On Properties of ClassA(n)andn-Paranormal Operators

Direct Sum ◽

Paranormal Operator ◽

Letnbe a positive integer, and an operatorT∈B(ℋ)is called a classA(n)operator ifT1+n2/1+n≥|T|2andn-paranormal operator ifT1+nx1/1+n≥||Tx||for every unit vectorx∈ℋ, which are common generalizations of classAand paranormal, respectively. In this paper, firstly we consider the tensor products for classA(n)operators, giving a necessary and sufficient condition forT⊗Sto be a classA(n)operator whenTandSare both non-zero operators; secondly we consider the properties forn-paranormal operators, showing that an-paranormal contraction is the direct sum of a unitary and aC.0completely non-unitary contraction.

Bipartite graphs with close domination and k-domination numbers

Open Mathematics ◽

10.1515/math-2020-0047 ◽

2020 ◽

Vol 18 (1) ◽

pp. 873-885

Author(s):

Gülnaz Boruzanlı Ekinci ◽

Csilla Bujtás

Keyword(s):

Positive Integer ◽

Dominating Set ◽

Domination Number ◽

Bipartite Graphs ◽

Sufficient Condition ◽

Minimum Cardinality ◽

Vertex Set ◽

Necessary And Sufficient ◽

The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

A study on distance in graph complement

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500457 ◽

2020 ◽

Vol 12 (03) ◽

pp. 2050045

Author(s):

A. Chellaram Malaravan ◽

A. Wilson Baskar

Keyword(s):

Positive Integer ◽

Complete Graph ◽

Sufficient Condition ◽

The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

Translatable radii of an operator in the direction of another operator II

Mathematica Slovaca ◽

10.2478/s12175-009-0171-y ◽

2010 ◽

Vol 60 (1) ◽

Author(s):

Kallol Paul

Keyword(s):

Eigenvalue Problem ◽

Unit Vector ◽

Generalized Eigenvalue Problem ◽

Sufficient Condition ◽

Generalized Eigenvalue ◽

Necessary And Sufficient ◽

Stationary Vector

AbstractOne of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.

On the Structure of G(H, k)

Algebra Colloquium ◽

10.1142/s1005386714000273 ◽

2014 ◽

Vol 21 (02) ◽

pp. 317-330 ◽

Cited By ~ 3

Author(s):

Guixin Deng ◽

Pingzhi Yuan

Keyword(s):

Abelian Group ◽

Positive Integer ◽

Explicit Formula ◽

Sufficient Condition ◽

Directed Edge ◽

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

Chaos for Cosine Operator Functions on Groups

Abstract and Applied Analysis ◽

10.1155/2014/603234 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Chung-Chuan Chen

Keyword(s):

Locally Compact Group ◽

Sufficient Condition ◽

Locally Compact ◽

Direct Sum ◽

Cosine Operator ◽

Weight Condition ◽

Operator Functions ◽

Cosine Operator Functions ◽

Let1≤p<∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

Direct-sum decomposition of atomic and orthogonally complete rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700006777 ◽

1970 ◽

Vol 11 (3) ◽

pp. 357-361 ◽

Cited By ~ 5

Author(s):

Alexander Abian

Keyword(s):

Natural Number ◽

Sufficient Condition ◽

Direct Sum ◽

Direct Sum Decomposition ◽

Image Position ◽

In this paper we give a necessary and sufficient condition for decomposition (as a direct sum of fields) of a ring R in which for every x ∈ R there exists a (and hence the smallest) natural number n(x) > 1 such that . We would like to emphasize that in what follows R stands for a ring every element x of which satisfies (1).

A generalization of the practical numbers

International Journal of Number Theory ◽

10.1142/s1793042118500902 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1487-1503

Author(s):

Nicholas Schwab ◽

Lola Thompson

Keyword(s):

Positive Integer ◽

Arithmetic Function ◽

Arithmetic Functions ◽

Asymptotic Density ◽

Sufficient Condition ◽

Necessary And Sufficient ◽

First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

Spectral properties and tensor products of quasi-*-A(n) operators

Filomat ◽

10.2298/fil1408559s ◽

2014 ◽

Vol 28 (8) ◽

pp. 1559-1565

Author(s):

Junli Shen ◽

Alatancang Chen

Keyword(s):

Spectral Properties ◽

Tensor Products ◽

Sufficient Condition ◽

In this paper, we prove that the spectrum is continuous on the class of all quasi-*-A(n) operators. And we obtain a sufficient condition for a quasi-*-A(n) operator to be normal. Finally, we consider the tensor products of quasi-*-A(n) operators, giving a necessary and sufficient condition for T S to be a quasi-*-A(n) operator when T and S are both non-zero operators.

Sifat-Sifat Representasi Indekomposabel The Properties of Indecomposable Representations

Natural Science Journal of Science and Technology ◽

10.22487/25411969.2019.v8.i3.14598 ◽

2019 ◽

Vol 8 (3) ◽

Author(s):

Vika Yugi Kurniawan

Keyword(s):

Vector Space ◽

Directed Graph ◽

Linear Mapping ◽

Sufficient Condition ◽

Paper Study ◽

Direct Sum ◽

Indecomposable Representations ◽

Finite Set ◽

A directed graph is also called as a quiver where is a finite set of vertices, is a set of arrows, and are two maps from to . A representation of a quiver is an assignment of a vector space to each vertex of and a linear mapping to each arrow. We denote by the direct sum of representasions and of a quiver . A representation is called indecomposable if is not ishomorphic to a direct sum of non-zero representations. This paper study about the properties of indecomposable representations. These properties will be used to investigate the necessary and sufficient condition of indecomposable representations.

Pairs of Consecutive Power Residues or Non-Residues

Canadian Journal of Mathematics ◽

10.4153/cjm-1964-030-6 ◽

1964 ◽

Vol 16 ◽

pp. 310-314 ◽

Cited By ~ 4

Author(s):

J. H. Jordan

Keyword(s):

Positive Integer ◽

Cyclic Group ◽

Proper Subgroup ◽

Multiplicative Group ◽

Sufficient Condition ◽

Root Of Unity ◽

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that