Acta Mathematica Sinica

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The Galois Correspondence in Field Algebra of G-spin Model

Li Ning JIANG, Mao Zheng GUO

Acta Mathematica Sinica. 2005, 21(4): 673-680. https://doi.org/10.1007/s10114-004-0513-1

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Suppose that G is a finite group and D(G) the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D(G;H) of D(G). This paper gives the concrete construction of a D(G;H)-invariant subspace ${\fancyscript A}$ _H in field algebra of G-spin model and proves that if H is a normal subgroup of G, then A_H is Galois closed.

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Additive Maps on Matrix Algebras Preserving Invertibility or Singularity

Ajda FOŠNER, PeterŠEMRL

Acta Mathematica Sinica. 2005, 21(4): 681-684. https://doi.org/10.1007/s10114-005-0519-3

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We characterize the additive singularity preserving almost surjective maps on M_n(F), the algebra of all n × n matrices over a field F with char F = 0. We also describe additive invertibility preserving surjective maps on M_n(F) and give examples showing that all the assumptions in these two theorems are indispensable.

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Periodic Solutions of Nonautonomous Second Order Hamiltonian Systems

Shi Xia LUAN, An Min MAO

Acta Mathematica Sinica. 2005, 21(4): 685-690. https://doi.org/10.1007/s10114-005-0532-6

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In this paper, we develop the local linking theorem given by Li and Willem by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems
Aü + A(t) u + ∇V (t, u) = 0, u ∈ R^{N, t ∈ R}.
We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem.

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Existence, Multiplicity and Infinite Solvability of Positive Solutions for One-Dimensional p-Laplacian

Qing Liu YAO

Acta Mathematica Sinica. 2005, 21(4): 691-698. https://doi.org/10.1007/s10114-004-0401-8

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The existence, multiplicity and infinite solvability of positive solutions are established for some two-point boundary value problems of one-dimensional p-Laplacian. In this paper, by multiplicity we mean the existence of m solutions, where m is an arbitrary natural number.

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Maluta’s Coefficient of Musielak-Orlicz Sequence Spaces

Hui Li YAO, Ting Fu WANG

Acta Mathematica Sinica. 2005, 21(4): 699-704. https://doi.org/10.1007/s10114-004-0429-9

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In this paper, we give the formula of Maluta's coefficient for calculation, and don't add any conditions to the generating function Φ.

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Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

Qi Kang RAN

Acta Mathematica Sinica. 2005, 21(4): 705-714. https://doi.org/10.1007/s10114-005-0569-6

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In this paper, we prove that the weak solutions $u \in W^{{1,p}}_{{{\text{loc}}}} {\left(\Omega \right)}{\left({1 < p < \infty } \right)}$ of the following equation with vanishing mean oscillation coefficients A(x): $-{\text{div}}{\left[ {{\left({A{\left(x \right)}\nabla u \cdot \nabla u} \right)}^{{\frac{{p-2}} {2}}} A{\left(x \right)}\nabla u + {\left| {F{\left(x \right)}} \right|}^{{p-2}} F{\left(x \right)}} \right]} = B{\left({x,u,\nabla u} \right)},$ belong to $W^{{1,q}}_{{{\text{loc}}}} {\left(\Omega \right)}{\left({\forall q \in {\left({p,\infty } \right)}} \right)}$ , provided $F \in L^{q}_{{{\text{loc}}}} {\left(\Omega \right)}$ and B(x, u, h) satisfies proper growth conditions, where Ω ⊂ R^N (N ≥ 2) is a bounded open set, A(x) = (A^ij(x))_N×N is a symmetric matrix function.

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Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

Yun Yan YANG, Xiao Xiang JIAO

Acta Mathematica Sinica. 2005, 21(4): 715-722. https://doi.org/10.1007/s10114-004-0426-z

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In this paper, curve shortening flow in Euclidian space Rⁿ(n ≥ 3) is studied, and S. Altschuler's results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.

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Compactness of Product Operators on the Bergman Space

Mao Fa WANG, Pei De LIU

Acta Mathematica Sinica. 2005, 21(4): 723-732. https://doi.org/10.1007/s10114-004-0441-0

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In this paper, we study the compactness of the product of a composition operator with another one's adjoint on the Bergman space. Some necessary and sufficient conditions for such operators to be compact are given.

