Set Of Nonnegative Integers Research Articles

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

Polynomial maps and polynomial sequences in groups

Abstract This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.

The sojourn times of one dimensional discrete-time quantum walks

In the existing literature, a sojourn time of a discrete-time quantum walk is not a random variable. To solve this problem, we redefine the sojourn time of a quantum walk where its coin evolution operator can be general. We first discuss a class of quantum walks governed by flip operators. We cumulatively calculate how much time a walker spends in the set of non-negative integers up to a fixed evolution time. Whether a walker makes a left or right evolution, we add up the staying times as long as it stays within the target set. We define a sojourn time as the total amount of the staying times. Compared with existing definitions, we show that this definition can satisfy the probability normalization. From this, we define a random variable about the sojourn time and discuss its probability distribution. We build a mathematical model to characterize a sojourn time that is embedded into a quantum walk. These results are also valid for a class of quantum walks governed by general coin operators. We also give a method for calculating the sojourn time and analyze the shape features of its probability distribution.

ON MONOTONE INCREASING REPRESENTATION FUNCTIONS

AbstractLet $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit [‘Counterexamples to a conjecture of Dombi in additive number theory’, Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon>0$ , then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$ . We also give an example to show that this bound is best possible.

Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix α ∈ ( 0 , 1 / 3 ) \alpha \in (0,1/3) . We show that, from a topological point of view, almost all sets A ⊆ N A\subseteq \mathbb {N} have the property that, if A ′ = A A^\prime =A for all but o ( n α ) o(n^{\alpha }) elements, then A ′ A^\prime is not a nontrivial sumset B + C B+C . In particular, almost all A A are totally irreducible. In addition, we prove that the measure analogue holds with α = 1 \alpha =1 .

A Flexible Semi-Poisson Distribution with Applications to Insurance Claims and Biological Data

In this paper, a discrete one-parameter distribution called the semi-Poisson distribution is introduced, which is based on a set of non-negative integers. It is seen that this distribution captures over-dispersion and zero-inflation scenarios well. A few properties of the proposed distribution, such as moments, the probability-generating function, index of dispersion, recurrence relation for the moments, and negative moments are presented. The distribution is applied to two real-life datasets related to insurance claims and parasite counts, where it is noted to perform better than many of the existing discrete distributions based on Z+, including some of the recently introduced ones.

Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions

In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the d-dimensional body-centered cubic (BCC) lattice $${\mathbb {L}^d}$$ and the set of non-negative integers $${{\mathbb {Z}}_+}$$ . Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on $${\mathbb {L}^d} \times {{\mathbb {Z}}_+}$$ in all dimensions $$d\ge 9$$ . As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value $$-1$$ .

DISCRETE LEFT-DEFINITE HAMILTONIAN SYSTEMS

In this paper we consider an even-dimensional discrete Hamiltonian system on the set of nonnegative integers in the left-definite form. Using the inertia indices of the hermitian form related with the solutions of the equation we construct some maximal subspaces of the solution space. After constructing some ellipsoids preserving nesting properties we introduce a lower bound for the number of Dirichlet-summable solutions of the equation. Moreover we introduce a limit-point criterion.

Probability solutions of the Sincov’s functional equation on the set of nonnegative integers

In this note, we establish when the bivariate discrete Schur-constant models possess the Sibuya-type aging property. It happens that the corresponding class is large, solving the counterpart of classical Sincov’s functional equation on the set of nonnegative integers.

A constant-time sampling algorithm for binary Gaussian distribution over the integers

Discrete Gaussian sampling over the integers is one of fundamental operations in lattice-based cryptography. The binary Gaussian distribution DZ+,σ2 is a special discrete Gaussian distribution with σ2=1/(2ln⁡2) and μ=0 over the set of non-negative integers Z+, and a sampling algorithm for DZ+,σ2 can be used as the base sampler in a generic algorithm based on rejection sampling for any discrete Gaussian distribution over the integers. We present a constant-time algorithm for sampling from the binary Gaussian distribution DZ+,σ2. It requires no precomputation storage and mainly relies on bitwise operations, which could be more hardware-friendly. Its computational complexity is lower than that of the algorithm based on the full-tree Knuth-Yao method, and its entropy consumption is smaller than that of the full-table access algorithm based on a cumulative distribution table. The Rényi-divergence based security analysis of our constant-time algorithm can also be simplified.

