Set Of Positive Integers Research Articles

An Inclusive Distance Irregularity Strength of n-ary Tree

An inclusive distance vertex irregular labelling of a simple graph G is a function of the vertex set of to positive integer set such that the sum of its vertex label and the labels of all vertices adjacent to the vertex are distinct. The minimum of maximum label of the vertices is said to be inclusive distance irregularity strength of G, denoted by dis(G). The purpose of this research is showing that dis(T_{n,2})= (n^2+2)/2 where T_{n,2} is a complete n-ary tree to level two.

Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

A Class of Bounded Iterative Sequences of Integers

In this note, we show that, for any real number τ∈[12,1), any finite set of positive integers K and any integer s1≥2, the sequence of integers s1,s2,s3,… satisfying si+1−si∈K if si is a prime number, and 2≤si+1≤τsi if si is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ,s1 and the maximal element of K only. In particular, if K is a singleton set and for each composite si the integer si+1 in the interval [2,τsi] is chosen by some prescribed rule, e.g., si+1 is the largest prime divisor of si, then the sequence s1,s2,s3,… is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K={1} and τ∈[12,34) or K={2} and τ∈[12,59).

On bounded basis with prescribed representation functions

Let [Formula: see text] be the set of integers and [Formula: see text] the set of positive integers. For a nonempty set [Formula: see text] of integers and any integers [Formula: see text], [Formula: see text] with [Formula: see text], define [Formula: see text] as the number of solutions of [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] A set [Formula: see text] of integers is defined as a basis of order [Formula: see text] for [Formula: see text] if [Formula: see text] for every integer [Formula: see text]. In 2004, Nešetřil and Serra considered the Erdős–Turán conjecture for a class of bounded bases. In this paper, we generalize the above result and obtain that: for any function [Formula: see text], there exists a bounded basis of order [Formula: see text] for [Formula: see text] such that [Formula: see text] for every integer [Formula: see text].

On the Number of Restricted One-to-One and Onto Functons Having Integral Coordinates

Let $N_m$ be the set of positive integers $1, 2, \cdots, m$ and $S \subseteq N_m$. In 2000, J. Caumeran and R. Corcino made a thorough investigation on counting restricted functions $f_{|S}$ under each of the following conditions:\begin{itemize}\item[(\textit{a})]$f(a) \leq a$, $\forall a \in S$;\item[(\textit{b})] $f(a) \leq g(a)$, $\forall a \in S$ where $g$ is any nonnegative real-valued continuous functions;\item[(\textit{c})] $g_1(a) \leq f(a) \leq g_2(a)$, $\forall a \in S$, where $g_1$ and $g_2$ are any nonnegative real-valued continuous functions.\end{itemize}Several formulae and identities were also obtained by Caumeran using basic concepts in combinatorics.In this paper, we count those restricted functions under condition $f(a) \leq a$, $\forall a \in S$, which is one-to-one and onto, and establish some formulas and identities parallel to those obtained by J. Caumeran and R. Corcino.

A note on the largest sum‐free sets of integers

AbstractGiven a set of positive integers, an old question in additive combinatorics asks that whether contains a sum‐free subset of size at least for some increasing unbounded function . The question is generally attacked in the literature by considering another conjecture, which asserts that as , . This conjecture, if true, would also imply that a similar phenomenon occurs for ‐sum‐free sets for every . In this note, we prove the latter result directly. The new ingredient of our proof is a structural analysis on the host set , which might be of independent interest.

Increasing sequence of T0-topologies on the set of positive integers

In this paper we introduce, similarly as it was done in [11] for Kirch's type topologies, a sequence of topologies Tm that are subtopologies of the common division topology T. The sum of all topologies Tm is the common division topology. We also show that all topologies Tm are T0 and they are not T1 on N, examine properties of prime numbers and characterize connectedness of arithmetic progressions in each of the topology Tm. An immediate consequence of these characterizations is local connectedness of the space (N,Tm) for each m∈N.

Remarks on Approximate Solutions to Difference Equations in Various Spaces

Quite often (e.g., using numerical methods), we are only able to find approximate solutions of some equations, and it is necessary to know the size of the difference between such approximate solutions and the mappings that satisfy the equation exactly. This issue is the main subject of the theory of Ulam stability, and it is related to other areas of research such as, e.g., shadowing, optimization, and approximation theory. In this expository paper, we present several selected outcomes on Ulam stability of difference equations, show possible extensions of them and indicate further directions for research. We also present and discuss some simple methods that allow improvement of several already known results concerning Ulam stability of some difference equations in normed or metric spaces and extend them to b-metric and 2-normed spaces. Our results show that the noticeable symmetry exists between the outcomes of this type in normed and metric spaces and those obtained by us for other spaces. In particular, we extend the result of Pólya and Szegö concerning the stability of equation xn+m=xn+xm for m,n∈T, where T means either the set of integers Z or the set of positive integers N. We also consider the stability of equation xn+p+a1xn+p−1+…+apxn+bn=0 (with a fixed positive integer p) and of two more general difference equations.

Random Sieves and Generalized Leader-election Procedures

A random sieve of the set of positive integers N is an infinite sequence of nested subsets N = S0 ⊃ S1 ⊃ S2 ⊃ · · · such that Sk is obtained from Sk−1 by removing elements of Sk−1 with the indices outside Rk and enumerating the remaining elements inthe increasing order. Here R1 , R2 , . . . is a sequence of independent copies of an infinite random set R ⊂ N. We prove general limit theorems for Sn and related functionals, as n → ∞.

Mod-φ convergence of Stirling distributions and limit theorems for zeros of their generating functions

We study mod-φ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when n balls are allocated equiprobably and independently among θ boxes in the regime when θ grows linearly with n.

