Sequence Of Positive Integers Research Articles

Deferred 𝜎-statistical summability in intuitionistic fuzzy 𝑟-normed linear spaces

Abstract In this paper, we define and study three novel summability concepts – strong deferred 𝜎-summability, deferred 𝜎-statistical summability, and 𝜎-statistical summability in intuitionistic fuzzy 𝑟-normed linear spaces (briefly called IF-𝑟-NLS) by using 𝜎-mean. We also provide an example in support of the new notions and uncover some interesting relationships. Additionally, we study deferred 𝜎-statistical summability in the context of two pairs of sequences of positive integers, namely, α n , γ n \alpha_{n},\gamma_{n} , and u n , v n u_{n},v_{n} satisfying α n ≤ u n < v n ≤ γ n \alpha_{n}\leq u_{n}<v_{n}\leq\gamma_{n} .

On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function $\zeta_{\mathbb{K}_3}(s)$ of a non-normal cubic extension $\mathbb{K}_3$ of $\mathbb{Q}$ over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ \sum_{{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}\atop{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}a_{k,\mathbb{K}_3} (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),\]where $(\zeta_{\mathbb{K}_3}(s))^{k}:=\sum_{n=1}^{\infty}\frac{a_{k,\mathbb{K}_3}(n)}{n^{s}}$.

On the Lindelöf hypothesis for general sequences

AbstractIn a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of “generic”: we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.

A threshold for the best two-term underapproximation by Egyptian fractions

Let G be the greedy algorithm that, for each θ∈(0,1], produces an infinite sequence of positive integers (an)n=1∞ satisfying ∑n=1∞1/an=θ. For natural numbers p<q, let Υ(p,q) denote the smallest positive integer j such that p divides q+j. Continuing Nathanson’s study of two-term underapproximations, we show that whenever Υ(p,q)⩽3, G gives the (unique) best two-term underapproximation of p/q; i.e., if 1/x1+1/x2<p/q for some x1,x2∈N, then 1/x1+1/x2⩽1/a1+1/a2. However, the same conclusion fails for every Υ(p,q)⩾4. Next, we study stepwise underapproximation by G. Let em=θ−∑n=1m1/an be the mth error term. We compare 1/am to a superior underapproximation of em−1, denoted by N/bm (N∈N⩾2), and characterize when 1/am=N/bm. One characterization is am+1⩾Nam2−am+1. Hence, for rational θ, we only have 1/am=N/bm for finitely many m. However, there are irrational numbers such that 1/am=N/bm for all m. Along the way, various auxiliary results are encountered.

Spectrality of random convolutions generated by finitely many Hadamard triples

Let {(Nj,Bj,Lj):1⩽j⩽m} be finitely many Hadamard triples in R . Given a sequence of positive integers {nk}k=1∞ and ω=(ωk)k=1∞∈{1,2,…,m}N , let μω,{nk} be the infinite convolution given by μω,nk=δNω1−n1Bω1∗δNω1−n1Nω2−n2Bω2∗⋯∗δNω1−n1Nω2−n2⋯Nωk−nkBωk∗⋯. In order to study the spectrality of μω,{nk} , we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if gcd(Bj−Bj)=1 for 1⩽j⩽m , then all infinite convolutions μω,{nk} are spectral measures. This implies that we may find a subset Λω,{nk}⊆R such that {eλ(x)=e2πiλx:λ∈Λω,{nk}} forms an orthonormal basis for L2(μω,{nk}) .

A characterization of 4-χS-vertex-critical graphs for packing sequences with s1=1 and s2≥3

If S=(s1,s2,…) is a non-decreasing sequence of positive integers, then the S-packing k-coloring of a graph G is a mapping c:V(G)→[k] such that if c(u)=c(v)=i for u≠v∈V(G), then dG(u,v)>si. The S-packing chromatic number of G is the smallest integer k such that G admits an S-packing k-coloring. A graph G is χS-vertex-critical if χS(G−u)<χS(G) for each u∈V(G). If G is χS-vertex-critical and χS(G)=k, then G is k−χS-vertex-critical. In this paper, 4−χS-vertex-critical graphs are characterized for sequences S=(1,s2,s3,…) with s2≥3. There are 28 sporadic examples and two infinite families of such graphs.

Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

AbstractFor a nondecreasing sequence of positive integers, an ‐packing edge‐coloring of a graph is a decomposition of edges of into disjoint sets such that for each the distance between any two distinct edges is at least . The notion of ‐packing edge‐coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not ‐packing (abbreviated to ‐packing) edge‐colorable and asked the question whether every subcubic graph is ‐packing edge‐colorable. Very recently, Hocquard, Lajou, and Lužar showed that every subcubic graph is ‐packing edge‐colorable and every 3‐edge‐colorable subcubic graph is ‐packing edge‐colorable. Furthermore, they also conjectured that every subcubic graph is ‐packing edge‐colorable.In this paper, we confirm the conjecture of Hocquard, Lajou, and Lužar, and extend it to multigraphs.

On a problem of Steinhaus

AbstractLet be a positive integer. A sequence of points in the unit interval [0,1) is piercing if holds for every and every . In 1958, Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most 74 elements. This was later improved to the best possible value of 17 by Warmus, and independently by Berlekamp and Graham. In this paper, we study a more general variant of piercing sequences. Let be an infinite nondecreasing sequence of positive integers. A sequence is ‐piercing if holds for every and every . A special case of , with a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each , the maximum length of any ‐piercing sequence is finite. Expressing this maximum length as , they obtained an exponential upper bound on the function , which was later improved to by Graham and Levy. Recently, Konyagin proved that holds for all sufficiently big . Using a different technique based on the Farey fractions and stick‐breaking games, we prove here that the function satisfies where and . We also prove that there exists an infinite ‐piercing sequence with if and only if .

An Extension of Sylvester’s Theorem on Arithmetic Progressions

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters: t1, being even or odd, and t2, congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester’s theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers.

Approximation by Egyptian fractions and the weak greedy algorithm

Let 0<θ⩽1. A sequence of positive integers (bn)n=1∞ is called a weak greedy approximation of θ if ∑n=1∞1/bn=θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (an) and (bn) such that(a) ∑n=1∞1/bn=θ;(b) 1/an+1<θ−∑i=1n1/bi<1/(an+1−1) for all n⩾1;(c) there exists t⩾1 such that bn/an⩽t infinitely often.We then investigate when a given weak greedy approximation (bn) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (an) with a1⩾2 and an→∞, there exist θ and (bn) such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence (an). Finally, we address the uniqueness of θ and (bn) and apply our framework to specific sequences.

Constructing bounded degree graphs with prescribed degree and neighbor degree sequences

Let $D = d_1, d_2, \ldots, d_n$ and $F = f_1, f_2,\ldots, f_n$ be two sequences of positive integers. We consider the following decision problems: is there a $i)$ multigraph, $ii)$ loopless multigraph, $iii)$ simple graph, $iv)$ connected simple graph, $v)$ tree, $vi)$ caterpillar $G = (V,E)$ such that for all $k$, $d(v_k) = d_k$ and $\sum_{w\in \mathcal{N}(v_k)} d(w) = f_k$ ($d(v)$ is the degree of $v$ and $\mathcal{N}(v)$ is the set of neighbors of $v$). Here we show that all these decision problems can be solved in polynomial time if $\max_{k} d_k$ is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.

Log-concavity of the restricted partition function [formula omitted] and the new Bessenrodt-Ono type inequality

Let A=(ai)i=1∞ be a non-decreasing sequence of positive integers and let k∈N+ be fixed. The function pA(n,k) counts the number of partitions of n with parts in the multiset {a1,a2,…,ak}. We find out a new Bessenrodt-Ono type inequality for the function pA(n,k). Further, we discover when and under what conditions on k, {a1,a2,…,ak} and N∈N+, the sequence (pA(n,k))n=N∞ is log-concave. Our proofs are based on the asymptotic behavior of pA(n,k) — in particular, we apply the results of Netto and Pólya-Szegő as well as the Almkavist's estimation.

Eggleston's dichotomy for characterized subgroups and the role of ideals

“Eggleston's dichotomy” is a “one of a kind” unique observation which broadly tells us that the characterized subgroups of the circle group (characterized by a sequence of positive integers (an)) are either countable or of cardinality c depending on the asymptotic behavior of the sequence of the ratios anan−1. One should note that these subgroups are generated by using the notion of usual convergence which is nothing but a special case of the more general notion of ideal convergence for the ideal Fin. It has been recently established that “Eggleston's dichotomy” fails in the case of modified versions of characterized subgroups when the ideal Fin is replaced by the natural density ideal Id, or more generally, by ideals which are now known as simple density and modular simple density ideals. As all the ideals mentioned above are analytic P-ideals, a natural question arises as to whether one can isolate some appropriate property of ideals which enforces the dichotomy or the failure of it. In this article we are able to isolate that particular feature of an ideal and come out with a new class of ideals which we call, “strongly non-translation invariant ideals” (in short snt-ideals). In particular, we are able to establish that for a sequence of positive integers (an), be it arithmetic or arising from the continued fraction expansion of an irrational number:(i)For non-snt analytic P ideals, the size of the corresponding characterized subgroups is always c even if the sequence (an) is b-bounded (i.e. the sequence of the ratios anan−1 is bounded) which implies the breaking down of “Eggleston's dichotomy”.(ii)For snt analytic P ideals, the corresponding characterized subgroups are always countable if the sequence (an) is b-bounded which means “Eggleston's dichotomy” holds.

