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International Conference on Mathematics and Its Applications in Science and Engineering(ICMASE 2020)

Preface

This abstract booklet includes the abstracts of the papers that have been presented at International

Conference on Mathematics and its Applications in Science and Engineering (ICMASE 2020)

which is held in Ankara Hacı Bayram Veli University, Ankara, Turkey between 9-10 July, 2020,

via Online because of Covid 19 pandemia. I hope this pandemic, which covers the whole world,

will lose its effect as soon as possible and we will go back to the days before the pandemic.

The aim of this conference is to exchange ideas, discuss developments in mathematics, develop

collaborations and interact with professionals and researchers from all over the world in with some

of the following interesting topics: Functional Analysis, Approximation Theory, Real Analysis,

Complex Analysis, Harmonic and non-Harmonic Analysis, Applied Analysis, Numerical Analy-

sis, Geometry, Topology and Algebra, Modern Methods in Summability and Approximation, Op-

erator Theory, Fixed Point Theory and Applications, Sequence Spaces and Matrix Transformation,

Modern Methods in Summability and Approximation, Spectral Theory and Diferantial Operators,

Boundary Value Problems, Ordinary and Partial Differential Equations, Discontinuous Differential

Equations, Convex Analysis and its Applications, Optimization and its Application, Mathemat-

ics Education, Application on Variable Exponent Lebesgue Spaces, Applications on Differential

Equations and Partial Differential Equations, Fourier Analysis, Wavelet and Harmonic Analysis

Methods in Function Spaces, Applications on Computer Engineering, Flow Dynamics. However,

the talks are not restricted to these subjects only. I am pleased to tell that this conference is also

organized as a final multiplier event of the Rules_Math Project, supported by the EU.

Many thanks to all committee members.

We wish everyone a fruitful conference and pleasant memories from ICMASE 2020.

Fatih YILMAZ

Chairman ICMASE 2020

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International Conference on Mathematics and Its Applications in Science andEngineering (ICMASE 2020)09-10 July 2020, Ankara

Honory and Advisory Board

Yusuf TEKIN, Rector of Ankara Hacı Bayram Veli University, (Turkey)

Gerardo Rodriguez SANCHEZ, Salamanca University, (Spain)

Organizing Committee

Fatih YILMAZ, Ankara Hacı Bayram Veli University, (Turkey) (Conference Chairman)

Deolinda RASTEIRO, Instituto Superior de Engenharia de Coimbra, (Portugal)

Dursun TASÇI, Gazi University, (Turkey)

Emel KARACA, Ankara Hacı Bayram Veli University, (Turkey)

Jesús Martin-VAQUERO, Salamanca University, (Spain)

Melek SOFYALIOGLU, Ankara Hacı Bayram Veli University, (Turkey)

Mücahit AKBIYIK, Beykent University, (Turkey)

Mustafa ÖZKAN, Gazi University, (Turkey)

Selçuk ÖZCAN, Karabük University, (Turkey)

Invited Speakers

Ayman BADAWI , American University of Sharjah, (UAE)

Araceli QUEIRUGA DIOS, Salamanca University, (Spain)

Carlos Martins da FONSECA, Kuwait College of Science and Technology, (Kuwait)

Scientific Committee

Agustín Martín MUNOZ, Spanish National Research Council, (Spain)

Ali Reza ASHRAFI, University of Kashan, (Iran)

Ascensión Hernández ENCINAS, University of Salamanca, (Spain)

Aynur Keskin KAYMAKÇI, Selçuk University, (Turkey)

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Carlos Martins da FONSECA, Kuwait College of Science and Technology, (Kuwait)

Cristina R. M. CARIDADE, Instituto Superior de Engenharia de Coimbra, (Portugal)

Cüneyt ÇEVIK, Gazi University, (Turkey)

Daniela RICHTARIKOVA, Slovak University of Technology in Bratislava, (Slovakia)

Daniela VELICHOVA, Slovak University of Technology, (Slovakia)

Engin ÖZKAN, Erzincan Binali Yıldırım University, (Turkey)

Gheorghe MOROSANU, Babes-Bolyai University, (Romania)

Ion MIERLUS-MAZILU, Technical University of Civil Engineering Bucharest, (Romania)

Ji-Teng JIA, Xidian University, (China)

Kadir KANAT, Ankara Hacı Bayram Veli University, (Turkey)

Luis Hernández ENCINAS, Spanish National Research Council, (Spain)

Maria Jesús Santos SANCHEZ, Salamaca University, (Spain)

Marie DEMLOVA, Czech Technical University in Prague, (Czech Republic)

Michael CARR, Technological University Dublin, (Ireland)

Moawwad E.A. EL-MIKKAWY, Mansura University, (Egypt)

Mohammad Sal MOSLEHIAN, Ferdowsi University of Mashad, (Iran)

Murat BEKAR, Gazi University, (Turkey)

Mustafa ÇALISKAN, Gazi University, (Turkey)

Nenad P. CAKIC, University of Belgrade, (Serbia)

Nihat AKGÜNES, Necmettin Erbakan University, (Turkey)

Seda YAMAÇ AKBIYIK, Gelisim University, (Turkey)

Snezhana GOCHEVA-ILIEVA, University of Plovdiv Paisii Hilendarski, (Bulgaria)

Tomohiro SOGABE, Nagoya University, (Japan)

Victor Gayoso MARTINEZ, Spanish National Research Council, (Spain)

Vildan ÖZTÜRK, Ankara Hacı Bayram Veli University, (Turkey)

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Contents

The Moore-Penrose Inverse of Symmetric Matrices with Nontrivial Equitable Partitions 1

ABDULLAH ALAZEMI, MILICA ANÐELIC, DRAGANA CVETKOVIC-ILIC

Entrepreneurial Framework Conditions: a PCA Approach in European Countries 2

ALDINA CORREIRA, ANA BORGES, ELIANA COSTA E SILVA, FÁBIO DUARTE

Longitudinal Analysis of Entrepreneurial Framework Conditions on Entrepreneurship Intent 5

ANA BORGES, ALDINA CORREIRA, ELIANA COSTA E SILVA, FÁBIO DUARTE

Training Open Courses to Assess Mathematical Competencies 7

ARACELI QUEIRUGA-DIOS, FATIH YILMAZ, DEOLINDA M.L.D. RASTEIRO, JESÚSMARTÍN VAQUERO

On Computing an Arbitrary Singular Value of a Tensor Sum 9

ASUKA OHASHI, TOMOHIRO SOGABE

On Gdc-Flat Modules 11

AYSE ÇOBANKAYA, ISMAIL SAGLAM

A New Approach for Computing the Mock-Chebyshev Nodes 13

B. ALI IBRAHIMOGLU

Application of Wendland’s Compactly Supported Functions for the Numerical Simulation ofthe Space Fractional NLS Equation 14

BAHAR KARAMAN, YILMAZ DERELI

Cyclic Codes over the Ring Z8 + uZ8 + vZ8 16

BASRI ÇALISKAN

On the Sedenions with Q−Integer Components 17

CAN KIZILATES, SELIHAN KIRLAK

Assessment with Mathematics Competencies-Based RULES_MATH’s Guides for Calculus I 19

CRISTINA M.R. CARIDADE, DEOLINDA M. L. D. RASTEIRO

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Assessment with Mathematics Competencies-Based RULES_MATH’s Guides for Linear Al-gebra 22

CRISTINA M.R. CARIDADE, DEOLINDA M. L. D. RASTEIRO

A Study on Constant Angle Ruled Surfaces Generated by Timelike Curve in Minkowski Space25

CUMALI EKICI, YASIN ÜNLÜTÜRK, GÜL UGUR KAYMANLI

Mathematical Competency Assessment-RULES_MATH Guides on Complex Numbers 27

DANIELA RICHTARIKOVA

Algebraic Structure of n-Dimensional Quaternionic Space 29

DENIZ ALTUN, SALIM YÜCE

Probability and Statistical Methods: Assessing Knowledge and Competencies – Case Study atISEC 31

DEOLINDA M. L. D. RASTEIRO, CRISTINA M.R. CARIDADE

Assessing Knowledge and Competencies: RULES_MATH Project’s effects at ISEC 33

DEOLINDA M. L. D. RASTEIRO, CRISTINA M.R. CARIDADE

A Novel Fractional-Order Model for COVID-19 Infectious with Incommensurate Orders 36

DIN PRATHUMWAN, KAMONCHAT TRACHOO, INTHIRA CHAIYA

Vehicle and Driver Scheduling Problem using ε-Constraint in Public Transportation 37

DURDU HAKAN UTKU, BURCU ALTUNOGLU

New Hyperbolic-Number Forms of the Euler Savary Equation: The Consideration of FuturePointing Timelike Pole Rays for Spacelike Pole Curves 39

DUYGU ÇAGLAR, NURTEN GÜRSES

Finite Difference Method for Fractional Order Pseudo-Parabolic Partial Differential Equation 42

ECEM GÖKTEPE, MAHMUT MODANLI

The Best Known Iteration Bound of Interior Point Methods for Linear Optimization Problem 43

EL AMIR DJEFFAL, BACHIR BOUNIBANE

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A Cluster Analysis on the European Entrepreneurial Framework Conditions over the Last TwoDecades 45

ELIANA COSTA E SILVA, ALDINA CORREIRA, ANA BORGES, FÁBIO DUARTE

Some Characterizations for Split Quaternions 47

EMEL KARACA, FATIH YILMAZ, MUSTAFA ÇALISKAN

A New Approach for Natural Lift Curves and Tangent Bundle of Unit 2-Sphere 48

EMEL KARACA, MUSTAFA ÇALISKAN

Comparison the Solutions between Vector and Set Approaches for Set-Valued OptimizationProblems 49

EMRAH KARAMAN

Generalized Interval-Valued Optimization Problems 51

EMRAH KARAMAN

On New Narayana Polynomials 53

ENGIN ÖZKAN, BAHAR KULOGLU

On k-Narayana Sequence 56

ENGIN ÖZKAN, BAHAR KULOGLU

Suborbital Graphs of Γ0,n(N) in Q(N) 59

ERDAL ÜNLÜYOL, AZIZ BÜYÜKKARAGÖZ

A Bi-Level Mathematical Model to Analyze the Risk at a Hazardous Material TransportationNetwork 61

FATIH KASIMOGLU, ÖZKAN BALI

The Intrinsic Metric on the Scale Irregular Sierpinski Gasket SG(2, 3) 63

FATMA DIGDEM KOPARAL, YUNUS ÖZDEMIR

Non-Rotating Frames and its Applications 65

FATMA KARAKUS

Fermi-Walker Parallelism and its Applications 67

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FATMA KARAKUS

Dynamics of a Modified a Discrete-Time Predator-Prey Model with Allee Effect 69

FIGEN KANGALGIL, SEVAL ISIK

Numerical Methods for a Nonlocal Peridynamic Model 71

GIUSEPPE MARIA COCLITE, ALESSANDRO FANIZZI, LUCIANO LOPEZ, FRANCESCOMADDALENA, SABRINA FRANCESCA PELLEGRINO

Double-Diffusion Instability: A Fingering Convection Model for Oceanic Staircase Formation72

GIUSEPPE MARIA COCLITE, FRANCESCO PAPARELLA, SABRINA FRANCESCA PEL-LEGRINO

Eigenvalues of the Selfadjoint Schrödinger Operator on Non-compact Star Graph 73

GÖKHAN MUTLU

Evolute Offset of Non-Cylindirical Ruled Surfaces with B-Darboux Frame 75

GÜL UGUR KAYMANLI

The Restrictive Sets and their Applications to Soft Group Theory 77

HAKAN AYKUT, AKIN OSMAN ATAGÜN

Mathematical Model of Extinction of Rastrelliger brachysoma, Harvesting and Control 79

INTHIRA CHAIYA, KAMONCHAT TRACHOO, DIN PRATHUMWAN

Some Applications of Eigenvalue and Eigenvectors in Engineering 80

ION MIERLUS MAZILU, FATIH YILMAZ

Generalized Quasi-Einstein Normal Metric Contact Pair Manifolds 81

INAN ÜNAL

A New Approach to Generalized Cantor Set in Fractal Geometry 84

IPEK EBRU KARAÇAY, SALIM YÜCE

Nonlinear Approximation in Discrete Operators of Sampling-type 86

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ISMAIL ASLAN

The Way of Assessment of Mathematical Competencies in RULES_MATH Project 88

JANA GABKOVÁ, PETER LETAVAJ

Nonlinear Analysis of Heart Rate Dynamics to Estimate the Risk of Cardiovaskular Events inHypertensive Patients 90

JOSÉ MARÍA LOPEZ-BELINCHON, MIGUEL ÁNGEL LOPEZ GUERRERO, RAÚL AL-CARAZ MARTINEZ

Natural Transform Adomian Decomposition Method(Ntadm) for Evaluation of Two Dime-nional Schrödinger Black Schloes Time Fractional Ordered Pde, an Application to Finan-cial Modelling 92

KAMRAN ZAKARIAR, SAEED HAFEEZ

The Study of Axial Load in the Lumbar Spine of Patients with Lumbar Spinal Stenosis byusing Finite Element Method 94

KAMONCHAT TRACHOO, INTHIRA CHAIYA, DIN PRATHUMWAN

Improving Engineering Thermodynamics Learning with Mathematica 95

MARÍA JESÚS SANTOS, ALEJANDRO MEDINA, JOSÉ MIGUEL MATEOS ROCO,ARACELI QUERIRUGA-DIOS

An Application of Double Sumudu Transform for Solving Telegraph Equation 96

MARIA AYDIN, HALDUN ALPASLAN PEKER

Triple Sumudu Transform and its Application for Solving Volterra Integro-Partial DifferentialEquation 98

MARIA AYDIN, HALDUN ALPASLAN PEKER

Understanding the Failure in Differential and Integral Calculus in the Degrees of Engineeringat a Higher Education School in Portugal 100

MARIA EMÍLIA BIGOTTE DE ALMEIDA, ARACELI QUEIRUGA-DIOS, MARÍA JOSÉCÁCERES

Analytical Investigation of a Two-Mass System Connected with Linear and Nonlinear Stiff-nesses 102

MD. ALAL HOSEN

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On Approximation Properties of Szász-Mirakyan Operators 105

MELEK SOFYALIOGLU, KADIR KANAT

Cardinalities and Ranks of Nilpotent Subsemigroups of Cn 106

MELEK YAGCI

The Number of m-Potent Elements in Cn,Y 108

MELEK YAGCI

Modelling Tap Water Consumer Ratio 110

MELTEM EKIZ, OSMAN UFUK EKIZ

Comparison of Nontraditional Optimization Techniques in Optimization of Shell and TubeHeat Exchanger 112

MERT AKIN INSEL, INCI ALBAYRAK, HALE GONCE KOCKEN

Fuzzy Modelling on Control of Heat Exchangers 114

MERT AKIN INSEL, INCI ALBAYRAK, HALE GONCE KOCKEN

On the New Families of Gauss k-Lucas Numbers and their Polynomials 116

MERVE TASTAN, ENGIN ÖZKAN

On Spaces Derivable from a Solid Sequence Space and a Non-Negative Lower TriangularMatrix 118

MERVE ULUDAG

Numerical Analysis of an Adaptive Non-linear Filter Based Time Regularization Model for theIncompressible Non-Isothermal Fluid Flows 119

MINE AKBAS, BUSE INGENÇ

Approximate and Analytical Solutions for Nonlinear Fractional Systems 120

MUAMMER AYATA, OZAN ÖZKAN

Runge-Kutta-Nystrom Method Based On Arithmetic Mean for Solving Special Second OrderOrdinary Differential Equations 122

MUKADDES ÖKTEN TURACI

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Refinements of Bullen-Type Inequalities for s−Convex Functions via Riemann-Liouville Frac-tional Integrals 124

MUSA ÇAKMAK

A Generalization of Hermite-Hadamard, Bullen, and Simpson Inequalities via h−Convexity 126

MUSA ÇAKMAK

The Agreement of the Conventional Defect Correction Method and the Novel Defect Correct-ing Extrapolation Technique for Linear Equations 127

MUSTAFA AGGUL

On Helicoidal Hypersurfaces in Euclidean 4-Space 128

MUSTAFA ALTIN, AHMET KAZAN

Periodic Point Results on Quasi Metric Spaces 130

MUSTAFA ASLANTAS

Some Fixed Point Results for Multivalued Mappings on M-Metric Spaces 132

MUSTAFA ASLANTAS, UGUR SADULLAH

k-Order Fibonacci Quaternions 134

MUSTAFA ASCI, SÜLEYMAN AYDINYÜZ

Rigid Body Motion of an Elastic Rectangle under the Effect of Sliding Boundary Condition 136

ONUR SAHIN

Stability Analysis and Topological Classification of Fixed Points of Dicrete-Time Prey-Predator Model 138

ÖZGÜR DEMIR, FIGEN KANGALGIL

The Roads We Choose: Ceitical Thinking Experiences in Mathematical Courses 140

PAULA CATARINO, MARIA M. NASCIMENTO, EVA MORAIS, PAULO VASCO

A Study on the Tangent Bundle of Riemannian Manifold Endowed with the Bronze Structure 142

RABIA ÇAKAN AKPINAR

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Estimation of Turkey and World Population in the Year 2050 in view of Local and NonlocalFractional Derivatives 143

RAMAZAN OZARSLAN

The Effect of Caputo Fractional Derivative Concerning Another Function on Deflection ofHorizontal Beams 145

RAMAZAN OZARSLAN

On Lightlike Submersions 147

RAMAZAN SARI

MMR Encryption Algorithm as an Alternative Encryption Algorithm to RSA Encryption Al-gorithm 148

REMZI AKTAY

Results on Set-valued Prešic Type Mappings 150

SEHER SULTAN YESILKAYA, CAFER AYDIN

New Results on Expansive Mappings 152

SEHER SULTAN YESILKAYA, CAFER AYDIN

On a Type-2 Fuzzy Approach to Solution of Second Order Initial Value Problem 154

SELAMI BAYEG, RAZIYE MERT, TAHIR KHANIYEV, ÖMER AKIN

Existence and Uniqueness Results for a Nonlinear Fractional Differential Equations of Orderσ ∈ (1, 2) 156

SINAN SERKAN BILGICI, MÜFIT SAN

Assessing Students’ Knowledge and Competences in Mathematics by Combined Tests UsingMachine Learning Methods 158

SNEZHANA GOCHEVA-ILIEVA, HRISTINA KULINA, ATANAS IVANOV

Generalized k-Order Fibonacci Hybrid and Lucas Hybrid Numbers 160

SÜLEYMAN AYDINYÜZ, MUSTAFA ASCI

k-Order Gaussian Fibonacci Polynomials and Applications to the Coding/Decoding Theory 163

SÜLEYMAN AYDINYÜZ, MUSTAFA ASCI

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Finite Difference Method for Fractional Partial Differential Equation Defined by Atangana-Baleanu Caputo (ABC) 165

SÜMEYYE EKER, MAHMUT MODANLI

On Vertex and Link Residual Domination Numbers of Generalized Caterpillar Graphs 166

TUFAN TURACI

Some Constant Angle Spacelike Surfaces Construct on a Curve in De-Sitter 3-Space 168

TUGBA MERT, MEHMET ATÇEKEN

Hamilton Matrices of Real Quaternions and Spinors 170

TÜLAY ERISIR, EMRAH YILDIRIM

Using Freeware Mathematical Software in Calculus Classes 172

VICTOR GAYOSO MARTINEZ, LUIS HERNÁNDEZ ENCINAS, AGUSTIN MARTÍNMUÑOZ, ARACELI QUEIRUGA DIOS

Characterizations and Integral Formula for Generalized Ricci Almost Solitons 174

YASEMIN SOYLU

On Reduction of a (2+1)-Dimensional Nonlinear Schrödinger Equation via Conservation Laws176

YESIM SAGLAM ÖZKAN

Rings whose g-Semiartinian Modules Have Maximal or Minimal Subprojectivity Domain 177

YILMAZ DURGUN, AYSE ÇOBANKAYA

On the ?-Congruence Sylvester Equation 179

YUKI SATAKE, TOMOHIRO SOGABE, TOMOYA KEMMOCHI, SHAO-LIANG ZHANG

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THE MOORE-PENROSE INVERSE OF SYMMETRIC MATRICESWITH NONTRIVIAL EQUITABLE PARTITIONS

Abdullah ALAZEMI1, Milica ANÐELIC2, Dragana CVETKOVIC-ILIC3

1,2Kuwait University, Kuwait3University of Niš, Serbia

Corresponding Author’s E-mail: [email protected]

ABSTRACT

We consider symmetric matrices that admit nontrivial equitable partitions. We de-termine some sufficient conditions for the quotient matrix of the Moore-Penroseinverse of the initial matrix to be equal to the Moore-Penrose inverse of its quo-tient matrix. We also study several particular cases when the computation of theMoore-Penrose inverse can be reduced significantly by establishing the formula forits computation based on the Moore-Penrose inverse of the quotient matrix. Amongothers we consider the adjacency matrix of a generalized weighted threshold graph.

Keywords Equitable partitions ·Moore-Penrose inverse · Stepwise matrices

References

[1] A. Alazemi, M. Andelic, S. K. Simic, Eigenvalue location for chain graphs,Linear Algebra Appl. 505 (2016), 194-210.

[2] R.B. Bapat, On the adjacency matrix of a threshold graph, Linear Algebra Appl.438 (2013), 3008-3015.

[3] A. Ben-Israel, T. N. E. Greville, Generalized Inverse: Theory and Applications,2nd Edition, Springer, New York, 2003.

[4] D.S. Cvetkovic-Ilic, Y. Wei, Algebraic Properties of Generalized Inverses, Se-ries: Developments in Mathematics, Vol. 52, Springer, New York, 2017.

[5] D. Cvetkovic, P. Rowlinson, S. Simic, An Introduction to the Theory of GraphSpectra, Cambridge University Press, Cambridge, 2010.

[6] W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995), 593-616.

[7] N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, Annals ofDiscrete Mathematics 56, North-Holland, New York, 1995.

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ENTREPRENEURIAL FRAMEWORK CONDITIONS: A PCAAPPROACH IN EUROPEAN COUNTRIES

Aldina CORREIRA1, Ana BORGES1, Eliana COSTA E SILVA1, Fábio DUARTE1

1ESTG - School of Management and Technology2P.PORTO - Polytechnic of Porto

3CIICESI – Center for Research and Innovation in Business Sciences and Information Systems


ABSTRACT

Entrepreneurial Framework Conditions (EFC), as Valliere (2010) [1] defined, arethe environmental conditions that encourage and support entrepreneurial activity atthe national level. Thereafter Valliere’s work, and until today as illustrate Orobiaet. al. (2020) [8], several authors focused on the study of countries EFC.In this work we consider the EFC indicators from Global Entrepreneurship Monitor(GEM), similar to Silva et. al. (2018) [9] and Sampaio et. al. (2016) [7]. The studyof Silva et. al. (2018) focus on the twelve indicators of the entrepreneurial ecosys-tem, defined by the GEM project as EFC and analysed how experts’ perceptionshave changed in Portugal between 2010 and 2016, by establishing a comparativeanalysis between the Portuguese reality and the reality in other European countries.GEM project began in 1999 and is conceived to analyse the phenomenon of en-trepreneurship conducting two tipes of surveys: the Adult Population Survey (APS)and the National Expert Survey (NES).Correia et. al. (2017) considered NES data-sets for 2011, 2012 and 2013 to studythe effects of different type of entrepreneurship expert specialization on the percep-tions about the EFC. These data-sets are also considered by Braga et. al. (2018)[4], Queiros et. al. (2019) [5], [10] and Correia et. al. (2016) [3].According to GEM 2019–20 Global Report, fifty economies participated in theGEM 2019 APS, including: 11 from the Middle East and Africa, 8 from Asia andPacific, 8 from Latin America and Caribbean, 23 from Europe and North Amer-ica. 5 of these economies are classified as low-income level, 12 as middle-incomeand the rest 33 as high-income. Over 150 000 individuals participated in extendedinterviews as part of the GEM research in 2019.Thus GEM is a large scale database for internationally comparative entrepreneur-ship that includes information about many aspects of entrepreneurship activities

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of a large number of countries. Because of that is largely study by the academiccommunity.In this work a Principal Component Analisys (PCA), with varimax rotation ap-proach (Factor Analysis), is implemented for EFC in European countries, duringtwo decades. Our goal is to analyse if the correlation structure of the EFC GEM in-dicators remains the same during this period or if there are changes in this structure.

Keywords Entrepreneurship · PCA · Factor Analysis · Global EntrepreneurshipMonitor (GEM) · Entrepreneurial Framework Conditions (EFC)

AcknowledgmentThis work has been partially supported by national funds through FCT – Fundaçãopara a Ciência e Tecnologia through project UIDB/04728/2020.

References

[1] Valliere, D. (2010). Reconceptualizing entrepreneurial framework conditions.International Entrepreneurship and Management Journal, 6(1), 97-112.

[2] Levie, J., Autio, E. (2007, October). Entrepreneurial framework conditions andnational-level entrepreneurial activity: Seven-year panel study. In Third GlobalEntrepreneurship Research Conference (pp. 1-39).

[3] Correia, A., e Silva, E. C., Lopes, I. C., Braga, A., Braga, V. (2017, Novem-ber). Experts’ perceptions on the entrepreneurial framework conditions. In AIPConference Proceedings (Vol. 1906, No. 1, p. 110004). AIP Publishing LLC.

[4] Braga, V., Queirós, M., Correia, A., Braga, A. (2018). High-Growth BusinessCreation and Management: A multivariate quantitative approach using GEMdata. Journal of the Knowledge Economy, 9(2), 424-445.

[5] Queirós, M., Braga, V., Correia, A. (2019). Cross-country analysis to high-growth business: Unveiling its determinants. Journal of Innovation Knowledge,4(3), 146-153.

[6] Correia, A., Costa e Silva, E., Lopes, I. C., Braga, A. (2016, December).MANOVA for distinguishing experts’ perceptions about entrepreneurship us-ing NES data from GEM. In AIP Conference Proceedings (Vol. 1790, No. 1, p.140002). AIP Publishing LLC.

[7] Sampaio, C., Correia, A., Braga, V., Braga, A. (2016). Environmental variablesand entrepreneurship–a study with GEM NES global individual level data. In-ternational Journal of Knowledge Based Development, 2040-4476.

[8] Orobia, L. A., Tusiime, I., Mwesigwa, R., Ssekiziyivu, B. (2020). En-trepreneurial framework conditions and business sustainability among the youthand women entrepreneurs. Asia Pacific Journal of Innovation and Entrepreneur-ship.

[9] Silva, E. C. E., Correia, A., Duarte, F. (2018, November). How Portugueseexperts’ perceptions on the entrepreneurial framework conditions have changedover the years: A benchmarking analysis. In AIP Conference Proceedings (Vol.2040, No. 1, p. 110005). AIP Publishing LLC.

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[10] Pilar, M. D. F., Marques, M., Correia, A. (2019). New and growing firms’ en-trepreneurs’ perceptions and their discriminant power in EDL countries. GlobalBusiness and Economics Review, 21(3-4), 474-499.

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LONGITUDINAL ANALYSIS OF ENTREPRENEURIALFRAMEWORK CONDITIONS ON ENTREPRENEURSHIP INTENT

Ana BORGES1, Aldina CORREIRA1, Eliana COSTA E SILVA1, Fábio DUARTE1

1CIICESI, ESTG, Politécnico do Porto


ABSTRACT

Entrepreneurship Intentions are defined as the entrepreneurial orientation, voca-tional aspirations and the desire to own a business by [5]. This study intents tocontribute to the understanding of this emerging concept since it is consensuallyaccepted the major role of entrepreneurship for economic development, job cre-ation and innovation ([4], [3]).Global Entrepreneurship Monitor (GEM) is a large-scale database for internation-ally comparative entrepreneurship that includes information about many aspectsof entrepreneurship activities of a large number of countries. It is based on col-lecting primary data through an Adult Population Survey (APS) of at least 2000randomly selected adults (18–64 years of age) in each economy. In addition, na-tional teams collect experts’ opinions about components of the entrepreneurshipecosystem through a National Expert Survey (NES) ([2]). We aim to understandwhich GEM Framework Conditions have impact on Entrepreneurship Intentions.The database used in this work contains the GEM the indicators, corresponding tothe Entrepreneurial Behaviour and Attitudes – EBA and Entrepreneurial FrameworkConditions – EFC. Subsequent to the study of both the state of the entrepreneurialmind-set, motivations, activities and ambition, and the national framework condi-tions required to allow entrepreneurship to flourish in an economy. In particular, weanalyze how twelve EFCs indicators relate do Entrepreneurial intentions, througha longitudinal study of an unbalance dataset on 28 European countries between2001 and 2019. Entrepreneurship is heterogeneous across countries, according sev-eral GEM data analyses (e.g. Braga et al., 2018, Pilar et al., 2019). Hence, itis important to take into account intra-country specificities when conducting thistype of analyses, since company performance can variate regarding the economiclevel of the country. For that, we tested different specification of mixed-effect lon-gitudinal models to understand which EFC have significant impact on countriesentrepreneurial intentions, thought time. Additionally, we also control, similarlyto the work of [1] for per-capita GDP, population growth, and industry structure,

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since country’s wealth has a significant effect on the character of its entrepreneurialactivity [6].

Keywords Entrepreneurship Intentions · GEM · Longitudinal models


References

[1] Levie, J., and Autio, E. , Entrepreneurial framework conditions and national-level entrepreneurial activity: Seven-year panel study. In Third Global En-trepreneurship Research Conference: 1-39, 2007.

[2] Herrington M. and Kew P. K., Global entrepreneurship monitor: 2016/17 globalreport, 2017.

[3] Pruett, M., Shinnar, R., Toney, B., Llopis, F., and Fox, J., Explaining en-trepreneurial intentions of university students: a cross-cultural study. Interna-tional Journalof Entrepreneurial Behavior & Research, 2009.

[4] Remeikiene, R., Startiene, G., and Dumciuviene, D., Explaining entrepreneurialintention of university students: The role of entrepreneurial education. In Inter-national conference: 299–307, 2013.

[5] Thompson, E. R., Individual entrepreneurial intent: Construct clarification anddevelopment of an internationally reliable metric.Entrepreneurship theory and-practice, 33(3):669–694, 2009.

[6] van Stel A., Carree M., Thurik R., The Effect of Entrepreneurial Activity onNational Economic Growth, Small Business Economics, 24 (3): 311-321, 2005.

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TRAINING OPEN COURSES TO ASSESS MATHEMATICALCOMPETENCIES

Araceli QUEIRUGA-DIOS1, Fatih YILMAZ 2, Deolinda M.L.D. RASTEIRO 3, Jesús MartínVAQUERO 1

1Universidad de Salamanca, Salamanca, Spain2Ankara Haci Bayram Veli University, Ankara, Turkey

3Coimbra Institute of Engineering, Coimbra Polytechnique Institute, Coimbra, Portugal


ABSTRACT

As part of Rules_Math project a group of 9 European institutions have been work-ing in a set of materials and resources to assess mathematical competencies. “NewRules for Assessing Mathematical Competencies” project gave them the opportu-nity of sharing their university situation in evaluating mathematics and to analyzeand discuss about their country’s current situation in secondary and high school forassessing competencies, and finally, they define the standards for assessing math-ematical competencies in engineering education [1, 2]. We present here one ofthe final outputs of this project: the training courses that have been developed andimplemented for training teachers in assessing competencies and for students toacquire the competencies and skills to be competent in mathematics.The use of mathematical software and tools, with different pedagogical approachesmake possible to involve students in this competencies-based teaching and learningprocess. During the project life several papers have been published that presentevidences of the work that have been developed by students and trainers: The useof Wolfram Mathematica through practicums for students [3], game-based learningas a new proposal [4], video-lessons [5], concept maps [6], etc.The knowledge and the acquisition of mathematical competencies by students wasmeasured with some surveys after using the material developed during the project[7, 8, 9] and the results agree on the usefulness of such a material.

Keywords Engineering Education · Competencies · Learning outcomes

References

[1] Araceli Queiruga-Dios, Ascensión Hernández Encinas, Marie Demlova, De-olinda Dias Rasteiro, Gerardo Rodríguez Sánchez, and María Jesús Sánchez

7


Santos. Rules_math: Establishing assessment standards. In International JointConference: 12th International Conference on Computational Intelligence inSecurity for Information Systems (CISIS 2019) and 10th International Confer-ence on EUropean Transnational Education (ICEUTE 2019), pages 235–244.Springer, 2019.

[2] Araceli Queiruga-Dios. Assessment standards and activities: Some results. InBook of abstracts. 19th conference on Appiled Mathematics (APLIMAT 2020)(Plenary session), page iii. STU, 2020.

[3] Hristina N Kulina, Snezhana G Gocheva-Ilieva, Desislava S Voynikova,Pavlina Kh Atanasova, Anton I Iliev, and Atanas V Ivanov. Integrating ofcompetences in mathematics through software–case study. In 2018 IEEEGlobal Engineering Education Conference (EDUCON), pages 1586–1590.IEEE, 2018.

[4] Araceli Queiruga-Dios, María Jesús Santos Sánchez, Marián Queiruga Dios,Víctor Gayoso Martínez, and Ascensión Hernández Encinas. A virus infectedyour laptop. let’s play an escape game. Mathematics, 8(2):166, 2020.

[5] Cristina MR Caridade and Deolinda MLD Rasteiro. Evaluate mathematicalcompetencies in engineering using video-lessons. In International Joint Con-ference: 12th International Conference on Computational Intelligence in Se-curity for Information Systems (CISIS 2019) and 10th International Confer-ence on EUropean Transnational Education (ICEUTE 2019), pages 245–251.Springer, 2019.

[6] Amalia Beatriz Orúe, Alina Maria Orúe, Liliana Montoya, Victor Gayoso,and Agustin Martín. Enhancing the learning of cryptography through conceptmaps. In International Joint Conference: 12th International Conference onComputational Intelligence in Security for Information Systems (CISIS 2019)and 10th International Conference on EUropean Transnational Education(ICEUTE 2019), pages 263–272. Springer, 2019.

[7] Cristina Caridade, Deolinda M.L.D. Rasteiro, and Daniela Richtarikova.Mathematics for engineers: Assessing knowledge and competencies. In Proc-cedings 19th conference on Appiled Mathematics (APLIMAT 2020), pages199–211. STU, 2020.

[8] Snezhana Gocheva-Ilieva, Hristina Kulina, Anton Iliev, Atanas Ivanov, andIliycho Iliev. A model for assessment of mathematical competencies in engi-neering education. In Proccedings 19th conference on Appiled Mathematics(APLIMAT 2020), pages 546–554. STU, 2020.

[9] Atanas Ivanov. Decision trees for evaluation of mathematical competencies inthe higher education: A case study. Mathematics, 8(5):748, 2020.

[10] Mogens Niss. Mathematical competencies and the learning of mathematics:The danish kom project. In 3rd Mediterranean conference on mathematicaleducation, pages 115–124, 2003.

8


ON COMPUTING AN ARBITRARY SINGULAR VALUE OF ATENSOR SUM

Asuka OHASHI1, Tomohiro SOGABE2

1Ritsumeikan University, Shiga, Japan2Nagoya University, Aichi, Japan


ABSTRACT

We consider computing a desired singular value of a tensor sum: T := In⊗ Im⊗A + In ⊗ B ⊗ I` + C ⊗ Im ⊗ I` ∈ R`mn×`mn, where A ∈ R`×`, B ∈ Rm×m,C ∈ Rn×n, In is the n × n identity matrix, and the symbol “⊗” denotes the Kro-necker product. The tensor sum T arises from a finite difference discretization ofthree-dimensional constant coefficient partial differential equations. The methodsto compute the maximum/minimum singular values of T were provided in [3, 4].

For computing a desired eigenvalue λ of a symmetric matrix M ∈ Rn×n, theLanczos method with the shift-and-invert technique, see, e.g., [1], is widely known.The method solves the shift-and-invert eigenvalue problem: (M − σIn)−1x =(λ − σ)−1x, where x is the eigenvector of M corresponding to λ, and σ is a shiftpoint which is set to the nearby λ (σ 6= λ). Since the eigenvalue problem hasthe eigenvalue (λ − σ)−1 as the maximum eigenvalue, the method is effective forcomputing the desired eigenvalue λ near σ.

Therefore, we obtain a method of computing a desired singular value of T basedon the shift-and-invert Lanczos method. The method solves a shift-and-invert eigen-value problem: (TTT−σ2I`mn)−1x = (σ2−σ2)−1x, where σ is the desired singularvalue of T , x is the corresponding right-singular vector, and σ is set to the nearbyσ (σ 6= σ). This shift-and-invert Lanczos method needs to solve large-scale linearsystems with the coefficient matrix TTT − σ2I`mn. Since the direct methods cannotbe applied due to the nonzero structure of the coefficient matrix, the preconditionedconjugate gradient (PCG) method, see, e.g., [1], is applied. However, it is difficultin terms of memory requirements to simply implement this shift-and-invert Lanc-zos method and the PCG method since the size of T grows rapidly by the sizes ofA, B, and C.

In this talk, we present the following two techniques: 1) efficient implementa-tions of the shift-and-invert Lanczos method for the eigenvalue problem of TTTand the PCG method for TTT − σ2I`mn using three-dimensional arrays (third-order

9


tensors) and the n-mode products, see, e.g., [2]; 2) a preconditioning matrix basedon a structure of T for faster convergence of the PCG method. Finally, we show theeffectiveness of the proposed method through numerical experiments.

Keywords Shift-and-invert Lanczos method · Tensor sum · Singular value

References

[1] Bai Z., Demmel J., Dongarra J., Ruhe A, and Van der Vorst H.A., Templatesfor the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM,Philadelphia, USA, 2000.

[2] Kolda T.G., Bader B.W., Tensor Decompositions and Applications, SIAM.Rev., 51(3): 455-500, 2009.

[3] Ohashi A. and Sogabe T., On computing maximum/minimum singular valuesof a generalized tensor sum, Electron. T. Numer. Ana., 43(1): 244-254, 2015.

[4] Ohashi A. and Sogabe T., On computing the minimum singular value of a tensorsum, Special Matrices, 7(1): 95-106, 2019.

