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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
Preface
This abstract booklet includes the abstracts of the papers that have been presented at III. Inter-
national Conference on Mathematics and its Applications in Science and Engineering (ICMASE
2021) which is held in Technical University of Civil Engineering of Bucharest (Romania) between
4-7 July, 2022.
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 about some
of the following interesting topics: Functional Analysis, Approximation Theory, Real Analysis,
Complex Analysis, Harmonic and non-Harmonic Analysis, Applied Analysis, Numerical Anal-
ysis, Geometry, Topology and Algebra, Modern Methods in Summability and Approximation,
Operator Theory, Fixed Point Theory and Applications, Sequence Spaces and Matrix Transfor-
mation, 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,
Mathematics Education, Applications 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.
Thanks to all committee members.
We wish everyone a fruitful conference and pleasant memories from ICMASE 2022.
Ion Mierlus-Mazilu
Chair, ICMASE 2022
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
International Conference on Mathematics and Its Applications in Science and
Engineering (ICMASE 2022)
4-7 July 2022, Technical University of Civil Engineering Bucharest, Romania
Honory and Advisory Board
Prof. Dr. José Miguel MATEOS ROCO, Vice Chancellor for Research and Transfer, University of
Salamanca, (Spain)
Prof. Dr. Eng. Radu Sorin VACAREANU, Technical University of Civil Engineering Bucharest,
(Romania)
Prof. Dr. Yusuf TEK˙
IN, Rector of Ankara Hacı Bayram Veli University, (Turkey)
Organizing Committee
Ion MIERLUS-MAZILU, Technical University of Civil Engineering Bucharest, (Romania) (Con-
ference Chair)
Fatih YILMAZ, Ankara Hacı Bayram Veli University, (Turkey) (Organizing Chair)
Araceli QUEIRUGA-DIOS, Salamanca University, (Spain)
Jesús Martin-VAQUERO, Salamanca University, (Spain)
María Jesús Santos SANCHEZ, Salamaca University, (Spain)
Mustafa ÖZKAN, Gazi University, (Turkey)
Mücahit AKBIYIK, Beykent University, (Turkey)
Local Organizing Committee
Leonard DAUS, Technical University of Civil Engineering, Bucharest, (Romania)
Narcisa TEODORESCU, Technical University of Civil Engineering, Bucharest, (Romania)
Mariana ZAMFIR, Technical University of Civil Engineering, Bucharest, (Romania)
Daniel TUDOR, Technical University of Civil Engineering, Bucharest, (Romania)
Stefania CONSTANTINESCU, Technical University of Civil Engineering, Bucharest, (Romania)
Alice ANGHELESCU, Technical University of Civil Engineering, Bucharest, (Romania)
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
Invited Speakers
Humberto BUSTINCE, Public University of Navarra, (Spain)
Zacharias ANASTASSI, De Montfort University, (UK)
Deolinda RASTEIRO, Coimbra Engineering Institute-ISEC, (Portugal)
Scientific Committee
Agustín Martín MUNOZ, Spanish National Research Council, (Spain)
Abdullah ALAZEMI, Kuwait University, (Kuwait)
Ángel Martín del REY, Universidad de Salamanca, (Spain)
Ascensión Hernández ENCINAS, University of Salamanca, (Spain)
Ayman BADAWI, American University of Sharjah, (UAE)
Aynur KESK˙
IN KAYMAKÇI, Selçuk University, (Turkey)
Carlos Martins da FONSECA, Kuwait College of Science and Technology, (Kuwait)
Cesar BENAVENTE-PECES, Technical University of Madrid, Madrid, (Spain)
Cristina R. M. CARIDADE, Instituto Superior de Engenharia de Coimbra, (Portugal)
Daniela RICHTARIKOVA, Slovak University of Technology in Bratislava, (Slovakia)
Daniela VELICHOVA, Slovak University of Technology, (Slovakia)
Dolores QUEIRUGA, Universidad de La Rioja, (Spain)
Dursun TA ¸SÇI, Gazi University, (Turkey)
Emel KARACA, Ankara Hacı Bayram Veli University, (Turkey)
Emília BIGOTTE, Instituto Superior de Engenharia de Coimbra, (Portugal)
Emily VELIKOVA, University of Ruse, Ruse, (Bulgaria)
Esa KUJANSUU, Tampere University of Applied Sciences, Tampere, (Finland)
Fatma KARAKU¸S, Sinop University, Sinop, (Turkey)
Gheorghe MOROSANU, Babes-Bolyai University, (Romania)
Hanna KINNARI-KORPELA, Tampere University of Applied Sciences, Tampere, (Finland)
Hari Mohan SRIVASTAVA, University of Victoria, (Canada)
Ji-Teng JIA, Xidian University, (China)
Juan José Bullón PEREZ, Universidad de Salamanca, (Spain)
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
Judit García FERRERO, Universidad de Salamanca, (Spain)
Kirsi-Maria RINNEHEIMO, Tampere University of Applied Sciences, Tampere, (Finland)
Lucian NITA, Technical University of Civil Engineering, Bucharest, (Romania)
Luis Hernández ENCINAS, Spanish National Research Council, (Spain)
Luis Hernández ÁLVAREZ, Spanish National Research Council, (Spain)
Marie DEMLOVA, Czech Technical University in Prague, (Czech Republic)
María José Cáceres GARCIA, Universidad de Salamanca, (Spain)
Melek SOFYALIO ˘
GLU, Ankara Hacı Bayram Veli University, (Turkey)
Michael CARR, Technological University Dublin, (Ireland)
Milica ANDJELIC, Kuwait University, (Kuwait)
Miguel Ángel González de la TORRE, Spanish National Research Council, (Spain)
Miguel Ángel QUEIRUGA-DIOS, Universidad de Burgos, (Spain)
Mustafa ÇALI¸SKAN, Gazi University, (Turkey)
Nenad P. CAKIC, University of Belgrade, (Serbia)
Praveen AGARWAL, Anand International College of Engineering, (India)
Seda YAMAÇ AKBIYIK, Geli¸sim University, (Turkey)
Selçuk ÖZCAN, Karabük University, (Turkey)
Serpil HALICI, Pamukkale University, (Turkey)
S. H. J. PETROUDI, Payame Noor University, (Iran)
Snezhana GOCHEVA-ILIEVA, University of Plovdiv Paisii Hilendarski, (Bulgaria)
Tomohiro SOGABE, Nagoya University, (Japan)
Vishnu Narayan MISHRA, Indira Gandhi National Tribal University, (India)
Víctor Gayoso MARTINEZ, Spanish National Research Council, (Spain)
Zhibin DU, South China Normal University, (China)
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International Conference on Mathematics and Its Applications in Science and Engineering
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Contents
De Moivre’s Formulas for Split Octonions 1
MÜCAHIT AKBIYIK
Sobolev Orthogonality of a Class of Finite Orthogonal Polynomials 3
RABIA A KTA ¸S, E SRA GÜLDO ˘
GAN LEKES˙
IZ
Derivative-Free Finite-Difference Homeier Method for Systems of Nonlinear Models 4
YANAL AL-SHORMAN, OBADAH SAID SOLAIMAN, ISHAK HASHIM
A Spectral Collocation Method with Convergence Analysis for Solving Nonlinear Fractional
Fredholm Integro-Differential Equations 6
A.Z. AMIN, I. HASHIM
Quaternion Algebras and the Role of Quadratic Forms in their Study 7
NECHIFOR ANA-GABRIELA
The Moore-Penrose Inverse in Rickart ∗-Rings 9
MEHSIN JABEL ATTEYA
Error Detection and Correction for Coding Theory on k-Order Gaussian Fibonacci Matrices 10
SULEYMAN AYDINYUZ, MUSTAFA ASCI
Fuzzy Clustering based Fuzzy Regression Function in Z-Environment 12
MÜKERREM BAHAR BA ¸SKIR, OLGA M. POLESHCHUK
An Individual Work Plan to Influence Educational Learning Paths in Engineering Undergrad-
uate Students 14
MARIA EMÍLIA BIGOTTE DE ALMEIDA, JOÃO RICARDO BRANCO, LUÍS MAR-
GALHO, MARÍA JOSÉ CÁCERES, ARACELI QUEIRUGA-DIOS
A Sequential Construction of Hypercomplex Algebras 16
REMUS BOBOESCU
Stability Analysis Of a Caputo Type Fractional Waterborne Infectious Disease Model 18
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
CEMIL BÜYÜKADALI
A note on k-Telephone and Incomplete k-Telephone Numbers 20
PAULA CATARINO, EVA MORAIS, HELENA CAMPOS
A Note on Special Matrices involving k-Bronze Fibonacci Numbers 22
PAULA CATARINO, SANDRA RICARDO
On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative 25
MEHMET NIYAZI ÇANKAYA
Multicovariance and Multicorrelation for p-variables 27
MEHMET NIYAZI ÇANKAYA
Experience in Teaching Mathematics to Engineers: Students v.s Teacher Vision 29
CRISTINA CARIDADE
The Effect (Impact) of Project-Based Learning through Augmented Reality on higher Math
Classes 31
CRISTINA CARIDADE
Some Spectral Properties of Operators Generated by Quantum Difference Equations 33
¸SERIFENUR CEBESOY ERDAL
The Theory of Dirac Equations and System of Difference Equations 35
¸SERIFENUR CEBESOY ERDAL
Renewed Neutrosophic Soft Graphs with Some New Operations 38
YILDIRAY ÇEL˙
IK
A Generalization of Multiple Zeta Values 40
ROUDY EL HADDAD
Some Results for Matrix Sturm-Liouville Equations with a Point Interaction 44
˙
IBRAHIM ERDAL
Weyl Theory for the Fractional Sturm-Liouville Equations 46
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(ICMASE 2022)
˙
IBRAHIM ERDAL
Approximation by a new modification of Bernstein-Durrmeyer operators 48
