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.     

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.     

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.     

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.     

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.     

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.     

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…     

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.     

Global optimization method for linear multiplicative programming

In this paper, a new global algorithm is presented to globally solve the linear multiplicative programming(LMP). The problem(LMP) is firstly converted into an equivalent programming problem(LMP(H))by introducing p auxiliary variables. Then by exploiting structure of(LMP(H)), a linear relaxation programming(LP(H)) of(LMP(H)) is obtained with a problem(LMP) reduced to a sequence of linear programming problems. The algorithm is used to compute the lower bounds called the branch and bound search by solving linear relaxation programming problems(LP(H)). The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Some examples are given to illustrate the feasibility of the proposed algorithm.     

Non-stationary subdivision for exponential polynomials reproduction

In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.     

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.     

The complexity and randomness of linear multi-secret sharing schemes with non-threshold structures

In a linear multi-secret sharing scheme with non-threshold structures, several secret values are shared among n participants, and every secret value has a specified access structure. The efficiency of a multi- secret sharing scheme is measured by means of the complexity a and the randomness . Informally, the com- plexity a is the ratio between the maximum of information received by each participant and the minimum of information corresponding to every key. The randomness is the ratio between the amount of information distributed to the set of users U = {1, …, n} and the minimum of information corresponding to every key. In this paper, we discuss a and of any linear multi-secret sharing schemes realized by linear codes with non-threshold structures, and provide two algorithms to make a and to be the minimum, respectively. That is, they are optimal.     

Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.     

Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function φn, a cascade sequence (φn)n∞=1 constructed by the iteration φn=Cnφn-1=1,2.. where Cαis defined by g∈Lp(R) In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e., ||φ 1-φn||Lp(R)=O((？)n)for some (？)∈(0.1i). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.     

A Unified Ternary Curve Subdivision Scheme

A unified $m\,(m>2)$-point ternary scheme with some parameter is proposed. The continuity of subdivision scheme is analyzed based on the relationship between the subdivision scheme and difference scheme. Moreover, the proposed subdivision is extended to asymmetric multi-parameter subdivision and the asymmetric schemes in four cases are presented in detail. Some examples are given to show that the presented scheme has better approximating effect.     

The Construction of Association Schemes by Parameter Amendment

Association scheme is a structure on a finite set that has some special relations among elements in the set. These relations are usually hidden in other relations, so how to derive them out is a problem. The paper gives a constructing method of mending parameters of the association schemes, and a new family of association schemes is obtained： quasiFmetric association scheme.     

Statistical and Geometrical Way of Model Selection for a Family of Subdivision Schemes

The objective of this article is to introduce a generalized algorithm to produce the m-point n-ary approximating subdivision schemes(for any integer m, n ≥ 2). The proposed algorithm has been derived from uniform B-spline blending functions. In particular, we study statistical and geometrical/traditional methods for the model selection and assessment for selecting a subdivision curve from the proposed family of schemes to model noisy and noisy free data. Moreover, we also discuss the deviation of subdivision curves generated by proposed family of schemes from convex polygonal curve. Furthermore, visual performances of the schemes have been presented to compare numerically the Gibbs oscillations with the existing family of schemes.     

一类新的细分曲线方法

Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements based on initial control polygon or grid.Usually the curve refinements is the basis of the corresponding surface rules. In this paper we analyze previous subdivision scheme according to theories about convergence of N.Dyn and M.F Hassan. In terms of binary and ternary subdivision schemes general construction about curve‘s refinements are studied.Two approximating curve subdivision schemes with neighboring four control points are derived,the generating limit curves can both reach the smoothness of C^1 over the initial polygon using the two schemes and the tolerances of them are given according to the method of [7].     

RESEARCH ANNOUNCEMENTS Dagger Formal Geometry and de Rham Cohomology

Let K be a finite extension of Q_p with R its ring of integers and k=F_q its residue field.Let π be a uniformizer of R. At first, let us recall some concepts. A K-linear map L:M→M is called nuclear, if the following two conditions hold. (ⅰ) For every λ≠0 in K~(ac) the algebraic closure of K with g the minimal polynomial of λ over K, ∪(Ker(g(L)~n)) is of finite dimension. (ⅱ) The nonzero eigenvalues of L, form a finite set or a sequence with a limit 0. Let us define     

STRUCTURES OF VALUES FOR SET GAMES RESTRICTED BY PARTITION SYSTEM

§1IntroductionA cooperative game with transferable utility(TU)is a pair(N,v),where N is anonempty,finite set and v∶2N→R is a characteristic function defined on the power set ofN satisfying v()∶=0.LetCGdenote the set of all cooperative TU-games with anarbitrary player set.An element of N(notation:i∈N)and a nonempty subset S of N(notation:S N or S∈2Nwith S≠)are called a player and coalition respectively,andthe associated real number v(S)is called the worth of coalition S to be in…