scholarly journals Semilattice strongly regular relations on ordered $ n $-ary semihypergroups

2021 ◽  
Vol 7 (1) ◽  
pp. 478-498
Author(s):  
Jukkrit Daengsaen ◽  
◽  
Sorasak Leeratanavalee ◽  
Keyword(s):  
Related Problem ◽  
Sufficient Condition ◽  
Semilattice Congruence ◽  
Previous Conclusion ◽  
Strongly Regular ◽  
Positive Integers

<abstract><p>In this paper, we introduce the concept of $ j $-hyperfilters, for all positive integers $ 1\leq j \leq n $ and $ n \geq 2 $, on (ordered) $ n $-ary semihypergroups and establish the relationships between $ j $-hyperfilters and completely prime $ j $-hyperideals of (ordered) $ n $-ary semihypergroups. Moreover, we investigate the properties of the relation $ \mathcal{N} $, which is generated by the same principal hyperfilters, on (ordered) $ n $-ary semihypergroups. As we have known from <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup> that the relation $ \mathcal{N} $ is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on $ n $-ary semihypergroups where $ n\geq 3 $. However, we provide a sufficient condition that makes the previous conclusion true on $ n $-ary semihypergroups and ordered $ n $-ary semihypergroups where $ n\geq 3 $. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their $ \mathcal{N} $-classes. As an application of the results, a related problem posed by Tang and Davvaz in <sup>[<xref ref-type="bibr" rid="b31">31</xref>]</sup> is solved.</p></abstract>

Matrix Transformations Based on Dirichlet Convolution

1997 ◽  
Vol 40 (4) ◽  
pp. 498-508
Author(s):  
Chikkanna Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Bounded Variation ◽  
Matrix Transformations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractThis paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

Varieties of involution monoids with extreme properties

2019 ◽  
Vol 70 (4) ◽  
pp. 1157-1180
Author(s):  
Edmond W H Lee
Keyword(s):  
The Other ◽  
Sufficient Condition ◽  
Finitely Based ◽  
Finitely Based Variety ◽  
Lattice Of Subvarieties ◽  
Power Set ◽  
Cross Variety ◽  
Positive Integers

Abstract A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These examples—all first of their kinds—include the following: finite involution monoids that generate huge varieties but whose reduct monoids generate Cross varieties; two finite involution monoids sharing a common reduct monoid such that one generates a huge, non-finitely based variety while the other generates a Cross variety; and two finite involution monoids that generate Cross varieties, the join of which is huge.

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

On a certain kind of reducible rational fractions

1980 ◽  
Vol 21 (3) ◽  
pp. 321-328
Author(s):  
Mordechai Lewin
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rational Fraction ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Positive Coefficients ◽  
Positive Integers ◽  
Exact Positions

The rational fractiona, c, p, q positive integers, reduces to a polynomial under conditions specified in a result of Grosswald who also stated necessary and sufficient conditions for all the coefficients to tie nonnegative.This last result is given a different proof using lemmas interesting in themselves.The method of proof is used in order to give necessary and sufficient conditions for the positive coefficients to be equal to one. For a < 2pq, a = αp + βq, α, β nonnegative integers, c > 1, the exact positions of the nonzero coefficients are established. Also a necessary and sufficient condition for the number of vanishing coefficients to be minimal is given.

scholarly journals Characterization of $[1,k]$-Bar Visibility Trees

10.37236/1116 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Guantao Chen ◽  
Joan P. Hutchinson ◽  
Ken Keating ◽  
Jian Shen
Keyword(s):  
Knapsack Problem ◽  
Unit Length ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Visibility Representation ◽  
Visibility Graphs ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to $[1,k]$-bar-visibility graphs by allowing the lengths of the bars to be between $1/k$ and $1$. We completely characterize these graphs for trees. We establish an algorithm with complexity $O(kn)$ to determine whether a tree with $n$ vertices has a $[1,k]$-bar-visibility representation. In the course of developing the algorithm, we study a special case of the knapsack problem: Partitioning a set of positive integers into two sets with sums as equal as possible. We give a necessary and sufficient condition for the existence of such a partition.

