Acta Mathematica Sinica

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A Weierstrass Representation Formula for Minimal Surfaces in H₃ and H²×R

Francesco MERCURI, Stefano MONTALDO, Paola PIU

Acta Mathematica Sinica. 2006, 22(6): 1603-1612. https://doi.org/10.1007/s10114-005-0637-y

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We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold.Then,we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line.

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The Center Problem for a Linear Center Perturbed by Homogeneous Polynomials

Jaume GINÉ

Acta Mathematica Sinica. 2006, 22(6): 1613-1620. https://doi.org/10.1007/s10114-005-0623-4

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The centers of the polynomial differential systems with a linear center perturbed by homogeneous polynomials have been studied for the degrees s=2,3,4,5.They are completely classified for s=2,3,and partially classified for s=4,5.In this paper we recall these results for s=2,3,4,5,and we give new centers for s=6,7

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Super-edge-graceful Labelings of Some Cubic Graphs

Wai Chee SHIU

Acta Mathematica Sinica. 2006, 22(6): 1621-1628. https://doi.org/10.1007/s10114-005-0672-8

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The notion of super-edge-graceful graphs was introduced by Mitchem and Simoson in 1994.However,few examples except trees are known.In this paper,we exhibit two classes of infinitely many cubic graphs which are super-edge-graceful.A conjecture is proposed.

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Continuous Selection, Collectively Fixed Points and System of Coincidence Theorems in Product Topological Spaces

Xie Ping DING

Acta Mathematica Sinica. 2006, 22(6): 1629-1638. https://doi.org/10.1007/s10114-005-0831-y

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Some new continuous selection theorems are first proved in noncompact topological spaces.As applications,some new collectively fixed point theorems and coincidence theorems for two families of set-valued mappings defined on product space of noncompact topological spaces are obtained under very weak assumptions.These results generalize many known results in recent literature.

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On the Non-Commutative Neutrix Product of the Distributions X₊^λ and X₊^μ

B. FISHER, K. TAŞ

Acta Mathematica Sinica. 2006, 22(6): 1639-1644. https://doi.org/10.1007/s10114-005-0762-7

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Let f and g be distributions and let g_n=(g*δ_n)(x),where δ_n(x) is a certain sequence converging to the Dirac delta function.The non-commutative neutrix product fog of f and g is defined to be the limit of the sequence {fg_n},provided its limit h exists in the sense that
${\mathop {{\text{N}}-\lim }\limits_{n \to \infty } }{\left\langle {f{\left( x \right)}g_{n} {\left( x \right)},\varphi {\left( x \right)}} \right\rangle }={\left\langle {h{\left( x \right)},\varphi {\left( x \right)}} \right\rangle },$
for all functions ¢ in D.It is proved that
$\begin{array}{*{20}c} {{{\left( {x^{\lambda }_{ + } \ln ^{p} x_{ + } } \right)} \circ {\left( {x^{\mu }_{ + } \ln ^{q} x_{ + } } \right)}=x^{{\lambda + \mu }}_{ + } \ln ^{{p + q}} x_{ + },}} & {{{\left( {x^{\lambda }_{-} \ln ^{p} x_{-} } \right)} \circ {\left( {x^{\mu }_{-} \ln ^{q} x_{-} } \right)}=x^{{\lambda + \mu }}_{-} \ln ^{{p + q}} x_{-},}} \\ \end{array}$
for λ +μ<-1;λ,μ,λ+μ ≠-1,-2,...and p,q=0,1,2.....

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Asymptotic of Eigenvalues and Lattice Points

Juan Pablo PINASCO

Acta Mathematica Sinica. 2006, 22(6): 1645-1650. https://doi.org/10.1007/s10114-005-0761-8

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In this work we study the spectral counting function for the p-Laplace operator in one dimension.We show the existence of a two-term Weyl-type asymptote.The method of proof is rather elementary,based on the Dirichlet lattice points problem,which enables us to obtain similar results for domains of infinite measure.

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Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Ran Chao WU, Jian Hua SUN

Acta Mathematica Sinica. 2006, 22(6): 1651-1656. https://doi.org/10.1007/s10114-005-0705-3

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Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied.By establishing a local coordinate system and a Poincarü map near the homoclinic orbit,the existence and uniqueness of 1-homoclinic orbit and 1-periodic orbit are given.Also considered is the existence of 2-homoclinic orbit and 2-periodic orbit.In additon,the corresponding bifurcation surfaces are given.

