Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article The Adjoint Reidemeister Torsion for Compact 3-Manifolds Admitting a Unique Decomposition(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2023) Erdal, Esma DiricanLet M be a triangulated, oriented, connected compact 3 -manifold with a connected nonempty boundary. Such a manifold admits a unique decomposition into △ -prime 3 -manifolds. In this paper, we show that the adjoint Reidemeister torsion has a multiplicative property on the disk sum decomposition of compact 3 -manifolds without a corrective term.Article Parity of an Odd Dominating Set(2022) Batal, AhmetFor a simple graph $G$ with vertex set $V(G)={v_1,...,v_n}$, we define the closed neighborhood set of a vertex $u$ as $N[u]={v in V(G) ; | ; v ; text{is adjacent to} ; u ; text{or} ; v=u }$ and the closed neighborhood matrix $N(G)$ as the matrix whose $i$th column is the characteristic vector of $N[v_i]$. We say a set $S$ is odd dominating if $N[u]cap S$ is odd for all $uin V(G)$. We prove that the parity of the cardinality of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.Article Discrete Fractional Integrals, Lattice Points on Short Arcs, and Diophantine Approximation(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2022) Temur, FarukRecently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in both subjects.Article Relaxation of Conditions of Lyapunov Functions(2021) Şahan, GökhanIn this study, stability conditions are given for nonlinear time varying systems using the classical Lyapunov 2nd Method and its arguments. A novel approach is utilized and so that uniform stability can also be proved by using an unclassical Lyapunov Function. In contrast with the studies in the literature, Lyapunov Function is allowed to be negative definite and increasing through the system. To construct a classical Lyapunov Function, we use a reverse time approach methodology for the intervals where the unclassical one is increasing. So we prove the stability using a new Lyapunov Function construction methodology. The main result shows that the existence of such a function guarantees the stability of the origin. Some numerical examples are also given to demonstrate the efficiency of the method we use.Article A Note on Points on Algebraic Sets(Hacettepe Üniversitesi, 2021) Çam Çelik, Şermin; Göral, HaydarIn this short note, we count the points on algebraic sets which lie in a subset of a domain. It is proved that the set of points on algebraic sets coming from certain subsets of a domain has the full asymptotic. This generalizes the first theorem of [E. Alkan and E.S. Yoruk, Statistics and characterization of matrices by determinant and trace, Ramanujan J., 2019] and also anwers some questions from the same article.Article Citation - WoS: 1Citation - Scopus: 1Stability in Commutative Rings(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2020) Ay Saylam, BaşakLet R be a commutative ring with zero-divisors and I an ideal of R. I is said to be ES-stable if JI = $I^2$ for some invertible ideal J ? I , and I is said to be a weakly ES-stable ideal if there is an invertible fractional ideal J and an idempotent fractional ideal E of R such that I = JE . We prove useful facts for weakly ES-stability and investigate this stability in Noetherian-like settings. Moreover, we discuss a question of A. Mimouni on locally weakly ES-stable rings: is a locally weakly ES-stable domain of finite character weakly ES-stable?Article Some Properties of Rickart Modules(Ankara Üniversitesi, 2012) Üngör, Burcu; Kafkas, Gizem; Halıcıoğlu, Sait; Harmancı, AbdullahR birimli bir halka, M saº g R-mod¸l ve M nin endomorÖzma halkas¨ S = EndR(M) olsun. Her f 2 S iÁin rM(f) = eM olacak biÁimde e2 = e 2 S varsa (denk olarakKerf,Mmod¸l¸n¸nbirdirekttoplanan¨ise)MyeRickartmod¸lad¨verilmi?stir[8]. BuÁal¨?smadaRickartmod¸llerinˆzellikleriincelenmeyedevamedilmi?stir. M birRickart mod¸l olmak ¸zere, M nin S-kat¨ (s¨ras¨yla S-indirgenmi?s, S-simetrik, S-yar¨ deºgi?smeli, S-Armendariz)mod¸l olmas¨ iÁin gerek ve yeter ?sart¨n S nin kat¨ (s¨ras¨yla indirgenmi?s, simetrik, yar¨ deºgi?smeli, Armendariz) halka olduºgu gˆsterilmi?stir. M[x], S[x] halkas¨na gˆre Rickart mod¸l iken M nin de Rickart mod¸l oldugu,tersinin M nin S-Armendariz olmas¨ durumunda doºgru olduºgu ispatlanm¨?st¨r. Ayrıca bir M mod¸l¸n¸n Rickart ol- mas¨iÁingerekveyeter?sart¨nhersaºgmod¸l¸nM-temelprojektifolduºgueldeedilmi?stir.Article Citation - WoS: 2Citation - Scopus: 2Rings With Few Units and the Infinitude of Primes(Hacettepe Üniversitesi, 2020) Özcan, Hikmet Burak; Taşkın, SedefIn this short note, our aim is to provide novel proofs for the infinitude of primes in an algebraic way. It’s thought that the first proof for the infinitude of primes was given by the Ancient Greek mathematician Euclid. To date, most of the proofs have been based on the fact that every positive integer greater than 1 can be written as a product of prime numbers. However, first we are going to prove a ring theoretic fact that if R is an infinite commutative ring with unity and the cardinality of the set of invertible elements is strictly less than the cardinality of the ring, then there are infinitely many maximal ideals. This fact leads to an elegant proof for the infinitude of primes. In addition, under the same cardinality assumption, we consider the special case in which R is a unique factorization domain (for short UFD) and establish another ring theoretic result. Thanks to it, we give a second proof of the infinitude of primes. © 2020, Hacettepe University. All rights reserved.Article The Frequency Function and Its Connections To the Lebesgue Points and the Hardy-Littlewood Maximal Function(TÜBİTAK, 2019) Temur, FarukThe aim of this work is to extend the recent work of the author on the discrete frequency function to the more delicate continuous frequency function tau, and further to investigate its relations to the Hardy-Littlewood maximal function M, and to the Lebesgue points. We surmount the intricate issue of measurability of tau f by approaching it with a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this, we give analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity of Mf for f simple to the zeros of tau f, and to the non-Lebesgue points of f.Article Citation - WoS: 5Citation - Scopus: 5Convergence Analysis and Numerical Solution of the Benjamin-Bona Equation by Lie-Trotter Splitting(TUBITAK, 2018) Zürnacı, Fatma; Gücüyenen Kaymak, Nurcan; Seydaoğlu, Muaz; Tanoğlu, GamzeIn this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting is applied to the equation. The local error bounds are obtained by using the approach based on the differential theory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global error estimate is obtained via Lady Windermere's fan argument. Finally, to confirm the expected convergence order, numerical examples are studied.