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DRAFT OF ABSTRACT BOOKLET OF MICOPAM 2019

Draft of Abstract Booklet of The 2nd Mediterranean

International Conference of Pure & Applied

Mathematics and Related Areas 2019

(MICOPAM 2019)

Contents

1 CONTRIBUTED SPEAKERS 1

The extension of some Diophantine triples 2Alan Filipina

Automated inequality proving and applications 3Branko Malesevica

Orthogonality on the Semicircle in the Complex Plane and QuadratureFormulas 5

Gradimir V. Milovanovica

Rank codes based cryptography 6Olivier Ruattaa

∆h−Gould-Hopper Appell Polynomials 7Mehmet Ali Ozarslana, Banu Yılmaz Yasarb

Accuracy tests on built-in algorithms from Mathematica applied to thechaotic system 9

M.C Kekanaa, M.Y Shatalovb, T.O Tongb, S.P Moshokoac

New Topological Results on Omega Invariant 10Ismail Naci Cangula, Aysun Yurttasa, Muge Togana, Sadik Delena

Mass-Preserving High-Order Characteristic Finite Volume Scheme forConvection Diffusion Equations and Environmental Computations 11

Dong Lianga, Kai Fub

The multiplicity of positive solutions for a class of quasilinear boundaryvalue problems 13

Hafidha Sebbagha , Mohammed Derhabb

i Paris - FRANCE

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Iterate problem in Roumieu spaces of systems of interior and boundarydifferential operators 14

Rachid Chailia

New constructions on a special type of General products over monoids 15Ahmet Sinan Cevika, Suha Ahmad Wazzanb, Firat Atesc

Finding a Quintic spline for solving Cauchy Initial Value Problem 16Kulbhushan Singh

Mathematical Analysis of a discrete SEIR epidemic model with treat-ment 17

Mahmoud H. DarAssia

On p-adic multiple Barnes-Euler zeta functions and the correspondinglog gamma functions 18

Su Hua , Min-Soo Kimb

Generalizations of Ostrowski type inequalities via Hermite polynomials 22Ljiljanka Kvesica , Josip Pecaricb , Mihaela Ribicic Penavac

Recurrence Relations and Difference Equation for the ω-Multiple Char-lier Polynomials 23

Mehmet Ali Ozarslana, Gizem Baranb

Stability analysis of a generalized TCP-RED algorithm for Internetcongestion control 25

A. Gimeneza, Jose M. Amigoa, G. Duranb, O. Martınez-Bonastrea, J. Valeroa

Jensen-type operator inequalities for bounded, Lipschitzian and higherorder convex functions 27

Mario Krnica, Rozarija Mikicb, Josip Pecaricc

Best Proximity Point Results for Proximal Nonexpansive MultivaluedMappings on starshaped sets 28

N. Bunluea,∗, Y. J. Chob , S. Suantaic

Linking Bernoulli and Euler Polynomials and their interpolation prob-lem 29

Mehmet Bozera, Mehmet Ali Ozarslana

Arc spaces and partition identities 30Hussein Mourtadaa

A note on q-associated polynomials 31Rahime Dere Pacina

A new approach to solve multi-attribute decision making problems bylogarithmic operational laws in interval neutrosophic environment 32

T. S. Haquea, S. Alama,∗, and S. Dalapatia

Bivariate h-Mittag-Leffler Functions with 2D-h-Laguerre-Konhauser Poly-nomials Corresponding h-Fractional Operators 34

Cemaliye Kurta, Mehmet Ali Ozarslana

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Generating Functions and Difference Equations for ω-Multiple MeixnerPolynomials 36

Sonuc Zorlu Ogurlua Ilkay Elidemirb

Kinematic Parallel Surfaces 37Erhan Ataa, Yasemin Yıldırıma, Derya Kahvecib, Yusuf Yaylıb

Anti–Gaussian quadrature rule for trigonometric polynomials 40Marija P. Stanica, Nevena Z. Petrovica, Tatjana V. Tomovica

A generalization of S-Noetherian rings 41Dong Kyu Kima, Jung Wook Lima

Star global dimension of power series rings over w-coherent rings 42Minjae Kwona, Jung Wook Lima

A generalization of Hurwiz polynomial rings- Gaussain polynomial rings 43Seung Min Leea , Jung Wook Lima

2 POSTER PRESENTATIONS 44

An enhanced algorithm for non-convex cost function in machine learn-ing 45

Sunyoung Bua

High-order time discretization schemes for simulation of the Burgers’equations 46

Soyoon Baka, Yonghyeon Jeona, Sunyoung Bub

A Note on Ruled Submanifolds in Minkowski Space with Gauss map 48Sun Mi Junga

Some annihilator conditions in generalized composite Hurwitz rings 49Dong Kyu Kwona, Jung Wook Lima

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1 CONTRIBUTED SPEAKERS

1 Paris - FRANCE

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The extension of some Diophantine triples

Alan Filipina

Abstract

Diophantine m-tuple is the set of m positive integers such that the productof any two of them increased by 1 is a perfect square. One of the most interestingquestion, and problem various mathematicians tried to solve, is how large thosesets can be. Recently, He, Togbe and Ziegler proved the folklore conjecture thatthere does not exist a Diophantine quintuple. There is a stronger version ofthat conjecture which states that every Diophantine triple can be extended toa quadruple with a larger element in a unique way. That is still an open problem.

In this talk, we consider the extension of the infinite two-parameter familyof Diophantine triples. More precisely, we prove the following: Let a and b bepositive integers defined by a = KA2, b = 4KA4 + 4εA with K, A positiveintegers and ε = ±1. Define an integer c = c±ν by

c±ν =1

4ab((√b±√a)2(r +

√ab)2ν + (

√b∓√a)2(r −

√ab)2ν − 2(a+ b)),

with ν positive integer. Then, there is a unique extension of the Diophantinetriple a, b, c to a quadruple with a larger element.

We also prove the stronger version of Diophantine quintuple conjecture forthe triples of the form a, b, c where

a

(a+

7

2− 1

2

√4a+ 13

)≤ b ≤ 4a2 + a+ 2

√a.

This is joint work with M. Cipu and Y. Fujita.

2010 Mathematics Subject Classifications : 11D09, 11B37, 11J86Keywords: Diophantine m-tuples, Pell equations, Linear forms in logarithms.

aFaculty of Civil Engineering, University of Zagreb, Croatia.E-mail : [email protected]

2 Paris - FRANCE

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Automated inequality proving and

applications

Branko Malesevica

Abstract

An overview of the development of automatic inequality proving and itsapplications is presented in this paper. The presented automatic prover is de-veloped for a class of mixed trigonometric polynomial inequalities. To illustratehow the prover works, we showcase the proofs of several open problems from thetheory of analytical in-equalities which were recently published and proven bythe unified method. By taking into consideration double sided Taylors approxi-mations, we give the basis of a significant expansion of the developed automatedprover over the class of real analytic inequalities on the real interval.

2010 Mathematics Subject Classifications : 68T15, 42A10, 26D05Keywords: Automated inequality proving, Taylor’s approximations, Double-

sided Taylor’s approximations.Acknowledgements: The work was partially supported by the Serbian Ministry of

Education, Science and Technological Development, under Projects ON 174032 & III 44006.

References

[1] D.S. Mitrinovic, Analytic Inequalities, Springer 1970.

[2] G.V. Milovanovic, M.Th. Rassias - ed, Analytic number theory, approximationtheory and special functions, Springer 2014.

[3] B. Malesevic, One method for proving inequalities by computer, J. Inequal. Appl.2007 Article ID 78691 (2007), 1–8.

[4] B. Banjac, M. Makragic, B. Malesevic, Some notes on a method for proving in-equalities by computer, Results Math. 69:1 (2016), 161–176.

[5] B. Malesevic, M. Makragic, A Method for Proving Some Inequalities on MixedTrigonometric Polynomial Functions, J. Math. Inequal. 10:3 (2016), 849–876.

[6] T. Lutovac, B. Malesevic, C. Mortici, The natural algorithmic approach of mixedtrigonometric-polynomial problems, J. Inequal. Appl. 2017:116 (2017), 1–16.

[7] B. Malesevic, I. Jovovic, B. Banjac, A proof of two conjectures of Chao-Ping Chenfor inverse trigonometric functions, J. Math. Inequal. 11:1 (2017), 151–162.

[8] B. Malesevic, T. Lutovac, B. Banjac, A proof of an open problem of YusukeNishizawa for a power-exponential function, J. Math. Inequal. 12:2 (2018), 473–485.

[9] B. Malesevic, M. Rasajski, T. Lutovac, Double-sided Taylor’s approximationsand their applications in Theory of analytic inequalities (in Th.M. Rassias, D.Andrica - ed: Differential and Integral Inequalities, Springer, Sepember 2019.arXiv:1811.10124)

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[10] B. Malesevic, T. Lutovac, M. Rasajski, B. Banjac, Double-sided Taylor’s approx-imations and their applications in theory of trigonometric inequalities (in M.Th.Rassias, A. Raigorodskii - ed: Trigonometric Sums and their Applications, toappear, Springer 2019. arXiv:1906.04641)

[11] B. Banjac, System for automatic proving of some classes of analytic inequalities,Doctoral dissertation (in Serbian), School of Electrical Engineering, Belgrade, May2019. Available via: http://baig.etf.rs/

a School of Electrical Engineering, University of Belgrade, Bule-var Kralja Aleksandra 73, 11000 Belgrade, SerbiaE-mail : [email protected]

4 Paris - FRANCE

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Orthogonality on the Semicircle in the

Complex Plane and Quadrature Formulas

Gradimir V. Milovanovica

Abstract

One nonstandard type of orthogonality, the so-called orthogonality on thesemicircle, was introduced by Gautschi and Milovanovic [1]. In general, theinner product can be introduced as

〈f, g〉 =

∫Γ

f(z)g(z)w(z)(iz)−1dz =

∫ π

0

f(eiθ)g(eiθ)w(eiθ)dθ,

where Γ is the semicircle Γ =z ∈ C : z = eiθ, 0 ≤ θ ≤ π

and w is

an appropriate complex weight function (see [2] and [3]). This product is notHermitian, but the corresponding (monic) orthogonal polynomials exist uniquelyand satisfy a three-term recurrence relation.

In this talk, a few new classes of polynomials and rational functions, orthog-onal in the complex plane with respect to the non-Hermitian inner products areconsidered, as well as the construction of the corresponding quadrature formu-las of Gaussian type. Distribution of zeros, special cases and applications innumerical analysis are presented.