10


ON GDC-FLAT MODULES

Ayse ÇOBANKAYA1, Ismail SAGLAM2

1Department of Mathematics, Cukurova University, Turkey2Adana Alparslan Türkes Science and Technology University, Turkey


ABSTRACT

Let E : 0 → A→fB → C → 0 be a short exact sequence of modules. It is calledgd-closed (respectively, s-closed, d-closed) if Im(f) is gd-closed (respectively, s-closed, d-closed) in B. The class of gd-closed sequences is a proper class and it isprojectively generated by g-semiartinian modules . An R-module M is said to beflat if the functor M ⊗R − is exact. It is well-known that M is flat if and only ifevery short exact sequence 0 → A → B → M → 0 of R-modules is pure-exact(see, for example, [6, Proposition 3.67]). Further studies had been arose from therelation between flatness and purity. For example, weakly flat modules and neatflat modules have been studied in [7] and [1]. Note that a module is weakly-flat(respectively, neat-flat), if any short exact sequence ending with M is included inClosed (respectively,N eat). This studies motivate us to study the modules M suchthat any short exact sequence ending withM is included in GD − Closed. A moduleM gd-closed flat (or shortly gdc-flat ) if the kernel of any epimorphism L → M isgd-closed in the module L. An obvious example of gdc-flat modules is a projectivemodules. Also, nonsingular modules and modules with zero socle are less obviousexamples of gdc-flat modules. It is a fact that a simple module is either singularor projective. We call a module M g-semiartinian if every non-zero homomorphicimage of M contains a singular simple submodule. The class of g-semiartinianmodules is a torsion class of G-Dickson torsion theory and this torsion theory isgenerated by singular simple modules. The class of modules with projective socleis the torsion-free class of the same torsion theory. modules M such that each shortexact sequence ending with M belongs to GD − Closed are investigated by us. Inthis study, we prove that if the torsion submodule τgd(E) of an injective module Eis projective, then E is gdc-flat; and the converse is true for C-rings. Moreover, forC-rings, we prove that all injective modules are gdc-flat if and only if the injectivehull of each g-semiartinian module is projective if and only if every semiartinianmodule embeds in a projective module . Finally, we prove that every GD − Closedsequence is a Pure sequence if and only if every module with a projective socle is

11


flat if and only if every gdc-flat module is flat if and only if every finitely generatedmodule with a projective socle is flat.

Acknowledgement: This work was supported by Research Fund of the CukurovaUniversity. Project Number: 12308

Keywords gdc-flat · g-semiartinian · PS-ring

References

[1] Büyükasık E., Durgun Y., Neat-flat Modules, Comm. Algebra,. 44 (1): 416-428,2016.

[2] Clark J., Lomp C., Vanaja N., and Wisbauer R., Lifting modules, BirkhäuserVerlag, Basel, 2006.

[3] Durgun Y., Özdemir S., On S-Closed Submodules, J. Korean Math. Soc., 54,1281-1299,2017.

[4] Durgun Y., Özdemir S., On D-Closed Submodules, Proc. Indian Acad. Sci(Math. Sci.), 130 (1), 14pp, 2020.

[5] Kara Y., Tercan A., When some complement of a z-closed submodule is a sum-mand, Comm. Algebra, 46 (7), 3071-3078, 2018.

[6] Rotman J., An Introduction to Homological Algebra, Universitext, Springer-Verlag, New York, 2009.

[7] Zöschinger H., Schwach-Flache Moduln., Comm. Algebra, 41 (12), 4393-4407,2013.

12


A NEW APPROACH FOR COMPUTINGTHE MOCK-CHEBYSHEV NODES

B. Ali IBRAHIMOGLU

Yıldız Technical University Department of Mathematical Engineering, Davutpasa Campus, 34210 Istanbul, Turkey


ABSTRACT

In the univariate polynomial interpolation, it is well known that equidistant pointsare unreliable due to the Runge phenomenon. Many different numerical methodshave been proposed to defeat this phenomenon. The mock-Chebyshev subset inter-polation is one of the best methods to overcome Runge’s phenomenon [1].The algorithm proposed in the recent paper [2] provides a robust method for com-puting the mockChebyshev nodes for a given set of (n + 1) Chebyshev-Lobattopoints. This talk discusses a modification of the algorithm with the same smallcomputational cost. Some numerical experiments are given to show the effective-ness of the proposed procedure.

Keywords Interpolation · Runge phenomenon ·Mock-Chebyshev interpolation

References

[1] J.P. Boyd, J.R. Ong, Exponentially-convergent strategies for defeating theRunge phenomenon for the approximation of non-periodic functions, Part 1:single-interval schemes. Comput. Phys. 5 (2–4) (2009) 484–497.

[2] B.A. Ibrahimoglu, A fast algorithm for computing the mock-Chebyshev nodes.J. Comput. Appl. Math. 373 (2020).

13


APPLICATION OF WENDLAND’S COMPACTLY SUPPORTEDFUNCTIONS FOR THE NUMERICAL SIMULATION OF THE SPACE

FRACTIONAL NLS EQUATION

Bahar KARAMAN1, Yılmaz DERELI2

1,2Department of Mathematics, Eskisehir Technical University,Eskisehir 26470, Turkey

Corresponding Author’s E-mail:[email protected]

ABSTRACT

This research describes an efficient numerical method based on Wendland’s com-pactly supported functions to obtain the numerical solution of space fractional non-linear Schrödinger (NLS) equation. Here, the Conformable derivative is used forthe fractional derivative according to the space of order α with 1 < α < 2. Thepresent numerical discussion is based on two following ways: Firstly, the Crank-Nicolson scheme is employed in the mentioned equation to discretize the spacefractional NLS equation, and second, a linear difference scheme is implemented toavoid solving nonlinear systems. In this way, we have a linear, suitable calcula-tion scheme. Then, Wendland’s compactly supported functions are established forthis scheme. We need to calculate the conformable fractional derivative of thesebasis functions because the scheme includes space fractional derivative based onConformable. To our knowledge, for computing the conformable derivative of afunction f(t), the function must be defined for all t > 0. The basis function can bescaled to have compact support on [0, δ] by replacing r with r

δfor δ > 0. Here δ is

called a scaling factor. Therefore, the Conformable fractional derivatives of thesefunctions can be computed. The algorithm of the presented scheme is accurate andeasy to apply on computers. In the numerical investigation, the stability analysisof the suggested scheme is examined in a similar way to the classic Von-Neumanntechnique for the governing equation. Two test problems are used to confirm theefficiency, capability, and simplicity of the method. In our results of computationsof approximation solutions, it is worth observing that the choice of order α changesboth the height and the width of the wave, and it also causes two turning points.This situation is expected due to the use of the fractional Schrödinger equation inphysics to adapt the shape of the wave without the switch of the nonlinearity anddispersion effects. When the fractional-order α is two, all obtained results are in

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accord with the proposed technique for the classical NLS equation. Finally, allobtained numerical experiments are presented in tables and figures.

Keywords Fractional NLS equation · Crank-Nicolson method · Wendland’scompactly supported functions · Fractional derivative operator · Von-Neumannstability.

AcknowledgementsThis work is supported by the Scientific Research Projects of Eskisehir TechnicalUniversity (No: 19ADP108).

References

[1] Khalil R., Al Horani M., Yousef A., and Sababheh, M., A new definition offractional derivative. J. Comput. Appl. Math. 264: 65-70, 2014.

[2] Schaback R., Error estimates and condition numbers for radial basis functioninterpolation. Advances in Comput. Math. 32(2): 251-264, 1995.

[3] Wendland H, Scattered data approximation. Cambridge Monographs on Ap-plied and Computational Mathematics, Cambridge University Press, Cam-bridge, 17, 2005.

[4] Zhang H., Jiang X., Wang C., and Chen S., Crank-Nicolson Fourier Spectralmethods for the space fractional nonlinear Schrödinger equation and its param-eter estimation. Int. J. Comput. Math. 96(2): 238-263, 2019.

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CYCLIC CODES OVER THE RING Z8 + uZ8 + vZ8

Basri ÇALISKAN

Osmaniye Korkut Ata University, Faculty of Arts and Science, Department of Mathematics, 80000, Osmaniye, Turkey


ABSTRACT

In this paper, we study the cyclic codes over the ring R = Z8 + uZ8 + vZ8 whereu2 = u, v2 = v, uv = vu = 0. We construct the generator polynomials of cycliccodes over R. Also, we introduce a Gray map from Rn to Z3n

8 and show that theGray image of the cyclic codes of odd length n over R is a quasi-cyclic code ofindex 3 and length 3n over Z8.

Keywords Cyclic codes · Quasi-cyclic code · Gray map

References

[1] Abualrub, T., Siap, I., Cyclic Codes over the Rings Z2 + uZ2 and Z2 + uZ2 +u2Z2, Designs Codes and Cryptography, 42(3):273-287, 2007.

[2] Dertli, A. and Cengellenmis, Y., On the Codes Over the Ring Z4 + uZ4 + vZ4

Cyclic, Constacyclic, Quasi-Cyclic Codes, Their Skew Codes, Cyclic DNA andSkew Cyclic DNA Codes, Prespacetime Journal 10(2): 196-213, 2019.

[3] Calderbank, A. R. and Sloane, N. J. A., Modular and p-adic Cyclic Codes,Designs, Codes and Cryptography, 6: 21-35, 1995.

[4] Yu, H., Wang Y. and Shi, M., (1+u)-Constacyclic codes over Z4+uZ4, SpringerPlus, 5:1325, DOI 10.1186/s40064-016-2717-0, 2016.

[5] Hammons, A.R., Kumar, V., Calderbank, A.R., Sloane, N.J.A., Solé, P., TheZ4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans.Inform. Theory, 40: 301-319, 1994.

[6] Dougherty, S.T., Gaborit, p., Harada, M., Solé, P., Type II codes over F2 + uF2,IEEE Trans.Inf.Theory 45: 32–45, 1999.

16


ON THE SEDENIONS WITH Q−INTEGER COMPONENTS

Can KIZILATES 1, Selihan KIRLAK 2

1,2Zonguldak Bulent Ecevit University, Department of Mathematics, Zonguldak, 67100, Turkey


ABSTRACT

The set of Sedenions, denoted by S, are 16−dimensional algebra. Sedenions arenoncommutative and nonassociative algebra over the set of real numbers, obtainedby applying the Cayley–Dickson construction to the octonions. Like octonions,multiplication of sedenions are not neither commutative and associative. Sedenionsappear in many areas of science, such as electromagnetic theory, linear gravity andthe field of quantum mechanics [1, 2, 3, 4, 5]. A sedenion is defined by

p =15∑i=0

qiei,

where q0, q1, ..., q15 ∈ R and e0, e1, ..., e15 are called unit sedenion such thate0 is the unit element and e1, e2, ..., e15 are imaginaries satisfying, for i, j, k =1, 2, . . . , 15 the following multiplication rules:

e0ei = eie0 = ei, (ei)2 = −e0, (1)

eiej = −ejei, i 6= j, (2)ei (ejek) = − (eiej) ek, i 6= j, eiej 6= ±ek. (3)

The addition of sedenions is defined as componentwise and for p1, p2 ∈ S, themultiplication is defined as follows:

p1p2 =

(15∑i=0

aiei

)(15∑j=0

bjej

)

=15∑

i,j=0

aibj (eiej) ,

where eiej satisfies the identities (1), (2) and (3). Several generalizations of thewell-known sedenions such as Fibonacci sedenions, Lucas sedenions, k−Pell andk−Pell-Lucas sedenions, Jacobshtal and Jacobshtal-Lucas sedenions, and so on

17


have been studied by several researchers. For example, in [6], the authors definedthe Fibonaccci and Lucas sedenions

Fn =15∑s=0

Fn+ses,

and

Ln =15∑s=0

Ln+ses.

Then they also obtained the generating functions, Binet-Like formulas and someinteresting identities related to Fibonacci and Lucas sedenions. In this paper, by thehelp of the q−integers, we define a new family of sedenions. Then we obtain somespecial cases for this type of sedenions studied by many researchers before. We alsoget a number of results for this type of sedenions included Binet-Like formulas, ex-ponential generating functions, summation formulas, Catalan’s identities, Cassini’sidentities and d’Ocagne’s identities.

Keywords Sedenion algebra · q−integer · Horadam number

References

[1] Baez, J.C., The octonions. Bull. Am. Math. Soc. 39, 145-205 2002.[2] Carmody, K., Circular and Hyperbolic Quaternions, Octonions and Sedenions,

Appl. Math.Comput., 28, 47-72, 1988.[3] Cawagas, R. E., On the structure and zero divisors of the Cayley-Dickson sede-

nion algebra, Discuss. Math. Gen. Algebra Appl. 24, 251-265, 2004.[4] Mironov, V. L. and Mironov, S. V. Associative space-time sedenions and their

application in relativistic quantum mechanics and field theory, Appl. Math. 6(1),46-56, 2015.

[5] Imaeda, K. and Imaeda, M., Sedenions: algebra and analysis, Appl. Math. Com-put. 115, 77-88, 2000.

[6] Bilgici, G., Tokeser, U. and Unal, Z., Fibonacci and Lucas Sedenions, J. IntegerSeq., Vol. 20, Article 17.1.8, 2017.

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ASSESSMENT WITH MATHEMATICS COMPETENCIES-BASEDRULES_MATH’S GUIDES FOR CALCULUS I

Cristina M.R. CARIDADE1, Deolinda M. L. D. RASTEIRO1

1Coimbra Institute of Engineering, Portugal


ABSTRACT

This work follows from the “New Rules for assessing Mathematical Competencies”(RULES_MATH) project which aims to change the educational paradigm and to geta common European teaching and learning system based on mathematical compe-tencies rather than contents [https://rules-math.com/]. For this, RULES_MATH de-scribes and analyses ways to assess mathematics taught to future engineers throughcompetencies. The eight mathematical competencies already identified (think-ing mathematically; reasoning mathematically; posing and solving mathematicalproblems; modelling mathematically; representing mathematical entities; handlingmathematical symbols and formalism; communicating in, with, and about math-ematics and make use of aids and tools for mathematical activity) are recognizedin the assessment made to students. The RULES_MATH project partners’ work-ing groups have developed a set of “Guide for a Problem” in the different areas ofmathematics that are intended to provide some examples of proposed forms of as-sessment and competence-based activities. The materials are available to all projectpartners to apply to different students from different courses and institutions. Oneof such materials is AC7, which aims to evaluate students about "Methods of inte-gration".The study was carried out in a group of 80 students of Calculus 1 of the degreein Electrotechnical Engineering at the Engineering Institute of Coimbra. 59 of thestudents attended the course unit on daytime schedule and 21 students attended onan after-work time.The results obtained were not satisfatory since at 5 questions (out of 9) the mean areless tha 50% and in one of them all the students have grades less than 50%. Relatingto the test itself as a tool to assess competencies, we may conclude that it coversmost of the competencies that we need to evaluate and its difficulty is adequate toour students. However, many students need a personal attention like office hours orextra work, because their results were clearly negative.

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Keywords Assessment · Significant Learning · Competencies · Mathematics ·Engineering

References

[1] Alpers, B., et al., “A Framework for Mathematics Curricula in Engineer-ing Education”, Proceedings of SEFI MWG Seminar, 2013. Available at:http://sefi.htw-aalen.de/

[2] Niss, M., “Mathematical competencies and the learning of mathematics: TheDanish KOM project”, Proceedings of the 3rd Mediterranean conference onmathematical education, 2003, 115-124.

[3] RULES_MATH project. https://rules-math.com/[4] Niss M., Bruder R., Planas N., Turner R., Villa-Ochoa J.A. (2017) Conceptual-

isation of the Role of Competencies, Knowing and Knowledge in MathematicsEducation Research. In: Kaiser G. (eds) Proceedings of the 13th InternationalCongress on Mathematical Education. ICME-13 Monographs. Springer, Cham

[5] Rasteiro D. D., Gayoso Martinez V. , Caridade C., Martin-Vaquero J.,Queiruga-Dios A. (2018), “Changing teaching: competencies versus contents.”,EDUCON 2018 - Emerging Trends and Challenges of Engineering Education”.Tenerife, Spain.

[6] Queiruga-Dios, A.; Sanchez, M.J.S.; Perez, J.J.B.; Martin-Vaquero, J.; Enci-nas, A.H.; Gocheva-Ilieva, S.; Demlova, M.; Rasteiro, D.D.; Caridade, C.;Gayoso-Martinez, V. Evaluating EngineeringCompetencies: A New Paradigm.In Proceedings of the Global Engineering Education Conference (EDUCON),Tenerife, Spain, 17–20 April 2018; IEEE: New York, NY, USA, 2018; pp.2052–2055.

[7] Caridade, C.M.; Encinas, A.H.; Martín-Vaquero, J.; Queiruga-Dios, A.;Rasteiro, D.D. Project-based teaching in calculus courses: Estimation of thesurface and perimeter of the Iberian Peninsula. Comput. Appl. Eng. Educ. 2018,26, 1350–1361.

[8] Caridade C.M.R., Rasteiro D.M.L.D. (2020) Evaluate Mathematical Competen-cies in Engineering Using Video-Lessons. In: Martínez Álvarez F., TroncosoLora A., Sáez Muñoz J., Quintián H., Corchado E. (eds) International JointConference: 12th International Conference on Computational Intelligence inSecurity for Information Systems (CISIS)

[9] Caridade C.M.R., Rasteiro D.M.L.D. , Richtarikova Daniela(2020) Mathemat-ics for Engineers: Assessing Knowledge and Competencies. In 19th Confer-ence on Applied Mathematics Aplimat 2020, Proceedings. First edition. Pub-lishing house Spektrum STU Bratislava, 2020. ISBN 978-80-227-4983-1. Edi-tors: Dagmar Szarkova, Daniela Richtarikova, Monika Prasilova.

[10] Rasteiro D.M.L.D., Caridade C.M.R.(2020) Probability and Statistical Meth-ods: Assessing Knowledge and Competencies. In 19th Conference on Ap-plied Mathematics Aplimat 2020, Proceedings. First edition. Publishing houseSpektrum STU Bratislava, 2020. ISBN 978-80-227-4983-1. Editors: DagmarSzarkova, Daniela Richtarikova, Monika Prasilova

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[11] Queiruga-Dios A. et al., "Evaluating engineering competencies: A newparadigm," 2018 IEEE Global Engineering Education Conference (EDUCON),Tenerife, 2018, pp. 2052-2055, doi: 10.1109/EDUCON.2018.8363490.

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ASSESSMENT WITH MATHEMATICS COMPETENCIES-BASEDRULES_MATH’S GUIDES FOR LINEAR ALGEBRA

Cristina M.R. CARIDADE1, Deolinda M. L. D. RASTEIRO1



ABSTRACT

This work follows from the “New Rules for Assessing Mathematical Competen-cies” (RULES_MATH) project which aims to change the educational paradigmand to get a common European teaching and learning system based on math-ematical competencies rather than contents [https://rules-math.com/]. For this,RULES_MATH describes and analyses ways to assess mathematics taught to fu-ture engineers through competencies. The eight mathematical competencies alreadyidentified (thinking mathematically; reasoning mathematically; posing and solvingmathematical problems; modelling mathematically; representing mathematical en-tities; handling mathematical symbols and formalism; communicating in, with, andabout mathematics and make use of aids and tools for mathematical activity) arerecognized in the assessment made to students. The RULES_MATH project part-ners’ working groups have developed a set of “Guide for a Problem” in the differ-ent areas of mathematics that are intended to provide some examples of proposedforms of assessment and competence-based activities. The materials are availableto all project partners to apply to different students from different courses and insti-tutions. Two of such materials are LA3 and LA4, which aims to evaluate studentsabout "Matrices and determinants" and "Solution of simultaneous linear equations",respectively.The study was carried out in a group of 143 students of Linear Algebra from thedegree in Biomedical Engineering (20 students), Electromechanics (39 students)and Mechanics (84 students) from the Engineering Institute of Coimbra.The results obtained were satisfactory since at almost questions (9 out of 11) stu-dents were able to obtain a positive grade. The performed assessment test alsopermits to identify level differences between the courses where it was applied. Re-lating the test itself as a tool to assess competencies, we may conclude that it coversmost of the competencies that we need to evaluate, and its difficulty is adequate toour students.

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References








[8] Caridade C.M.R., Rasteiro D.M.L.D. (2020) Evaluate Mathematical Competen-cies in Engineering Using Video-Lessons. In: Martínez Álvarez F., TroncosoLora A., Sáez Muñoz J., Quintián H., Corchado E. (eds) International JointConference: 12th International Conference on Computational Intelligence inSecurity for Information Systems (CISIS)

[9] Caridade C.M.R., Rasteiro D.M.L.D. , Richtarikova Daniela(2020) Mathemat-ics for Engineers: Assessing Knowledge and Competencies. In 19th Confer-ence on Applied Mathematics Aplimat 2020, Proceedings. First edition. Pub-lishing house Spektrum STU Bratislava, 2020. ISBN 978-80-227-4983-1. Edi-tors: Dagmar Szarkova, Daniela Richtarikova, Monika Prasilova.

[10] Rasteiro D.M.L.D., Caridade C.M.R.(2020) Probability and Statistical Meth-ods: Assessing Knowledge and Competencies. In 19th Conference on Ap-plied Mathematics Aplimat 2020, Proceedings. First edition. Publishing houseSpektrum STU Bratislava, 2020. ISBN 978-80-227-4983-1. Editors: DagmarSzarkova, Daniela Richtarikova, Monika Prasilova

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A STUDY ON CONSTANT ANGLE RULED SURFACESGENERATED BY TIMELIKE CURVE IN MINKOWSKI SPACE

Cumali EKICI1, Yasin ÜNLÜTÜRK2, Gül Ugur KAYMANLI3

1Department of Mathematics and Computer Science, Eskisehir Osmangazi University Eskisehir, 26040, Turkey2Department of Mathematics, Kirklareli University, Kirklareli, 39100, Turkey

3Department of Mathematics, Cankiri Karatekin University, Cankiri, 18100, Turkey


ABSTRACT

In geometry, a constant angle surface in three dimensional space is a surface whosetangent planes make a constant angle with a fixed vector field of the ambient spaceand a ruled surface in three dimensional space is a surface which can be describedas the set of points swept by a moving straight line. These constant angle ruled sur-faces generalize the concept of general helix, that is, curves whose tangent vectorsmake a constant angle with a fixed vector. General helix is characterized by thefact that the ratio of the torsion and curvatures of the curve is constant. In additionto this, the slant helix is a curve whose normal vectors make a constant angle witha fixed direction and it is characterized by the fact that the geodesic curvature ofthe principal image of the principal normal indicatrix is a constant function. In thisstudy, we work on constant angle ruled surfaces generated by timelike curve withrespect to Frenet frame {t, n, b} in three dimensional Minkowski space E3

1 . Wealso deal with constant angle ruled surfaces parallel to both tangent of general helixand normal of slant helix. That is, we take the tangent vector and normal vector ofa ruled surface are linearly dependent to the tangent vector t and normal vector nof the timelike base curve α(t), respectively. We then give some characterizationssuch as minimality, developability (i.e. flatness) by using both Gaussian and meancurvatures for these surfaces using Frenet frame in three dimensional Minkowskispace from the point of view the constant angle property. In this setting we are ableto show that the constant angle ruled surfaces generated by both tangent and nor-mal vectors are developable (i.e. these constant angle ruled surfaces have vanishingGaussian curvature) in three dimensional Minkowski space.

Keywords Constant Angle Surface · Ruled Surface · Timelike Curve

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References

[1] Ali A.T., A constant angle ruled surfaces, International Journal of Geometry,7(1): 69-80, 2018.

[2] Ali A.T., Non-lightlike ruled surfaces with constant curvatures in Minkowski3-space, International Journal of Geometric Methods in Modern Physics, 15:1850068, 2018.

[3] Atalay G.S., Güler F. and Kasap E., Spacelike constant angle surfaces inMinkowski 3-space. J. Math. Comput. Sci., 2(3): 451-461, 2012.

[4] Bayard P., Monterde J. and Volpe R., Constant angle surfaces in 4-dimensionalMinkowski space, 2019.

[5] Güler F., Saffak G. and Kasap E., Timelike constant angle surfaces inMinkowski Space R31, Int. J. Contemp. Math. Sciences, 6(44): 2189-2200,2011.

[6] Karakus S.O., Certain Constant Angle Surfaces Constructed on Curves inMinkowski 3-Space, International Electronic Journal of Geometry, 11: 37-47,2018.

[7] Lopez R., Differential geometry of curves and surfaces in Lorentz-Minkowskispace, International Electronic Journal of Geometry, 7: 44–107, 2014.

[8] O’Neill B., Semi-Riemannian Geometry, New York Academic press, 1983.[9] Rafael L. and Munteanu M. I., Constant angle surfaces in Minkowski space,

Bulletin of The Belgian Mathematical Society-simon Stevin, 18: 2009.

26


MATHEMATICAL COMPETENCY ASSESSMENT –RULES_MATH GUIDES ON COMPLEX NUMBERS

Daniela RICHTARIKOVA

Slovak University of Technology in Bratislava Institute of Mathematics and Physics, Faculty of Mechanical Engineering, Slovakia


ABSTRACT

Assessment makes an integral part in education. Generally, it is considered to bethe representative of a level of mastering some specialisation, and gives a promisefor future that the person will be able to use the achieved knowledge and skills infurther study, occupation or common life. Assessment comprises several very im-portant roles. It provides the feedback of study process and its results for the persononeself, and for his or her educator as well. It reveals strengths and weaknesses,gives motivation for further study, or decides about admission to following gradesof study or to future employment. Speaking about assessment of a student withina course, it gives the information about the level of student’s knowledge and skillson some curricula or a curricula unit. On one hand, it reflects the level of master-ing the content, what is in general took into consideration predominantly, and onanother hand, it is usually automatically supposed to reflect also the level of corre-sponding competency. Regard to competency education becomes very important inthese days and the international team of Rules_Math Erasmus + project developedthe Guides focussed especially for training and assessment of mathematical compe-tency for students of technical tertiary education. With respect to the SEFI MWGGroup document “A Framework for Mathematics Curricula in Engineering Educa-tion” (Alpers et al., 2013), the Guides were build upon Core 1 contents – math-ematics curricula units studied mostly at bachelor technical degree, distinguishingeight main mathematical competencies proposed by the Danish KOM project in twothousands (Niss, 2003).In the paper, we deal with the Guide on Complex Numbers prepared in the Depart-ment of Mathematics and Physics at the Faculty of Mechanical Engineering STU,the partner of the project, and introduce a competencies evaluation methodology.The results of testing, and the teaching methods of best practice for competencydevelopment are presented too.

Keywords Mathematical competency · Assessment · Technical university ·Complex numbers

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References

[1] Alpers, B. et al., A Framework for Mathematics Curricula in Engineering Edu-cation. SEFI, Brussels, 2013. ISBN: 978-2-87352-007-6.

[2] Niss, M., Mathematical competencies and the learning of mathematics: TheDanish KOM project. In A. Gagatsis, S. Papastravidis (Eds.), 3rd MediterraneanConference on Mathematics Education, Athens, Greece: Hellenic Mathemati-cal Society and Cyprus Mathematical Society, 115- 124, 2003.

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ALGEBRAIC STRUCTURE OF n-DIMENSIONAL QUATERNIONICSPACE

Deniz ALTUN 1, Salim YÜCE 2

1,2Yildiz Technical University, Faculty of Arts and Sciences, Department of Mathematics Istanbul, 34220, Turkey

Corresponding Author’s E-mail: [email protected] (https://orcid.org/0000-0001-8327-1161),

[email protected] (https://orcid.org/0000-0002-8296-6495)

ABSTRACT

Real quaternions, first described by W. R. Hamilton in 1843, are a 4-dimensionalnumber system [7, 8, 9]. Real quaternions have been studied in many areas suchas algebra, geometry, physics, computer-aided design (CAD), put into practice andhave provided technological improvements.Just as the elements of an n-dimensional real numbers space are called vectors, theelements of n-dimensional real quaternion space can also be called n-vectors. Then-dimensional quaternionic space is a 4n-dimensional vector space over the realnumbers. Regarding this space, studies are carried out in areas such as symplecticgeometry, Clifford algebra, etc [4, 2, 1, 6].In the literature, various ordering types are given on the n-dimensional real num-bers space and the component-wise product is defined [5, 3]. In this study, first ofall, a metric is defined depending on the order relation of the n-dimensional realnumbers set. At the same time, ring structure was established in n-dimensionalquaternionic space by considering the component-wise product which is defined inthe n-dimensional real numbers set.Following that, the set of n-dimensional quaternions is also defined by a differentrepresentation of n-vectors. Using this notation, formulations corresponding to thebasic operations in n-dimensional quaternionic space are obtained. In addition, byadhering these representations we mentioned, vector space and module structuresof n-dimensional quaternionic space over the set of real ordered n-tuples and overthe set of n-dimensional real numbers were obtained.The definitions, theorems and results we provide in this study are preliminary infor-mation in terms of our research in the field of differential geometry in n-dimensionalquaternionic space. In the light of this study, it is aimed to add the theory of curvesin the n-dimensional quaternionic space to the literature.

Keywords Symplectic geometry · n-dimensional quaternionic space ·Vector space

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References

[1] Chevalley, C., Theory of Lie groups, Courier Dover Publications, 2018.[2] Da Silva, A.C., and Da Salva, A.C., Lectures on symplectic geometry, Berlin:

Springer, 2001.[3] Davey, B.A., and Priestley, H.A., Introduction to lattices and order, Cambridge

university press, 2002.[4] Farenick, D.R., and Pidkowich, B.AF., The spectral theorem in quaternions,

Linear algebra and its applications, 371: 75-102, 2003.[5] Ghorpade, S.R., and Limaye, B.V., A course in multivariable calculus and anal-

ysis, Springer Science & Business Media, 2010.[6] Girard, P.R., Quaternions, Clifford algebras and relativistic physics, Springer

Science & Business Media, 2007.[7] Hamilton, W.R., XI. On quaternions; or on a new system of imaginaries in alge-

bra, The London, Edinburgh, and Dublin Philosophical Magazine and Journalof Science, 33.219: 58-60, 1848.

[8] Hamilton, W.R., Elements of quaternions, Longmans, Green & Company, 1866.[9] Ward, J.P., Quaternions and Cayley numbers: Algebra and applications, Vol.

403. Springer Science & Business Media, 2012.

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PROBABILITY AND STATISTICAL METHODS: ASSESSINGKNOWLEDGE AND COMPETENCIES – CASE STUDY AT ISEC

Deolinda M. L. D. RASTEIRO1, Cristina M.R. CARIDADE1



ABSTRACT

The concepts taught during a Statistical Methods course make use of different math-ematical skills and competencies. The idea of presenting a real problem to studentsand expect them to solve it from beginning to end is, for them, a harder task thenjust to obtain the value of a probability given a known distribution. Much has beensaid about teaching mathematics related to day life problems. In fact, we all seem toagree that this is the way for students to get acquainted of the importance of the con-tents that are taught and how they may be applied in the real world. The definitionof mathematical competence as was given by Niss (Niss, 2003) means the ability tounderstand, judge, do, and use mathematics in a variety of intra– and extra – math-ematical contexts and situations in which mathematics plays or could play a role.Necessarily, but certainly not sufficient, prerequisites for mathematical competenceare lots of factual knowledge and technical skills, in the same way as vocabulary,orthography, and grammar are necessary but not sufficient prerequisites for liter-acy. In the OEDC PISA document (OECD, 2009), it can be found other possibilityof understanding competency which is: reproduction, i.e, the ability to reproduceactivities that were trained before; connections, i.e, to combine known knowledgefrom different contexts and apply them do different situations; and reflection, i.e,to be able to look at a problem in all sorts of fields and relate it to known theoriesthat will help to solve it. The competencies that were identified in the KOM project(Niss, 2003, Niss & Hojgaard, 2011) together with the three “clusters” describedin the OECD document referred above were considered and adopted will slightlymodifications by the SEFI MWG (European Society for Engineering Education), inthe Report of the Mathematics Working Group (Alpers, 2013). At Statistical Meth-ods courses often, students say that assessment questions or exercises performedduring classes have a major difficulty that is to understand what is asked meaningthe ability to read and comprehend the problem and to translate it into mathemat-ical language. The study presented in this paper reflects an experience performedwith second year students of Mechanical Engineering graduation of Engineering

31


Institute of Coimbra, where the authors assessed statistical methods contents taughtduring the first semester of 2017/2018 academic year. The questions assessmenttests were separated into two types: ones that referred only to problem comprehen-sion and its translation into what needed to be calculated and others where studentsneed only to apply mathematical techniques in order to obtain the results. This pa-per is one of the results of RULES_MATH project which aims to develop tools toassess mathematical competencies. Eight mathematical competencies identified arerecognized in the assessment made to students in what concerns learning probabilitytheory concepts. Since 2017 a study was carried out with Mechanical Engineeringstudents at Engineering Institute of Coimbra. The results obtained cover the test asa tool to assess competencies and, also its fitness to our students.


References






[6] Rasteiro, D. D., Gayoso Martinez, V., Caridade, C., Martin-Vaquero, J. andQueiruga-Dios, A., "Changing teaching: Competencies versus contents," 2018IEEE Global Engineering Education Conference (EDUCON), Tenerife, 2018,pp. 1761-1765, doi: 10.1109/EDUCON.2018.8363447.


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ASSESSING KNOWLEDGE AND COMPETENCIES:RULES_MATH PROJECT’S EFFECTS AT ISEC

Deolinda M. L. D. RASTEIRO1, Cristina M.R. CARIDADE1



ABSTRACT

In higher education, mathematics has an important role in engineering courses.From any engineering course curriculum there are, in the first year, curricular units(CU) in the area of mathematics that are fundamental for students to acquire thenecessary basic knowledge to the most specific CU of their course. Without thiswell-established mathematical foundation, success in the engineering CUs is seri-ously compromised. During an engineering course, students learn and consolidatethe basic principles of mathematics to solve practical problems, reinforcing theirmathematical concepts knowledge. However, although mathematics is a basic disci-pline in admission to engineering courses, difficulties are identified by engineeringstudents in CUs related to mathematic basic core. In this context, it seems perti-nent to identify the mathematics competencies attained by engineering students sothat they can use these skills in other CU and take them as tool support to theirprofessional activities. In this talk we present some results obtained while partici-pating in project RULES_MATH, namely ways to assess competencies in calculus,algebra and statistical methods curricular units. The study presented in this pa-per reflects an experience performed with second year students of Mechanical En-gineering graduation and with first year students of Electrotechnical Engineering,Biomedical Engineering, Electromechanics and Mechanics of Engineering Instituteof Coimbra, where the authors assessed curricular units contents taught during thefirst semester of 2017/2018 to 2019/2020 academic years. This paper is one of theresults of RULES_MATH project which aims to develop tools to assess mathemat-ical competencies. The eight mathematical competencies already identified (think-ing mathematically; reasoning mathematically; posing and solving mathematicalproblems; modelling mathematically; representing mathematical entities; handlingmathematical symbols and formalism; communicating in, with, and about mathe-matics and make use of aids and tools for mathematical activity) are recognized inthe assessment made to students in what concerns learning concepts. The authors

33


will present the changes performed in their curricular units as a result of the projectdevelopment and show how the reaction of students was.


References







[7] Rasteiro, D. D., Gayoso Martinez, V., Caridade, C., Martin-Vaquero, J. andQueiruga-Dios, A., "Changing teaching: Competencies versus contents," 2018IEEE Global Engineering Education Conference (EDUCON), Tenerife, 2018,pp. 1761-1765, doi: 10.1109/EDUCON.2018.8363447.


[9] Kulina, H.; Gocheva-Ilieva, S.; Voynikova, D.; Atanasova, P.; Ivanov, A.; Iliev,A. Integrating of Competences in Mathematics through Software-Case Study.In Proceedings of the Global Engineering Education Conference (EDUCON),Tenerife, Spain, 17–20 April 2018; IEEE: New York, NY, USA, 2018; pp.1586–1590.

[10] Gocheva-Ilieva, S.; Teofilova, M.; Iliev, A.; Kulina, H.; Voynikova, D.; Ivanov,A.; Atanasova, P. Data mining for statistical evaluation of summative andcompetency-based assessments in mathematics. In Advances in Intelligent Sys-tems and Computing; Martínez Álvarez, F., Troncoso, L.A., Sáez Muñoz, J.,

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Quintián, H., Corchado, E., Eds.; Springer: Cham, Switzerland, 2020; Volume951, pp. 207–216.

[11] Queiruga-Dios, A.; Santos Sánchez, M.J..; Queiruga-Dios, M.; Gayoso-Martínez, V.; Encinas, A.H. A virus infected your laptop. Let’s play an escapegame. Mathematics 2020, 8(2), 166.

[12] Caridade C.M.R., Rasteiro D.M.L.D. (2020) Evaluate Mathematical Compe-tencies in Engineering Using Video-Lessons. In: Martínez Álvarez F., TroncosoLora A., Sáez Muñoz J., Quintián H., Corchado E. (eds) International JointConference: 12th International Conference on Computational Intelligence inSecurity for Information Systems (CISIS


[14] Richtarikova, D., Training and Assessment with Respect to Competencies –Forms. In Aplimat 2019, Proceedings of the 18th conference on applied math-ematics 2019. Bratislava, 5.-7. 2. 2019. STU, 2019, pp. 989-991, CD ROM.ISBN 978-80-227-4884-1.

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A NOVEL FRACTIONAL-ORDER MODEL FOR COVID-19INFECTIOUS WITH INCOMMENSURATE ORDERS

Din PRATHUMWAN1, Kamonchat TRACHOO2, Inthira CHAIYA2

1Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand2Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand


ABSTRACT

It is well known that the spread of the novel coronavirus (COVID-19) outbreak hasaffected people and economics around the world. This outbreak was first found inWuhan, China. The number of cases is increasing every day. The understanding ofthe dynamics of infectious disease becomes an important way to control the pan-demic. Mathematical modeling is an essential role to predict and understand thebehavior of the disease spreading. One of the most powerful tools for forecasting isa fractional-order model because the fractional-order model provides more realisticthan the traditional integer order model. In this research, we propose the fractional-order system of differential equations with incommensurate orders to predict thetrend of disease spreading, and the model examines the dynamics of disease trans-mission for various orders theoretically and numerically.

Keywords Incommensurate fractional order derivative · COVID-19 · Equilibriumpoints · Epidemic model

References

[1] Rajagopal K., Hasanzadeh N., Parastesh F., et al. A fractional-order model forthe novel coronavirus (COVID-19) outbreak, Nonlinear Dyn, 2020, 8 pages,2020. doi.org/10.1007/s11071-020-05757-6

[2] Miller K.S., Ross B., An Introduction Fractional Calculus Functional Differen-tial Equation, Johan Willey and Sons Inc., New York, NY, USA, 2003.

[3] Alkahtani B.S.T., Alzaid S.S., Numerical solution of the time fractionalBlack–Scholes model governing European options, Chaos, Solitons & Fractals,138, 110006, 2020.

[4] Diethelm K., An algorithm for the numerical solution of differential equationsof fractional order, Electron. Trans. Numer. Anal. 5, 1–6, 1997.