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An Embedding for the E₂-term of the Adams Spectral Sequence at Odd Primes

Maurizio BRUNETTI, Adriana CIAMPELLA, Luciano A. LOMONACO

Acta Mathematica Sinica. 2006, 22(6): 1657-1666. https://doi.org/10.1007/s10114-005-0704-4

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Let p be an odd prime.In this paper we introduce a quadratic linear Fp -algebra Q₁ obtained by suitably changing the generators of Q,the homogeneous quadratic algebra of cohomology operations in the category of H_∞-ring spectra,and study the map induced on cohomology by the quotient $\ifmmode\expandafter\bar\else\expandafter\=\fi{\pi }:Q_{1} \to {\fancyscript A}_{p}.$

Like in the case p=2,it turns out that $\ifmmode\expandafter\bar\else\expandafter\=\fi{\pi }*$ is injective.Thus,its target contains the E₂-term of the classical Adams spectral sequence as subalgebra.An explicit description of ${\text{EXT}}_{{Q_{1} }} {\left( {{\Bbb F}_{p},{\Bbb F}_{p} } \right)}$ is given under the reasonable assumption on Q to be a Koszul algebra.

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N-Widths and Average Widths of Besov Classes in Sobolev Spaces

Gui Qiao XU, Yong Sheng SUN, Yong Ping LIU

Acta Mathematica Sinica. 2006, 22(6): 1667-1678. https://doi.org/10.1007/s10114-005-0692-4

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In this paper,we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces.The weak asymptotic results concerning the Kolmogorov n-widths,the linear n-widths,the Gel'fand n-widths,in the Sobolev spaces on T^d,and the infinite-dimensional widths and the average widths in the Sobolev spaces on R^dare obtained,respectively.

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Asymptotic Properties of the Maximum Likelihood Estimate in Generalized Linear Models with Stochastic Regressors

Jie Li DING, Xi Ru CHEN

Acta Mathematica Sinica. 2006, 22(6): 1679-1686. https://doi.org/10.1007/s10114-005-0693-3

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For generalized linear models (GLM),in case the regressors are stochastic and have different distributions,the asymptotic properties of the maximum likelihood estimate (MLE)

of the parameters are studied.Under reasonable conditions,we prove the weak,strong consistency and asymptotic normality of

.

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Fractional-Linear Transformations of Operator Balls; Applications to Dynamical Systems

V. A. KHATSKEVICH, V. A. SENDEROV

Acta Mathematica Sinica. 2006, 22(6): 1687-1694. https://doi.org/10.1007/s10114-005-0694-2

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The operator sets,which are the subject of this paper,have been studied in many papers where,under different restrictions on the generating operators,convexity,compactness in the weak operator topology,and nonemptiness were proved for sets of different classes under study.Then the results obtained were used in these papers to solve several applied problems.Namely,they played the key role in establishing the dichotomy of nonautonomous dynamical systems,with either continuous or discrete time.In the present paper,we generalize and sharpen the already known criteria and obtain several new criteria for convexity,compactness,and nonemptiness of several special operator sets.Then,using the assertions obtained,we construct examples of sets of the form under study which are nonconvex,noncompact in the weak operator topology,as well as empty,and are generated by “smooth” operators of a special class.The existence problem for such sets remained open until the authors of this paper announced some of its results.

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Some Results on Subelliptic Equations

Li MA, De Zhong CHEN, Yang YANG

Acta Mathematica Sinica. 2006, 22(6): 1695-1704. https://doi.org/10.1007/s10114-005-0882-0

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In this paper,we consider the principal eigenvalue problem for Hormander's Laplacian on Rⁿ,and we find a comparison principle for such principal eigenvalues.We also study a related semi-linear sub-elliptic equation in the whole Rⁿand prove that,under a suitable condition,we have infinitely many positive solutions of the problem.

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Skew Pairs of Idempotents in Transformation Semigroups

T. S. BLYTH, M. H. ALMEIDA SANTOS

Acta Mathematica Sinica. 2006, 22(6): 1705-1714. https://doi.org/10.1007/s10114-005-0703-5

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An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent.We have shown previously that there are four distinct types of skew pairs of idempotents.Here we investigate the ubiquity of such skew pairs in full transformation semigroups.