2010 Mathematics Subject Classifications : 30C15, 42C05, 65D30.Keywords: Complex orthogonal systems, Moments, Recurrence relations, Quadra-

ture formula, Zeros

References

[1] W. Gautschi, G. V. Milovanovic, Polynomials orthogonal on the semicircle, J.Approx. Theory, 46 (1986),230–250.

[2] W. Gautschi, H. J. Landau, G. V. Milovanovic, Polynomials orthogonal on thesemicircle. II, Constr. Approx., 3 (1987), 389–404.

[3] G. V. Milovanovic, Complex orthogonality on the semicircle with respect to Gegen-bauer weight: theory and applications, In: Topics in Mathematical Analysis (Th.M. Rassias, ed.), pp. 695–722, Ser. Pure Math., 11, World Sci. Publ., Teaneck,NJ, 1989.

aSerbian Academy of Sciences and Arts, Belgrade, Serbia

E-mail : [email protected]

5 Paris - FRANCE

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Rank codes based cryptography

Olivier Ruattaa

Abstract

In this talk, we will introduce rank metric codes and show how they canbe used to design cryptosystems. We will focus on the cryptosystem ROLLObased on Low Rank Parity Check codes which is still on competition for theNIST post-quantum cryptography contest.

aMATHIS-XLIM UMR 7252 Universite de Limoges-CNRSE-mail : [email protected]

6 Paris - FRANCE

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∆h−Gould-Hopper Appell Polynomials

Mehmet Ali Ozarslana, Banu Yılmaz Yasarb

Abstract

In the present paper, we introduce the ∆h−Gould-Hopper Appell polynomi-als An(x, y;h) via h−Gould-Hopper polynomials Ghn(x, y). First, we list mainproperties of the h−Gould-Hopper polynomials. Then, we obtain an explicitform of the ∆h−Gould-Hopper Appell polynomials. Moreover, we obtain deter-minantal form of theof them in terms of the h− Gould-Hopper polynomials. Asa special case of ∆h−Gould-Hopper Appell polynomials, we present the explicitforms and determinants satisfied by ∆h− Appell polynomials, 2D ∆− Appellpolynomials, ∆− Appell polynomials, 2D Appell polynomials and Appell poly-nomials. Furthermore, we find explicit form, recurrence relation, shift operatorsand difference equation satisfied by ∆h−Gould-Hopper Appell polynomials. Inthe special cases, we present the corresponding explicit forms, determinantalforms, recurrence relations, difference equations, differential equations satis-fied by the degenerate ∆h−Gould-Hopper Bernoulli, Euler, Genocchi, Boole,Bernoulli polynomials of the second kind; degenerate ∆h− Bernoulli, Euler,Genocchi, Booole polynomials, Bernoulli polynomials of the second kind.

2010 Mathematics Subject Classifications : 11B68, 33E20, 39A70, 26C05,33E30,47B39

Keywords: Degenerate Hermite polynomials, discrete heat equation, determi-nantal form, recurrence relation, difference equation, differential equation.

References

[1] P. Appell, J. Kampe de Feriet, Fonctions hypergeometriques et hyperspheriques.Polynomes d’ Hermite, Gauthier-Villars, Paris, 1926.

[2] G. Bretti, P.E. Ricci, Multidimensional extension of the Bernoulli and Appellpolynomials. Taiwan. J. Math 8(3), 415-428 (2004).

[3] M. Brickenstein, A. Dreyer, Grobner-free normal forms for Boolean polynomials,ISSAC, 2008, 55-62, ACM, New York, 2008.

[4] L. Carlitz, A degenerate Staudt-Clausen theorem. Arch. Math. (Basel) 7, 28-33(1956).

[5] F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, Jour-nal of Computational and Applied Mathematics 234 (2010) 1528-1542.

[6] F. A. Costabile, E. Longo, ∆h− Appell sequences and related interpolation prob-lem, Numer Algor (2013) 63:165-186.

[7] M. X He, P.E. Ricci, Differential equation of Appell polynomials via the factor-ization method. J. Comput. Appl. Math. 139 (2), 231-237 (2002).

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[8] W. A. Khan, A Note on Degenerate Hermite Poly-Bernoulli Numbers and Poly-nomials, Journal of Classical Analysis, Volme 8, Number1 (2016), 65-76.

[9] S. Khan, T. Nahid, M. Riyasat, On degenerate Apostol-type polynomials andapplications, Boletin de la Sociedad Matematica Mexicana, 2018.

[10] H. Ozden, Y. Simsek, H. M. Srivastava, A unified presentation of the generatingfunctions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput-ers and Mathematics with Applications 60 (2010) 2779-2787.

[11] M. A. Ozarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchipolynomials, Adv. Difference Equ., 116 (2013), 13 pages.

[12] M. A. Ozarslan, B. Yılmaz, A set of finite order differential equations for theAppell polynomials, J. Comp and Appl. Math., 259 (2014), 108-116.

[13] H. M. Srivastava, M. A. Ozarslan, Difference equations for a class of twice iterated∆h−Appell sequences of polynomials, RACSAM (2019), 113, 1851-1871.

[14] H. M. Srivastava, M. A. Ozarslan, B. Yılmaz, Some families of differential equa-tions associated with the Hermite based Appell polynomials, Filomat, 28 (4) 2014,695-708.

[15] S. Varma, B.Y. Yasar, M.A. Ozarslan, Hahn-Appell polynomials and their d-orthogonality, RACSAM (2019), 113, 2127-2143.

[16] B. Yılmaz, M. A. Ozarslan, Differential Equations of the Extended 2D Bernoulliand Euler Polynomials, Advances in Difference Equations, 2013, 107.

aDepartment of Mathematics, Faculty of Arts and Science Univer-sity of Eastern Mediterranean, Famagusta , Cyprus.bDepartment of Mathematics, Faculty of Arts and Science Universityof Eastern Mediterranean, Famagusta , Cyprus.E-mail : [email protected] ,[email protected]

8 Paris - FRANCE

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Accuracy tests on built-in algorithms from

Mathematica applied to the chaotic system

M.C Kekanaa, M.Y Shatalovb, T.O Tongb, S.P Moshokoac

Abstract

In this presentation, A mathematical framework of checking accuracy ofbuilt-in algorithms by residuals is developed and investigated. Lorenz systemand Duffing with periodic excitation are used as case studies on built-in algo-rithms from Mathematica. The algorithms used include Adams, Backward dif-ferential formula and Explicit Runge-Kutta with difference-order. The graphicalresults are showed.

2010 Mathematics Subject Classifications : 74H15, 45G10, 65D32, 65D30,65G99

Keywords: Lorenz system, Mathematica software, Runge-Kutta, residual func-tion.

References

[1] S.V.Joubert and J.C. Greeff, Accuracy estimates for computer algebra systeminitial value problem(IVP) solvers, S.A. J. Sci. 102 (2006), 46–50.

[2] M.C.Kekana, M.Y.Shatalov, S.P. Moshokoa and E.L. Voges, Accuracy monitoringof initial value problem solution by means of residual function using Mathematicasoftware, G. J. for Pure. and App 14 (2018), 81–90.

aDepartment of Mathematics, Faculty of Science Tshwane Universityof Technology, Pretoria 0001, South Africa.E-mail : [email protected]

9 Paris - FRANCE

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New Topological Results on Omega

Invariant

Ismail Naci Cangula, Aysun Yurttasa, Muge Togana, Sadik Delena

Abstract

A new graph invariant for a given degree sequence has recently been intro-duced by Delen and Cangul. It can be calculated for a given graph, too. Thisnew invariant called omega resembles to the well-known Euler characteristic andcyclomatic number of a graph. It gives a new realization theorem for the degreesequences and also helps one to find many algebraic, graph theoretical, numbertheoretical, topological and combinatorial properties of all the realizations ofthe given degree sequence including cyclicity, connectivity, numbers of compo-nents, cycles, loops, chords, multiple edges, pendant and support vertices, etc.Since 2018, several applications of this invariant have been found. In this work,we shall introduce the omega invariant together with some fundamental com-binatoric and topologic properties and obtain new applications to some graphindices, matching number, connectivity, etc.

2010 Mathematics Subject Classifications : 05C07, 05C30, 05C40, 94C15Keywords: degree sequence, omega invariant, realizability, connectedness, match-

ing number.

References

[1] S. Delen, I. N. Cangul, A New Graph Invariant, Turkish Journal of Analysis andNumber Theory, 6 (1), 30-33 (2018).

[2] S. Delen, I. N. Cangul, Extremal Problems on Components and Loops in Graphs,Acta Mathematica Sinica, English Series, 35 (2), 161-171 (2019).

aDepartment of Mathematics, Faculty of Arts and Science, BursaUludag University, Bursa 16059, Turkey.E-mail : [email protected], [email protected], [email protected],[email protected]

10 Paris - FRANCE

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Mass-Preserving High-Order Characteristic

Finite Volume Scheme for Convection

Diffusion Equations and Environmental

ComputationsDong Lianga, Kai Fub

Abstract

For solving convection diffusion equations, the methods of characteristics(MOC) have been developed, which incorporate the fixed Eulerian grids withLagrangian tracking along the characteristics to treat the advective part of theequations. Douglas and Russell first proposed a modified method of character-istics for solving one dimensional convection diffusion equations in [1]. Lately,developments of the characteristics methods were carried out to solve convec-tion diffusion equations in high dimensions. They can not only reduce the non-physical oscillation and excessive numerical dispersion but also have no stabilityconstraints required on the time step. However, these MOCs fail to preservemass. Recently, Arbogast and Huang [2, 3] developed a modified characteristics-mixed method that solves the problem of inaccurate concentration densitiescaused by incorrect volume of characteristic trace-back region. The method canpreserve mass balance identity well, but is of first order accuracy in time.

In this talk, we develop a mass-preserving temporal second order and spatialfourth order characteristic finite volume scheme for two dimensional convectiondiffusion equations. While the characteristics tracking is applied to treat theconvection term, we use conservative interpolation to treat the convective inte-grals over the irregular tracking volume cells at the previous time level. A timesecond order discretization by averaging along the characteristics is proposed forthe diffusion term, where the diffusion fluxes are approximated by high orderspatial discrete operators. The developed characteristic finite volume methodpreserves mass and has fourth order accuracy in space and second order accuracyin time. Numerical experiments show its excellent performance. Atmosphericenvironmental simulations are further carried out by scheme. The developedalgorithm can be used to solve the large scale environmental problems in realapplications.

