Set Of Nonnegative Integers Research Articles

Approximation theorem for principle eigenvalue of discrete p-Laplacian

For the principle eigenvalue of discrete weighted p-Laplacian on the set of nonnegative integers, the convergence of an approximation procedure and the inverse iteration is proved. Meanwhile, in the proof of the convergence, the monotonicity of an approximation sequence is also checked. To illustrate these results, some examples are presented.

The computation power of tissue P systems with flip-flop channel states

Tissue P systems with channel states are a class of parallel computing models, where the communication between two regions is regulated by a channel state, and at a computation step the state of a channel can evolve to one of the arbitrarily many states. In this work, we limit the “arbitrarily many states of a channel” to “two states of a channel”. Such a variant of P systems is called tissue P systems with flip-flop channel states (TPFFCSs). The computation power of TPFFCSs is studied. We show that TPFFCSs with arbitrarily many cells and antiport rules of length 2 are able to compute only finite sets of non-negative integers. However, TPFFCSs with two cells, antiport rules of length 3, or symport rules of length 2, or symport rules of length 1 and antiport rules of length 2 are proved to be Turing universal.

Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern p we provide a bivariate generating function where the coefficient of xnyk in its series expansion is the number of length np-avoiding Catalan words with k descents. As a byproduct, we enumerate the set of Catalan words avoiding p, and we provide the popularity of descents on this set.

On the structure of some sets which have the same representation functions

Let $$\mathbb {N}$$ be the set of nonnegative integers. For a given set $$S\subset \mathbb {N}$$ the representation function $$R_S(n)$$ is defined as the number of solutions of the equation $$n=s+s', s<s', s,s'\in S$$ . In this paper, we characterize some sets which have the same representation functions.

A NOTE ON THE ERDŐS–GRAHAM THEOREM

Let ${\mathcal{A}}=\{a_{1}&lt;a_{2}&lt;\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.

Switched signed graphs of integer additive set-valued signed graphs

Let [Formula: see text] denote a set of non-negative integers and [Formula: see text] be its power set. An integer additive set-labeling (IASL) of a graph [Formula: see text] is an injective set-valued function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is the sumset of [Formula: see text] and [Formula: see text]. An IASL of a signed graph [Formula: see text] is an IASL of its underlying graph [Formula: see text] together with the signature [Formula: see text] defined by [Formula: see text]. A marking of a signed graph is an injective map [Formula: see text], defined by [Formula: see text] for all [Formula: see text]. Switching of signed graph is the process of changing the sign of the edges in [Formula: see text] whose end vertices have different signs. In this paper, we discuss certain characteristics of the switched signed graphs of certain types of integer additive set-labeled signed graphs.

A note on dispensing number algorithm of graphs

Let X denote a finite set of non-negative integers and 𝒫(X) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function f : V(G) → 𝒫(X) − {∅}, where induced function f+ : E(G) → 𝒫(X) − {∅} is defined by f+ (uv) = f (u) + f (v), the sumset of f (u) and f (v). An arithmetic integer additive set-labeling (AIASL) of a graph is an IASL under which the set-label of every element of G is an arithmetic progression. A prime AIASL is an AIASL in which the common difference of the set label of any vertex is a prime multiple (or divisor) of the common difference of the set-labels of its adjacent adjacent vertices. The dispensing number of an AIASL-graph G is the minimum number of edges to be removed from G so that it admits a prime AIASL. In this note, we discuss an algorithm for finding the dispensing number of arbitrary graphs.

Formula omitted]- and [formula omitted]-labeling problems on interval graphs

For a given graph G=(V,E), the L(3,2,1)- and L(4,3,2,1)-labeling problems assign the labels to the vertices of G. Let Z∗ be the set of non-negative integers. An L(3,2,1)- and L(4,3,2,1)-labeling of a graph G is a function f:V→Z∗ such that |f(x)−f(y)|≥k−d(x,y), for k=4,5 respectively, where d(x,y) represents the distance (minimum number of edges) between the vertices x and y, and 1≤d(x,y)≤k−1. The L(3,2,1)- and L(4,3,2,1)-labeling numbers of a graph G, are denoted by λ3,2,1(G) and λ4,3,2,1(G) and they are the difference between highest and lowest labels used in L(3,2,1)- and L(4,3,2,1)-labeling respectively. In this paper, for an interval graph G, it is shown that λ3,2,1(G)≤6Δ−3 and λ4,3,2,1(G)≤10Δ−6, where Δ represents the maximum degree of the vertices of G. Also, two algorithms are designed to label an interval graph by maintaining L(3,2,1)- and L(4,3,2,1)-labeling conditions. The time complexities of both the algorithms are O(nΔ2), where n represent the number of vertices of G.

An improved lower bound for finite additive 2-bases

A set of non-negative integers A is an additive 2-basis with range n, if its sumset A+A contains 0,1,…,n but not n+1. Explicit bases are known with arbitrarily large size |A|=k and n/k2≥2/7>0.2857. We present a more general construction and improve the lower bound to 85/294>0.2891.

Distance labelings: a generalization of Langford sequences

A Langford sequence of order m and defect d can be identified with a labeling of the vertices of a path of order 2 m in which each label from d up to d + m − 1 appears twice and in which the vertices that have been labeled with k are at distance k . In this paper, we introduce two generalizations of this labeling that are related to distances. The basic idea is to assign nonnegative integers to vertices in such a way that if n vertices ( n > 1 ) have been labeled with k then they are mutually at distance k . We study these labelings for some well known families of graphs. We also study the existence of these labelings in general. Finally, given a sequence or a set of nonnegative integers, we study the existence of graphs that can be labeled according to this sequence or set.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

On the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---

In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4. As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.