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A Note on Commutators with Power Central Values on Lie Ideals

Yu WANG, Hong YOU

Acta Mathematica Sinica. 2006, 22(6): 1715-1720. https://doi.org/10.1007/s10114-005-0805-0

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This note proves that,if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d(u),u]ⁿis central,for all u∈L,then R must satisfy s₄,the standard identity in 4 variables.The case where R is a semiprime ring is also examined by the authors.The results of the note improve Carini and Filippis's results.

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The Isometric Extension of an Into Mapping from the Unit Sphere S(l₍₂₎^∞) to S(L¹(μ))

Guang Gui DING

Acta Mathematica Sinica. 2006, 22(6): 1721-1724. https://doi.org/10.1007/s10114-005-0695-1

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This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces.We prove that,under some conditions,an into-isometric mapping from the unit sphere $S{\left( {{\ell }^{\infty }_{{{\left( 2 \right)}}} } \right)}$ to $S{\left( {L^{1} {\left( \mu \right)}} \right)}$ can be (real) linearly isometrically extended.

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Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

Yue Bao WANG, Kai Yong WANG, Dong Ya CHENG

Acta Mathematica Sinica. 2006, 22(6): 1725-1734. https://doi.org/10.1007/s10114-005-0745-8

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In this paper,we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function.We discover that,under certain conditions,three precise large-deviation probabilities with different centering numbers are equivalent to each other.Furthermore,we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences.The obtained results extend and improve the corresponding results of Ng,Tang,Yan and Yang (J.Appl.Prob.,41,93-107,2004).

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On the Edge-forwarding Indices of Frobenius Graphs

Yan WANG, Xin Gui FANG, D. F. HSU

Acta Mathematica Sinica. 2006, 22(6): 1735-1744. https://doi.org/10.1007/s10114-005-0833-9

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A G-Frobenius graph Γ,as defined by Fang,Li,and Praeger,is a connected orbital graph of a Frobenius group G=K

H with Frobenius kernel K and Frobenius complement H.Γ is also shown to be a Cayley graph,Γ=Cay(K,S) for K and some subset S of the group K.On the other hand,a network N with a routing function R,written as (N,R),is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph.The edge-forwarding index π(N) of a network (N,R),defined by Heydemann,Meyer,and Sotteau,is a parameter to describe the maximum load of edges of N.In this paper,we study the edge-forwarding indices of Frobenius graphs.In particular,we obtain the edge-forwarding index of a G-Frobenius graph Γ with rank(G)≤50.

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Positive and Nontrivial Solutions for the Urysohn Integral Equation

Daniel FRANCO, Gennaro INFANTE, Donal O’REGAN

Acta Mathematica Sinica. 2006, 22(6): 1745-1750. https://doi.org/10.1007/s10114-005-0782-3

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We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation.We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator.Among others,we employ techniques based on fixed point index theory for compact maps,which are new for this type of equation.

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On π-Subpairs of π-Blocks

Jin Ke Hai, Yi Xin Zhu

Acta Mathematica Sinica. 2006, 22(6): 1751-1756. https://doi.org/10.1007/s10114-005-0866-0

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Alperin and Brouá have given the p-subpairs in a finite group,and proved that there is a Sylow theorem for p-subpairs.For a π-separable group with π-Hall subgroup nilpotent,we prove that there is a π-Sylow theorem for π-subpairs.Note that our π-subpairs are different from what Robinson and Staszewski gave.

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Regularity for P-Harmonic Type Systems with the Gradients below the Controllable Growth

Shen Zhou ZHENG, Shu Le ZHAO

Acta Mathematica Sinica. 2006, 22(6): 1757-1766. https://doi.org/10.1007/s10114-005-0804-1

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In this paper,we shall establish that each weak solution of p-harmonic type systems with the gradients below the controllable growth belongs to,H?lder continuity spaces with any Hölder exponent α∈[0,1).Furthermore,we can obtain that the gradients of the corresponding weak solutions also belong to locally Hölder continuity spaces with some Hölder exponent.

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Functional Limit Theorems for d-dimensional FBM in Hölder Norm

Zheng Yan LIN, Wen Sheng WANG, Kyo Shin HWANG

Acta Mathematica Sinica. 2006, 22(6): 1767-1780. https://doi.org/10.1007/s10114-005-0744-9

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In this paper,we obtain functional limit theorems for d-dimensional FBM in Hölder norm via estimating large deviation probabilities for d-dimensional FBM in Hölder norm.

