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.

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.

Automorphism Groups of Algebras of Finite Type

1972 ◽  
Vol 24 (6) ◽  
pp. 1065-1069 ◽  
Author(s):  
Matthew Gould
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Finite Type ◽  
Permutation Group ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

scholarly journals Embedding subshifts of finite type into the Fibonacci–Dyck shift

2016 ◽  
Vol 37 (6) ◽  
pp. 1862-1886
Author(s):  
TOSHIHIRO HAMACHI ◽  
WOLFGANG KRIEGER
Keyword(s):  
Finite Type ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Subshift Of Finite Type ◽  
Subshifts Of Finite Type ◽  
Necessary And Sufficient

A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci–Dyck shift.

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 On the geometry of generalized Chaplygin systems

2002 ◽  
Vol 132 (2) ◽  
pp. 323-351 ◽  
Author(s):  
FRANS CANTRIJN ◽  
JORGE CORTÉS ◽  
MANUEL DE LEÓN ◽  
DAVID MARTÍN DE DIEGO
Keyword(s):  
Invariant Measure ◽  
Type A ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Affine Connections ◽  
Chaplygin System ◽  
Necessary And Sufficient

Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

scholarly journals a*-families of analytic functions

1984 ◽  
Vol 7 (3) ◽  
pp. 435-442 ◽  
Author(s):  
G. P. Kapoor ◽  
A. K. Mishra
Keyword(s):  
Analytic Function ◽  
Taylor Series ◽  
Analytic Functions ◽  
Extreme Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Family ◽  
Symmetric Points ◽  
Necessary And Sufficient

Using convolutions, a new family of analytic functions is introduced. This family, calleda*-family, serves in certain situations to unify the study of many previously well known classes of analytic functions like multivalent convex, starlike, close-to-convex or prestarlike functions, functions starlike with respect to symmetric points and other such classes related to the class of univalent or multivalent functions. A necessary and sufficient condition on the Taylor series coefficients so that an analytic function with negative coefficients is in ana*-family is obtained and sharp coefficents bound for functions in such a family is deduced. The extreme points of ana*-family of functions with negative coefficients are completely determined. Finally, it is shown that Zmorvic conjecture is true if the concerned families consist of functions with negative coefficients.

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 Support Neighbourly Edge Irregular Graphs

2019 ◽  
Vol 8 (3) ◽  
pp. 5329-5332
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
New Family ◽  
Necessary And Sufficient ◽  
Irregular Graphs

In this paper, we introduce a new family of irregular graphs called support neighbourly edge irregular graphs based on degree in edge sense. In any graph, the support of an edge is the sum of the edge degrees of its neighbours. A graph is said to be support neighbourly edge irregular(or simply SNEI), if any two adjacent edges have different support. Basic properties of theses graphs are studied. A necessary and sufficient condition for a graph to be SNEI has been obtained and several methods to construct SNEI graph from other graphs have been discussed.

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