15 August 2006, Volume 22 Issue 4

    

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  Interval Valued Intuitionistic (ST)-fuzzy H_v

  Jian Ming ZHAN, W. A. DUDEK

  Acta Mathematica Sinica. 2006, 22(4): 963-970. https://doi.org/10.1007/s10114-005-0699-x

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  On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K.Atanassov,the notion of interval valued intuitionistic fuzzy H_v -submodules of an H_v -module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described.The homomorphic image and the inverse image are investigated.In particular,the connections between interval valued intuitionistic (S,T)-fuzzy H_v -submodules and interval valued intuitionistic (S,T)-fuzzy submodules are discussed.
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  Volume-preserving Fields and Reeb Fields on 3-manifolds

  Hong Jun LI

  Acta Mathematica Sinica. 2006, 22(4): 971-988. https://doi.org/10.1007/s10114-005-0543-3

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  Volume-preserving field X on a 3-manifold is the one that satisfies L_X Ω ≡ 0 for some volume Ω.The Reeb vector field of a contact form is of volume-preserving,but not conversely.On the basis of Geiges-Gonzalo’s parallelization results,we obtain a volume-preserving sphere,which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields.From many aspects,we discuss the distinction between volume-preserving fields and Reeb-like fields.We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction.We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.
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  Triangular Matrix Representations of Rings of Generalized Power Series

  Zhong Kui LIU

  Acta Mathematica Sinica. 2006, 22(4): 989-998. https://doi.org/10.1007/s10114-005-0555-z

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  Let R be a ring and S a cancellative and torsion-free monoid and≤a strict order on S.If either (S,≤) satis.es the condition that 0≤s for all s∈S,or R is reduced,then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R.Furthermore,if R is a PWP ring,then so is [[R^S ,^≤]].
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  Two Kinds of Numbers and Their Applications

  Zhi Zheng ZHANG, Hong FENG

  Acta Mathematica Sinica. 2006, 22(4): 999-1006. https://doi.org/10.1007/s10114-005-0749-4

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  C.Radoux (J.Comput.Appl.Math. 115 (2000) 471-477) obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials,Bell polynomials,Hermite polynomials,Derangement polynomials etc.From a pair of matrices this paper introduces two kinds of numbers.Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences,i.e.,to consider a general representation of Hankel matrices on the first kind of numbers.It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.
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  On Nonlinear Rayleigh-Taylor Instabilities

  B. DESJARDINS, E. GRENIER

  Acta Mathematica Sinica. 2006, 22(4): 1007-1016. https://doi.org/10.1007/s10114-005-0559-8

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  Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field g are investigated.The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system.The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.
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  Multiplicative Mappings of Rings

  Fang Yan LU, Jin Hai XIE

  Acta Mathematica Sinica. 2006, 22(4): 1017-1020. https://doi.org/10.1007/s10114-005-0620-7

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  Let R and S be arbitrary associative rings.A mapping φ of R onto S is called a multiplicative isomorphism if φ is bijective and satisfies φ(xy)=φ(x)φ(y) for all x,y∈R.In this short note,we establish a condition on R,in the case where R may not contain any non-zero idempotents,that assures that φ is additive,which generalizes the famous Martindale's result.As an application,we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.
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  Time-like Linear Weingarten Surfaces in Lorentzian Space Forms

  Da Feng ZUO

  Acta Mathematica Sinica. 2006, 22(4): 1021-1026. https://doi.org/10.1007/s10114-005-0657-7

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  In this paper we study Bäcklund transformations of time-like linear Weingarten surfaces with negative discriminant in Lorentzian space forms.
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  Weak Tensor Category and Related Generalized Hopf Algebras

  Fang LI, Gong Xiang LIU

  Acta Mathematica Sinica. 2006, 22(4): 1027-1046. https://doi.org/10.1007/s10114-005-0659-5

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  There are at least two kinds of generalization of Hopf algebra,i.e.pre-Hopf algebra and weak Hopf algebra.Correspondingly,we have two kinds of generalized bialgebras,almost bialgebra and weak bialgebra.Let C=(C,,I,a,l,r) be a tensor category.By giving up I,l,r and keeping ,a in ${\fancyscript C}$,the first author got so-called pre-tensor category C=(C,,a) and used it to characterize almost bialgebra and pre-Hopf algebra in Comm.in Algebra,32(2):397-441 (2004).Our aim in this paper is to generalize tensor category C=(C,,I,a,l,r) by weakening the natural isomorphisms l,r,i.e.exchanging the natural isomorphism ll^-1=rr^-1=id into regular natural transformations llll=l,rrrr=r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra.The relations between these generalized (bialgebras) Hopf algebras and two kinds generalized tensor categories will be described by using of diagrAms.Moreover,some related concepts and properties about weak tensor category will be discussed.
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  On Polynomial Functions over Finite Commutative Rings