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Stability Problem for Jensen-type Functional Equations of Cubic Mappings

Kil-Woung JUN, Hark-Mahn KIM

Acta Mathematica Sinica. 2006, 22(6): 1781-1788. https://doi.org/10.1007/s10114-005-0736-9

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In this paper,we establish the general solution and the generalized Hyers-Ulam-Rassias stability problem for a cubic Jensen-type functional equation,
4ƒ(3x+y/4)+4ƒ(x+3y/4)=6ƒ(x+y/2)+ƒ(x)+ƒ(y),
9ƒ(2x+y/3)+9ƒ(x+2y/3)=16ƒ(x+y/2)+ƒ(x)+ƒ(y)
in the spirit of D.H.Hyers,S.M.Ulam,Th.M.Rassias and P.G?vruta.

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Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Choonkil BAAK

Acta Mathematica Sinica. 2006, 22(6): 1789-1796. https://doi.org/10.1007/s10114-005-0697-z

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Let X,Y be vector spaces.It is shown that if a mapping ƒ:X→Y satisfies
ƒ(2/x+y+z)+ƒ(2/x-y+z=ƒ(x)+2ƒ(z),(0.1)
ƒ(2/x+y+z)-ƒ(2/x-y+z=ƒ(y),(0.2)
or
2ƒ(2/x+y+z)+ƒ(x)+ƒ(y)2ƒ(z)(0.3)
for all X,Y,Z∈X,then the mapping ƒ:X→Y is Cauchy additive.Furthermore,we prove the Cauchy-Rassias stability of the functional equations (0.1),(0.2) and (0.3) in Banach spaces.The results are applied to investigate isomorphisms between unital Banach algebras.

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Positive Solutions for Second-Order m-Point Boundary Value Problems on Time Scales

Wan Tong LI, Hong Rui SUN

Acta Mathematica Sinica. 2006, 22(6): 1797-1804. https://doi.org/10.1007/s10114-005-0748-5

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Let T be a time scale such that 0,T∈T.By means of the Schauder fixed point theorem and analysis method,we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales

where a∈C_ld((0,T),[0,∞)),ƒ∈C_ld([0,∞)×[0,∞),[0,∞)),β,γ∈[0,∞),ξ_i∈[0,p∞)),b,a_i∈(0,∞)(forem=1,...,m-2)are given constants satisfying some suitable hypotheses.We show that this problem has at least one positive solution for sufficiently small b > 0 and no solution for sufficiently large b.Our results are new even for the corresponding differential equation (T=R) and difference equation (T=Z).

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Vector Spaces of Non-measurable Functions

Francisco J. GARCÍA-PACHECO, Juan B. SEOANE-SEPÚLVEDA

Acta Mathematica Sinica. 2006, 22(6): 1805-1808. https://doi.org/10.1007/s10114-005-0726-y

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We show that there exists an infinite dimensional vector space every non-zero element of which is a non-measurable function.Moreover,this vector space can be chosen to be closed and to have dimension β for any cardinality β.Some techniques involving measure theory and density characters of Banach spaces are used.

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Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Lei DENG, Ming Ge YANG

Acta Mathematica Sinica. 2006, 22(6): 1809-1818. https://doi.org/10.1007/s10114-005-0743-x

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Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces.As applications,some new minimax inequalities,section theorem,best approximation theorem,existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.

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Extension of Isometries Between the Unit Spheres of Normed Space E and C(Ω)

Xi Nian FANG, Jian Hua WANG

Acta Mathematica Sinica. 2006, 22(6): 1819-1824. https://doi.org/10.1007/s10114-005-0725-z

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In this paper,we study the extension of isometries between the unit spheres of normed space E and C (Ω).We obtain that any surjective isometry between the unit spheres of normed space E and C (Ω) can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley's problem (where Ω be a compact metric space).

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Existence of Positive Solutions to Fourth Order Quasilinear Boundary Value Problems

Zhan Bing BAI, Wei Gao GE

Acta Mathematica Sinica. 2006, 22(6): 1825-1830. https://doi.org/10.1007/s10114-005-0806-z

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In this paper,we generalize the fixed point theorem of cone expansion and compression of norm type to the theorem of functional type.As an application,the existence of positive solutions for some fourth-order beam equation boundary value problems is obtained.The emphasis is put on that the nonlinear term is dependent on all lower order derivatives.