2010 Mathematics Subject Classifications : 65M08, 76R99, 86A10Keywords: Fourth order in space; second order in time; mass-preserving; char-

acteristic FV; convection diffusion.

References

[1] J. Douglas, Jr and T. F. Russell, Numerical methods for convection-dominateddiffusion problems based on combining the method of characteristics with finiteelement or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.

[2] T. Arbogast and C.-S. Huang, A fully mass and volume conserving implementationof a characteristic method for transport problems, SIAM J. Sci. Comput., 28(2006), 2001-2022.

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[3] T. Arbogast and C. Huang, A fully conservative eulerian-lagrangian method for aconvection-diffusion problem in a solenoidal field, J. Comput. Phys., 229 (2010),3415-3427.

aDepartment of Mathematics and Statistics, York University, Toronto,ON, M3J 1P3, Canada

bSchool of Mathematical Sciences, Ocean University of China, Qing-dao, Shandong, 266100, ChinaE-mail : [email protected]

12 Paris - FRANCE

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The multiplicity of positive solutions for a

class of quasilinear boundary value

problems

Hafidha Sebbagha , Mohammed Derhabb

Abstract

By using the time-mapping approach we study the exact number of positivessolutions for the following quasilinear boundary value problem. − (ϕp (uα)ϕp (u′))

′= λf(u)in (0, 1) ,

u > 0in (0, 1) ,u (0) = u (1) = 0,

(1)

where ϕp (y) = |y|p−2 y, y ∈ R, α > 0, p > 1, a > 0, λ > 0 and f(u) =up−1(1− up−1)(up−1 − a).

This problem with p = 2 occur in models. If α > 0 this models a substancewhose particles have a little movement if there are very few of them and whosediffusion velocity increases with the number of particles present in the spaceelement, modelling for example flows through porous media. The case α < 0models a situation where particles have a very high velocity if there are few anda very low one if their density is large, thus modelling some effect of stickiness.

2010 Mathematics Subject Classifications : 34B15-34C10Keywords: p-Laplacian, positive solutions, quadrature method.

References

[1] I. Addou, S. M. Bouguima, M. Derhab & Y. S. Raffed, On the number of solutionsof a quasilinear elliptic class of B. V. P. with jumping nonlinearities, Dynamic Syst.Appl. 7 (4) (1998), 575-599.

[2] 1. D. Aronson, M. G. Crandall, L. A. Peletier, Stabilization of solutions of degen-erate nonlinear diffusion problem, Nonlinear Anal. 6 10 (1982) 1001-1022.

[3] 2. M. Guedda & L. Veron, Bifurcation phenomena associate to the P-Laplacianoperator, Trans. Amer. Math. Soc. 310 (1988), 419-431.

[4] 3. Renate Schaaf, Global solution branches of two-point boundary value problems,Lecture Notes in Mathematics 1458, Springer-Verlag Berlin Heidelberg 1990.

aHigh school on applied sciences,Tlemcen, Algeria.bUniversity Abou Bekr Belkaid, Tlemcen, Algeria.

E-mail : [email protected] ,[email protected]

13 Paris - FRANCE

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Iterate problem in Roumieu spaces of

systems of interior and boundary

differential operators

Rachid Chailia

Abstract

The purpose of this work is to show the iterate theorem in Roumieu classesof both systems of differential operators, one of interior differential operatorsand the other of boundary differential operators.

Let (Mk) be a Roumieu sequence of positive real numbers.

Definition 1. Let K be a compact set of Rn, we call Roumieu class in K, andwe denote RM (K), the space of the restrictions over K of C∞ functions u in aneighborhood of K such that

∃C > 0,∀α ∈ Zn+ : ‖Dαu‖L2(K) ≤ C|α|+1M|α|

Let Ω be an open bounded set of Rn, with smooth boundary Γ, and let(Pj)1≤j≤N [resp. (Bλ)1≤λ≤p ] be a system of differential operators in Ω, of

order m with coefficients in RM(Ω), [resp. in Γ with coefficients in RM (Γ) and

of order νλ ≤ m− 1. ]

Definition 2. We call Roumieu vector of the systems (Pj)1≤j≤N and (Bλ)1≤λ≤p

in Ω every function u ∈ C∞(Ω)such that

∃L ≥ 0, ∀l ∈ N, 1 ≤ j ≤ l, 1 ≤ ij ≤ N, k ∈ N :‖Pi1 ...Pilu‖L2(Ω) ≤ Ll+1Mlm

‖BλPi1 . . . Pilu‖Hm−νλ−1/2+km(Γ)≤ Ll+k+1M(l+k+1)m, 1 ≤ λ ≤ p

The space of these functions is denoted RM(

Ω, (Pj)Nj=1 , (Bλ)pλ=1

).

Theorem 3. If (Pj)Nj=1 is elliptic then

RM(

Ω, (Pj)Nj=1 , (Bλ)pλ=1

)⊂ RM

(Ω).

References

[1] N. Burger, Powers and Gevrey’s regularity for a system of interior and boundarydifferential operators, Bollettino U.M.I., (6) 1-B, (1982), 1-16.

[2] R. Chaıli, Systems of differential operators in anisotropic Roumieu classes, Rend.Circ. Mat. Palermo, (2013) 62, 189-198.

aDepartment of Mathematics, Faculty of Mathematics and ComputerScience, USTMB, Oran DZ-31000, Algeria.E-mail : [email protected]

14 Paris - FRANCE

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New constructions on a special type of

General products over monoidsAhmet Sinan Cevika, Suha Ahmad Wazzanb, Firat Atesc

Abstract

In this talk, for arbitrary monoids A and B, we will first recall the definitionof the generalization General (or Zappa) product of A⊕B by B⊕A (cf. [2]). Afterthat, we will give necessary and sufficient conditions for this generalization Gen-eral product to be a finitely generated. Moreover we will discuss necessary andsufficient conditions on the generalization General product of A⊕B by B to befinitely presented after giving to be finitely generated of this product. Further-more, we will define the periodicity and local finiteness for the generalizationGeneral product of A⊕B by B.

2010 Mathematics Subject Classifications : 20E22, 20F05, 20L05, 20M05Keywords: Zappa products, Wreath products, Periodicity, Local finiteness.

References

[1] M. G. Brin, On the Zappa-Szep product, Communications in Algebra 33 (2005),393-424.

[2] A. S. Cevik, S. A. Wazzan, F. Ates, Generalization of general products of monoids,Hacettepe Journal of Mathematics and Statistics (in review).

[3] A. S. Cevik, The efficiency of standard wreath product, Proceedings of the Edin-burgh Mathematical Society 43(2) (2000), 415-423.

[4] N. D. Gilbert, S. Wazzan, Zappa-Szep products of bands and groups, SemigroupForum 77 (2008), 438455.

[5] T. G. Lavers, Presentations of general products of monoids, Journal of Algebra204 (1998), 733-741.

[6] J. Szep, On the structure of groups which can be represented as the product oftwo subgroups, Acta Sci. Math. Szeged 12 (1950), 57-61.

[7] G. Zappa, Sulla construzione dei grappi prodotto di due dati sottogruppi per-mutabili traloro, Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, EdizioniCremonense, Rome, 119-125, 1942.

a Department of Mathematics, KAU King Abdulaziz University, Sci-ence Faculty, 21589, Jeddah-Saudi Arabia.

b Department of Mathematics, KAU King Abdulaziz University, Sci-ence Faculty, Girls Campus, 21589, Jeddah-Saudi Arabia.

c Department of Mathematics, Balikesir University, Science and ArtFaculty, Campus, 42075, Balikesir-Turkey.

E-mail : [email protected] (A.S. Cevik), [email protected] (S.A. Waz-zan), [email protected] (F. Ates)

15 Paris - FRANCE

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Finding a Quintic spline for solving Cauchy

Initial Value Problem

Kulbhushan Singh

Abstract

In the present work a special lacunary interpolation problem is solved, herefunction value, first derivatives and fourth derivatives are prescribed at nodesof the unit interval I = [0, 1]. A special spline function has been found for it,and then theorem of unique existence and convergence for this spline functionare proved. In this paper firstly a spline function using the given set of data isdeveloped, and then the theorem of convergence is proved. Later we have shownthat this special function can be used to solve Cauchys Initial Value Problem.

2010 Mathematics Subject Classifications : 65D07Keywords: Cauchy Initial Value problem, Spline functions :acunary Interpola-

tion

References

[1] Kim F. R. Loscalzo, T.D. Talbot, Spline and approximation for solutions of ordi-nary differential equations, SIAM J. Numer. Anal. 4(1967), 433-445.

[2] F. R. Loscalzo, On the use of spline functions for the numerical solution of ordinarydifferential equations,Doctoral thesis, Univ. of Wisconsin, Madison, W.S. 1968.

[3] Gh. Micula,Approximate solution of the differential equation y”(x) = f(x,y) withspline functions, Math. of comput. 27 (1973), 807-816.

[4] A. Meir, A. Sharma, Lacunary interpolation by splines, SIAM J. Numer. Anal.,10(1973) No.3, 433-442.

[5] T. Fawzy, Spline functions and the Cauchy’s problem II, Acta Math. Hung.29(1977), (3-4), 259-271.

[6] T. C. Joshi, R.B. Saxena, On quantic Splines interpolation,Ganita, 33(1982), 97-111.

[7] J. Gyorvari, Lakunare spline funktion un das Cauchy problem, Acta Math Hung.,44(1984) (3-4), 327-335.

[8] K. B. Singh, Ambrish Kumar Pandey, Qazi Shoeb Ahmad, Solution of a BirkhoffInterpolation Problem by a Special Spline Function, International J. of Comp.App., June 48(2012), 22-27.

Faculty of Mathematical and Statistical Sciences, Sri RamswaroopMemorial University, Lucknow India.E-mail : kul b [email protected]

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Mathematical Analysis of a discrete SEIR

epidemic model with treatment

Mahmoud H. DarAssia

Abstract

In this talk, we will study the global dynamics of a discrete SEIR epidemicmodel with treatment. A unique positive solution for the proposed model withthe positive initial conditions is obtained. The stability analysis of the disease-free equilibrium and endemic equilibrium have been investigated. It has beenproved that the DFE is globally asymptotically stable when the basic reproduc-tion number R0 ≤ 1. The proposed model has a unique endemic equilibriumthat is globally asymptotically stable whenever R0 > 1. The theoretical resultsare illustrated by a numerical simulation.