  Jian Jun JIANG,Guo Hua PENG, Qi SUN, Qi Fan ZHANG

  Acta Mathematica Sinica. 2006, 22(4): 1047-1050. https://doi.org/10.1007/s10114-005-0655-9

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  Let R be an arbitrary finite commutative local ring.In this paper,we obtain a necessary and sufficient condition for a function over R to be a polynomial function.Before this paper,necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.
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  A Class of Bidimensional FMRA Wavelet Frames

  Yun Zhang LI

  Acta Mathematica Sinica. 2006, 22(4): 1051-1062. https://doi.org/10.1007/s10114-005-0653-y

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  This paper addresses the construction of wavelet frame from a frame multiresolution analysis (FMRA) associated with a dilation matrix of determinant ±2.The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity.In this paper,for four classes of them,the construction of wavelet frame from an FMRA is obtained,and,as examples,Shannon type wavelet frames are constructed,which have an independent value for their optimality in some sense.
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  A Scoring Criterion For Learning Chain Graphs

  Zhong Guo ZHENG, Jing XU, Xing Wei TONG

  Acta Mathematica Sinica. 2006, 22(4): 1063-1068. https://doi.org/10.1007/s10114-005-0651-0

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  A chain graph allows both directed and undirected edges,and contains the underlying mathematical properties of the two.An important method of learning graphical models is to use scoring criteria to measure how well the graph structures fit the data.In this paper,we present a scoring criterion for learning chain graphs based on the Kullback-Leibler distance.It is score equivalent,that is,equivalent chain graphs obtain the same score,so it can be used to perform model selection and model averaging.
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  Optimized Local Trigonometric Bases with Nonuniform Partitions

  Qiao Fang LIAN, Yong Ge WANG, Dun Yan YAN

  Acta Mathematica Sinica. 2006, 22(4): 1069-1084. https://doi.org/10.1007/s10114-005-0647-9

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  The authors provide optimized local trigonometric bases with nonuniform partitions which efficiently compress trigonometric functions.Numerical examples demonstrate that in many cases the proposed bases provide better compression than the optimized bases with uniform partitions obtained by Matviyenko.
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  Global Existence of Small Solutions for Cubic Quasi-linear Klein-Gordon Systems in One Space Dimension

  Dao Yuan FANG, Ru Ying XUE

  Acta Mathematica Sinica. 2006, 22(4): 1085-1102. https://doi.org/10.1007/s10114-005-0668-4

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  In this paper,we consider a system of two cubic quasi-linear Klein-Gordon equations with different masses for small,smooth,compactly supported Cauchy data in one space dimension.We show that such a system has global existence when the nonlinearities satisfy a convenient null condition.Our results extend the global existence proved by Sunagawa recently under the non-resonance assumption to that under the resonance assumption.
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  Quasi-sure Product Variation of Two-parameter Smooth Martingales on the Wiener Space

  Ji Cheng LIU(Jicheng LIU), Jia Gang REN(Jiagang REN)

  Acta Mathematica Sinica. 2006, 22(4): 1103-1114. https://doi.org/10.1007/s10114-005-0661-y

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  In this paper,we prove that the process of product variation of a two-parameter smooth martingale admits an ∞ modification,which can be constructed as the quasi-sure limit of sum of the corresponding product variation.
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  On 2-Factors with Prescribed Properties in a Bipartite Graph

  Jin YAN, Gui Zhen LIU

  Acta Mathematica Sinica. 2006, 22(4): 1115-1120. https://doi.org/10.1007/s10114-005-0666-6

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  Liu and Yan gave the degree condition for a balanced bipartite graph G=(V₁,V₂;E) to have k vertex-disjoint quadrilaterals containing any given k independent edges e₁,...,e_k of G,respectively.They also conjectured that for any k independent edges e₁,...,e_k of G,G has a 2-factor with k cycles C₁,C₂,...,C_k with respect to {e₁,e₂,...,e_k } such that k-1 of them are quadrilaterals.In this paper,we prove this conjecture.

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  Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

  Ai Xia QIAN, Shu Jie LI

  Acta Mathematica Sinica. 2006, 22(4): 1121-1126. https://doi.org/10.1007/s10114-005-0665-7

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  Under simple conditions,we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem.For the resonance case at infinity,we do not need to assume any more conditions to ensure the boundedness of the (PS) sequence of the corresponding functional.
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  Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

  Ya Ping FANG, Nan Jing HUANG

  Acta Mathematica Sinica. 2006, 22(4): 1127-1132. https://doi.org/10.1007/s10114-005-0664-8

