Acta Mathematica Sinica

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Fountain-Gould Left Orders for Associative Pairs

José A. ANQUELA, Teresa CORTÉS, Miguel GÓMEZ LOZANO, Mercedes SILES MOLINA

Acta Mathematica Sinica. 2006, 22(3): 641-652. https://doi.org/10.1007/s10114-005-0691-5

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In this paper, we introduce a notion of weak Fountain-Gould left order for associative pairs and give a Goldie-like theory of associative pairs which are weak Fountain-Gould left orders in semiprime pairs coinciding with their socles.

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Pointwise Convergence and Uniform Convergence of Wavelet Frame Series

Zhi Hua ZHANG

Acta Mathematica Sinica. 2006, 22(3): 653-658. https://doi.org/10.1007/s10114-005-0602-9

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Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.

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Congruence Permutable Symmetric Extended de Morgan Algebras

Jie FHANG

Acta Mathematica Sinica. 2006, 22(3): 659-664. https://doi.org/10.1007/s10114-005-0706-2

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An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, K _n,0-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras.

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Existence and Multiplicity Results for a Degenerate Elliptic Equation

Wei DONG, Jian Tao CHEN

Acta Mathematica Sinica. 2006, 22(3): 665-670. https://doi.org/10.1007/s10114-005-0696-0

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The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f(x, s), we show the following problem: $$ \begin{array}{*{20}c} {{ - \Delta _{p} u=\lambda u^{\alpha } - a{\left(x \right)}u^{q} \;{\text{in}}\;\Omega,}} & {{\left. u \right|_{{\partial \Omega }} }} \\ \end{array}=0 $$ possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N≥2, with C² boundary, λ is a positive parameter, Δ_p is the p-Laplacian operator with p > 1, α, q are given constants such that p ? 1 < α < q, and a(x) is a continuous positive function in $ \ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega } $ Ω.

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Characterizations and Extensions of Lipschitz-α Operators

Huai Xin CAO, Jian Hua ZHANG, Zong Ben XU

Acta Mathematica Sinica. 2006, 22(3): 671-678. https://doi.org/10.1007/s10114-005-0727-x

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In this work, we prove that a map F from a compact metric space K into a Banach space X over ${\Bbb F}$ is a Lipschitz-α operator if and only if for each σ in X* the map σ°F is a Lipschitz-α function on K. In the case that K=[a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ ? (σ ° f)' is a bounded linear operator from X* into L^∞([a, b]). When K is a compact subset of a finite interval (a, b) and 0 < α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with L_α(f) ≤ L_α(F) ≤ 3^1?α L_α(f). A similar extension theorem for a little Lipschitz-α operator is also obtained.

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Lattice Structure for Paraunitary Linear-phase Filter Banks with Accuracy

Hong Ying XIAO

Acta Mathematica Sinica. 2006, 22(3): 679-688. https://doi.org/10.1007/s10114-005-0618-1

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Multivariate filter banks with a polyphase matrix built by matrix factorization (lattice structure) were proposed to obtain orthonormal wavelet basis. On the basis of that, we propose a general method of constructing filter banks which ensure second and third accuracy of its corresponding scaling function. In the last part, examples with second and third accuracy are given.

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On the Seneta Sequences

Dragan Djur?i?, Aleksandar Torga?ev

Acta Mathematica Sinica. 2006, 22(3): 689-692. https://doi.org/10.1007/s10114-005-0684-4

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In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SO_c.

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Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

Jiang Feng WHANG, Feng Bin LU

Acta Mathematica Sinica. 2006, 22(3): 693-700. https://doi.org/10.1007/s10114-005-0601-x

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In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^?-mixing random fields. As its applications we get the Hájeck-Rènyi inequality and weak convergence of sums of ρ^?-mixing sequence. These results extend related results for NA sequence and ρ*-mixing random fields.

