Vector subdivision schemes in (L_p(R~5))~r(1≤p≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the formwhere the vector of functions = (1, … ,r)T is in (LP(R8))T,1 ≤ p ≤∞, α(α),α ∈ Z5, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s x a integer matrix such that limn→ ∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions (0 ∈ (LP(R8))r and use the iteration schemes fn := Qan0,n = 1,2,…, where Qa is the linear operator defined on (Lp(R8))r given byThis 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 some linear operators determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group Z8/MZ8 containing 0.     

Lp-Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.     

Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))α∈Zs is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that limn→∞M-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φj,j=1, . . . ,r, belonging to L2(Rs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ0.     

Convergence of subdivision schemes in L p spaces

Abstract. In this paper it is proved that Lp solutions of a refinement equation exist if and only ifthe corresponding subdivision scheme with suitable initial function converges in Lp without anyassumption on the stability of the solutions of the refinement equation. A characterization forconvergence of subdivision scheme is also given in terms of the refinement mask. Thus a com-plete answer to the relation between the existence of Lp solutions of the refinement equation andthe convergence of the corresponding subdivision schemes is given.     

Biorthogonal multiple wavelets generated by vector refinement equation

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis.In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form (?)(x)=(?)a(α)(?)(Mx-α),x∈R~s, where the vector of functions(?)=((?)_1,...,(?)_r)~T is in(L_2(R~s))~r,a=:(a(α))_(α∈Z~s)is a finitely supported sequence of r×r matrices called the refinement mask,and M is an s×s integer matrix such that lim_(n→∞)M~(-n)=0.Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.     

The independence of initial vectors in the subdivision schemes

Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ.     

Solutions of multiple vector refinement equations with infinite mask

This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.     

Multivariate refinement equation with nonnegative masks

This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.     

Maximal integral over observable measures

Let f be a continuous transformation on a compact,finite-dimensional manifold M,andψa continuous function on M.This paper establishes the following formula:ess sup lim sup n→∞ 1/n ψ_n(x)=sup{∫ψdμ︱μ∈O_f}≤lim sup n→∞ 1/n ess sup ψ_n(x),where ess sup denotes the essential supremum taken against the Lebesgue measure,ψ_n(x)=n-1 ∑ i=0 ψ(f~ix) and O_f is the set of observable measures.Examples are provided to illustrate that the inequality could be an equality or strict.Moreover,if μ is the unique maximizing observable measure for ψ,it is weakly statistical stable.     

Lp空间中重分算法的收敛性

§ 1　IntroductionThe nontrivial solution of the following refinement equation = αa(α)( 2 .-α) ( 1 )is called refinable.The sequence a is called the refinementmask.When the mask a is finite-ly supported on Zand αa(α) =2 ,itis well known thatthe refinementequation( 1 ) hasa unique compactly supported distribution solution satisfies^( 0 ) =1 ,where^denotes theFourier transform of.This solution is called the normalized solution of( 1 ) .In order to study the solution of the equ…     

Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

Let G  C be a simply connected domain whose boundary L := αG is a Jordan curve and 0 ∈ G.Let w = ψ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| r0}, satisfying ψ(0) = 0,ψ′(0) = 1. We consider the following extremal problem for p 0:∫∫G|ψ′(z)- P ′n(z) pdσz→ min in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, P ′n(0) = 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair(G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the ψ(z) on G with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their "touch".     

SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD ∑r OF B(E,F) AND A DIMENSION FORMULA IN B(IRn,IRm)

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear oper-ators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑#r the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submani-fold in B(E, F) with given expression of its tangent space at each A ∈∑r. In this paper, the equality ∑#r = 1 is proved. Consequently, the following theorem is obtained: for any non-negative integer r, ∑r is a smooth and path connected Banach submanifold in B(E, F) with the tangent space TA∑r={B∈B(E,F): BN(A) R(A)} at each A∈∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = IRn and F = IRm, then ∑r is a smooth and path connected submanifold of B(IRn,IRm) and its dimension is dim ∑r = (m + n)r- r2 for each r, 0 ≤ r < min{n,m}.     

