur (Balıkesir University, Turkey)

Prof. Dr. Cristina Flaut (Ovidius Univesity, Romania)

Prof. Dr. Cumali Ekici (Eskisehir Osmangazi University, Turkey)

Prof. Dr. Daniyal Israfilzade (Balıkesir University, Turkey)

297

Prof. Dr. Elgiz Bairamov (Ankara University, Turkey)

Prof. Dr. Elimhan Mahmudov (Istanbul Technical University, Turkey)

Prof. Dr. Etibar Panakhov (Baku State University, Azerbaijan)

Prof. Dr. Eugeny Smirnov (Yaroslavl State Pedagogical University, Russia)

Prof. Dr. Fikret Aliev (Baku State University, Azerbaijan)

Prof. Dr. Galina Mehdiyeva (Baku State University, Azerbaijan)

Prof. Dr. Gholam Reza Hojjati (University of Tabriz, Iran)

Prof. Dr. H. Hilmi Hacısalihoglu (Bilecik Seyh Edebali University, Turkey)

Prof. Dr. Hatice Gul Ince Ilarslan (Gazi University, Turkey)

Prof. Dr. Hossein Kheiri Estiar (Tabriz University, Iran)

Prof. Dr. Idzhad Sabitov (Lomonosov Moscow State University, Russia)

Prof. Dr. Inta Volodko (Riga Technical University, Latvia)

Prof. Dr. Jose Martınez-Aroza (Granada University, Spain)

Prof. Dr. Josef Mikes (Palacky University Olomouc, Czech Republic)

Prof. Dr. Katarzyna Horbacz (Silesian University, Poland)

Prof. Dr. Kamile Sanlı Kula (Kırsehir Ahi Evran University, Turkey)

Prof. Dr. Kazem Ghanbari (Sahand University of Technology, Iran)

Prof. Dr. Kazım Ilarslan (Kırıkkale University, Turkey)

Prof. Dr. Levent Kula (Kırsehir Ahi Evran University, Turkey)

Prof. Dr. Mahmut Ergut (Tekirdag Namık Kemal University, Turkey)

Prof. Dr. Mammad H. Yagubov (Baku State University, Azerbaijan)

Prof. Dr. Mariam Avalishvili (University of Georgia, Georgia)

Prof. Dr. Masoumeh Zeinali (University of Tabriz, Iran)

Prof. Dr. Mehmet Ali Gungor (Sakarya University, Turkey)

Prof. Dr. Mehmet Ali Sarıgol (Pamukkale University, Turkey)

Prof. Dr. Mehriban Imanova (Baku State University, Azerbaijan)

Prof. Dr. Mikail Et (Fırat University, Turkey)

Prof. Dr. Mohammed Saeed Khan (Sultan Qaboos University, Oman)

Prof. Dr. M. Rashid Kamal Ansari (Sir Syed Uni. of Engr. and Tech., Pakistan)

Prof. Dr. Murat Tosun (Sakarya University, Turkey)

298

Prof. Dr. Musa Cakir (Van Yuzuncu Yıl University, Turkey)

Prof. Dr. Nazım Mahmudov (Eastern Mediterranean University, Turkey)

Prof. Dr. Necip Simsek (Istanbul Ticaret University, Turkey)

Prof. Dr. Nico Groenewald (Nelson Mandela University, SouthAfrica)

Prof. Dr. Nihal Yılmaz Ozgur (Balıkesir University, Turkey)

Prof. Dr. Nihan A. Aliyev (Baku State University, Azerbaijan)

Prof. Dr. Nirmal Sacheti (Sultan Qaboos University, Oman)

Prof. Dr. Nizameddin Iskenderov (Baku State University, Azerbaijan)

Prof. Dr. Pallath Chandran (Sultan Qaboos University, Oman)

Prof. Dr. Pranas Katauskis (Vilnius University, Lithuania)

Prof. Dr. Predrag Vukovic (Zagreb University, Croatia)