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  Let X,Y be two finite-dimensional topological vector spaces,Z a Hausdorff topological vector space,K⊂X and D⊂Z be two nonempty sets,C be a pointed,closed,and convex cone in Y with int C≠ø.Let S:K→2^K and T:K→2^D be two multivalued mappings,and φ:K×D×K→Y be a trifunction.In this paper,we consider the generalized vector quasi-equilibrium problem,which is formulated by finding ∈K and ?∈T( ) such that ∈S( ) and φ( ,?,u)-int C for all u∈S().We establish an existence result in which T is not supposed to have any continuity property.Our results extend and improve the corresponding results of Cubiotti,Yao and Guo.
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  The Threshold for the Erd?s, Jacobson and Lehel Conjecture to Be True

  Jiong Sheng LI, Jian Hua YIN

  Acta Mathematica Sinica. 2006, 22(4): 1133-1138. https://doi.org/10.1007/s10114-005-0676-4

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  Let σ(k,n) be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ(k,n) can be realized by a graph containing a clique of k+1 vertices.Erd?s et al.(Graph Theory,1991,439-449) conjectured that σ(k,n)=(k-1)(2n-k)+2.Li et al.(Science in China,1998,510-520) proved that the conjecture is true for k≥5 and n ≥ (?k/2) + 3,and raised the problem of determining the smallest integer N(k) such that the conjecture holds for n≥N(k).They also determined the values of N(k) for 2≤k≤7,and proved that for k≥8.In this paper,we determine the exact values of σ(k,n) for n≥2k+3 and k≥6.Therefore,the problem of determining σ(k,n) is completely solved.In addition,we prove as a corollary that for k≥6.
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  Regularity of Solutions of the Cahn-Hilliard Equation with Non-constant Mobility

  Chang Chun LIU, Yuan Wei QI, Jing Xue YIN

  Acta Mathematica Sinica. 2006, 22(4): 1139-1150. https://doi.org/10.1007/s10114-005-0711-5

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  In this paper,we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility.Basing on the L^p type estimates and Schauder type estimates,we prove the global existence of classical solutions.
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  Applications of k-covers

  G. DI MAIO, Ljubiša D. R. KOČINAC, Enrico MECCARIELLO

  Acta Mathematica Sinica. 2006, 22(4): 1151-1160. https://doi.org/10.1007/s10114-005-0753-8

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  We investigate a role of k-covers in selection principles theory.Some results about relationships between RAmsey theory and game theory and selection principles that involve k-covers are also given.
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  Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

  Marina GHISI, Massimo GOBBINO

  Acta Mathematica Sinica. 2006, 22(4): 1161-1170. https://doi.org/10.1007/s10114-005-0649-7

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  We consider the Cauchy problem εu_ε"+δu_ε'+Au_ε=0,u_ε(0)=u₀=u₁,where ε 0,δ0,H is a Hilbert space,and A is a self-adjoint linear non-negative operator on H with dense domain D(A).We study the convergence of {u_ε} to the solution of the limit problem δu_ε'+Au_ε=0,u(0)=u₀.For initial data (u₀)∈D(A^1/2)×Hwe prove global-in-time convergence with respect to strong topologies.Moreover,we estimate the convergence rate in the case where (u₀)∈D(A^3/2)×∈D(A^1/2),and we show that this regularity requirement is sharp for our estimates.We give also an upper bound for |u_ε'(t)|which does not depend on ε.
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  A Note on Weakly Isotone Maps and Common Solutions for Differential Systems

  Giuseppe MARINO, Vittorio COLAO, Luigi MUGLIA

  Acta Mathematica Sinica. 2006, 22(4): 1171-1174. https://doi.org/10.1007/s10114-005-0643-0

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  In this short note we examine the connection between weakly isotone maps and common solutions for first order Cauchy problems in Rⁿ and,as a rule,in Banach lattices.
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  Hilbert Space of Probability Density Functions Based on Aitchison Geometry

  J. J. EGOZCUE, J. L. DÍAZ-BARRERO, V. PAWLOWSKY-GLAHN

  Acta Mathematica Sinica. 2006, 22(4): 1175-1182. https://doi.org/10.1007/s10114-005-0678-2

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  The set of probability functions is a convex subset of L¹ and it does not have a linear space structure when using ordinary sum and multiplication by real constants.Moreover,difficulties arise when dealing with distances between densities.The crucial point is that usual distances are not invariant under relevant transformations of densities.To overcome these limitations,Aitchison’s ideas on compositional data analysis are used,generalizing perturbation and power transformation,as well as the Aitchison inner product,to operations on probability density functions with support on a finite interval.With these operations at hand,it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space.A Hilbert space of densities,whose logarithm is square-integrable,is obtained as the natural completion of the pre-Hilbert space.
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  Fractional Brownian Motion and Sheet as White Noise Functionals

  Zhi Yuan HUANG, Chu Jin LI, Jian Ping WAN, Ying WU

  Acta Mathematica Sinica. 2006, 22(4): 1183-1188. https://doi.org/10.1007/s10114-005-0734-y