2010 Mathematics Subject Classifications : 05A10, 05A15, 11B68, 11S80,26C05, 65D17

Keywords: Discrete model, treatment, backward difference, equilibria, globalstability.

References

[1] DarAssi, MH, Safi, MA, Al-Hdaibat, B: A delayed SEIR epidemic model withpulse vaccination and treatment. Nonlinear Studies 25, (3) : 1-16 (2018).

[2] Wang, L, Cui, Q, Teng, Z: Global dynamics in a class of discrete-time epidemicmodels with disease courses. Advances in Difference Equations 2013, 57 (2013).

[3] Wang, Y, Teng, Z, Rehim, M: Lyapunov functions for a class of discrete SIRS mod-els with nonlinear incidence rate and varying population sizes. Discrete Dynamicsin Nature and Society 2014, 1-10 (2014).

aDepartment of Basic Sciences, Faculty of Engineering Princess SumayaUniversity for Technology, Amman, Jordan.E-mail : [email protected]

17 Paris - FRANCE

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On p-adic multiple Barnes-Euler zeta

functions and the corresponding log

gamma functions

Su Hua , Min-Soo Kimb

Abstract

Suppose that ω1, . . . , ωN are positive real numbers and x is a complex numberwith positive real part. The Barnes-Euler multiple zeta function ζE,N (s, x; ω)with parameter x and ω = (ω1, . . . , ωN ) is defined Re(s) > 0 by a deformationof the Barnes multiple zeta function as follows

ζE,N (s, x; ω) =

∞∑m1,...,mN=0

(−1)m1+···+mN

(x+ ω1m1 + · · ·+ ωNmN )s.

In this paper, based on the fermionic p-adic integral, we define the p-adicanalogue of Barnes-Euler multiple zeta function ζE,N (s, x; ω) which we denoteby ζp,E,N (s, x; ω). We prove several properties of ζp,E,N (s, x; ω), including theconvergent Laurent series expansion, the distribution formula, the differenceequation, the reflection functional equation and the derivative formula. Bycomputing the values of this kind of p-adic zeta function at nonpositive integers,we show that it interpolates the N -th order Euler polynomials EN,n(x; ω) p-adically.

Furthermore, we consider the corresponding multiple p-adic Diamond-EulerLog Gamma function. We also show that the multiple p-adic Diamond-EulerLog Gamma function Log ΓD,E,N (x; ω) has an integral representation by themultiple fermionic p-adic integral, and it satisfies the distribution formula, thedifference equation, the reflection functional equation, the derivative formulaand also the Stirling’s series expansions.

2010 Mathematics Subject Classifications : 11B68, 11S80, 11M35, 11M417Keywords: p-adic Barnes-Euler multiple zeta function, Multiple p-adic Diamond-

Euler Log Gamma function, Fermionic p-adic integral, p-adic analysis.

References

[1] N. Aoki, On Tate’s refinement for a conjecture of Gross and its generalization, J.Theor. Nombres Bordeaux 16 (2004), no. 3, 457–486.

[2] I.N. Cangul, H. Ozden and Y. Simsek, A new approach to q-Genocchi numbersand their interpolation functions, Nonlinear Anal. 71 (2009), no. 12, e793–e799.

[3] M.W. Coffey, Series representation of the Riemann zeta function and other results:complements to a paper of Crandall, Math. Comp. 83 (2014) 1383–1395.

[4] H. Cohen, Number theory Vol. I: Tools and Diophantine equations, Graduate Textsin Mathematics, 239, Springer, New York, 2007.

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[5] H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Textsin Mathematics, 240, Springer, New York, 2007.

[6] H. Cohen and E. Friedman, Raabe’s formula for p-adic gamma and zeta functions,Ann. Inst. Fourier (Grenoble) 58 (2008) 363–376.

[7] J. Choi and H.M. Srivastava, The multiple Hurwitz zeta function and the multipleHurwitz-Euler eta function, Taiwanese J. Math. 15 (2011), no. 2, 501–522.

[8] J. Diamond, The p-adic log gamma function and p-adic Euler constants, Trans.Amer. Math. Soc. 233 (1977), 321–337.

[9] J. Diamond, On the values of p-adic L-functions at positive integers, Acta Arith.35 (1979), 223–237.

[10] R. Ernvall, Generalized Bernoulli numbers, generalized irregular primes, andclass number, Ann. Univ. Turku. Ser. AI 178 (1979), 1–72.

[11] G.J. Fox, A method of Washington applied to the derivation of a two-variablep-adic L-function, Pacific J. Math. 209 (2003), 31–40.

[12] B. Gross, On the values of abelian L-functions at s = 0, J. Fac. Sci. Univ. Tokyo35 (1988), 177–197.

[13] S. Hu and M.-S. Kim, The (S, 2)-Iwasawa theory, J. Number Theory 158(2016), 73–89.

[14] S. Hu, D. Kim and M.-S. Kim, The p-adic Arakawa-Kaneko-Hamahata zeta func-tions and poly-Euler polynomials, J. Number Theory 177 (2017), 73–90.

[15] T. Kashio, On a p-adic analogue of Shintani’s formula, J. Math. Kyoto Univ. 45(2005), 99–128.

[16] T. Kashio and H. Yoshida, On p-adic absolute CM-periods I, Amer. J. Math. 130(2008), no. 6, 1629–1685.

[17] N.M. Katz, p-adic L-functions via moduli of elliptic curves, Algebraic geometry(Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974),pp. 479–506, Amer. Math. Soc., Providence, R.I., 1975.

[18] T. Kim, On the analogs of Euler numbers and polynomials associated with p-adicq-integral on Zp at q = −1, J. Math. Anal. Appl. 331 (2007), 779–792.

[19] M.-S. Kim, On Euler numbers, polynomials and related p-adic integrals, J. Num-ber Theory 129 (2009), no. 9, 2166–2179.

[20] M.-S. Kim and S. Hu, Sums of products of Apostol-Bernoulli numbers, Ramanu-jan J. 28 (2012), no. 1, 113–123.

[21] M.-S. Kim and S. Hu, On p-adic Hurwitz-type Euler zeta functions, J. NumberTheory 132 (2012), no. 12, 2977–3015.

[22] M.-S. Kim and S. Hu, On p-adic Diamond-Euler log gamma functions J. NumberTheory 133 (2013), no. 12, 4233–4250.

[23] S. Lang, Cyclotomic Fields I and II, Combined 2nd ed., Springer-Verlag, NewYork, 1990.

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[24] M. Lerch, Note sur la fonction K(w, x, s) =∑∞

k=0e2kπix

(w+k)s , Acta Mathematica 11

(1887), 19–24 (in French).

[25] H. Maıga, Some identities and congruences concerning Euler numbers and poly-nomials, J. Number Theory 130 (2010), no. 7, 1590–1601.

[26] N.E. Norlund, Vorlesungen uber Differenzenrechnung, Berlin, 1924.

[27] Ju.V. Osipov, p-adic zeta functions, Uspekhi Mat. Nauk 34 (1979), 209–210 (inRussian).

[28] A.M. Robert, A course in p-adic analysis, Graduate Texts in Mathematics, 198,Springer-Verlag, New York, 2000.

[29] S.N.M. Ruijsenaars, On Barnes’ multiple zeta and gamma functions, Adv. Math.156 (2000), no. 1, 107–132.

[30] M.A. Stern, Zur Theorie der Eulerschen Zahlen, J. Reine Angew. Math. 79(1875), 67–98.

[31] W.H. Schikhof, Ultrametric calculus, An introduction to p-adic analysis, Cam-bridge Studies in Advanced Mathematics 4, Cambridge University Press, Cam-bridge, 1984.

[32] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci.Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167–199.

[33] K. Shiratani and S. Yamamoto, On a p-adic interpolation function for the Eulernumbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Ser. A 39 (1985), 113–125.

[34] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Seriesand Integrals, Elsevier Science Publishers, Amsterdam, London and New York,2012.

[35] Z.W. Sun, On Euler numbers modulo powers of two, J. Number Theory 115(2005), no. 2, 371–380.

[36] B.A. Tangedal and P.T. Young, On p-adic multiple zeta and log gamma functions,J. Number Theory 131 (2011), no. 7, 1240–1257.

[37] B.A. Tangedal and P.T. Young, Explicit computation of Gross-Stark units overreal quadratic fields, J. Number Theory 133 (2013), no. 3, 1045–1061.

[38] E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed. Oxford Sci-ence Publications, 1986.

[39] I. Vardi, Determinants of Laplacians and multiple Gamma functions, SIAM J.Math. Anal. 19 (1988), 493-507.

[40] D. Vallieres, On a generalization of the rank one Rubin-Stark conjecture, Ph.D.Thesis, University of California, San Diego, 2011.

[41] A. Volkenborn, Ein p-adisches Integral und seine Anwendungen I, ManuscriptaMath. 7 (1972), 341–373.

[42] A. Volkenborn, Ein p-adisches Integral und seine Anwendungen II, ManuscriptaMath. 12 (1974), 17–46.

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[43] S.S. Wagstaff Jr., Prime divisors of the Bernoulli and Euler numbers, in: Numbertheory for the millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002,pp. 357–374.

[44] P.T. Young, Congruences for Bernoulli, Euler, and Stirling numbers, J. NumberTheory 78 (1999), 204–227.

[45] P.T. Young, The p-adic Arakawa-Kaneko zeta functions and p-adic Lerch tran-scendent, J. Number Theory 155 (2015), 13–35.

[46] P.T. Young, Congruences for Bernoulli-Lucas sums, Fibonacci Quart 55 (2017),no. 5, 201–212.

[47] P.T. Young, Polylogarithmic zeta functions and their p-adic analogues, Int. J.Number Theory 13 (2017), no. 10, 2751–2768.

aDepartment of Mathematics, South China University of Technol-ogy, Guangzhou, Guangdong 510640, China

bDivision of Mathematics, Science, and Computers, Kyungnam Univer-sity, 7(Woryeong-dong) kyungnamdaehak-ro, Masanhappo-gu, Changwon-si, Gyeongsangnam-do 51767, Republic of KoreaE-mail : [email protected], [email protected]

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Generalizations of Ostrowski type

inequalities via Hermite polynomials

Ljiljanka Kvesica , Josip Pecaricb , Mihaela Ribicic Penavac

Abstract

The classical Ostrowski inequality states:∣∣∣∣f(x)− 1

b− a

∫ b

a

f(t)dt

∣∣∣∣ ≤[

1

4+

(x− a+b

2

)2(b− a)2

](b− a)

∥∥f ′∥∥∞ ,for all x ∈ [a, b], where f : [a, b] → R is continuous on [a, b] and differentiableon (a, b) with bounded derivative. Ostrowski type inequalities are largely in-vestigated in the literature since they are very useful in Numerical analysis andProbability.