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Horizontal Connection and Horizontal Mean Curvature in Carnot Groups

Kang Hai TAN, Xiao Ping YANG

Acta Mathematica Sinica. 2006, 22(3): 701-710. https://doi.org/10.1007/s10114-005-0572-y

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In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli-Garofalo-Nhieu and Pauls who recently introduced sub-Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction (projection) of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and (intrinsic) regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.

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Phantom Maps from a Strong Homology Viewpoint

Dae Woong LEE

Acta Mathematica Sinica. 2006, 22(3): 711-718. https://doi.org/10.1007/s10114-005-0597-2

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Let X and Y be connected nilpotent CW-complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.

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The Fractional Derivatives of a Fractal Function

Kui YAO, Wei Yi SU, Song Ping ZHOU

Acta Mathematica Sinica. 2006, 22(3): 719-722. https://doi.org/10.1007/s10114-005-0644-z

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The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.

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Algebraic Structure on Dirichlet Spaces

Xing FHANG, Ping HE, Jian Gang YING

Acta Mathematica Sinica. 2006, 22(3): 723-728. https://doi.org/10.1007/s10114-005-0583-8

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In this short note, we shall give a few equivalent conditions for a closed form to be Markovian, and prove that the closure of a sub-algebra of bounded functions in a Dirichlet space must be Markovian. We also study the regular representation of Dirichlet spaces and the classification of Dirichlet subspaces.

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Asymptotic Approximations of Jacobi Functions and Their Zeros with Error Bounds

Xiao Feng ZHU, Xiu Chun LI

Acta Mathematica Sinica. 2006, 22(3): 729-740. https://doi.org/10.1007/s10114-005-0593-6

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The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variablex(t), we've got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions $\varphi ^{{{\left({\alpha,\beta } \right)}}}_{\mu } {\left(t \right)}{\left({\alpha > - 1} \right)}{\text{as}}\;\mu \to+\infty$ . Besides, an accurate approximation with error bounds is also constructed correspondingly for the zerost_μ,s of $\varphi ^{{{\left({\alpha,\beta } \right)}}}_{\mu } {\left(t \right)}{\left({\alpha \geqslant 0} \right)}{\text{as}}\;\mu \to+\infty$ , uniformly with respect tos=1, 2,....

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On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Chong LI, Genaro LOPEZ

Acta Mathematica Sinica. 2006, 22(3): 741-750. https://doi.org/10.1007/s10114-005-0595-4

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LetB(resp.K,BC,KC) denote the set of all nonempty bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach spaceX, endowed with the Hausdorff metric, and letG be a nonempty relatively weakly compact closed subset ofX. LetB^ostand for the set of allF∈B such that the problem (F,G) is well-posed. We proved that, ifX is strictly convex and Kadec, the setKC∩B^o is a denseG_δ-subset ofKC\G. Furthermore, ifX is a uniformly convex Banach space, we will prove more, namely that the setB\B^o (resp.K\B^o,BC\B^o,KC\B^o) isσ-porous inB (resp.K,BC,KC). Moreover, we prove that for most (in the sense of the Baire category) closed bounded subsetsGofX, the setK\B^o is dense and uncountable inK.

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On Hereditarily Indecomposable Banach Spaces

Li Xin CHENG, Huai Jie ZHONG

Acta Mathematica Sinica. 2006, 22(3): 751-756. https://doi.org/10.1007/s10114-005-0614-5

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This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.