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  In this short note,we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises,and the fractional white noise calculus developed by Hu and Øksendal follows directly from the classical white noise functional calculus.As examples we prove that the fractional Girsanov formula,the Itôtype integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts.An extension to the fractional Brownian sheet is also briefly discussed.
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  Strongly Irreducible Submodules of Modules

  A. KHAKSARI, M. ERSHAD H. SHARIF

  Acta Mathematica Sinica. 2006, 22(4): 1189-1196. https://doi.org/10.1007/s10114-005-0681-7

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  Strongly irreducible submodules of modules are defined as follows:A submodule N of an R-module M is said to be strongly irreducible if for submodules L and K of M,the inclusion L∩K N implies that either L N or K N.The relationship among the families of irreducible,strongly irreducible,prime and primary submodules of an R-module M is considered,and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.
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  Lagrangian Submanifolds Foliated by (n-1)-spheres in R²ⁿ

  Henri ANCIAUX, Ildefonso CASTRO

  Acta Mathematica Sinica. 2006, 22(4): 1197-1124. https://doi.org/10.1007/s10114-005-0690-6

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  We study Lagrangian submanifolds foliated by (n-1)-spheres in R²ⁿ for n≥3.We give a general parametrization for such submanifolds,and refine that description when the submanifold is special Lagrangian,self-similar,Hamiltonian stationary or has mean curvature vector of constant length.In all these cases,the submanifold is centered,i.e.invariant under the action of SO(n).It suffices then to solve a simple ODE in two variables to describe the geometry of the solutions.
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  Equivariant Lagrangian Minimal S³ in CP³

  Zhen Qi LI, Yong Qian TAO

  Acta Mathematica Sinica. 2006, 22(4): 1215-1220. https://doi.org/10.1007/s10114-005-0796-x

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  The equivariant Lagrangian minimal immersion of S³ into CP³ is studied.The complete classification and analytic expression for such kinds of immersions are given.
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  The Compact Quantum Group U_q(2) (Ⅱ)

  Xiao Xia ZHANG

  Acta Mathematica Sinica. 2006, 22(4): 1221-1226. https://doi.org/10.1007/s10114-005-0785-0

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  In this paper,we first prove that the θ-deformation U_θ (2) of U (2) constructed by Connes and Violette is our special case of the quantum group U_q (2) constructed in our previous paper.Then we will show that the set of traces on the C^*-algebra U_θ,θ irrational,is determined by the set of the traces on a subalgebra of U_θ.
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  Towards an L^p Potential Theory for Sub-Markovian Semigroups: Kernels and Capacities

  Niels JACOB, RenéL. SCHILLING

  Acta Mathematica Sinica. 2006, 22(4): 1227-1250. https://doi.org/10.1007/s10114-005-0758-3

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  We study in fairly general measure spaces (X,μ) the (non-linear) potential theory of L^p sub-Markovian semigroups which are given by kernels having a density with respect to the underlying measure.In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities.We give various (dual) representations for several associated capacities and,in the corresponding abstract Bessel potential spaces,we study the role of the truncation property.Examples are discussed in the case of Rⁿ,where abstract Bessel potential spaces can be identified with concrete function spaces.
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  A Characterization of Generalized Frame MRAs Deriving Orthonormal Wavelets

  Zhi Hua ZHANG

  Acta Mathematica Sinica. 2006, 22(4): 1251-1260. https://doi.org/10.1007/s10114-005-0771-6

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  In 2000,Papadakis announced that any orthonormal wavelet must be derived by a gener-alized frame MRA (GFMRA).In this paper,we give a characterization of GFMRAs which can derive orthonormal wavelets,and show a general approach to the constructions of non-MRA wavelets.Finally we present two examples to illustrate the theory.
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  Long Bialgebras, Dimodule Algebras and Quantum Yang-Baxter Modules over Long Bialgebras

  Liang Yun ZHANG

  Acta Mathematica Sinica. 2006, 22(4): 1261-1270. https://doi.org/10.1007/s10114-005-0683-5

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  This paper gives a sufficient and necessary condition for a bialgebra to be a Long bialgebra,and proves a braided product to be a Long bialgebra under some conditions.It also gives a direct sum decomposition of quantum Yang-Baxter modules over Long bialgebras.
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  Error Bounds for Glimm Difference Approximations for Scalar Conservation Laws Without Convexity

  Xiao Ping YE, Long Wei LIN

  Acta Mathematica Sinica. 2006, 22(4): 1271-1282. https://doi.org/10.1007/s10114-005-0713-3

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  In last century,D.Hoff and J.Smoller derived the error bounds for the Glimm difference approximations of the solutions to scalar conservation laws with convexity.Our work is to extend the corresponding result of them to the case without convexity.