Prof. Dr. Ramiz Aslanov (Azerbaijan National Acad. of Sci., Azerbaijan)

Prof. Dr. S. Amir M. Ghannadpour (Shahid Beheshti University, Iran)

Prof. Dr. Sadi Bayramov (Baku State University, Azerbaijan)

Prof. Dr. Salah Eddine Rebiai (University of Batna 2, Algeria)

Prof. Dr. Shakir Yusubov (Baku State University, Azerbaijan)

Prof. Dr. Shao-Ming Fei (Capital Normal University, China)

Prof. Dr. Snezhana Hristova (Plovdiv University, Bulgaria)

Prof. Dr. Soley Ersoy (Sakarya University, Turkey)

Prof. Dr. Syam Prasad Kuncham (MIT, India)

Prof. Dr. V. R. Lakshmi Gorty (SVKM’s NMIMS University, India)

Prof. Dr. Valeriy Bardakov (Sobolev Institute of Mathematics, Russia)

Prof. Dr. Vaqif Ibrahimov (Baku State University, Azerbaijan)

Prof. Dr. Victor Martinez-Luaces (The Republic of Uruguay University, Uruguay)

Prof. Dr. Vladas Skakauskas (Vilnius University, Lithuania)

Prof. Dr. Xinhua Yang (Huazhong Uni. of Sci. and Tech., China)

Prof. Dr. Yashar Mehraliyev (Baku State University, Azerbaijan)

Prof. Dr. Yusif Gasimov (Azerbaijan University, Azerbaijan)

Assoc. Prof. Dr. Alexander Skaliukh (Southern Federal University, Russia)

Assoc. Prof. Dr. Arzu Q. Aliyeva (Azerbaijan National Acad. of Sci., Azerbaijan)

299

Assoc. Prof. Dr. Aynur Yalcıner (Selcuk University, Turkey)

Assoc. Prof. Dr. Aytekin Cıbık (Gazi University, Turkey)

Assoc. Prof. Dr. Boris Semenov (Saint Petersburg State University, Russia)

Assoc. Prof. Dr. Celil Nebiyev (Ondokuz Mayıs University, Turkey)

Assoc. Prof. Dr. Ebrahim Ghorbani (K. N. Toosi University of Technology, Iran)

Assoc. Prof. Dr. Elvin I. Azizbayov (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Emrah Evren Kara (Duzce University, Turkey)

Assoc. Prof. Dr. Erol Yılmaz (Bolu Abant Izzet Baysal University, Turkey)

Assoc. Prof. Dr. Eva Voronkova (St. Petersburg State University, Russia)

Assoc. Prof. Dr. Faig Namazov (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Fuat Usta (Duzce Univesity, Turkey)

Assoc. Prof. Dr. Furkan Yıldırım (Ataturk University, Turkey)

Assoc. Prof. Dr. Gulnar Salmanova (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Gurhan Icoz (Gazi University, Turkey)

Assoc. Prof. Dr. Habil Fattayev (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Harikrishnan Panackal (MIT, India)

Assoc. Prof. Dr. Ismail Kınacı (Selcuk University, Turkey)

Assoc. Prof. Dr. Jafar Azami (UMA, Iran)

Assoc. Prof. Dr. Jasbir Singh Manhas (Sultan Qaboos University, Oman)

Assoc. Prof. Dr. K. Babushri Srinivas (MIT, India)

Assoc. Prof. Dr. Kazem Haghnejad Azar (University of Mohaghegh Ardabili, Iran)

Assoc. Prof. Dr. Mahmut Akyigit (Sakarya University, Turkey)

Assoc. Prof. Dr. Messaoud Boulbrachene (Sultan Qaboos University, Oman)

Assoc. Prof. Dr. Metanet Mursalova (Baku State University, Azerbaijan)

Assoc. Prof. Dr. M. Bagher Moghimi (UMA, Iran)

