First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

On cardinal sequences of scattered spaces

It was proved by Dow and Simon that there are 2^ω₁ (as many as possible) pairwise nonhomeomorphic compact, T₂, scattered spaces of height ω₁ and width ω. In this paper, we prove that if is an ordinal withω₁ < ω₂ and θ = κ_ξ: ξ < is a sequence of cardinals such that either κ_ξ = ω or κ_ξ = ω₁ for every ξ < , then there are 2^ω₁ pairwise nonhomeomorphic compact, T₂, scattered spaces whose cardinal sequence is θ.     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

素变量混合幂丢番图逼近

设λ_1,λ_2,λ_3,λ_4为不全为负的非零实数,λ_1/λ_2是无理数和代数数.■是具有良好间隔的序列,δ>0.本文证明了:对于任意ε>0及v∈■,v≤X,使得不等式|λ_1p_1~2+λ_2p_2~2+λ_3p_3~3+λ_4p_4~3-v|相似文献   

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ₁. We consider the power method, i.e. that of choosing a vector v₀ and setting v_k = A^kv₀; then the Rayleigh quotients R_k = (Av_k, v_k)/(v_k, v_k) usually converge to λ₁ as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close R_k is to λ₁. They are both based on a bound on λ₁ − R_k involving the difference of two consecutive Rayleigh quotients and a quantity ω_k. While we do not know how to directly calculate ω_k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ₁ − R_k which is proportional to (λ₂/λ₁)^2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ).     

随机Hamilton系统同胚流的大偏差原理

在非Lipschitz条件下得到随机微分方程同胚流的大偏差原理.作为应用,本文同时给出了随机Hamilton系统同胚流的大偏差原理.特别地,以下二阶非线性随机振荡方程同胚流的大偏差原理也同样成立:Z_t=C_0Z_t-Z_t~3+Θ(Z_t)W_t,(Z_0,Z_0)=(z,u)∈R~2,其中C_0为任意常数,Θ为一阶导数有界的二阶连续可微函数,W_t是一维Brown白噪声.     

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x₀,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence (i_j)_{0 j n} of ordinals such that i₀ = β, i_n = and x_ij+1 is adjacent with x_ij and with all neighbors of x_ij in the subgraph ofG induced by {x_y: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.     

混合幂次为2和3的整数变量非线性型的整数部分

证明了:假设λ_1,…,λ_6是正实数,λ_1/λ_2是无理数,Dirichlet L函数满足黎曼猜想,x_1,…,x_6是正整数,那么,λ_1x_1~2+λ_2x_2~2+λ_3x_3~3+λ_4x_4~3+λ_5x_2~3+λ_3x_3~3的整数部分可表示无穷多素数.     

Cycle decompositions of crowns

For an integer n3, the crown S_n⁰ is defined to be the graph with vertex set {x₀,x₁,…,x_n−1,y₀,y₁,…,y_n−1} and edge set {x_iy_j: 0i,jn−1, i≠j}. In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.     

Asymptotic behavior of a nonlinear delay difference equation

This paper considers a class of nonlinear difference equations

Δ₃y_n + ƒ(n, y_n, y_n−r) = 0, n N (n₀)

. A necessary and sufficient condition for the existence of a bounded nonoscillatory solution is given.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Inverse geophysical problems for some noncompactly supported inhomogeneities

Let[²+k²n(x₁,x₃)]u(x₁,x₂,x₃)=−δ(x₁,y₁δ(x₂,y₂)δ(x₃,y₃) in R³₊. Assume that u(x₁,x₂,x₃=0,y₁,y₂=0,y₃=0,k) is measured at the plane P {x:x₃=0} for all positions of the source on the line y = (y₁,y₂ = 0,y₃ = 0), -∞ < y₁ < ∞, and receiver on the plane(x₁,x₂,x₃ − <x₁,x₂ < ∞, and for low-frequencies 0 < k <k₀, k₀ > 0 is an arbitrary small wave number. Assume thatn(x₁,x₃) is an arbitrary bounded piecewise-continuous function. The basic result is: the above low-frequency surface data determinen(x₁,x₃)uniquely.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

Research Problem

We define an arithmetic invariant for the congruence subgroups Γ₀(N), denoted by

b(Γ₀(N)) and pose the problem of finding good asymptotic upper bonds for b(Γ₀(N)) as N approaches X especially when N is prime or the product of two (not necessarily distinct) primes.     

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes

in the space , where fL²([0,T]ⁿ) is a given function, and {η(t)}_>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t₁,…,t_n)=1_{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t₁,…,t_n)=f₁(t₁)f_n(t_n)1_{t1<t2<<tn}, with f_iL²([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.     

Distributional chaos for triangular maps

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2^∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smítal [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φ_xy and Φ_xy^* generated by F such that Φ_xy^*≡1 and Φ_xy(0₊)<1, and no distribution functions Φ_uv, and Φ_uv^* such that Φ_uv^*≡1 and Φ_uv(t)=0 whenever 0<t<ε, for some ε>0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.     

幂次为2,3,4,5的素变量非线性型的整数部分

考虑了一个混合幂次为2,3,4,5的素变量非线性型的整数部分表示无穷多素数的问题.运用Davenport-Heilbronn方法证明了:如果λ_1,λ_2,λ_3,λ_4是正实数,至少有一个λ_i/λ_j(1≤ij≤4)是无理数,那么存在无穷多素数p_1,p_2,p_3,p_4,p,使得[λ_1p_1~2+λ_2p_2~3+λ_3p_3~4+λ_4p_4~5]=p.