MELEK SOFYALIO ˘
GLU, KADIR KANAT, SELIN ERDAL
On Quaternions with Gaussian Oresme Numbers 49
AYBÜKE ERTA ¸S, FATIH YILMAZ
Existence and Multiplicity of Positive Steady States for Classes of Reaction Diffusion Equa-
tions 50
NALIN FONSEKA, RATNASINGHAM SHIVAJI, KERI SPETZER, BYUNGJAE SON
A Class of Multivariate Orthogonal Functions Associated with Fourier Transforms of Orthog-
onal Polynomials on the Simplex 52
ESRA GÜLDO ˘
GAN LEKES˙
IZ, RABIA A KTA ¸S, I VAN AREA
Rho-Statistical Convergence of Interval Numbers 54
HAFIZE GUMUS
Statistical Convergence of Multiset Sequences From A Different Perspective 55
HAFIZE GUMUS
On k-Oresme Numbers with Negative Indices 56
SERPIL HALICI, ELIFCAN SAYIN, ZEHRA BETÜL GÜR
k-Oresme Polynomials and their Derivatives 58
On New Families of Bicomplex Jacobsthal Numbers with q-Integer Components 60
SERPIL HALICI, SULE CURUK
Quantum Calculus Approach to the Dual Bicomplex Jacobsthal Numbers 62
SERPIL HALICI, SULE CURUK
Extrapolated IMEX Runge-Kutta Methods to Solve Nonlinear Parabolic PDEs 64
ALBERTO ALONSO IZQUIERDO, JESÚS MARTÍN VAQUERO
A Survey On Slant Ruled Surfaces 66
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(ICMASE 2022)
EMEL KARACA
On Roots of Some Quaternionic Polynomials 67
GONCA KIZILASLAN, ILKER AKKUS
The Extended Exponential-Weibull Accelerated Failure Time Model with Applications to Can-
cer Data Set 69
ADAM BRAIMA MASTOR, OSCAR NGESA, JOSEPH MUNGATU, AHMED Z. AFIFY
Sid Sackson’s Mathematical Games 71
JIND ˇ
RICH MICHALIK
A Monge-Kantorovich-type Norm on a Vector Measures Space 73
ION MIERLUS-MAZILU, LUCIAN NITA
On Vector Spaces and Some Applications 74
ION MIERLUS-MAZILU, FATIH YILMAZ
Convergence and Error Estimation for the Infinite System of Volterra–Fredholm Integral Equa-
tions Involving Erd´
elyi-Kober Fractional Operator 75
LAKSHMI NARAYAN MISHRA
Mappings on Rings with Idempotents 77
AMIRHOSSEIN MOKHTARI, PARISA SAADATI
Amoud-G Family of Lifetime Distributions: Properties, Hazard-Based Regression Models and
Applications to Survival Data 78
ABDISALAM HASSAN MUSE, SAMUEL MWALILI, OSCAR NGESA
Quantum Graph Realization of Transmission Problems 80
GÖKHAN MUTLU
Malmquist-Takenaka System and Equilibrium Condition on the Unit Disc and Upper Half-
plane 82
ZSUZSANNA NAGY-CSIHA, MARGIT PAP
Neutrosophic Multi-Hypergroups 84
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
SERKAN ONAR
Six Sigma Application in Leather Textitle Company 86
SELÇUK ÖZCAN, HAZAL ÖZDEM˙
IR
Submanifolds of Almost Complex Metallic Manifolds 87
MUSTAFA ÖZKAN, AY ¸SE TORUN
Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor
Equations 89
OKTAY K PASHAEV
Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s Theorem for
Tetrahedron 90
OKTAY K PASHAEV
Approximation of Solutions for Nonlinear Functional Integral Equations 92
VIJAI KUMAR PATHAK, LAKSHMI NARAYAN MISHRA
Application of Double Kashuri Fundo Decomposition Method to Goursat Problem 94
HALDUN ALPASLAN PEKER, FATMA AYBIKE ÇUHA
Kashuri Fundo Decomposition Method for Solving Michaelis-Menten Nonlinear Biochemical
Reaction Model 96
HALDUN ALPASLAN PEKER, FATMA AYBIKE ÇUHA
RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles Triangle
under the Effect of Inclined Periodic Magnetic Field 98
BENGISEN PEKMEN GERIDONMEZ
From Paths to Vector Fields. Application in the Descriptive Proximity of Optical Flows in
Video Frame Sequences 100
JAMES FRANCIS PETERS, TANE VERGILI
Leonardo-Mersenne Sequence, Binomial Transform and some Properties 102
SEYYED HOSSEIN JAFARI PETROUDI, MARYAM PIROUZ, FATIH YILMAZ
On Certain Vertex Operator Algebras 104
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International Conference on Mathematics and Its Applications in Science and Engineering
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GORDAN RADOBOLJA
Social Interactions and Mathematical Competencies Development 105
DANIELA RICHTARIKOVA
Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for some Particular Phase
Transformations in Materials Science 107
PAULO R. RIOS, ELENA VILLA
Influence of the Collaboration among Predators and the Allee Effect on Prey in a Leslie-Gower-
type Predation Model 109
ALEJANDRO ROJAS-PALMA, EDUARDO GONZÁLEZ-OLIVARES
A Fast Algorithm for Inversing a Toeplitz Heptadiagonal Matrix Based on the CL Factorization
of a Tridiagonal Matrix 111
PAULA CATARINO, EVA MORAIS, HELENA CAMPOS
On The Neimark-Sacker Bifurcation of a Certain Second Order Difference Equation 115
ERKAN TA ¸SD EM˙
IR,YÜKSEL SOYKAN
Performance of Machine Learning Methods Using Tweets 117
˙
ILKAY TU ˘
G, BETÜL KAN-KILINÇ
Timelike Ruled Surfaces in 3−Dimensional a Walker Manifold 119
AYSEL TURGUT VANLI, ALEV ABE ¸S
On the Solution of an Integral Geometry Problem Over Surfaces of Revolution 121
ZEKERIYA USTAOGLU
A Discretization Approach and Inversion of Radon Transform via Fuzzy Basic Functions 122
ZEKERIYA USTAOGLU
On Hybrid Numbers with Gaussian-Mersenne Coefficients 123
SERHAT YILDIRIM, FATIH YILMAZ
Further Fixed Point Results for Rational Suzuki Contractions in B-Metric-Like Spaces 125
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
KASTRIOT ZOTO, ILIR VARDHAMI
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DEMOIVRE’SFORMULAS FOR SPLIT OCTONIONS
Mücahit AKBIYIK1
1Department of Mathematics, Beykent University, Istanbul, Turkey
Corresponding Author’s E-mail: mucahitakbiyik@beykent.edu.tr
ABS TR ACT
In this talk, we calculate De Moivre’s formulas of split octonions. We examine the
nth−roots of split octonions. Also, we give an illustrative example. In additon to
this, we present polars form for two type of split octionions.
Keywords Split Octonions; Matrix representation; De Moivre’s formula.
References
[1] Cho, E., De Moivre’s Formula For Quaternions. Appl. Math. Lett. Vol. 11,
No. 6, pp. 33-35, 1998.
[2] Kabadayi, H.; Yayli, Y. De Moivre’s Formula for Dual Quaternions. Kuwait
J. Sci. 2011, 38, 15–23.
[3] Bekta¸s, Ö., Yüce, S. De Moivre’s and Euler’s Formulas for the Matrices
of Octonions. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 89, 113–127
(2019).
[4] Kösal, H.H.; Bilgili, T. Euler and De Moivre’s Formulas for Fundamental
Matrices of Commutative Quaternions. Int. Electron. J. Geom. 2020,13,
98–107.
[5] Özdemir, M. Finding nth−Roots of a 2 ×2 Real Matrix Using De Moivre’s
Formula. Adv. Appl. Clifford Algebr. 2019, 29, 1–25.
[6] Akbıyık, M., Yamaç Akbıyık, S., Karaca, E., Yılmaz, F., De Moivre’s and
Euler Formulas for Matrices of Hybrid Numbers, Axioms 10 (3), 213, (2021).
[7] Akbıyık, M., Yamaç Abıyık, S., Yılmaz, F., The Matrices of Pauli Quater-
nions, Their De Moivre’s and Euler’s Formulas, International Journal of Ge-
ometric Methods in Modern Physics (accepted 2022, Jun 3).
[8] M. Gogberashvili, “Observable algebra,” http://arxiv.org/abs/ hep-
th/0212251.
[9] M. Gogberashvili, “Octonionic geometry,” Advances in Applied Clifford Al-
gebras, vol. 15, no. 1, pp. 55–66, 2005.
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International Conference on Mathematics and Its Applications in Science and Engineering
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[10] M. Gogberashvili, “Octonionic electrodynamics,” Journal of Physics A, vol.
39, no. 22, pp. 7099–7104, 2006.
[11] M. Gogberashvili, “Octonionic version of Dirac equations,” International
Journal of Modern Physics A, vol. 21, no. 17, pp. 3513–3524, 2006.
[12] M. Gogberashvili,O. Sakhelashvili1 “Geometrical Applications of Split Oc-
tonions,” Advances in Mathematical Physics Volume 2015, Article ID
196708, 14 pages.
[13] Bektas O. Split-type octonion matrix. Math Methods Appl Sci. 2018; 42(16).
[14] Carmody K. Circular and hyperbolic quaternions, octonions, and sedenions.
Appl Math Comput. 1988;28:47-72.
[15] Tan¸slı M, Kansu ME, Demir S. A new approach to Lorentz invariance in elec-
tromagnetism with hyperbolic octonions. Eur Phys J Plus. 2012;127(69):1-
12.
[16] Demir S, Tan¸slı M. Hyperbolic octonion formulation of the fluid maxwell
equations. J Korean Phys Soc. 2016;68(5):616-623.
[17] Candemir N, Tan¸slı M, Özdas K, Demir S. Hyperbolic octonionic Proca-
Maxwell equations. Z Naturforsch. 2008;63:15-18.
[18] Demir S, Tan¸slı M, Kansu ME. Generalized hyperbolic octonion formula-
tion for the fields of massive Dyons and Gravito-Dyons. Int J Theor Phys.
2016;52:3696-3711.
[19] Köplinger J. Dirac equation on hyperbolic octonions. Appl Math Comput.
2006;182(1):443-446.
[20] Cariow A, Cariowa G, Knapi´
nski J. Derivation of a low multiplicative
complexity algorithm for multiplying hyperbolic octonions. 2015:1-15.
arXiv:1502.06250.