ON THE ITERATES OF THE SHIFTED EULER’S FUNCTION

AbstractLet $\varphi $ be Euler’s function and fix an integer $k\ge 0$ . We show that for every initial value $x_1\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

Hausdorff and packing dimensions of homogeneous product Moran sets

Let be the collection of homogeneous product Moran sets determined by two sequences of positive integers , and two sequences of positive numbers , . We obtain the maximal and minimal values of the Hausdorff and packing dimensions of elements in .

Forcibly bipartite and acyclic (uni-)graphic sequences

The paper concerns the question of which properties of a graph are already determined by its degree sequence. The classic degree realization problem asks to characterize graphic sequences, i.e., sequences of positive integers, which are the degree sequence of some simple graph. Erdős and Gallai [11] solved this problem. Havel and Hakimi [14,17] provide a different characterization implying an algorithm to generate a realizing graph (if one exists). It is known that a graphic sequence can have several non-isomorphic realizations.We characterize graphic sequences where every realization has some given graph property P. Such sequences are called forcibly P-graphic. In particular, we present complete results characterizing forcibly (connected) bipartite, forcibly acyclic, and forcibly tree-graphic sequences. In those four models, we also characterize the sequences with a unique realizing graph, called unigraphic sequences. Finally, we address the problem of counting the number of sequences in each model.

The Ergodicity and Sensitivity of Nonautonomous Discrete Dynamical Systems

Let (E,h1,∞) be a nonautonomous discrete dynamical system (briefly, N.D.D.S.) that is defined by a sequence (hj)j=1∞ of continuous maps hj:E→E over a nontrivial metric space (E,d). This paper defines and discusses some forms of ergodicity and sensitivity for the system (E,h1,∞) by upper density, lower density, density, and a sequence of positive integers. Under some conditions, if the rate of convergence at which (hj)j=1∞ converges to the limit map h is “fast enough” with respect to a sequence of positive integers with a density of one, it is shown that several sensitivity properties for the N.D.D.S. (E,h1,∞) are the same as those properties of the system (E,h). Some sufficient conditions for the N.D.D.S. (E,h1,∞) to have stronger sensitivity properties are also presented. The conditions in our results are less restrictive than those in some existing works, and the conclusions of all the theorems in this paper improve upon those of previous studies. Thus, these results are extensions of the existing ones.

On [formula omitted]-packing colourings of distance graphs [formula omitted] and [formula omitted]

For a non-decreasing sequence of positive integers S=(s1,s2,…), the S-packing chromatic numberχS(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into subsets Xi, i∈{1,2,…,k}, where vertices in Xi are pairwise at distance greater than Si. By an infinite distance graph with distance set D we mean a graph with vertex set Z in which two vertices i,j are adjacent whenever |i−j|∈D. In this paper we investigate the S-packing chromatic number of infinite distance graphs with distance set D={1,t}, t≥2, and D={1,2,t}, t≥3, for sequences S having all elements from {1,2}.

A quantitative Erdős-Fuchs type result for multivariate linear forms

Let A be an infinite sequence of positive integers and let m={(k1,m1),⋯,(kl,ml)} be a set of integer tuples such that gcd⁡(m1,m2,⋯,ml)>1. Denote by Rm(A,n) the number of different solutions of the equation k1(a1,1+⋯+a1,m1)+⋯+kl(al,1+⋯+al,ml)≤n with ai,j∈A. In 2013, Rué proved that for every ε>0 the function Rm(A,n)=cn+O(n1/4−ε) cannot hold for any constant c>0. Recently, we improved this bound to o(n1/4). In this paper, we obtain a stronger version about this result.

The sequence of prime gaps is graphic

Let us call a simple graph on n⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\geqslant 2$$\\end{document} vertices a prime gap graph if its vertex degrees are 1 and the first n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} prime gaps. We show that such a graph exists for every large n, and in fact for every n⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\geqslant 2$$\\end{document} if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object.