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The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

E. M. E. ZAYED

Acta Mathematica Sinica. 2005, 21(4): 733-752. https://doi.org/10.1007/s10114-004-0514-0

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The trace of the wave kernel $\hat{\mu }{\left(t \right)} = {\sum\nolimits_{\omega = 1}^\infty {\exp {\left({-{\text{\bf i}}tE^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}_{\omega } } \right)}} },\;{\text{where}}\;{\left\{ {E_{\omega } } \right\}}^{\infty }_{{\omega = 1}}$ are the eigenvalues of the negative Laplacian $-\nabla ^{2} =-{\sum\nolimits_{k = 1}^3 {{\left({\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2}$ in the (x¹, x², x³)-space, is studied for a variety of bounded domains, where -∞ < t < ∞ and ${\text{\bf i}} = {\sqrt {-1} }$ . The dependence of û(t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in R³ surrounded by simply connected bounded domains Ω_j with smooth bounding surfaces S_j (j = 1, . . ., n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components $S^{ * }_{i}$ (i = 1+k_j-1, . . ., k_j) of the bounding surfaces S_j are considered, such that $S_{j} = \cup ^{{k_{j} }}_{{i = 1 + k_{{j-1}} }} S^{ * }_{i}$ , where k₀ = 0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion of û (t) for small |t|.

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On the Growth and Fixed Points of Solutions of Second Order Differential Equations with Meromorphic Coefficients

Zong Xuan CHEN, Kwang Ho SHON

Acta Mathematica Sinica. 2005, 21(4): 753-764. https://doi.org/10.1007/s10114-004-0434-z

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In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain that the exponents of convergence of these fixed points are all equal to the order of growth.

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The (L^P.F_P^β∞)-Boundedness of Commutators of Multipliers

Pu ZHANG, Jie Cheng CHEN

Acta Mathematica Sinica. 2005, 21(4): 765-772. https://doi.org/10.1007/s10114-004-0457-5

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In this paper, we study the commutator generalized by a multiplier and a Lipschitz function. Under some assumptions, we establish the boundedness properties of it from L^p(Rⁿ) into ${\dot{F}^{{\beta,\infty }}_{p} }$ (Rⁿ), the Triebel-Lizorkin spaces.

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Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

Yan WANG, Rong Quan FENG

Acta Mathematica Sinica. 2005, 21(4): 773-778. https://doi.org/10.1007/s10114-004-0455-7

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A Cayley map is a Cayley graph embedded in an orientable surface such that the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified.

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A Useful Extension of Itô’s Formula with Applications to Optimal Stopping

Gerold ALSMEYER, Markus JAEGER

Acta Mathematica Sinica. 2005, 21(4): 779-786. https://doi.org/10.1007/s10114-004-0524-y

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Given a continuous semimartingale M = (M_t)_t≥0 and a d-dimensional continuous process of locally bounded variation V = (V¹, . . ., V^d), the multidimensional Itô Formula states that

if f(x₀, . . ., x_d) is of C²-type with respect to x₀ and of C¹-type with respect to the other arguments. This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x₀ is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.

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Analytic Characterization for Hilbert-Schmidt Operators on Fock Space

Cai Shi WANG, Zhi Yuan HUANG, Xiang Jun WANG

Acta Mathematica Sinica. 2005, 21(4): 787-796. https://doi.org/10.1007/s10114-004-0456-6

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In this paper we first introduce and investigate some special classes of nonlinear maps and get several useful results. Then, by using these results, we obtain analytic criteria for checking whether or not an operator defined only on the exponential vectors of a symmetric Fock space becomes a Hilbert-Schmidt operator on the whole space. Additionally, as an application, we also get an analytic criterion for Hilbert-Schmidt operators on a Gaussian probability space through the Wiener-Itô-Segal isomorphism.

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On Two Theorems of Finite Solvable Groups

Shi Rong LI

Acta Mathematica Sinica. 2005, 21(4): 797-802. https://doi.org/10.1007/s10114-004-0357-8

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For a finite group G, let τ(G) denote a set of primes such that a prime p belongs to τ(G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ(G); (2) |τ(G)|=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ(G); then G either is solvable or G/Sol(G)

(2, 7) where Sol(G) is the largest solvable normal subgroup of (G).

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A Note on Hermite and Subelliptic Operators

Der Chen CHANG, Jing Zhi TIE

Acta Mathematica Sinica. 2005, 21(4): 803-818. https://doi.org/10.1007/s10114-004-0336-0

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In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈Rⁿ. We also apply this result to obtain the fundamental solutions for the Grushin operator in R² and the sub-Laplacian in the Heisenberg group H_n.

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Relatively Compact Sets on Abstract Wiener Space

Xi Cheng ZHANG

Acta Mathematica Sinica. 2005, 21(4): 819-822. https://doi.org/10.1007/s10114-005-0529-1

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In this note, we obtain a sufficient and necessary condition for a set in an abstract Wiener space (X, H, μ) to be relatively compact in L²(X, μ). Meanwhile, we give a sufficient condition for relative compactness in L^p(X, μ) for p > 1. We also provide an example of Da Prato-Malliavin-Nualart to show the result.

