The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a₁,a₂,...,a_n} and B = {b₁,b₂,...,b_m}, in which the numbers t_ai,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_ai,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_c,d,A, the number of vertices in A with degree between c and d of G_n,m∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_n,m∈ζ（n,m;{pk}） has maximum degree D（n）in A? under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A?     

NA序列的重对数矩收敛的精确渐近性

假设{X_n,n≥1}为一列严平稳的NA随机变量,期望为零,方差有限.设S_n=∑_(i=1)~n∑X_i,M_n=max_(1≤i≤n)|S_i|.在适当的条件下,得到了一类NA序列部分和部分和最大值重对数矩收敛的精确渐近性.     

三角代数上 Jordan 积为幂等元处的高阶ξ-Lie 可导映射

设U=Tri（A，M，B）是三角代数，{φ_n}_n∈N：U→U是一列线性映射.本文利用代数分解的方法，证明了如果对任意U，V∈U且U?V=P为标准幂等元，有φ_n（[U，V]_ξ=Σ_i+j=n（φ_i（U）φ_j（V）-ξφ_i（V）φ_j（U））（ξ≠1），则{φ_n}_n∈N是一个高阶导子，其中φ₀=id为恒等映射，U?V=UV+VU为Jordan积，[U，V]_ξ=UV-ξVU为ξ-Lie积.     

Maps preserving peripheral spectrum of generalized Jordan products of operators

Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let(i1, i2,..., im) be a finite sequence such that {i1, i2,..., im} = {1, 2,..., k} and assume that at least one of the terms in(i1,..., im) appears exactly once. Define the generalized Jordan product T1 o T2 o ··· o Tk= Ti1Ti2··· Tim+ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 + A2A1, and the Jordan triple A1 A2 A3 + A3 A2 A1. Let Φ : A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ(Φ(A1) o ··· o Φ(Ak)) = σπ(A1 o ··· o Ak) for all A1,..., Ak,where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.     

关于Neyman-Pearson基本引理的几个注记

本文探讨了Neyman-Pearson基本引理.通过论证总体参数θ只有θ₀或θ₁两种可能时最优检验功效函数的唯一性,得到了两种假设T₁:θ=θ₀←→θ=θ₁和T₂:θ=θ₁←→θ=θ₀各自对应最优检验的两类错误概率可以互换的结论.     

The fractional metric dimension of permutation graphs

Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 ε. We give examples showing that neither is there a function h1 such that dimf(G) h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.     

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

On Asymptotic Behavior of Increments of Sums Along Head Runs

Let be a sequence of independent equidistributed random vectors with . Let , where and denotes the indicator function of the event in brackets. If, for example, are the gains and are the indicators of success in repetitions of a game of chance, then is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values , , where is the length of the longest head run in . We show that the asymptotics of the values depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the ErdsRéenyi law of large numbers) to the strong invariance (as in the CsöorgRévész strong approximation laws). We also consider the Shepp-type statistics. Bibliography: 17 titles.     

Convergence Rates of Wavelet Estimators in Semiparametric Regression Models Under NA Samples

Consider the following heteroscedastic semiparametric regression model:yi =XTiβ + g(ti) + σiei, 1 ＜ i ≤ n,where {Xi,1 ＜ i ＜ n} are random design points,errors {ei,1 ＜ i ＜ n} are negatively associated (...     

Gowers norms and pseudorandom measures of subsets

Let $A \subset {{\Bbb Z}_N}$, and ${f_A}(s) = \left\{ {\begin{array}{*{20}{l}}{1 - \frac{{|A|}}{N},}&{{\rm{for}}\;s \in A,}\\{ - \frac{{|A|}}{N},}&{{\rm{for}}\;s \notin A.}\end{array}} \right.$ We define the pseudorandom measure of order k of the subset A as follows, P_k(A, N) = $\begin{array}{*{20}{c}}{\max }\\D\end{array}$|$\mathop \Sigma \limits_{n \in {\mathbb{Z}_N}}$f_A(n + c₁)f_A(n + c₂) … f_A(n + c_k)|, where the maximum is taken over all D = (c₁, c₂, . . . , c_k) ∈ ${\mathbb{Z}^k}$ with 0 ≤ c₁ < c₂ < … < c_k ≤ N - 1. The subset A ⊂ ${{\mathbb{Z}_N}}$ is considered as a pseudorandom subset of degree k if P_k(A, N) is “small” in terms of N. We establish a link between the Gowers norm and our pseudorandom measure, and show that “good” pseudorandom subsets must have “small” Gowers norm. We give an example to suggest that subsets with “small” Gowers norm may have large pseudorandom measure. Finally, we prove that the pseudorandom subset of degree L(k) contains an arithmetic progression of length k, where L(k) = 2·lcm(2, 4, . . . , 2|$\frac{k}{2}$|), for k ≥ 4, and lcm(a₁, a₂, . . . , a_l) denotes the least common multiple of a₁, a₂, . . . , a_l.     