Exceptional Hahn and Jacobi orthogonal polynomials

Using Casorati determinants of Hahn polynomials (hnα,β,N)n, we construct for each pair F=(F1,F2) of finite sets of positive integers polynomials hnα,β,N;F, n∈σF, which are eigenfunctions of a second order difference operator, where σF is certain set of nonnegative integers, σF⊊︀N. When N∈N and α, β, N and F satisfy a suitable admissibility condition, we prove that the polynomials hnα,β,N;F are also orthogonal and complete with respect to a positive measure (exceptional Hahn polynomials). By passing to the limit, we transform the Casorati determinant of Hahn polynomials into a Wronskian type determinant of Jacobi polynomials (Pnα,β)n. Under suitable conditions for α, β and F, these Wronskian type determinants turn out to be exceptional Jacobi polynomials.

A class of adding machines and Julia sets

In this work we define a stochastic adding machine associated to the Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$. We obtain a Markov chain whose states are the set of nonnegative integers. We study probabilistic properties of this chain, such as transience and recurrence. We also prove that the spectrum associated to this Markov chain is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$.

Numerical semigroups in a problem about cost-effective transport

Abstract Let ℕ ${{\mathbb{N}}}$ be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients, a 1 ⁢ x 1 + ⋯ + a p ⁢ x p &lt; n &lt; b 1 ⁢ x 1 + ⋯ + b p ⁢ x p $a_{1}x_{1}+\cdots+a_{p}x_{p}&lt;n&lt;b_{1}x_{1}+\cdots+b_{p}x_{p}$ has at least one solution in ℕ p ${{\mathbb{N}}^{p}}$ . We will see that T ∪ { 0 } ${T\cup\{0\}}$ is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if { a + b - 1 , a + b + 1 } ⊆ S ${\{a+b-1,a+b+1\}\subseteq S}$ , for all a , b ∈ S ∖ { 0 } ${a,b\in S\setminus\{0\}}$ . In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Scaling limits for a family of unrooted trees

We introduce weights on the unrooted unlabelled plane trees as follows: for each p ≥ 1, let μp be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the μp-weight of a plane tree t is defined as Π μ_p(degree(v) − 1), where the product is over the set of vertices v of t. We study the random plane tree T_p which has a fixed diameter p and is sampled according to probabilities proportional to these μ_p-weights. We prove that, under the assumption that the sequence of laws μ_p, p ≥ 1, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of (T_p , p ≥ 1) are random compact real trees called the unrooted Levy trees, which have been introduced in Duquesne and Wang (2016+).

On Complete S-Reachable Graphs

We consider a variant of pebble motion game on a simple connected undirected graph. A configuration in such a graph indicates the placement of a robot and a hole at two different vertices and obstacles at the remaining vertices. A graph is said to be complete S-reachable if starting from any configuration the robot can be taken to any other vertex by a sequence of moves consisting of simple moves of the obstacles and mRJ moves of the robot for m ∈ S, where S is a finite non-empty set of non-negative integers. An mRJ move is the process of moving the robot from a vertex to another vertex along a path by jumping over m obstacles. The 0RJ move is known as a simple move. We identify some classes of graphs that are complete {m}-reachable.

On Mittag-Leffler distributions and related stochastic processes

Random variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. They can be of two different types, one (type-1) heavy-tailed with index α∈(0,1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching process, the discrete-space continuous-time Neveu branching process, the continuous space/time Neveu branching process (CSBP) and renewal processes with rare events. Its relation to (discrete or continuous) self-decomposability and branching processes with immigration is emphasized. Special attention will be paid to the Neveu CSBP for its connection with the Bolthausen–Sznitman coalescent. In this context, and following a recent work of Möhle (2015), a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu’s CSBP block-counting process arising in sampling from PD(e−t,0). Further combinatorial developments of this model are investigated.

Binomial ideal associated to a lattice and its label code

In coding theory the study of the binomial ideal derived from an arbitrary code is currently of great interest; see for example [5]. This is mainly because of a known relation between binomial ideals and lattices or codes. Also, studying the relation between binomial ideals associated to a lattice and its label code helps to solve the closest vector problem in lattices as well as decoding binary and non-binary codes [1, 3] and finding a label code of a lattice, as we do in this work. Every lattice Λ can be described in terms of a label code L and an orthogonal sublattice Λ′ such that Λ/Λ′ ∼= L [2]. We assign binomial ideals IΛ and IL to an integer lattice Λ and its label code L, respectively. In this work, we identify the binomial ideal associated to an integer lattice and then establish the relation IΛ = IΛ′ + IL between the ideal of the lattice and its label code. In this work, we define a binomial ideal for an integer lattice and its label code slightly different from [1, 3, 4, 7]. Let K[X] = K[x1, . . . , xn] denote the polynomial ring, where K is an arbitrary field. Consider ≺ as a fixed total degree compatible term order with x1 x2 · · · xn. The monomials in K[X] are denoted by X = x1 1 . . . x bn n where b = (b1, . . . , bn) is an element of N0 and N0 is the set of non-negative integers. We use the notation