The main purpose of this talk is to present some new generalizations ofweighted Ostrowski type inequalities for differentiable functions of class Cn us-ing weighted Montgomery identity and Hermite interpolating polynomials. Byapplying those results we also derive certain inequalities for the class of n-convexfunctions.

2010 Mathematics Subject Classifications : 26D15, 26D20, 26A51Keywords: Hermite interpolating polynomial, Montgomery identity, Ostrowski

inequality, Gruss inequality.

References

[1] R. P. Agarwal, P. J. Y. Wong, Error Inequalities in Polynomial Interpolation andTheir Applications, Dordrecht: Kluwer Academic Publishers; 1993.

[2] A. Aglic Aljinovic, A. Civljak, S. Kovac, J. Pecaric, M. Ribicic Penava, GeneralIntegral Identities and Related inequalities, Zagreb: Element; 2013.

[3] G. Aras-Gazic, J. Pecaric, A. Vukelic, Integral Error Representation of Hermite In-terpolating Polynomial and Related Inequalities for Quadrature Formulae, Math-ematical Modelling and Analysis 21 (2016), 836–851.

[4] P. Cerone and S. S. Dragomir, Some new Ostrowsky-type bounds for the Cebysevfunctional and applications, J. Math. Inequal. 8(1) (2014), 159–170.

[5] A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion vonihrem Integralmittelwert, Comment. Math. Helv. 10 (1938), 226–227.

aFaculty of Science and Education, University of Mostar, Maticehrvatske bb, 88 000 Mostar, Bosnia and Herzegovina.

bRUDN University, Miklukho-Maklaya str.6, 117198 Moscow, Russia.cDepartment of Mathematics, University of Osijek, Trg Ljudevita

Gaja 6, 31 000 Osijek, Croatia

E-mail : [email protected], [email protected], [email protected]

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Recurrence Relations and Difference

Equation for the ω-Multiple Charlier

Polynomials

Mehmet Ali Ozarslana, Gizem Baranb

Abstract

In this paper, the generating function of the ω−multiple Charlier polyno-mials is given. We obtained the recurrence relation and the lowering operatorof these polynomials [7]. Furthermore, (r + 1) th order difference equation forω−multiple Charlier polynomials is given. The particular case ω = 1, gives themultiple Charlier polynomials. The results obtained in this paper coincide withthe theorems given in [2] and [6] for multiple Charlier polynomials when ω = 1.

2010 Mathematics Subject Classifications : 33C45, 33D50, 39A13, 42C05Keywords: Multiple orthogonal polynomials, ω−multiple Charlier polynomials,

Rodrigues type formula, generating function, recurrence relation, difference equation.

References

[1] A. I. Aptekarev, Multiple orthogonal polynomials, J. Comput. Appl. Math. 99(1998), 423-447.

[2] J. Arvesu, J. Coussement, W. Van Assche, Some discrete multiple orthogonalpolynomials, J. Comput. Appl. Math. 153 (2003) 19-45.

[3] E. Coussement, W. Van Assche, Some classical multiple orthogonal polynomials,J. Comput. Appl. Math. 127 (2001), 317-347.

[4] M. Haneczok, W. Van Assche, Interlacing properties of zeros of multiple orthogonalpolynomials, J. Math. Anal. Appl. 339 (2012), 429-438.

[5] R. Koekoek, P. A. Lesky, R. F. Swarttouw: Hypergeometric Orthogonal Polyno-mials and their q-Analogues, Springer, Berlin, 2010.

[6] D. W. Lee, Difference equations for discrete classical multiple orthogonal polyno-mials, J. Approx Theory 150 (2008) 132-152.

[7] M. A. Ozarslan, G. Baran, On the ω−multiple Charlier polynomials, (2019).Manuscript submitted for publication.

[8] W. Van Assche, Nearest neighbor recurrence relations for multiple orthogonalpolynomials, J. Approx Theory 163 (2011) 1427-1448.

[9] W. Van Assche, Difference equations for multiple Charlier and Meixner polyno-mials, in: S. Elaydi et al. (Eds.). New Progress in Difference Equations, Taylorand Francis, London, 2003, pp. 547-555.

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aDepartment of Mathematics, Faculty of Art and Science Univer-sity of Eastern Mediterranean, Gazimagusa, TRNC, Mersin 10, Turkey.bDepartment of Mathematics, Faculty of Art and Science University ofEastern Mediterranean, Gazimagusa, TRNC, Mersin 10, Turkey.E-mail : [email protected], [email protected]

24 Paris - FRANCE

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Stability analysis of a generalized

TCP-RED algorithm for Internet

congestion control

A. Gimeneza, Jose M. Amigoa, G. Duranb, O. Martınez-Bonastrea, J. Valeroa

Abstract

IP networks include memory buffers into routers to manage the networktraffic congestion by implementing a variety of different queue algorithms. Ob-viously, the length of this queue is limited to the buffer size, and therefore in ascenario of network congestion some actions have to be taken to avoid overflowor network collapse. An Active Queue Management (AQM) system is an algo-rithm acting on the queue length to achieve a efficient control of the networkcongestion. The Random Early Detection (RED) algorithm was the first formaland complete proposal AQM system implemented in TCP/IP networks and ithas subsequently been one of the most intensively studied algorithm in the lit-erature of the Internet development. From a mathematical point of view REDcan be seen as a one-dimensional discrete-time non linear dynamical systemin which the states correspond to the average queue size of the incoming datapackets. One of the major problem is the high sensitivity of RED parameters,which hinder a suitable tuning to obtain high performance and stability. In thistalk we propose a generalized RED algorithm by introducing two new controlparameters. As a first step a theoretical analysis of stability is performed. In astep further, the above theoretical results are used to find robust ranges of thenew control parameters that ensure stability. Finally some numerical experi-ments are displayed to confirm that appropriate choice of the new parametersmakes it possible to improve stability properties for robust domains.

2010 Mathematics Subject Classifications : 93C55, 93D09, 93D20, 68M11,68M12

Keywords: Congestion control in the Internet, Adaptive queue management,Random early detection, Dynamical systems, Global stability.

References

[1] G. Duran, J. Valero, J.M. Amigo, A. Gimenez, and O. Martınez-Bonastre, Sta-bilizing chaotic behavior of RED. 2018 IEEE 26th International Conference onNetwork Protocols, 241-242, Sept. 2018.

[2] G. Duran, J. Valero, J.M. Amigo, A. Gimenez, and O. Martınez-Bonastre, Bifur-cation analysis for the internet congestion, 2019 IEEE Infocom.

[3] R. Adams, Active queue management: A survey, IEEE Communications Surveysand Tutorials 15, 1425-1476, 2013.

[4] S. Athuraliya, S. Low, V. H. Li, and Q. Yin, REM: active queue management,IEEE Network 15, 48–53, 2001.

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[5] H.A. El-Morshedy and E. Liz, Globally attracting fixed points in higher orderdiscrete population models, J. Math. Biol. 53, 365-384, 2006.

[6] S. Floyd and V. Jacobson, Random early detection gateways for congestion avoid-ance, IEEE/ACM Trans. Networking 1, 397-413 (1993).

[7] T. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82, 985-992, 1975.

[8] M. Mathis, J. Semke, J. Mahdavi, and T. Ott, The macroscopic behavior of theTCO congestion avoidance algorithm, Computer Communication Review 27 (3),1997.

[9] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer Verlag,Berlin, 1993.

[10] L.J. Pei, X.W. Mu, R.M. Wang, and J.P. Yang, Dynamics of the Internet TCP-RED congestion control system, Nonlinear Analysis: Real World Applications 12,947-955 (2011).

[11] P. Ranjan, E.H. Abed and R.J. La, Nonlinear Instabilities in TCP-RED,IEEE/ACM Transactions on Networking 12, 1079-1092, 2004.

[12] A.N. Sharkovsky, Coexistence of cycles of a continuous map of the line into itself,Ukranian Mathematical Journal 16, 61-71, 1964.

[13] D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl.Math. 35, 260-267, 1978.

aCentro de Investigacion Operativa, Universidad Miguel Hernandez,Avda. de la Universidad s/n, 03202 Elche, Spain.E-mail : a.gimenez@umh, jm.amigo@umh, oscar.martinez@umh, [email protected]

bFragile Technologies, 30007 Murcia, Spain.E-mail : [email protected]

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Jensen-type operator inequalities for

bounded, Lipschitzian and higher order

convex functions

Mario Krnica, Rozarija Mikicb, Josip Pecaricc

Abstract

The main objective of this talk is a study of mutual bounds for the Jensenoperator inequality and the Lah-Ribaric operator inequality for the classes ofbounded real-valued functions and Lipschitzian functions. The connection withthe classical convexity is also discussed. As an application, our main results arethen applied to quasi-arithmetic operator means, with a particular emphasis topower operator means. In particular, we obtain some new reverse relations thatcorrespond to quasi-arithmetic and power operator means.

2010 Mathematics Subject Classifications : 47A63, 47A64Keywords: Jensen operator inequality, Lah-Ribaric operator inequality, convex-

ity, operator convexity, bounded function, Lipschitzian function

References

[1] S.S. Dragomir, Trapezoid type inequalities for isotonic functionals with applica-tions, RGMIA Res. Rep. Coll. 20 Art. 4 (2017), 1–24.

[2] F. Hansen, G. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc. 35(2003), 553–564.

[3] F. Hansen, J. Pecaric, I. Peric, Jensen’s operator inequality and its converses,Math. Scand. 100 (2007), 61–73.

[4] R. Jaksic, M. Krnic, J. Pecaric, More precise estimates for the Jensen operator in-equality obtained via the Lah-Ribaric inequality, Appl. Math. Comput. 249 (2014),346–355.