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International Conference on Mathematics and Its Applications in Science and Engineering
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SOBOLEV ORTHOGONALITY OF A CLASS OF FINITE
ORTHOGONAL POLYNOMIALS
Rabia AKTA ¸S 1, Esra GÜLDO ˘
GAN LEKES˙
IZ
1Ankara University
Corresponding Author’s E-mail: esragldgn@gmail.com
ABS TR ACT
In this paper, a class of the finite Sobolev orthogonal polynomials is considered.
The aim is to set an orthogonal structure in the Sobolev space and investigate the
structure for negative parameters in some special cases.
Keywords Sobolev space ·finite orthogonal polynomials ·partial differential
operators
References
[1] Masjed-Jamei M., Three Finite Classes of Hypergeometric Orthogonal Polyno-
mials and Their Application in Functions Approximation, Integral Transforms
and Special Functions, 13:2, 169-190, 2002.
[2] Akta¸s R., Xu Y., Sobolev Orthogonal Polynomials on a Simplex, International
Mathematics Research Notices, 2013(13), p. 3087–3131, 2013.
[3] Alfaro M., Marcellan F. and Rezola M.L., Estimates for Jacobi-Sobolev type
orthogonal polynomials, Appl. Anal. 67:157–174, 1997.
[4] Alfaro M., Alvarez de Morales M. and Rezola M.L., Orthogonality of the Jacobi
polynomials with negative integer parameters, J. Comput. Appl. Math. 145,
379-386, 2002.
[5] Arvesú J., Álvarez-Nodarse R., Marcellán F. and Pan K., Jacobi-Sobolev-type
orthogonal polynomials: Second-order differential equation and zeros, Journal
of Computational and Applied Mathematics, 90:2, 135-156, 1998.
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DERIVATIVE-FREE FINITE-DIFFERENCE HOMEIER METHOD
FOR SYSTEMS OF NONLINEAR MODELS
Yanal AL-SHORMAN1, Obadah Said SOLAIMAN1, Ishak HASHIM1
1School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor,
Malaysia
Corresponding Author’s E-mail: P114391@siswa.ukm.edu.my
ABS TR ACT
An efficient derivative free method for finding roots of nonlinear equations was
implemented in this paper. The third-order Homeier’s method has been taken as
the basis for this work, which can be derived by using Newton’s theorem for the
inverse function and derive a new class of cubically convergent Newton-type meth-
ods. Several nonlinear problems, including nonlinear equations, complex equations,
and nonlinear systems of equations, have been taken to compare the efficiency of
the proposed method to other popular derivative-free schemes. Results show that
the proposed method outperformed the considered published methods. The pre-
sented scheme needs fewer iterations to achieve the desired solution, with an order
of convergence of about 2.5, which is higher than the convergence order of the com-
pared methods, and one of the popular nonlinear equation solvers used to compare
with our proposed method is secant method with order of convergence 1.618 in the
absence of derivative. When using the proposed method for solving systems of non-
linear equations, the Jacobian problem can be avoided by following the procedure
in Broyden’s method. Thus, the proposed method can be considered as an upper-
most method giving faster convergence to find the roots of nonlinear equations in
the absence of the derivative for uni-variate nonlinear equations with complex roots
as well as for the multivariate systems of nonlinear equations. We expect that our
proposed method will undoubtedly be very useful and effective to the scientific and
industrial community.
Keywords Homeier method ·Secant method ·Nonlinear equations ·Derivative-
free methods ·Iterative methods ·Broyden’s method ·Order of convergence
References
[1] Amat S., Busquier S., Gutiérrez J.M., Geometric constructions of iterative func-
tions to solve nonlinear equations, J. Comput. Appl. Math. 157: 197–205, 2003.
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[2] Burden R.L. & Faires J.D. Numerical Analysis, 8th Edition, Bob Pirtle, USA,
2005.
[3] Frontini M., Sormani E., Modified Newton’s method with third-order conver-
gence and multiple roots, J. Comput. Appl. Math. 156: 345–354, 2003.
[4] Frontini M., Sormani E., Some variant of Newton’s method with third-order
convergence, J. Comput. Appl. Math. 140: 419–426, 2003.
[5] Homeier H.H.H., On Newton-type methods with cubic convergence, J. Comput.
Appl. Math. 176: 425–432, 2005.
[6] Homeier H.H.H., A modified Newton method for root finding with cubic con-
vergence, J. Comput. Appl. Math. 157: 227–230, 2003.
[7] Homeier H.H.H., A modified Newton method with cubic convergence: the mul-
tivariate case, J. Comput. Appl. Math. 169: 161–169, 2003.
[8] Heenatigala S.L., Weerakoon S., Fernando T. G. I., Finite Difference
Weerakoon-Fernando Method to solve nonlinear equations without using
derivatives, University of Sri Jayewardenepura, Gangodawila, Nugegoda, Sri
Lanka, 2021.
[9] Nishani H. P. S., Weerakoon S., Fernando T.G.I. Liyanage M. Third order con-
vergence of Improved Newton’s method for systems of nonlinear equations,
502/E1, Proceedings of the annual sessions of Sri Lanka association for the
Advancement of Science, Sri Lanka, 2014.
[10] Said Solaiman O., Abdul Karim S.A., Hashim I., Dynamical comparison of
several third-order iterative methods for nonlinear equations, Computers, Ma-
terials & Continua, 67(2): 1951-1962, 2021.
[11] Said Solaiman O., Hashim I., Optimal eighth-order solver for nonlinear equa-
tions with applications in chemical engineering, Intelligent Automation & Soft
Computing, 27(2) :379-390, 2021.
[12] Said Solaiman O., Hashim I., An iterative scheme of arbitrary odd order and
its basins of attraction for nonlinear systems, Computers, Materials & Continua,
66(2): 1427-1444, 2021.
[13] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with acceler-
ated third-order convergence, Appl. Math. Lett.13: 87–93, 2000.
[14] Young T. & Martin J. Introduction to Numerical Methods and Matlab Pro-
gramming for Engineers, Department of Mathematics, University of Ohio,
2021.
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International Conference on Mathematics and Its Applications in Science and Engineering
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A SPECTRAL COLLOCATION METHOD WITH CONVERGENCE
ANALYSIS FOR SOLVING NONLINEAR FRACTIONAL FREDHOLM
INTEGRO-DIFFERENTIAL EQUATIONS
A.Z. AMIN1, I. HASHIM1
1Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi,
Selangor, Malaysia
Corresponding Author’s E-mail:azm.amin@yahoo.com
ABS TR ACT
In this article, an efficient and accurate spectral numerical method is presented for
solving nonlinear fractional Fredholm integro-differential equations (FFIDEs) with
the initial condition. The proposed method is based on the shifted Legendre-Gauss-
Lobatto collocation (SL-GL-C) method for fractional derivative, described in the
Caputo derivative sense. We adapt the SL-GL-C algorithm to solve the nonlinear
FFIDEs. Moreover, we provide a framework for studying the rate of convergence
of the proposed algorithm. The effectiveness and validity of the method have been
proved by solving four numerical examples. Besides, we give a numerical test
example to show that the approach can preserve the non-smooth solution of the
underlying problems.
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QUATERNION ALGEBRAS AND THE ROLE OF QUADRATIC
FORMS IN THEIR STUDY
Nechifor ANA-GABRIELA
1Romania
Corresponding Author’s E-mail: nechifor.ana96@gmail.com
ABS TR ACT
For a long time, Hamilton tried to model a 3-dimensional space with a structure
similar to that of complex numbers, whose addition and multiplication are found in
bidimensional space. In this sense, Hamilton realized that it would need a fourth
dimension and thus invented the term of quaternions for real space, generated by
the elements 1, i, j, k, relative to multiplication. (John Voight, Quaternion algebras,
March 27, 2021, p.1)
But further, Dickson was the first which considered quaternion algebras over an
arbitrary field. He started it by generalizing those algebras in which each element
satisfies a quadratic equation, he set out a diagonalizable basis for such an algebra
and analyzed the conditions to be a division algebra. (John Voight, Quaternion
algebras, March 27, 2021, p.9). This led him to what he afterwards called the
generalized quaternion algebra, for which:
i2=α,j2=β,k2=−αβ
ij =−ji =k,ik =−ki =αj,kj =−jk =βi
One way to classify and also, characterize quaternion algebras is by using quadratic
forms. In this sense, we will recall some basic notions related to the theory of
quadratic forms and we will emphasize their connection with quaternion algebras.
We will not include the proofs, but they can be found in any of the references
mentioned.
Keywords Bilinear forms ·Quadratic forms ·Quaternion algebras ·Norm form ·
Pure quaternions
References
[1] Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, New York,
CRC Press, Taylor & Francis Group, 2017, pp. 1-5
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[2] C. N˘
ast˘
asescu, C. Ni¸t˘
a,C. Vraciu. Bazele Algebrei, Bucure¸sti, Editura
Academiei Republicii Socialiste România, 1986,p. 18, pp. 130, 153-157, 242-
259
[3] D. Fetcu. Elemente de algebr˘a liniar ˘a, geometrie analitic ˘a ¸si geometrie difer-
en¸tial˘a , Ia¸si, Casa Editorial˘
a Demiurg, 2009, pp.65-71
[4] Winfried Scharlau. Quadratic and Hermitian Forms, Berlin, Heidelberg, New
York, Spring-Verlag,1985 pp. 1-3, pp.5-6, p. 8
[5] David W. Lewis, Quaternion Algebras and the Algebraic Legacy of Hamilton’s
Quaternions, Irish Math, Soc. Bulletin 57,2006, pp. 43-46
[6] John Voight,Quaternion algebras, v.0.0.26, March 27, 2021, p. 1
[7] T.Y. Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in
Mathematics, vol. 67, American Mathematical Society Providence, Rhode Is-
land, 2004, pp. 51
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THE MOORE-PENROSE INVERSE IN RICKART ∗-RINGS
Mehsin Jabel ATTEYA
Department of Mathematics, College of Education,
Al-Mustansiriyah University, Baghdad, Iraq. Corresponding Author’s E-mail: mehsinatteya88@uomustansiriyah.edu.iq
ABS TR ACT
The main purpose of this paper is to introduce several new necessary and sufficient
conditions for the existence of the Moore-Penrose inverse of an element in a ring R
are obtained. In addition, the formulae of the Moore-Penrose inverse of an element
in a ring are presented.