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations ∑in, j=1 aijuxixj = f,if f belongs to the generalized Morrey spaces Lp,ψ(ω), then uxixj ∈ Lp,ψ(ω), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp,ψ (ω).     

Bers型空间和复合算子

For α∈（0,∞),let Hα^∞（or Hα^∞,0)denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|^2)^α=O(1)(or|f(z)|(1-|z|^2)^α=o(1) as |z|→1).Hα^∞,0)is called a Bers-type space (or a little Bers-type space).In this paper,we give some basic properties of Hα^∞，Cψ，the composition operator associated with a symbol function ψ which is an analytic self map of D,is difined by Cψf=f o ψ,We characterize the boundedness,and compactness of Cψ which sends one Bers-type space to another function space.     

Shrinking target problems for beta-dynamical system

For any β>1,let([0,1],Tβ) be the beta dynamical system.For a positive function ψ:N→R+ and a real number x0 ∈[0,1],we define D(Tβ,ψ,x0) the set of ψ-well approximable points by x0as {x∈[0,1]:|Tβnx-x0|<ψ(n) for infinitely many n∈N}.In this note,by proving a structure lemma that any ball B(x,r) contains a regular cylinder of comparable length with r,we determine the Hausdorff dimension of the set D(Tβ,ψ,x0) completely for any β>1 and any positive function ψ.     

具有多项式衰减面具的向量细分格式

具有多项式衰减面具的向量细分方程在刻画小波Riesz基和双正交小波等方面有着重要作用.本文主要研究这类方程解的性质.向量的细分方程具有形式：Ф=∑α∈Zsa（α）（2·-α）,其中Ф=（Ф1,...,Фr）T是定义在Rs上的向量函数,a：=（a（α））α∈Zs是一个具有多项式衰减的r×r矩阵序列称为面具.关于面具a定义一个作用在（Lp（Rs））r上的线性算子Qa,Qaf：=∑α∈Zsa（α）f（2·α）.迭代格式（Qanf）n=1,2,...称为向量细分格式或向量细分算法.本文证明如果具有多项式衰减面具的向量细分格式在（L2（Rs））r中收敛,那么其收敛的极限函数将自动具有多项式衰减.另外,给出了当迭代的初始函数满足一定的条件时的向量细分格式的收敛阶.     

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.     

Positive Solutions for Singular Quasilinear Schrodinger Equations with One Parameter,II

We establish the existence of positive bound state solutions for the singular quasilinear Schrodinger equation iψ/t=-div（ρ（｜ψ｜^2） ψ）＋ω（｜ψ｜62）ψ-λρ（｜ψ｜^2）ψ,x∈Ω,t〉0,where ω（τ^2）τ→∞ as τ→ 0 and, λ 〉 0is a parameter and Ω is a ball in ^RN. This problem is studied in connection with the following quasilinear eigenvalue problem with Dirichlet boundary condition -div（ρ（｜ ψ｜^2） ψ）=λ1ρ（｜ψ｜^2）ψ=λ1ρ（｜ψ｜^2）ψ,x∈Ω.Indeed, we establish the existence of solutions for the above Schrodinger equation when A belongs to a certain neighborhood of the first eigenvahie λ1 of this eigenvalue problem. The main feature of this paper is that the nonlinearity ω｜ψ｜^2 is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter A combined with the nonlinear nonhomogeneous term div （ρ（｜ ψ｜^2） ψ） which leads us to treat this prob- lem in an appropriate Orlicz space. The proofs are based on various techniques related to variational methods and implicit function theorem.     

2-(v,k,1)设计和PSL(3,q)(q是奇数)

§ 1　 IntroductionA2 -(v,k,1 ) design D=(S,B) consists ofa finite set Sof v points and a collection Bof some subsets of S,called blocks,such that any two points lie on exactly one blockand each block contains exactly k points.A flag of Dis a pair(α,B) such thatα∈S,B∈Bandα∈B,the set of all flags is denoted by F.We assume that2≤k≤v.An automorphism of Dis a permutation of the points which leaves the set Binvari-ant,all the automorphisms form a group Aut D.Let G be a subgroup of A…