Unified characteristic numbers and solutions of equations for birth and death processes with barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.     

Functional Equations and Weierstrass Transforms

The purpose of this article is to illustrate the utility of the Weierstrass transform in the study of functional equations (and systems) of the form 1 $${\mathop \sum^N\limits_{k=0}}\alpha_{k}f(x+r_{k})=f_{0}(x)\ \ \ \, x\in\ {\rm R}.$$ One may think of α₀, α₁,…, α_N as given complex numbers, r₀, r₁,…, r_N as given real numbers, ?₀: ? → C as a given function and ? as the unknown.     

Rank 1 real Wishart spiked model

In this paper, we consider N‐dimensional real Wishart matrices Y in the class \input amssym $W_{\Bbb R} (\Sigma ,M)$ in which all but one eigenvalue of Σ is 1. Let the nontrivial eigenvalue of Σ be 1+τ; then as N, M → ∞, with M/N → γ² finite and nonzero, the eigenvalue distribution of Y will converge into the Marchenko‐Pastur distribution inside a bulk region. When τ increases from 0, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko‐Pastur density. As this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish‐Chandra Itzykson‐Zuber integral $\int_{O(N)} {e^{{\rm tr}(XgYg^{\rm T} )} } g^{\rm T} dg$ when X and Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large‐ N limit characterized by Painlevé transcendents. The approach used in this paper is very different from a recent paper by Bloemenal and Virág, in which the largest eigenvalue distribution was obtained using a stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new. © 2012 Wiley Periodicals, Inc.     

Some universal limits for nonhomogeneous birth and death processes

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP X_N. Finally we present some examples where these bounds are used in order to approximate the double mean. AMS Subject Classification: Primary: 60J27 Secondary: 60K25 34A30     

On a linear iterative equation

We consider the following iterative equation $$ \sum_{i=0}^{k}a_{i}f^{i}(x)=0, $$ where a₀,…, a _k are given real numbers and ? is an unknown function. Assuming some conditions on the coefficients a₀,…, a _k we prove that this equation has exactly one solution and that the solution depends continuously on the coefficients.     

On the powers of 3/2 and other rational numbers

Letp > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ (p /q)ⁿ }, n = 0, 1, 2, …, for certain non‐zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ (3/2)ⁿ }, n ∈ N . The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ (3/2)ⁿ } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Virtual Eigenvalues of the High Order Schrödinger Operator I

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion.     

Principal eigenvalue in an unbounded domain with indefinite potential

In this work, we discuss the existence and the non-existence of principal eigenvalue in an unbounded domain of \mathbbR^N{\mathbb{R}^N} for some potentials which change sign. We also give certain properties of this principal eigenvalue.     

A nonsymmetric matrix with integer eigenvalues

A nonsymmetric N?×?N matrix with elements as certain simple functions of N distinct real or complex numbers r ₁, r ₂, …, r_N is presented. The matrix is special due to its eigenvalues???the consecutive integers 0,1,2, …, N?1. Theorems are given establishing explicit expressions of the right and left eigenvectors and formulas for recursive calculation of the right eigenvectors. A special case of the matrix has appeared in sampling theory where its right eigenvectors, if properly normalized, give the inclusion probabilities of the conditional Poisson sampling design.     

Continuum limits for classical sequential growth models

A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,…,n ? 1} and putting i below j if there is a path i = i₁…i_k = j in the graph with i₁ < … < i_k. Rideout and Sorkin 14 provide computational evidence that suitably normalized sequences of random graph orders have a “continuum limit.” We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large‐scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,…,n ‐ 1} such that the random partial order on {0,…,n ‐ 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

Let Λ={λ ₁,…,λ _p } be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

$x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0$

admits a nonzero solution x∈? ⁿ .

     

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

Quadratic forms on symmetry classes of tensors

Let V:₁,…,V_m be inner product spaces, and let L be a linear transformation on V₁ ?…?V_m which satisfies (Lz,z)=0 for every decomposable tensor z. It is known that if the field is the complex numbers, then (Lz,z)=0 for every z. This paper contains a short proof of this result, an extension of it to arbitrary symmetry classes of tensors, and an analysis of its failure when the field is the real numbers.     

Preservation of reliability classes associated with the mean residual life by a renewal process stopped at a random time

In this paper, we show that some ageing classes of a random time T related to the mean residual life are preserved by the discrete random count variable N(T), where {N(t) : t ?0}is a renewal process independent from T under suitable conditions. In the particular case of the Poisson process, we extend the results to more reliability classes. We also consider real examples of N(T) and apply the results to queuing systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Cantor sets arising from continued radicals

If a ₁,a ₂,a ₃,… are nonnegative real numbers and $f_{j}(x) = \sqrt{a_{j}+x}$ , then lim _n→∞ f ₁°f ₂°?°f _n (0) is a continued radical with terms a ₁,a ₂,a ₃,…. The set of real numbers representable as a continued radical whose terms a _i are all from a set S={a,b} of two natural numbers is a Cantor set. We investigate the thickness, measure, and sums of such Cantor sets.     

On multiple counting processes with the Markov property

We compute the distributions of the size of the jumps of an increasing Markov process on $\text{N}{}_0\text{= {0, 1,…}}$, and we give a necessary and sufficient condition in order to have only jumps of size one.     

Some results on odd factors of graphs

A {1, 3, …,2n ? 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n ? 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n ? 1}-factor.     

Critical Pairs of Sequences of a Mixed Frame Potential

The classical frame potential in a finite-dimensional Hilbert space has been introduced by Benedetto and Fickus, who showed that all finite unit-norm tight frames can be characterized as the minimizers of this energy functional. This was the starting point of a series of new results in frame theory, related to finding tight frames with determined lengths. The frame potential has been studied in the traditional setting as well as in the finite-dimensional fusion frame context. In this work we introduce the concept of mixed frame potential, which generalizes the notion of the Benedetto-Fickus frame potential. We study properties of this new potential, and give the structure of its critical pairs of sequences on a suitable restricted domain. For a given sequence {α _m } _{m=1,…, N} in K, where K is ? or ?, we obtain necessary and sufficient conditions in order to have a dual pair of frames {f _m } _{m=1,…, N} , {g _m } _{m=1,…, N} such that ? f _m , g _m  ? = α _m for all m = 1,…, N.     

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies the inequality , where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T ²×N, where T ² is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.     

A unique arithmetic labeling of hexagonal lattices

Consider an r-layer hexagon consisting of 3r² + 3r + 1 hexagonal cells. Can one label the cells by the set N_r = {0,1,…, 3r² + 3r} such that each line of adjacent cells are labeled by numbers forming an arithmetic progression modulo (3r² + 3r + 1) (in proper ordering)? We show that for each r there exists such a labeling unique up to equivalence. We also study some other related issues. © 1995 John Wiley & Sons, Inc.