Keywords The Moore-Penrose inverse · ∗-ring ·A Rickart ∗-ring ·Projection
element ·Weak-supported element
References
[1] Moore E. H., On the reciprocal of the general algebraic matrix, Abstract, Bull.
Amer. Math. Sot., 26:394-395, 1920.
[2] Penrose R., A generalized inverse of matrices, Mathematical Proceedings of the
Cambridge Philosophical Society, 51(3): 401-413, 1955.
[3] Mosic D., and Djordjevic D.S., New characterizations of EP, generalized nor-
mal and generalized Hermitian elements in rings, Appl. Math. Comput. 218,
no. 12: 6702-6710, 2012.
[4] Tian Y.G., and Wang H.X., Characterizations of EP matrices and weighted-EP
matrices,Linear Algebra Appl., 434: 1295-1318, 2011.
[5] Zhu H.H., Chen J.L., and Patricio P., Further results on the inverse along an
element in semigroups and rings,Linear and Multilinear Algebra, 64(3): 393-
403, 2016.
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
ERROR DETECTION AND CORRECTION FOR CODING THEORY
ON K-ORDER GAUSSIAN FIBONACCI MATRICES
Suleyman AYDINYUZ1, Mustafa ASCI2
1Pamukkale University Science and Arts Faculty Department of Mathematics Kınıklı Denizli TURKEY
2Pamukkale University Science and Arts Faculty Department of Mathematics Kınıklı Denizli TURKEY
Corresponding Author’s E-mail: aydinyuzsuleyman@gmail.com
ABS TR ACT
In this study, we consider the coding theory for Gaussian Fibonacci numbers of
order-k. This coding method is based on the Qk, Rkand E(k)
nmatrices. In this
respect, it differs from classical encryption methods. Unlike classical algebraic
coding methods, this method theoretically allows for the correction of matrix ele-
ments that can be infinite integers. Error detection criterion is examined for the case
of k= 2 and this method is generalized to k and error correction method is given.
In the simplest case, for k= 2, the correct capability of the method is essentially
equal to 93,33%, exceeding all well known correction codes. It appears that for a
sufficiently large value of k, the probability of decoding error is almost zero.
Keywords Gaussian Fibonacci numbers ·k-order Gaussian Fibonacci numbers ·k-
order Gaussian Fibonacci matrices ·k-order Gaussian Fibonacci ·Error Detection,
Correction coding/decoding.
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 and
Applications", Ellis Harwood Limitted, (1989).
[3] Stakhov A. P., "A generalization of the Fibonacci Q-matrix", Rep. Natl. Acad.
Sci. Ukraine 9 (1999), 46-49.
[4] Stakhov A. P., Massinggue V., Sluchenkov A., "Introduction into Fibonacci
Coding and Cryptography", Osnova, Kharkov (1999).
[5] Jordan J. H., "Gaussian Fibonacci and Lucas numbers", Fibonacci Quart. 3
(1965), 315-318.
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(ICMASE 2022)
[6] Stakhov A. P., "Fibonacci matrices, a generalization of the Cassini formula and
a new coding theory", Chaos, Solitions and Fractals 30 (2006), no. 1, 56-66.
[7] Basu M., Prasad B., "The generalized relations among the code elements
for Fibonacci Coding Theory", Chaos, Solitons and Fractals 41(5) (2009),
2517,2525.
[8] Basu M., Das M., "Coding theory on Fibonacci n-step numbers", Discrete Math.
Algorithms Appl., 6(2), (2014) article ID: 145008.
[9] Asci M., Gurel E., "Some Properties of k-order Gaussian Fibonacci and Lucas
Numbers", Ars Combin., 135, (2017), 345-356.
[10] Esmaeili M., Esmaeili M., "A Fibonacci-polynomial based coding method
with error detection and correction", Computers and Mathematics with Appli-
cations 60 (2010), 2738-2752.
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(ICMASE 2022)
FUZZY CLUSTERING BASED FUZZY REGRESSION FUNCTION IN
Z-ENVIRONMENT
Mükerrem Bahar BA¸SKIR1, Olga M. POLESHCHUK2
1Bartın University, Bartın 74100, Turkey
2Bauman Moscow State Technical University, Moscow, Russia
Corresponding Author’s E-mail: mbaskir@bartin.edu.tr
ABS TR ACT
Real-life applications involve various epistemic uncertainties. These uncertainties
arise from the limitations in data or measurement, and modeling approximations.
As known, fuzzy theory (Zadeh, 1965) deals with all these uncertainty resources.
Besides, fuzzy set theory has been developed to analyze and reduce its own un-
certainty due to the precise nature of primary membership. There have been pro-
posed numerous versions of type-1 fuzzy sets, such as interval or general type-2
fuzzy sets, intuitionistic fuzzy sets, neutrosophic fuzzy sets, Z-numbers, etc. One
of the remarkable proposals is Z-numbers to improve perception-based decisions.
Zadeh (2011) introduced Z-numbers to measure the reliability of information in
decision-making. A Z-number is an ordered pair fuzzy number, (A,R). The first
component, A, is a restriction on a random variable, and the second, R, is the reli-
ability of the first one. Hybrid methods including Z-numbers are needed to make
reliable information-based decisions. The most important issue in decision prob-
lems is to model any given system by identifying the relationships between its
components (input-output variables). Türk¸sen (2008) proposed fuzzy regression
function as a fundamental of system modeling in fuzzy environment. The orig-
inal inputs and the memberships of any given system are used as new inputs in
Türk¸sen’s fuzzy regression models. Thus, an effective model structure is created
for the system and its uncertainty. There are limited studies regarding the regres-
sion models in Z-environment (i.e. Zeinalova et.al., 2017; Ezadi and Allahviranloo,
2017; Poleshchuk, 2020). Fuzzy system modeling based on Z-numbers can handle
discrepancies in judgments by defining functional relationships between linguistic
input-output variables. Thus, in this study, we propose a Z-number valued fuzzy re-
gression function with fuzzy clustering. Fuzzy c-means algorithm (Bezdek, 1981) is
used in order to form a mathematical model using initial Z-numbered input-output
variables and their memberships. The performance of the proposed approach is
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examined using an illustrative example related to the maturity level of digital
transformation.
Keywords Z-numbers ·Fuzzy clustering ·Fuzzy regression function
References
[1] Bezdek J.C., Pattern recognition with fuzzy objective function algorithms,
Plenum Press, New York, 1981.
[2] Ezadi S., and Allahviranloo T., Numerical solution of linear regression based
on Z-numbers by improved neural network, Intell. Autom. Soft Comput., 24:
1-11, 2017.
[3] Türk¸sen I.B., Fuzzy functions with LSE, Applied Soft Computing,
8: 1178–1188, 2008.
[4] Poleshchuk O., Multiple Z-regression with fuzzy coefficients, In book: 14th
International Conference on Theory and Application of Fuzzy Systems and
Soft Computing ICAFS-2020, pp.63-70, 2020.
[5] Zadeh L.A., Fuzzy sets, Information Control, 8: 338-353, 1965.
[6] Zadeh L.A., A Note on Z-numbers, Information Sciences, 14(181): 2923-2932,
2011.
[7] Zeinalova L., Huseynov O., and Sharghi P . , A Z-number valued regression
model and its application, Intell. Autom. Soft Comput., 24: 1-5, 2017.
[8] Farahani H., Ebadi M. J., and Jafari H., Finding inverse of a fuzzy matrix
using eigenvalue method. International Journal of Innovative Technology and
Exploring Engineering, 9(2): 3030-3037, 2019. DOI: 10.35940/
ijitee.B6295.129219.
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ANINDIVIDUAL WORK PLAN TO INFLUENCE EDUCATIONAL
LEARNING PATHS IN ENGINEERING UNDERGRADUATE
STUDENTS
Maria Emília Bigotte de ALMEIDA1, João Ricardo BRANCO1, Luís MARGALHO1, María José
CÁCERES2, Araceli QUEIRUGA-DIOS3
1Department of Physics and Mathematics, Coimbra Institute of Engineering, Polytechnic Institute of Coimbra, Coimbra, Portugal
2Department of Didactics of Mathematics and Didactics of Experimental Sciences, Universidad de Salamanca, Salamanca, Spain
3Department of Applied Mathematics, Universidad de Salamanca, Salamanca, Spain
Corresponding Author’s E-mail: ebigotte@isec.pt
ABSTRACT
Issues related to the failure of mathematics in the engineering education and the
negative impact of these difficulties in the success of the Differential and Integral
Calculus curricular units taught in engineering courses are a problem to which we
have devoted our attention and research [1]. Most students entering higher educa-
tion have a weak mathematics preparation. It’s even more aggravated due to the
different areas of knowledge of its experience when entering engineering courses.
The Support Center for Mathematics in Engineering (CeAMatE) in Coimbra Engi-
neering Institute is a space dedicated to accompanying students to overcome diffi-
culties in basic and elementary Mathematics knowledge, which is essential for full
integration into engineering courses. In this Center, there is a set of activities and
resources based in Primary and Secondary Education programs in Portugal, and in
the Core Zero outcomes from the SEFI guidelines, Mathematics for the European
Engineer – A Curriculum for the Twenty-First Century [2]. These outcomes are
grouped taking into account the different topics from Algebra, Analysis and Calcu-
lus, Discrete Mathematics, Geometry and Trigonometry, and Statistics and Proba-
bility. The Center also incorporates an e-learning component, adapting to learning
styles and students’ knowledge levels.
Diagnostic test results provide information about the mathematical contents that
should be worked with students. These results make possible to define individ-
ual working plans that allow autonomous work in overcoming the difficulties in
mathematics [3, 4]. Moreover, these individual plans will describe the evolution of
students’ learning, through the monitoring and reformulation of knowledge acqui-
sition.