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Singular Integrals and Commutators in Generalized Morrey Spaces

Lubomiea SOFTOVA

Acta Mathematica Sinica. 2006, 22(3): 757-766. https://doi.org/10.1007/s10114-005-0628-z

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We consider a singular an integral operatorK with a variable Calderón-Zygmund type kernelk(x; ξ),x∈Rⁿ, ξ∈Rⁿ\{0}, satisfying a mixed homogeneity condition of the form $k{\left({x;\mu ^{{\alpha _{1} }} \xi _{1}, \ldots,\mu ^{{\alpha _{n} }} \xi _{n} } \right)}=\mu ^{{ - {\sum\nolimits_{i=1}^n {\alpha _{i} } }}} k{\left({x;\xi } \right)},\alpha _{i} \geqslant 1$ and μ > 0. The continuity of this operator inL^p(Rⁿ) is well studied by Fabes and Rivière. Our goal is to extend their result to generalized Morrey spacesL^p,ω(Rⁿ),p∈(1,∞) with a weight ω satisfying suitable dabbling and integral conditions. A special attention is paid to the commutatorC[a, k]=Ka-aK with the operator of multiplication by BMO functions.

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Strongly Singular Integral Operators on Weighted Hardy Space

Jun Feng LI, Shan Zhen LU

Acta Mathematica Sinica. 2006, 22(3): 767-772. https://doi.org/10.1007/s10114-005-0604-7

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In this paper, we obtain that a strongly singular integral operator is bounded onL_w^p space for 1<p<∞. We also obtain that a strongly singular integral operator is a bounded operator fromH_w^p toL_w^p for some weightw and 0<p≤1. And by an atomic decomposition, we obtain that a strongly singular integral operator is a bounded operator onH_w^p for some w and 0<p≤1.

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Existence Results for Functional Differential Inclusions with Infinite Delay

Shi Huang HONG

Acta Mathematica Sinica. 2006, 22(3): 773-780. https://doi.org/10.1007/s10114-005-0600-y

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The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.

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Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Ye JIANG, Li Xin ZHANG

Acta Mathematica Sinica. 2006, 22(3): 781-792. https://doi.org/10.1007/s10114-005-0615-4

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Let {X,X_n;n≥1} be a sequence of i.i.d. random variables, EX=0, EX²=σ² < ∞. SetS_n=X₁+X₂+…+X_n,M_n=max_k≤n ∣S_k∣,n≥1. Leta_n=O(1/log logn). In this paper, we prove that, forb >-1,
$\begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \searrow 0} }\varepsilon ^{{2{\left({b+1} \right)}}} {\sum\limits_{n=1}^\infty {\frac{{{\left({\log \;\log \;n} \right)}^{b} }} {{n\log n}}} }n^{{ - 1 \mathord{\left/{\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\rm E}{\left\{ {M_{n} - \sigma {\left({\varepsilon+a_{n} } \right)}{\sqrt {2n\log \;\log \;n} }} \right\}}+\\ &=\frac{{\sigma 2^{{ - b}} }} {{{\left({b+1} \right)}{\left({2b+3} \right)}}}{\rm E}{\left| N \right|}^{{2b+3}} {\sum\limits_{k=0}^\infty {\frac{{{\left({ - 1} \right)}^{k} }} {{{\left({2k+1} \right)}^{{2b+3}} }}} } \\ \end{aligned}$
holds if and only if EX=0 and EX²=σ² < ∞.

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Hölder Continuity of Weak Solutions for Parabolic Equations with Nonstandard Growth Conditions

Meng XU, Ya Zhe CHEN

Acta Mathematica Sinica. 2006, 22(3): 793-806. https://doi.org/10.1007/s10114-005-0582-9

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In this paper, we investigate the interior regularity including the local boundedness and the interior Hölder continuity of weak solutions for parabolic equations of thep(x, t)-Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.

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On the Empty Essential Spectrum for Markov Processes in Dimension One

Yong Hua MAO

Acta Mathematica Sinica. 2006, 22(3): 807-812. https://doi.org/10.1007/s10114-005-0587-4

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This paper gives characterizations for diffusion processes on the line and birth-death processes whose generators admit the empty essential spectra. Some equivalent conditions for empty essential spectra for general Markov generators are also discussed.

