scholarly journals Extremal Problems for Subset Divisors

10.37236/3438 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tony Huynh
Keyword(s):  
Extremal Problem ◽  
Extremal Problems ◽  
Number Of Divisors ◽  
Positive Integers ◽  
Special Case

Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$, what is the maximum number of divisors a set of $n$ positive integers can have? We determine this function exactly for all values of $n$. Moreover, for each $n$ we characterize all sets that achieve the maximum. We also prove results for the $k$-subset analogue of our problem. For this variant, we determine the function exactly in the special case that $n=2k$. We also characterize all sets that achieve this bound when $n=2k$.  

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

scholarly journals Factorizations of outer functions and extremal problems

1996 ◽  
Vol 39 (3) ◽  
pp. 535-546 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Extremal Problems ◽  
The Other ◽  
Outer Function ◽  
Inner Functions ◽  
Special Case ◽  
Class Of Functions

The author has proved that an outer function in the Hardy space H1 can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case. Using it, we describe the solutions of extremal problems in the Hardy spaces Hp for 1≦p<∞.

scholarly journals Detachments Preserving Local Edge-Connectivity of Graphs

1999 ◽  
Vol 6 (35) ◽  
Author(s):  
Tibor Jordán ◽  
Zoltán Szigeti
Keyword(s):  
Edge Connectivity ◽  
Common Generalization ◽  
Connectivity Augmentation ◽  
Local Edge ◽  
Positive Integers ◽  
Special Case

Let G = (V +s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with<br />Sum dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with<br />d(sj) = dj, 1 <= j <= p. Given a requirement function r(u, v) on pairs of vertices of V , an<br />S-detachment is called r-admissible if the detached graph G' satisfies lambda_G' (x, y) >= r(x, y)<br />for every pair x, y in V . Here lambda_H(u, v) denotes the local edge-connectivity between u and v<br />in graph H.<br />We prove that an r-admissible S-detachment exists if and only if (a) lambda_G(x, y) >= r(x, y),<br />and (b) lambda_G−s(x, y) >= r(x, y) − Sum |dj/2| hold for every x, y in V .<br />The special case of this characterization when r(x, y) = lambda_G(x, y) for each pair in V was conjectured by B. Fleiner. Our result is a common generalization of a theorem of W. Mader on edge splittings preserving local edge-connectivity and a result of B. Fleiner on detachments preserving global edge-connectivity. Other corollaries include previous results of L. Lov´asz and C.J.St.A. Nash-Williams on edge splittings and detachments, respectively. As a new application, we extend a theorem of A. Frank on local edge-connectivity augmentation to the case when stars of given degrees are added.

The Bracket Function and Complementary Sets of Integers

1969 ◽  
Vol 21 ◽  
pp. 6-27 ◽  
Author(s):  
Aviezri S. Fraenkel
Keyword(s):  
Irrational Numbers ◽  
Image Position ◽  
Positive Integers ◽  
Special Case

The following result is well known (as usual, [x]denotes the integral part of x):(A) Let α and β be positive irrational numbers satisfying1Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.The theorem has a converse, and the following holds:(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

Two Extremal Problems in Graph Theory

10.37236/1182 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Richard A. Brualdi ◽  
Stephen Mellendorf
Keyword(s):  
Graph Theory ◽  
Extremal Problems ◽  
Turan's Theorem ◽  
Turán’S Theorem ◽  
Positive Integers

We consider the following two problems. (1) Let $t$ and $n$ be positive integers with $n\geq t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that contains neither $K_t$ nor $K_{t,t}$ as a subgraph. (2) Let $r$, $t$ and $n$ be positive integers with $n\geq rt$ and $t\geq 2$. Determine the maximum number of edges of a graph of order $n$ that does not contain $r$ disjoint copies of $K_t$. Problem 1 for $n < 2t$ is solved by Turán's theorem and we solve it for $n=2t$. We also solve Problem 2 for $n=rt$.

The Average Number of Divisors in an Arithmetic Progression

1981 ◽  
Vol 24 (1) ◽  
pp. 37-41 ◽  
Author(s):  
R. A. Smith ◽  
M. V. Subbarao
Keyword(s):  
Arithmetic Progression ◽  
Divisor Function ◽  
Number Of Divisors ◽  
Image Position ◽  
Positive Integers

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.

scholarly journals Juggler's Exclusion Process

2012 ◽  
Vol 49 (01) ◽  
pp. 266-279
Author(s):  
Lasse Leskelä ◽  
Harri Varpanen
Keyword(s):  
Markov Process ◽  
Gibbs Measure ◽  
Equilibrium Distribution ◽  
Exclusion Process ◽  
Jump Height ◽  
The Family ◽  
System Of Particles ◽  
Unit Speed ◽  
Positive Integers ◽  
Special Case

Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.

Weight-Equitable Subdivision of Red and Blue Points in the Plane

2018 ◽  
Vol 28 (01) ◽  
pp. 39-56 ◽  
Author(s):  
Jude Buot ◽  
Mikio Kano
Keyword(s):  
General Position ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Weight ◽  
Blue Point ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

The fundamental principle of counting, tree diagrams, and the number of divisors of a number (the nu-function)

1969 ◽  
Vol 16 (4) ◽  
pp. 308-310
Author(s):  
Elvin Rasof
Keyword(s):  
Prime Number ◽  
Fundamental Principle ◽  
Number Concept ◽  
Number Of Divisors ◽  
Positive Integers

An article in the January 1968 issue of THE ARITHMETIC TEACHER demonstrates an introduction to the teaching of the prime-number concept. Towards the end of this article, mention is made of the number of divisors of a number (we restrict ourselves to the positive integers).In particular, the table shown here.1

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