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The objective of this study is to understand the relationship between the Diagnos-
tic Test results, the attendance at CeAMATE and the evaluations obtained in the
Curricular Unit of Differential and Integral Calculus.
Keywords Mathematics Knowledge ·Diagnosis Test ·Engineering ·Differential
and Integral Calculus ·Individual Working Plan
References
[1] Bigotte de Almeida, M.E., Queiruga-Dios, A., Cáceres, M.J., Differential and
Integral Calculus in First-Year Engineering Students: A Diagnosis to Under-
stand the Failure, Mathematics, 9(1), 2021.
[2] Alpers, B.A., Demlova, M., Fant, C.H., Gustafsson, T., Lawson, D., Mustoe,
L., and Velichova, D. A framework for mathematics curricula in engineering
education: a report of the mathematics working group, Brussels, SEFI, 2013.
[3] Bigotte, E., Gomes, A., Branco, J.R., Pessoa, T., The influence of educational
learning paths in academic success of mathematics in engineering undergradu-
ate, IEEE Front. Educ. Conf., 1-6, 2016.
[4] Bigotte de Almeida, M.E., Branco, J.R., Margalho, L., Queiruga-Dios, A.,
Cáceres, M.J. Understanding the Level of Mathematics Knowledge of Stu-
dents Who Joined ISEC. In: Yilmaz, F., Queiruga-Dios, A., Santos Sánchez,
M.J., Rasteiro, D., Gayoso Martínez, V., Martín Vaquero, J. (eds) Mathe-
matical Methods for Engineering Applications. ICMASE 2021. Springer Pro-
ceedings in Mathematics & Statistics, 384. Springer, Cham, 2022.
https:
//doi.org/10.1007/978-3-030-96401-6_23
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A SEQUENTIAL CONSTRUCTION OF HYPERCOMPLEX
ALGEBRAS
Remus BOBOESCU1
1Ovidius University of Constanta
Corresponding Author’s E-mail: remus_boboescu@yahoo.com
ABS TR ACT
The structure of algebra over a vector space is a way of obtaining an extension of
sets of numbers.Entering complex-split numbers (doubled numbers) is a simple way
to get an extension of real numbers. This has the square of the imaginary element
1.Similar to the introduction of quaternions, octonions and sedenions, the structure
of complex-split numbers can be doubled in size. A series of commutative algebras
is obtained. The multiplication table of the basis elements for these algebras can
be easily generated using a binary write of the index of the basis elements. These
algebras can be generalized by introducing parameters in defining the squares of
the imaginary elements of the base similar to generalized quaternions and octo-
nions.Thus, the introduction of hypercomplex algebras into the known form can be
achieved.Thus, the study of the simplest way to introduce hypercomplex algebras
is proposed. The properties of the introduced algebras and the objectives of their
introduction are discussed. These refer to the trace and norm functions, given by the
addition of a hypercomplex number and its conjugate.For the introduced algebras
the norm function is different from the square of the Euclidean norm on the consid-
ered linear space. It is investigated how to perform operations in a non-associative
algebra. It is shown that in this there must be only one way of associating the
factors.
Keywords complex-split numbers ·hypercomplex algebras ·binary index ·bitwise
XOR ·generalized quaternions
References
[1] John W. Bales A tree for computing the Cayley-Dickson Twist, Misouri Journal
of Mathematical Sciences Volume 21 Number 2, 2009, SCIENCES VOLUME
21, NUMBER 2, 2009.
[2] R. D. Schafer AN INTRODUCTION TO NONASSOCIATIVE ALGEBRAS
Stillwater, Oklahoma, 1961 p.23
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[3] I. P. Shestakov Asosociative identities of octonions Algebra and Logic, Vol. 49,
No. 6, 2011.
[4] John Baez The Octonions Bulletin of the American Mathematical Society
39(2)May 2001.
[5] Diana Savin About Special Elements in Quaternion Algebras Over Finite Fields
Adv. Appl. Clifford Algebras 27 (2017), 1801-1813.
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STABILITY ANALYSIS OF A CAPUTO TYPE FRACTIONAL
WATERBORNE INFECTIOUS DISEASE MODEL
Cemil BÜYÜKADALI1
1Depatment of Mathematics, Faculty of Sciences, Van Yüzüncü Yıl University, Van, Türkiye
Corresponding Author’s E-mail: cbuyukadali@yyu.edu.tr
ABS TR ACT
In this presentation we propose a Caputo type fractional waterborne bacterial in-
fection model with saturation effect of infectious population on the transmission of
disease caused by waterborne bacteria. The total population size N(t)is divided
into two compartments: susceptible individual S(t)and infectious with symptoms
I(t)at time t≥0.Furthermore, we consider a compartment B(t)that reflects the
bacterial concentration at time t. We assume positive natural death rate µ. Suscep-
tible individuals can become infected by contact with infected individuals at rate
ρI(t)
1+m1I(t).Susceptible individuals can become infected with waterborne disease, like
cholera by contact with contaminated sources at rate βB(t)
1+m2I(t),where β > 0is in-
gestion rate of the bacteria through contaminated sources. We assume nonnegative
death rate δcaused by infection. Each infected individual contributes to the increase
of the bacterial concentration at rate ξ. On the other hand, the bacterial concentra-
tion can decrease at mortality γ. With these assumptions we have the following
Caputo type fractional waterborne disease model
DαS(t) = Λ −ρI(t)
1+m1I(t)+βB(t)
1+m2I(t)S(t)−µS(t),
DαI(t) = ρI(t)
1+m1I(t)+βB(t)
1+m2I(t)S(t)−δI(t)−µI(t)
DαB(t) = ξI (t)−γB(t).
(1)
For this model with initial conditions
S(0) ≥0, I(0) ≥0, B(0) ≥0,(2)
we first find existence and uniqueness of the solution of system (1) with initial
conditions (2) which remains in a positively invariant region
Ω = (S, I, B)∈R3
+:S+I≤Λ
µ,0≤B≤ξΛ
µγ .(3)
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Secondly, we find the existence of disease-free equilibrium point E0= (Λ/µ, 0,0)
and endemic epidemic equilibrium point E∗= (S∗, I∗, B∗)and define basic re-
production number R0of this system. Next, we investigate the local stabilities of
equilibriums E0and E∗by using Jacobian matrix of system (1) at these equilibri-
ums and global stability of the disease-free equilibrium using Lyapunov’s method.
Keywords Fractional calculus ·Caputo derivatives ·Epidemiology ·Equilibrium ·
Stability
References
[1] Codeço C.T., Endemic and epidemic dynamics of cholera: the role of the
aquatic reservoir, BMC Infectious Diseases, 1, 2001.
[2] Zhou X., Cui J., Global Stability Of The Viral Dynamics With Crowley-Martin
Functional Response, Bull. Korean Math. Soc. 48: 555–574, 2011.
[3] Wang Y., Cao J., Global stability of general cholera models with nonlinear in-
cidence and removal rates, J. Franklin Inst. 352: 2464–2485, 2015.
[4] Lemos-Paião A.P., Silva C.J., Torres D.F.M., An epidemic model for cholera
with optimal control treatment, J. Comput. Appl. Math. 318: 168–180, 2017.
[5] Ammi M. R. S., Tahiri M., Torres D. F. M., Global Stability of a Caputo
Fractional SIRS Model with General Incidence Rate, Math. Comput. Sci. 15:
91–105, 2021.
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(ICMASE 2022)
ANOTE ON k-TELEPHONE AND INCOMPLETE k-TELEPHONE
NUMBERS
Paula CATARINO1,2,3, Eva MORAIS1,2, Helena CAMPOS1,3
1Department of Mathematics, School of Science and Technology, University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
2Research Centre of Mathematics, University of Minho-Polo CMAT-UTAD
3Research Centre on Didactics and Technology in the Education of Trainers-CIDTFF, University of Aveiro
Corresponding Author’s E-mail:emorais@utad.pt
ABS TR ACT
The telephone numbers, also known as involution numbers, are given by
Tn=Tn−1+ (n−1)Tn−2, n ≥2(4)
with initial terms T0=T1= 1. The recurrence relation (4) of the sequence {Tn}n
was found by Heinrich August Rothe in 1800 ([5]) when counting the involutions
(that is, permutations that are their own inverse) in a set of nelements.
This sequence can also be seen as the number of possible patterns of connections
between the nsubscribers of a telephone service, therefore the designation tele-
phone numbers. Another application of the telephone numbers is to graph theory,
with Tngiven by the number of matchings (Hosoya index) in a complete graph. In
recreational mathematics, the nth telephone number Tnis the number of ways to
place nrooks on an n×nchessboard such that no two rooks attack each other and
such that the configuration of the rooks is symmetric under a diagonal reflection of
the board.
In this work, the k-telephone and the incomplete telephone numbers are introduced
using the same methodology that was applied to other sequences, such as the Fi-
bonacci sequence ([3, 4, 6]), the Pell sequence ([5, 6, 7, 8]) or the Leonardo se-
quence ([8]) .
Similarly to the works with the Fibonacci sequence, for any positive real number k,
the k-telephone sequence {Tk,n}n∈IN is defined by
Tk,0= 1, Tk,1=k, Tk,n =kTk,n−1+ (n−1)Tk,n−2
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and it is proved that the explicit formula for this sequence is given by
Tk,n =bn/2c
X
i=0
n!
2i(n−2i)!i!kn−2i.
Furthermore, the incomplete telephone numbers are defined by
Tl
n=
l
X
i=0
n!
2i(n−2i)!i!0≤l≤jn
2k
and the incomplete k-telephone numbers are defined by
Tl
k,n =
l
X
i=0
n!
2i(n−2i)!i!kn−2i,0≤l≤jn
2k.
The recurrence relations and relevant properties for the k-telephone numbers and
the incomplete k-telephone numbers are presented in this work.
Keywords Telephone numbers ·k-telephone numbers ·Incomplete telephone
numbers ·Incomplete k-telephone numbers
References
[1] Catarino P., and Borges A., A note on incomplete Leonardo numbers, Integers,
20(7), 2020.
[2] Catarino P., and Campos H., Incomplete k−Pell, k−Pell-Lucas and Modified
k−Pell Numbers, Hacet. J. Math. Stat., 46(3): 361-372, 2017.
