The bifurcation of limit cycles in Zn-equivariant vector fields

In this paper, we study the bifurcation of limit cycles from fine focus in Z_n-equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z₈-equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z₈-equivariant systems, our results are good and interesting.     

Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Z_n) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population W_n=Z_n/E[Z_n|ξ] to its limit W_∞: we show a central limit theorem for W_∞−W_n with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for W_n+k−W_n for each fixed k∈N^∗.     

The oscillating random walk

{Y_n;n=0, 1, …} denotes a stationary Markov chain taking values in R^d. As long as the process stays on the same side of a fixed hyperplane E⁰, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′_n = |Y′_n?1?Z_n| (with the Z_n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y_n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.     

Homogeneous random fields and statistical mechanics

We illustrate the connection between homogeneous perturbations of homogeneous Gaussian random fields over Rⁿ or Zⁿ, with values in R^m, and classical as well as quantum statistical mechanics. In particular we construct homogeneous non-Gaussian random fields as weak limits of perturbed Gaussian random fields and study the infinite volume limit of correlation functions for a classical continuous gas of particles with inner degrees of freedom. We also exhibit the relation between quantum statistical mechanics of lattice systems (anharmonic crystals) at temperature β^?1 and homogeneous random fields over Zⁿ × S_β, where S_β is the circle of length β, which then provides a connection also with classical statistical mechanics. We obtain the infinite volume limit of real and imaginary times Green's functions and establish its properties. We also give similar results for the Gibbs state of the correspondent classical lattice systems and show that it is the limit as h → 0 of the quantum statistical Gibbs state.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

On the range of a regenerative sequence

For a given countable partition of the range of a regenerative sequence {X_n: n ? 0}, let R_n be the number of distinct sets in the partition visited by X up to time n. We study convergence issues associated with the range sequence {R_n: n ? 0}. As an application, we generalize a theorem of Chosid and Isaac to Harris recurrent Markov chains.     

The regular graph of a commutative ring

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2 ⁿ , where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.     

On endomorphism-regularity of zero-divisor graphs

The paper studies the following question: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ(R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z₂×Z₂×Z₂; Z₂×Z₄; Z₂×(Z₂[x]/(x²)); F₁×F₂, where F₁,F₂ are fields. In addition, we determine all positive integers n for which Γ(Z_n) has the property.     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

Some properties of random walk paths

For random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the random walk in question. Some special cases are considered, among them the following: for d = 2, what sequences (n_j) have the property that with probability one the random walk visits the origin for infinitely many n_j. A related problem, which is however not a special case of the recurrence problem, is to decide for what sequences (n_j) the states visited by the random walk at times n_j are all distinct, with only a finite number of exceptions. This problem is dealt with in the final part of the paper.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

On the sequence of partial maxima of some random sequences

Let {X_n, n ? 1} be a sequence of identically distributed random variables, Z_n = max {X₁,…, X_n} and {u_n, n ? 1 } an increasing sequence of real numbers. Under certain additional requirements, necessary and sufficient conditions are given to have, with probability one, an infinite number of crossings of {Z_n} with respect to {u_n}, in two cases: (1) The X_n's are independent, (2) {X_n} is stationary Gaussian and satisfies a mixing condition.     

Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈M_n(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice M_n(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=M_n(R), in which case a presentation for M_n(R) in terms of C and D is given.     

Reverse processes and some limit theorems of multitype Galton-Watson processes

We classify the reverse process {X_n} of a multitype Galton-Watson process {Z_n}. In the positive recurrent cases we give the stationary measure for {X_n} explicitly, and in the critical case, supposing that all the second moments of Z₁ are finite, we establish the convergence in law to a gamma distribution. Limit distributions of {Z_cn}, 0 < c < 1, conditioned on Z_n, are also given in the subcritical, supercritical and critical cases, respectively. These extend the previous one-type work of W. W. Esty.     

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let (Z_n) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C_n=E^ω[Z_n∣Z_n>0] be the conditional expectation of Z_n given Z_n>0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z_n/C_n, conditional on Z_n>0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Hitting Time Distributions for Denumerable Birth and Death Processes

For an ergodic continuous-time birth and death process on the nonnegative integers, a well-known theorem states that the hitting time T _0,n starting from state 0 to state n has the same distribution as the sum of n independent exponential random variables. Firstly, we generalize this theorem to an absorbing birth and death process (say, with state ?1 absorbing) to derive the distribution of T _0,n . We then give explicit formulas for Laplace transforms of hitting times between any two states for an ergodic or absorbing birth and death process. Secondly, these results are all extended to birth and death processes on the nonnegative integers with ?? an exit, entrance, or regular boundary. Finally, we apply these formulas to fastest strong stationary times for strongly ergodic birth and death processes.     

Mesh geometries of root orbits of integral quadratic forms

Integral quadratic forms q:Zⁿ→Z, with n≥1, and the sets R_q(d)={v∈Zⁿ;q(v)=d}, with d∈Z, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications b_A:Zⁿ×Zⁿ→Z of q, with Z-regular matrices A∈M_n(Z). Mesh geometries of roots of positive definite quadratic forms q:Zⁿ→Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams A_n,D_n,E₆,E₇,E₈ and the Auslander-Reiten quivers of the derived category D^b(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification b_A of a quadratic form q, a weighted Φ_A-mesh of vectors in Zⁿ, and a weighted Φ_A-mesh orbit translation quiver Γ(R_q,Φ_A) of vectors in Zⁿ, where R_q?R_q(1) and Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. The existence of mesh geometries on R_q is discussed. It is shown that, under some assumptions on the morsification b_A:Zⁿ×Zⁿ→Z, the set admit a Φ_A-orbit mesh quiver , where Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver Z⋅Δ, with Δ an extended Dynkin quiver, and has the shape of a sand-glass tube.     

A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Z_n,α=Y₁+2^αY₂+?+n^αY_n, with α∈R and Y₁,Y₂,… independent and exponentially distributed random variables with mean one. The distribution function of Z_n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z_n,α<x) that remains valid inside (α≥−1/2) and outside (α<−1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α=1, for which we sharpen some of the results in Kingman and Volkov (2003).     

The rate of escape for anisotropic random walks in a tree

Summary Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z _n on G or T which applies to the distance of Z _n from a fixed point, as well as to the distribution of the last R letters in Z _n . For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z _n with a finitely supported step distribution is derived and discussed.Partially supported by grant NSF MCS85-04315     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.