[5] M. Krnic, R. Mikic, J. Pecaric, Double precision of the Jensen-type op-erator inequalities for bounded and Lipschitzian functions, Aequat. Math.https://doi.org/10.1007/s00010-018-0599-7.

aUniversity of Zagreb, Faculty of Electrical Engineering and Com-puting, Unska 3, 10000 Zagreb, Croatia

bUniversity of Zagreb, Faculty of Textile Technology, Prilaz barunaFilipovica 28a, 10000 Zagreb, Croatia

cRUDN University, Miklukho-Maklaya str. 6, Moscow, Russia

E-mail : [email protected], [email protected], [email protected]

27 Paris - FRANCE

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Best Proximity Point Results for Proximal

Nonexpansive Multivalued Mappings on

starshaped sets

N. Bunluea,∗, Y. J. Chob , S. Suantaic

Abstract

It is well known that the concept of a fixed point is a special case of bestproximity point. In this paper, we introduce the new concept of proximal mul-tivalued mapping and prove the existence of best proximity points for proximalmultivalued contractions in metric spaces, and for proximal nonexpansive multi-valued mappings on starshape sets in Banach spaces with examples to illustrateour results. Moreover, we study projections of multivalued mappings, and re-lation between fixed points and best proximity points. The results will becomeimportant tools used in the approximation of fixed points and best proximitypoints problems.

2010 Mathematics Subject Classifications : 41A29, 90C26, 47H09.Keywords: best proximity point; fixed point, proximal multivalued mapping,

multivalued projection, approximation, starshaped set.

References

[1] S.S. Basha, Best proximity points: optimal solutions. J. Optim. Theory Appl.151(1) (2011), 210-216.

[2] J. Chen , S. Xiao, H. Wang and S. Deng, Best proximity point for the proxi-mal nonexpansive mapping on the starshaped sets. Fixed Point Theory Appl. 19(2015).

[3] M. Gabeleh, Best Proximity Point Theorems via Proximal Non-self Mappings. J.Optim. Theory Appl. 164 (2015), 565-576.

[4] P. Sarnmetal, S. Suantai, Existence and Convergence Theorems for Best Proxim-ity Points of Proximal Multi-valued Nonexpansive Mappings. Communications inMathematics and Applications (2019)(Submitted)

aPhD Degree Program in Mathematics, Faculty of Science, ChiangMai University, Chiang Mai 50200, Thailand

bDepartment of Mathematics Education and the RINS, GyeongsangNational University, Jinju 660-701, Korea

cResearch Center in Mathematics and Applied Mathematics, Depart-ment of Mathematics, Faculty of Science, Chiang Mai University, Chi-ang Mai 50200, Thailand.∗Speaker

E-mail : [email protected] , [email protected] , [email protected]

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Linking Bernoulli and Euler Polynomials

and their interpolation problem

Mehmet Bozera, Mehmet Ali Ozarslana

Abstract

In this paper, we define the linking Bernoulli-Euler polynomials via theirlinear functionals, which are used in the general Appell interpolation problems.We obtain their generating relation, which helps to define the correspondinglinking numbers. We give the recurrence relation for the computation of thosenumbers. Furthermore, the corresponding interpolation problem is consideredfor the linking Bernoulli and Euler polynomials. More precisely, we obtain theirinterpolation polynomial together with the error term. Finally, we discuss somenumerical examples.

2010 Mathematics Subject Classifications : 11B68, 30E05.Keywords: Bernoulli polynomials, Euler polynomials, generating function, in-

terpolation problem.

References

[1] F.A.Costabile, E. Longo, The Appel interpolation problem, Journal of Comp andApp Sci.236 (2011), 1024–1032.

aDeparment of Mathematics, Faculty of Arts and Sciences, EasternMediterranean University Mersin 10 Turkey

E-mail : [email protected], [email protected]

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Arc spaces and partition identities

Hussein Mourtadaa

Abstract

We will show a link between the arc space (which is an algebro-geometricobject) and the identities of partitions of integer numbers: a partition of apositive integer is simply a way of writing it as a sum of positive integers.Integer partitions have a long and beautiful history in number theory. The linkthat we will describe comes from an invariant of singularities and gives a newpoint of view on known results and new identities. The talk is accessible to awide audience. Joint work with Pooneh Afsharijoo (2019), and with ClemensBruschek and Jan Schepers (2013).

aInstitut de Mathmatiques de Jussieu-Paris Rive Gauche Equipe Gomtrieet Dynamique, Universit Paris 7, UFR de Mathmatiques, Batiment SophieGermain, case 7012, 75205 Paris Cedex 13, France


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A note on q-associated polynomials

Rahime Dere Pacina

Abstract

The Sheffer polynomials play very important role for the umbral calculustheory. In this work, we investigate a q-generalization of the associated poly-nomials which are also Sheffer polynomials. Then we give some relations andspecial examples of these polynomials by using the methods of umbral calculusand quantum calculus.

2010 Mathematics Subject Classifications : 05A40, 05A30, 11B83.Keywords: q-umbral calculus, q-calculus, q-Sheffer polynomials.

References

[1] R. Dere, q-Hermite Base Euler polynomials based upon the q-umbral algebra, AIPConference Proceedings 1978, 040011 (2018).

[2] R. Dere, Some Hermite Base Polynomials on q-Umbral Algebra, Filomat, Filomat,30:4 (2016), 961-967.

[3] R. Dere, Some Identities of the q-Laguerre Polynomials on q-Umbral Calculus, Nu-merical Analysis and Applied Mathematics ICNAAM 2016: International Confer-ence of Numerical Analysis and Applied Mathematics. AIP Conference Proceed-ings (2016)

[4] R. Goldman, P. Simeonov and Y. Simsek, Generating Function for the q-BernsteinBases. Siam J. Discrete Math. 28(3) (2014), 1009-1025.

[5] V. Kac and P. Cheung, Quantum Calculus, Springer, 2002.

[6] S. Roman, More on the Umbral Calculus, with Emphasis on the q-Umbral Calcu-lus, J. Math. Anal. Appl. 107(1) (1985) 222-254.

[7] S. Roman, The Umbral Calculus, Dover Publ. Inc. New York, 2005.

aDepartment of Science and Mathematics Education, Faculty of Edu-cation, Alanya Alaaddin Keykubat University Alanya/Antalya, TURKEY.


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A new approach to solve multi-attribute

decision making problems by logarithmic

operational laws in interval neutrosophic

environment

T. S. Haquea, S. Alama,∗, and S. Dalapatia

Abstract

In current era, multi-attribute decision making (MADM) process has beenattracted much attention to several researchers as it can nicely handle manyreal-life challenging problems in many front-line areas like financial investment,recruitment policies, clinical diagnosis of disease, design of complex circuit etc. Itis not an over stated fact that fuzzy set theory plays very crucial role in decisionmaking problems specially when decision makers work in uncertain environment.The theories of uncertainty have geared up dramatically after introduction offuzzy set by Professor Zadeh [1] and intuitionistic fuzzy set by Atanassov [2]. Re-cently, Smarandache [3] introduced the concept of neutrosophic set (NS) whichis designated by truth, indeterminacy and falsity-membership functions thatare independent to each other. Moreover, Wang et al. [4] introduced inter-val neutrosophic set (INS) as a generalization of interval-valued intuitionisticfuzzy sets. In this article, we defined, analysed and established new logarith-mic operational laws (LOLs) of INSs and interval neutrosophic numbers (INNs)where the logarithmic base δ is a positive real number. Then, we proposed theweighted average and geometric aggregation operators namely, logarithmic inter-val neutrosophic weighted average (LINWA), logarithmic interval neutrosophicweighted geometric (LINWG), logarithmic interval neutrosophic order weightedaverage (LINOWA) and logarithmic interval neutrosophic order weighted geo-metric (LINOWG) based on the LOLs of INNs. The idempotency, boundednessand monotonicity properties of our proposed operators have been investigated.Finally, by using these aggregation operators, we developed a multi-attributedecision making (MADM) method for solving numerical problems under inter-val neutroshopic environment. The method has been clearly demonstrated witha particular real-life problem. Moreover, the influence of the logarithmic opera-tor and the choice of the logarithmic base δ is described in a careful manner sothat any expert can select the best alternatives by choosing a different real baseδ according to his or her priority. Lastly, we make a comparison study betweenour proposed operator with the existing weighted averaging and geometric ag-gregation operators to illustrate the advantages and validity of our proposedmethod.

2010 Mathematics Subject Classifications : 90C70Keywords: Interval neutrosophic numbers, logarithmic operational laws, aggre-

gation operators, multi-attribute decision making.

References

[1] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3)(1965), 338-353.

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[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1)(1986),8796.

[3] F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Proba-bility, Set and Logic, American Research Press, Rehoboth, Mass, USA, 1999.

[4] H. Wang, F. Smarandache, Y. Q. Zhang and R. Sunderraman, Interval neutro-sophic sets and logic: Theory and applications in computing, Hexis, Phoenix, AZ,2005.

a Department of Mathematics, Indian Institute Engineering Scienceand Technology, Shibpur, B. Garden, Howrah - 711103, India.E-mail : ∗[email protected], Tel: +91-9874452995.

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Bivariate h-Mittag-Leffler Functions with

2D-h-Laguerre-Konhauser Polynomials

Corresponding h-Fractional Operators

Cemaliye Kurta, Mehmet Ali Ozarslana

Abstract

In this paper, we first introduce new class of 2D-h-Laguerre-Konhauser poly-nomials, κL

(α,β)n,h (x, y), which generalizes the 2D-Laguerre-Konhauser polynomi-

als (see [1]). We give operational representation and Rodrigues-type relation ofthe above mentioned class. Furthermore, we obtain the linear generating func-tion for the polynomials κL

(α,β)n,h (x, y). After, by introducing a new family of

h-Mittag-Leffler functions, E(γ)h,α,β,κ(x, y), we discuss the convergence condition

of the mentioned functions. We establish the h-Riemann-Liouville double frac-tional integral and derivative of the functionsE

(γ)h,α,β,κ(x, y). Moreover, we introduce an integral operator

hε(γ)

α,β,κ;ω1,ω2;a+,c+which contains the h-Mittag-Leffler functions in the kernel

and study its boundness in the Lebesgue summable and continuous functionspaces. Furthermore, the composition of two integral operators is analysed andfinally, the left inverse operator is constructed.

2010 Mathematics Subject Classifications : 33C45, 33B15, 33E12, 26A33,44A10, 45E10

Keywords: Laguerre and Konhauser polynomials, k -Gamma function, k -Mittag-Leffler function, k -fractional integral, k -fractional derivative, double Laplace trans-form, convolution integral equation.