Factorizations in evaluation monoids of Laurent semirings

For let be the semiring of real numbers with all where is the set of nonnegative integers and is the semiring of Laurent polynomials with coefficients in In this paper, we study various factorization properties of the additive structure of We characterize when is atomic. Then we characterize when satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when satisfies the unique factorization property and show that, when this is not the case, has infinite elasticity.

Universal bounds for fixed point iterations via optimal transport metrics

We present a self-contained analysis of a particular family of metrics over the set of non-negative integers. We show that these metrics, which are defined through a nested sequence of optimal transport problems, provide tight estimates for general Krasnosel'skii-Mann fixed point iterations for non-expansive maps. We also describe some of their very special properties, including their monotonicity and the so-called convex quadrangle inequality that yields a greedy algorithm to compute them efficiently.

On some relationships between records in the transition from continuous to discrete distributions

In the present work, we consider discretization of continuous distributions and study relations between continuous record values, weak record values and discrete record values obtained from continuous record values by discretization. We first compare the numbers of continuous record values registered in real intervals and the numbers of discrete record values and weak record values located on the sets of non-negative integers. Further in the paper, we introduce the so-called strong continuous record values that hold out after discretization and we discuss their distributional properties. We also analyze what happens with record values after discretization of the standard exponential distribution. Finally, we present a simulation experiment, which supports the theoretical results of the paper.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.
 We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

Hyper-positive definite functions I: Scalar case, branching-type stationary stochastic processes

We propose a definition of branching-type stationary stochastic processes on rooted trees and related definitions of hyper-positivity for functions on the unit circle and functions on the set of non-negative integers. We then obtain (1) a necessary and sufficient condition on a rooted tree for the existence of non-trivial branching-type stationary stochastic processes on it, (2) a complete criterion of the hyper-positive functions in the setting of rooted homogeneous trees in terms of a variant of the classical Herglotz-Bochner Theorem, (3) a prediction theory result for branching-type stationary stochastic processes.As an unexpected application, we obtain natural hypercontractive inequalities for Hankel operators with hyper-positive symbols.

On the structure of the h-fold sumsets

Let A be a set of nonnegative integers. Let (hA) (t) be the set of all integers in the sumset hA that have at least t representations as a sum of h elements of A. In this paper, we prove that, if k≥2, and A=a 0 ,a 1 ,⋯,a k is a finite set of integers such that 0=a 0 <a 1 <⋯<a k and gcda 1 ,a 2 ,⋯,a k =1, then there exist integers c t ,d t and sets C t ⊆[0,c t -2], D t ⊆[0,d t -2] such that

Density of sets with missing differences and applications

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

Positive Solution of Discrete BVPs Involving the Mean Curvature Operator on the Set of Nonnegative Integers

Positive Solution of Discrete BVPs Involving the Mean Curvature Operator on the Set of Nonnegative Integers

A Curious Identity on Multiple Sums over Fields with Applications

Let [Formula: see text] be a field, and let [Formula: see text] be integers such that [Formula: see text] and [Formula: see text]. We show that for any subset [Formula: see text], the curious identity [Formula: see text] holds with [Formula: see text] being the set of nonnegative integers. As an application, we prove that for any subset [Formula: see text] with [Formula: see text] being the finite field of [Formula: see text] elements and [Formula: see text] being integers such that [Formula: see text] and [Formula: see text], [Formula: see text] Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996, we show that if [Formula: see text] is an integer with [Formula: see text], then for any subset [Formula: see text] we have [Formula: see text] This implies [Formula: see text].

L(2,1,1)-Labeling of Circular-Arc Graphs

Graph labeling problem has been broadly studied in recent past for its wide applications, in mobile communication system for frequency assignment, radar, circuit design, X-ray crystallography, coding theory, etc. An L211-labeling (L211L) of a graph G = (V, E) is a function γ : V → Z∗ such that |γ(u) − γ(v)| ≥ 2, if d(u, v) = 1 and |γ(u) − γ(v)| ≥ 1, if d(u, v) = 1 or 2, where Z∗ be the set of non-negative integers and d(u, v) represents the distance between the nodes u and v. The L211L numbers of a graph G, are denoted by λ2,1,1(G) which is the difference between largest and smallest labels used in L211L. In this article, for circular-arc graph (CAG) G we have proved that λ2,1,1(G) ≤ 6∆ − 4, where ∆ represents the degree of the graph. Beside this we have designed a polynomial time algorithm to label a CAG satisfying the conditions of L211L. The time complexity of the algorithm is O(n∆2), where n is the number of nodes of the graph G.