Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 1Citation - Scopus: 1Irreducibility and Primality in Differentiability Classes(Michigan State University Press, 2023) Batal, Ahmet; Eyidoğan, S.; Göral, HaydarIn this note, we give criteria for the irreducibility of functions in Cm [0, 1], where m ∈ {1, 2, 3, ...} ∪ {∞} ∪ {ω}. We also discuss irreducibility in multivariable differentiability classes. Moreover, we characterize irreducible functions and maximal ideals in C∞ [0, 1]. In fact, irreducible and prime smooth functions are the same, and every maximal ideal of C∞ [0, 1] is principal. © 2023 Michigan State University Press. All rights reserved.Article Citation - WoS: 7Citation - Scopus: 9Parity, Virtual Closure and Minimality of Knotoids(World Scientific Publishing, 2021) Güğümcü, Neslihan; Kauffman, Louis H.In this paper, we study parity in planar and spherical knotoids in relation to virtual knots. We introduce a planar version of the parity bracket polynomial for planar knotoids. We show that the virtual closure map (a map from the set of knotoids in S-2 to the set of virtual knots of genus at most one) is not surjective, by utilizing the surface bracket polynomial of virtual knots. We give specific examples of virtual knots that are not in the image of the virtual closure map. Turaev conjectured that minimal diagrams of knot-type knotoids have zero height. We prove this conjecture by using the results of Nikonov and Manturov induced by parities of virtual knots.Article Citation - WoS: 1Citation - Scopus: 1Rings Whose Nonsingular Right Modules Are R-Projective(Mathematical Institute of Charles University, 2021) Alagöz, Yusuf; Benli Göral, Sinem; Büyükaşık, EnginA right R-module M is called R-projective provided that it is projective relative to the right R-module R-R. This paper deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that R-R is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Sigma-CS and fiat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(R-R). In this case, the injective hull E(R-R) has the decomposition E(R-R) = U-R circle plus V-R, where U is projective and Hom(V, R/I) = 0 for each right ideal I of R. Finally, we focus on the right orthogonal class N-perpendicular to of the class IV of nonsingular right modules.Article Citation - WoS: 2Citation - Scopus: 3Biquandle Brackets and Knotoids(World Scientific Publishing, 2021) Güğümcü, Neslihan; Nelson, Sam; Oyamaguchi, NatsumiBiquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper, we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in (N. Gügümcü and S. Nelson, Biquandle coloring invariants of knotoids, J. Knot Theory Ramif. 28(4) (2019) 1950029). We provide examples to illustrate the method of calculation and to show that the new invariants are stronger than the previous ones. As an application we show that the trace of the biquandle bracket matrix is an invariant of the virtual closure of a knotoid.Article Citation - WoS: 20Citation - Scopus: 25Vertex-Decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuoco-Mumford Regularity(Electronic Journal of Combinatorics, 2014) Biyikoglu, Turker; Civan, YusufWe call a vertex x of a graph G = (V, E) a codominated vertex if N-G[y] subset of N-G[x] for some vertex y is an element of V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G - x is codismantlable. We show that (C-4, C-5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C-4, C-5, C-7)-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity reg(G) of such graphs, and show that reg(G) = im(G) whenever G is a (C-4, C-5)-free vertex-decomposable graph, where im(G) is the induced matching number of G. Furthermore, we prove that H must be a codismantlable graph if im(H) = reg(H) = m(H), where m(H) is the matching number of H. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family H-n (n >= 4) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and reg(H-n) = im(H-n) if n >= 6. This answers a recent question of Mahmoudi, et al [12].