36


VEHICLE AND DRIVER SCHEDULING PROBLEM USINGε-CONSTRAINT IN PUBLIC TRANSPORTATION

Durdu Hakan UTKU1, Burcu ALTUNOGLU1

1University of Turkish Aeronautical Association, Bahçekapı Mah. Okul Sk. No:11, 06790 Etimesgut, Ankara, Turkey


ABSTRACT

In this study, a multiple objective mixed integer programming model for vehicleand driver scheduling problem is applied to a public transportation system. Thefirst objective function has the operational objective and tries to determine the op-timal number of vehicles and drivers to be used to provide the minimum cost. Thesecond objective minimizes the transfer times which related with the satisfaction ofthe customers. The ε-constraint method is used for the multi-objective public trans-portation model. In the ε-constraint method, while one of the objectives reachesthe optimum value, the other objective function is solved within the bounds of theε-constraint. The function that minimizes the cost in the proposed model is desig-nated as the objective function and the other objective function, which minimizesthe waiting time at the transfer centers, is included in the model as a constraint. Thecosts include the total fixed costs of drivers and vehicles, as well as the variablecosts caused by deadhead and d-trips. Each d-trips must be assigned to the appro-priate vehicle and driver without any conflict. D-trips belonging to the same routemust be assigned to the same vehicle. The other objective function that provides theminimum waiting time is restricted by the ε-constraint method. The waiting timeis calculated by considering the time the vehicles reach these transfer points andspend at the transfer points. The model considers the specified waiting times as theexpected waiting times. In the first phase, the model determines the cost by havingto assign a driver to each vehicle after determining the appropriate number of vehi-cles. Drivers are selected by considering the salary they get. These salaries includefixed and overtime component. In the second phase, the objective function, whichprovides minimum cost, determines the number of buses to be kept in stock andconsiders their holding costs. This part, which makes the waiting time minimumin the transfer centers, achieves its objective without paying attention to the cost.The transfer centers at the intersection of the d-trips are areas that many passengersuse and sometimes must wait a lot. The waiting time is calculated by consideringthe time the vehicles reach these transfer points and spend at the transfer points.

37


The test problems have been solved with GAMS/CPLEX solver to determine theperformance of the model.

Keywords Vehicle and Driver Scheduling · Multi-Objective Mixed IntegerProgramming · ε-constraint Method

References

[1] Aghaei J., Amjady N., Shayanfar H. A., Multi-objective electricity marketclearing considering dynamic security by lexicographic optimization and aug-mented epsilon constraint method. Elsevier, Applied Soft Computing, Vol. 11,pp. 3846 – 3858, 2011.

[2] Aghaei J., Shayanfar H., Amjady N., Multi Objective Market Clearing of JointEnergy and Reserves Auctions Ensuring Power System Security. Energy Con-version and Management, Vol. 50, pp. 899 – 906, 2009.

[3] Amberg B., Amberg B., Kliewera N., Robust Efficiency in Urban Public Trans-portation: Minimizing Delay Propagation in Cost-Efficient Bus and DriverSchedules. TRANSPORTATION SCIENCE Vol. 53, No. 1, pp. 89–112, 2019.

[4] Babicheva T., Burghout W., Empty Vehicle Redistribution in Autonomous TaxiServices. EURO J Transp Logist, 2019.

[5] Bach L., Dollevoet T., Huisman D., Integrating Timetabling and Crew Schedul-ing at a Freight Railway Operator. Transportation Science, Vol. 50, No. 3, pp.878–891, 2016.

[6] Boland N. L., Savelsbergh M.W. P., Perspectives on Integer Programming ForTime-Dependent Models. Springer, TOP (2019), Vol. 27, pp.147–173, 2019.

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NEW HYPERBOLIC-NUMBER FORMS OF THE EULER SAVARYEQUATION: THE CONSIDERATION OF FUTURE POINTING

TIMELIKE POLE RAYS FOR SPACELIKE POLE CURVES

Duygu ÇAGLAR 1, Nurten GÜRSES 1

1Yildiz Technical University, Faculty of Art and Sciences, Department of Mathematics, 34220, Istanbul, Turkey


[email protected] (https://orcid.org/0000-0001-8407-854X)

ABSTRACT

In 2-dimensional Euclidean plane E2, one-parameter planar motions are introducedby Blaschke and Müller [1]. Also in [1], the relation between velocities and acceler-ations is examined and the Euler Savary Equation is calculated during the motion.In addition, in the complex plane C, one parameter planar motions are examinedin [1] and Euler Savary Equation is obtained by presenting the relation between thecurvatures of the trajectory curves in [2]. In analogy with complex motions, one pa-rameter motions in the hyperbolic plane considering hyperbolic numbers (the num-bers of the form z = x + jy where x and y are real numbers and j2 = 1, j 6= ±1,see in [3]-[11]) is defined by [12]. Also, Euler Savary Equation in the hyperbolicplane is determined by [13].Euler Savary Equation gives the radius of curvature and the center of curvature ofthe path traced by a point in moving plane. In [14]-[16], the new complex-numberforms of the Euler Savary Equation are presented in a computer-oriented format. Byusing these forms, if any three of the following four points: "pole point, arbitrarypoint of moving plane, inflection point and center of curvature of the path tracedby the point moving plane" are known, the fourth one can be found. This complexnumber technique has the advantage of eliminating the need for the traditional signconventions and is suitable for digital computation, [14]-[16].By inspired of the complex number approach, and using the basic concepts in [17]-[19], we examine new hyperbolic number forms of the Euler Savary Equationsin this study for the following case: We firstly categorize all of the situations inhyperbolic plane and examine the calculations for future pointing timelike pole raysby considering spacelike pole curves in analog with the studies [14]-[16]. Thesenew hyperbolic forms give not only the relationship between a point of movingplane and center of curvature of the path traced by the point moving plane but

39


also the relation between the above mentioned four-points. This hyperbolic numbermethod based on vector calculations and provides a detailed examination of thelocations of the points in the hyperbolic plane. Finally, an example is presented toillustrate this case.

Keywords One-parameter planar motion · The Euler Savary Equation · HyperbolicNumber

References

[1] Blaschke W. and Müller H.R., Ebene Kinematic, Verlag Oldenbourg,München, 1956.

[2] Masal M., Tosun M. and Pirdal A.Z., Euler Savary Formula for the One Pa-rameter Motions in the Complex Plane C, International Journal of PhysicalSciences, 006-010, 5(2010).

[3] Catoni F., Cannata R., Catoni V. and Zampetti P., Hyperbolic Trigonometry intwo-dimensional space-time geometry, N. Cim. B, 475–491, 118(2003).

[4] Catoni F. and Zampetti P., Two-dimensional space-time symmetry in hyper-bolic functions, N. Cim. B, 1433–1440, 115(2000).

[5] Cockle J., On a new imaginary in algebra, Phil. Mag., 37–47, 3(34)(1847).[6] Fjelstad P., Extending special relativity via the perplex numbers, Amer. J.

Phys., 416–422, 54(1986).[7] Fjelstad P. and Gal S. G., n-dimensional hyperbolic complex numbers, Adv.

appl. Clifford alg., 47–68, 8(1)(1998).[8] Fjelstad P. and Gal S. G., Two-dimensional geometries, topologies, trigonome-

tries and physics generated by complex-type numbers, Adv. appl. Clifford alg.,81–107, 11(1)(2001).

[9] Rochon D. and Shapiro M., On algebraic properties of bicomplex and hyper-bolic numbers, Anal. Univ. Oradea, fasc. math., 71–110, 11(2004).

[10] Sobczyk G., Complex and Hyperbolic Numbers, In: New Foundations inMathematics, Birkhäuser, Boston, 2013.

[11] Sobczyk, G., Hyperbolic number plane, College Math. J., 268–280,26(4)(1995).

[12] Yüce S. and Kuruoglu N., One-Parameter Plane Hyperbolic Motions, Adv.Appl. Clifford Algebras, 279-285, 2008.

[13] Ersoy, S., Akyigit, M. One-Parameter Homothetic Motion in the HyperbolicPlane and Euler-Savary Formula. Adv. Appl. Clifford Algebras, 297–313,21(2011).

[14] Sandor G. N., Erdman A.G., Hunt L. and Raghavacharyulu E., New Complex-Number Forms of the Euler-Savary Equation in a Computer-Oriented Tret-ment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact ,ASMEJ. Mech. Des., 104:227–232, 1982.

[15] Sandor G. N., Xu Y. and Weng T. C., A Graphical Method for Solvingthe Euler-Savary Equation, Mechanism and Machine Theory, 25(2):141-147,1990.

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[16] Sandor G. N., Arthur G. E. and Raghavacharyulu, E., Double Valued Solutionsof the Euler-Savary Equation and Its Counterpart in Bobillier’s Construction,Mechanism and Machine Theory, 20(2): 145-178, 1985.

[17] Birman G. S., Nomizu K., Trigonometry in Lorentzian geometry, Amer. Math.Monthly, 543–549, 91(9)(1984).

[18] Nesovic E., Hyperbolic Angle Function in the Lorentzian Plane, KragujevacJournal of Mathematics, 28(2005).

[19] Nesovic E. and Petrovic–Torgasev M., Some Trigonometric Relations in theLorentzian Plane, Kragujevac J. Math., 219–225, 25(2003).

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FINITE DIFFERENCE METHOD FOR FRACTIONAL ORDERPSEUDO-PARABOLIC PARTIAL DIFFERENTIAL EQUATION

Ecem GÖKTEPE 1, Mahmut MODANLI 2

1,2Harran University, Turkey


ABSTRACT

In this paper fractional pseudo-parabolic equation with initial-boundary conditionsis investgated. Finite difference schemes are constructed for this differential equa-tion. Stability estimaties are proved for these difference schemes. Exact solution ofthis differential equation is calculated by Laplace transform method. Error analysisare made by comparing the exact solution and the approximate solutions. Figuresshowing the physical properties of the exact and approximate solutions are given.From the error analysis table and figures, it is clearly seen that this applied methodis an effective and good method for this equation.

Keywords Finite difference method · Fractional order pseudo-parabolic partialdifferential equation · Stability

References

[1] Jachimaviciene, J., Sapagovas, M., Štikonas, A., & Štikoniene, O. (2014). Onthe stability of explicit finite difference schemes for a pseudoparabolic equationwith nonlocal conditions. Nonlinear Anal. Model. Control, 19(2), 225-240.

[2] Modanli, M. (2018). Two numerical methods for fractional partial differentialequation with nonlocal boundary value problem. Advances in Difference Equa-tions, 2018(1), 333.

[3] Eltayeb, H., Mesloub, S., T Abdalla, Y., & Kılıçman, A. (2019). A Note on Dou-ble Conformable Laplace Transform Method and Singular One DimensionalConformable Pseudohyperbolic Equations.Mathematics, 7(10), 949.

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THE BEST KNOWN ITERATION BOUND OF INTERIOR POINTMETHODS FOR LINEAR OPTIMIZATION PROBLEM

El Amir DJEFFAL1, Bachir BOUNIBANE2

1LEDPA Laboratory, University of Batna 2, Batna, Algeria2University of Batna 2, Batna, Algeria


ABSTRACT

Kernel functions play an important role in the complexity analysis of the interiorpoint methods for linear optimization. In this paper, we present a primal dual in-terior point method for linear optimization based on a new parameterized kernelfunction.The proposed kernel function is a generalization of the one used recentlyin Bai et al. (SIAM J Optim 15:101128, 2004) for LO, and they showed that theiteration bound for the corresponding algorithm is O(

√n(log n)2 log n

ε). By using

several new technical lemmas, the iteration complexity bound as O(√n log n) log n

εis obtained, which coincides with the currently best iteration complexity bounds forlarge-update methods. The numerical tests confirm that the proposed algorithm isrobust

Keywords Kernel function · Linear optimization · Primal-dual interior-pointmethods · Large-and small update methods

References

[1] Amini, K., Haseli, A.: A new proximity function generating the best knowniteration bounds for both large-update and small-updatebinterior point methods.ANZIAM 49, 259–270 (2007)

[2] El Ghami, M., Ivanov, I., Melissen, J.B.M., Roos, C., Steihaug, T.: Apolynomial-time algorithm for linear optimization based on a new class of ker-nel functions. J. Comput. Appl. Math. 224, 500–513 (2009)

[3] Bai, Y. Q., Ghami, M. E., Roos, C.: A comparative study of kernel functionsfor primal-dual interior-point algorithms in linear optimization. SIAM Journalon Optimization, 15(1), 101–128 (2004)

[4] Bai, Y.Q., Wang, G.Q., Roos, C.: A new kernel function yielding the bestknown iteration bounds for primal-dual interior point method. Acta Math.Sinica 25(12), 2169–2178 (2009)

43


[5] Bai, Y. Q., Ghami, M. E., Roos, C.: A new efficient large-update primal-dualinterior-point method based on a finite barrier. SIAM Journal on Optimization,13(3), 766–782 (electronic) (2003)

44


A CLUSTER ANALYSIS ON THE EUROPEAN ENTREPRENEURIALFRAMEWORK CONDITIONS OVER THE LAST TWO DECADES

Eliana COSTA E SILVA1, Aldina CORREIRA1, Ana BORGES1, Fábio DUARTE1

1CIICESI, ESTG, Politécnico do Porto


ABSTRACT

Most policymakers and academics agree that entrepreneurship is critical to the de-velopment and well-being of the society [1]. In fact, entrepreneurs create jobs, anddrive and shape innovation. Therefore, entrepreneurship is a catalyst for economicgrowth and national competitiveness. The Global Entrepreneurship Monitor (GEM)research project, founded in 1997, is the largest ongoing study of entrepreneurialdynamics in the world [2]. According to GEM 2019/2020 Global Report, fiftyeconomies participated in the GEM 2019 Adult Population Survey (APS), includ-ing 21 European countries. The present study focus on the twelve indicators of theentrepreneurial ecosystem, defined by GEM, i.e. the Entrepreneurial FrameworkConditions (EFCs). Specifically, the aim is to study the changes that have occurredin the Portuguese experts’ perceptions over the last two decades, while comparingthem to the perceptions of experts on other European countries. Many studies haveused GEM data for their research [4], [5], [6], [7], [8]. In [3], the period from2010 to 2016 is analysed. Substantial changes on the clusters of European throughthese years were observed. In particular, it was found that despite the economic andfinancial similarities between Portugal, Italy, Greece and Spain, that faced a dra-matic period between 2010-2012, Portugal took off from the remaining countriesafter 2012, and only in 2016 it was caught up by Spain. The present study aims atextending that work by considering the period before the crisis and after 2016, inorder to obtain a wider view on the European entrepreneurs’ perceptions. Severalmultivariate cluster analysis techniques are used to group the European economiesaccording to the experts’ perceptions on the EFCs of their country.

Keywords Cluster Analysis · Entrepreneurial Framework Conditions · GlobalEntrepreneurship Monitor


45


References

[1] Kelley, D., Singer, and Herrington, M., Global entrepreneurship monitor 2011global report,2012.

[2] Singer, S., Herrington, M. and Menipaz, E., Global entrepreneurship monitor:Global report 2017/18, 2018.

[3] Costa e Silva, E., Correia, A., and Duarte, F., How Portuguese experts’ percep-tions on the entrepreneurial framework conditions have changed over the years:A benchmarking analysis. In AIP Conference Proceedings, 2040(1):110005,2018.

[4] E. Autio, E., Kenney, M., Mustar, P., Siegel, D. and Wright, M., ResearchPolicy, 43:1097–1108, 2014.

[5] Martínez-Fierro, S., Biedma-Ferrer, J. M., and Ruiz-Navarro, J., Revista deEnonomía Mundial, 41:181–212, 2015.

[6] Acosta, Y. A. C., Adu-Gyamfi, R., Nabi, M. N. U. and Dornberger, U. En-trepreneurial Business and Economics Review, 5:9–29, 2017.

[7] Valliere, D., International Entrepreneurship and Management Journal 6:97–112, 2010.

[8] Costa e Silva, E., Correia, A., Braga, A. and Braga, V., WEAS Transactions onBusiness and Economics, 15, 2018.

46


SOME CHARACTERIZATIONS FOR SPLITQUATERNIONS

Emel KARACA1, Fatih YILMAZ2, Mustafa ÇALISKAN3

1,2Department of Mathematics, Ankara Hacı Bayram Veli University, Ankara, Turkey3Department of Mathematics, Gazi University, Ankara, Turkey


ABSTRACT

Quaternions, introduced by W. Hamilton, plays a significant role in many areas ofscience; such as differential geometry, theory of relativity, engineering, etc. Al-though they have been studied by many researchers in recent years, they yield in-teresting results every time. In this study, some properties of split quaternions withquaternion coefficients are investigated. Moreover, some different properties be-tween split quaternions with quaternion coefficients and quaternions with complexcoefficients are compared.

Keywords Quaternions · Split quaternions · Quaternions with quaternion coeffi-cients

References

[1] Alagöz Y. and Özyurt G., Some properties of complex quaternion and complexsplit quaternion, Miskolc Mathematical Notes, (20): 45-58, 2019.

[2] Jafari M. and Yaylı Y., Generalized quaternions and their algebraic properties,Commun. Fac. Sci. Univ. Ank. Series A1, (64): 15-27, 2015.

[3] Erdogdu M. and Özdemir M., On complex split quaternion matrices, Adv. Appl.Clifford Algebr., (23): 625-638, 2013.

[4] Alagöz Y., Oral K. H. and Yüce S., Split quaternion matrices, Miskolc Mathe-matical Notes, (13): 223-232, 2012.

47


A NEW APPROACH FOR NATURAL LIFT CURVES ANDTANGENT BUNDLE OF UNIT 2-SPHERE

Emel Karaca 1, Mustafa Çalıskan 2

1Ankara Hacı Bayram Veli University, Department of Mathematics, Ankara, Turkey2 Gazi University, Department of Mathematics, Ankara, Turkey


ABSTRACT

In this study, some characterizations of Bertrand, involute-evolute curve couples fornatural lift curves were considered. Obtained results were restricted to the subsetof tangent bundle of unit 2-sphere. Moreover, conditions for these curves to beBertrand and involute-evolute curve couples were given. Finally, some exampleswere represented to support the main results.

Keywords Natural lift curve · tangent bundle · Involute-evolute curve couple ·Bertrand curve

References

[1] Ergün, E. and Çalı¸skan, M., On natural lift of a curve, Pure Mathematical Sci-ences (2) (2012), 81-85.

[2] Karaca, E. and Çalı skan, M., Dual spherical curves of natural lift curve andtangent bundle of unit 2-sphere, Journal of Science and Arts (3) (2019), 561-574.

[3] Hathout, F., Bekar, M. and Yaylı, Y., Ruled surfaces and tangent bundle of unit2-sphere, International Journal of Geometric Methods in Modern Physics (2)(2017).

[4] Karaka¸s, B. and Gündogan, H., A relation among DS2, TS2 and non-cyclindirical ruled surfaces, Mathematical Communications (8) (2003), 9-14.

[5] Ergün, E. and Çalı skan, M., The Frenet vector fields and the curvatures ofthe natural lift curve, The Buletin of Society for Mathematical Services andStandarts (2) (2012), 38-43.

[6] Ekmekci, N. and Ilarslan, K., On Bertrand curves and their characterization,Differ. Geo. Dyn. Syst. (3) (2001), 17-24.

48


COMPARISON THE SOLUTIONS BETWEEN VECTORAND SET APPROACHES FOR SET-VALUED

OPTIMIZATION PROBLEMS

Emrah KARAMAN

Karabük University, Faculty of Science, Department of Mathematics,78050, Karabük, Turkey


ABSTRACT

We encounter the optimization problems (mathematical programming) in our life.When these problems are expressed mathematically, an objective function emerges.Naturally, solution methods of these problems have attracted the attention to re-searchers working in the field such as mathematics, engineering, computer science,economics, business, etc. for years. Optimization problems are classified accordingto the objective function. For example, the problem is called a set-valued optimiza-tion problem (shortly, set optimization problem) when the objective function is aset-valued map. Set optimization problems are a generalization of the scalar, vec-tor, and interval optimization problems because the set-valued maps are generaliza-tions of the real-valued, vector-valued, and interval-valued functions, respectively.There are some types of solution concepts in the set-valued optimization problemsas vector approach and set optimization approach. For further information aboutthese approaches, one can see in [6, 1, 2, 3, 4, 5]. In the vector approach, the solu-tions that give efficient (minimal or maximal) points of image set of the objectiveset-valued map are looked at. Set optimization was introduced by Kuroiwa [6].This approach depends on comparisons among values of the set-valued map andthis concept requires order relations to compare sets. Kuroiwa et al. [9] introducedsix order relations for sets. Later, Jahn and Ha [8] defined new order relations andgave some properties of them for set optimization problems.In this study, solutions are compared between vector and set approaches for set-valued optimization problems, where solutions are obtained by lower set less, upperset less, and set fewer order relations for set approach. Because these order relationsare pre-order relations, we use the definition of the minimal and maximal elementsfor given pre-order relations. Also, according to these approaches, the relationshipsbetween the solutions of a set optimization problem are shown in the examples.Moreover, it has been shown on the examples that a set optimization problem, which

49


doesn’t have any solution according to the vector approach, has some solutionsaccording to the set approach, and vice versa.

Keywords Vector optimization · Set optimization · Set-valued optimization ·Pre-order relation

References

[1] Karaman E. , Atasever Güvenç I., Soyertem M., Tozkan D., Küçük M. andKüçük, Y., A vectorization for nonconvex set-valued optimization, Turkish. J.Math., 42(4): 1815-1832, 2018.

[2] Karaman E., Soyertem M., Atasever Güvenç I., Tozkan D., Küçük M. andKüçük Y., Partial order relations on family of sets and scalarizations for setoptimization, Positivity, 22: 783-802, 2018.

[3] Karaman E., Soyertem M. and Atasever Güvenç I., Optimality conditions inset-valued optimization problem with respect to a partial order relation via di-rectional derivative, Taiwan. J. Math., 24(3): 709-722, 2020.

[4] Karaman E., Gömme fonksiyonu kullanılarak küme optimizasyonuna göre ver-ilen küme degerli optimizasyon problemlerinin optimallik kosulları, SüleymanDemirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi, 14: 105-111, 2019.

[5] Karaman E., Atasever Güvenç I. and Soyertem M., Optimality conditions in set-valued optimization problems with respect to a partial order relation by usingsubdifferentials, Optimization, 2020, doi:10.1080/02331934.2020.1728270.

[6] Kuroiwa D., The natural criteria in set-valued optimization, RIMS Kokyuroku,1031: 85-90, 1998.

[7] Khan A.A., Tammer C. and Zalinescu C., Set-Valued Optimization: An Intro-duction with Applications, Berlin, Springer-Verlag, 2015.

[8] Jahn J., Ha T.X.D., New order relations in set optimization, J. Optimiz. Theory.App., 148: 209-236, 2011.

[9] Kuroiwa D., Tanaka T. and Ha T.X.D., On cone convexity of set-valued maps,Nonlinear. Anal-Theor., 30(3): 1487-1496, 1997.

50


GENERALIZED INTERVAL-VALUED OPTIMIZATIONPROBLEMS

Emrah KARAMAN

Karabük University, Faculty of Science, Department of Mathematics,78050, Karabük, Turkey


ABSTRACT

We encounter the optimization problems (mathematical programming) in our life.When these problems are expressed mathematically, an objective function emerges.Naturally, solution methods of these problems have been attracted the attention toresearchers working in the field such as mathematics, engineering, computer sci-ence, economics, business, etc. for years.Optimization problems are classified according to the objective function. For exam-ple, we say that the problem is the interval-valued optimization problem (shortly,interval optimization problem) when the objective function is interval-valued. Inter-val optimization problems are a generalization of the scalar optimization problemsbecause the interval-valued function is a generalization of the real-valued functions.Also, the interval optimization problems are a special form of the set-valued opti-mization problems. Order relations defined on intervals are used to solve intervaloptimization problems. In the literature, these order relations are obtained by usingnatural order relation on the real numbers [2, 2, 2, 4, 7, 5, 6].In any vector space, there is no natural order relation such as in real numbers. So,an order relation is defined with the help of a subset of the space in order to comparethe vectors. This set is called ordering cone. This order relation is either preorderrelation, partial order relation or total order relation according to ordering cone.So, all definitions and results used in vector optimization problems depend on theordering cone. Also, all order relations, which are defined on vector space, inducean ordering cone. Thus, each order relation in the vector space corresponds to anordering cone and vice versa. Naturally, minimal and maximal element definitionscan also change when ordering cone or order relation change. More informationabout the vector optimization can be found in [8].The purpose of this study is to get a general version of interval optimization prob-lems on any real vector space. Thus, intervals are defined by using the orderingcone. So, generalized interval optimization problems will be constructed. Also,

51


two order relations are introduced to obtain the solution of the interval optimizationproblems.

Keywords Interval-valued optimization · Vector optimization

References

[1] Ishibuch H. and Tanaka H., Multiobjective programming in optimization of theinterval objective function, Eur. J. of Oper. Res., 48: 219-225, 1990.

[2] Costa T.M., Chalco-Cano Y., Lodwick W.A. and Silva G.N., Generalized inter-val vector spaces and interval optimization, Information Sciences, 311: 74-85,2015.

[3] Moore R., Interval Analysis, Englewood Cliffs, New Jerseym: Prentice-Hall,1966.

[4] Bhurjee A. and Panda G., Efficient solution of interval optimization problem,Math. Method. of Oper. Res., 76: 273-288, 2012.

[5] Chalco-Cano Y, Lodwick WA, Rufian-Lizan A. Optimality conditions of typeKKT for optimization problem with interval-valued objective function via gen-eralized derivative, Fuzzy. Optim. Decis. Ma., 2013; 12: 305-322.

[6] Jiang C, Han X, Liu GR, Liu GP. A nonlinear interval number programmingmethod for uncertain optimization problems, Eur. J. of Oper. Res., 188 (2008)1-13.

[7] Bhurjee A.K. and Pandhan S.K., Optimality conditions and duality results fornon-differentiable interval optimization problems, Journal of Applied Mathe-matics and Computing, 50: 59-71, 2016.

[8] Jahn J., Vector Optimization, Heidelberg, Springer, 2004.

52


ON NEW NARAYANA POLYNOMIALS

Engin ÖZKAN 1, Bahar KULOGLU 2

1Department of Mathematics, Erzincan Binali Yıldırım University, Faculty of Arts and Sciences, Erzincan, Turkey2 Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan, Turkey


ABSTRACT

Some number sequences have a great importance in mathematics and they havemany great applications such as calculus, applied mathematics and so on. It isnot well known whether Narayana numbers have similar applications. But if weestablish a close relationship between the studies and the features of the Fibonaccisequences, we can confidently talk about the application area in many fields, justlike Fibonacci sequence.

The Narayana numbers come up as a result of solving the problem of the Narayana’scow which was given by the Indian mathematician Narayana. This problem is that:A cow produces one calf every year. Beginning in its fourth year, each calf producesone calf at the beginning of each year. How many calves are there altogether after20 years? [1].

Let n show the year. So, the Narayana problem can be given by the recurrencerelation:

Nn+1 = Nn +Nn−2 , n ≥ 2

where N0 = 0, N1 = 1and N2 = 1[1].

The first few terms of the Narayana numbers are 0,1,1,1,2,3,4,6,9,13,. . . . This se-quence is called Narayana sequence [1]. It is well known that the Narayana numbersare given by Binet’s formula

Nn =αn+1

(α− β)(α− γ)+

βn+1

(β − α)(β − γ)+

γn+1

(γ − α)(γ − β)

where

α =1

3

1 + 3

√2

29 + 3√

93+

3

√29 + 3

√93

2

53


β =1

3

1− w 3

√2

29 + 3√

93+ w2 3

√29 + 3

√93

2

and

γ =1

3

1 + w2 3

√2

29 + 3√

93+ w

3

√29 + 3

√93

2

where w = 1+i

√3

2is the primitive cube root of unity [1].

The Fibonacci polynomials were studied in 1883 by E. C. Catalan. The Fibonaccipolynomials Fn(x) are defined by the recurrence relation:

Fn (x) = xFn+1 (x) + Fn (x) , n ≥ 1

with the initial condition F1 (x) = 1 and F2 (x) = x.When x = 1, we get the Fibonacci numbers [7].We can see some important properties of the Narayana sequence in [2, 10, 1]. ThePell polynomials are defined by

[Pn (x) = 2xPn−1 (x) + Pn−2 (x) , n ≥ 2]

where P0 (x) = 0 and P1 (x) = 1 [7, 9]. They are related to the Fibonacci polyno-mials by Pn (x) = Fn(2x).In this study we gave a new definiition of Narayana. We showed that there is a rela-tionship between the coefficient of the Narayana polynomials and Pascal’s triangleand Pell numbers. Afterwards we also gave some definitions, properties and theo-rems related to the numbers and the polynomials. We define the Hankel Transformof them.

Keywords Narayana sequence · Narayana polynomials · Pascal triangle.

References

[1] Allouche J. P. and J. Johnson, Narayana’s cows and delayed morphism. In: Ar-ticles of 3rd Computer Music Conference JIM96, France. (1996).

[2] Didkivska T. V., Stopochkina M. V., Properties of Fibonacci-Narayana num-bers, In the World of Mathematics, 9 (2003), 29–36. [inUkrainian]

[3] Falcon, S. Plaza, A. The k- Fibonacci sequence and Pascal 2-triangle 33 (2007)38-49

[4] Halici S., On the Pell polynomials. Applied Mathematical Sciences, vol, 5, 37(2011) 1833-1838.

[5] Hoggatt V.E., Lndand JR. and D. A, Symbolic substitutions into Fibonacci poly-nomials. . (1968).

[6] Koshy, T. (2001) Finonacci and Lucas Numbers with Applications, John WileySons, Inc., Canada

[7] Marin M., Generalized solutions in elasticity of micropolor bodies with voids.Revista de la Academia Canaria de Ciencias, vol. 8 (1996) 01-106.

[8] Özkan, E. Tastan, M. On Gauss Fibonacci polynomials, on Gauss Lucas poly-nomials and their applications 10.1080/00927872.2019.1670193.

54


[9] Pell polynomials. http://mathworld.wolfram.com/PellPolynomial.html.2020(accessed 10 February 2020).

[10] Ramirez, Jose L. Sirvent, Victor F. (2015) A note on the k-Narayana sequence,45, pp. 91-105.

55


THE k-NARAYANA SEQUENCE

Engin ÖZKAN 1, Bahar KULOGLU 2

1Department of Mathematics, Erzincan Binali Yıldırım University, Faculty of Arts and Sciences, Erzincan, Turkey2Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan, Turkey


ABSTRACT

Narayana was an outstanding Indian mathematician of the XIV century. From himcame to us the manuscript ”Bidzhahanity” (incomplete), written in the middle ofthe XIV century. For Narayana was interesting summation of arithmetic series andmagic squares. [2]

The Narayana numbers come up as a result of solving the problem of the Narayana’scow which was given by the Indian mathematician Narayana. This problem is that:A cow produces one calf every year. Beginning in its fourth year, each calf producesone calf at the beginning of each year. How many calves are there altogether after20 years? [1].Let n show the year. So, the Narayana problem can be given by the recurrencerelation [9]:

Nn+1 = Nn +Nn−2 , n ≥ 2

where N0 = 0, N1 = 1and N2 = 1

The first few terms of the Narayana numbers are 0,1,1,1,2,3,4,6,9,13,. . . . This se-quence is called Narayana sequence [8].The generalized Narayana sequence especially, for any nonzero integer k, the k-Narayana sequence is defined in [8] by the recurrence relation:

Nk,n = kNk,n−1 +Nk,n−3

with the initial values Nk,0 = 0, Nk,1 = 1 and Nk,2 = k.The first few terms are:

0, 1, k, k2, k3 + 1, k4 + 2k, k5 + 3k2, . . . .

It is well known that the k− Narayana numbers are given by Binet’s formula

Nk,n =αn+1k

(αk − βk)(αk − γk)+

βn+1k

(βk − αk)(βk − γk)+

γn+1k

(γk − αk)(γk − βk)

56


where

αk =1

3

k + k2 3

√2

27 + 2k3 + 3√

81 + 12k3+

3

√27 + 2k3 + 3

√81 + 12k3

2

βk =

1

3

k − wk2 3

√2

27 + 2k3 + 3√

81 + 12k3+ w2 3

√27 + 2k3 + 3

√81 + 12k3

2

and

γk =1

3

k + w2k2 3

√2

27 + 2k3 + 3√

81 + 12k3+ w

3

√27 + 2k3 + 3

√81 + 12k3

2

.

w = 1+i√3

2is the primitive cube root of unity.[8]

Characteristic equation of the k-Narayana numbers is that

x3 − kx2 − 1 = 0

For any integer number k, (k 6= 0) , we have the following matrixes

Qk =

k 0 11 0 00 1 0

and for n ≥ 3,

(Qk)n =

Nk,n+1 Nk,n−1 Nk,n

Nk,n Nk,n−2 Nk,n−1Nk,n−1 Nk,n−3 Nk,n−2

Then Qk is a generating matrix of the k-Narayana sequence.[8]The general k-Fibonacci sequence generalizes, between others, both the classicalFibonacci sequence and the Pell sequence.We give some properties and theorems related to the k- Narayana numbers. Weprove the Cassini identity and a limit of the k-Narayana sequence. We calculate thedeterminant of the k-Narayana numbers and show that their determinant are fixedterms. Further, we give sum of the first terms of the k-Narayana sequence and wededuce many properties of the k-Narayana sequence.

Keywords k-Narayana sequence · Catalan Transform · Hankel Transform

References

[1] Allouche J. P. and J. Johnson, Narayana’s cows and delayed morphism. In: Ar-ticles of 3rd Computer Music Conference JIM96, France. (1996).

[2] Didkivska T. V., Stopochkina M. V., Properties of Fibonacci-Narayana num-bers, In the World of Mathematics, 9(2003), 29–36. [inUkrainian]

[3] Hoggatt V.E., Lındand JR. and D. A, Symbolic substitutions into Fibonaccipolynomials. . (1968).

57


[4] Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley & Sons,Inc., Canada. (2001).

[5] Özkan E., 3-Step Fibonacci Sequences in Nilpotent Groups, Applied Mathe-matics and Computation, 144, (2003) 517-527

[6] Özkan E., H. Aydın and Dikici R., 3-Step Fibonacci Series Modulo m, AppliedMathematics and Computation, 143, (2003) 165-172. DOI: 10.1016/S0096-300300360-0

[7] Özkan E. and , Altun I. Generalized Lucas polynomials and relationships be-tween the Fibonacci polynomials and Lucas polynomials, Communications inAlgebra, 47, (2019) 4020-4030. DOI: 10.1080/00927872.2019.1576186.

[8] Ramirez L. Jose and Sirvent Victor F., A note on the k-Narayana sequence.Annales Mathematicae et Informaticae. 45.(2015) pp. 91-105.

[9] A. Stakhov, B. Rozin, Theory of Binet formulas for Fibonacci and Lu-cas p-numbers. Chaos, Solitons and Fractals 27.(2006),1162-1177. Doi:10.1016/j.chaos.2005.04.106.

58


SUBORBITAL GRAPHS OF Γ0,n(N) IN Q(N)

Erdal ÜNLÜYOL1, Aziz BÜYÜKKARAGÖZ1

1Ordu University, Faculty of Arts and Sciences, Department of Mathematics, Ordu, TURKEY


ABSTRACT

In this paper, it is studied at the group of Γ0,n(N). It is generated by the groupΓ0,n(N) with respect to reflection transformation R(z) = −z. Firstly, it is obtainthat the group does not contain second and third degree elliptic element under thesome conditions. Secondly, it is defined the set of Q(N). Thirdly, it is establishedan equivalence relation for the group of Γ0,n(N) in Q(N). Then, it is researchedthe edge conditions in the graph G∗u,N , which it is obtained by the equivalencerelation. Fourth, it is proved that the graph F ∗u,N does not contain a triangle. F ∗u,Nis a subgraph which it is edges of graph G∗u,N in the block [∞] =

[10

]=[−

10

]. Finally, it is expressed and proved that the graph F ∗u,N is a forest under some

conditions.

Keywords Suborbital graph · Elliptic element · Edges of graph · Forest

References

[1] Akbas M., The normalizer of modular subgroup, Ph. D. Thesis, Faculty ofMathematical Studies, University of Southompton, Southompton, England,1989.

[2] Akbas M., On suborbital graphs for the modular group, Bull. London Math.Soc., 33: 647-652, 2001.

[3] Büyükkaragöz A., Signatures and graph connections of some subgroups of ex-tended modular group, Ph. D. Thesis, Ordu University, Science Institute, Ordu,Turkey, 2019.

[4] Jones G. A., Singerman D., and Wicks, K., The modular group and generalizedFarey graphs, London Math. Soc. Lecture Notes Series 160, CUP, Cambridge1991.

[5] Sims C. C., Graphs and finite permutation groups, Math. Zeitschr., 95: 76-86,1967.

59


[6] Sanlı Z., The normalizer of Γ2,Γ3 and Hecke groups and graphs, K.T.U. Sci.Inst., Ph. D. Thesis, Trabzon, Turkey 2019.

60


A BI-LEVEL MATHEMATICAL MODEL TO ANALYZE THE RISK ATA HAZARDOUS MATERIAL TRANSPORTATION NETWORK

Fatih KASIMOGLU1, Özkan BALI2

1University of Turkish Aeronautical Association, Bahçekapı Mah. Okul Sk. No:11, 06790 Etimesgut,Ankara,Turkey2Turkish Military Academy Defense Sciences Institute, 06654, Bakanlıklar, Ankara, Turkey


ABSTRACT

Transportation of a hazardous material such as explosives and ammunition can bea target for an enemy attack. While a competent authority for hazardous materialtransportation (HMT) aims to minimize the total risk to public or environment,an attacker might attempt to maximize it by destroying some specific segments(arcs) on the transportation network. The problem can be designed as a bi-levelnetwork interdiction model [1,2,3]. Generally speaking, there are two players ina network interdiction problem. Network owner, who naturally tries to run thenetwork optimally, is the defending side (defender). The opponent, trying to impedethe network process on the other side, is the attacking one (interdictor/attacker).With limited resources at hand, the attacker inflicts some negative effects on thenetwork arcs so that the objective of the network owner is maximally deteriorated.In this study, we first develop a bi-level max-min HMT attack model (Bi-HMTA)considering that the network is liable to one or more enemy attacks depending onthe attack resources available. Then to solve the problem, we convert the model Bi-HMTA into an ordinary mixed integer programming (MIP) model (HMTA) usingduality theorem, integer and combinatorial optimization techniques [4,5,6]. Finally,we test the resulting model HMTA using different size networks with various attackresources. The results reveal out the arcs that are most liable to an enemy attackas well as the ensuing increase in the total risk of transportation. In our specificapplication, we observe that roads between train unload stations and demand pointsare most likely targets for an attack. The total risk substantially increases dependingon the available attack resources. The decision makers can get use of such resultsin analyzing the risk of transportation through a specific path and develop theirown strategies accordingly to decrease the risk they encounter in transporting thehazardous materials.

61


Keywords Bi-Level Modeling · Integer Programming · Network Modeling ·Network Interdiction · Hazardous Materials Transportation · Risk Analysis

References

[1] Wen U.-P., Hsu S.-T. (1991). Linear bi-level programming problems—A re-view. Journal of the Operational Research Society, 42(2), 125–133, 1991.

[2] Kalashnikov, V.V., Dempe, S., Pérez-Valdés, G.A., Kalashnykova, N.I,Camacho-Vallejo, J.F. Bilevel Programming and Applications. MathematicalProblems in Engineering. Vol. 2015, Article ID 310301, 16 Pages, 2015.

[3] Smith, J. C., Lim, C., Sudargho, F. Survivable network design under optimaland heuristic interdiction scenarios. Journal of global optimization, 38(2), 181-199, 2007.

[4] Ahuja, R. K., Magnanti, T. L., Orlin, J. B. Network flows: theory, algorithms,and applications. New Jersey : Prentice Hall, 1993.

[5] Bazaraa, M. S., Jarvis, J. J., Sherali, H. D. Linear programming and networkflows. John Wiley and Sons, 2005.