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Generalized Differential Identities of (Semi-)Prime Rings

Feng WEI

Acta Mathematica Sinica. 2005, 21(4): 823-832. https://doi.org/10.1007/s10114-005-0563-z

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Let R be a semiprime ring with characteristic p ≥ 0 and R_Fbe its left Martindale quotient ring. If

is a reduced generalized differential identity for an essential ideal of R, then Φ(Z_ije(Δ_j)) is a generalized polynomial identity for R_F, where e(Δ_j) are idempotents in the extended centroid of R determined by Δ_j. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If

is a reduced generalized differential identity for a noncommutative Lie ideal of R, then Φ(Z_ij) is a generalized polynomial identity for[R,R]. Moreover, if

is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then Φ(Z_ij) is a generalized polynomial identity for Q.

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Sums of Five Almost Equal Prime Squares

Claus BAUER

Acta Mathematica Sinica. 2005, 21(4): 833-840. https://doi.org/10.1007/s10114-004-0506-0

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Let P_i, 1 ≤ i ≤ 5, be prime numbers. It is proved that every integer N that satisfies N ≡ 5(mod24) can be written as $N = p^{2}_{1} + p^{2}_{2} + p^{2}_{3} + p^{2}_{4} + p^{2}_{5},{\text{where}}{\left| {{\sqrt N }5-p_{i} } \right|} \leqslant N^{{\frac{1} {2}-\frac{{19}} {{850}} + \in }}$ .

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Canonical Foliations of Certain Classes of Almost Contact Metric Structures

Tae Wan KIM, Hong Kyung PAK

Acta Mathematica Sinica. 2005, 21(4): 841-846. https://doi.org/10.1007/s10114-004-0520-2

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The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation F defined by the contact distribution is Riemannian and tangentially almost Köhler of codimension 1 and that F is tangentially K?hler if the manifold M is normal. Furthermore, we show that a semi-invariant submanifold N of such a manifold M admits a canonical foliation F_Nwhich is defined by the antiinvariant distribution and a canonical cohomology class c(N) generated by a transversal volume form for F_N. In addition, we investigate the conditions when the even-dimensional cohomology classes of N are non-trivial. Finally, we compute the Godbillon-Vey class for F_N.

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A Conjecture Concerning the Pure Exponential Diophantine Equation a^x+b^y=c^z

Mao Hua LE

Acta Mathematica Sinica. 2005, 21(4): 843-848. https://doi.org/10.1007/s10114-004-0436-x

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Let a, b, c, r be fixed positive integers such that a²+b²=c^r, min(a, b, c, r) > 1 and 2|r. In this paper we prove that if a≡2 (mod 4), b ≡ 3 (mod 4) c > 3.10³⁷ and r > 7200, then the equation a^x+b^y=c^z only has the solution (x, y, z) = (2, 2, r).

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Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

Wen Jun SUN, Shu WANG

Acta Mathematica Sinica. 2005, 21(4): 847-854. https://doi.org/10.1007/s10114-004-0512-2

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In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive (weak) solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition.

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Characterization of Compact Support of Fourier Transform for Orthonormal Wavelets of L² (R^d)

Zhi Hua ZHANG (Zhihua ZHANG)

Acta Mathematica Sinica. 2005, 21(4): 855-864. https://doi.org/10.1007/s10114-004-0419-y

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Let {ψ _μM } be an orthonormal wavelet of L² (R^d) and the support of a whole of its Fourier transform be

Under the weakest condition that each |

_μM| is continuous for

, a characterization of the above support of a whole is given.

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Some Representations of Unified Voigt Functions

M. KAMARUJJAMA, Dinesh SINGH

Acta Mathematica Sinica. 2005, 21(4): 865-868. https://doi.org/10.1007/s10114-004-0431-2

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The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.

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Fredholm Composition Operators on Riemann Surfaces

Guang Fu CAO

Acta Mathematica Sinica. 2005, 21(4): 869-872. https://doi.org/10.1007/s10114-004-0430-3

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It is proved that the invertibility of a composition operator on the differential form space for a Riemann surface is equivalent to its Fredholmness. In addition, the Fredholmness of weighted composition operators is discussed.

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Topological Structure of Non-wandering Set of a Graph Map

Rong Bao GU, Tai Xiang SUN, Ting Ting ZHENG

Acta Mathematica Sinica. 2005, 21(4): 873-880. https://doi.org/10.1007/s10114-004-0432-1

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Let G be a graph (i.e., a finite one-dimensional polyhedron) and f : G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.