scholarly journals Friezes and continuant polynomials with parameters

2018 ◽  
Vol 36 (2) ◽  
pp. 57-81
Author(s):  
Véronique Bazier-Matte ◽  
David Racicot-Desloges ◽  
Tanna Sánchez McMillan
Keyword(s):  
Finite Type ◽  
Type A ◽  
Cluster Algebras ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Family ◽  
Laurent Phenomenon ◽  
Necessary And Sufficient ◽  
Positive Integers

Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c-friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c-friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c-friezes of integers and of positive integers. We also show some specific properties of c-friezes.

scholarly journals A congruence involving harmonic sums modulo pαqβ

2017 ◽  
Vol 13 (05) ◽  
pp. 1083-1094 ◽  
Author(s):  
Tianxin Cai ◽  
Zhongyan Shen ◽  
Lirui Jia
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Factors ◽  
Necessary And Sufficient ◽  
Positive Integers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

scholarly journals Dynamics of Boundary Graphs

2013 ◽  
Vol 5 (3) ◽  
pp. 447-455
Author(s):  
G. Mariumuthu ◽  
M. S. Saraswathy
Keyword(s):  
Simple Graph ◽  
The Other ◽  
Boundary Vertex ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Periodic Graph ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Boundary Graph

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic.  Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013) 

scholarly journals On the modular functions arising from the Theta constants

2003 ◽  
Vol 34 (1) ◽  
pp. 77-86
Author(s):  
Ugur S. Kirmaci
Keyword(s):  
Positive Integer ◽  
Modular Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Modular Functions ◽  
Necessary And Sufficient ◽  
Positive Integers

Some modular functions arising from the theta constants $ \vartheta_2(\tau)$, $ \vartheta_3(\tau)$, $ \vartheta_4(\tau)$ are investigated. Let $n$ be an odd square-free positive integer as in [4,7]. It is obtained a necessary and sufficient condition that $ \varphi_{\delta,\rho,3}(\tau)=\prod_{\delta|n,\rho|n}\Big({\vartheta_3(\delta\tau) \over\vartheta_3(\rho\tau)}\Big)^{r_\delta}$  is invariant with respect to transformations in $ \theta(n)$. Also, It is deduced that $ \varphi_{\delta,\rho,i}(\tau)$ is a modular function on $ P^{-2}\theta(n)P^2$, $ \theta(n)$, $P^{-1}\theta(n)P$, for $ i=2,3,4$, respectively. Thus, the result of L. Wilson's paper [7] is generalized. Furthermore, let $ m$ and $ n$ denote positive integers. Let $ r$, $ r_1$, $ r_2$ be integers such that $ r(m-1)(n+1)\equiv 0({\rm mod}~8)$, $ r_1(m-1)(n-1)\equiv 0({\rm mod}~ 8)$, $ r_2^2(n-m)(nm-1)\equiv 0({\rm mod}~8)$, it is shown that $ T_{m,n,i}^r(\tau)=\Big({\disp{\vartheta_i(\tau)\vartheta_i(n\tau)\over\vartheta_i(m\tau) \vartheta_i (mn\tau)}}\Big)^r$, $ H_{m,n,i}^{r_1}(\tau)=\Big({\disp{\vartheta_i(m\tau)\vartheta_i(n\tau)\over \vartheta_i(\tau)\vartheta_i(mn\tau)}}\Big)^{r_1}$ and $ \Phi_{m,n,i}^{r_2}(\tau)=\Big({\disp{\vartheta_i(m\tau)\over\vartheta_i(n\tau)}\Big)^{ r_2 }} $ are modular functions on $ \theta(mn)$, when $ i=3$. Similar results are deduced for $ P^{-2}\theta(mn)P^2$ and $ P^{-1}\theta(mn)P$, the suffixes 3 being replaced by 2 and 4, respectively. Therefore, the modular functions used in B. C. Berndt's paper [1] is rewritten for theta constants.

scholarly journals Generic point equivalence and Pisot numbers

2019 ◽  
Vol 40 (12) ◽  
pp. 3169-3180
Author(s):  
SHIGEKI AKIYAMA ◽  
HAJIME KANEKO ◽  
DONG HAN KIM
Keyword(s):  
Linear Transformation ◽  
Piecewise Linear ◽  
Pisot Number ◽  
Sufficient Condition ◽  
Generic Point ◽  
Pisot Numbers ◽  
Uncountable Family ◽  
Piecewise Linear Transformation ◽  
Generic Points ◽  
Positive Integers

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.

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