[6] Nemhauser G.L. Wolsey L.A. (1988). Integer and Combinatorial Optimization.New York: John Wiley and Sons, 1988.

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THE INTRINSIC METRIC ON THE SCALE IRREGULARSIERPINSKI GASKET SG(2, 3)

Fatma Digdem KOPARAL1, Yunus ÖZDEMIR1

1Eskisehir Technical University, Department of Mathematics, 26470, Eskisehir, Turkey


This work is supported by the Eskisehir Technical University Research Fund Under Contract 19ADP113.

ABSTRACT

The Sierpinski Gasket is the well-known self-similar set in fractal geometry. Thereexist several generalizations of this set in the literature such as regular SierpinskiGaskets, mod-p Sierpinski Gaskets (or Pascal-Sierpinski Gaskets), the Sierpinskipedal triangles and etc. A scale irregular Sierpinski Gasket formed by regular Sier-pinski Gaskets also can be thought as an another generalization (see [1] for moredetails).On the other hand, expressing the intrinsic metric and determining the geodesicsof a self-similar set such as the (classical) Sierpinski Gasket (SG(2)), the Vicsekfractal, the Sierpinski carpet etc. has been studied in the last decades by usingdifferent tools and techniques (see [2] for example). In [3], the intrinsic metric ofthe (classical) Sierpinski Gasket (SG(2)) is expressed via the code representationsof the points, and the authors show that there exist at most five geodesics betweentwo different points. In [4], for the regular Sierpinski Gasket SG(3), the authorgives an expression of the intrinsic metric via code representation of the points andshow that there can be infinitely many geodesics between two different points.In this work, we express the intrinsic metric formula of the scale irregular SierpinskiGasket SG(2, 3) formed by SG(2) and SG(3). To express the intrinsic metric ofSG(2, 3), we use the code representations of the points (as done in the cases SG(2)and SG(3)). Using the previous results, we also investigate the geodesics in thisirregular case.

Keywords Sierpinski Gasket · Scale irregular Sierpinski Gasket · Intrinsic metric ·Geodesic

References

[1] Hambly B.M., Brownian motion on a random recursive Sierpinski gasket, Ann.Probab. 25 : 1059–1102, 1997.

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[2] Strichartz R.S., Isoperimetric estimates on Sierpinski gasket type fractals,Trans. Amer. Math. Soc., 351: 1705-1752, 1999.

[3] Saltan M., Özdemir Y., Demir B., Geodesics of the Sierpinski gasket, Fractals,26 (3): 1850024, 2018.

[4] Özdemir Y., The intrinsic metric and geodesics on the Sierpinski gasket SG(3),Turk. J. Math. 43: 2741-2754, 2019.

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NON-ROTATING FRAMES AND ITS APPLICATIONS

Fatma KARAKUS

Sinop University, Faculty of Sciences and Arts, Department of Mathematics, Sinop, Turkey


ABSTRACT

In this paper, I will explain Fermi-Walker derivative and non-rotating frame con-cepts for some frames in Euclidean space, Minkowski space, Dual Space, DualLorentzian space. All non-rotating frame conditions will be analyzed for Frenetframe, Bishop frame and Darboux frame in Euclidean space, Minkowski space,Dual space, Dual Lorentzian space. Then, I will explain non-rotating frame no-tions for three-dimensional Lie group with left-invariant metric. I will generalizethe results known for the case of three-dimensional Lie group with the bi-invariantmetric. I will define Bishop frame (relatively parallel adapted frame accordancewith dot-Frenet frame). After that, I will explain that the Bishop frame and Frenetframe whether they are a non-rotating frame or not for three-dimensional Lie groupwith left-invariant metric. I will give some examples for the non-rotating framesapplications.

Keywords Fermi-Walker Derivative, Non-Rotating Frame, Euclidean space, DualSpace, Minkowski Space, 3-Dimensional Lie Group

Acknowledgements. This research has been supported by the Scientific and Tech-nological Research Council of Turkey (TUBITAK).

References

[1] Milnor, J., Curvatures of Left Invariant Metrics on Lie Groups, Advances inMathematics 21: 293-329, 1976.

[2] Pripoae, G.T.: Generalized Fermi-Walker Transport, Libertas Math., XIX.: 65-69, 1999.

[3] Pripoae, G.T.: Generalized Fermi-Walker Parallelism Induced by GeneralizedSchouten Connections, Geometry Balkan Press, 117-125, 2000.

[4] Hawking, S.W. and Ellis, G.F.R.: The large scale structure of spacetime, Cam-bridge Univ. Press, 1973.

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[5] Karakus, F. and Yaylı, Y.: On the Fermi-Walker derivative and Non-rotatingframe, Int. Journal of Geometric Methods in Modern Physics, 9(8): 1250066-1250077, 2012.

[6] Karakus, F. and Yaylı, Y., The Fermi-Walker derivative in Lie Groups, Int. Jour-nal of Geometric Methods in Modern Physics, 10(7): 1320011-1320022, 2013.

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FERMI-WALKER PARALLELISM AND ITSAPPLICATIONS

Fatma KARAKUS

Sinop University, Faculty of Sciences and Arts, Department of Mathematics, Sinop, Turkey


ABSTRACT

In this paper, I will explain Fermi-Walker derivative and Fermi-Walker parallelismconcepts in Euclidean space, Minkowski space, Dual Space, Dual Lorentzian space.I will explain the necessary conditions to be Fermi-Walker parallelism along anycurve for any vector field in Euclidean space, Minkowski space, Dual space, DualLorentzian space. Also, I will explain Fermi-Walker parallelism concept for three-dimensional Lie group with left-invariant metric and I will generalize the resultsknown for the case of three-dimensional Lie group with the bi-invariant metric. Iwill give some examples for the Fermi-Walker parallel vector fields applications.

Keywords Fermi-Walker Derivative, Fermi-Walker Parallelism, Space Curves,Plane Curves, Line of Curvature, Geodesic Curves

Acknowledgements. This research has been supported by the Scientific and Tech-nological Research Council of Turkey (TUBITAK).

References

[1] Milnor, J., Curvatures of Left Invariant Metrics on Lie Groups, Advances inMathematics 21: 293-329, 1976).

[2] Pripoae, G.T.: Generalized Fermi-Walker Transport, Libertas Math., XIX.: 65-69, 1999.

[3] Pripoae, G.T.: Generalized Fermi-Walker Parallelism Induced by GeneralizedSchouten Connections, Geometry Balkan Press, 117-125, 2000.

[4] Hawking, S.W. and Ellis, G.F.R.: The large scale structure of spacetime, Cam-bridge Univ. Press, 1973.

[5] Karakus, F. and Yaylı, Y.: On the Fermi-Walker derivative and Non-rotatingframe, Int. Journal of Geometric Methods in Modern Physics, 9(8): 1250066-1250077, 2012.

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[6] Karakus, F. and Yaylı, Y., The Fermi-Walker derivative in Lie Groups, Int. Jour-nal of Geometric Methods in Modern Physics, 10(7): 1320011-1320022, 2013.

68


DYNAMICS OF A MODIFIED A DISCRETE-TIMEPREDATOR- PREY MODEL WITH ALLEE EFFECT

Figen KANGALGIL1, Seval ISIK2

1Dokuz Eylul University, Bergama Vocational High School, 35700, Izmir, Turkey2Sivas Cumhuriyet University, Faculty of Education, Department of Mathematics and Science Education, 58140, Sivas, Turkey


ABSTRACT

The dynamics of interaction between predators and preys have great interested inecology and mathematical biology. Differential and difference equations have beenused to investigate a wide range of population models. The populations modelsgoverned by differential equations studied extensively by many researchers [1,2]and the reference therein. However, in recent years considerable number articlesin literature discussed the dynamic of the discretization of predator prey modelsgoverned by differential equations [3,4]. Because the discrete-time models presentdynamics consistency according to continuous-time. In addition, the discrete-timepopulation models are suitable for non-overlapping generations and are appropriateto describe the nonlinear dynamics and possibly their chaotic behaviour.Bifurcation theory is research field that analyzes the change of dynamical modelswith respect to a control parameter. So, the behaviour the model according to acontrol parameter is observed. There are many studies on bifurcation analysis [5].The Allee effect, a reduction of the per capita growth rate of a population of bi-ological species at densities smaller than a critical value, was first introduced byAllee in 1931 [7]. Although there are many extensive researches on Allee effect,many articles have not addressed the Allee effect with focus on bifurcation analysisof discrete-time predator prey model [8]. Thus, in this work it is investigated thatbifurcation and stability analysis of a modified a discrete-time predator prey modelwhich developed from the inclusion of Allee effect in predator population.The objective of this work is to investigate a discrete-time predator- prey modelwith Allee effect in predator species. The Euler method is constructed to trans-form the continuous time predator prey model with Allee effect into the discrete-time model. The existence of coexistence fixed point of the considered model isstudied. The parametric conditions for local asymptotic stability of fixed point ofthe discrete-time model is presented. Moreover, it is proved that model undergoes

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Neimark-Sacker bifurcation for small range of parameters by using bifurcation the-ory. Finally, some numerical simulations have been carried out in order to supportof our analytical findings.

Keywords Stability · Bifurcation · Prey-Predator Model

References

[1] He, X. Z., Stability and delays in a predator-prey system. Journal of Mathemat-ical Analysis and Applications, 198(2), 355-370, 1996.

[2] Liu, X., & Xiao, D., Complex dynamic behaviors of a discrete-time predator–prey system. Chaos, Solitons & Fractals, 32(1), 80-94, 2007.

[3] Salman, S. M., Yousef, A. M., & Elsadany, A. A. (2016). Stability, bifurcationanalysis and chaos control of a discrete predator-prey system with square rootfunctional response. Chaos, Solitons & Fractals, 93, 20-31.

[4] Rana, Sarker Md Sohel,Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model, Journal of the Egyptian Mathematical Soci-ety 27.1 2019.

[5] Selvam, A. George Maria, S. Britto Jacob, and R. Dhineshbabu. "Bifurcationand Chaos Control for Prey Predator Model with Step Size in Discrete Time."Journal of Physics: Conference Series. Vol. 1543. 2020.

[6] Allee, W. C., Co-operation among animals. American Journal of Sociology,37(3), 386-398, 1931.

[7] Zhou, S. R., Liu, Y. F., & Wang, G., The stability of predator–prey systemssubject to the Allee effects. Theoretical Population Biology, 67(1), 23-31, 2005.

70


NUMERICAL METHODS FOR A NONLOCAL PERIDYNAMIC MODEL

Giuseppe Maria COCLITE1, Alessandro FANIZZI2, Luciano LOPEZ2, FrancescoMADDALENA1, Sabrina Francesca PELLEGRINO2

1Department of Mechanics, Mathematics and Management, Polytechnic of Bari, Via E. Orabona 4, 70125 Bari, Italy2Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy


ABSTRACT

We consider the peridynamic equation of motion, which is described by a secondorder in time partial integro-differential equation. This equation has recently re-ceived great attention in several fields of Engineering because seems to providean effective approach to modeling mechanical systems avoiding spatial discontinu-ous derivatives and body singularities. In particular, we consider the linear modelof peridynamics in a one-dimensional spatial domain. We review some numericaltechniques to solve this equation and propose some new computational methodsof higher order in space. We discuss the implementation of a spectral method forthe spatial discretization of the linear problem. Several numerical tests are given inorder to validate our results.

Keywords Peridynamics · Quadrature formula · Spectral methods · Trigonometrictime discretization

References

[1] Coclite G.M., Fanizzi A., Lopez L., Maddalena F., Pellegrino S.F., Numericalmethods for the nonlocal wave equation of the peridynamics, Applied Numeri-cal Mathematics, 155: 119-139, 2020.

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DOUBLE-DIFFUSION INSTABILITY: A FINGERING CONVECTIONMODEL FOR OCEANIC STAIRCASE FORMATION

Giuseppe Maria COCLITE1, Francesco PAPARELLA2, Sabrina Francesca PELLEGRINO3

1Department of Mechanics, Mathematics and Management, Polytechnic of Bari, Via E. Orabona 4, 70125 Bari, Italy2Departement of Mathematics, New York University in Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab

Emirates3Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy


ABSTRACT

Fingering convection is a mechanism of double-diffusion which occurs in fluidswhere density is controlled by two buoyancy-changing scalars, temperature andsalinity, that diffuse at different rates. This set-up allows the convective cells toassume the shape of tall finger-like structures called “salt fingers”.We consider a system of nonlinear degenerate parabolic equations describing thephenomenon in terms of horizontally averaged kinetic energy field and buoyancy.We prove the existence of weak solutions for such system and discuss their stabilityand asymptotic properties.

Keywords Salt fingers · Water waves · Degenerate parabolic equations · Weaksolutions

References

[1] Coclite G.M., Paparella F., Pellegrino S.F., On a salt fingers model, NonlinearAnalysis, 176: 100-116, 2018.

72


EIGENVALUES OF THE SELFADJOINT SCHRÖDINGER OPERATORON NON-COMPACT STAR GRAPH

Gökhan MUTLU

Gazi University, Faculty of Science, Department of Mathematics


ABSTRACT

The point spectrum of the non-compact quantum star graph Γ with n edges of infi-nite length has been investigated. We assume the Hamiltonian on this non-compactgraph acts as Schrödinger operator −f ′′ +Q(x)f on each edge where the potentialQ(x) is a real-valued function which is Lebesgue integrable on (0,∞) and∫∞

0(1 + x) |Q(x)| dx <∞, (1)

holds. We assume that the central vertex v possesses the most general selfadjointvertex conditions

AvF (v) +BvF′(v) = 0,

Rank (Av | Bv) = n,

AvB∗v −BvA

∗v = 0,

where Av and Bv are n× n matrices and "∗" denotes the matrix adjoint.

Quantum graphs have many applications in physics and engineering. Quantumgraphs enable us to model systems in mathematics, physics, chemistry, and en-gineering to analyze phenomena such as the free-electron theory of conjugatedmolecules, quantum wires, dynamical systems, photonic crystals, thin waveguides,and many other. As a result, there is a remarkable interest in this field nowa-days. Most of the studies concern the negative second derivative operator −f ′′i.e. Q(x) ≡ 0 acting on the edges of graphs [1, 2, 3]. The Schrödinger operator oncompact quantum graphs have been also considered [4, 5].

Obviously, the Hamiltonian on Γ is selfadjoint. Therefore, all eigenvalues arereal. We relate this scattering model on non-compact star graph with the matrixSchrödinger operator on the half-line

HA,B := −y′′ + V (x)y, x ∈ (0,∞) ,

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where the potential V (x) is an n×nmatrix-valued function such that V (x)∗ = V (x)for each x ∈ (0,∞) and ∫ ∞

0

(1 + x) ‖V (x)‖dx <∞,

holds where ‖, ‖ denotes any of the equivalent matrix norms. The boundary condi-tion at the origin is assumed to be the most general selfadjoint boundary conditions

Ay(0) +By′(0) = 0,

Rank (A | B) = n,

AB∗ −BA∗ = 0,

where A and B are n× n matrices.We observe that the Hamiltonian on non-compact star graph and the matrixSchrödinger operator HA,B on the half-line are isospectral. We find the resolvent ofHA,B and show that the eigenvalues of the non-compact star graph coincides withthe zeros of the Jost function in the upper half-plane. We show that the multiplicityof each eigenvalue is finite and this multiplicity is equal to the multiplicity of thecorresponding zero of the Jost function. Finally, we prove that the non-compact stargraph has a finite number of eigenvalues under the condition (1).

Keywords Non-compact star graph · Eigenvalues ·Matrix Schrödinger operator

References

[1] Berkolaiko, G., An elementary introduction to quantum graphs,arXiv:1603.07356 [math-ph], 2016.

[2] Berkolaiko G. and Kuchment P., Introduction to Quantum Graphs (Mathemati-cal Surveys and Monographs vol 186). American Mathematical Society, RhodeIsland, 2013.

[3] Kuchment, P., Quantum graphs: an introduction and a brief survey,arXiv:0802.3442 [math-ph], 2008.

[4] Kurasov P., Schrödinger operators on graphs and geometry I: Essentiallybounded potentials, Journal of Functional Analysis, 254: 934–953, 2008.

[5] Kurasov P. and Suhr R., Schrödinger operators on graphs and geometry. III.General vertex conditions and counterexamples, J. Math. Phys., 59: 102104,2018.

74


EVOLUTE OFFSET OF NON-CYLINDIRICAL RULED SURFACESWITH B-DARBOUX FRAME

Gül Ugur KAYMANLI

Department of Mathematics, Cankiri Karatekin University, Cankiri, 18100, Turkey


ABSTRACT

A ruled surface in three dimensional space is a surface which can be described as theset of points swept by a moving straight line. By using the directions, many offset ofthis surface (or curve) such as Bertrand, involute-evolute, Mannheim and Smaran-dache have been defined. Both offsets surfaces and ruled surfaces are widely usedin kinematics, mechanism, Computer-Aided Geometric Design (CAGD) and geo-metric modelling. Offsets of ruled surfaces are generally more complex than theirprogenitor ruled surface. Because of this offsets surfaces (curves) are examined interms of the properties of the base surface (curve) is important. In this work, we in-troduce the evolute offset of non-cylindirical ruled surfaces with B-Darboux frame.Some geometric properties of this evolute offset of non-cylindirical ruled surfaceare studied. That is, we examine the striction curve, distribution parameter and thedevelopability of evolute offset of given ruled surfaces in terms of B-Darboux framein three dimensional Euclidean space.

Keywords B-Darboux Frame · Curvatures · Evolute Offset · Ruled Surface

References

[1] Bektas O. and Yuce S., Special involute-evolute partner D-curves in E3, Euro-pean Journal of Pure and Applied Mathematics, 6(1): 20-29, 2013.

[2] Dede M., Ekici C. and Gorgulu A., Directional q-frame along a space curve.IJARCSSE, 5(12): 775–780, 2015.

[3] Kasap E., Yuce S. and Kuroglu N., The involute-Evolute Offsets of Ruled Sur-face, Iranian Journal of Science and Technology, Transaction A, 33(2): 195-201, 2009.

[4] Kaymanli G.U., Ekici C. and Dede M., Bertrand Offsets of Ruled Surfaces withB-Darboux Frame, submitted.

[5] O’Neill B., Elementary differential geometry. Academic Press Inc, New York,1996.

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[6] Senturk G. and Yuce S., On The Evolute Offsets of Ruled Surface Using theDarboux Frame, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2):1256-1264, 2019.

[7] Senturk G.Y. and Yuce S., Characteristic properties of the ruled surface withDarboux frame in E3, Kuwait J. Sci., 42(2): 14–33, 2015.

[8] Yoon D.W. On the evolute offsets of ruled surfaces in Minkowski 3-space, Turk-ish Journal of Mathematics, 40: 594-604, 2016.

76


THE RESTRICTIVE SETS AND THEIR APPLICATIONSTO SOFT GROUP THEORY

Hakan AYKUT1, Akın Osman ATAGÜN2

1,2Kırsehir Ahi Evran University, Turkey


ABSTRACT

Molodtsov [12] introduced the soft set theory to give a new approach to solve prob-lems including uncertainties. He gave many applications of this theory in manyareas such as information systems, probability and measurement theory. Applica-tions of soft set theory are progressing rapidly also in the areas decision making, op-timization theory, algebraic structures. This study includes a novel algebraic appli-cation of soft set theory using a nonempty subset of initial universe set. We call thisset " generator set". In this paper, firstly four different restrictions, α − including,α− covered, α− intersection and α− union sets of a soft set over an initial uni-verse U obtained by using a non-empty subset α of U are examined, in detail. Weinvestigate some useful novel properties of these four restrictive sets according tosoft set operations throughout. Firstly, we give some properties of the restrictive setsaccording to soft subset ⊆. Then, some properties of the restrictive sets accordingto the operation soft union "∪", soft restricted union "∪R", soft extended intersec-tion "uε", AND operation "∧", OR operation "∨" and restricted difference "∼R" aregiven. Finally, some properties of soft sets that are restricted by α − intersectionand α − union are investigated. We also give examples of the properties of re-strictive sets. Then an application of α − intersection sets over a soft intersectiongroups is given. Here, the novel concept of the soft int-subgroup generated byα ⊆ U of a group is introduced and some properties are given. Then we obtained arelationship between a subgroup of a group and a soft subgroup of the same group.Also we examined count of generators of a soft intersection groups. Furthermorewe investigated connections of generated sets of same group. The generators of softintersections and products of soft int-subgroups are given. Some related propertiesabout generators of soft int-subgroups are investigated and illustrated by severalexamples.

Keywords Soft set · Soft int-group · Restrictive set · Generator of a group

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References

[1] Acar U., Koyuncu F. and Tanay B., Soft sets and soft rings, Computers andMathematics with Applications, 59: 3458-3463, 2010.

[2] Aktas H. and Çagman N., Soft sets and soft groups, Information Sciences, 177:2726-2735, 2007.

[3] Ali M.I., Feng F., Liu X., Min W.K., Shabir M., On some new operations in softset theory, Computers and Mathematics with Applications, 57(9): 1547-1553,2009.

[4] Atagün A.O. and Kamacı H., Decomposition of soft sets and soft matrices withapplications in group decision making, submitted.

[5] Atagün A. O., Set-generated soft subrings of a ring, submitted.[6] Atagün A.O. and Sezgin Sezer A., Soft substructures of rings fields and mod-

ules, Computers and Mathematics with Applications, 61: 592–601, 2011.[7] Çagman N. and Enginoglu S., Soft set theory and uni-int decision making, Eu-

ropean Journal of Operational Research, 207: 848-855, 2010.[8] Çagman N., Çıtak F. and Aktas H., Soft int-group and its applications to group

theory, Neural Computing and Applications, 21: (Issue 1-Supplement), 151-158, 2012.

[9] Feng F., Jun Y.B. and Zhao X., Soft semirings, Computers and Mathematicswith Applications, 56: 2621-2628, 2008.

[10] Maji P.K., Biswas R. and Roy A.R., Soft set theory, Computers and Mathe-matics with Applications, 45: 555-562, 2003.

[11] Maji P.K., Biswas R. and Roy, A.R., An application of soft sets in a decisionmaking problem, Computers and Mathematics with Applications, 44: 1077-1083, 2002.

[12] Molodtsov D., Soft set theory-first results, Computers and Mathematics withApplications, 37: 19-31, 1999.

[13] Pei D., Miao D., From soft sets to information systems, in:Hu, X. , Liu, Q.,Skowron, A. ; Lin, T.L., Yager, R.R., Zhang, B. (Eds:), Proceedings of GranularComputing, IEEE(2): 617-621, 2005.

[14] Sezgin A., Çagman, N. and Çıtak, F. α-inclusions applied to group theoryvia soft set and logic, Communications Series A1: Mathematics And Statistics,68(1): 334-352, 2019.

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MATHEMATICAL MODEL OF EXTINCTION OFRastrelliger brachysoma, HARVESTING AND CONTROL

Inthira CHAIYA1, Kamonchat TRACHOO1, Din PRATHUMWAN2

1Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand2Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand


ABSTRACT

In this work, we proposed a mathematical model of the population density of Indo-Pacific mackerel and the population density of small fishes based on impulsive fish-ery. The developed impulsive mathematical model was analyzed theoretically in or-der to obtain the conditions on the parameters of the model for which Indo-Pacificmackerel (Rastrelliger brachysoma) conservation might be expected. Numeri-cal results were also carried out to confirm our theoretical predictions. The resultsshowed that the duration and quantity of fisheries were the keys to prevent the ex-tinction of Indo-Pacific mackerel.

Keywords Implusive model · Indo-Pacific mackerel · Stability · Harvesting control

References

[1] Raw S.N., Tiwari B., and Mishra P. Analysis of a plankton-fish model withexternal toxicity and nonlinear harvesting. Ricerche di Matematica. 1-29, 2019.

[2] Lakmeche A. and Arini O. Bifurcation of non trivial periodic solu-tions of impulsive differential equations arising chemotherapeutic treatment.Dyn.Contin.Discrete Impuls.Syst.,Ser.A.Math.Anal. 7: 265-287, 2000.

[3] Leah E.K. Mathematical Models in Biology. New York, Random House. 1988.[4] Liu B., Zhi, Y., and Chen, L. The dynamics of a predator-prey model

with lvlev’s functional response concerning integrated pest management. ActaMath.Appl.Sin., 20(1): 133–146, 2004.

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SOME APPLICATIONS OF EIGENVALUE AND EIGENVECTORS INENGINEERING

Ion MIERLUS MAZILU 1, Fatih YILMAZ 2

1Technical University of Civil Engineering Bucharest, Bucharest,Romania2Ankara Haci Bayram Veli University, Ankara, Turkey


ABSTRACT

In engineering and science, many matrix applications benefit from eigenvalue andeigenvector. For any n-square matrix A, the special numbers which are called aseigenvalues and some special vectors which are called as eigenvectors have greatimportance. Here we examplified their applications in stretching of an elastic mem-brane, population growth, using eigenvalues and eigenvectors to study vibrations,Google’s page rank.

Keywords Eigenvalue · Eigenvector · Vibration

References

[1] K.Fukunaga, W.L.Koontz, Eigenvalues and eigenvectors, Lafayette, Indiana,1969.

[2] Philip H. Dybvig, Mathematical Foundations for Finance, Saint Louis, Mis-souri, 2010.

[3] Gilbert Strang. Linear algebra and its applications.Belmont, CA: Thomson,Brooks/Cole, 2006.

[4] Y. Wu, M. Schuster, Z. Chen, Q. V. Le, M. Norouzi, W. Macherey, M. Krikun,Y. Cao, Q. Gao, K. Macherey, and Others,Google’s neural machine translationsystem, 2016.

[5] Sergey Brin and Lawrence Page, The anatomy of a large-scale hypertextual websearchengine, Computer Networks and ISDN Systems, 30, 1998.

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GENERALIZED QUASI-EINSTEIN NORMAL METRIC CONTACTPAIR MANIFOLDS

Inan ÜNAL

Department of Computer Engineering, Munzur University, Tunceli, Turkey


ABSTRACT

In this study, we investigate generalized quasi-Einstein structure for normal met-ric contact pair manifolds. We obtain some elementary properties and we give thecharacterizations of generalized quasi-Einstein normal metric contact pair mani-folds. We consider this structure under some curvature conditions and we givesome related results.

Keywords generalized quasi-Einstein · metric contact pairs · Generalized quasi-constant curvature · M−projective curvature tensor

References

[1] Ayar, G., and Chaubey, S. K. M-projective curvature tensor over cosymplecticmanifolds. Differential Geometry-Dynamical Systems 2019; 21: 23-33.

[2] Bande, G., Hadjar, A. Contact pairs. Tohoku Mathematical Journal 2005; Sec-ond Series:57(2): 247-260. doi:10.2748/tmj/1119888338

[3] Bande, G., Hadjar, A. Contact pair structures and associated metrics. Differen-tial Geometry - Proceedings of the V III International Colloquium 2009; (pp.266-275).

[4] Bande, G., Hadjar, A. On normal contact pairs. International Journal of Mathe-matics 2010; 21(06): 737-754. doi: 10.1142/S0129167X10006197

[5] Bande, G., and Blair, D. E. Symmetry in the geometry of metric con-tact pairs. Mathematische Nachrichten 2013; 286(17-18): 1701-1709.doi:10.1002/mana.201100300

[6] Bande, G., Blair, D. E., Hadjar, A. Bochner and conformal flatness of normalmetric contact pairs. Annals of Global Analysis and Geometry 2015; 48(1):47-56. doi:10.1007/s10455-015-9456-2

[7] Bejan, C. L. Characterization of quasi-Einstein manifolds. An. Stiint. Univ. Al.I. Cuza Iasi. Mat.(NS) 2007; 53: 67-72.

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[8] Blair, D. E., Ludden, G. D., and Yano, K. Geometry of complex manifolds sim-ilar to the Calabi-Eckmann manifolds. Journal of Differential Geometry 1974;9(2),: 263-274.

[9] Catino, G. Generalized quasi-Einstein manifolds with harmonic Weyl tensor.Mathematische Zeitschrift 2012; 271(3-4):. 751-756. doi: 10.1007/s00209-011-0888-5

[10] Chaki, M. C. On quasi Einstein manifolds. Publicationes Mathematicae De-brecen, 2000; 57:297-306.

[11] Chaki, M.C.: On Generalized quasi-Einstein manifold. Publ. Math. Debrecen2001; 58: 638–691

[12] Chen, B. Y., and Yano, K. Hypersurfaces of a conformally flat space. Tensor(NS) 1972; 26: 318-322.

[13] Chen, B. Y. Some new obstructions to minimal and Lagrangian isometric im-mersions. Japanese journal of mathematics. 2000; New series: 26(1): 105-127.

[14] Chen, B. Y., Dillen, F., Verstraelen, L., and Vrancken, L. Characterizations ofRiemannian space forms, Einstein spaces and conformally flat spaces. Proceed-ings of the American Mathematical Society 2000; 128(2): 589-598.

[15] De, U., and Ghosh, G. On quasi Einstein manifolds. Peri-odica Mathematica Hungarica 2004; 48(1-2); 223-231. Doi:10.1023/B:MAHU.0000038977.94711.ab

[16] De, U. C., and Ghosh, G. C. On generalized quasi Einstein manifolds. Kyung-pook mathematical journal 2004; 44(4): 607-607.

[17] De, U. C., and Mallick, S. On the existence of generalized quasi-Einstein man-ifolds. Arch. Math.(Brno) 2011; 47(4): 279-291.

[18] De, U. C., and Mallick, S. On generalized quasi-Einstein manifolds admittingcertain vector fields. Filomat 2015; 29(3): 599-609. doi:10.2298/FIL1503599D

[19] De, U. C., and Shenawy, S. Generalized quasi-Einstein GRW space-times.International Journal of Geometric Methods in Modern Physics 2019; 16(08):1950124. doi: 10.1142/S021988781950124X

[20] Dumitru, D. On Einstein spaces of odd dimension. Bul. Univ. TransilvaniaBrasov 2007; 14(49): 95-97.

[21] Dumitru, D. A characterization of generalized quasi-Einstein manifolds. NoviSad J. Math, 2012; 42.1: 89-94.

[22] Ghosh, G. C., De, U. C., and Binh, T. Q. Certain curvature restrictions on aquasi Einstein manifold. Publicationes Mathematicae-Debrecen 2006; 69(1-2):209-217.

[23] Güler, S., and Demirbag, S. A. A study of generalized quasi Einstein space-times with applications in general relativity. International Journal of TheoreticalPhysics 2016; 55(1): 548-562. doi: 10.1007/s10773-015-2692-1

[24] Mallick, S., and De, U. C. On a Class of Generalized quasi-Einstein Manifoldswith Applications to Relativity. Acta Universitatis Palackianae Olomucensis.Facultas Rerum Naturalium. Mathematica 2016; 55(2): 111-127.

[25] Perrone, D. Contact Riemannian manifolds satisfying R(X, ξ)R = 0, Yoko-hama Math. J. 1992; 39 : 141–149.

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[26] Pokhariyal, G. P. and Mishra, R. S. Curvature tensors and their relativisticssignificance, Yokohama Mathematical Journal 1970; vol. 18: pp. 105–108.

[27] Prakasha, D. G., and Venkatesha, H. Some results on generalized quasi-Einstein manifolds. Chinese Journal of Mathematics; 2014: Article ID 563803doi: 10.1155/2014/563803

[28] Prakasha, D. G., Fernandez, L. M., and Mirji, K. The M-projective curva-ture tensor field on generalized (κ, µ)-paracontact metric manifolds. GeorgianMathematical Journal 2020; 27(1): 141-147. doi:10.1515/gmj-2017-0054.

[29] Sular, S., and Özgür, C. Characterizations of generalized quasi-Einstein mani-folds. Analele Universitatii" Ovidius" Constanta-Seria Matematica 2012; 20(1):407-416.

[30] Tripathi, M. M., and Kim, J. On N(k)−quasi Einstein manifolds.Communications-Korean Mathematical Society 2007; 22(3): 411.

[31] Ünal, I., Sarı, R., and Vanlı, A. T. Concircular curvature tensor on generalizedkenmotsu manifolds. Gümüshane Üniversitesi Fen Bilimleri Enstitüsü Dergisi2018; 99-105.doi: 10.17714/gumusfenbil.437302

[32] Ünal, I. Some Flatness Conditions on Normal Metric Contact Pairs. arXivpreprint 2019, arXiv:1902.05327.

[33] Turgut Vanli, A., and Unal, I. On Complex η-Einstein Normal ComplexContact Metric Manifolds. Communications In Mathematics And Applicationn2017, 8(3); 301 - 313. doi:10.26713/cma.v8i3.509

[34] Turgut Vanli, A., and Unal, I. Conformal, concircular, quasi-conformal andconharmonic flatness on normal complex contact metric manifolds. Inter-national Journal of Geometric Methods in Modern Physics 2017; 14(05):1750067. doi: 10.1142/S0219887817500670

[35] Yıldırım, Ü., Atçeken, M., and Dirik, S. A Normal Paracontact Metric Mani-fold Satisfying Some Conditions on the M -Projective Curvature Tensor. Konu-ralp Journal of Mathematics 2019; 7(1): 217-221.

[36] Zengin, F. Ö. On m-projectively flat LP-Sasakian manifolds. Ukrainian Math-ematical Journal 2014; 65(11): 1725-1732.

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A NEW APPROACH TO GENERALIZED CANTOR SET IN FRACTALGEOMETRY

Ipek Ebru KARAÇAY1, Salim YÜCE1

1Yildiz Technical University, Faculty of Art and Sciences, Department of Mathematics, 34220, Istanbul, Turkey


[email protected] (https://orcid.org/0000-0002-8296-6495)

ABSTRACT

Many studies have been done about the fractal geometry from past to present. How-ever, the meaningful development of fractal geometry was taken place with increas-ing of the computer science. Fractal geometry emerged when French mathemati-cian Mandelbrot discovered the Mandelbrot set by giving complex number valuesto quadratic polynomials, which are described by his name and applied this methodat each step continuously infinitely many times, [1]. Before the development ofcomputer technologies, fractal structures such as Cantor Set and Sierpinski Gaskethad been identified without being called fractals. Cantor set was first introduced byG.Cantor in 1883, [2]. It is defined by [0, 1] closed interval, which is a subset ofthe set of real numbers, being reduced by 1

3reduction rate infinitely, [2, 3]. When

the length calculation of this structure is made, the result is zero and it is concludedthat the concept of length does not indicate a characteristic feature for this fractalstructure, [3]. Then, the concepts of self-similarity and fractal dimension, which aretwo features specific to fractal structures, were defined. Fractals are created withiterated function systems by making use of self-similarity feature, [3, 4]. General-ized Cantor set is defined for [0, 1] closed interval, which is a subset of the set ofreal numbers, [5]. For this structure, length and size calculations were made andalso fractal structure was created with iterated function systems, [5, 6, 7, 8]. Indaily life, fractal geometry has been used in many fields such as architecture, artand medicine.In this study, Cantor set is defined for the [a, b] closed interval, which is a subsetof the set of real numbers. The interval of [a, b] is divided into (2n + 1) parts,creating a series that decreases by 1

(2n+1)and this series goes to infinitely, morever

the result is zero when the length is calculated for the interval [a, b] . Dimensioncalculation was made for this interval and fractal structure that was obtained withiterated function systems. In addition, [a, b] interval was examined on the curve.The length of the fractal curve obtained in this range was found to be zero and thestudy was completed by giving the circle example.

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Keywords Cantor set · Fractal · Iterated Function System

References

[1] Mandelbrot B.B. , Fractal Geometry of Nature, W. H. Freeman and CompanyISBN 0-7167-1186-9,1983.

[2] Cantor G. , Über unendliche, lineare Punktmannigfaltigkeiten V. Mathematis-che Annalen 21, 1883.

[3] Edgar G.A. , Measure, Topology and Fractal Geometry, Springer-Verlag,Newyork, 1990.

[4] Barnsley M.F. and Demko S., iterated function systems and the global construc-tion of fractals, Proc.R.Soc. London A 399, (243-275), 1985.

[5] Chovanec, F. , Cantor sets, Sci. Military J., Vol. 1, No. 1, 5-11, 2010.[6] Islam J. and Islam S., Lebesgue Measure of Generalized Cantor Set, Annals of

Pure and App. Math. 10(1), 75-86., 2015.[7] Islam J. and Islam S. , Generalized Cantor Set and its Fractal Dimension,

Bangladesh J. Sci.Ind. Res. 46(4), 499-506, 2011.[8] Islam J. and Islam S. , Invariant measures for Iterated Function System of Gen-

eralized Cantor Sets, German J. Ad. Math. Sci. 1(2), 41-47, 2016.

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NONLINEAR APPROXIMATION IN DISCRETE OPERATORS OFSAMPLING-TYPE

Ismail ASLAN

Hacettepe University, Department of Mathematics, Ankara, Turkey


ABSTRACT

In this presentation, our aim is to construct a nonlinear form of sampling type dis-crete operators, given in [1]. These types of operators have close relationships withgeneralized sampling series. Taking some specific kernels, it is possible to obtainclassical generalized sampling series and nonlinear form of these ones from theseoperators. On the other hand, summability methods are a common method to getover the lack of convergence. Especially A-summability (see [4, 5]) is fairly gen-eral. Beside the classical convergence, it includes many well-known methods, suchas Cesaro method, almost convergence and order summability. It is also possibleto accelerate the rate of convergences. In the literature, although there exist manyapplications of this method onto the linear operators [7], there are very few workson nonlinear ones ([2, 3]).

In this study, we also improve discrete operators by using summability method un-der some generalizations of approximate identities. Considering both the usualsupremum norm and convergence inϕ-variation, introduced by Musielak and Orliczin [6], approximation properties of these operators are investigated. While doingthis, uniformly and absolutely continuous functions are taken into account respec-tively. In addition, rates of approximation are evaluated by means of some suitableLipschitz classes of continuous functions. Approximation in N-dimensional caseunder supremum norm is also obtained. Thanks to summability method, we showthat these approximations are still valid under different convergence methods suchas Cesaro’s method, almost convergence method and etc. Finally, we illustrate ourtheorems and estimations with nontrivial applications. Hence, we clearly show whywe need this study.

Acknowledgements. This study is supported by TÜBITAK 3501 Career Develop-ment Program, Project ID: 119F262.

Keywords Sampling Operators · Nonlinear Operators · Summability Method

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References

[1] Angeloni, L. Vinti, G. (2018). Discrete operators of sapmling type and approx-imation in φ-variation. Math. Nachr., 291 (4), 546-555.

[2] Aslan, I. Duman, O. (2019). Summability on Mellin-type nonlinear integralopeartors. Integral Transforms Spec. Funct. 30 (3), 492-511.

[3] Aslan, I. Duman, O. (2020). Approximation by nonlinear integral operators viasummability process. Math. Nachr. 293 (3), 430-448.

[4] Aslan, I. (2020). Approximation by Sampling type Discrete Operators. Com-munications Faculty of Sciences University of Ankara Series A1 Mathematicsand Statistics. 69 (1), 969-980.

[5] Bell, H. T. (1971). A-summability. Dissertation, (Lehigh University, Bethle-hem, Pa.).