Assoc. Prof. Dr. Mustafa Fahri Aktas (Gazi University, Turkey)

Assoc. Prof. Dr. Ozlem Acar (Selcuk University, Turkey)

Assoc. Prof. Dr. Osman Zeki Okuyucu (Bilecik Seyh Edebali University, Turkey)

Assoc. Prof. Dr. Resat Yılmazer (Fırat University, Turkey)

Assoc. Prof. Dr. Samed Aliyev (Baku State University, Azerbaijan)

300

Assoc. Prof. Dr. Serpil Halıcı (Pamukkale University, Turkey)

Assoc. Prof. Dr. Seyda Kılıcoglu (Baskent University, Turkey)

Assoc. Prof. Dr. Sevda Isayeva (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Shiv Kaushik (Delhi University, India)

Assoc. Prof. Dr. Suzan Cival Buranay (Eastern Mediterranean University, Cyprus)

Assoc. Prof. Dr. Tofig Huseynov (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Tuncer Acar (Selcuk University, Turkey)

Assoc. Prof. Dr. Vaqif Gasimov (Baku State University, Azerbaijan)

Assoc. Prof. Dr. Yusif Sevdimaliyev (Baku State University, Azerbaijan)

Assist. Prof. Dr. Akbar Paad (University of Bojnord, Iran)

Assist. Prof. Dr. Andrej Novak (University of Zagreb, Croatia)

Assist. Prof. Dr. Ayse Yılmaz Ceylan (Akdeniz University, Turkey)

Assist. Prof. Dr. Azizeh Jabbari (Tabriz University, Iran)

Assist. Prof. Dr. Basar Yılmaz (Kırıkkale University, Turkey)

Assist. Prof. Dr. Bulent Altunkaya (Ahi Evran University, Turkey)

Assist. Prof. Dr. Davood Ahmadian (University of Tabriz, Iran)

Assist. Prof. Dr. Didem Aydın Arı (Kırıkkale University, Turkey)

Assist. Prof. Dr. Elif Segah Oztas (Karamanoglu Mehmetbey University, Turkey)

Assist. Prof. Dr. Erdal Bayram (Tekirdag Namık Kemal University, Turkey)

Assist. Prof. Dr. G. Canan Hazar Gulec (Pamukkale University, Turkey)

Assist. Prof. Dr. Gulhan Ayar (Karamanoglu Mehmetbey University, Turkey)

Assist. Prof. Dr. Hasim Cayır (Giresun University, Turkey)

Assist. Prof. Dr. Harun Barıs Colakoglu (Mediterranean University, Turkey)

Assist. Prof. Dr. Hatice Aslan Hancer (Kırıkkale University, Turkey)

Assist. Prof. Dr. Hidayet Huda Kosal (Sakarya University, Turkey)

Assist. Prof. Dr. Khole Timothy Poumai (Delhi University, India)

Assist. Prof. Dr. Kemal Taskopru (Bilecik Seyh Edebali University, Turkey)

Assist. Prof. Dr. Mehmet Guner (Sakarya University, Turkey)

Assist. Prof. Dr. Mehmet Solgun (Bilecik Seyh Edebali University, Turkey)

Assist. Prof. Dr. Merve Ilkhan (Duzce University, Turkey)

301

Assist. Prof. Dr. M. Talat Sarıaydın (Selcuk University, Turkey)

Assist. Prof. Dr. Nikolay Buyukliev (Sofia University, Bulgaria)

Assist. Prof. Dr. Nuray Eroglu (Tekirdag Namık Kemal University, Turkey)

Assist. Prof. Dr. Nurkut Nuray Urgan (Tekirdag Namık Kemal University, Turkey)

Assist. Prof. Dr. Onder Gokmen Yıldız (Bilecik Seyh Edebali University, Turkey)

Assist. Prof. Dr. Pembe Sabancıgıl (Eastern Mediterranean University, Cyprus)

