• Title/Summary/Keyword: d-regular

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FUZZY D-CONTINUOUS FUNCTIONS

  • Akdag, Metin
    • East Asian mathematical journal
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    • v.17 no.1
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    • pp.1-17
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    • 2001
  • In this paper, fuzzy D-continuous function is defined. Some basic properties of this continuity are summarized; and sufficient conditions on domain and/or ranges implying fuzzy D-continuity of fuzzy D-continuous functions are given. Also fuzzy D-regular space is defined and by using fuzzy D-continuity, the condition which is equivalent to fuzzy D-regular space, is given.

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A Relationship between the Second Largest Eigenvalue and Local Valency of an Edge-regular Graph

  • Park, Jongyook
    • Kyungpook Mathematical Journal
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    • v.61 no.3
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    • pp.671-677
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    • 2021
  • For a distance-regular graph with valency k, second largest eigenvalue r and diameter D, it is known that r ≥ $min\{\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2},\;a_3\}$ if D = 3 and r ≥ $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ if D ≥ 4, where λ = a1. This result can be generalized to the class of edge-regular graphs. For an edge-regular graph with parameters (v, k, λ) and diameter D ≥ 4, we compare $\frac{{\lambda}+\sqrt{{\lambda}^2+4k}}{2}$ with the local valency λ to find a relationship between the second largest eigenvalue and the local valency. For an edge-regular graph with diameter 3, we look at the number $\frac{{\lambda}-\bar{\mu}+\sqrt{({\lambda}-\bar{\mu})^2+4(k-\bar{\mu})}}{2}$, where $\bar{\mu}=\frac{k(k-1-{\lambda})}{v-k-1}$, and compare this number with the local valency λ to give a relationship between the second largest eigenvalue and the local valency. Also, we apply these relationships to distance-regular graphs.

ON KERNELS AND ANNIHILATORS OF LEFT-REGULAR MAPPINGS IN d-ALGEBRAS

  • Ahn, Sun-Shin;So, Keum-Sook
    • Honam Mathematical Journal
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    • v.30 no.4
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    • pp.645-658
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    • 2008
  • In this paper, left-regular maps on d-algebras are defined. These mappings show behaviors reminiscent of homomorphisms on d-algebras which have been studied elsewhere. In particular for these mappings kernels, annihilators and co-annihilators are defined and some of their properties are investigated, especially in the setting of positive implicative d-algebras.

ASYMPTOTIC NUMBERS OF GENERAL 4-REGULAR GRAPHS WITH GIVEN CONNECTIVITIES

  • Chae, Gab-Byung
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.1
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    • pp.125-140
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    • 2006
  • Let $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ be the number of general4-regular graphs on n labelled vertices with $l_1+2l_2$ loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of $g(n,\;l_1,\;l_2,\;d,\;t,\;q)$ and hence that of g(2n), the total number of general 4-regular graphs where $l_1,\;l_2,\;d,\;t\;and\;q\;=\;o(\sqrt{n})$, respectively. We show that almost all general 4-regular graphs are 2-connected. Moreover, we determine the asymptotic numbers of general 4-regular graphs with given connectivities.

CONSTRUCTIONS OF REGULAR SPARSE ANTI-MAGIC SQUARES

  • Chen, Guangzhou;Li, Wen;Xin, Bangying;Zhong, Ming
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.3
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    • pp.617-642
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    • 2022
  • For positive integers n and d with d < n, an n × n array A based on 𝒳 = {0, 1, …, nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n, d), if each non-zero element of X occurs exactly once in A, and its row-sums, column-sums and two main diagonal-sums constitute a set of 2n + 2 consecutive integers. An SAMS(n, d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order n ≡ 1, 5 (mod 6), and prove that there exists a regular SAMS(n, d) for any n ≥ 5, n ≡ 1, 5 (mod 6) and d with 2 ≤ d ≤ n - 1.