[6] Bell, H. T. (1973). Order summability and almost convergence. Proc. Amer.Math. Soc., 38, 548—552.

[7] Musielak, J., Orlicz, W. (1959). On generalized variations (I). Studia Math., 18,11-41.

87


THE WAY OF ASSESSMENT OF MATHEMATICAL COMPETENCIESIN RULES_MATH PROJECT

Jana GABKOVÁ1, Peter LETAVAJ2

1,2Slovak University of Technology, Faculty of Mechanical Engineering


ABSTRACT

The main goal of the project RULES_MATH was to design the standards for as-sessment of mathematical competence at technical bachelor studies via contentsmodels. FME STU as one of the partners of the project prepared testing models inthe topics Analysis and Calculus and Probability and Statistics. The models weredesigned with respect to main pedagogical aim, to develop and to assess the math-ematical competency through mutually overlapping eight competencies elaboratedby SEFI MWG in the Framework [1] and previously introduced by Danish KOMproject [2]. We will deal with two guidelines covering the learning outcomes in eachmodel. The mathematical competencies were determined for each learning outcomein tables: Learning outcomes with degree of coverage of competencies involved inthis assessment activity. With respect to scope and curriculla, we prepared tests, bywhich the problems in guidelines were verified at FME STU: in the topic Analysisand Calculus at bachelor study programmes in subject Mathematics I for the firstyear of the study and in the topic Probability and Statistics in compulsorily elec-tive subject Basics of Statistical Analysis for the second year of the study. Levelsof acquirement of mathematical competencies were determined by means of givenassessment tests. Since not every competence was present in the test with the sameimportance, it was necessary to determine all competencies covered by the test. Inthis article, we introduce the process of testing the acquirement of mathematicalcompetencies covered by the test and its results. The main idea for assessment ofacquirement of the mathematical competencies is based on division of the pointsamount achieved by a student in the test into problems/subproblems/subtopics cov-ering the corresponding competencies. After that, these points are distributed onparticular competencies in given learning outcome following the “table of compe-tencies” built by teacher for each test. It was needed to norm the resulting valuesof acquirement of competence in order to analyse it correctly. It means we weresupposed to compare these values with maximal possible achievable value for each

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competence. Data we obtained were quantitatively processed by program devel-oped in Wolfram Mathematica software. The two input tables and one vector weretransformed into one resulting table. The first input table contained the points ac-quired by each student in each test problem. The second input table contained theimportance of each competence in each test problem. Entries of the input vectorswere the maximal point ammount, that one could earn in each test problem. En-tries in the output table give the acquirement of each competence for each student.Consequently, these results (this output table) were statistically processed by soft-ware Statgraphics, in order to obtain quantitative analysis of acquired mathematicalcompetencies.

Keywords Engineering ·Mathematics ·Mathematical competence · Assessment ·A competencies-based methodology · Rules_Math project

References

[1] Alpers B. et al, A framework for mathematics curricula in engineering educa-tion, SEFI, 2013, http://sefi.htw-aalen.de/

[2] Niss M., Mathematical Competencies and the Learning of Mathematics: TheDanish KOM Project, In Proceedings of the 3rd Mediterranean, 2003.

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NONLINEAR ANALYSIS OF HEART RATE DYNAMICSTO ESTIMATE THE RISK OF CARDIOVASCULAR

EVENTS IN HYPERTENSIVE PATIENTS

José María LOPEZ-BELINCHON1, Miguel Ángel LOPEZ GUERRERO2, Raúl AlcarazMARTINEZ3

1Research Group in Dynamical Systems, University of Castilla-La Mancha (UCLM), 16071 Cuenca, Spain2Polytechnic School of Cuenca, Department of Mathematics and Institute of Applied Mathematics in Science and Engineering

(IMACI), University of Castilla-La Mancha (UCLM), 16071 Cuenca, Spain3Research Group in Electronic, Biomedical and Telecommunication Engineering, University of Castilla-La Mancha (UCLM),

16071 Cuenca, Spain


ABSTRACT

In the last years many theoretical and experimental studies have suggested that mostphysiological systems are complex and can be better characterized as complex dy-namical processes [1]. In fact, it is nowadays well-established the hypothesis thatwhen a physiological system became less complex, it is less adaptable to externalchanges, and hence is prone to suffering from a disease [1, 2]. Accordingly, a broadvariety of physiological time series have been recently analyzed from a nonlinearpoint of view to improve current understanding of the mechanisms triggering dif-ferent pathologies [3].Although nonlinear analysis has been widely applied to characterize cardiac dy-namics in a variety of scenarios [3], this kind of study has not still been consideredto estimate the risk of cardiovascular event in hypertensive patients [4]. Hence,the main goal of the present study is to analyze the ability of different complexity-based metrics to anticipate patients at high risk of suffering a cardiovascular event(i.e., myocardial infarction, stroke, syncopal event, etc.) during a follow-up of oneyear. Indices such as sample entropy, fuzzy entropy, spectral entropy, and C0 com-plexity have been computed from the heart rate time series derived from 24-hourelectrocardiogram (ECG) recordings acquired for 139 hypertensive patients.Some of the analyzed nonlinear indices have been able to successfully identifyabout 70% of those patients who suffered one or more events within the follow-up. Moreover, this diagnostic accuracy has increased to about 75% by combiningseveral metrics with different matching learning classifiers. These results overcome

90


those reported by common clinical indicators, such as echocargraphic parameters,thus opening a new door to improve the early diagnosis of hypertensive patients athigh-risk to develop future vascular and cardiovascular events.

Keywords Nonlinear analysis · Heart rate variability · Classification

References

[1] Goldberger A.L., Peng C.K., and Lipsitz L.A., What is physiologic complexityand how does it change with aging and disease? Neurobiol Aging 23: 23-26,2002.

[2] Goldberger, A.L. Non-linear dynamics for clinicians: Chaos theory, fractals,and complexity at the bedside. Lancet 347: 1312-1314, 1996.

[3] Spasic S.Z., and Kesic S., Nonlinearity in living systems: Theoretical and prac-tical perspectives on metrics of physiological signal complexity, Front Physiol10: 298, 2019.

[4] Solaro N., Malacarne M., Pagani, M., and Lucini, D., Cardiac baroreflex, HRV,and statistic: An interdisciplinary approach in hypertension. Front Physiol 10:478, 2019.

91


NATURAL TRANSFORM ADOMIAN DECOMPOSITIONMETHOD(NTADM) FOR EVALUATION OF TWO DIMENIONAL

SCHRÖDINGER BLACK SCHLOES TIME FRACTIONAL ORDEREDPDE, AN APPLICATION TO FINANCIAL MODELLING

Kamran ZAKARIAR1, Saeed HAFEEZ2

1,2Department of Mathematics, NED University, Karachi, Pakistan


ABSTRACT

This paper represents the new novel technique of Natural Transform Adomian De-composition Method (NTADTM) to get real and Imaginary option prices of twostocks in form of analytic infinite series by solving Schrödinger Black Schloes timefractional ordered PDE consisting two different stocks. For that reason, the ap-proach is discovered to be appropriate for the models that can be expressed as par-tial differential equations of integer and fractional orders, subjected to preliminaryor boundary situations. This technique is the combination of Natural Transform andAdomian Decomposition Method.

Keywords Natural Integrals Transform1 · Adomian Decomposition Method ·Time fractional order Schrödinger black Scholes PDE

References

[1] Misrana, M lub, Fractional Black sholes Model with application of Fractionaloption Pricing; International Conference of optimization and control 2010,17,99-111 (Cross reference) pp-573-588

[2] Tracho; Two dimensional Black Sholes model will European call option. Mathcompute.2017,2223(eross reference)

[3] Black Sholes Pricing Model; Black Sholes option Pricing Model Bardley Uni-versity, n.d web 2016.

[4] Khan Z.H and Khan W.A. Natural Transform, properties and applications, NustJournal of Engineering Science 1, (2008) 127-133

[5] Deshna Loonker, Solution of Fractional ordinary D.E by Natural Transform,International Journal of Mathematical engineering and science volume 2 IS-SUE12,Dec.2013..

92


[6] 12. Deshna Loonker, Solution of Fractional ordinary D.E by Natural Trans-form, International Journal of Mathematical engineering and science volume 2ISSUE12,Dec.2013.

93


THE STUDY OF AXIAL LOAD IN THE LUMBAR SPINE OFPATIENTS WITH LUMBAR SPINAL STENOSIS BY USING FINITE

ELEMENT METHOD

Kamonchat TRACHOO1, Inthira CHAIYA1, Din PRATHUMWAN2

1Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand2Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand


ABSTRACT

Lumbar spinal stenosis is caused by stenosis of the vertebral cavity which may becaused by thicker bone, tendon thicker, etc. In this paper, a mathematical model isdeveloped to simulate a two-dimensional Lumbar spine (L3), taking into accountthe stress distribution and total displacement during the axial load of the lumbarspine in lumbar spinal stenosis patients and people without the disease. A realisticdomain is created based on CT scan data. Using the finite element method, the totaldisplacement and the Von Misses stress is carried out. The influences of lumbarspinal stenosis patients and people without disease during three different cases ofbending down are investigated. The advantage of this research has utilized surgeryinvolved with lumbar spinal stenosis.

Keywords Lumbar spinal stenosis · Finite element method · Numerical simulation

References

[1] Chen C.S., Cheng C.K., Liu C.L., Lo W.H., Stress analysis of the disc adjacentto interbody fusion in lumbar spine, Medical Engineering & Physics, 23: 485-493, 2001.

[2] Coobm D.J., Rullkoetter P.J., Laz P.J., Efficient probabilistic finite element anal-ysis of a lumbar motion segment, Journal of Biomechanics, 61: 65-74, 2017.

[3] Guan W., Sun Y., Qi X., Hu Y., Duan C., Tao H., Yang X., Spinal biomechanicsmodeling and finite element analysis of surgical instrument interaction, Com-puter Assisted Surgery, 24: 151-159, 2019.

94


IMPROVING ENGINEERING THERMODYNAMICSLEARNING WITH MATHEMATICA

María Jesús SANTOS1, Alejandro MEDINA1, José Miguel Mateos ROCO1, AraceliQUEIRUGA-DIOS2

1Dpt. of Applied Physics and Research Institute for Fundamental Physics and Mathematics, University of Salamanca, Spain2Dpt. of Applied Mathematics, University of Salamanca, Spain


ABSTRACT

Sophomore students from the Chemical Engineering Degree are involved in aMathematics course during the first semester and an Engineering Thermodynamicscourse during the second one. When they participate in the latter they are alreadyfamiliar with mathematical software to solve numerical methods problems, includ-ing non-linear equations, interpolation, or differential equations. We present in thispaper, on the one hand, some of the materials elaborated in both courses with theMathematica® tool, and on the other, the didactic organization of the EngineeringThermodynamics course. The objective of the experience is to increase the inter-relationship between different subjects, to promote transversal skills, and to makethe subject closer to the real working procedure the students will find in their futureworking life [1]. The satisfactory results of the experience are exposed in this work.

Keywords Engineering Education · Engineering Thermodynamics · Numericaland computational Methods

References

[1] Mogens, N. Mathematical competencies and the learning of mathematics: TheDanish KOM project, 3rd Mediterranean Conference on Mathematical Educa-tion, Athens, Greece, 115-124, 2003.

95


AN APPLICATION OF DOUBLE SUMUDU TRANSFORMFOR SOLVING TELEGRAPH EQUATION

Maria AYDIN 1, Haldun Alpaslan PEKER2

1Graduate School of Natural and Applied Sciences, Selcuk University, Konya, Turkey2Department of Mathematics, Faculty of Science, Selcuk University, Konya, Turkey

Corresponding Author’s E-mail: [email protected], ORCID: 0000-0002-1862-2911

ABSTRACT

The modelling of the most of the physical and biological phenomena, engineeringmodels, real life problems are expressed mathematically in terms of differentialequations.There are so many different techniques to solve differential equations. Integraltransform techniques such as Laplace, Fourier, etc. are extensively applied in the-ory and application [1]. There are a lot of work on the theory and applications ofthe integral transformations. Sumudu transform, which is one of the integral trans-forms, is not only having very special and useful properties but also be able to helpwith intricate applications in science and engineering [2]. Some of these applica-tions can be found in [3]. Although the Sumudu transform is the theoretical dual tothe Laplace transform, the Sumudu transform rivals it in problem solving. Besidesthis, the Sumudu transform is also widely and extensively used for solving differ-ential equations. This transform is one of the very convenient mathematical toolsfor solving differential equations. The definition, properties and applications of theSumudu transform to differential equations were described in [4].The main advantage of the Sumudu transform is the fact that it may be used to solvethe problems without resorting to a new frequency domain [5].In recent years, many various methods such as double Laplace transform, doubleSumudu transform introduced by many researchers to find the solution of partialdifferential equations. The definition, properties and applications of the doubleSumudu transform were described in [5].In this study, we believe for the first time, the exact solution of general linear tele-graph equation which appear in the propagation of electrical signals along a tele-graph line, digital image processing, telecommunication, signals and systems isobtained by double Sumudu transform. On the other hand, in addition to these, thisstudy is a part of the M.Sc. Thesis of the first author [6].

96


Keywords Sumudu Transform · Double Sumudu Transform · Telegraph Equation

References

[1] Debnath L., and Bhatta D., Integral Transforms and Their Applications, SecondEdition, Chapman and Hall/CRC, Boca Raton, Florida, USA, 2006.

[2] Watugala G.K., Sumudu transform: a new integral transform to solve differen-tial equations and control engineering problems, International Journal of Math-ematical Education in Science and Technology, 24 (1): 35–43, 1993.

[3] Belgacem F.B.M., Karaballi A.A., and Kalla S.L., Analytical investigations ofthe Sumudu transform and applications to integral production equations, Math-ematical Problems in Engineering, 3: 103-118, 2003.

[4] Belgacem F.B.M., and Karaballi A.A., Sumudu transform fundamental prop-erties investigations and applications, Journal of Applied Mathematics andStochastic Analysis, Article ID 91083, 1-23, 2006.

[5] Tchuenche J.M., and Mbare N.S., An application of the double Sumudu trans-form, Applied Mathematical Sciences, 1 (1): 31-39, 2007.

[6] Aydın, M., M.Sc. Thesis, Selcuk University, Konya, Turkey, 2020 (to appear).

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TRIPLE SUMUDU TRANSFORM AND ITS APPLICATIONFOR SOLVING VOLTERRA INTEGRO-PARTIAL

DIFFERENTIAL EQUATION

Maria AYDIN 1, Haldun Alpaslan PEKER2

1Graduate School of Natural and Applied Sciences, Selcuk University, Konya, Turkey2Department of Mathematics, Faculty of Science, Selcuk University, Konya, Turkey

Corresponding Author’s E-mail: [email protected], ORCID: 0000-0002-1862-2911

ABSTRACT

The theory and application of ordinary and partial differential equations playan important role in the modelling of most of the physical phenomena, biologicalmodels, engineering sciences, real life problems. Thus, ordinary and partial differ-ential equations are significant both in mathematics and other applied sciences. Inthe light of this fact, finding the analytical solutions of models expressed in termsof ordinary or partial differential equations has a great importance from the pointof scientific progression. Therefore, new methods are always being developed. In-tegral methods are one of the most known methods [1]. The Laplace transform, themost well-known integral method, has been intensively used to solve the nonlinearand linear equations [1-3].

Many various methods such as double Laplace transform [4], double Sumudutransform [5], double Mellin transform [6], introduced by many researchers to findthe solution of partial differential equations. The definition, properties and applica-tions of the double Sumudu transform were described in [5]. In addition to these,triple Laplace transform was defined, related theorems and properties were pre-sented in [7]. Thakur and Panda [8] had gave various properties of the triple Laplacetransform.

The aim of this study, we believe for the first time, is to extend the Sumudutransform to triple Sumudu transform. Therefore, firstly, the triple Sumudu trans-form is defined and then various useful properties of the triple Sumudu transformare given. We believe that the triple Sumudu transform will be very useful in thesolution of many partial differential equations and this relatively new transform willbe used as a very effective tool in simplifying the calculations in many fields of en-gineering, mathematics and other applied sciences. In order to illustrate the ability

98


of the method, Volterra integro-partial differential equation [9] solved by the tripleSumudu transform. On the other hand, in addition to these, this study is a part ofthe M.Sc. Thesis of the first author [10].

Keywords Sumudu Transform · Triple Sumudu Transform · Volterra integro-partial differential equations

References

[1] Duffy D.G., Transforms Methods for Solving Partial Differential Equations,CRC Press, New York, NY, USA, 2004.

[2] Myint U.T., Partial Differential Equations of Mathematical Physics, AmericanElsevier Pub., New York, NY, USA, 1973.

[3] Constanda C., Solution Techniques for Elementary Partial Differential Equa-tions, Chapman & Hall / CRC, New York, NY, USA, 2002.

[4] Eltayeb H., and Kilicman A., A note on double Laplace transform and tele-graphic equations, Abstract and Applied Analysis, Vol. 2013, Article ID932578, 6 pages, 2013.

[5] Eltayeb H., and Kilicman A., On double Sumudu transform and double Laplacetransform, Malaysian Journal of Mathematical Sciences, 4(1): 17-30, 2010.

[6] Krapivsky P.L., and Ben-Naim E., Scaling and multiscaling in models of frag-mentation, Phys. Rev. E, 50(5), 1994.

[7] Atangana A., A note on the triple Laplace transform and its applications tosome kind of third-order differential equation, Abstract and Applied Analysis,Vol. 2013, Article ID 769102, 10 pages, 2013.

[8] Thakur A.K., and Panda S., Some properties of triple Laplace transforms, Jour-nal of Mathematics and Computer Appliations Research, 2(1):23-28, 2015.

[9] Elzaki T.M., and Mousa A., On the convergence of triple Elzaki transform, SNApplied Sciences, 1, Article Number 275, 2019.

[10] Aydın M., M.Sc. Thesis, Selcuk University, Konya, Turkey, 2020 (to appear).

99


UNDERSTANDING THE FAILURE IN DIFFERENTIALAND INTEGRAL CALCULUS IN THE DEGREES OF

ENGINEERING AT A HIGHER EDUCATION SCHOOL INPORTUGAL

Maria Emília Bigotte de ALMEIDA1, Araceli QUEIRUGA-DIOS2,María José CÁCERES 2

1Coimbra Institute of Engineering, Polytechnic Institute of Coimbra, Coimbra, Portugal2University of Salamanca, Salamanca, Spain


ABSTRACT

Major difficulties are often found among students of engineering degrees in theCurricular Units (CU) of Mathematical Science area, particularly those related toDifferential and Integral Calculus (DIC). The failure and drop out rates in CU-DIChave evidenced the need to question what methodologies and teaching approachesare applied and which assessment practices are related to their school success andlead to significant learning [1, 2, 3, 4]. A research work was developed with theobjective of finding the reasons that lead the students to fail in the CU-DIC, taughtin the 1st year of the engineering undergraduate degrees, in the Polytechnic Insti-tute of Coimbra (ISEC). Applying a case study methodology, this paper presentsa current diagnosis of CU-DIC in the ISEC, with the objective of defining a setof actions that will guide a research on the causes of failure in mathematics in en-gineering degrees. We intend to establish relationships between teaching methodsand how students learn, and besides, build learning environments that lead to highersuccess with the co-responsibility of all actors in the educational process. The anal-ysis of collected data allows us to conclude that the CU-DIC in ISEC maintains anidentical distribution in the hourly load in several engineering degrees, contents areadjusted to each context taking into account CU of each degree. The data analysisfound better results in the process that includes two examination moments withoutno relationship between class attendance, dropout and pass rates. The low partici-pation of the students in the curricular assessment process, together with the passrate obtained in the distributed evaluation, can lead us to question which set of ba-sic and elementary level knowledge students need to master upon entering higher

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education. This allow us the definition of a structured intervention in overcominggaps.

Keywords Differential and Integral Calculus · Mathematical Knowledge inEngineering · Teaching and Learning

References

[1] Abdulwahed M., Jaworski B. and Crawford A., Innovative approaches to teach-ing mathematics in higher education: a review and critique, 2012, Loughbor-ough University

[2] Bigotte M.E., Gomes A., Branco J.R. and Pessoa T., The influence of educa-tional learning paths in academic success of mathematics in engineering under-graduate, 2016 IEEE Frontiers in Education Conference (FIE), 1-6, 2016

[3] Davis M., Hunter G., Thalaal L., Ba V.T. and Wooding-Olajorin A., Develop-ing" Smart" Tutorial Tools to Assist Students Learn Calculus, Taking Accountof Their Changing Preferred Approaches to Learning, Intelligent Environments(Workshops), 227-238,2019

[4] Ní F., E. and Carr M., Formative assessment in mathematics for engineeringstudents, European Journal of Engineering Education, 42(4), 458-470, 2017,Taylor & Francis

101


ANALYTICAL INVESTIGATION OF A TWO-MASS SYSTEMCONNECTED WITH LINEAR AND NONLINEAR STIFFNESSES

Md. Alal HOSEN

Department of Mathematics, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh


ABSTRACT

An iteration procedure has been developed based on the Mickens iteration method.This procedure also offers the angular frequencies and corresponding periodic so-lutions to the nonlinear vibration of a two-mass system connected with linear andnonlinear stiffnesses. A real-world case of this system is analysed and introduced.In this paper, the truncated terms of the Fourier series have been used and utilized inevery step of iterations. The approximated results are compared with existing andcorresponding numerical (considered to be exact) results. The obtained results arevalid for whole ranges of vibration amplitude of the oscillations. The error analysishas carried out and shown acceptable results for the proposed iteration procedure.Effectiveness of the proposed iteration procedure found from comparison with otherexisting methods. The method is demonstrated by an example.

Keywords Nonlinear stiffnesses · Mickens iterative method · Two-mass system ·Two-degree-of-freedom oscillation systems · Duffing equation

References

[1] Moochhala Y.E., and Raynor S., Free vibration of multi-degree-of-freedomnon-linear systems, Int. J. Non-Linear Mech., 7(6): 651-661, 1972.

[2] Huang T.C., Harmonic oscillations of nonlinear two-degree-of-freedom sys-tems, J. Appl. Mech., 22: 107-110, 1995.

[3] Gilchrist A.O., The free oscillations of conservative quasilinear systems withtwo degrees of freedom, Int. J. Mech. Sci., 3: 286-311, 1961.

[4] Efstathiades G.J., Combination tones in single mode motion of a class of non-linear systems with two degrees of freedom, J. Sound & Vib., 34: 379-397,1974.

[5] Alexander F.V., and Richard H.R., Non-linear dynamics of a system of coupledoscillators with essential stiffness nonlinearities, Int. J. Non-linear Mech., 39:1079-1091, 2004.

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[6] Chen G., Applications of a generalized Galerkin’s method to non-linear oscilla-tions of two-degree-of-freedom systems, J. Sound & Vib., 119: 225-242, 1987.

[7] Ladygina Y.V., and Manevich A.I., Free oscillations of a non-linear cubic sys-tem with two degrees of freedom and close natural frequencies, J. Appl. Math.Mech., 57: 257-266, 1993.

[8] Cveticanin L., Vibrations of a coupled two-degree-of-freedom system, J. Sound& Vib., 247: 279-292, 2001.

[9] Cveticanin L., The motion of a two-mass system with non-linear connection, J.Sound & Vib., 252: 361-369, 2002.

[10] Dimarogonas A.D., and Haddad S., Vibration for Engineers, Prentice-Hall,Englewood Cliffs, New Jersey, USA, 1992.

[11] Telli S., and Kopmaz O., Free vibrations of a mass grounded by linear andnon-linear springs in series, J. Sound & Vib., 289: 689-710, 2006.

[12] Lai S.K., and Lim C.W., Nonlinear vibration of a two-mass system with non-linear stiffnesses, Nonlinear Dynamics, 49: 233-249, 2007.

[13] Hashemi Kachapi S.H.A., Dukkipatic R.V., Hashemi S.G., HashemiS.M.,Hashemi S.M., and Hashemi S.K., Analysis of the nonlinear vibration ofa two-mass-spring system with linear and nonlinear stiffness, Nonlinear Anal-ysis: Real World Appl., 11: 1431-1441, 2010.

[14] Bayata M., Shahidia M., Bararib A., and Ganji D.D., Analytical evaluationof the nonlinear vibration of coupled oscillator systems, Z. Naturforsch, 66a:67-74, 2011.

[15] Ganji S.S., Bararib A., and Ganji D.D., Approximate analysis of two-mass-spring systems and buckling of a column, Comp. Math. Appl., 61: 1088-1095,2011.

[16] Bayat M., Pakar I., and Shahidi M., Analysis of nonlinear vibration of coupledsystems with cubic nonlinearity, Mechanika, 17(6): 620-629, 2011.

[17] Hosen M.A., and Chowdhury M.S.H., Accurate approximations of thenonlinear vibration of couple-mass-spring systems with linear and non-linear stiffnesses, J. Low Freq. Noise, Vib. Act. Cont., 2019: DOI:10.1177/1461348419854625.

[18] Wang Y., and An J.Y., Amplitude-frequency relationship to a fractional Duff-ing oscillator arising in microphysics and tsunami motion, J. Low Freq. Noise,Vib. Act. Cont., 2018: DOI: 10.1177/1461348418795813.

[19] Zhang Z., Wang Y., Wang W., and Tian R., Periodic solution of the stronglynonlinear asymmetry system with the dynamic frequency method, Symmetry,11(5): 676, 2019.

[20] Hosen M.A., Chowdhury M.S.H., Ali M.Y., and Ismail A.F., An analyticalapproximation technique for the Duffing oscillator based on the energy balancemethod, Ital. J. Pure & Appl. Math., 37: 455-466, 2017.

[21] Nayfeh A.H., Perturbation Methods, John Wiley, New York, USA, 1973.[22] Mickens R.E., Fourier representations for periodic solutions of odd-parity sys-

tems, J. Sound & Vib., 258(2): 398-401, 2002.[23] Mickens R.E., Iteration procedure for determining approximate solutions to

non-linear oscillator equations, J. Sound & Vib., 116(1): 185-187, 1987.

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[24] Mickens R.E., A generalized iteration procedure for calculating approxima-tions to periodic solutions of “truly nonlinear oscillators”, J. Sound & Vib.,287(4): 1045-1051, 2005.

[25] Mickens R.E., Iteration method solutions for conservative and limit-cyclex(1/3) force oscillators, J. Sound & Vib., 292(3): 964-968, 2006.

[26] Hu H., Solutions of the Duffing-harmonic oscillator by an iteration procedure,J. Sound & Vib., 298(1): 446-452, 2006.

[27] Hu H., Solutions of a quadratic nonlinear oscillator: Iteration procedure, J.Sound & Vib., 298(4): 1159-1165, 2006.

[28] Zheng M., Zhang B., Zhang N., Shao X., and Sun G., Comparison of twoiteration procedures for a class of nonlinear jerk equations, Acta Mechanica,224(1): 231-239, 2013.

[29] Ikramul Haque B.M., Selim Reza A.K.M., Rahman M.M., On the AnalyticalApproximation of the Nonlinear Cubic Oscillator by an Iteration Method, J.Adv. Math. & Comp. Sci., 33(3): 1-9, 2019.

104


ON APPROXIMATION PROPERTIES OF SZÁSZ-MIRAKYANOPERATORS

Melek SOFYALIOGLU1, Kadir KANAT2

1,2Ankara Hacı Bayram Veli University, Polatlı Faculty of Science and Arts, Mathematics, Ankara, Turkey

Corresponding Author’s E-mail: [email protected] (https://orcid.org/0000-0001-7837-2785)

ABSTRACT

At this talk, we initially give a brief historical background of the Szász-Mirakyanoperators. Then we construct a genaralization of the Szász-Mirakyan operators andinvestigate the approximation properties of these operators. Moreover, we studyuniform convergence and give quantitative estimation for the newly constructedoperators. In order to show the rate of convergence, we use modulus of continuity.

Keywords Szász-Mirakyan operators · Linear positive operators · Modulus ofcontinuity

References

[1] Aral A., Inoan D., Rasa I., Approximation properties of Szász-Mirakyan oper-ators preserving exponential functions, Positivity, 23, 233-246, 2019.

[2] Boyanov B.D., Veselinov V.M., A note on the approximation of functions in aninfinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. Roum.,14(62), 9–13, 1970.

[3] Holhos A., The rate of approximation of functions in an infinite interval bypositive linear operators, Stud. Univ. Babes-Bolyai Math., 2, 133-142, 2010.

[4] Mazhar S.M.,Totik V., Approximation by modified Szász operators, Acta Sci.Math., 49, 257-269, 1985.

105


CARDINALITIES AND RANKS OF NILPOTENT SUBSEMIGROUPSOF Cn

Melek YAGCI

Department of Mathematics, Çukurova University, Adana, 01330, Turkey


ABSTRACT

For n ∈ N, let Tn be the (full) transformation semigroup (under composition) onXn = {1, . . . , n} under its natural order. A transformation α ∈ Tn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y ∈ Xn and decreasing (increasing)if xα ≤ x (xα ≥ x) for all x ∈ Xn. The subsemigroup of all order-preserving trans-formations in Tn is denoted by On, and the subsemigroup of all order-preservingand decreasing (increasing) transformations in Tn is denoted by Cn (C+n ). It is a wellknown fact [5, Corollary 2.7] that Cn and C+n are isomorphic semigroups.Let S be a semigroup, and let A be any non-empty subset of S. Then the subsemi-group generated byA is denoted by 〈A〉. The rank of a finitely generated semigroupS is defined by

rank(S) = min{ |A| : 〈A〉 = S}.An element a of a semigroup S is called indecomposable if a 6= xy for all x, y ∈ Sthat is, a ∈ S\S2. It is clear that every generating set of S must contain all indecom-posable elements of S. Thus, if S = 〈A〉 andA consists entirely of indecomposableelements of S, then it is clear that A is the minimum generating set of S.The image, kernel and the fix of any transformation α ∈ Tn are defined by

im(α) = {xα : x ∈ Xn},ker(α) = {(x, y) : xα = yα for all x, y ∈ Xn} andfix(α) = {x ∈ Xn : xα = x},

respectively. An element e of a semigroup S is called idempotent if e2 = e, andthe set of all idempotents in S is denoted by E(S). It is literally immediate that forα ∈ Tn, α is an idempotent if and only if xα = x for all x ∈ im(α). Equivalently,α is an idempotent if and only if fix(α) = im(α). Moreover, an element a of asemigroup S with zero, denoted by 0, is called a nilpotent element if am = 0 forsome positive integer m. The set of all nilpotent elements of S is denoted by N(S).A semigroup S with 0 is called nilpotent if there exists m ∈ N such that Sm = 0.

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As in other algebraic theories it is an important problem to investigate the structureof nilpotent elements and nilpotent subsemigroups of semigroups. In this paper, wedetermine the structure of all the nilpotent subsemigroups of Cn.

Keywords Order-preserving/decreasing transformation · Rank · Nilpotent sub-semigroup

References

[1] Al-Kharousi, F., Kehinde, R., Umar, A., On the semigroup of partial isometriesof a finite chain. Comm. Algebra, 44: 639–647 (2016)

[2] Ayık G., Ayık H., Koç, M., Combinatorial results for order-preserving andorder-decreasing transformations, Turk. J. Math., 35: 1-9, 2011.

[3] Laradji, A., Umar, A., Combinatorial results for semigroups of order-preservingfull transformations, Semigroup Forum, 72: 51–62 (2006)

[4] Umar A., Semigroups of order-decreasing transformations: the isomorphismtheorem, Semigroup Forum 53: 220-224, 1996.

107


THE NUMBER OF m-POTENT ELEMENTS IN Cn,Y

Melek YAGCI

Department of Mathematics, Çukurova University, Adana, 01330, Turkey


ABSTRACT

For n ∈ N, let Tn be the (full) transformations semigroup (under composition) onXn = {1, . . . , n} under its natural order. A transformation α ∈ Tn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y ∈ Xn and decreasing (increasing)if xα ≤ x (xα ≥ x) for all x ∈ Xn. The semigroup of all order-preserving trans-formations in Tn is denoted by On and the semigroup of all decreasing (increasing)transformations in On is denoted by Cn (C+n ). It is a well known fact from [5,Corollary 2.7] that Cn and C+n are isomorphic semigroups. The problem of findingcertain combinatorial properties of On and Cn, has been an important research areain Semigroup Theory. The index and period of an element a of a finite semigroupare the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element withindex m and period 1 is called an m-potent element. In [2], the authors obtained aformulae for the number of m-potent and (m, r)-potent elements in Tn. An elementa of a finite semigroup S with a zero, denoted by 0, is called nilpotent if am = 0 forsome positive integer m, and moreover, if am−1 6= 0 then a is called an m-nilpotentelement of S. The set of all nilpotent elements of S is denoted by N(S). For anyα ∈ Cn, it is clear that α is an m-potent element for some m ∈ Z+.

Recall that nth Catalan number Cn is defined by

C0 = 1

Cn =1

n+ 1

(2n

n

)=

1

n

(2n

n− 1

)for n ≥ 1,

(see for example p38 of [3]). From [4, Theorem 2.1 and Proposition 2.3] we knowthat

|Cn| = |C+n | = Cn,|N(Cn)| = |N(C+n )| = Cn−1.

In general a transformation α ∈ Tn is represented by the following tabular form:

α =

(1 · · · n

1α · · · nα

).

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For any α ∈ Tn and for any non-empty subset Y of Xn we define

Fix(α) = {x ∈ Xn : xα = x} andCn,Y = {α ∈ Cn : Fix(α) = Y },

the set Cn,Y is first defined in [1].The number of nilpotent elements in Cn have been computed by Laradji and Umar in[4]. The number of m-potent and (m, r)-potent elements in Tn have been computedby Ayık, Ayık, Ünlü and Howie in [2]. However we have not seen any informationabout the number of m-potent elements in Cn,Y in the literature. The aim of thispaper is to find a formulae for any subset Y of Xn, we obtain a formulae for thenumber of m-potent elements of Cn,Y = {α ∈ Cn : Fix(α) = Y }.

Keywords Order-preserving/decreasing transformation ·m-nilpotent ·m-potent

References

[1] Ayık G., Ayık H., Koç, M., Combinatorial results for order-preserving andorder-decreasing transformations. Turk. J. Math., 35: 1-9 2011.

[2] Ayık G., Ayık H., Ünlü Y., Howie, J.M., The structure of elements in finite fulltransformation semigroups, Bull. Austral. Math. Soc., 71: 69-74, 2005.

[3] Grimaldi R.P., Discrete and Combinatorial Mathematics. Pearson EducationInc, USA, 2003.

[4] Laradji A., Umar A., Combinatorial results for semigroups of order-preservingfull transformations, Semigroup Forum, 72: 51-62, 2006.

[5] Umar A., Semigroups of order-decreasing transformations: the isomorphismtheorem, Semigroup Forum 53: 220-224, 1996.

109


MODELLING TAP WATER CONSUMER RATIO

Meltem EKIZ1 , Osman Ufuk EKIZ2

1,2Department of Statistics, Faculty of Science, Gazi University, Ankara, Turkey


ABSTRACT

Unplanned population is known as a factor that causes water depletion in the worldbasins. Therefore, it is important to accurately predict the future ratios of tap waterconsumers using the same watershed to the population living in the specified area,in order to produce better water policies and take the necessary measures. Turkey, acountry having land both in Asia and Europe and approximately living eighty-twomillion people, is exposed to dry seasons. European Environment Agency reportedthat Turkey will encounter moderate and high level water scarcity in many areas[1]. Thus, it is obvious that Turkey is a candidate country to experience problemson water scarceness. Considering that the population is predicted to be close to 100million at 2030, [2], it is crucial to take precautions to avoid water shortages and toproduce better water policies. Hence, it is important for a country to both followand also predict the tap water consumer rates (TWCR) particularly to take measureson decreasing the negative effects of reduction in tap water that could occur in thevery near future. TWCR are defined as the ratios of number of people using tapwater to the number of population for each 26 grouped cities and are obtained forchosen years between 2001 and 2016. This study attempted to model tap water con-sumer ratios (TWCR) of watersheds in Turkey with growth curve models (GCMs).GCMs are preferred to model growth problems and are used to make short-termpredictions [3, 4]. However, presence of outliers has an impact on parameter esti-mations and future predictions. Therefore, robust least median square (LMS) andM estimators are proposed to construct the GCMs. It is observed that the use ofrobust M estimator, which is resistant to outliers, gives more successful results. Thepredictions with the estimated third order GCM based on M estimator for 2020 and2021 are founded to be approximately at the rate of 90 and 95, respectively.

Keywords Growth curve model · Outlier · Prediction · Robust estimators

References

[1] EEA, European Environment Agency Report: European Environment Outlook(Report No. 4). ISSN 1725-9177, Copenhagen, 2005.

110


[2] TSI, Turkish Statistical Institute: accessed February 18, 2018. Retrieved fromhttp://www.tuik.gov.tr/PreHaberBultenleri.do?id=30567, 2018.

[3] Pan J.X., and Fang K.T., Growth curve models and statistical diagnostics,Springer-Verlag, New York, USA, 2002.

[4] Pendergast J.F., and Broffitt J.D., Robust estimation in growth curve models,Commun. Stat. Theory Methods, 14(8): 1919-1939, 1985.

111


COMPARISON OF NONTRADITIONAL OPTIMIZATIONTECHNIQUES IN OPTIMIZATION OF SHELL AND TUBE HEAT

EXCHANGER

Mert Akin INSEL1, Inci ALBAYRAK2, Hale Gonce KOCKEN2

1Yildiz Technical University, Chemical Engineering Department, Turkey2Yildiz Technical University, Mathematical Engineering Department, Turkey


ABSTRACT

Heat exchangers, which are the systems that allow heat transfer between two ormore fluids, are widely used in industry, and they are an indispensable part of chem-ical processes. Thus, a considerable amount of research has been conducted in theoptimization of these systems, especially regarding cost optimization. There aretwo main parameters affecting the cost of a heat exchanger. One is the surface areaof the exchanger, which mainly affects the capital investment cost. The other oneis the pressure drop, which mainly affects the operating cost. These parameters areinterlinked; however, since the change in the surface area may result in a changein the pressure drop and vice versa due to the physical laws. In the case which isanalyzed in this study, there are four main variables for the heat exchanger, whichwill be adjusted in order to obtain the optimum values for surface area and pressuredrop: the number of tube side passages, shell inside diameter, baffles spacing, andtube outside diameter. In this study, we compare the success of the nontraditionaloptimization techniques such as generic algorithms, particle swarm optimization,artificial bee colony, and biography-based optimization for a shell and tube heat ex-changer. In the considered three processes in which these optimization methods areutilized, we see that biography-based optimization has given the minimum cost foreach process.

Keywords Optimization · Heat Exchanger · Cost Minimization

References

[1] Martins G.F.: “Tuning PID Controllers Us ing the ITAE Criterion,” Interna-tional Journal of Engineering Education, vol. 21, no. 3, 2005.

[2] Parmar N., Khan A.: Shell and Tube Heat Exchanger Thermal Design Opti-mization with flow pressure drop due to fouling, 2012.