References

[1] M.G. Bin-Saad, Associated Laguerre-Konhauser polynomials, quasi-monomialityand operational identities, J. Math. Anal. Appl. 324 (2006), 1438–1448.

[2] R. Diaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol,Divulg. Mat. 17 (2007), 179–192.

[3] G. A. Dorrego, Generalized Riemann-Liouville Fractional Operators Associatedwith a Generalization of the Prabhakar Integral Operator, Progr. Fract. Differ.Appl. 2 (2) (2016), 131–140.

[4] G. A. Dorrego, R. A. Cerutti, The k-Mittag-Leffler function, Int. J. Contemp.Math. Sci. 7 (2012), 705–716.

[5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Frac-tional Differential Equations, North Holland Mathematics Studies vol. 204: Else-vier, Amsterdam; 2006.

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[6] S. Mubeen, G. M. Habibullah, k-fractional integrals and application, Int. J. Con-temp. Math. Sci. 7 (2012), 89–94.

[7] M. A. Ozarslan, C. Kurt, Bivariate Mittag-Leffler functions arising in the solutionsof convolution integral equation with 2D-Laguerre-Konhauser polynomials in thekernel, Appl. Math. Comput. 347 (2019), 631–644.

aDepartment of Mathematics, Faculty of Arts and Science East-ern Mediterranean University, Famagusta Mersin 10 Turkey, NorthCyprus.E-mail : [email protected] and [email protected]

35 Paris - FRANCE

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Generating Functions and Difference

Equations for ω-Multiple Meixner

Polynomials

Sonuc Zorlu Ogurlua Ilkay Elidemirb

Abstract

There are two types ω-multiple Meixner polynomials, where ω is positivereal number. In this study, we first give generating function and by use ofthe generating function we obtain several results and then we find a loweringoperator for these two types of ω-multiple Meixner polynomials. We derivea (r + 1)th order difference equation for ω-multiple Meixner polynomials ofthe first and second kind by combining the lowering operator with the raisingoperator. As a corollary we give an explicit difference equation for two typesof ω-multiple Meixner polynomials for the case of r = 2.Finally, we prove thatwhen ω = 1, the obtained results coincide with the existing results for multipleMeixner polynomials of the first and second kind.

2010 Mathematics Subject Classifications : 42C05, 33E50, 33B15Keywords: ω-Multiple Meixner Polynomials,orthogonal polynomials, generating

function, difference equation.

References

[1] A.I.Aptekarev, Multiple orthogonal polynomials, J.Comput.Appl.Math. 99 (1998)423–447.

[2] J. Arvesu, J. Coussement, W. Van Assche, Some discrete multiple orthogonalpolynomials, J.Comput.Appl.Math. 153 (2003) 19-45.

[3] T.S Chihara, An introduction to orthogonal polynomials,Mathematics and itsApplications 13, Gordon and Breach, New York, 1978.

[4] F.N Dayiragije, W.Van Assche, Multiple Meixner polynomials and non-Hermitianoscillator Hamiltonians,2013 (math.CA/[email protected]).

[5] D.W.Lee, Difference equations for discrete classical multiple orthogonal polyno-mials, J. of Approximation Theory. 150 (2008) 132-152.

[6] W.Van Assche, Difference equations for multiple Charlier and Meixner polynomi-als,in: S. Elaydi et al. (Eds.), New Proress in Difference Equations, Taylor andFrancis, London, 2003, pp. 547-555.

aDepartment of Mathematics, Faculty of Arts and Science Univer-sity of Eastern Mediterranean, Famagusta , Cyprus.bDepartment of Mathematics, Faculty of Arts and Science Universityof Eastern Mediterranean, Famagusta , Cyprus.E-mail : [email protected] ,[email protected]

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Kinematic Parallel Surfaces

Erhan Ataa, Yasemin Yıldırıma, Derya Kahvecib, Yusuf Yaylıb

Abstract

In this study, kinematic parallel surfaces MK of a surface M in 3-dimensionalEuclidean space E3 by the help of screw motion defined by quaternions. As spe-cial case of screw motion; if the rotation angle is taken as θ = 0 then the parallelsurfaces Mr of M are obtained. If it is taken that r = 0 then the kinematicsurfaces obtained by rotation of points of the surface M around the unit normalvector field of this surface without translation. If for r = 0 the rotation axis istaken as constant then the rotational surface obtained by rotating the points ofM around this constant axis. The basic differential geometric properties such asshape operator, Gaussian and mean curvatures of the kinematic parallel surfaceMK are computed. These obtained concepts are compared with the conceptssuch as shape operator, Gaussian and mean curvatures of M . Thence, it isobserved that how the screw motion changes the properties of the surface M .Finally, the concepts such as shape operator, Gaussian and mean curvature arecomputed for the surfaces obtained from the surface M for the special cases ofscrew motion.

2010 Mathematics Subject Classifications : 53A05, 53A17, 70B05Keywords: Screw motion, parallel surface, shape operator, curvatures, kine-

matic surface.

References

[1] Yasin Unluturk and Erdal Ozusaglam, On parallel surfaces in Minkowski 3-space,TWMS Journal of Applied and Engineering Mathematics 3.2 (2013), 214–222.

[2] Yasin Unluturk and Cumali Ekici, Parallel surfaces of spacelike ruled Weingartensurfaces in Minkowski 3-space, New Trends in Mathematical Sciences 1.1 (2013),85–92.

[3] A. Ceylan Coken, Unver Ciftci and Cumali Ekici, On parallel timelike ruled sur-faces with timelike rulings, Kuwait Journal of Science and Engineering 35.1A(2008), 21–31.

[4] Thomas Craig, Note on Parallel Surfaces, Journal fr die reine und angewandteMathematik 94 (1883), 162–170.

[5] L. P. Eisenhart, A treatise on the differential geometry of curves and surfaces,Ginn and Company, Boston, New York, 1909.

[6] A. Gorgulu and A. C. Coken, The Euler theorem for parallel pseudo-Euclideanhypersurfaces in pseudo-Euclidean space En+1

1 , J. Inst. Math. Comp. Sci.(Math.Ser.) 6 (1993), 161–165.

[7] A. Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press Inc.,(1993).

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[8] H. H. Hacısalihoglu, Diferensiyel Geometri, Cilt 1-2, Ankara niversitesi Fen Fakl-tesi Yayınları, 2000.

[9] N. J. Hicks, Notes on differential geometry, Van Nostrand Reinhold Company,London, 1974.

[10] W. Khnel, Differential Geometry, Curves-Surfaces-Manifolds, American Mathe-matical Society, 2002.

[11] R. Lpez, Differential geometry of curves and surfaces in Lorentz-Minkowski space,Mini-course taught at the Instituto de Mathematica e Estatistica (IME-USP) Uni-versity of Sao Paulo, Brasil, 2008.

[12] S Nizamoglu, Surfaces rgles paralleles, Ege niv. Fen Fak. Derg 9 (Ser. A), (1986),37–48.

[13] A. M. Patriciu, On some 1,3H3-helicoidal surfaces and their parallel surfaces ata certain distance in 3-dimensional Minkowski space, Annals of the University ofCraiova-Mathematics and Computer Science Series V. 37.4 (2010), 93–98.

[14] S. Roberts, On Parallel Surfaces, Proc. London Math. Soc., 1-4(1), (1871), 218–236.

[15] M. Dede, C. Ekici and A. Ceylan Cken., On the parallel surfaces in Galileanspace, Hacettepe J. Math. Stat 42.6 (2013), 605–615.

[16] M. Dede and C. Ekici, On parallel ruled surfaces in Galilean space, KragujevacJournal of Mathematics 40.1 (2016), 47–59.

[17] F. M. E. Dimentberg, The screw calculus and its applications in mechanics,Foreign Technology Div Wright-Pattersonafb oh, 1968.

[18] A. T. Yang, Calculus of screws, in basic quaternions of design theory, WilliamR. Spillers (ed.), Elsevier, 266–281.

[19] R. S. Ball, A Treatise on the Theory of Screws, Cambridge university press, 1998.

[20] J. M. Selig, Rational interpolation of rigid-body motions, Advances in the The-ory of Control, Signals and Systems with Physical Modeling. Springer, Berlin,Heidelberg, (2010), 213-224.

[21] M. Hiller and C.Woernle, A unified representation of spatial displacements,Mechanism and Machine Theory 19.6 (1984), 477–486.

[22] J. S. Dai, An historical review of the theoretical development of rigid body dis-placements from Rodrigues parameters to the finite twist, Mechanism and MachineTheory 41.1 (2006), 41–52.

[23] J. P. Ward, Quaternions and Cayley numbers: Algebra and applications. Vol.403 of Mathematics and its Applications, Kluvver, Dordrecht, 1997.

[24] R. Mukundan, Quaternions: From classical mechanics to computer graphics, andbeyond. In Proceedings of the 7th Asian Technology conference in Mathematics,(2002), 97–105.

[25] H. Pottmann and J. Wallner, Computational Line Geometry, Springer-Verlag,Berlin Heidelberg, 2001.

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[26] B. Juttler and M. G. Wagner, Computer-Aided Design With Spatial RationalB-Spline Motions, Journal of Mechanical Design, Transactions of the ASME, vol.118.2 (1996), 193–201.

[27] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Prac-tical Guide, Academic Press, Boston, MA, 1989.

aDepartment of Mathematics, Faculty of Science and Art, DumlupınarUniversity, 43100 Kutahya, Turkey.

bDepartment of Mathematics, Faculty of Science, Ankara Univer-sity, 06100 Ankara, Turkey.E-mail : [email protected], [email protected], [email protected],[email protected]

39 Paris - FRANCE

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Anti–Gaussian quadrature rule for

trigonometric polynomials

Marija P. Stanica, Nevena Z. Petrovica, Tatjana V. Tomovica

Abstract

In this paper we introduce anti–Gaussian quadrature rules for trigonometricpolynomials. Special attention is paid to even weight functions on [−π, π).We prove the main properties of such quadrature rules and present numericalmethod for their construction. Some numerical examples are included.