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Strict Stability of Impulsive Differential Equations

Yu ZHANG, Ji Tao SUN

Acta Mathematica Sinica. 2006, 22(3): 813-818. https://doi.org/10.1007/s10114-005-0634-1

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In this paper, we will extend the strict stability to impulsive differential equations. By using Lyapunov functions, we will get some criteria for the strict stability of impulsive differential equations, and we can see that impulses do contribute to the system's strict stability behavior. An example is also given in this paper to illustrate the efficiency of the obtained results.

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Positive Solutions of a Class of Quasilinear Elliptic Equations in Two-dimensional Exterior Domains

Zhao Yang YIN

Acta Mathematica Sinica. 2006, 22(3): 819-826. https://doi.org/10.1007/s10114-005-0646-x

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Sufficient conditions for the existence of positive solutions to a class of quasilinear elliptic equations in two-dimensional exterior domains are given.

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An Upper and Lower Solution Theory for the Problem (G'(y))'+f(t, y)=0 on Finite and Infinite Intervals

Ravi P. AGARWAL, Donal O'REGAN, Svatoslav STANĚK

Acta Mathematica Sinica. 2006, 22(3): 827-832. https://doi.org/10.1007/s10114-005-0635-0

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A new upper and lower solution theory is presented for the second order problem (G'(y))'+f(t, y)=0 on finite and infinite intervals. The theory on finite intervals is based on a Leray-Schauder alternative, whereas the theory on infinite intervals is based on results on the finite interval and a diagonalization process.

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Gauss Sum of Index 4: (2) Non-cyclic Case

Jing YANG, Shi Xin LUO, Ke Qin FENG

Acta Mathematica Sinica. 2006, 22(3): 833-844. https://doi.org/10.1007/s10114-005-0645-y

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Assume that/m≥2,p is a prime number, (m, p(p-1))=1, -1∈〈p〉⊂(Z/mZ)^* and [(Z/mZ)^*: 〈p〉]=4. In this paper, we calculate the value of Gauss sum $G{\left(\chi \right)}={\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left(x \right)}\zeta ^{{T{\left(x \right)}}}_{p} } }$ over F_q, whereq=p^f,f=φ(m)/4,,χ is a multiplicative character of F_q andT is the trace map from F_q to F_p. Under our assumptions,G(χ) belongs to the decomposition field K ofp in Q(ζ_m) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/Q) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non-cyclic case.

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Normal Bases and Their Dual-Bases over Finite Fields

Qun Ying LIAO, Qi SUN

Acta Mathematica Sinica. 2006, 22(3): 845-848. https://doi.org/10.1007/s10114-005-0613-6

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In this paper, we prove the following results: 1) A normal basisN over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2) The normal basisN is self-dual if and only if its multiplication table is symmetric and Tr(α²)=1, where α generates N; 3) An optimal normal basis N is self-dual if and only if N is a type-I optimal normal basis withq=n=2 or N is a type-Ⅱ optimal normal basis.

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On the Category of Koszul Modules of Complexity One of an Exterior Algebra

Jin Yun GUO, Bing Jun LI, Qiu Xian WU

Acta Mathematica Sinica. 2006, 22(3): 849-854. https://doi.org/10.1007/s10114-005-0589-2

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In this paper, we prove that there is a natural equivalence between the categoryF(x) of Koszul modules of complexity 1 with filtration of given cyclic modules as the factor modules of an exterior algebra Λ=∧V of anm-dimensional vector space, and the category of the finite-dimensional locally nilpotent modules of the polynomial algebra ofm-1 variables.

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Homoclinic Bifurcation of Orbit Flip with Resonant Principal Eigenvalues

Tian Si ZHANG, De Ming ZHU

Acta Mathematica Sinica. 2006, 22(3): 855-864. https://doi.org/10.1007/s10114-005-0581-x

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Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigen- values are studied for a four-dimensional system. The existence, number, co-existence and non- coexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are ob- tained. The bifurcation surfaces and existence regions are also given.

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15 June 2006, Volume 22 Issue 3

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15 June 2006, Volume 22 Issue 3