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[3] S¸ahin A. S ., Kiliç B., Kiliç U.: “Design and economic optimization of shelland tube heat exchangers using Artificial Bee Colony (ABC) algorithm”, En-ergy Conversion and Management, 52(11), 3356–3362, 2011.

[4] Incropera F.P., DeWitt D.P.: “Fundamentals of heat and mass transfer”, USA:John Wiley and Sons Inc.; 1996.

[5] Genceli O.: “Heat exchangers”, Istanbul, Turkey: Birsen Book Company; 1999.[6] Kakaç, S.: “Boilers, Evaporators, and Condensers”, 58, 1991.[7] Valadi J., Siarry P.: “Applications of Metaheuristics in Process Engineering”,

250, 2014.[8] Caputo A.C., Pelagagge P.M., Salini P.: “Heat exchanger design based on eco-

nomic optimisation”, Applied Thermal Engineering, 2008.[9] Hadidi,A.,Nazari,A.: Designandeconomicoptimizationofshell-and-

tubeheatexchangersusingbiogeographybased (BBO) algorithm, AppliedThermal Engineering, 2013.

[10] Sinnot R.K.: “Coulson and Richardson’s Chemical Engineering, ChemicalEngineering Design”, vol. 6, Butterworth-Heinemann, 2005.

[11] Patel V.K., Rao R.V.: “Design optimization of shell-and-tube heat exchangerusing particle swarm optimization technique, Applied Thermal Engineering,2010.

[12] Padleckas, H.: https : //commons.wikimedia.org/wiki/F ile : U −tubeheatexchanger.PNG Access Date: 19.04.2020.

[13] Meyer-Baese, A., Schmid, V., Pattern Recognition and Signal Analysis inMedical Imaging (Second Edition), 2014.

[14] Michalewicz, Z.: Genetic Algorithms, Springer Verlag, 1995.[15] KennedyJ.,EberhartR.: Particleswarmoptimiza-

tion,IEEEInternationalConferenceonNeuralNetworks,1995.[16] Martínez, C.M., Cao, D.: Ihorizon-Enabled Energy Management for Electri-

fied Vehicles, 2019.[17] Karaboga, N.: A new design method based on Artificial Bee Colony algorithm

for digital IIR filters, J Franklin Inst, 2009.[18] Garg, H.: An efficient biogeography based optimization algorithm for solving

reliability optimization problems, Swarm and Evolutionary Computation Vol-ume 24, 2015.

113


FUZZY MODELLING ON CONTROL OF HEAT EXCHANGERS

Mert Akin INSEL1, Inci ALBAYRAK2, Hale Gonce KOCKEN2

1Yildiz Technical University, Chemical Engineering Department, Turkey2Yildiz Technical University, Mathematical Engineering Department, Turkey


ABSTRACT

Heat exchangers are an indispensable part of many chemical processes. These arethe systems that allow the transfer of heat between two or more fluids. The dynam-ics of heat exchangers are highly nonlinear; thus, many methods have been usedto control these systems. Feedback, feedforward and cascade control loops can bepresented as traditional methods. The type of controller used in these systems isalso crucial, and controllers such as P, PI, PID are widely used in industry. Becauseof the complexity of the control, intelligent controllers like fuzzy logic controllers(FLCs) are also considered to provide an alternative to existing controllers, sinceFLCs provide ability to implement human knowledge and experience. In this study,control of a heat exchanger with process gain Kp = 50 and process time constant Tp= 30 was investigated by using a fuzzy-PID controller with feedback control loop.By comparing the conventional membership function for the fuzzy variables withproposed membership function, the response of the heat exchanger system to a stepchange of 10 units applied to the setpoint was interpreted.

Keywords Control · Heat Exchanger · Fuzzy Control

References

[1] Genceli O.F.: “Heat exchangers”, Istanbul, Turkey: Birsen Book Company;1999

[2] Parmar N., Khan A.: Shell and Tube Heat Exchanger Thermal Design Opti-mization with flow pressure drop due to fouling, 2012.

[3] Martins G.F., “Tuning PID Controllers Using the ITAE Criterion,” InternationalJournal of Engineering Education, vol. 21, no. 3, 2005.

[4] Duka A.V., Grif H.St., Oltean S.E., A Fuzzy-PD Design Method for a Class ofUnstable Systems, Proceedings of the International Scientific Conference Inter-Ing, 2007

114


[5] Padleckas, H.: https : //commons.wikimedia.org/wiki/F ile : U −tubeheatexchanger.PNG Access Date: 19.04.2020

[6] Padhee, S., Controller design for temperature control of heat exchanger sys-tem: Simulation studies, WSEAS Transactions on Systems and Control, 9(1),485–491., 2014.

115


ON THE NEW FAMILIES OF GAUSS K-LUCASNUMBERS AND THEIR POLYNOMIALS

Merve TASTAN1, Engin ÖZKAN2

1Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan, Turkey2Department of Mathematics, Erzincan Binali Yıldırım University, Faculty of Arts and Sciences, Erzincan, Turkey


ABSTRACT

In [1], Mikkawy and Sogabe gave a new family of k- Fibonacci numbers. In [2],Özkan et.all. defined a new family of k- Lucas numbers and give some propertiesabout the family of these numbers. There are some works on polynomials of thefamilies of k- Fibonacci numbers and k- Lucas numbers [3, 4]. In [5], Falcon andPlaza defined general k- Fibonacci numbers and showed that properties of thesenumbers were related with elementary matrix algebra. Finally [6], Tas presentedk- Gauss Fibonacci numbers and give some properties related to these numbers. Inthis paper, we define the new families of Gauss k- Lucas numbers. We give the rela-tionships between the family of the Gauss k- Lucas numbers and the known GaussLucas numbers. We also define the generalized polynomials for these numbers.We also obtain some interesting properties of the polynomials. We also give therelationships between the generalized Gauss k- Lucas polynomials and the knownGauss Lucas polynomials. Furthermore, we find the new generalizations of thesefamilies and the polynomials in matrix representation. Then we prove Cassini’sIdentities for the families and their polynomials.

Keywords Lucas Numbers · Lucas polynomials · Gauss Lucas Numbers · GaussLucas polynomials · Cassini’s Identity

References

[1] M. Mikkawy and Sogabe, T., A new family of k-Fibonacci numbers, AppliedMathematics and Computation, 215: 4456-4461, 2010.

[2] E. Özkan, I. Altun and A. Göçer, On Relationship among A New Family ofk-Fibonacci, k-Lucas Numbers, Fibonacci and Lucas Numbers, Chiang MaiJournal of Science, 44: 1744-1750, 2017.

116


[3] A. A. Öcal, N., Tuglu and E. Altinisik, On the representation of k-generalizedFibonacci and Lucas numbers, Applied Mathematics and Computation. 170:584-596, 2005.

[4] E. Özkan and M. Tastan, k-Fibonacci Polynomials in The Family of FibonacciNumbers, Research and Reviews: Discrete Mathematical Structure, 6(3): 19-22, 2019.

[5] S. Falcon, A. Plaza, On the Fibonacci k-numbers, Chaos, Solitons and Fractals32(5): 1615-1624, 2007. doi:10.1016/j.chaos.2006.09.022.

[6] S. Tas, A new family of k-Gaussian Fibonacci numbers, J. BAUN Inst. Sci.Technol., 21(1): 184-189, 2019.

117


ON SPACES DERIVABLE FROM A SOLID SEQUENCESPACE AND A NON–NEGATIVE LOWER TRIANGULAR

MATRIX

Merve ULUDAG

Department of Mathematics, Faculty of Science, Çankırı Karatekin University, Tr-18100, Çankırı, Turkey


ABSTRACT

In this talk, we will give some recent results on the sequence of solid sequencespaces, each term of it generated by an infinite lower triangular matrix obtained byJohnson and Polat [1]. The scalar field will be either the real or complex numbers.Suppose that λ is a solid sequence space over the scalar field and A is an infinitelower triangular matrix with non-negative entries and positive entries on the maindiagonal such that each of its columns is in λ. For each positive integer k, the kth

predecessor of λ with respect to A is the solid vector space of scalar sequencesx such that Ak|x| is an element of λ. We denote this space by Λk and λ itselfwill be denoted by Λ0. Under reasonable assumptions, these spaces inherit sometopological properties from λ. We are interested in a projective limit of the infiniteproduct of the Λk consisting of sequences of sequences (x(k)) satisfying Ax(k) =x(k−1) for each k > 0. We show that for interesting classes of situations includingthe cases when λ = lp for some p > 1 and A is the Cesa ro matrix, the space of ourinterest can be non-trivial.

Keywords Solid sequence space · Infinite lower triangular matrix · Projective limit

References

[1] Polat, Faruk and Johnson, Peter, On spaces derivable from a solid sequencespace and a non-negative lower triangular matrix, Operators and matrices, 2015,Vol. 9, 813-819.

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NUMERICAL ANALYSIS OF AN ADAPTIVENON-LINEAR FILTER BASED TIME REGULARIZATION

MODEL FOR THE INCOMPRESSIBLENON-ISOTHERMAL FLUID FLOWS

Mine AKBAS1, Buse INGENÇ2

1Department of Mathematics, Düzce University, 81620, Düzce, Turkey2Graduate School of Natural and Applied Sciences, Düzce University, 81620, Düzce, Turkey


ABSTRACT

This report proposes a modular filter based regularization model for the incom-pressible non-isothermal fluid flows, and a numerical method for solving that. Theproposed model plugs a time relaxation term into Leray model which uses a decon-volution based indicator function. The task of this relaxation term is to drive thesmall scales to zero exponentially. The paper provides a complete finite elementanalysis of the model consisting of the stability and convergence analysis. Numer-ical experiments verify theoretical convergence rates and compare the approximatesolutions of the model on a benchmark problem with the non-regularized finite ele-ment discretization of the flow problem.

Keywords Time relaxation · Finite element method · Non-linear filtering

References

[1] Adams N. A. and Stolz S., Deconvolution methods for subgrid-scale approxi-mation in large eddy simulation, In Modern Simulation Strategies for TurbulentFlow, R.T. Edwards, 2001.

[2] Adams N. A. and Stolz S., A subgrid-scale deconvolution approach for shockcapturing, J.C.P., 178: 391–426, 2002.

[3] Ervin V. J., Layton W. and Neda M., Numerical analysis of a higher order timerelaxation model of fluids, Int. J. Numer. Anal. Mod., 4(3-4): 648-670, 2007.

[4] Takhirov A., Neda M. and Waters J., Modular nonlinear filter based time re-laxation scheme for high reynolds number flows, Int. J. Numer. Anal. Mod.,15(4-5): 699-714, 2018.

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APPROXIMATE AND ANALYTICAL SOLUTIONS FOR NONLINEARFRACTIONAL SYSTEMS

Muammer AYATA1, Ozan ÖZKAN1

1Department of Mathematics, Faculty of Science, Selcuk University, Konya, 42003, Turkey


ABSTRACT

It is the first time in this study that we apply newly presented conformable Laplacedecomposition method (CLDM) to fractional Drinfeld-Sokolov-Wilson equation(DSWE) and coupled viscous Burgers’ equation (CVBE) systems. While Drinfeld-Sokolov-Wilson equation system is considered as a mathematical model for shallowwater waves, coupled viscous Burgers’ equation system is considered as a mathe-matical model for hydrodynmic turbulence. The results obtained in the applicationsare compared to exact solutions by the help of 2 and 3 dimensional figures and atable. It is seen that, CLDM is easy to use and it gives quite good results in even 2steps. Also it can be said that the method will give researchers a new perspectivefor finding approximate and analytical solutions of many mathematical and physicalproblems.

Keywords Conformable fractional derivative · Drinfeld-Sokolov-Wilson system ·coupled viscous Burgers’ equation · Shock wave theory · Adomian DecompositionMethod

References

[1] Miller, K. S., Ross, B.. An introduction to the fractional calculus and fractionaldifferential equations, Wiley, 1993

[2] Ortigueira, M. D.. Fractional calculus for scientists and engineers, SpringerScience Business Media, 2011

[3] Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.. A new definition of frac-tional derivative, Journal of Computational and Applied Mathematics, 264: 65-70, 2014

[4] Atangana, A., Baleanu, D., and Alsaedi, A. New properties of conformablederivative, Open Mathematics, 13: 889-898, 2015

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[5] Özkan, O., and Kurt, A. On conformable double Laplace transform, Optical andQuantum Electronics, 50: 103-111, 2018

[6] Jaradat, H. M., Al-Shara, S., and Khan, Q. J., Alquran, M., Al-Khaled, K. Ana-lytical solution of time fractional Drinfeld Sokolov Wilson system using resid-ual power series method, IAENG International Journal of Applied Mathemat-ics, 46: 64-70, 2016

[7] Prakash, A., Kumar, and M., Sharma, K. K. Numerical method for solving frac-tional coupled Burgers equations, Applied Mathematics and Computation, 260:314-320, 2015

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RUNGE-KUTTA-NYSTROM METHOD BASED ON ARITHMETICMEAN FOR SOLVING SPECIAL SECOND ORDER ORDINARY

DIFFERENTIAL EQUATIONS

Mukaddes ÖKTEN TURACI

Department of Computer Programming, Yenice Vocational School, Karabük University, 78700, Karabük, Turkey


ABSTRACT

The Runge-Kutta methods are widely used by the researchers for solving ordinarydifferential equations. Recently, there has been a great deal of interest in the Runge-Kutta methods based on various means formulation for the numerical solution ofordinary differential equations. In this study, inspired by the Runge-Kutta methodbased on arithmetic mean, a third-order three-stage explicit Runge-Kutta-Nyströmmethod based on arithmetic mean is presented for solving directly special second-order initial value problem of the form y′′ = f(x, y), y(x0) = y0, y

′(x0) = y′0 . Thestability properties of the proposed method are discussed and numerical examplesare given to show the efficiency of the proposed methods compared to the existingRunge-Kutta and Runge-Kutta-Nystrom methods. The numerical results indicatethat the proposed method is more efficient in terms of accuracy compared to otherwell known methods in literature.

Keywords Runge-Kutta method · Runge-Kutta-Nyström method · Arithmeticmean · Stability

References

[1] Evans D. J., New Runge-Kutta Methods for Initial Value Problems, Appl. Math.Letters, 2:25-28, 1989.

[2] Evans D. J., and Yaakub A. R., Weighted fifth order Runge-Kutta formulasfor second order differential equations, International Journal of ComputationalMathematics, 70(2):233–239, 1998.

[3] Evans D. J., and Jayes, M. I., New numerical methods for a special class ofsecond order odes, International Journal of Computer Mathematics, 48(3-4):191-201, 1993.

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[4] Hairer E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I:Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg, 1993.

[5] Sanugi B.B., and Yaacob N.B., A new fifth-order five-stage runge kutta methodfor initial value type problems in ODEs, International Journal of ComputerMathematics, 59(3-4): 187-207, 1996.

[6] van der Houwen P.J., and Sommeijer B.P., Explicit Runge-Kutta(-Nyström)methods with reduced phase errors for computing oscillating solutions, SIAMJ. Numer. Anal. 24(3):595-617, 1987.

[7] Wazwaz A., A comparison of Modified Runge-Kutta Formulas Based On AVariety of Means, Intern. J.Computer Math, 50: 105 – 112, 1994.

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REFINEMENTS OF BULLEN-TYPE INEQUALITIES FORs−CONVEX FUNCTIONS VIA RIEMANN-LIOUVILLE

FRACTIONAL INTEGRALS

Musa ÇAKMAK

Hatay Mustafa Kemal University, Reyhanlı Vocational School of Social Sciences, Turkey


ABSTRACT

In this paper, the author establishes a new identity for differentiable functions andobtain some new inequalities for s−Convex Functions via Riemann-Liouville Frac-tional Integrals, afterwords applies these inequalities to construct inequalities forspecial means of two positive numbers.

Keywords Bullen’s inequality · s−convexity · Riemann-Liouville FractionalIntegrals

References

[1] Dahmani, Z., “New inequalities in fractional integrals,” International Journal ofNonlinear Scinece, vol. 9, no. 4, pp. 493-497, 2010.

[2] Dahmani, Z., “On Minkowski and Hermite-Hadamard integral inequalities viafractional integration,”Ann. Funct. Anal., vol. 1, no. 1, pp. 51–58, 2010.

[3] Dragomir,S.S. and Fitzpatricks,S., The Hadamard’s inequality for s-convexfunctions in the second sense, Demonstratio Math., 32 (4), 687-696, 1999.

[4] Gorenflo, R. and Mainardi, F. , “Fractional calculus: integral and differentialequations of fractional order,” Springer Verlag, Wien, pp. 223-276, 1997.

[5] H. Hudzik, L. Maligranda, Some remarks on s-convex functions, AequationesMath., 48(1994), 100-111.

[6] Miller, S. and Ross, B. , “On introduction to the fractional calculus and frac-tional differential equations,”John Wiley & Sons, USA„ vol. 13, p. 2, 1993.

[7] Mitrinovic, D.S., Pecaric, J., Fink, A.M., Classical and New Inequalities inAnalysis, KluwerAcademic, Dordrecht, 1993.

[8] Podlubni, I. , “Fractional differential equations,” Academic Press, San Diego,1999.

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[9] Sarikaya, M. Z., Set, E., Yaldiz, H., and Basak, N., “Hermite-Hadamard’sinequalities for fractional integrals and related fractional inequalities,” Math-ematical and Computer Modelling, vol. 57, pp.2403–2407, 2013, doi:10.1016/j.mcm.2011.12.048.

[10] Wang, J., Li, X., Feckan, M. and Zhou, Y., “Hermite-Hadamard-type inequali-ties for Riemann-Liouville fractional integrals via two kinds of convexity,” Ap-plicable Analysis, pp. 1–13, 2012, doi: 10.1080/00036811.2012.727986.

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A GENERALIZATION OF HERMITE-HADAMARD, BULLEN, ANDSIMPSON INEQUALITIES VIA h−CONVEXITY

Musa ÇAKMAK

Hatay Mustafa Kemal University, Reyhanlı Vocational School of Social Sciences, Turkey


ABSTRACT

In this paper, the author established a new identity for differentiable functions, af-terward, he obtained some new general inequalities for functions whose first deriva-tives in absolute value at certain powers are h-convex by using the identity. On theother hand, a general inequality is studied, which gives Hermite-Hadamard, Bullen,and Simpson inequalities. Also, he gave an application for special means for arbi-trary positive numbers.

Keywords h-convex function · Bullen inequality · Simpson inequality

References

[1] Burai P. and Házy A., On approximately h−convex functions, J. Convex Anal.18 (2) (2011) 447-454.

[2] Dragomir S. S., Pecaric J. and Persson L.E., Some inequalities of Hadamardtype, Soochow J.Math., 21, 335-241, 1995.

[3] Dragomir S. S., Agarwal R. P. and Cerone P., On Simpson’s inequality andapplications, J. Inequal. Appl. 5 (2000), no. 6, 533-579; Available online athttp://dx.doi.org/10.1155/S102558340000031X.

[4] Sarıkaya M.Z., Set E. and Özdemir M.E., On some new inequalities ofHadamard-type involving h−convex functions, Acta Math. Univ. ComenianLXXIX (2) (2010) 265-272.

[5] Tunç M., On new inequalities for h−convex functions via Riemann-Liouvillefractional integration, Filomat 27 (4) (2013), 559-565.

[6] Varošanec S., On h−convexity, J. Math. Anal. Appl., Volume 326, Issue 1(2007), 303-311.

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THE AGREEMENT OF THE CONVENTIONAL DEFECTCORRECTION METHOD AND THE NOVEL DEFECT CORRECTING

EXTRAPOLATION TECHNIQUE FOR LINEAR EQUATIONS

Mustafa AGGUL

Hacettepe University, Ankara, Turkey


ABSTRACT

The defect correcting extrapolation technique has been developed for, and appliedto Oseen and Navier-Stokes flows successfully. This model solves equations withvarying artificial viscosities (AV) and then employs an extrapolation idea to recovera corrected solution of the target problem. This way, each AV solution can besolved fully parallel, and hence reduce computational time dramatically comparingthe conventional defect correction methods, which relies on the previously foundAV step each time for a corrected solution. On the other hand, these two methodsagree both in terms of theory and computational results for linear equations. Inthis talk, the narrator will address how these models coincide and exemplify theirsimilarities with some linear equations.

Keywords Defect correction · Defect correcting extrapolation · Artificial viscos-ity · Artificial diffusivity

References

[1] M. Aggul, Defect Correcting Extrapolation Technique for Oseen and Navier-Stokes Flows, Computers and Mathematics with Applications, accepted, 2020.

[2] A. LABOVSKY, A Defect Correction Method for the Time-Dependent Navier-Stokes Equations, Numerical Methods for Partial Differential Equations, 2008,25: 1-25.

[3] M. AGGUL, A. LABOVSKY, A high accuracy minimally invasive regulariza-tion technique for Navier-Stokes equation at high Reynolds Number, Numeri-cal Methods for Partial Differential Equations, 2017, 33: 814-839.

[4] M. AGGUL, J. M. CONNORS, D. ERKMEN, A. LABOVSKY, A Defect-Deferred Correction Method for Fluid-Fluid Interaction, SIAM Journal onNumerical Analysis, 2018 56:4, 2484-2512

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ON HELICOIDAL HYPERSURFACES IN EUCLIDEAN 4-SPACE

Mustafa ALTIN1, Ahmet KAZAN2

1Technical Sciences Vocational School, Bingöl University, Bingöl, Turkey2Department of Computer Technologies, Dogansehir Vahap Küçük Vocational School, Malatya Turgut Özal University, Malatya,

Turkey


ABSTRACT

It is known that a rotation hypersurface is defined as a hypersurface rotating a curvearound an axis. In this context, if α : I ⊂ R −→ π is a curve in a plane πof E4 and l is a straight line in E4, then a rotation hypersurface is defined by ahypersurface rotating the profile curve α around the axis l. Furthermore, if the pro-file curve α rotates around the axis l and it simultaneously displaces parallel linesorthogonal to the axis l, then the obtained hypersurface is called helicoidal hyper-surface with axis l. With the aid of these definitions, the differential geometry ofrotation (hyper)surfaces and helicoidal (hyper)surfaces in 3 or higher-dimensionalEuclidean, Minkowskian, Galilean and pseudo-Galilean spaces have been studiedby scientists. For instance, finite type surfaces of revolution in a Euclidean 3-spacehave been classified in [1] and some properties about surfaces of revolution in fourdimensions have been given in [2]. The third Laplace-Beltrami operator and theGauss map of the rotational hypersurface in Euclidean 4-space and the general ro-tation surfaces in Minkowski 4-space have been studied in [3] and [4], respectively.Also, Dini-type helicoidal hypersurface in E4 and Dini-type helicoidal hypersur-faces with timelike axis in E4

1 have been studied in [5] and [6], respectively. In thepresent study, we deal with general helicoidal hypersurfaces in Euclidean 4-space.Firstly, we obtain the Gaussian and mean curvatures of helicoidal hypersurfaces inE4 and give some results about the minimality and flatness of these hypersurfaces.After that, we study on the Gauss map of helicoidal hypersurfaces by obtainingthe Gaussian and mean curvatures of it after long calculations. Finally, we givesome visualizations for projections of some special helicoidal hypersurfaces andthe Gauss map by fixing one of the three parameters of a hypersurface into 3-space.

Keywords Helicoidal hypersurface · Gauss map ·Mean curvature

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References

[1] Chen B.Y. and Ishikawa S., On classification of some surfaces of revolution offinite type, Tsukuba J. Math., 17(1): 287-298, 1993.

[2] Moore C., Surfaces of rotation in a space of four dimensions, Ann. Math., 21:81-93, 1919.

[3] Güler E., Hacısalihoglu H.H. and Kim Y.H., The Gauss map and the thirdLaplace-Beltrami operator of the rotational hypersurface in 4-Space, Symmetry,10(9): 1-11, 2018.

[4] Ganchev G. and Milousheva V., General rotational surfaces in the 4-dimensional Minkowski space, Turkish J. Math., 38: 883-895, 2014.

[5] Güler E. and Karaalp A.G., Dini Type Helicoidal Hypersurface in 4-Space, Iko-nian Journal of Mathematics, 1(1): 26-34, 2019.

[6] Güler E. and Kisi Ö., Dini-type helicoidal hypersurfaces with timelike axis inMinkowski 4-space E4

1 , Mathematics: Comp. Alg. Sci. Comp., 7(2): 1-8, 2019.

129


PERIODIC POINT RESULTS ON QUASI METRIC SPACES

Mustafa ASLANTAS

Department of Mathematics, Science Faculty, CankiriKaratekin University, Cankiri, Turkey


ABSTRACT

Fixed point theory has been broadly advanced and applied to various problems dur-ing last century. The applications of fixed point theorems comprise differing disci-plines of mathematics, statistics, engenering and mathematical economics In 1922,Banach proved an important theorem which is known as Banach contraction princi-ple considered the beginning of the fixed point theory on metric spaces. Because ofthis principle’s applicability, this result has been extended and generalized in manyvarious ways. In this sence, Boyd and Wong have been obtained a generalization afBanach contraction principle via nonlinear functions. After that, Ciric introducedthe concept of periodic point. Since each fixed point is a periodic point, but theconverse is not true. Also,a periodic point is a natural generalization of fixed point.Hence, periodic point results have been obtained by many authors in the abstractspaces. On the other hand, it was defined a new concept so called quasi metricspaces by removing the symmetry condition on metric spaces. This new concepthas attracted interest of many authors, since the results obtained on standart metricspaces is not clear on this concept. Recently, Al Homidian et al introduced a no-tation which is called Q-function on quasi metric spaces to generalize ω-distanceand τ -function. Hence, there are many studies in the literature obtained via the Q-functions. In this paper, we investigate some periodic point results on quasi metricspaces via Q-funtion. Hence, we generalize and extend some fixed point resultsexisting in the literature. Finally, we present illustrative example to support ourresults.

Keywords Fixed point · Q-function · Quasi metric spaces

References

[1] Al-Homidan, Q. H. Ansaria, J.-C. Yaoc, Some generalizations of Ekeland-typevariational principle with applications to equilibrium problems and fixed pointtheory, Nonlinear Analysis: Theory, Methods and Applications. 69 (2008),126–139.

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[2] S. Banach, Sur les oprations dans les ensembles abstraits et leur applicationsaux quations intgrales, Fund. Math. 3 (1922), 133–181.

[3] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Am. Math.Soc.,20 (1969), 458-464.S.

[4] L. B. Ciri´c, On some maps with a nonunique fixed point, Institut Mathmatique.17 (1974), 52–58

[5] J. Marín, S. Romaguera, and P. Tirado, Q-functions on quasimetric spaces andfixed points for multivalued maps, Fixed Point Theory and Applications,2011(2011), 1-10.

[6] J. Marín, S. Romaguera and P. Tirado, Weakly contractive multivalued mapsand w-distances on complete quasi-metric spaces, Fixed Point Theory and Ap-plications, 2011 (2011), 1-9.

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SOME FIXED POINT RESULTS FOR MULTIVALUED MAPPINGSON M-METRIC SPACES

Mustafa ASLANTAS1, Ugur SADULLAH2

1,2Department of Mathematics, Science Faculty, CankiriKaratekin University, Cankiri, Turkey


ABSTRACT

In 1922, Banach proved an important theorem which is known as Banach contrac-tion principle on metric space. This principle has a great number of applicationsin fields such as mathematics, computer and economic. Thus, this result was ex-tended in many various ways. In this sence, Nadler proved a fixed point theorem forclosed and bounded multivalued mappings on metric spaces. Then, this result wasgeneralized by many authors by using Pompei Hausdorff metric(MT). However,Feng and Liu obtained a fixed point result for closed multivalued mapping with-out Pompei Hausdorf metric. After that, taking into account MT functions, Klimand Wardowski extended a fixed point result of Feng and Liu. On the other hand,Mathews introduced the concept of partial metric spaces and proved a fundamentalfixed point theorem on these spaces. After, many authors studied fixed point theoryboth single valued and multivalued mappings on partial metric spaces. Recently,Asadi et al. give a new definiton of M-metric spaces to generalize the concept ofpartial metric spaces. Moreover, they proved the M-metric version of the Banachcontraction principle. Then, Hakan et al. obtained fixed point results for multival-ued mappings of Feng-Liu type on M-metric spaces. In this paper, we prove somefixed point results for multivalued mappings of Klim-Wardowski type on M-metricspaces. Hence, our result generalizes a number of fixed point result for multivaluedmappings in the literature. Finally, we present illustrative example to support ourresults.

Keywords Fixed point ·Multivalued mappings ·M-metric spaces

References

[1] Altun, I., Sahin, H., and Turkoglu, D. (2018). Fixed point results for multivaluedmappings of Feng-Liu type on M-metric spaces. J. Nonlinear Funct. Anal, 7,2018.

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[2] M. Asadi, E. Karapınar, and P. Salimi, New extension of p-metric spaces withsome fixed point results on M-metric spaces, J. Inequal. Appl. 2014 (2014),Article ID 18.

[3] Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappingsand multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006), 103-112.

[4] Klim, D., and Wardowski, D. (2007). Fixed point theorems for set-valued con-tractions in complete metric spaces. Journal of Mathematical Analysis and Ap-plications, 334(1), 132-139.

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K-ORDER FIBONACCI QUATERNIONS

Mustafa ASCI 1, Süleyman AYDINYÜZ2

1,2Pamukkale University Science and Arts Faculty Department of Mathematics, Kinikli, Denizli, Turkey


ABSTRACT

In this paper we define and study another interesting generalization of Fibonacciquaternions is called k−order Fibonacci quaternions. Then we obtain for k = 2Fibonacci quaternions, for k = 3 Tribonacci quaternions and for k = 4 Tetranacciquaternions. We give generating function, the summation formula and some prop-erties about k−order Fibonacci quaternions. Also, we identify and prove the ma-trix representation for k−order Fibonacci quaternions. The Qk matrix given fork−order Fibonacci numbers is defined for k−order Fibonacci quaternions and af-ter the matrices with the k−order Fibonacci quaternions is obtained with help ofauxiliary matrices, important relationships and identities were established.

Keywords Fibonacci Numbers · k-Order Fibonacci Numbers · Quaternions ·Fibonacci Quaternions · k-order Fibonacci Quaternions · Matrix representations ·Q-matrix

References

[1] Koshy T., "Fibonacci and Lucas Numbers with Applications", A Wiley-Interscience Publication, (2001).

[2] Vajda S., "Fibonacci and Lucas Numbers and the Golden Section Theory andApplications", Ellis Harwood Limitted, (1989).

[3] Gould H. W., "A history of the Fibonacci Q-matrix and a Higher-DimensionalProblem", Fibonacci Quart. 19 (1981), no.3, 250-257.

[4] Hoggat V. E., "Fibonacci and Lucas Numbers", Houghton-Mifflin, Palo Alto(1969).

[5] Stakhov A. P., "A generalization of the Fibonacci Q-matrix", Rep. Natl. Acad.Sci. Ukraine 9 (1999), 46-49.

[6] Stakhov A. P., Massinggue V., Sluchenkov A., "Introduction into FibonacciCoding and Cryptography", Osnova, Kharkov (1999).

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[7] Er, M. C., "Sums of Fibonacci numbers by matrix methods.", Fibonacci Quart22.3 (1984): 204-207.

[8] Kilic E., Dursun T., "On the generalized order-k Fibonacci and Lucas numbers",Rocky Mountain J. Math., 36, no:6 (2006): 1915-1926.

[9] Lee G-Y., Lee S-G., Kim J-S., Shin H-K., "The Binet Formula and Representa-tions of k-Generalized Fibonacci Numbers", The Fibonacci Quart. (2001) 158-164.

[10] Lee G-Y., "k-Lucas numbers and Associated bipartite graphs", Linmear Alge-bra and Appl. 320.1 (2000): 51-61.

[11] Lee G-Y., Lee S-G., " A note on generalized Fibonacci numbers", The Fi-bonacci Quart. 33 (1995) 273-8.

[12] Gürlebeck K., Sprössig W., "Quaternionic and Clifford Calculus for Physicistsand Engineers", Wiley, New York (1997).

[13] Hamilton W. R., "Elements of Quaternions Longmans", Green and Co., Lon-don, (1866).

[14] Horadam A. F., "Complex fibonacci Numbers and Fibonacci Quaternions",American Math. Monthly 70 (1963) 289-291.

[15] Horadam A. F., "Quaternion Recurrence Relations", Ulam Quaterly 2, (1993)23-33.

[16] Iyer M. R., "A note on Fibonacci Quaternions", The Fibonacci Quart. 3, (1969)225-229.

[17] Halici S., "On Fibonacci Quaternions", Adv. Appl. Clifford Algebras 22,(2012) 321-327.

[18] Cimen B. C., Ipek A., "On Pell Quaternions and Pell-Lucas Quaternions",Adv. Appl. Clifford Algebras 26(1), (2016) 39-51.

[19] Szynal-Liana A., Wloch I., "A note on Jacobsthal Quaternions", Adv. Appl.Clifford Algebras 26 (2016), 441-447.

[20] Gamaliel C-M., "On a Generalization for Tribonacci Quaternions", Mediterr.J. Math. (2017) 14:239.

[21] Keçilioglu, O., Akkus I., "The Fibonacci Octanions", Adv. Appl. Clifford Al-gebras 25 (2015), 151-158.

[22] Polatli E., Kızılates E., Kesim S., "On split k-Fibonacci and k-Lucas quater-nions", Adv. Appl. Clifford Algebras 26(1), (2015), 37-52.

[23] Tasci D., Yalcin F., "Fibonacci p-quaternions", Adv. Appl. Clifford Algebras25,(2015), 245-254.

[24] Aydın T. F., Koklu K., Yuce S., "Generalized dual Pell Quaternions", Notes onNumber Theory and Discrete Mathematics, 23(4), (2017), 66-84.

[25] Tasci D., " Padovan and Pell-Padovan Quaternions", Journal of Science andArts, No. I(42), (2018), 125-132.

135


RIGID BODY MOTION OF AN ELASTIC RECTANGLEUNDER THE EFFECT OF SLIDING BOUNDARY

CONDITION

Onur SAHIN

Department of Mathematics, Giresun University, Giresun, Turkey


ABSTRACT

Dynamics of elastic structures are of interest for numerous studies in modern indus-trial applications, see [1]. One of the important points of this topic is developing themathematical models which reveal the eigenfrequencies of the system from the tra-ditional equations of rigid body dynamics, see [2]. However, the exact formulationof the eigenfrequencies and corresponding eigenforms of such problems are com-monly given by lengthy transcendental relations. A considerable number of studiesin recent years, therefore, have focused on developing perturbation approaches al-lowing further insight into the dynamic response of considered elastic structures,e.g. [3] and [4]. It is well known that the conventional equations of rigid bodymotion are an application of Newtonian mechanics to elastic solids. We mention[5] which suggests a new methodology leading a better approximation to Newton’ssecond law of motion for rigid macroscopic bodies. A distinguished elastodynamichomogenization theory for periodic and random inhomogeneous media was pre-sented in [6]-[9], including effective constitutive relations which are non-local inspace and time.In this work, the in-plane dynamic problem for an elastic rectangle with the slidingboundary condition in case of low-frequency motion related to a typical time scaleis considered. The governing equations and the boundary conditions are given interms of the scaled variables and a perturbation scheme is derived by expandingthe stresses and displacements in asymptotic series through the associated smallparameter related to low frequency. As might be expected the leading order termsin these expansions correspond to rigid boy motions. It should also be emphasizedthat the asymptotic study of boundary-value problems usually agrees with the Saint-Venant principle which states that statically self-equilibrated loads cause only localdisturbances that do not propagate far away from a loaded area, see [2] and [10].At the next order of the asymptotic expansion, we have a boundary value problem

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for a system of partial differential equations. The aim of further study is to find thesolution to the next order problem and present the asymptotic solution of the elasticrectangle.

Keywords Low-frequency · Perturbation · Elastic rectangle · Rigid body motion

References

[1] Kaplunov J.D., Kossovitch, L. Y., Nolde, E. V., Dynamics of thin walled elasticbodies, Academic Press, 1998.

[2] Babenkova E. and Kaplunov J., Low–frequency decay conditions for a semi-infinite elastic strip, Proceedings of the Royal Society of London. Series A:Mathematical, Physical and Engineering Sciences, 460(2048): 2153–2169,2004.

[3] Kaplunov J., Prikazchikov D.A., Prikazchikova L.A., Sergushova O., The low-est vibration spectra of multi-component structures with contrast material prop-erties, Journal of Sound and Vibration, 445, 132–147, 2019.

[4] Sahin O., The effect of boundary conditions on the lowest vibration modes ofstrongly inhomogeneous beams, Journal of Mechanics of Materials and Struc-tures, 14(4): 569–585, 2019.

[5] Milton G.W., and Willis J.R., On modifications of Newton’s second law andlinear continuum elastodynamics, Proceedings of the Royal Society A: Mathe-matical, Physical and Engineering Sciences, 463(2079): 855–880, 2007.

[6] Willis J., Properties of composites, Advances in applied mechanics, 21, 1, 1982.[7] Willis J.R., Variational principles for dynamic problems for inhomogeneous

elastic media, Wave Motion, 3(1): 1–11, 1981.[8] Capdeville Y., Guillot L. and Marigo,J.J., 1-D non-periodic homogenization for

the seismic wave equation, Geophysical Journal International, 181(2): 897–910, 2010.

[9] Nassar H., He Q.C. and Auffray N., Willis elastodynamic homogenizationtheory revisited for periodic media. Journal of the Mechanics and Physics ofSolids, 77, 158–178, 2015.

[10] Babenkova Y.V., Kaplunov Y.D., Ustinov Y.A., Saint-venant’s principle in thecase of the low-frequency oscillations of a half-strip, Journal of applied mathe-matics and mechanics, 69(3): 405–416, 2005

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STABILITY ANALYSIS AND TOPOLOGICALCLASSIFICATION OF FIXED POINTS OF

DISCRETE-TIME PREY-PREDATOR MODEL

Özgür DEMIR1, Figen KANGALGIL2

1Sivas Cumhuriyet University, Faculty of Education, Department of Mathematics and Science Education, 58140, Sivas, Turkey2Dokuz Eylul University, Bergama Vocational High School, 35700, Izmir, Turkey


ABSTRACT

Prey-predator interaction has always been important issue in mathematical mod-elling of ecological processes. Lotka-Volterra model is the simplest model ofpredator-prey interactions. The model was developed independently by Lotka(1925) and Volterra (1926) [1,2]. The Lotka-Volterra equations, also known asthe predator-prey equations, are a pair of first-order nonlinear differential equa-tions, frequently used to describe the dynamics of biological systems in which twospecies interact, one as a predator and the other as prey. The model of Lotka andVolterra is not very realistic. It does not consider any competition among prey orpredators. As a result, prey population may grow infinitely without any resourcelimits. Predators have no saturation: their consumption rate is unlimited. The rateof prey consumption is proportional to prey density. Thus, it is not surprising thatmodel behavior is unnatural showing no asymptotic stability. However numerousmodifications of this model exist which make it more realistic. One of these modi-fications is to introduce the Allee effect [3,4,5]. In 1931, an ecologist Warder ClydeAllee described that per capita growth rate of prey is negative or an increasing func-tion at low population density. This type of mechanism is known as Allee effect.The Allee effect is a nonlinear phenomenon exhibited in the population dynamicsof sparse populations in which the per capita population growth rate increases withincreasing population density. In sufficiently sparse populations, the Allee effectmay lead to extinction and is known to generate a threshold in the probability of es-tablishment when presented as a function of introduced population size or density.Recently, many researchers have paid attention to study the dynamics of discretetime prey-predator models with Allee effect.In this study, we consider discrete-time prey-predator model with Allee effect. Weshow existence of the fixed point of the presented discrete-time model. We analyze

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the topological classifications of these fixed points of the model and investigatestability analyses of the all fixed points. We give numerical simulations to supportobtained theoretical finding.

