Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

保Taylor联合谱的线性映射

设L(H),Lncom(H)分别是HilbertH上有界算子及n个两两交换的算子组的集合.设T∈Lncom(H),sp(T)表示Taylor联合谱,φi(i=1,2,…,n)是L(H)上满的线性映射且满足φi(Tl)φj(Tk)=φj(Tk)φi(Tl)当且仅当TlTk=TkTl,i,j=1,2,…,n.设T=(T1,T2,…,Tn)∈Lncom(H),φ=(φ1,φ2,…,φn),φ(T)=(φ1(T1),φ2(T2),…,φn(Tn)).文章证明了如果dimH<∞,对任意T=(T1,T2,…Tn)∈Lncom(H),sp(φ(T))=sp(T),则φi=φj,i,j=1,2,…,n.如果dimH=∞,T=(T1,T2,…Tn)∈Lncom(H),sp(φ(T))=sp(T),则φ是自同构或反自同构.     

一个变分不等式的奇异摄动

对变分不等式的奇异摄动问题进行了探索,证明了解的重合集Iε={x∈Ωuε（x)=φ}在Hausdorff距离意义下收敛到ε＝0时解的重合集。     

从Apα到A∞(ψ)的加权复合算子

Suppose u ∈ H(D) and φ is an analytic map of the unit disk D into itself. Definethe weighted composition operator uCφ : uCφ(f) = uf o φ, for all f ∈ H(D). In this paper,we get necessary and sufficient conditions for the bounded and compact weighted compositionoperators from the weighted Bergman spaces Apα to A∞(ψ) (A∞0(ψ)) spaces.     

带约束广义变分不等式问题的一般分解算法的收敛性分析

本文考虑如下带约束广义变分不等式问题的增广Lagrangian对偶理论:寻找一点x∈Γ使满足,〈F(x),y-x〉 φ(x,y)-φ(x,x)≥0,y∈Γ,其中,Γ={y∈X｜Θ(y)∈-C}.对于求解这类一般变分不等式问题的基于增广Lagrangian对偶理论分解算法,本文给出了算法的收敛性分析.     

定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.     

Nonaffine differential-algebraic curves do not exist

The paper outlines why the spectrum of maximal ideals Spec _? A of a countable-dimensional differential ?-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ?-homomorphism ψ _M: A → ? (M ∈ Spec _? A) and any a ∈ A the Taylor series \(\widetilde {{\psi _M}}{\left( a \right)^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} \frac{{{z^m}}}{{m!}}\) has nonzero radius of convergence depending on the element a ∈ A.     

Littlewood-Paley g-函数交换子的加权估计

设g_(φ,b)是Littlewood-Paley g-函数与b生成的交换子,ω∈A_1.证明了若b属于加权BMO空间BMO(ω),则g_(φ,b)是L~p(ω)到L~p(ω~(1-p))(1p∞)有界的;若b属于加权Lipschitz空间Lip_β(ω)(0β1),则g_(φ,b)是L~p(ω)到L~q(ω~(1-q))的有界算子,其中1pq∞,1/q=1/p-β/n.     

Global minimum and orthogonality in Cp ‐classes

In this paper we use recent results [14] to establish various characterizations of the global minimum of the map F_ψ : U → ?⁺ defined by F_ψ (X) = ‖ψ (X)‖_p (1 < p < ∞) where ψ: U → C_p is a map defined by ψ (X) = S +? (X), with ?: B (H) → B (H) a linear map and S ∈ C_p , and U = {X ∈ B (H): ? (X) ∈ C_p }. Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

与线性变换的完全环同构的环理论(Ⅰ)

<正> 熟知地,满足极小条件的单纯环只与一个有限维向量空间的线性变换的完全环同构.并且此向量空间如取为左向量空间的话,那末R的任一极小右理想均可取为此左向量空间.在没有有限条件情况下,Jacobsoo用本原环来取代这种单纯环.接着Wolfson研     

Cauchy Problem for Semilinear Wave Equations in Four Space Dimensions with Small Initial Data

In this paper, we consider the Cauchy problem ◻u(t,x) = |u(t,x)|^p, (t,x) ∈ R^+ × R^4 t = 0 : u = φ(x), u_t = ψ(x), x ∈ R^4 where ◻ = ∂²_t - Σ^4_{i=1}∂²_x_i, is the wave operator, φ, ψ ∈ C^∞_0 (R^4). We prove that for p > 2 the problem has a global solution provided tile initial data is sufficiently small.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

Hermite Expansion of the Riemann Zeta Function

Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction, it is crucial to regard the zeta function as Fourier transfomations of Schwartz' distributions.     

广义Mellin变换Ⅰ

<正> 推广至广义函数.古典 Mellin 变换作用于半直线(0,∞).因此我们将以半直线上的广义函数类(定义1)为 Mellin 变换的定义域.古典理论中 Mellin 像函数一般为解析函数,因此将以某种“解析”的广义函数类(定义2)为像域.作指数变换后,Mellin 变     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MATHEMATICAL ANALYSIS OF THE COLLAPSE IN BOSE-EINSTEIN CONDENSATE

In this article, the authors consider the collapse solutions of Cauchy problem for the nonlinear Schrdinger equation iψt + 1/2 △ψ - 1/2 ω2|x|2ψ + |ψ|2ψ = 0, x ∈ R2, which models the Bose-Einstein condensate with attractive interactions. The authors establish the lower bound of collapse rate as t → T . Furthermore, the L2-concentration property of the radially symmetric collapse solutions is obtained.     

一类线性流形上矩阵方程X^TAX=B的反问题

设Ω={A∈ASR^nxn｜Ax=C，↓Ax∈RT（S），SS^＋C=0，T2^TC2=-C2^TT2，C2T2^＋72=C2}，考虑问题Ⅰ：给定X∈R^nxm，B∈R^mxm求A∈Ω，使得f（A）=｜｜X^TAX—B｜｜=min；问题Ⅱ：给定A^＋∈R^nxm，求A∈SE，使得｜｜A^＋-A｜｜=minA∈SE｜｜A^＋-A｜｜，SE是问题Ⅰ的解集。本文给出了问题Ⅰ、Ⅱ的解的通式，并给出了问题Ikf（A）=0成立的充分必要条件。     

A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space $$H^n: sup_{u∈W^{1,n}(H^n),∫_{H^n}|∇_H^nu|^ndμ ≤ 1}∫_{H^n}\frac{e^{α|u|\frac{n}{n-1}}-Σ^{n-2}_{k=0}\frac{α^k|u|^\frac{nk}{n-1}}{k!}}{ρ^β}dμ‹∞ ⇔ \frac{α}{α_n}+\frac{β}{n} ≤ 1,$$ where α>0,α ∈ [0,n), ρ and dμ are the distance function and volume element of $H^n$ respectively.