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Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

Shuang Ping TAO, Shang Bin CUI

Acta Mathematica Sinica. 2005, 21(4): 881-892. https://doi.org/10.1007/s10114-004-0433-0

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This paper is devoted to studying the initial value problems of the nonlinear Kaup- Kupershmidt equations ∂u/∂t + a₁u∂²u/∂x² + β∂³u/∂x³ + γ∂⁵u/∂x⁵= 0, (x, t) ∈ R², and ∂u-∂t + a₂∂u/∂x∂²u∂x² + β∂³ u/∂x³ + γ∂⁵u/∂x⁵= 0, (x, t) ∈ R². Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup-Kupershmidt equations. The results show that a local solution exists if the initial function u₀ (x) ∈ H^s (R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation.

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Ideals in Morita Rings and Morita Semigroups

Yu Qun CHEN, Yun FAN, Zhi Feng HAO

Acta Mathematica Sinica. 2005, 21(4): 893-898. https://doi.org/10.1007/s10114-004-0427-y

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We characterize the lattice of all ideals of a Morita ring (semigroup) when the corresponding pair of rings (semigroups) in the Morita context are Morita equivalent s-unital (like-unity) rings (semigroups).

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The Characterization of Finite Simple Groups, L₃(3^2m-1)(m≥2), by Their Element Orders

Ming Chun XU

Acta Mathematica Sinica. 2005, 21(4): 899-902. https://doi.org/10.1007/s10114-004-0448-6

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In this paper the following theorem is proved: Every group L₃(3^2m-1)(m≥2) is characterized by its set of element orders.

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Normality of Tetravalent Cayley Graphs of Odd Prime-cube Order and Its Application

Yan Quan FENG, Ming Yao XU

Acta Mathematica Sinica. 2005, 21(4): 903-912. https://doi.org/10.1007/s10114-004-0500-6

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Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p³ are normal. As an application, a classification of tetravalent symmetric graphs of odd prime-cube order is given.

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Isomorphic Commutative Group Algebras of p-Mixed Warfield Groups

Peter V. DANCHEV

Acta Mathematica Sinica. 2005, 21(4): 913-916. https://doi.org/10.1007/s10114-004-0526-9

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Let G be a p-mixed Warfield Abelian group and F a field of char F = p ≠ 0. It is proved that if for any group H the group algebras FH and FG are F-isomorphic, then H is isomorphic to G. This presentation enlarges a result of W. May argued when G is p-local Warfield Abelian and published in Proc. Amer. Math. Soc. (1988).

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On the Uniqueness of the ADS Spacetime

Xiao Dong WANG (Xiaodong Wang)

Acta Mathematica Sinica. 2005, 21(4): 917-922. https://doi.org/10.1007/s10114-004-0489-x

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The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n ≤ 7. For dimension n > 7, the same result is established under the spin assumption.

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Global Existence and Blow-up of Solutions for Parabolic Systems Involving Cross-Diffusions and Nonlinear Boundary Conditions

Xiu Hui YANG, Fu Cai LI, Chun Hong XIE

Acta Mathematica Sinica. 2005, 21(4): 923-928. https://doi.org/10.1007/s10114-004-0439-7

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In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions:

Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.

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On the Countable Tightness of Product Spaces

Chuan LIU, Shou LIN

Acta Mathematica Sinica. 2005, 21(4): 929-936. https://doi.org/10.1007/s10114-004-0438-8

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In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent: (1) b = ω₁; (2) t(S_ω ×S_ω1) > ω; (3) For any pair (X, Y), which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t(X × Y) ≤ ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.

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Decompositions of the Hardy space H_z¹(Ω)

Zeng Jian LOU, Shou Zhi YANG, Dao Jin SONG

Acta Mathematica Sinica. 2005, 21(4): 949-954. https://doi.org/10.s10114-004-0499-8

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We give a decomposition of the Hardy space H_z¹(Ω) into "div-curl" quantities for Lipschitz domains in Rⁿ. We also prove a decomposition of H_z¹(Ω) into Jacobians det Du, u ∈ W₀^{1, 2} (Ω, R²) for Ω in R². This partially answers a well-known open problem.

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Bounds for the Least Laplacian Eigenvalue of a Signed Graph

Yao Ping HOU

Acta Mathematica Sinica. 2005, 21(4): 955-960. https://doi.org/10.1007/s10114-004-0437-9

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A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.

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15 August 2005, Volume 21 Issue 4

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15 August 2005, Volume 21 Issue 4