Keywords Allee effect · Stability · Fixed Point

References

[1] Lotka, AJ., Elements of Mathematical Biology, Williams & Wilkins, Baltimore(1925).

[2] Volterra, V., Variazioni e fluttuazioni del numero di’ individui in specie animaliconviventi. Mem. R. Accad. Naz. Dei Lincei, Ser. VI 2, 31-113 (1926).

[3] Lai, X., Liu, S., Lin, R., "Rich Dynamical Behaviours for Predator-Prey Modelwith Weak Allee", Applicable Analysis, 89 (8) 1271-1292 (2010).

[4] Kangalgil, F., "Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey", Advances in DifferenceEquations, 2019 (92) (2019).

[5] Feng, P. and Kang, Y., "Dynamics of a modified Leslie-Gower model with dou-ble Allee effects", Nonlinear Dyn , 80: 1051-1062 (2015).

139


THE ROADS WE CHOOSE: CEITICAL THINKING EXPERIENCESIN MATHEMATICAL COURSES

Paula CATARINO1, Maria M. NASCIMENTO1, Eva MORAIS1, Paulo VASCO1

1Department of Mathematics, University of Trás-os-Montes e Alto Douro, Vila Real, Portugal


ABSTRACT

The constant technological development has changed profoundly the way we inter-act with each other, think and work, so the skills required by employers for new pro-fessionals graduated from universities also changed. This new paradigm requireseducators, especially higher education teachers, to improve teaching and learningmethodologies in order to promote critical thinking for solutions to complex anddecision-making problems. Education should also take into account communica-tion and collaboration, as well as the ability to master new technologies, avoidingthe risks of their misuse. Along with them, skills such as critical thinking, creativ-ity, originality, initiative, persuasion and negotiation should also be considered intraining and they are in line with European Union (2018) documents along withthe need to support the development of key skills in the acquisition of science,technology, engineering and mathematics (STEM) skills, taking into account theirlinks with the arts, creativity and innovation and motivating young people to en-gage in STEM(/STEAM) careers. According to the report of the World EconomicForum (2018), politicians, educators and workers’ representatives will gain if thereis a deeper understanding of the labour market, as well as preparation for the on-going changes. As teachers of higher education in science and technology andin particular in engineering, and as participants in the Erasmus+ CRITHINKEDUproject (based on the knowledge and experience of European Higher Education In-stitutions, Enterprises and Non-Governmental Organisations), we believe that thereshould exist a constant concern to improve the quality of learning in higher edu-cation institutions in order to address the need for well-trained and well-educatedcitizens with skills tailored to labour market requirements. The present work arisesfrom this concern in the Mathematics field and from the changes that the authorsexperienced fostered by the teacher-training course in Education for Critical Think-ing of the Erasmus+ CRITHINKEDU project. This is a qualitative work, and wepresent a description of three different courses that were altered after the trainingcourse. The changes are presented at the level of planning, teaching and learning

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strategies used, as well as the learning assessment. Despite the introduced alter-ations, we know that we still have to continue the path started with training, as thereare still aspects to improve to achieve all the aspects of the CRITHINKEDU projectframework, in the teaching practices described, namely those that cover the criticalthinking assessment.

Keywords Mathematics · Critical thinking · Teaching methodology

References

[1] Cruz, G., Nascimento, M., Payan, R., Dominguez, C., Silva, H., and Morais, F.,From labor market needs to critical thinking across the European High Educa-tion Curricula, EAIR 39th Annual Forum, Porto, Portugal, 2017.

[2] Dominguez et al., Adding value to the learning process by online peer reviewactivities: towards the elaboration of a methodology to promote critical thinkingin future engineers, European J. of Engineering Ed., 40(5): 573-591, 2015.

[3] Dominguez, C. (coord.), A European collection of the Critical Thinking skillsand dispositions needed in different professional fields for the 21st century, VilaReal, Portugal, 2018.

[4] Dominguez, C. (coord.), The CRITHINKEDU European course on criticalthinking education for university teachers: from conception to delivery, VilaReal, Portugal, 2018.

[5] Dominguez, C. (coord.), A European review on Critical Thinking educationalpractices in Higher Education Institutions, Vila Real, Portugal, 2018.

[6] Silva, H., Lopes, J., Dominguez, C., Payan-Carreira, R., Morais, E., Nasci-mento, M., and Morais, F., Fostering critical thinking through peer review be-tween cooperative learning groups, Rev. Lusófona de Educação, 32: 31-45,2016.

[7] van Merriënboer, J., Clark, R., and Croock, M. de., Blueprints for ComplexLearning: The 4C/ID-Model, ETR&D, 50(2): 39-64, 2002.

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A STUDY ON THE TANGENT BUNDLE OF RIEMANNIANMANIFOLD ENDOWED WITH THE BRONZE STRUCTURE

Rabia ÇAKAN AKPINAR

Kafkas University, Faculty of Arts and Sciences, Department of Mathematics, Turkey


ABSTRACT

In this work, the bronze Riemannian structures are determined. An integrabilitycondition for bronze Riemannian structure is investigated via the Tachibana opera-tor. An example for the tangent bundle of Riemannian manifold endowed with theSasaki metric and the bronze structure is studied.

Keywords Bronze structure · Pure tensor · Riemannian manifold · Tangent bundle ·Sasaki metric

References

[1] Gezer A., and Karaman Ç., On metallic Riemannian structures, Turk. J. Math.,39(6): 954-962, 2015.

[2] Pandey P.K., and Sameer, Bronze Differantial Geometry, Journal Science andArts, 4(45): 973-980, 2018.

[3] Salimov A.A., and Agca, F., Some Properties of Sasakian Metrics in CotangentBundles, Mediterr. J. Math., 8(2): 243-255, 2011.

[4] Spinadel V.W., The metallic means family and multifractal spectra, NonlinearAnal. Ser. B: Real World Appl., 36(6): 721-745, 1999.

[5] Tachibana S., Analytic tensor and its generalization, Tohoku Math. J., 12: 208-221, 1968.

[6] Yano K., and Ishihara S., Tangent and Cotangent Bundles, Marcel Dekker, NewYork, 1973.

142


ESTIMATION OF TURKEY AND WORLD POPULATION IN THEYEAR 2050 IN VIEW OF LOCAL AND NONLOCAL FRACTIONAL

DERIVATIVES

Ramazan OZARSLAN

Department of Mathematics, Firat University, Elazig, Turkey


ABSTRACT

In this work , the study aims to estimate Turkey and the world population moreaccurately in 2050 by using World Bank real population data. In view of local andnon-local fractional derivatives, which are conformable, Caputo, Caputo Katugam-pola, population growth model is evaluated. The results are analyzed by using leastsquares method and compared to the World Bank’s estimates for 2050.

Keywords Population growth · Local derivative · Fractional calculus ·Mathemati-cal Modeling

References

[1] Jarad, F., Abdeljawad, T. A modified Laplace transform for certain generalizedfractional operators. Results in Nonlinear Analysis. 1 (2) (2018) 88-98.

[2] Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definitionof fractional derivative. Journal of Computational and Applied Mathematics,264, 65-70.

[3] Abdeljawad, T. (2015). On conformable fractional calculus. Journal of compu-tational and Applied Mathematics, 279, 57-66.

[4] World Bank (accessed June 17, 2020).World Development Indicators,https://databank.worldbank.org/source/population-estimates-and-projections

[5] Katugampola, U. N. New approach to a generalized fractional integral. AppliedMathematics and Computation. 218 (3) (2011) 860-865.

[6] Katugampola, U. N. A new approach to generalized fractional derivatives. Bul-letin of Mathematical Analysis and Applications. 6 (4) (2014) 1-15.

[7] Bas, E., Acay, B., & Ozarslan, R. (2019). The price adjustment equation withdifferent types of conformable derivatives in market equilibrium.

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[8] Ozarslan, R., Bas, E., Baleanu, D., & Acay, B. (2020). Fractional physical prob-lems including wind-influenced projectile motion with Mittag-Leffler kernel.AIMS Mathematics, 5(1), 467.

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THE EFFECT OF CAPUTO FRACTIONAL DERIVATIVECONCERNING ANOTHER FUNCTION ON DEFLECTION OF

HORIZONTAL BEAMS

Ramazan OZARSLAN

Department of Mathematics, Firat University, Elazig, Turkey


ABSTRACT

This study investigates the effect of Caputo fractional derivative concerning anotherfunction on beam deflection under the vertical load for determining transverse de-flection. The standard model includes a fourth order differential model and is anal-ysed in this study in a broader way through the fractional case while the orderbetween 3 < α < 4 and the fractional derivative concerning another functions liketranscendental, rational functions, etc. analytically under various boundary condi-tions.

Keywords ψ−Caputo fractional derivative · Beams · Buckling load

References

[1] Rajasekaran, S., Gimena, L., Gonzaga, P., & Gimena, F. N. (2009). Solutionmethod for the classical beam theory using differential quadrature. StructuralEngineering and Mechanics, 33(6), 675-696.

[2] Yavari, A., Sarkani, S., & Moyer Jr, E. T. (2000). On applications of general-ized functions to beam bending problems. International Journal of Solids andStructures, 37(40), 5675-5705.

[3] Almeida, R. (2017). A Caputo fractional derivative of a function with respect toanother function. Communications in Nonlinear Science and Numerical Simu-lation, 44, 460-481.

[4] Almeida, R. (2017). What is the best fractional derivative to fit data?. Applica-ble Analysis and Discrete Mathematics, 11(2), 358-368.

[5] Osler, T. J. (1970). The fractional derivative of a composite function. SIAMJournal on Mathematical Analysis, 1(2), 288-293.

[6] Jarad, F., Abdeljawad, T. (2018). A modified Laplace transform for certain gen-eralized fractional operators. Results in Nonlinear Analysis, 1(2), 88-98.

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[7] Acay, B., Ozarslan, R., & Bas, E. (2020). Fractional physical models based onfalling body problem. AIMS Math, 5, 2608-2628.

[8] Ozarslan, R., Bas, E., Baleanu, D., & Acay, B. (2020). Fractional physical prob-lems including wind-influenced projectile motion with Mittag-Leffler kernel.AIMS Mathematics, 5(1), 467.

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ON LIGHTLIKE SUBMERSIONS

Ramazan SARI

Amasya University, Turkey


ABSTRACT

We introduce semi-slant lightlike submersions from indefinite Hermitian manifoldonto a lightlike manifold. We obtain characterizations; investigate the geometryof foliations which arise from the definition of this new submersion. After weinvestigate the geometry of foliations, we obtain necessary and sufficient conditionfor base manifold to be a locally product manifold and proving new conditions tobe totally geodesicness, respectively.

Keywords Lightlike submersions · Slant lightlike submersions · Lightlikemanifolds

References

[1] M.A. Akyol, Conformal semi-slant submersions. Int. J. Geom. Methods Mod.Phys., 14(7), 1750114, 2017.

[2] M. Falcitelli, S. Ianus, and A. M. Pastore, Riemannian Submersions and RelatedTopics, World Scientific, Singapore, 2004.

[3] B. O’Neill, “The fundamental equations of a submersion,” Mich. Math. J., 13,459–469, 1966.

[4] K.S. Park, R. Prasad, Semi-slant submersions. Bull. Korean Math. Soc. 50(3),951–962, 2013.

[5] B. Sahin, “On a submersion between Reinhart lightlike manifolds and semi-Riemannian manifolds,” Mediterr. J. Math., 5, 273–284, 2008.

[6] B. Sahin and Y. Gündüzalp, “Submersion from semi-Riemannian manifoldsonto lightlike manifolds,” Hacet. J. Math. Stat., 39,41–53, 2010.

[7] B. Sahin, “Slant submersions from almost Hermitian manifolds,” Bull. Math.Soc. Sci. Math. Roumanie, 54, No. 102, 93–105 (2011).

147


MMR ENCRYPTION ALGORITHM AS AN ALTERNATIVEENCRYPTION ALGORITHM TO RSA ENCRYPTION ALGORITHM

Remzi AKTAY

Keçiören Sehit Halil Isılar Middle School, Mathematics Teacher, Turkey


ABSTRACT

In this study we aimed to try to develop an alternative algorithm with the purposeof (key encryption algorithm) RSA’s current disadvantages and possible problemswhen we switch to the future quantum computers. Because of the usage methodof the RSA, the values during the key making process are limited. Owing to thisfeature of this method, may side channel attacks have been made and some of themhave had good results. Especially at the attack of Shor algorithm and Acousticcrypto analysis; Shamir and his team had good results in 2013. We also know atthe error analysis attack and Bellcore attack in 2010. Firstly in the algorithm thathas planned to be composed; the keys will take infinite values. And also besidesthe closed keys there will be observational keys which are worked only in thesealgorithms. In the event of the closed keys are found; these observational keys willstep in.While composing this algorithm at first, an equation has been made. This equationis a10(n−1) − b10(n−1) ≡ 0(mod10n). In this equation we started to the algorithm viamaking a and b as twin prime numbers. With this equation we composed the closedkeys by means of modular arithmetic rules and Euclid Algorithm. And we make aworking algorithm principle using Euler Theory, Euler Function and factorizing.The first difference of this algorithm from the open keys algorithm is the obser-vational keys. In the event that the open keys are stolen or broken, without theobservational the keys data and secret text cannot be reached. The second differ-ence is that the open key, closed key, and the observational key can have an infinitevalue. The third difference is this algorithm has its own character code chart. Andalso this algorithm is safer against the side channel attacks. While the algorithm’scontrol key changes, the closed and open key of the message sender also changes.Besides, the twin prime numbers of the equation can be changed to make an up-date. This algorithm can be used for secure e-mail, e-commerce, mobile bankingand secure data transfer.

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Keywords Encryption · closed key · Observation key · Euclid Algorithm · Euler’sTheorem · Side Channel Attacks · Open Key encryption

References

[1] Akben, B., Subası, A., “RSA ve Eliptik Egri Algoritmasının Karsılastırılması”,Fen ve Mühendislik Fakültesi Dergisi, Kahramanmaras, s. 36-40, 2005.

[2] Bulus, E., Yerlikaya, T., Bulus, N., “Asimetrik Sifreleme AlgoritmalarındaAnahtar Degisim Sistemleri”, Kütahya: Dumlupınar Üniversitesi s. 1-2, 2007.

[3] Akçam, N., “RSA Algoritması Kullanılarak Hata Düzeltme ve Dijital ImzalıProtokol Gelistirme”, Ankara: Politeknik Dergisi, s. 41-51, 2007.

[4] Levi, A., Özcan, M., “Açık Anahtar Tabanlı Sifreleme Neden Zordur?”, Is-tanbul: Sabancı Üniversitesi Mühendislik ve Doga Bilimleri Fakültesi s. 1-6 ,2010.

[5] Kodaz, H., Botsalı, F., “Simetrik ve Asimetrik Sifreleme AlgoritmalarınınKarsılastırılması”, Konya: Selçuk Teknik Dergisi, s. 10-23, 2010.

[6] Kartal , S., “Açık Anahtarlı Kriptografi” https://www.savaskartal.com ›2010/04/13 › acik-anahtarli-kriptografi, 2010.

[7] Beskirli, A., Beskirli, M., Özdemir, D., “Sifreleme Yöntemleri ve RSA Algo-ritması Üzerine Bir Inceleme”, Istanbul: Avrupa Bilim ve Teknoloji Dergisi s.284-291, 2019.

[8] Hatun, E., “Raspberry Pı Üzerine Gerçeklenmis RSA Algoritmasına Yan KanalAnalizi”, Istanbul Teknik Üniversitesi Fen Bilimleri Enstitüsü Yüksek LisansTezi, s. 11-24, 2018.

[9] Yerlikaya, T., Bulus, E., Bulus, H. N., “RSA Sifreleme Algoritmasının Pol-lard RHO Yöntemi Ile Kriptanalizi” Akademik Bilisim Konferansı, 31 Ocak-2Subat 2007, Trakya Üniversitesi Dergisi, Edirne, 2007.

149


RESULTS ON SET-VALUED PREŠIC TYPE MAPPINGS

Seher Sultan YESILKAYA1, Cafer AYDIN2

1Institute of Science and Technology, Kahramanmaras Sütçü Imam University, Kahramanmaras, 46040, Turkey2Department of Mathematics, Kahramanmaras Sütçü Imam University, Kahramanmaras, 46040, Turkey


ABSTRACT

Fixed point theory concern itself with a very basic mathematical setting. It is alsowell known that one of the fundamental and most useful results in fixed point theoryis Banach fixed point theorem. In 1922 Banach [1], introduced the famous fixedpoint theorem, also known as the Banach contraction principle. Throughout theyears, several extensions and generalizations of this principle have appeared in theliterature. Furthermore, among the different generalization of Banach contractionprinciple, Prešic [2] in 1965 gave a contractive condition on finite product of metricspaces and proved a fixed point theorem. Also Ciric and Prešic [3], Abbas et al. [4],Shukla and Sen [5] and Yesilkaya et al.[6] extended and generalized these results.On the other hand, there are several applications of Prešic type contractions, e.g.,in the convergence of sequences in solving the nonlinear difference equations andthe nonlinear inclusion problems in the convergence problems of nonlinear matrixdifference equations etc.. Set-valued analysis is an important extension of the gen-eral concepts studied in mathematical analysis. Fixed point theory for set-valuedoperators is an important topic of set-valued analysis. Nadler [7] extended the Ba-nach contraction principle to set-valued mappings by using the Hausdorff metric.Inspired by the results of Nadler, the fixed point theory of set-valued contraction us-ing this Hausdorff metric has been further developed in different directions by manyauthors. Recently, Wardowski [8] has introduced a new concept of contraction, andproved a fixed point theorem which generalizes the Banach contraction principle ina different direction than in the known results from the literature in complete metricspaces.In this work, we introduce a new concept of contraction called set-valued Prešictype F -contraction mapping on metric spaces and prove some fixed point resultsfor a mapping M : Xr → CB(X). These results extend the main results of manycomparable results from the current literature. Also, we provide example showingthat our main theorem are applicable.

Keywords Fixed point · metric spaces · Prešic type contraction

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References

[1] Banach S., Sur les operations dans les ensembles abstracits et leur applicationaux equations integrales. Fund. Math., 3: 133-181, 1922.

[2] Prešic S.B., Sur une classe d’in equations aux difference finite et. sur la conver-gence de certains suites. Publ de L’Inst Math. 5: 75-78, 1965.

[3] Ciric L.B., Prešic S.B., On Prešic type generalization of the Banach contractıonmapping principle. Acta Math Univ Comenianae 76: 143-147, 2007.

[4] Abbas M., Berzig M., Nazır T., Karapınar E., Iterative approximation of fixedpoints for Prešic type F-Contraction operators. UPB Sci Bull Series A. 78:1223-7027, 2016.

[5] Shukla, S., Sen, R., Set-valued Prešic-Reich type mappings in metric spaces.Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. doi:10.1007/s13398-012-0114-2, 2012.

[6] Yesilkaya S.S., Aydın C., Eroglu A., Fixed Point Results on Ordered PrešicType Mappings. AIMS Mathematics, doi:10.3934/math.2020330, 5(5): 5140-5156, 2020.

[7] Nadler, S.B. Multivalued contraction mappings. Pasific J. Math., 30,2: 475-488,1969.

[8] Wardowski D., Fixed Point of a new type of contactive mappings in completemetric spaces. Fixed Point Theory and Applications, 94, 2012.

151


NEW RESULTS ON EXPANSIVE MAPPINGS

Seher Sultan YESILKAYA1, Cafer AYDIN2

1Institute of Science and Technology, Kahramanmaras Sütçü Imam University, Kahramanmaras, 46040, Turkey2Department of Mathematics, Kahramanmaras Sütçü Imam University, Kahramanmaras, 46040, Turkey


ABSTRACT

The study of expansive mappings is a very interesting research area in fixed pointtheory. Wang, et al.[1] proved some fixed point theorems for expansion mappings,which correspond to some contractive mappings in metric spaces. Rhoades [2] andTaniguchi [3] generalized the results of Wang for pair of mappings. Thereafter,several authors obtained many fixed point theorems for expansive mappings. Formore details see [4, 5]. Fixed point theory in metric spaces is an important branchof nonlinear analysis, which is closely related to the existence and uniqueness ofsolutions of differential and integral equations. There are many generalizations ofthe concept of metric spaces in the literature. Ran and Reurings [6] proved a fixedpoint theorem on a partially ordered metric space. Then several authors consideredthe problem of existence (and uniqueness) of a fixed point for contraction typeoperators on partially ordered sets. Recently, Jleli and Samet [7] introduced a newtype of contraction which is called the θ-contraction and established some new fixedpoint theorems for such a contraction in the context of generalized metric spaces.The purpose of this study is to present some fixed point results for self-generalizedcontractions in partially ordered metric spaces. Also, we introduce a new ap-proach to expansion mappings in fixed point theory. We introduce the conceptof θ-expansive mappings in metric spaces and prove fixed point theorems for suchmappings. Following, we provide examples to show the significance of such map-pings. Secondly, we give some fixed point results for θ-expansive mappings espe-cially on a partially ordered metric spaces. These results extend the main resultsof many comparable results from the current literature. Finally, some examples arepresent to support the new theorems and results. Also, these examples show thenew class of θ-expanding mapping is not included in expanding mapping known inliterature.

Keywords Fixed point ·Metric spaces · Expansion mappings

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References

[1] Wang S. Z., Li B.Y., Gao Z. M., and Iseki K., Some fixed point theorems onexpansion mappings, Mathematica Japonica, 29: 631-636, 1984.

[2] Rhoades B. E., Some fixed point theorems for pairs of mappings, Jnanabha, 15:151-156, 1985.

[3] Taniguchi T., Common fixed point theorems on expansion type mappings oncomplete metric spaces, Mathematica Japonica, 34: 139-142, 1989.

[4] Khan M.A., Khan M.S., Sessa S., Some theorems on expansion mappings andtheir fixed points. Demonstr. Math., 19: 673-683, 1986.

[5] Daffer P. Z., Kaneko H., On expansive mappings, Math. Japonica. 37: 733-735,1992.

[6] Ran A.C.M., Reurings M.C.B., A fixed point theorem in partially ordered setsand some application to matrix equations, Proc Amer Math Soc., 132: 1435-1443, 2004.

[7] Jleli M., Samet B., A new generalization of the Banach contraction principle, J.Inequal. Appl., 38: 2014.

153


ON A TYPE-2 FUZZY APPROACH TO SOLUTION OF SECONDORDER INITIAL VALUE PROBLEM

Selami BAYEG1, Raziye MERT1, Tahir KHANIYEV2, Ömer AKIN2

1Turkish Aeronautical Association University2TOBB University of Economics and Technology


ABSTRACT

Type-2 fuzzy sets (T2FS) were firstly introduced by Zadeh in 1975 as a naturalextension of fuzzy sets [1]. A T2FS is defined by a fuzzy membership function,the membership of which is a fuzzy set in the unit interval [0, 1] rather than a pointin [0, 1]. Hence type-2 fuzzy sets are helpful in models where the exact form ofmembership functions cannot be determined or where membership functions arethemselves imprecise in nature [2]. Practically, type-2 fuzzy sets are complex bynature and their logic is characterised by expensive computational cost. Many rep-resentations are developed to overcome this obstacle, and still the applicability oftype-2 fuzzy sets is very limited in applications. A differential equation with type-2fuzzy initial values is called a type-2 fuzzy initial value problem (T2FIVP). Al-though there are many studies on fuzzy initial value problems, there are only a fewstudies on type-2 fuzzy initial value problems [3, 4, 5].

In our work we have developed an algorithm to find the solutions of the secondorder linear type-2 fuzzy initial value problems by using the (α, β)-cut representa-tion of type-2 fuzzy numbers (T2FNs) given in [6]. These type-2 fuzzy solutionsare firstly constructed from the solution of the crisp problem using the Zadeh’s ex-tension principle. Then type-2 fuzzy solutions are obtained using the (α, β)-cutrepresentations of T2FNs. In this representation, type-2 fuzzy solutions are repre-sented by lower and upper solutions such that each of these solutions has anotherlower and upper parts corresponding to (α, β)-cut solutions. To prevent the switch-ing points of these lower and upper solutions we have used Heaviside step function.Since the second order linear differential equations have a variety of applicationssuch as the motion of a particle, the vibration of a spring and the flow of electronsin an electric circuit, as an application of the method we have applied the method tomechanical vibrations and RLC-circuits as applications

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Keywords Type-2 fuzzy sets · Second order type-2 fuzzy differential equations ·Zadeh’s extension principle

References

[1] Zadeh L. A., The concept of a linguistic variable and its application to approxi-mate reasoning, Learning systems and intelligent robots. Springer, Boston, MA,1-10, 1974.

[2] Mizumoto M., Kokichi T., Fuzzy sets and type 2 under algebraic product andalgebraic sum, Fuzzy Sets and Systems 5(3), 277-290, 1981.

[3] Bandyopadhyay A., Samarjit K., System of type-2 fuzzy differential equationsand its applications, Neural Computing and Applications 31(9), 5563-5593,2019.

[4] Bandyopadhyay A., Samarjit K., On type-2 fuzzy partial differential equationsand its applications, Journal of Intelligent Fuzzy Systems 34(1), 405-422, 2018.

[5] Kardan I., Akbarzadeh M., Akbarzadeh A., Kalani H., Quasi type 2 fuzzy dif-ferential equations, Journal of Intelligent Fuzzy Systems 32(1), 551-563, 2017.

[6] Khaniyev T., Baskır M. B., Gokpınar F., Mırzayev F., Statistical distributionsand reliability functions with type-2 fuzzy parameter, Eksploatacja ı nıeza-wodnosc 21(2), 268, 2019.

155


EXISTENCE AND UNIQUENESS RESULTS FOR A NONLINEARFRACTIONAL DIFFERENTIAL EQUATIONS OF ORDER σ ∈ (1, 2)

Sinan Serkan BILGICI1, Müfit SAN2

1,2Department of Mathematics, Faculty of Science, Çankırı Karatekin University, Tr-18100, Çankırı, Turkey


ABSTRACT

The importance of existence and uniqueness theorems in the study of initial valueproblems is unquestionable because, without them, one cannot understand modelledsystems correctly and cannot make predictions about the behaviour of them. Re-cently, with the popularity of fractional derivative operators, the equations involvingthese operators have begun to be studied in detail (See, [1]-[3]). In this research,we investigate the existence and uniqueness of solutions of an initial value prob-lem involving a nonlinear fractional differential equation of order σ ∈ (1, 2) in thesense of Riemann-Liouville derivative. Unlike the previous studies (See,[2],[7]),initial values of the problem we considered will be taken as nonhomogenous. Asshown in [4]-[6], an initial value problem including a nonlinear fractional differen-tial equation of order σ ∈ (0, 1) has no continuous solution when the problem has anonhomogenous initial value and the right-hand side of the equation is continuouson [0, T ] × R. Since the similar problem arises in our investigation, we will revealsome appropriate necessary and sufficient conditions under which the problem hasat least one solution. Moreover, we will study on the uniqueness of the problemas well. The existence and uniqueness of solutions will be investigated in BanachspaceCβ([0, T ]),where 0 < β < 1 and T > 0, and it is endowed with the followingnorm

||ω||β = ||ω||∞ + ||Dβω||∞,where ‖.‖∞ stands for the supremum norm defined on the class of continuous func-tions and Dβ represents the Riemann-Liouville fractional derivative. Moreover,Schauder fixed point theorem and some well known theorems will be used to proveour claims.

Keywords Fractional differential equations · Existence theorem · Riemann-Liouville derivative · Initial-value problem.

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References

[1] Kilbas, A. A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and appli-cations of fractional differential equations (Vol. 204). Elsevier Science Limited.

[2] Lakshmikantham, V., & Vatsala, A. S. (2008). Basic theory of fractional differ-ential equations. Nonlinear Analysis: Theory, Methods & Applications, 69(8),2677-2682.

[3] Podlubny, I. (1998). Fractional differential equations: an introduction to frac-tional derivatives, fractional differential equations, to methods of their solutionand some of their applications. Elsevier.

[4] San, M., & Soltanov, K. N. (2015). The New Existence and Uniqueness Re-sults for Complex Nonlinear Fractional Differential Equation. arXiv preprintarXiv:1512.04780.

[5] San, M. (2018). Complex variable approach to the analysis of a fractional dif-ferential equation in the real line. Comptes Rendus Mathematique, 356(3), 293-300.

[6] Ugur, Sert., & Müfit, San. (2020) Some analysis on a fractional differentialequation with a right-hand side which has a discontinuity at zero. HacettepeJournal of Mathematics and Statistics, 1-8.

[7] Yoruk, F., Bhaskar, T. G., & Agarwal, R. P. (2013). New uniqueness results forfractional differential equations. Applicable Analysis, 92(2), 259-269.

157


ASSESSING STUDENTS’ KNOWLEDGE AND COMPETENCES INMATHEMATICS BY COMBINED TESTS USING MACHINE

LEARNING METHODS

Snezhana GOCHEVA-ILIEVA1, Hristina KULINA1, Atanas IVANOV1

1University of Plovdiv Paisii Hilendarski, Plovdiv, Bulgaria


ABSTRACT

A major part of any training process in education is the assessment of the knowledgeand skills achieved by students at a given stage. Traditional methods of teaching,learning and testing are still essentially related to the content of the relevant teach-ing material, according to university programs. In recent years, this model has beengradually replaced by many new methods, mostly computer-assisted. Requirementsare formed for a greater connection of the studied material with its possible appli-cations in practice, in the sense of the concept of competence. The aim of this workis to evaluate the student’s performance in mathematics using a competency-basedassessment approach. The assessment methodology applies eight well-specifiedmathematical competencies, developed in the literature, intended for engineeringhigher education [1,2]. In addition, some new assessment tools obtained as a resultof the RULES_MATH project, recently published in [3-5], are applied. This pa-per presents an example assessment test, which combines questions on elements ofmathematical analysis, numerical methods and applied statistics, as well as an addi-tional small practical project. The practical project requires students to apply a setof mathematical knowledge to formulate problems, solve them with or without spe-cialized software, and interpret the resulting solutions. The developed assessmenttest was used as a final exam in Applied mathematics with a course of first-yearstudents in the specialty of Business Information Technology at Plovdiv UniversityPaisii Hilendarski, Bulgaria. The accumulated empirical data are processed by us-ing powerful machine learning techniques, such as Classification and RegressionTrees (CART), capable to identify the influence of different factors and their im-portance on the aquired level of knowledge and competencies of the students. Thetest results show that the mastery of the content-based material during the schoolyear and the level of the mathematical competence, measured by the outcome ofthe small practical project, differ in favor of the formal training. This requires a

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change in teaching methods and strengthening the competency-oriented componentto increase the overall level of competencies in mathematics of students.

Keywords Mathematical competence · Assessment test · Student performance

References

[1] Alessio O., Differential geometry of intersection curves in of three implicit sur-faces, Comput. Aided Geom. Design, 26: 455-471, 2009.Niss M., Mathemati-cal competencies and the learning of mathematics: The Danish KOM project,Proceedings of the 3rd Mediterranean Conference on Mathematical Education,Athens, Greece, 3-5 January 2003, The Hellenic mathematical society, Athens,Greece, 115-124.

[2] Alpers B.A. et al., A framework for mathematics curricula in engineering ed-ucation: A report of the mathematics working group, European Society forEngineering Education (SEFI), Brussels, Belgium, 2013..

[3] Queiruga-Dios A. et al., Evaluating engineering competencies: A newparadigm, Proceedings of the 2018 IEEE Global Engineering Education Con-ference (EDUCON), 17-20 April 2018, Tenerife, Spain, IEEE, 2052-2055.

[4] Queiruga-Dios A. et al., Rules_Math: Establishing assessment standards, Ad-vances in Intelligent Systems and Computing, Springer, Cham, Switzerland,951: 235-244, 2020.

[5] Gocheva-Ilieva S. et al., Data mining for statistical evaluation of summative andcompetency-based assessments in mathematics, Advances in Intelligent Sys-tems and Computing, Springer, Cham, Switzerland, 951: 207-216, 2020.

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GENERALIZED K-ORDER FIBONACCI HYBRID AND LUCASHYBRID NUMBERS

Süleyman AYDINYÜZ 1, Mustafa ASCI 2



ABSTRACT

In this study, we define the generalized k−order Fibonacci and Lucas numbers.We also give some important result about the generalized k−order Fibonacci andLucas numbers with special choices. The aim of this study is to introduce a newsequence of numbers called the generalized k−order Fibonacci Hybrid and LucasHybrid numbers. Then we obtain for k = 2 the Horadam Hybrid numbers andwith the special choices usual Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal andJacobsthal-Lucas Hybrid numbers. For k = 3 and some special choices we get theTribonacci Hybrid numbers. We give the generating functions and some proper-ties about the generalized k−order Fibonacci Hybrid and Lucas Hybrid numbers.Also, we identify and prove the matrix representation for the generalized k−orderFibonacci Hybrid and Lucas Hybrid numbers. TheQk matrix given for k−order Fi-bonacci numbers is defined. Important relationships and identities are established.

Keywords Fibonacci and Lucas numbers · k-order Fibonacci and Lucas Numbers ·Generalized k-order Fibonacci and Lucas numbers · Hybrid numbers · Generalizedk-order Fibonacci Hybrid and Lucas Hybrid numbers · Matrix representations ·Q-matrix

References

[1] Gürlebeck K., Sprössig W., "Quaternionic and Clifford Calculus for Physicistsand Engineers", Wiley, New York (1997).

[2] Hamilton W. R., "Elements of Quaternions Longmans", Green and Co., Lon-don, (1866).

[3] Horadam A. F., "Complex Fibonacci Numbers and Fibonacci Quaternions",American Math. Monthly 70 (1963) 289-291.

[4] Horadam A. F., "Quaternion Recurrence Relations", Ulam Quaterly 2, (1993)23-33.

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[5] Iyer M. R., "A note on Fibonacci Quaternions", The Fibonacci Quart. 3, (1969)225-229.

[6] Szynal-Liana A., Wloch I., "The Pell Quaternions and the Pell Octanions", Adv.Appl. Clifford Algebras, 2016; 26(1): 435-40.

[7] Szynal-Liana A., Wloch I., "A note on Jacobsthal Quaternions", Adv. Appl.Clifford Algebras 26 (2016), 441-447.

[8] Cimen B. C., Ipek A., "On Pell Quaternions and Pell-Lucas Quaternions", Adv.Appl. Clifford Algebras 26(1), (2016) 39-51.

[9] Halici S., "On Fibonacci Quaternions", Adv. Appl. Clifford Algebras 22, (2012)321-327.

[10] Catarino P. A, "A note on h(x)-Fibonacci Quaternion polynomials", ChaosSolitons Fractals, 2015; 77: 1-5.

[11] Akkus I., Kizilaslan G., "Quaternions: quantum calculus approach with appli-cations", Kuwait J Sci 2019. In press.

[12] Keçilioglu, O., Akkus I., "The Fibonacci Octanions", Adv. Appl. Clifford Al-gebras 25 (2015), 151-158.

[13] Er, M. C., "Sums of Fibonacci numbers by matrix methods.", Fibonacci Quart22.3 (1984): 204-207.

[14] Kilic E., Dursun T., "On the generalized order-k Fibonacci and Lucas num-bers", Rocky Mountain J. Math., 36, no:6 (2006): 1915-1926.

[15] Lee G-Y., Lee S-G., Kim J-S., Shin H-K., "The Binet Formula and Repre-sentations of k-Generalized Fibonacci Numbers", The Fibonacci Quart. (2001)158-164.

[16] Lee G-Y., "k-Lucas numbers and Associated bipartite graphs", Linmear Alge-bra and Appl. 320.1 (2000): 51-61.

[17] Lee G-Y., Lee S-G., " A note on generalized Fibonacci numbers", The Fi-bonacci Quart. 33 (1995) 273-8.

[18] Özdemir M., "Introduction to the Hybrid numbers", Adv. Appl. Clifford Alge-bras, 2018; 28:11. doi:10.1007/s00006-018-0833-3.

[19] Szynal-Liana A., "The Horadam Hybrid numbers", Discuss Math Gen AlgebraAppl., 2018;38(1):91-8.

[20] Szynal-Liana A., Wloch I., "On Jacobsthal and Jacobsthal-Lucas Hybrid num-bers", Ann Math Sil 2018. doi:10.2478/amsil-2018-0009.

[21] Catarino P., " On k-Pell Hybrid numbers", J Discrete Math Sci Cryptography,2019;22(1):83-9.

[22] Kızılates C., "A new generalization of Fibonacci Hybrid and Lucas Hybridnumbers", Chaos, Solitons and Fractals, 130 (2020), 109449.

[23] Tasyurdu Y., "Tribonacci and Tribonacci-Lucas Hybrid numbers", Int. Journalof Contemporary Math. Sci., Vol. 14, 2019, no. 4, 245-254.

[24] Lee G. Y., Kim J. S., " The linear algebra of the k-Fibonacci matrix", LinearAlgebra Appl., 373: 75-87 (2003).

[25] Asci M., Tasci D., Tuglu N., "Generalized F-nomial Matrix and Factoriza-tions", Ars Combin. 108, (2013), 81-95.

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[26] Tasci D., Tuglu N., Asci M., "On Fibo-Pascal Matrix involving k-Fibonacciand k-Pell matrices", Arab. J. Sci. Eng. 36, (2011), No. 6, 1031-1037.

[27] Asci M., Tasci D., El-Mikkawy M., "On Determinants and Permanents of k-Tridiaagonal Toeplitz Matrices", Util. Math., 89, (2012), 97-106.


[29] Stakhov A. P., Rozin B., " On a new class of Hyperbolic Functions", Chaos,Solitons and Fractals, 23:(2), 379-389, (2005).

[30] Stakhov A. P., Rozin B., " The Golden Shofar", Chaos, Solitons and Fractals,26:(3), 677-684, (2005).

[31] Stakhov A. P., Rozin B., " The continuous functions for the Fibonacci andLucas p-numbers", Chaos, Solitons and Fractals, 28:(4), 1014-1025, (2006).

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K-ORDER GAUSSIAN FIBONACCI POLYNOMIALS ANDAPPLICATIONS TO THE CODING/DECODING THEORY

Süleyman AYDINYÜZ 1, Mustafa ASCI 2



ABSTRACT

In this paper we define k-order Gaussian Fibonacci polynomials with boundaryconditions and give the generating function, explicit formula and some identities fork-order Gaussian Fibonacci polynomials. We introduce the matrix representationand we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a newcoding theory called k-order Gaussian Fibonacci Polynomials coding theory andestablish the code elements for values of k. This coding/decoding method boundto the Qk(x), Rk(x) and Ek,n(x) matrices. So, this method is different from theclassical algebraic coding. Consequently, with this method, we move the codingtheory onto a complex space which is a different field. Therefore, new areas arecreated.