2010 Mathematics Subject Classifications : 65D32Keywords: Anti–Gaussian quadrature rules, recurrence relation, averaged Gaus-

sian formula

aUniversity of Kragujevac, Faculty of Science, Department of Math-ematics and Informatics, Kragujevac, Serbia.E-mail : [email protected], [email protected], [email protected]

40 Paris - FRANCE

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A generalization of S-Noetherian rings

Dong Kyu Kima, Jung Wook Lima

Abstract

LetR be a commutative ring with identity and S a (not necessarily saturated)multiplicative subset of R. In this talk, we define R to be a weakly S-Noetherianring if every S-finite proper ideal of R is an S-Noetherian R-module. Thisconcept is a generalization of an S-Noetherian ring. We give some properties ofweakly S-Noetherian rings.

2010 Mathematics Subject Classifications : 13A15, 13B10, 13B25, 13B30,13C05, 13F99

Keywords: S-finite, weakly S-Noetherian ring, amalgamation, idealization.

References

[1] D.D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra 30 (2002)4407-4416.

[2] J. Baeck, G. Lee, and J.W. Lim, S-Noetherian rings and their extensions, Tai-wanese J. Math. 20 (2016) 1231-1250.

[3] N. Mahdou and A.R. Hassani, On weakly-Noetherian rings, Rend. Sem. Mat.Univ. Politec. Torino 70 (2012) 289-296.

[4] J.W. Lim and D.Y. Oh, S-Noetherian properties on amalgamated algebras alongan ideal, J. Pure Appl. Algebra 218 (2014) 1075-1080.

[5] J.W. Lim and D.Y. Oh, S-Noetherian properties of composite ring extensions,Comm. Algebra 43 (2015) 2820-2829.

a Department of Mathematics, Kyungpook National University, Daegu41566, Republic of Korea.


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Star global dimension of power series rings

over w-coherent rings

Minjae Kwona, Jung Wook Lima

Abstract

Star operation is one of the useful tools to classify the class of integral do-mains. Using star operations, we can classify the important class of rings, UFD,GCD, PvMDs and so on. Recently, some people have been studying star op-erations over modules. w-modules are one of the remarkable examples. In thistalk, I will introduce well known definitions related with w-modules containingw-coherent, w-flat, w-homological dimension. Main results are inequalities ofw-flat dimensions of power series rings over w-coherent rings.

2010 Mathematics Subject Classifications : 13A15, 13C11, 13C12, 13C15Keywords: Star Operation, Weak global dimension, w-operation, w-flat

References

[1] F.G, Wang, H.K, Kim, Foundations of commutatvie rings and their modules,Springer.

[2] Jondrup, S, Power series over coherent rings, Math. Scand. 35(1974), 21-24

[3] F.G, Wang, H.K, Kim, w-injective modules and w-semi-hereditary rings, J. KoreanMath. Soc. 51 (2014), no. 3, 509-525

[4] F.G, Wang, Lei, Qiao, The w-weak global dimension of commutative rings, Bull.Korea Math. Soc. 52(2015), no. 4, 1327-1338

aDepartment of Mathematics, Kyungpook National University, Daegu41566,


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A generalization of Hurwiz polynomial

rings- Gaussain polynomial rings

Seung Min Leea , Jung Wook Lima

Abstract

Gaussian polynomial ring is defined similar to Hurwitz polynomial ring usingGaussian coefficient. In fact, Gaussian polynomial rings are considered to bea generalized of Hurwitz polynomial rings. In this talk, we show the Gaussianpolynomial ring, and we study some properties.

2010 Mathematics Subject Classifications : 11C08Keywords: Hurwiz rings, Hurwtiz polynomial rings, Gaussian coefficient, Gaus-

sian polynomial rings, q-number

References

[1] A. Benhissi, and F. Koja, Basic properties of Hurwitz series rings, Ric Mat., 61(2012),255-273.

[2] P. J. Cameron, (1994). Combinatorics: Topics, Techniques, Algorithms. Cam-bridge: Cambridge Univ. Press; 1994

[3] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.

[4] I. Kaplansky, Commutative rings, Rivised edition, The University of ChicagoPress, Chicago, 1947.

[5] P. T. Toan and B. Y. Kang, Krull dimension and unique factorization in Hurwitzpolynomial rings, Rocky Mountain J. Math., 47(2017), no. 4, 1317-1332.

aDepartment of Mathematics, Kyungpook National University, Daegu41566, Republic of Korea.E-mail : [email protected] , [email protected]

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2 POSTER PRESENTATIONS

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An enhanced algorithm for non-convex cost

function in machine learning

Sunyoung Bua

Abstract

A learning process of machine learning is a process to find values of unknownweights in a given cost function. The existing methods to find the minimumvalues usually use the first derivative of the cost function. However, the costfunction is non-convex, the existing techniques are no longer applied. In thistalk, we provide a modified technique to calculate the weights of the given non-convex cost function.

2010 Mathematics Subject Classifications : 65K05, 65K10, 68Txx, 94-04Keywords: Machine learning, Adam, Optimization, Non-convex cost function

Acknowledgement: This work was supported by the National Research Foundationof Korea(NRF) grant funded by the Korea government(MSIT) (grant number: NRF-2019R1F1A1058378).

References

[1] C. T. Kelley, Iterative methods for linear and nonlinear equations. Frontiers inApplied Mathematics, vol. 16, 1995, SIAM: Philadelphia, PA.

[2] D. Kingma, and J. Ba, ADAM: A method for stochastic optimization. InternationalConference for Learning Representations, San Diego, 2015.

[3] T. Tieleman, and G. E. Hinton, Lecture 6.5 - RMSProp, COURSERA: NeuralNetworks for Machine Learning. Technical report, 2012.

[4] M. Zinkevich, Online convex programming and generalized infinitesimal gradientascent. 2003.

aDepartment of Liberal Arts, Faculty of Mathematics Hongik Uni-versity, Sejong 30016, South Korea.E-mail : [email protected]

45 Paris - FRANCE

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High-order time discretization schemes for

simulation of the Burgers’ equations

Soyoon Baka, Yonghyeon Jeona, Sunyoung Bub

Abstract

In this talk, high-order characteristic-tracking schemes in backward semi-Lagrangian method are developed to solve nonlinear advection-diffusion typeproblems. The proposed characteristic-tracking strategies are second-order L-stable and third-order L(α)-stable methods, which are based on an implicit typemultistep method combined with the error-correction method. To demonstratethe adaptability and efficiency of these time-discretization strategies, we applythese methods to nonlinear advection-diffusion type problems such as the viscousBurgers’ equation. Through simulations, not only the temporal and spatialaccuracies are numerically evaluated but also the proposed methods are shownto be superior to the compared existing characteristic-tracking methods underthe same rates of convergence in terms of accuracy and efficiency.

2010 Mathematics Subject Classifications : 65M25, 65M06Keywords: High-order time-discretization, Semi-Lagrangian method, Characteristic-

tracking method, Burgers’ equation.

References

[1] S. Bak, High-order characteristic-tracking strategy for simulation of a nonlin-ear advection-diffusion equation, Numer Methods Partial Differential Eq. 35(5)(2019), 1756-1776.

[2] F. Filbet and C. Prouveur, High order time discretization for backward semi-Lagrangian methods, J Comput Appl Math. 303 (2016), 171-188.

[3] R. Jiwari, R. C. Mittal, K. K. Sharma, A numerical scheme based on weightedaverage differential quadrature method for the numerical solution of Burgers’ equa-tion, Appl Math Comput. 219 (2013), 6680-6691.

[4] R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear Burgers’ equationwith modified cubic B-splines collocation method, Appl Math Comput. 218 (2012),7839-7855.

[5] H. P. Bhatt and A. Q. M. Khaliq, Fourth-order compact schemes for the numericalsimulation of coupled Burgers’ equation, Comput Phys Commun. 200 (2016), 117-138.

a Department of Mathematics, Kyungpook National University, Daegu,Republic of Korea.

b Department of Liberal Arts, Hongik University, Sejong, Republicof Korea.E-mail : [email protected] , [email protected] , [email protected]

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Acknowledgements

It was supported by basic science research program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (grant number 2016R1D1A1B03930734)

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A Note on Ruled Submanifolds in

Minkowski Space with Gauss map

Sun Mi Junga

Abstract

In this talk, we study ruled submanifolds in Minkowski space in regard to theGauss map satisfying some partial differential equation. As a generalization ofusual cylinders, cones and null scrolls in Minkowski 3-space, we introduce someruled submanifolds in Minkowski space and characterize those with the partialdifferential equation regarding the Gauss map.

2010 Mathematics Subject Classifications : 53B25, 53B30Keywords: finite-type immersion; pointwise 1-type Gauss map of the second

kind; generalized B-scroll kind.

References

[1] C. Baikoussis, Ruled submanifolds with finite-type Gauss map, J. Geom. 49 (1994),42-45.

[2] B.-Y. Chen, Finite-type submanifolds and generalizations, Instituto ”GuidoCastelnuovo”, Rome, 1985.

[3] M. Choi, D.-S. Kim, Y. H. Kim and D. W. Yoon, Circular cone and its Gaussmap, Colloq. math. 129 (2012), no 2.

[4] S. M. Jung, D.-S. Kim and Y. H. Kim, Minimal ruled submanifolds associatedwith Gauss map, Taiwanese J. Math. 22 (2018), 567-605.

[5] D.-S. Kim, Y. H. Kim and D. W. Yoon, Extended B-scrolls and their Gauss maps,Indian J. Pure Appl. Math. 33 (2002), 1031-1040.

[6] Y. H. Kim and D. W. Yoon, Ruled surfaces with finite type Gauss map inMinkowski spaces, Soochow J. Math. 26 (2000), 85-96.

aDepartment of Mathematics, Kyungpook National University, Daegu41566, Rep. of KoreaE-mail : [email protected] author was supported by the National Research Foundation of Korea(NRF) grantfunded by the Korea government(MSIT)(2019R1C1C1006370).

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Some annihilator conditions in generalized

composite Hurwitz rings

Dong Kyu Kwona, Jung Wook Lima

Abstract

In this talk, we study some annihilator conditions on generalized compos-ite Hurwitz rings. Especially, we study PP-rings, PF-rings and PS-rings viageneralized composite Hurwitz rings.

2010 Mathematics Subject Classifications : 13A15, 13B25, 13F25Keywords: Annihilator, PP-ring, PF-ring, PS-ring

References

[1] A. Benhissi, F. Koja, Basic properties of Hurwitz series rings, Ric. Mat. 61 (2012)255-273.

[2] S. Hizem, Power series over an ascending chain of rings, Comm. Algebra 40(2012) 4263-4275.

aDepartment of Mathematics, Kyungpook National University, Daegu41566,


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