On the \(D(4)\)-pairs \(\{a, ka\}\) with \(k\in \{2,3,6\}\)

Let \(a\) and \(b=ka\) be positive integers with \(k\in \{2, 3, 6\},\) such that \(ab+4\) is a perfect square. In this paper, we study the extensibility of the \(D(4)\)-pairs \(\{a, ka\}.\) More precisely, we prove that by considering families of positive integers \(c\) depending on \(a,\) if \(\{a, b, c, d\}\) is a set of positive integers which has the property that the product of any two of its elements increased by \(4\) is a perfect square, then \(d\) is given by d=a+b+c+1/2(abc±√((ab+4)(ac+4)(bc+4))). As a corollary, we prove that any \(D(4)\)-quadruple tht contains the pair \(\{a, ka\}\) is regular.

Exceptional Jacobi polynomials which are deformations of Jacobi polynomials

In [10] we constructed new examples of exceptional Jacobi polynomials. The difference with the so called standard families is that the examples constructed in [10] depend on an arbitrary number of continuous parameters. Besides these continuous parameters, denoted by M={M0,M1,…}, each one of the new families of exceptional Jacobi polynomials is associated to two negative integers a≤b≤−1 and a finite set of positive integers F satisfying that {−b,…,−a−b−1}⊂F. In this paper we prove that when F={u,u+1,…,−a−b−1}, with 1≤u≤−b, the exceptional Jacobi polynomials are deformations of Jacobi polynomials (when the continuous parameters Mi→1). Otherwise, the exceptional Jacobi polynomials are deformations of standard exceptional Jacobi polynomials (again when the continuous parameters Mi→1).

Some remarks on the Erdős Distinct subset sums problem

Let [Formula: see text] be a set of positive integers, [Formula: see text] denoting the largest element, so that for any two of the [Formula: see text] subsets the sum of all elements is distinct. Erdős asked whether this implies [Formula: see text] for some universal [Formula: see text]. We prove, slightly extending a result of Elkies, that for any [Formula: see text], [Formula: see text] with equality if and only if all subset sums are [Formula: see text]-separated. This leads to a new proof of the currently best lower bound [Formula: see text]. The main new insight is that having distinct subset sums and [Formula: see text] small requires the random variable [Formula: see text] to be close to Gaussian in a precise sense.

Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}

ABSTRACT Let M be a set of positive integers. We study the maximal density μ(M) of the sets of nonnegative integers S whose elements do not differ by an element in M. In 1973, Cantor and Gordon established a formula for μ(M) for |M| ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case |M| ≥ 3, including some results in the case when M is an infinite set. In this paper, we study the maximal density problem for the families M = {a, a+1, 2a+1, n} and M = {a, a+1, 2a+1, 3a+1, n}, where a and n are positive integers and n is sufficiently large. In most of the cases, we find bounds for the parameter kappa, denoted by κ(M), which actually serves as a lower bound for μ(M). The parameter κ(M) has already got its importance due to its rich connection with problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = {a, a +1, 2a + 1,…, (s – 2)a + 1, n}, where s ≥ 5 and mention related problems in the remaining cases for future work.

Integer colorings with forbidden rainbow sums

For a set of positive integers A⊆[n], an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of [n] with the maximum number of rainbow sum-free r-colorings. We show that for r=3, the interval [n] is optimal, while for r≥8, the set [⌊n/2⌋,n] is optimal. We also prove a stability theorem for r≥4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

N-hyponormality and n-contractivity of generalized Bergman weighted shifts

Several bridges of operator properties between subnormal and hyponormal operators on Hilbert spaces have been developed well. In the class of contractive operators, we may consider two bridges consisting of n-hyponormal and n-contractive operators for n∈N, where N is the set of positive integers. Several operator theorists have studied the one-step backward extension of generalized Bergman weighted shifts such as Agler shifts, Bergman-like shifts, and Bergman-type weighted shifts to detect gap for these two bridges. Obviously the class of homographic shifts S(a,b,c,d) contains these all generalized Bergman weighted shifts above. In this paper we obtain some equivalent conditions for the n-hyponormality and n-contractivity of the one-step backward extension of S(a,b,c,d), which may induce the known results about Agler shifts, Bergman-like shifts, and Catalan-like weighted shifts etc.

On the restricted order of two

A set of positive integers is an asymptotic basis of order h if every sufficiently large integer can be expressed as a sum of h or fewer elements of the set and if h is the smallest number for which this is true. The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. In this paper, we prove that if B is a set with lower asymptotic density d_(B)>1/2, then the restricted order of B is 2. Furthermore, we will prove that 1/2 is optimal.

Variations on the Erdős distinct-sums problem

Let { a 1 , . . . , a n } be a set of positive integers with a 1 < ⋯ < a n such that all 2 n subset sums are distinct. A famous conjecture by Erdős states that a n > c ⋅ 2 n for some constant c , while the best result known to date is of the form a n > c ⋅ 2 n / n . In this paper, inspired by an information-theoretic interpretation, we extend the study to vector-valued elements a i ∈ Z k and we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to λ n be distinct. For this case, we derive lower and upper bounds on the smallest possible value of a n .

Even partition functions and $2$-adic analysis

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

SOME RESULTS ON r-TRUNCATED DEGENERATE POISSON RANDOM VARIABLES

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. Recently, as a natural extension of those distributions, Kim–Kim studied the zero-truncated degenerate Poisson distributions. In this paper, we introduce the [Formula: see text]-truncated degenerate Poisson random variable with parameter [Formula: see text], whose probability mass function is given by [Formula: see text], and investigate various properties of this random variable.