Assist. Prof.Dr. Rabia Cakan Akpınar (Kafkas University, Turkey)

Assist. Prof. Dr. Ramazan Kama (Siirt University, Turkey)

Assist. Prof. Dr. Sayed Masih Ayat (Zabol University, Iran)

Assist. Prof. Dr. Tulay Erisir (Erzincan Binali Yıldırım University, Turkey)

Assist. Prof. Dr. Yunus Akdogan (Selcuk University, Turkey)

Assist. Prof. Dr. Zehra Pınar (Namık Kemal University, Turkey)

Lecturer Dr. Azime Cetinkaya (Piri Reis University, Turkey)

Lecturer Dr. Meryem Odabasi (Ege University, Turkey)

Lecturer Dr. Pınar Zengin Alp (Duzce University, Turkey)

Lecturer Abdullah Ahmetoglu (Gazi University, Turkey)

Lecturer Suheda Guray (Baskent University, Turkey)

Lecturer Suleyman Cengizci (Antalya Bilim University, Turkey)

Rsc. Assist. Dr. Neslihan Aysen Ozkiriici (Yıldız Technical University, Turkey)

Rsc. Assist. Dr. Nesrin Manav (Erzincan Binali Yıldırım University, Turkey)

Rsc. Assist. Dr. Gul Ugur Kaymanlı (Cankırı Karatekin University, Turkey)

Dr. Alfred Witkowski (UTP University, Poland)

Dr. Anuradha Mahasinghe (University of Colombo, SriLanka)

Dr. Aygun T. Huseynova (Baku State University, Azerbaijan)

Dr. Canan Ciftci (Ordu University, Turkey)

Dr. Fahimeh Sultanzadeh (Islamic Azad University, Iran)

Dr. Fatih Say (Ordu University, Turkey)

Dr. Hanna Wojewodka (Silesia in Katowice University, Poland)

Dr. Irakli Chitaia (Ivane Javakhishvili Tbilisi St. Uni., Georgia)

Dr. Kamilla A. Alimardanova (Azerbaijan National Acad. of Sci., Azerbaijan)

302

Dr. Kemal Eren (Sakarya University, Turkey)

Dr. Nikhil Khanna (University of Delhi, India)

Dr. Ovgu Cidar Iyikal (Eastern Mediterranean University, Cyprus)

Dr. Salah Eid (Paris Diderot University, France)

Dr. Sumit Kumar Sharma (Delhi University, India)

Rsc. Assist. Aziz Yazla (Selcuk University, Turkey)

Rsc. Assist. Busra Aktas (Kırıkkale University, Turkey)

Rsc. Assist. Caner Tanıs (Selcuk University, Turkey)

Rsc. Assist. Kadir Karakaya (Selcuk University, Turkey)

Rsc. Assist. Kemale Veliyeva (Baku State University, Azerbaijan)

Rsc. Assist. Sebuhi Abdullayev (Baku State University, Azerbaijan)

Anna Rudak (National Research University, Russia)

Eldost U. Ismailov (Baku State University, Azerbaijan)

Gurbanali J. Valiyev (Baku State University, Azerbaijan)

Hatice Altın Erdem (Kırıkkale University, Turkey)

Hatice Eryigit (Gazi University, Turkey)

Mariana Geanina Zaharia (Ovidius University of Constanta, Romania)

Maryam Boyukzade (Baku State University, Azerbaijan)

Nezakat Javanshir (Hacettepe University, Turkey)

Nihal Kilic Aslan (Kırıkkale University, Turkey)

Nikita Vakhtanov Vakhtanov (HSE University, Russia)

Radu Vasile (Ovidius University of Constanta, Romania)

Reyhan S. Akbarly (Baku State University, Azerbaijan)

Shima Baharlouei (Isfahan University of Technology, Iran)

Azadeh Hosseinpour (Islamic Azad University, Iran)

Paugam Frederic (Sorbonnes University, France)

303

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