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Ginzburg-Landau Vortex and Mean Curvature Flow with External Force Field

Huai Yu JIAN, Yan Nan LIU

Acta Mathematica Sinica. 2006, 22(6): 1831-1842. https://doi.org/10.1007/s10114-005-0698-y

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This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg-Landau system which simulates inhomogeneous type Ⅱ superconducting materials and three-dimensional superconducting thin films having variable thickness.We will prove that the vortex of the problem is moved by a codimension k mean curvature flow with external force field.Besides,we will show that the mean curvature flow depends strongly on the external force,having completely different phenomena from the usual mean curvature flow.

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A Note on Certain Block Spaces on the Unit Sphere

Xiao Feng YE, Xiang Rong ZHU

Acta Mathematica Sinica. 2006, 22(6): 1843-1846. https://doi.org/10.1007/s10114-005-0781-4

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In this note,we clarify a relation between block spaces and the Hardy space.We obtain that block spaces B_q^0,v(Sⁿ-1)⊂H¹(Sⁿ-1)+L(ln+L)^1+v(Sⁿ-1),v>-1,q1.Furthermore,if v≥0,q1,we verify that block spaces B_q^0,v(Sⁿ-1) are proper subspaces of H¹(Sⁿ-1)

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On a Measure of Non-compactness for Some Classical Operators

David E. EDMUNDS, Alberto FIORENZA, Alexander MESKHI

Acta Mathematica Sinica. 2006, 22(6): 1847-1862. https://doi.org/10.1007/s10114-005-0674-6

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The measure of non-compactness is estimated from below for various operators,including the Hardy-Littlewood maximal operator,the fractional maximal operator and the Hilbert transform,all acting between weighted Lebesgue spaces.The identity operator acting between weighted Lebesgue spaces and also between the counterparts of these spaces with variable exponents is similarly analysed.These results enable the lack of compactness of such operators to be quantified.

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The N-Isometric Isomorphisms in Linear N -Normed C^*-Algebras

Chun-Gil PARK, Themistocles M. RASSIAS

Acta Mathematica Sinica. 2006, 22(6): 1863-1890. https://doi.org/10.1007/s10114-005-0878-9

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We prove the Hyers–Ulam stability of linear N-isometries in linear N -normed Banach modules over a unital C^*-algebra. The main purpose of this paper is to investigate N-isometric C^*-algebra isomorphisms between linear N-normed C^*-algebras, N-isometric Poisson C^*-algebra isomorphisms between linear N-normed Poisson C^*-algebras, N-isometric Lie C^*-algebra isomorphisms between linear N-normed Lie C^*-algebras, N-isometric Poisson J C^*-algebra isomorphisms between linear N-normed Poisson JC^*-algebras, and N-isometric Lie J C^*-algebra isomorphisms between linear N -normed Lie JC^*-algebras.Moreover, we prove the Hyers–Ulam stability of their N-isometric homomorphisms.

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Existence of Positive Solutions to Semipositone Singular Dirichlet Boundary Value Problems

Svatoslav STANĚK

Acta Mathematica Sinica. 2006, 22(6): 1891-1914. https://doi.org/10.1007/s10114-005-0843-7

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The paper presents the conditions which guarantee that for some positive value of μ there are positive solutions of the differential equation (Φ(x'))'+μQ(t,x,x')=0 satisfying the Dirichlet boundary conditions x(0)=x(T)=0.Here Q is a continuous function on the set [0,T ]×(0,∞)×(R\{0}) of the semipositone type and Q is singular at the value zero of its phase variables.

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Lie Bialgebras of Generalized Virasoro-like Type

Yue Zhu WU, Guang Ai SONG, Yu Cai SU

Acta Mathematica Sinica. 2006, 22(6): 1915-1922. https://doi.org/10.1007/s10114-005-0742-y

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In this paper,Lie bialgebra structures on generalized Virasoro-like algebras are studied.It is proved that all such Lie bialgebras are triangular coboundary.

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15 December 2006, Volume 22 Issue 6

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15 December 2006, Volume 22 Issue 6