[3] Falcón S., and Plaza A., On the Fibonacci k−numbers, Chaos Solitons Fractals,
32(5): 1615-1624, 2007.
[4] Filipponi P., Incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat.
Palermo, 45(2): 37-56, 1996.
[5] Knuth D.E., The Art of Computer Programming, Vol 3, Sorting and Searching,
Addison-Wesley, Reading, 1973.
[6] Ramírez J.L., Incomplete k−Fibonacci and k−Lucas numbers, Chinese J. of
Math., 7 pages, 2013.
[7] Ramírez J.L., Incomplete generalized Fibonacci and Lucas polynomials, Hacet.
J. Math. Stat., 44(2): 363-373, 2015.
[8] Ramírez J.L., and Sirvent V., Incomplete tribonacci numbers and polynomials,
J. Integer Seq., 17: article 14.4.2, 2014.
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A NOTE ON SPECIAL MATRICES INVOLVING k-BRONZE
FIBONACCI NUMBERS
Paula CATARINO1, Sandra RICARDO2
1Department of Mathematics, University of Trás-os-Montes e Alto Douro
2Department of Mathematics, University of Trás-os-Montes e Alto Douro
Corresponding Author’s E-mail: sricardo@utad.pt
ABS TR ACT
Numerical sequences are a source of very interesting and challenging mathemat-
ical problems and have attracted the attention of many researchers. Many devel-
opments have been done concerning the well-known Fibonacci sequence or the
Lucas sequence, but also many works have emerged on other sequences of num-
bers, polynomials, quaternions, octonions, sedenions, etc. We refer, for example,
to the works on Hybrid numbers [4, 12], applications of Fibonacci and Lucas num-
bers [13], Leonardo’s numbers [6], k-Pell generalized numbers of order m, where
m is a non-negative integer [7], Gaussian Fibonacci sequences [3], Gaussian Lu-
cas sequences [10], Gaussian Pell sequences and Gaussian Pell-Lucas sequences
[9], Gaussian Jacobsthal sequences [2], and Gaussian Bronze Fibonacci sequences
[11]. Newer developments appeared recently concerning third-order Bronze Fi-
bonacci numbers [1], on Vietoris’numbers sequence [8], on generalized sequences
of numbers known as k-Fibonacci numbers, k-Jacobsthal numbers, k-Pell numbers,
balancing numbers, k-telephone numbers, hyper k-pell numbers and incomplete
numbers [5].
In 1985, Levesque [3] deduced an important formula, known as the Binet formula,
which is obtained in terms of the roots of the characteristic equation associated with
the recurrence relation defining the considered sequence. Binet’s formula allows
one to find the general term of a sequence, without having to resort to other terms of
the sequence, thus being an important tool to study some properties of the sequence.
In the present work, we take as our starting point the Bronze Fibonacci sequence,
{BFn}n≥0, listed in the Online Encyclopedia of integers sequences [15] as the se-
quence A006190, and defined by the following recurrence relation
BFn+2 = 3BFn+1 +BFn,(5)
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with the initial conditions BF0= 0 and BF1= 1. We consider a generalization
of Bronze Fibonacci sequence, in which the recurrence formula depend on one
parameter k. We call this new sequence, the k-Bronze Fibonacci sequence, denoted
by {BFk,n}n≥0, and defined by the following recurrence relation
BFk,n+2 = 3BFk,n+1 +kBFk,n , k ∈Z+,(6)
with the initial conditions BFk,0= 0 and BFk,1= 1,
Our goal is to give alternative ways to determine the general term of the k-Bronze
Fibonacci sequence involving some special tridiagonal matrices and their determi-
nants.
Keywords Bronze Fibonacci numbers ·Tridiagonal matrices ·general term
References
[1] Akbiyik, M., and Alo, J., On Third-Order Bronze Fibonacci Numbers, Mathe-
matics, 9(20): 2606, 2021.
[2] Asc M. and Gurel E., Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Num-
bers, Ars Combin., 111, 53–63, 2013.
[3] Berzsenyi G., Gaussian Fibonacci Numbers, Fibonacci Quart., 15(3), 233–236,
1977.
[4] Catarino, P., On k-Pell hybrid numbers, J. Discrete Math. Sci. Cryptogr., 22(1),
83–89, 2019.
[5] Catarino, P., and Campos, H., From Fibonacci Sequence to More Recent Gen-
eralisations, In: Yilmaz, F., Queiruga-Dios, A., Santos Sánchez, M.J., Rasteiro,
D., Gayoso Martínez, V., Martín Vaquero, J. (eds), Mathematical Methods for
Engineering Applications, ICMASE 2021, Springer Proceedings in Mathemat-
ics & Statistics, 384, 259–269, 2022.
[6] Catarino, P., and Borges, A., On Leonardo numbers, Acta Mathematica Univer-
sitatis Comenianae, 89(1), 75–86, 2019.
[7] Catarino, P., and Vasco, P., The Generalized order−m(k-Pell)numbers, Analele
Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica, 20(1), 55–65, 2020.
[8] Catarino, P., and Almeida, R., A Note on Vietoris’ Number Sequence, Mediter-
ranean Journal of Mathematics, 19(41), 2022.
[9] Halici, S. and Öz, S., On some Gaussian Pell and Pell-Lucas numbers, Ordu
Univ. J. Sci. Tech., 6(1), 8–18, 2016.
[10] Jordan J.H., Gaussian Fibonacci and Lucas Numbers, Fibonacci Quart., 3,
315–318, 1965.
[11] Kartal, M. Y., Gaussian Bronze Fibonacci Numbers, Ejons International Jour-
nal on Mathematics, Engineering - Natural Sciences, 13, 19–25, 2020.
[12] Kizilates, C., A new generalization of Fibonacci hybrid and Lucas hybrid num-
bers, Chaos Solitons Fractals, 130, 5pp., 2020.
[13] Koshy, T., Fibonacci and Lucas Numbers with Applications, Wiley-
Interscience, New York, 2001.
[14] Levesque, C., On m-th order linear recurrences, Fibonacci Quart., 23(4), 290–
29, 1985.
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International Conference on Mathematics and Its Applications in Science and Engineering
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[15] Sloane N. J. A., The on-line encyclopedia of integer sequences, Available in
http://oeis.org/
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
ON THE STATISTICAL PROPERTIES OF THE DEFORMED
ALGEBRAS ON THE JACKSON q-DERIVATIVE
Mehmet Niyazi ÇANKAYA1
1Department of International Trading and Finance, Faculty of Applied Sciences, U¸sak University, U¸sak,
1Department of Statistics, Faculty of Arts and Sciences, U¸sak University, U¸sak
Corresponding Author’s E-mail: mehmet.cankaya@usak.edu.tr
ABS TR ACT
The property of Tsallis q-entropy is introduced. The q-deformed logarithm pro-
duced by Tsallis q-entropy is examined. In this study, this logarithm is extended
to the correlation coefficient type. Since a new deformed difference based on the
statistical properties such as correlation coefficient and the M-estimation method,
a new derivative and its corresponding integral can be proposed. The simulation
study is performed to observe the results of the proposed algebras.
Keywords Algebras ·Statistical inference ·Tsallis entropy ·q-calculus
References
[1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduc-
tion with Applications, (Birkh¨auser Basel, 2001).
[2] C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a
Complex World, (Springer, New York, 2009).
[3] T. Wada and H. Suyari, A two-parameter generalization of Shannon–Khinchin
axioms and the uniqueness theorem, Phys. Lett. A 368 (2007) 199-205.
[4] M.R. Ubriaco, Entropies based on fractional calculus, Phys. Lett. A 373 (2009)
2516-2519.
[5] G.B. Thomas, M.D. Weir, J. Hass and F.R. Giordano, Thomas’ Calculus,
(Addison-Wesley, 2005).
[6] M.N. Çankaya and J. Korbel, (2018). Least informative distributions in maxi-
mum q-log-likelihood estimation, Physica A 509, 140-150.
[7] E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive
thermostatistics, Physica A 340 (2004) 95-101.
[8] V.P. Godambe, An optimum property of regular maximum likelihood estima-
tion, Ann. Math. Statist. 31 (1960) 1208-1211.
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[9] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys.
52 (1988) 479-487.
[10] S. Abe, A note on the q-deformation-theoretic aspect of the generalized en-
tropies in nonextensive physics, Phys. Lett. A 224 (1997) 326-330.
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MULTICOVARIANCE AND MULTICORRELATION FOR
p-VARIABLES
Mehmet Niyazi ÇANKAYA1
1Department of International Trading and Finance, Faculty of Applied Sciences, U¸sak University, U¸sak,
1Department of Statistics, Faculty of Arts and Sciences, U¸sak University, U¸sak
Corresponding Author’s E-mail: mehmet.cankaya@usak.edu.tr
ABS TR ACT
The covariance and correlation are important indicators to measure the linear de-
pendence between two variables. The multivariables as p-variables in the research
are commonly observed in the engineering, social science, medical examination,
etc. when p-variables has a negative and positive dependence. In this study, The M-
function in the M-estimation method is used to define a multivariate generalization
for the correlation. A linear dependence among p-variables are generated by use
of the artificial data sets which have a linear dependence and normal distribution.
The different sample sizes and different number of p-variables are used while the
simulation study is performed. Even if the number of p-variables is greater than
the number of sample size, the dependence coefficient shows better performance.
Thus, the numerical results have shown that the proposed multicorrelation coeffi-
cient show better performance to detect the value of degree of dependence among
p-variables.
Keywords Correlation ·M-estimation ·Multicorrelation ·Dependence
References
[1] Prakasa Rao, B. L. S. (1998). Hoeffding identity, multivariance and multicorre-
lation. A Journal of theoretical and applied statistics, 32(1), 13-29.
[2] Díaz, W., Cuadras, C. M. (2017). On a multivariate generalization of the covari-
ance. Communications in Statistics-Theory and Methods, 46(9), 4660-4669.
[3] Rao, B. P., Dewan, I. (2001). Associated sequences and related inference prob-
lems. Handbook of statistics, 19, 693-731.
