scholarly journals Rank 2 preservers on symmetric matrices with zero trace

2021 ◽  
Vol 36 ◽  
pp. 03002
Author(s):  
Wai Keong Kok
Keyword(s):  
Positive Integer ◽  
Rank Function ◽  
Vector Spaces ◽  
Symmetric Matrices ◽  
Linear Map ◽  
Rank 2 ◽  
Fixed Positive Integer

Let F be a field, V1 and V2 be vector spaces of matrices over F and let ρ be the rank function. If T :V1 → V2 is a linear map, and k a fixed positive integer, we say that T is a rank k preserver if for any matrix Aϵ, V1 ρ(A) = k implies ρ(T( A))= k . In this paper, we characterize those rank 2 preservers on symmetric matrices with zero trace under certain conditions.

scholarly journals On the Stability of Quadratic Functional Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Jung Rye Lee ◽  
Jong Su An ◽  
Choonkil Park
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Quadratic Functional ◽  
The Stability ◽  
Fixed Positive Integer ◽  
Rassias Stability

LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

A Combinatorial Theorem

1966 ◽  
Vol 9 (4) ◽  
pp. 515-516
Author(s):  
Paul G. Bassett
Keyword(s):  
Positive Integer ◽  
Combinatorial Theorem ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

scholarly journals Hyperstability of the k -Cubic Functional Equation in Non-Archimedean Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Youssef Aribou ◽  
Mohamed Rossafi
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
Banach Spaces ◽  
Functional Equations ◽  
Cubic Functional Equation ◽  
Fixed Point Approach ◽  
Fixed Positive Integer

Using the fixed point approach, we investigate a general hyperstability results for the following k -cubic functional equations f k x + y + f k x − y = k f x + y + k f x − y + 2 k k 2 − 1 f x , where k is a fixed positive integer ≥ 2 , in ultrametric Banach spaces.

scholarly journals On the sum of digits of some sequences of integers

2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca ◽  
Juanjo Rué ◽  
Ana Zumalacárregui
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Integer Sequences ◽  
Sum Of Digits ◽  
Fixed Positive Integer ◽  
Almost All

AbstractLet b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

scholarly journals On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

2021 ◽  
Vol 6 (10) ◽  
pp. 10596-10601
Author(s):  
Yahui Yu ◽  
◽  
Jiayuan Hu ◽  
Keyword(s):  
Positive Integer ◽  
Unsolved Problem ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Elementary Methods ◽  
Fixed Positive Integer ◽  
Almost All ◽  
Terai’S Conjecture ◽  
Positive Integers

<abstract><p>Let $ k $ be a fixed positive integer with $ k &gt; 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

An optimal algorithm to find minimum k-hop dominating set of interval graphs

2019 ◽  
Vol 11 (02) ◽  
pp. 1950016 ◽  
Author(s):  
Sambhu Charan Barman ◽  
Madhumangal Pal ◽  
Sukumar Mondal
Keyword(s):  
Positive Integer ◽  
Optimal Algorithm ◽  
Dominating Set ◽  
Interval Graphs ◽  
Fixed Positive Integer

For a fixed positive integer [Formula: see text], a [Formula: see text]-hop dominating set [Formula: see text] of a graph [Formula: see text] is a subset of [Formula: see text] such that every vertex [Formula: see text] is within [Formula: see text]-steps from at least one vertex [Formula: see text], i.e., [Formula: see text]. A [Formula: see text]-hop dominating set [Formula: see text] is said to be minimal if there does not exist any [Formula: see text] such that [Formula: see text] is a [Formula: see text]-hop dominating set of G. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop dominating set, if it is minimal as well as it is [Formula: see text]-hop dominating set. In this paper, we present an optimal algorithm to find a minimum [Formula: see text]-hop dominating set of interval graphs with [Formula: see text] vertices which runs in [Formula: see text] time.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

An optimal algorithm to find minimum k-hop connected dominating set of permutation graphs

2020 ◽  
pp. 2150049
Author(s):  
Amita Samanta Adhya ◽  
Sukumar Mondal ◽  
Sambhu Charan Barman
Keyword(s):  
Positive Integer ◽  
Optimal Algorithm ◽  
Dominating Set ◽  
Time Algorithm ◽  
Connected Dominating Set ◽  
Permutation Graphs ◽  
Fixed Positive Integer

A set [Formula: see text] is said to be a [Formula: see text]-hop dominating set ([Formula: see text]-HDS) of a graph [Formula: see text] if every vertex [Formula: see text] is within [Formula: see text]-distances from at least one vertex [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is a fixed positive integer. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop connected dominating set of a graph [Formula: see text], if it is minimal as well as it is [Formula: see text]-HDS and the subgraph of G made by [Formula: see text] is connected. In this paper, we present an [Formula: see text]-time algorithm for computing a minimum [Formula: see text]-hop connected dominating set of permutation graphs with [Formula: see text] vertices.

Rings with xn + x or xn − x nilpotent

2021 ◽  
Author(s):  
Adel N. Abyzov ◽  
Peter V. Danchev ◽  
Daniel T. Tapkin
Keyword(s):  
Positive Integer ◽  
Fixed Positive Integer

Let [Formula: see text] be a ring and let [Formula: see text] be an arbitrary but fixed positive integer. We characterize those rings [Formula: see text] whose elements [Formula: see text] satisfy at least one of the relations that [Formula: see text] or [Formula: see text] is a nilpotent whenever [Formula: see text]. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with [Formula: see text] nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with [Formula: see text] nilpotent, J. Algebra Appl. 21 (2022)], respectively.

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