Keywords Fibonacci Numbers · Gaussian Fibonacci numbers · k-order Gaus-sian Fibonacci numbers · Fibonacci polynomials · k-order Gaussian FibonacciPolynomials · Coding/decoding method · k-order Gaussian Fibonacci polynomialcoding/decoding theory

References

[1] Koshy T., "Fibonacci and Lucas Numbers with Applications", A Wiley-Interscience Publication, (2001).

[2] Vajda S., "Fibonacci and Lucas Numbers and the Golden Section Theory andApplications", Ellis Harwood Limitted, (1989).

[3] Gould H. W., "A history of the Fibonacci Q-matrix and a Higher-DimensionalProblem", Fibonacci Quart. 19 (1981), no. 3, 250-257.

[4] Hoggat V. E., "Fibonacci and Lucas Numbers", Houghton-Mifflin, Palo Alto(1969).


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[6] Stakhov A. P., Massinggue V., Sluchenkov A., "Introduction into FibonacciCoding and Cryptography", Osnova, Kharkov (1999).

[7] Horadam A. F., "A Generalized Fibonacci Sequence", American Math. Monthly68 (1961), 455-459.

[8] Horadam A. F., "Complex Fibonacci Numbers and Fibonacci Quaternions",American Math Monthly 70 (1963), 289-291.

[9] Jordan J. H., "Gaussian Fibonacci and Lucas numbers", Fibonacci Quart. 3(1965), 315-318.

[10] Gurel E., Asci M., "Some Properties of k-Order Gaussian Fibonacci and LucasNumbers", Ars Combin., 135 (2017), 345-356.

[11] Horadam A. F., Horadam E. M., "Roots of Recurrence-Generated Polynomi-als", Fibonacci Quart., 20 (1982), no. 3, 219-226.

[12] Harman C. J., "Complex Fibonacci Numbers", Fibonacci Quart., 19 (1981),no. 1, 82-86.

[13] Berzsenyi G., "Gaussian Fibonacci Numbers", Fibonacci Quart., 15 (1977),no. 3, 233-236.

[14] Catalini M., "Some Formulae for Bivariate Fibonacci and Lucas Polynomials",arXiv:math.Co/0406323 v1, (2004).

[15] Djordjevic G. B., Some Properties of Partial Derivatives of GeneralizedFibonacci and Lucas Polynomials", The Fibonacci Quarterly, 39 (2): 138-14(2001).

[16] Frei G., "Binary Lucas and Fibonacci Polynomials I.", MathematischeNachrichten, 96: 83-112, (1980).

[17] Yu, H., Liang C., "Identities involving Partial Derivatives of Bivariate Fi-bonacci and Lucas polynomials", Fibonacci Quart. 35 (1997), no. 1, 19-23.

[18] Asci M., Gurel E., "Bivariate Gaussian Fibonacci and Lucas Polynomials",Ars Combin., 109 (2013), 461-472.

[19] Stakhov A. P., "Fibonacci matrices, a generalization of the Cassini formula anda new coding theory", Chaos, Solitions and Fractals 30 (2006), no. 1, 56-66.

[20] Basu M., Prasad B., "The generalized relations among the code elementsfor Fibonacci Coding Theory", Chaos, Solitons and Fractals 41(5) (2009),2517,2525.

[21] Basu M., Das M., "Tribonacci matrices and a new coding theory", DiscreteMath. Algorithms Appl., 6(1), (2014 ) article ID: 1450008.

[22] Basu M., Das M., "Coding theory on Fibonacci n-step numbers", DiscreteMath. Algorithms Appl., 6(2), (2014) article ID: 145008.

[23] Basu M., Das M., "Coding Theory on Generalized Fibonacci n-step Polyno-mials", Journal of Information & Optimization Sciences, Vol. 38 (2017), no. 1,83-131.

[24] Philippou A. N., Georghiou C., Philippou G., "Fibonacci Polynomials of Or-der k, Multinomial Expansions and probability", International Journal of Math-ematics and Mathematical Sciences, 6: 545-550 (1983).

164


FINITE DIFFERENCE METHOD FOR FRACTIONAL PARTIALDIFFERENTIAL EQUATION DEFINED BY ATANGANA-BALEANU

CAPUTO (ABC)

Sümeyye EKER1, Mahmut MODANLI 2

1,2Harran University, Turkey


ABSTRACT

In this paper fractional partial differential equation defined Atangana- BaleanuCaputo equation is viewed.The exact solution is calculated for partial differentialequation defined by Atangana- Baleanu Caputo fractioanl derivative operator de-pend on initial boundary value problems.Laplace transformation method is used forthe exact solution.Finite difference schemes are constructed and stability estimatesare obtained for this equation.For this problem, the stability of difference schemesis shown.This technique has been applied at certain levels of the ABC partial par-tial derivative.Approximation solution confirm the accuracy and effectiveness of thetechnique.

Keywords Finite difference method · Fractional order Atangana-Baleanu Caputopartial differential equation · Stability

References

[1] Atangana, A. Koca, I. (2016). Chaos in a Simple Nonlinear System withAtangana- Baleanu Derivatives with Fractional Order, Chaos, Solitons andFractals, 89, 447- 454..

[2] Modanli, M. (2018). Two numerical methods for fractional partial differentialequation with nonlocal boundary value problem. Advances in Difference Equa-tions, 2018(1), 333.

[3] Akgül, A., Modanli, M. (2019). Cranck- Nicholson Difference Scheme Methodand Reproducing Kernel Function for Third Order Differential Equations in theSense of Atangana- Baleanu Caputo Derivative, Chaos, Solitons and Fractals,127, 10-16.

165


ON VERTEX AND LINK RESIDUAL DOMINATION NUMBERS OFGENERALIZED CATERPILLAR GRAPHS

Tufan TURACI

Department of Mathematics, Faculty of Science, Karabük University, 78050 Karabük,Turkey


ABSTRACT

Graph theory are studied different areas such as mathematics, computer science,information and chemistry sciences. Furthermore, it becomes one of the most pow-erful mathematical tools in the analysis and study of network architecture. Net-works are often represented by graphs, which are identified by their vertices andedges where vertices are nodes and edges are links. The stability and reliability ofa network are of prime importance to network designers dealing with the nodes andlinks. The vulnerability value of a communication network shows the resistance ofthe network after the disruption of some centers or connection lines before a com-munication breakdown occurs. Several existing parameters have been proposedin the literature to measure network vulnerability, such as the domination numberγ(G), the average lower domination number γav(G), the residual domination num-ber γR(G), the average lower residual domination number γRav(G), the link residualdomination number γLR(G) and the average lower link residual domination numberγLRav (G).

Generalized caterpillar graphs are very important tree network topologies. Inthis study, the residual domination number, the average lower residual dominationnumber, the link residual domination number and the average lower link residualdomination number of two classes of generalized caterpillar graphs are obtained,and also these values are compared.

Keywords Graph vulnerability · Domination in graphs · Trees · Generalizedcaterpillar graphs

References

[1] Frank H., and I.T. Frisch, Analysis and design of survivable Networks, IEEETransactions on Communications Technology, 18(5): 501-519, 1970.

[2] Haynes T.W., Hedetniemi S.T., and Slater P.J., Fundementals of Domination inGraphs, Advanced Topic, Marcel Dekker, Inc, New York, 1998.

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[3] Nazeer S., Khan S.K., Kousar I., and Nazeer W., Radio labeling and radio num-ber for generalized caterpillar graphs, J. Appl. Math. and Infor., 34(5-6): 451-465, 2016.

[4] Turaci T., and Aytac A., Combining the Concepts of Residual and Dominationin Graphs, Fundamenta Informaticae, 166(4): 379-392, 2019.

[5] Turaci T., On Combining the Methods of Link Residual and Domination inNetworks, Fundamenta Informaticae, 174(1): 43-59, 2020.

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SOME CONSTANT ANGLE SPACELIKE SURFACESCONSTRUCT ON A CURVE IN DE-SITTER 3-SPACE

Tugba MERT1, Mehmet ATÇEKEN2

1Department of Mathematics, Sivas Cumhuriyet University, Sivas, Turkey2Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey


ABSTRACT

In this study, some special spacelike surfaces in de-Sitter 3- space are discussed.Spacelike ones of normal, binormal and Darboux surfaces were investigated in De-Sitter 3-space. Due to the variation in the causal character of a vector, a curve,and a surface in this space, this space has a rich structure. This rich structure givesthe opportunity to diversify the surfaces we consider. The situation of normal, bi-normal and Darboux surfaces that are spacelike are constant angle surfaces wereinvestigated. Since these surfaces are built on a curve, the causal character of thecurve is also very important for these types of surfaces. The concept of constantangle surface, similar to the Helisoid surfaces in Lorentz space, which has manyapplications in the technique, occupies a huge place in the establishment of thesespacelike surfaces in the de-Sitter space. Also, since these special surfaces will beconstructed as constant angle surfaces, attention should be paid to the causal char-acter of the constant vector area of the de-Sitter space and the causal character ofthe fixed angle when creating these surfaces. In this study, with all these importantsituations, using the rich structure of de-Sitter space, the spacelike ones of normal,binormal and Darboux surfaces will be introduced. In recent years, similar typesof surface types well known in Euclidean space are also being investigated in Sitterspace. De-Sitter space is a model for physical events and many physical events canbe interpreted in this space. Since the surface types in these spaces will guide the ar-eas related to our daily life, this type of surface is of great importance. It is possibleto see this from the structures used in the history of architecture. The importance ofsuch surfaces in this space is clear and this study will contribute to geometry.

Keywords Normal Surface · Binormal Surfaces · Darboux Surfaces

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References

[1] Munteanu, M.I., Nistor, A.I., A new approach on constant angle surfaces in E3,Turk T.Math. Vol.33 169-178, 2009.

[2] Di Scala, A.J., Ruiz-Hernandez, G., Helix submanifolds of Euclidean space,Monatsh. Math. DOI 10.1007 / s00605-008-0031-9.

[3] Ruiz-Hernandez, G., Helix, shadow boundary and minimal submanifolds, Illi-nois J. Math. Vol.52, 1385-1397, 2008.

[4] Cermelli, P., Di Scala, A.J., Constant angle surfaces in liquid crystals, Phylos.Magazine,Vol.87, 1871-1888, 2007.

[5] Dillen, F., Fastenakels, J., Van der Veken, J., Vrancken, L., Constant anglesurfaces in S2 × IR , Monaths. Math. , Vol.152, 89-96, 2007.

[6] Dillen, F., Munteanu, M.I., Constant angle surfaces in H2 × IR , Bull. Braz.Math. soc. , Vol.40, 85-97, 2009.

[7] Fastenakels, J., Munteanu, M.I., Van der Veken, J., Constant angle surfacesin the Heisenberg group, Acta Math. Sinica (English Series) Vol.27, 747-756,2011.

[8] Lopez, R., Munteanu, M.I., Constant angle surfaces in Minkowski space, Bul-letin of the Belgian Math. So. Simon Stevin, Vo.18, 271-286, 2011.

[9] Mert, T., Constant Angle Surfaces in Hyperbolic and de Sitter Spaces, GaziUniversity Graduate School of Natural and Applied Sciences, Ph. D. Thesis,2014.

[10] Nistor A., Certein Constant Angle Surfaces Constructed on Curves, Interna-tional Electronic Journal of Geometry Vol.4, No.1, 79-87, 2011.

[11] Karakus, S., Certain Constant Angle Surfaces Constructed on Curves inMinkowski 3-Space, International Electronic Journal of Geometry, Vol.11,No.1, 37-47, 2018.

[12] Kasedou, M., Spacelike Submanifolds in De Sitter Space ,DemonstratricaMathematica , Vol.2, 2010.

[13] Karlıga, B., On The Generalized Stereographic Projection, Beitrege zur Al-gebra and Geometrie Contributions to Algebra and Geometry, Vol.37, No.2,329-336, 1996.

[14] O ’Neill, B., Semi-Riemannian Geometry with applications to relativity, Aca-demic Press, New York, 1983.

[15] J.G.Ratclife, Foundations of Hyperbolic Manifolds, Springer.[16] C.Thas, A gauss map on hypersurfaces of submanifolds in Euclidean spaces,

J.Korean Math.Soc., Vol.16, No.1, 1979.

169


HAMILTON MATRICES OF REAL QUATERNIONS ANDSPINORS

Tülay ERISIR1, Emrah YILDIRIM2

1,2Erzincan Binali Yıldırım University, Turkey


ABSTRACT

The theory of spinors, especially used in applications to electron spin and theory ofrelativity in quantum mechanics, was expressed by B. L. van der Waerden in 1929.The introduction of spinors is one of the most difficult topics in quantum mechan-ics. Even if the spin-1/2 is considered, some fundamental aspects of spinors, such asthe effects of rotation on spinors, turn out to be difficult to explain. Spinors appearto be closely related to the electromagnetic theory. According to physicists spinorsare multilinear transformations. Thanks to this feature, spinors are mathematicalentities somewhat like tensors and allow a more general treatment of the notionof invariance under rotation and Lorentz boosts. For mathematicians, spinors arevectorial objects and their multilinear features do not play any role. Also, spinorshave one-index. In discussing vectors and tensors there are two ways in which wecan proceed: the geometrical and analytical. To use the geometrical approach, wedescribe each kind of quantity in terms of its magnitudes and directions. In theanalytical treatment, we use components. Spinors were first studied by Cartan in ageometrical sense. Cartan was one of the founders of Lie group theory which is oneof the most important topics of mathematics and has many physical applications.So, this study is a very impressive reference in terms of the geometry of the spinorssince this gives the spinor representation of the basic geometric definitions [2]. Ingeometrical meaning, another study was made by Vivarelli. In this study, Vivarelliestablished a one-to-one linear relationship between the quaternions and spinors.In addition, using the relationship between the rotations in quaternions and three-dimensional Euclidean space, Vivarelli actually obtained the spinor representationof the rotations in Euclidean space [4]. In this paper, we have studied on spinorsQ = (q3 + iq0, q1 + iq2) ∈ C2 and real quaternions q = q0 + iq1 + jq2 + kq3 ∈ H .Firstly, we have introduced spinors given by Cartan algebraically [1, 2]. Then,considering the Hamilton matrices of real quaternions we have given spinor repre-sentations of these Hamilton matrices. Moreover, we have obtained eigenvalues andeigenvectors of these spinor matrices. In addition that, we have proved some prop-

170


erties of spinor matrices. Finally, we have expressed some theorems and corollaries.So, we have brought a new perspective to real quaternions.

Keywords Spinors · Real quaternions · Hamilton matrices

References

[1] Cartan É., Les groupes projectifs qui ne laissent invariante aucune multipliciteplane, Bulletin de la Société Mathématique de France, 41: 53-96, 1913.

[2] Cartan É., The theory of spinors, Dover Publications, New York, 1966.[3] Hamilton W.R., Elements of quaternions, Chelsea, Newyork, 1899.[4] Vivarelli M.D., Development of spinors descriptions of rotational mecahanics

from Euler’s rigid body displacament theorem, Celestial Mechanics, 32: 193-207, 1984.

[5] Hernandez -Galeana A., Lopez -Vazquez R., Lopez-Bonilla J., Perez-TeruelG.R., Faraday tensor and Maxwell spinor (Part I), Prespacetime Journal, 6: 88-97, 2015.

[6] Grob J., Trenkler G., Troschke S.O., Quaternions: further contributions to amatrix oriented approach, Lineer Algebra and its Applications, 326: 205-2013,2001.

171


USING FREEWARE MATHEMATICAL SOFTWARE IN CALCULUSCLASSES

Victor GAYOSO MARTINEZ1, Luis HERNÁNDEZ ENCINAS1, Agustin MARTÍN MUÑOZ1,Araceli QUEIRUGA DIOS2

1Institute of Physical and Information Technologies (ITEFI), Spanish National Research Council (CSIC)2Applied Mathematics Department, University of Salamanca


ABSTRACT

Due to the Bologna Accord, the teaching of mathematics has undergone importantchanges. Some of the most visible modifications have been the need to complementthe traditional teaching-learning process with practical, real-life cases and the possi-bility to reinforce the introduction and usage of key concepts through mathematicalsoftware. Nowadays, there exist many computational packages dealing with mathe-matics, some of the best-known being Mathematica and Matlab. However, althoughthey are very complete and powerful, they demand the use of commercial licences,which can be a problem for some education institution or in the cases where stu-dents desire to use the software in an unlimited number of devices or to access fromseveral of them simultaneously.In this contribution, we show how to apply GeoGebra andWolframAlpha to theteaching of Calculus for first-year university students. While GeoGebra is an in-teractive geometry, algebra, statistics, and calculus application available both as anonline resource and a native application in Windows, macOS, and Linux systems,WolframAlpha is a computational knowledge engine developed by a subsidiary ofWolfram Research, the company behind Mathematica. However, unlike that prod-uct, WolframAlpha can be accessed by any individual as a web service free ofcharge. One of the key aspects of WolframAlpha is the possibility to use naturallanguage and Mathematica syntax for requesting computations, which allows usersto benefit from a large amount of Mathematica resources.Being able to use GeoGebra andWolframAlpha as web services without download-ing and installing software is another important advantage, as it avoids the need tohave adminstrator rights to use those computational engines, which typically rep-resents a problem in education centres where lab computers are locked so studentscannot inadvertently install malware that can compromise the university’s network.

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As the best way to show a topic in mathematics is to provide usage examples, thiscontribution reviews the main topics associated to a first-year Calculus class (limits,continuity, derivatives, curve interpolation and integrals), providing the commandsused in GeoGebra and WolframAlpha for the computations and concrete examplesused in actual Calculus classes.

Keywords Calculus · Freeware ·Mathematics · Software

References

[1] European Comission, The Bologna Process and the European Higher Ed-ucation Area, ec.europa.eu/education/policies/higher-education/bologna-process-and-european-higher-education- area\_en, last ac-cessed: June 12th, 2020.

[2] Larson R. and Edwards B.H., Calculus, Cengage Learning, Boston, USA, 2018.[3] GeoGebra, GeoGebra Math Apps, www.geogebra.org, last accessed: June

12th, 2020.[4] University of Salamanca, Universidad de Salamanca, www.usal.es/, last ac-

cessed: June 12th, 2020.[5] U-tad, Centro Universitario de Tecnología y Arte Digital, www.u-tad.com/

en/, last accessed: June 12th, 2020.[6] Wolfram Alpha LLC, Wolfram Alpha computational intelligence, www.

wolframalpha.com, last accessed: June 12th, 2020.

173


CHARACTERIZATIONS AND INTEGRAL FORMULAFOR GENERALIZED RICCI ALMOST SOLITONS

Yasemin SOYLU

Giresun University, Department of Mathematics, 28100, Giresun, Turkey


ABSTRACT

The concept of Ricci solitons was introduced by Hamilton [6] in 1988 that is natu-ral generalizations of Einstein metrics. Ricci solitons also correspond to self-similarsolutions of Hamilton’s Ricci flow [5] and often arise as limits of dilations of sin-gularities in the Ricci flow. Ricci solitons are of interests to physicists as well andare called quasi-Einstein metrics in physics literature (see, [2]). They also play cru-cial roles in the affirmative resolution of the Poincaré conjecture. In a recent study,Pigola et al. [7] introduced a natural extension of the concept of gradient Ricci soli-tons. They replaced the constant λ in Ricci soliton definition with a smooth functionλ ∈ C∞(M), called soliton function, then they said that manifold is a Ricci almostsoliton. They also provided existence and rigidity results and investigate some topo-logical properties. In [8], the authors showed that if a compact Riemannian mani-fold admits a gradient η-Einstein soliton such that the gradient Einstein potential isa non-trivial conformal vector field, then the manifold is isometric to the Euclideansphere. An other concept h-almost Ricci soliton which extends naturally the almostRicci soliton was investigated by Gomes et al. [4]. They showed that a compactnontrivial h-almost Ricci soliton of dimension no less than three with h having de-fined signal and constant scalar curvature is isometric to a standard sphere with thepotential function well determined. In [3], Ghahremani-Gol was investigated someequations of structure for h-almost Ricci soliton which are a natural generalizationfor almost Ricci solitons. As a result they obtained that a compact nontrivial h-almost Ricci soliton is isometric to a Euclidean sphere with some conditions usingthe Hodge–de Rham decomposition theorem. Moreover, they acquired an integralformula for the compact h-almost Ricci solitons. Later, some results were obtainedfor the solitons of the Ricci-Bourguignon flow by Dwivedi [1], generalizing corre-sponding results for Ricci solitons. Taking motivation from Ricci almost solitons,the author then introduced the notion of Ricci-Bourguignon almost solitons andproved some results about them which generalize previous results for Ricci almostsolitons. He also derived integral formulas for compact gradient Ricci-Bourguignon

174


solitons and compact gradient Ricci-Bourguignon almost solitons. Finally, usingthe integral formula he showed that a compact gradient Ricci-Bourguignon almostsoliton is isometric to an Euclidean sphere. Considering the Ricci-Bourguignon al-most solitons together with h almost solitons, we present a new generalized Riccialmost soliton and investigate some aspects of this generalized Ricci almost soli-tons in a complete Riemannian manifold. First, we prove that the compact gradientgeneralized Ricci almost soliton is isometric to the Euclidean sphere by showingthat the scalar curvature becomes constant. Second, we show that in a compactgeneralized gradient Ricci almost soliton satisfying an integral condition.

Keywords Generalized Ricci almost soliton · Riemannian manifold · Standardsphere

References

[1] Dwivedi S., Some results on Ricci-Bourguignon and Ricci-Bourguignon almostsolitons, arXiv:1809.11103, 2018.

[2] Friedan D., Nonlinear models in 2+ε dimensions, Ann. Physics, 163: 318–419,1985.

[3] Ghahremani-Gol H., Some results on h-almost Ricci solitons, J. Geom. Phys.,137: 212–216, 2019.

[4] Gomes J.N., Wang Q., and Xia C., On the h-almost Ricci soliton, J. Geom.Phys., 114: 216–222, 2017.

[5] Hamilton R.S., Three manifolds with positive Ricci curvature, J. Diff. Geom.,17: 255-306, 1982.

[6] Hamilton R.S., The Ricci flow on surfaces, Contemporary Mathematics, 71:237-261,1988.

[7] Pigola S., Rigoli M., Rimoldi M., and Setti A., Ricci almost solitons, Ann.Scuola Norm. Sup. Pisa Cl. Sci., 5: 757–799, 2011.

[8] Shaikh A.A., and Mondal C.K., Some results in η-Ricci Soliton and gradient ρ-Einstein soliton in a complete Riemannian manifold, arXiv:1808.04789, 2018.

175


ON REDUCTION OF A (2+1)-DIMENSIONAL NONLINEARSCHRÖDINGER EQUATION VIA CONSERVATION LAWS

Yesim SAGLAM ÖZKAN

Bursa Uludag University, Art and Science Faculty Department of Mathematics, Bursa, Turkey


ABSTRACT

In this work we consider a (2+1)-dimensional nonlinear Schrödinger equationwhich is one of the important models in branches of plasma physics, nonlinearoptics, fluid dynamics, etc. Considering the Lie point symmetries and conservationlaws obtained with the help of methods available in the literature, the association be-tween symmetries and conservation laws leads to a reduction in the equation. Thistheory reduces both the order and the number of independent variables involved inunderlying equation and quite useful to obtain new solutions.

Keywords Conservation laws · Lie symmetries · Schrödinger equation

References

[1] Sjöberg, A., Double reduction of PDEs from the association of symmetries withconservation laws with applications. Applied Mathematics and Computation,184(2), 608-616, 2007.

[2] Sjöberg, A., On double reductions from symmetries and conservation laws.Nonlinear Analysis: Real World Applications, 10(6), 3472-3477,2009.

[3] Bokhari, A. H., Al-Dweik, A. Y., Zaman, F. D., Kara, A. H., and Mahomed, F.M. , Generalization of the double reduction theory. Nonlinear Analysis: RealWorld Applications, 11(5), 3763-3769, 2010.

[4] Du, X. X., Tian, B., Yuan, Y. Q., and Du, Z., Symmetry Reductions, Group-Invariant Solutions, and Conservation Laws of a (2+ 1)-Dimensional NonlinearSchrödinger Equation in a Heisenberg Ferromagnetic Spin Chain. Annalen derPhysik, 531(11), 1900198, 2019.

[5] Zaidi, A. A., Khan, M. D., and Naeem, I. Conservation Laws and Exact So-lutions of Generalized Nonlinear System and Nizhink-Novikov-Veselov Equa-tion. Mathematical Problems in Engineering, 2018.

176


RINGS WHOSE G-SEMIARTINIAN MODULES HAVE MAXIMAL ORMINIMAL SUBPROJECTIVITY DOMAIN

Yılmaz DURGUN 1, Ayse ÇOBANKAYA2

1,2Department of Mathematics, Cukurova University, Turkey


ABSTRACT

G-semiartinian modules were studied and introduced in [1]. A module M is calledg-semiartinian if every non-zero homomorphic image of M has a singular simplesubmodule. In this study, we introduce the modules in which g-semiartinian mod-ules are subprojective. By Holston and et al. in [4], the subprojectivity domainwere introduced and determine how far a non-projective module is from being pro-jective. Following [4], given modules X1 and X2, X1 is X2-subprojective if foreach epimorphism α : P → X2 and each morphism h : X1 → X2, there existsa morphism f : X1 → P with αf = h. The necessary and sufficient conditionsfor a module X1 to be X2-subprojective are given in [4, 2]. For any module X ,Br−1(X) = {L ∈Mod−R | X is L-subprojective} is called the subprojectivitydomain ofX . Clearly, X is projective if and only if Br−1(X) = Mod−R. In otherwords, X is projective if its subprojectivity domain is as large as it can be. A g-semiartinian module whose domain of subprojectivity as small as possible is calledgsap-indigent. We investigated the structure of rings whose (simple, coatomic) g-semiartinian right modules are gsap-indigent or projective. Furthermore, over rightPS rings, necessary and sufficient condition to be gsap-indigent module was deter-mined. Finally, inspired by similar ideas and problems in [4, 3], gsap-indigent mod-ules are studied. Over right PS ring, we prove that a g-semiartinian right module Yis gsap-indigent if and only if Hom(Y, S) 6= 0 for every singular simple right R-module S. To keep in line with [5] and [3], we say R has no subprojective (simple)gsa-middle class if every (simple) g-semiartinian module is either gsap-indigent orprojective. It is proven that (1) If R has no subprojective simple gsa-middle class,then R is either right PS or right Kasch; (2) If R is not right Kasch, then R hasno subprojective simple gsa-middle class if and only if all non-projective coatomicg-semiartinian module is gsap-indigent if and only if all non-projective modules offinite length are gsap-indigent. Also, it is shown that if R is a commutative GD-extending ring, then R has no subprojective simple gsa-middle class if and only if

177


there is a ring direct sum R ∼= S× T , where S is semisimple Artinian ring and T isan indecomposable ring which is either a local QF-ring or a local nonsingular ring.

Keywords Subprojectivity domain · PS ring · Kasch ring

Acknowledgement: This work was supported by the Scientific and TechnologicalResearch Council of Turkey (TUBITAK) (Project number: 119F176).

References

[1] Durgun Y. and Çobankaya A., G-Dickson Torsion Theory, International Sci-ence, Mathematics and Engineering Sciences Congress (2019) 978-983.

[2] Durgun Y., On subprojectivity domains of pure-projective modules, J. AlgebraAppl. 19: 2020, https://doi.org/10.1142/S0219498820500917.

[3] Durgun Y., Rings whose modules have maximal or minimal subprojectivity do-main, J. Algebra Appl. 14(6): 1550083-2015.

[4] Holston C., Lopez-Permouth S. R., Mastromatteo J. and Simental- RodriguezJ.E., An alternative perspective on projectivity of modules, Glasg. Math., J.57:83-99, 2016.

[5] Holston C., Lopez-Permouth S. R.and Ertas N.O. , Rings whose modules havemaximal or minimal projectivity domain, J. Pure Appl. Algebra, 216: 673-678,2012.

[6] Kasch F., Modules and rings, London Mathematical Society Monographs, Aca-demic Press, Inc. London-New York, 1982.

178


ON THE ?-CONGRUENCE SYLVESTER EQUATION

Yuki SATAKE1, Tomohiro SOGABE1, Tomoya KEMMOCHI1, Shao-Liang ZHANG1

1Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan


ABSTRACT

We consider the following matrix equation

AX +X?B = C, (1)

where A ∈ Cm×n, B ∈ Cn×m, and C ∈ Cm×m are given, and X ∈ Cn×m is to bedetermined. The operator (·)? denotes the transpose (·)T or the conjugate transpose(·)H of a matrix, and then equation (1) is called the ?-congruence Sylvester equation.The ?-congruence Sylvester equation appears in palindromic eigenvalue problemsarising from some realistic applications such as the vibration analysis of fast trains,see, e.g., [1].

In recent paper [2], when ? = T and the given matrices are square (m = n),it was shown that the ?-congruence Sylvester equation is mathematically equiva-lent to the Lyapunov equation under certain conditions. The Lyapunov equation iswidely known in control theory and has been much studied. Therefore, the study [2]indicates that it can be possible to utilize the rich results on the Lyapunov equationfor the ?-congruence Sylvester equation (1). Indeed, it was shown that the directmethod for the Lyapunov equation can be used to obtain a numerical solution of(1) with computational cost lower than that of the conventional method [4] for (1).One of the important approaches in the previous study is to use an equivalent linearsystem that is obtained by vectorization. Oozawa et al. succeeded in reducing (1)to the Lyapunov equation by applying an appropriate linear operator to the linearsystem and returning it into a matrix [2].

The theoretical result was extended to the case where A and B are rectangular(m 6= n) [3]. In this case, it was shown that (1) is equivalent to the generalizedSylvester equation. However, when ? = H, the same transformation cannot beapplied to (1) because (1) is a nonlinear equation.

In this talk, we consider a linearization of (1) with ? = H, and construct anappropriate operator to obtain an equivalent linear matrix equation. We will showthat (1) is equivalent to the generalized Sylvester equation under certain conditions,and introduce numerical solvers utilizing our results.

179


Keywords ?-Congruence Sylvester equation ·Matrix equation · Linear operator

References

[1] D. Kressner et al., Numer. Algorithms, 51(2): 209–238, 2009.[2] M. Oozawa et al., J. Comput. Appl. Math., 329: 51–56, 2018.[3] Y. Satake et al., Appl. Math. Lett., 96: 7–13, 2019.[4] F. D. Terán and F. M. Dopico, Electron. J. Linear Algebra, 22: 849–863, 2011.

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Participant List

ARACELI QUEIRUGA-DIOS, (SPAIN), INVITED SPEAKER

AYMAN BADAWI, (UAE), INVITED SPEAKER

CARLOS MARTINS DA FONSECA, (KUWAIT), INVITED SPEAKER

ABDISALAM HASSAN MUSE, (SOMALIA), LISTENER

AGUSTIN MARTIN, (SPAIN), LISTENER

ALDINA CORREIA, (PORTUGAL), ORAL PRESENTATION

ALEJANDRO MEDINA, (SPAIN), ORAL PRESENTATION

ALICE ANGELESCU, (ROMANIA), LISTENER

ANA BORGES, (PORTUGAL), ORAL PRESENTATION

ASCENSIÓN HERNÁNDEZ, (SPAIN), LISTENER

ASUKA OHASHI, (JAPAN), ORAL PRESENTATION

AYSE ÇOBANKAYA, (TURKEY), ORAL PRESENTATION

B. ALI IBRAHIMOGLU, (TURKEY), ORAL PRESENTATION

BAHAR KARAMAN, (TURKEY), ORAL PRESENTATION

BAHAR KULOGLU, (TURKEY), ORAL PRESENTATION

BASRI ÇALISKAN, (TURKEY), ORAL PRESENTATION

BUSE INGENÇ, (TURKEY), ORAL PRESENTATION

CRISTINA CARIDADE, (PORTUGAL), ORAL PRESENTATION

CUMALI YOLDAS, (TURKEY), LISTENER

DANIELA RICHTARIKOVA, (SLOVAKIA), ORAL PRESENTATION

DENIZ ALTUN, (TURKEY), ORAL PRESENTATION

DEOLINDA MARIA DIAS RASTEIRO, (PORTUGAL), ORAL PRESENTATION

DIN PRATHUMWAN, (THAILAND), ORAL PRESENTATION

DURDU HAKAN UTKU, (TURKEY), ORAL PRESENTATION

DURSUN TASÇI, (TURKEY), LISTENER

DUYGU ÇAGLAR, (TURKEY), ORAL PRESENTATION

EL AMIR DJEFFAL, (ALGERIA), ORAL PRESENTATION

ELIANA COSTA E SILVA, (PORTUGAL), ORAL PRESENTATION

181


EMEL KARACA, (TURKEY), ORAL PRESENTATION

EMRAH KARAMAN, (TURKEY), ORAL PRESENTATION

EMRAH YILDIRIM, (TURKEY), ORAL PRESENTATION

ENGIN ÖZKAN, (TURKEY), LISTENER

ERDAL ÜNLÜYOL, (TURKEY), ORAL PRESENTATION

FATIH KASIMOGLU, (TURKEY), ORAL PRESENTATION

FATMA KARAKUS, (TURKEY), ORAL PRESENTATION

FATMA DIGDEM KOPARAL, (TURKEY), ORAL PRESENTATION

FIGEN KANGALGIL, (TURKEY), ORAL PRESENTATION

GERARDO RODRIGUEZ SANCHEZ, (SPAIN), LISTENER

GÖKHAN MUTLU, (TURKEY), ORAL PRESENTATION

GÜL UGUR KAYMANLI, (TURKEY), ORAL PRESENTATION

HAKAN AYKUT, (TURKEY), ORAL PRESENTATION

INTHIRA CHAIYA, (THAILAND), ORAL PRESENTATION

ION MIERLUS MAZILU, (ROMANIA), ORAL PRESENTATION

INAN ÜNAL, (TURKEY), ORAL PRESENTATION

IPEK EBRU KARAÇAY, (TURKEY), ORAL PRESENTATION

ISMAIL ASLAN, (TURKEY), ORAL PRESENTATION

J.M. LÓPEZ BELINCHÓN, (SPAIN), ORAL PRESENTATION

JANA GABKOVÁ, (SLOVAKIA), LISTENER

JESUS MARTIN VAQUERO, (SPAIN), LISTENER

JOSE MARIA CHAMOSO, (SPAIN), LISTENER

KAMRAN ZAKARIA, (PAKISTAN), ORAL PRESENTATION

LUCIAN NITA, (ROMANIA), LISTENER

LUIS HERNÁNDEZ ENCINAS, (SLOVAKIA), LISTENER

MAHMUT MODANLI, (TURKEY), ORAL PRESENTATION

MARIA AYDIN, (TURKEY), ORAL PRESENTATION

MARIA EMÍLIA BIGOTTE DE ALMEIDA, (PORTUGAL), ORAL PRESENTATION

MARIA JESUS SANTOS, (SPAIN), LISTENER

182


MARÍA JOSÉ CÁCERES, (SPAIN), LISTENER

MARIE DEMLOVA, (CZECH REPUBLIC), LISTENER

MD. ALAL HOSEN HOSEN, (BANGLADESH), ORAL PRESENTATION

MELEK SOFYALIOGLU, (TURKEY), ORAL PRESENTATION

MELEK YAGCI, (TURKEY), ORAL PRESENTATION

MELTEM EKIZ, (TURKEY), ORAL PRESENTATION

MERT AKIN INSEL, (TURKEY), ORAL PRESENTATION

MERVE ULUDAG, (TURKEY), ORAL PRESENTATION

MERVE TASTAN, (TURKEY), ORAL PRESENTATION

MICHAEL CARR, (IRELAND), LISTENER

MIGUEL ANGEL LOPEZ GUERRERO, (SPAIN), LISTENER

MILICA ANDELIC, (KUWAIT), ORAL PRESENTATION

MUAMMER AYATA, (TURKEY), ORAL PRESENTATION

MUKADDES ÖKTEN TURACI, (TURKEY), ORAL PRESENTATION

MUSA ÇAKMAK, (TURKEY), ORAL PRESENTATION

MUSTAFA AGGUL, (TURKEY), ORAL PRESENTATION

MUSTAFA ASÇI, (TURKEY), ORAL PRESENTATION

MUSTAFA ÇALISKAN, (TURKEY), LISTENER

MUSTAFA ÖZKAN, (TURKEY), LISTENER

MUSTAFA ALTIN, (TURKEY), ORAL PRESENTATION

MUSTAFA ASLANTAS, (TURKEY), ORAL PRESENTATION

MÜCAHIT AKBIYIK, (TURKEY), LISTENER

NURTEN GÜRSES, (TURKEY), LISTENER

ONUR SAHIN, (TURKEY), ORAL PRESENTATION

PAUL ROBINSON, (IRELAND), LISTENER

PETER LETAVAJ, (SLOVAKIA), ORAL PRESENTATION

RABIA ÇAKAN AKPINAR, (TURKEY), ORAL PRESENTATION

RAMAZAN ÖZARSLAN, (TURKEY), ORAL PRESENTATION

RAMAZAN SARI, (TURKEY), ORAL PRESENTATION

183


RAUL ALCARAZ, (SPAIN), LISTENER

REMZI AKTAY, (TURKEY), ORAL PRESENTATION

SABA INAM, (PAKISTAN), LISTENER

SABRINA FRANCESCA PELLEGRINO, (ITALY), ORAL PRESENTATION

SEDA YAMAÇ AKBIYIK, (TURKEY), LISTENER

SEHER SULTAN YESILKAYA, (TURKEY), ORAL PRESENTATION

SELAMI BAYEG, (TURKEY), ORAL PRESENTATION

SELÇUK ÖZCAN, (TURKEY), LISTENER

SELIHAN KIRLAK, (TURKEY), ORAL PRESENTATION

SEVAL ISIK, (TURKEY), ORAL PRESENTATION

SINAN SERKAN BILGICI, (TURKEY), ORAL PRESENTATION

SNEZHANA GOCHEVA-ILIEVA, (BULGARIA), ORAL PRESENTATION

STEFANIA CONSTANTINESCU, (ROMANIA), LISTENER

SÜLEYMAN AYDINYÜZ, (TURKEY), ORAL PRESENTATION

SÜMEYYE EKER, (TURKEY), ORAL PRESENTATION

TUFAN TURACI, (TURKEY), ORAL PRESENTATION

TUGBA MERT, (TURKEY), ORAL PRESENTATION

VÍCTOR GAYOSO MARTÍNEZ, (SPAIN), ORAL PRESENTATION

VILDAN ÖZTÜRK, (TURKEY), LISTENER

YASEMIN SOYLU, (TURKEY), ORAL PRESENTATION

YESIM SAGLAM ÖZKAN, (TURKEY), ORAL PRESENTATION

YUKI SATAKE, (JAPAN), ORAL PRESENTATION

184

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