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials P _n that are biorthogonal to exponentials , in the sense that

Uniform Mean Value Estimates and Discrete Hilbert Inequalities VIA Orthogonal Dirichlet Series

Let \({\{\lambda_j\}^\infty_{j=0}}\) be a strictly increasing sequence of positive numbers with λ₀ = 0 and λ₁ = 1. We use orthogonal Dirichlet polynomials associated with the arctangent density, to observe that for r > 0, $$\begin{array}{ll}\int^\infty_0\left |\sum\limits^\infty_{n=1}(-1)^{n-1}a_n\lambda^{-irt}_n\right |^2 \frac{dt}{\pi(1 + t^2)}\\ = \sum\limits^\infty_{n=1}(\lambda^{2r}_n - \lambda^{2r}_{n-1})\left |\sum\limits^\infty_{k=n}(-1)^{k-1}\frac{ak}{\lambda^r_k}\right |^2,\end{array}$$ when the right-hand side converges. As a consequence, we obtain uniform mean value estimates, discrete Hilbert type inequalities, and asymptotics as r → ∞ for classes of Dirichlet series.     

Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation

Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C＊-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces：f（2∑j=1^n-1 xj＋xn）＋f（2∑j=1^n-1 xj-xn）＋4∑j=1^n-1f（xj）=16f（∑j=1^n-1 xj）＋2∑j=1^n-1（f（xj＋xn）＋f（xj-xn）     

三维Boussinesq方程关于水平速度的一个爆破准则

本文主要讨论三维Boussinesq方程当扩散系数κ=0时光滑解的爆破准则.利用Littlewood-Paley分解和能量方法证明了如果方程关于水平速度场ū=（u₁,u₂,0）的水平导数满足▽_hū=（₁ū,₁ū,0）∈L¹（0,T;B0∞,∞（R³））.则解（u,θ）可以连续到T₁>T.     

变量核奇异积分和分数次微分加权范不等式

用T和D^γ（0 ≤ γ ≤ 1）分别表示变量核奇异积分和分数次微分算子.T^*和T^#分别为T的共轭算子及拟共轭算子.利用球调和多项式展式，本文得到了TD^γ-D^γT和（T^*-T^#）D^γ在?^q，λ_ω（Rⁿ）上的有界性.同时也得到了变量核奇异积分的积T₁T₂和拟积T₁°T₂的加权范不等式.     

Large deviations for random walks with nonidentically distributed jumps having infinite variance

Let = _i = 1^n _i , S_n = _k n S_k

XXZ Spin Chain with the Asymmetry Parameter Δ = −1/2: Evaluation of the Simplest Correlators

We consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. It appears that for the special value of the asymmetry parameter = –1/2, the ground state of this system described by the Hamiltonian has the energy E ₀ = –3N/2. Although the ground state is antiferromagnetic, we can find the corresponding solution of the Bethe equations. Specifically, we can explicitly construct a trigonometric polynomial Q(u) of degree n = (N–1)/2, whose zeros are the parameters of the Bethe wave function for the ground state of the system. As is known, this polynomial satisfies the Baxter T–Q equation. This equation also has a second independent solution corresponding to the same eigenvalue of the transfer matrix T. We use this solution to find the derivative of the ground-state energy of the XXZ chain with respect to the crossing parameter . This derivative is directly related to one of the spin–spin correlators, which appears to be . In turn, this correlator gives the average number of spin strings for the ground state of the chain, . All these simple formulas fail if the number N of chain sites is even.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Wave recursive interpolation

In this paper it is studied that the generated theory of wave recursive interpolation of uniform T-subdivi-ston scheme include wave parameter.The paper analyses the convergence of sequences of control polygons produced by wave recursive interpolation T-subdivision scheme of the formj=l,2,…,T-1;m=O,l,…,nTk;k=0,l,2,…,and differentiability of the limit curve.