[4] Gut, A., Gut, A. (2005). Probability: a graduate course (Vol. 200, No. 5). New
York: Springer.
[5] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduc-
tion with Applications, (Birkh¨auser Basel, 2001).
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(ICMASE 2022)
[6] C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a
Complex World, (Springer, New York, 2009).
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(ICMASE 2022)
EXPERIENCE IN TEACHING MATHEMATICS TO ENGINEERS:
STUDENTS V.STEACHER VISION
Cristina M.R. Caridade
ISEC - Coimbra Institute of Engineering, Coimbra, Portugal
Centre For Research in Geo-Space Science (CICGE), Porto, Portugal
Corresponding Author’s E-mail: caridade@isec.pt
ABS TR ACT
Classroom experience is very important for developing the skills necessary for ef-
fective teaching. Even when educational success is defined by test scores, research
shows that teachers’ effectiveness increases with experience and, under favourable
circumstances, continues to increase throughout their entire career. Many experi-
ments have been made in the application of new teaching/learning methodologies
in mathematics. In higher education, more specifically in the teaching of math-
ematics in engineering, these experiences are quite scarce. Collaborative learning
(CL) allows for meaningful learning in which the student takes an active role in their
teaching/learning process, developing a wide range of skills [1]. The CL experience
was carried out in the Calculus I course, with 25 Electrical Engineering students, on
academic year 2021/2022. Differential and Integral Calculus to calculate areas and
volumes of solids of revolution was the chosen theme, using GeoGebra to dynami-
cally model 3D objects and GeoGebra AR to visualize them directly in augmented
reality [2]. The experience had the participation of 6 groups of students and lasted 3
hours. It was necessary to reflect, think and plan the class so that it could play an im-
portant role in improving the quality of teaching and learning at this academic level.
The class was administered by the teacher as a guide, observing and writing down
some information [3]. The students, in addition to developing the proposed activity,
answered an initial and final quiz. This paper intends to present the opinions and
reflections of the students and the teacher about the effect of their participation in a
CL experience. The final observations suggest that CL is an important methodology
in the context of training, with a view to reflection and improvement of pedagogi-
cal practice. As such, it should be included, whenever possible, in the planning of
mathematics lessons for engineering and for higher education in general.
Keywords Teaching and Learning experiences ·Mathematics for engineering ·
Collaborative learning ·Exploratory approach
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References
[1] Herrera-Pavo, M. A., Collaborative learning for virtual higher education, Learn-
ing, Culture and Social Interaction, Elsevier, 28 (2021).
[2] Adhikari G.P., Effect of using GeoGebra software on students’ achieve-
ment at university level, Scholars’ Journal, vol. 3, 47–60, 2020.
https://doi.org/10.3126/SCHOLARS.V3I0.37129
[3] S. A. Lesik, S.A., Do developmental mathematics programs have a causal
impact on student retention? An application of discrete-time survival
and regression-discontinuity analysis, Research in Higher Education, 48(5),
583–608, 2007. https://doi.org/10.1007/s11162-006-9036-1
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International Conference on Mathematics and Its Applications in Science and Engineering
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THE EFFECT (IMPACT)OF PROJECT-BASED LEARNING
THROUGH AUGMENTED REALITY ON HIGHER MATH CLASSES
Cristina M.R. CARIDADE
ISEC - Coimbra Institute of Engineering, Coimbra, Portugal
Centre For Research in Geo-Space Science (CICGE), Porto, Portugal
Corresponding Author’s E-mail: caridade@isec.pt
ABS TR ACT
Mathematics is essential in the training of engineers. A weak mastery in this area
can affect other course subjects that require math skills. The lack of motivation
that students feel in relation to this subject influences their poor academic perfor-
mance [1]. It is necessary that the teaching of Mathematics be more creative and
stimulating, considering modern society and the interests of students. Project-based
learning can help to increase engagement, improve student interaction, and promote
student success in Mathematics. In this paper, project-based learning and its study
are presented to qualitatively assess students’ motivation and performance in this
type of teaching methodologies. During a 3-hour class of the second semester of
Calculus I in Electrical Engineering degree, students were proposed to develop a
project using augmented reality. Different views, examples, and applications for
3D object modelling as well as area and volume calculation using integral calcula-
tion were implemented [2]. Augmented reality was then exposed focusing on the
competencies that are considered in the learning project. To develop the referred
experience, the interactions established between students and student-teacher were
analysed through direct observation [3], the analysis of documents delivered by the
students, the satisfaction questionnaires carried out and the analysis of the eval-
uation grids. The analysis of the collected data allowed us to conclude that the
experience proposed in a classroom context, with the aim of motivating students to
learn Mathematics, proved to be effective and productive.
Keywords Project-based learning ·Mathematics for engineering ·Students’
motivation ·Students’ performance
References
[1] Eltahir, M.E., Alsalhi, N.R., Al-Qatawneh, S. et al. The impact of game-based
learning (GBL) on students’ motivation, engagement and academic perfor-
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(ICMASE 2022)
mance on an Arabic language grammar course in higher education. Educ Inf
Technol 26, 3251–3278 (2021). https://doi.org/10.1007/s10639-020-10396-w
[2] Caridade, C.M.R. GeoGebra augmented reality: ideas for teaching & learning
math. II Internacional Conference on Mathematics and its applications in sci-
ence and Engineering (ICMASE 2021) 01-12 July 2021, Universidad de Sala-
manca.
[3] Gunn, B., Smolkowski, K., Strycker, L.A. et al. Measuring Explicit Instruc-
tion Using Classroom Observations of Student–Teacher Interactions (COSTI).
Perspect Behav Sci 44, 267–283 (2021). https://doi.org/10.1007/s40614-021-
00291-1
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SOME SPECTRAL PROPERTIES OF OPERATORS GENERATED BY
QUANTUM DIFFERENCE EQUATIONS
¸Serifenur CEBESOY ERDAL1, Nuray ORHAN2
1Cankırı Karatekin University
2Cankırı Karatekin University
Corresponding Author’s E-mail: scebesoy@karatekin.edu.tr
ABS TR ACT
In this talk, we first get the q–analogous of the Sturm–Liouville equation which can
be written as
Dqpt
qDqyt
q+r(t)y(t) = 0, t ∈qZ,(7)
where the functions pand rare defined on qZwith p(t)6= 0 for all t∈qZ[6,7].
Equation (1) is called the second order selfadjoint linear homogenous q–difference
equation. After making required calculations using q-derivative, it can be seen that
(1) turns out to be
qγ(t)y(qt) + β(t)y(t) + γt
qyt
q= 0, t ∈qZ,(8)
then we consider the q–difference expression
qγ(t)y(qt) + β(t)y(t) + γt
qyt
q=λy(t), t ∈qZ,(9)
which was intensively studied in [1,2,3,4], where γ(t)6= 0 for all t∈qZ. The
spectral parameter λcan be chosen exponential function as λ:= 2√qcos zor poly-
nomial function as λ:= √q(z+z−1). Over the years, Equation (3) has been
handled in special case under some impulsive conditions and investigated in [5].
The main aim of this talk is to investigate some spectral properties of Equation
(3) such as getting the analytic properties and asymptotic behaviours of the Jost
solution and obtain the continuous spectrum of the related operator.
Keywords q–difference equation ·q–difference operator ·Jost solution ·eigen-
value ·spectral singularity ·asymptotic
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International Conference on Mathematics and Its Applications in Science and Engineering
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References
[1] Adıvar M. and Bohner M., Spectral analysis of q–difference equations with
spectral singularities, Math Comput Modelling, 43(7-8): 695-703, 2006.
[2] Adıvar M. and Bohner M., Spectrum and principal vectors of second order q–
difference equations, J Indian Math, 48(1): 17-33, 2006.
[3] Aygar Y. and Bohner M., On the spectrum of eigenparameter-dependent quan-
tum difference equations, Appl. Math Inf Sci., 9(4): 1725-1729, 2015.
[4] Aygar Y. and Bohner M., Polynomial-type Jost solution and spectral proper-
ties of a selfadjoint quantum difference operator, Complex Anal. Oper. Theory,
10(6): 1171-1180, 2016.
[5] Bohner M. and Cebesoy S., Spectral analysis of an impulsive quantum differ-
ence operator, Math. Meth. Appl. Sci. 42: 5331–5339,2019.
[6] Bohner M. and Peterson A., Dynamic Equations on Time Scales. An Introduc-
tion with Applications, Boston, MA: Birkhäuser Boston, Inc, 2001.
[7] Kac V. and Cheung P., Quantum Calculus, New York: Universitext Springer-
Verlag, 2002.
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(ICMASE 2022)
THE THEORY OF DIRAC EQUATIONS AND SYSTEM OF
DIFFERENCE EQUATIONS
¸Serifenur CEBESOY ERDAL1
1Cankırı Karatekin University
Corresponding Author’s E-mail: scebesoy@karatekin.edu.tr
ABS TR ACT
Spectral analysis, a sub–branch of applied mathematics and functional analysis, ex-
amines the solutions of boundary and initial value problems using "operator theory".
Operator theory is no doubt a diverse area that has grown out of linear algebra and
complex analysis, and is often described as the branch of functional analysis that
deals with bounded linear operators and their spectral properties. In mathematical
models, an operator is determined compatible with the differential equation used
and the properties of the operator are investigated. Differential operators generated
by the differential equations are the first operator type handled in the literature. In
particular, Sturm–Liouville operator, known as one dimensional Schrödinger oper-
ator is the most widely studied operator having a tremendous potential for applica-
tions in quantum theory.
Since the beginning of 18th. century, various investigations have been done by
various authors, but with the pioneering of Naimark. In his first study, Naimark
considered the Sturm–Liouville operator generated by the Sturm–Liouville equa-
tion
−y00 +q(x)y=λy, x ∈R+:= [0,∞)(10)
and the initial condition
y0(0) −hy(0) = 0,(11)
where qis a complex potential and λis a spectral parameter [6]. Also, the mod-
ellings of certain problems in engineering, physics, control theory and quantum
mechanics and other areas have led to a rapid development of the theory of Dirac
equations and discrete Dirac equations.