A case study on high school students' mental image in the process of solving regular polyhedron problems (정다면체 문제 해결 과정에서 나타나는 고등학교 학생들의 심상에 관한 사례연구)

  • Hong, Gap Lyung;Kim, Won Kyung
    • The Mathematical Education
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    • v.53 no.4
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    • pp.493-507
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    • 2014
  • The purpose of this study is to analyze how high school students form and interpret the mental image in the process of solving regular polyhedron problems. For this purpose, a set of problems about the regular polyhedron's vertex is developed on the base of the regular polyhedron's duality and circulation. and applied to 2 students of the 12th graders in D high school. After 2 hours of teaching and learning and another 2 hours of mental image-analysis process, the following research findings are obtained. Fisrt, a student who recorded medium high-level grade in the national scholastic test can build the dynamic image or the patten image in the process of solving regular polyhedron's vertex problems by utilizing the 3D geometry program. However, the other student who recorded low-level grade can build the concrete-pictorial image. Second, pattern image or dynamic image can help students solve the regular polyhedron's vertex problems by proper transformation of informations and the mental images while the concrete-pictorial image does not help. Hence, it is recommended that the mathematics teachers should develop teaching and learning materials about the regular polyhedron's duality and circulation and also give students suitable questions to build the various mental images.

EAKIN-NAGATA THEOREM FOR COMMUTATIVE RINGS WHOSE REGULAR IDEALS ARE FINITELY GENERATED

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
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    • v.18 no.3
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    • pp.271-275
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    • 2010
  • Let R be a commutative ring with identity, T(R) be the total quotient ring of R, and D be a ring such that $R{\subseteq}D{\subseteq}T(R)$ and D is a finite R-module. In this paper, we show that each regular ideal of R is finitely generated if and only if each regular ideal of D is finitely generated. This is a generalization of the Eakin-Nagata theorem that R is Noetherian if and only if D is Noetherian.

CLASSIFICATION OF TWO-REGULAR DIGRAPHS WITH MAXIMUM DIAMETER

  • Kim, Byeong Moon;Song, Byung Chul;Hwang, Woonjae
    • Korean Journal of Mathematics
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    • v.20 no.2
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    • pp.247-254
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    • 2012
  • The Klee-Quaife problem is finding the minimum order ${\mu}(d,c,v)$ of the $(d,c,v)$ graph, which is a $c$-vertex connected $v$-regular graph with diameter $d$. Many authors contributed finding ${\mu}(d,c,v)$ and they also enumerated and classied the graphs in several cases. This problem is naturally extended to the case of digraphs. So we are interested in the extended Klee-Quaife problem. In this paper, we deal with an equivalent problem, finding the maximum diameter of digraphs with given order, focused on 2-regular case. We show that the maximum diameter of strongly connected 2-regular digraphs with order $n$ is $n-3$, and classify the digraphs which have diameter $n-3$. All 15 nonisomorphic extremal digraphs are listed.

CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

  • Volkmann, Lutz;Winzen, Stefan
    • Journal of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.683-695
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    • 2007
  • A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with $r{\geq}2$ vertices in each partite set contains a cycle with exactly r-1 vertices from each partite set, with exception of the case that c=4 and r=2. Here we will examine the existence of cycles with r-2 vertices from each partite set in regular multipartite tournaments where the r-2 vertices are chosen arbitrarily. Let D be a regular c-partite tournament and let $X{\subseteq}V(D)$ be an arbitrary set with exactly 2 vertices of each partite set. For all $c{\geq}4$ we will determine the minimal value g(c) such that D-X is Hamiltonian for every regular multipartite tournament with $r{\geq}g(c)$.

G-REGULAR SEMIGROUPS

  • Sohn, Mun-Gu;Kim, Ju-Pil
    • Bulletin of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.203-209
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    • 1988
  • In this paper, we define a g-regular semigroup which is a generalization of a regular semigroup. And we want to find some properties of g-regular semigroup. G-regular semigroups contains the variety of all regular semigroup and the variety of all periodic semigroup. If a is an element of a semigroup S, the smallest left ideal containing a is Sa.cup.{a}, which we may conveniently write as $S^{I}$a, and which we shall call the principal left ideal generated by a. An equivalence relation l on S is then defined by the rule alb if and only if a and b generate the same principal left ideal, i.e. if and only if $S^{I}$a= $S^{I}$b. Similarly, we can define the relation R. The equivalence relation D is R.L and the principal two sided ideal generated by an element a of S is $S^{1}$a $S^{1}$. We write aqb if $S^{1}$a $S^{1}$= $S^{1}$b $S^{1}$, i.e. if there exist x,y,u,v in $S^{1}$ for which xay=b, ubv=a. It is immediate that D.contnd.q. A semigroup S is called periodic if all its elements are of finite order. A finite semigroup is necessarily periodic semigroup. It is well known that in a periodic semigroup, D=q. An element a of a semigroup S is called regular if there exists x in S such that axa=a. The semigroup S is called regular if all its elements are regular. The following is the property of D-classes of regular semigroup.group.

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