In this presentation, at first, we introduce the matrix equation
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International Conference on Mathematics and Its Applications in Science and Engineering
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Kdψ
dx +P(x)ψ=µψ, ψ(x) = ψ1(x)
ψ2(x), x ∈[0, π],(12)
where
K=0 1
−1 0, P (x) = P11 (x)P12(x)
P21(x)P22(x), P12(x) = P21(x),
Pij are real valued functions which are continuous on the interval [0, π]for i, j =
1,2and µis a spectral parameter. It follows from (3) that
0 1
−1 0ψ0
1(x)
ψ0
2(x)+P11(x)P12(x)
P21(x)P22(x)ψ1(x)
ψ2(x)=µψ1(x)
ψ2(x).
Using the last equality, we have
ψ0
2(x)
−ψ0
1(x)+P11(x)ψ1(x) + P12 (x)ψ2(x)
P21(x)ψ1(x) + P22 (x)ψ2(x)=µψ1(x)
µψ2(x).
Then, we arrive at that the Equation (3) is equivalent to the system of two simulta-
neous first order ordinary differential equations
(ψ0
2+P11(x)ψ1+P12(x)ψ2=µψ1,
−ψ0
1+P21(x)ψ1+P22(x)ψ2=µψ2.(13)
In the case that P12(x) = P21(x)=0, P11(x) = s(x) + m, P22(x) = s(x)−m,
where sis a potential function and mis the mass of a particle, the system (4) is
called a one dimensional stationary Dirac system in relativistic quantum theory
[5]. Many studies exist about the spectral theory of Equation (4) [4,5,7]. Over the
years, similar spectral properties have been obtained for the discrete case of Dirac
equations and extensively studied in [1-3].
Keywords Dirac equation ·Discrete Dirac equation ·Jost solution ·Wronskian ·
eigenvalue
References
[1] Bairamov E. and Celebi A. O., Spectrum and spectral expansion for a non-
selfadjoint discrete Dirac operators, Quart. J. Math. Oxford Ser., 50(2): 371-
384, 1999.
[2] Bairamov E. and Coskun C., Jost solutions and the spectrum of the system of
difference equations, Appl. Math. Lett., 17:1039-1045, 2004.
[3] Bairamov E. and Coskun C., The structure of the spectrum of a system of dif-
ference equations, Appl. Math. Lett., 18:387-394, 2005.
[4] Gasymov M. G. and Levitan B. M., Determination of the Dirac system from
scattering phase, Sov. Math. Dokl., 167: 1219-1222, 1966.
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[5] Levitan B. M. and Sargsjan I. S., Sturm–Liouville and Dirac Operators, Vol 59
of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht,
The Netherlands, 1991.
[6] Naimark M. A., Linear differential operators II, Ungar, New York, 1968.
[7] Roos B. W. and Sangren W. C., Spectral theory of Dirac’s radial relativistic
wave equation, J. Math. Physics, 3: 882-890, 1962.
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International Conference on Mathematics and Its Applications in Science and Engineering
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RENEWED NEUTROSOPHIC SOFT GRAPHS WITH SOME NEW
OPERATIONS
Yıldıray ÇEL˙
IK
Department of Mathematics, University of Ordu, 52200, Ordu, Turkey
E-mail: ycelik61@gmail.com
ABS TR ACT
It is celarly state that uncertainty arises from various areas cannot be captured within
a single mathematical approach. Mathematical modelling of problems with uncer-
tainty and development of solutions accordingly is one of the most important issue
in interdisciplinary research. For this reason, many theory have been developed
for solving problems involving uncertainty. Some of these are fuzzy set theory,
intuitionistic fuzzy set theory and fuzzy soft set theory. On the other hand, the neu-
trosophic set is a new mathematical approach which is developed for dealing with
incomplete and indeterminate information. Neutrosophic set is a generalization of
the intuitionistic fuzzy set theory. The neutrosophic sets are expressed with the help
of three membership functions named truth, indeterminacy and falsity membership
function. As compared to the other fuzzy models, the neutrosophic soft models
provide more sensitive evaluation for the complex systems. Graph theory which
is used to solve the complicated problems in many different fields is an important
mathematical tool. Graphs are used to put forth a relationship between elements in
a given set. Graph theory and fuzzy graph theory are finding an many number of
applications in modeling complicated systems because of its provide conveniences.
Theoretical point of view, graph structures have been many times evaluated on dif-
ferent sets especially soft sets, fuzzy soft sets, neutrosophic sets, neutrosophic soft
sets etc. This study is designed with the renewed concept of neutrosophic soft graph
structures which is a combination of graphs and neutrosophic soft sets. We redefine
notions of neutrosophic soft graphs and neutrosophic soft subgraphs from a differ-
ent perspective taking into account the shortcomings of previous studies. Also, we
introduce some new operations on neutrosophic soft graphs and elaborate them with
suitable examples by using neutrosophic soft sets. Moreover, we investigate some
remarkable properties of neutrosophic soft graphs via concepts given.
Keywords Neutrosophic soft set ·Graph ·Neutrosophic soft graph
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International Conference on Mathematics and Its Applications in Science and Engineering
(ICMASE 2022)
References
[1] Akram M, and Nawaz S., Operations on Soft Graphs, Fuzzy Information and
Engineering, 7(4): 423-449, 2015.
[2] Akram M, and Nawaz S., Fuzzy Soft Graphs with Applications, Journal of In-
telligent and Fuzzy Systems, 30(6): 3619-3632, 2016.
[3] Akram M, and Sundas S., Neutrosophic soft graphs with application, Journal of
Intelligent and Fuzzy Systems, 32(1): 841-858, 2017.
[4] Euler L., Solutio problematis ad geometriam situs pertinentis, Commentarii
Academiae Scientiarum Imperialis Petropolitanae, 8: 128-140, 1736.
[5] Kandasamy W.B, Ilanthenral K, and Smarandache F., Neutrosophic Graphs: A
New Dimension to Graph Theory, EuropaNova, USA, 2015.
[6] Maji P.K., Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics,
5(1): 157-168, 2013.
[7] Mordeson J.N, and Peng C.S., Operations on fuzzy graphs, Information Sci-
ences, 79: 159-170, 1994.
[8] Smarandache F., Neutrosophic set-a generalization of the intuitionistic fuzzy
set, Granular Computing, IEEE International Conference, 38-42, 2006.
[9] Zadeh L.A., Fuzzy sets, Information and Control, 8: 338-353, 1965.
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A GENERALIZATION OF MULTIPLE ZETA VALUES
Roudy EL HADDAD
Engineering Department, Sorbonne University, France
Corresponding Author’s E-mail: roudy1581999@live.com
ABS TR ACT
This study consists of two parts: Part 1: Recurrent Sums [23] and Part 2: Multiple
Sums [24]. A combined abstract for both is as below:
Multiple zeta star values (MZSV) and Multiple zeta values (MZV) have become of
great interest due to their numerous applications in mathematics and physics. In
this study, we propose a generalization of MZSVs and MZVs, which we will refer
to as recurrent sums and multiple sums, where the reciprocals are replaced by arbi-
trary sequences. We introduce a toolbox of formulas for the manipulation of such
sums. We begin by developing variation formulas that express the variation of such
sums in terms of lower order recurrent/multiple sums. We then proceed to derive
theorems (which we will call inversion formulas) which show how to interchange
the order of summation in a multitude of ways. Additionally, we derive a set of par-
tition identities that we use to prove a reduction theorem that expresses such sums
as a combination of simple non-recurrent sums. We use these theorems to derive
new results for multiple zeta (star) values and recurrent/multiple sums of powers.
Later, we present a variety of applications including applications concerning poly-
nomials and MZVs such as generating functions and expressions for ζ({2p}m)and
ζ?({2p}m). Finally, we establish the connection between multiple sums and recur-
rent sums. By exploiting this connection, we provide additional partition identities
for odd and even partitions.
Keywords Recurrent sums, Multiple sums, Partitions, Multiple zeta star values,
Multiple zeta values, Riemann zeta function, Viète’s formula, Polynomials,
Generating function, Faulhaber formula.
References
[1] Andrews, G. E. (1998). The Theory of Partitions, Vol. 2. Cambridge University
Press.
[2] Aoki, T., Ohno, Y., & Wakabayashi, N. (2011). On generating functions of
multiple zeta values and generalized hypergeometric functions. Manuscripta
Mathematica, 134(1), 139–155.
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[3] Arfken, G. B., & Weber, H. J. (2000). Mathematical Methods for Physicists,
5th Edition. Academic Press.
[4] Bernoulli, J. (1689). Propositiones arithmeticae de seriebus infinitis earumque
summa finita [Arithmetical propositions about infinite series and their finite
sums]. Basel, J. Conrad.
[5] Bernoulli, J. (1713). Ars Conjectandi, Opus Posthumum; Accedit Tractatus De
Seriebus Infinitis, Et Epistola Gallicè scripta De Ludo Pilae Reticularis [Theory
of inference, posthumous work. With the Treatise on infinite series. . . ]. Thur-
nisii.
[6] Bernoulli, J. (1742). Corollary III of de Seriebus Varia. Opera Omnia. Lausanne
& Basel: Marc-Michel Bousquet & co. 4:8.
[7] Blümlein, J., Broadhurst, D., & Vermaseren, J. A. (2010). The multiple zeta
value data mine. Computer Physics Communications, 181(3), 582–625.
[8] Blümlein, J., & Kurth, S. (1999). Harmonic sums and Mellin transforms up to
two-loop order. Physical Review D, 60(1), 014018.
[9] Broadhurst, D. (1986). Exploiting the 1,440-fold symmetry of the master two-
loop diagram. Zeitschrift für Physik C Particles and Fields, 32(2), 249–253.
[10] Broadhurst, D. (2013). Multiple zeta values and modular forms in quantum
field theory. In Computer Algebra in Quantum Field Theory, Springer, 33–73.
[11] Bruinier, J. H., & Ono, K. (2013). Algebraic formulas for the coefficients of
half-integral weight harmonic weak Maass forms. Advances in Mathematics,
246, 198–219.
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SOME RESULTS FOR MATRIX STURM-LIOUVILLE EQUATIONS
WITH A POINT<