Acta Mathematica Sinica

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Approximate Augmented Lagrangian Functions and Nonlinear Semidefinite Programs

X. X. Huang, K. L. Teo, X. Q. Yang

Acta Mathematica Sinica. 2006, 22(5): 1283-1296. https://doi.org/10.1007/s10114-005-0702-6

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In this paper,an approximate augmented Lagrangian function for nonlinear semidefinite programs is introduced.Some basic properties of the approximate augmented Lagrange function such as monotonicity and convexity are discussed.Necessary and sufficient conditions for approximate strong duality results are derived.Conditions for an approximate exact penalty representation in the framework of augmented Lagrangian are given.Under certain conditions,it is shown that any limit point of a sequence of stationary points of approximate augmented Lagrangian problems is a KKT point of the original semidefinite program and that a sequence of optimal solutions to augmented Lagrangian problems converges to a solution of the original semidefinite program.

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Global Existence and Nonexistence for a Strongly Coupled Parabolic System with Nonlinear Boundary Conditions

You Peng CHEN, Chun Hong XIE

Acta Mathematica Sinica. 2006, 22(5): 1297-1304. https://doi.org/10.1007/s10114-005-0654-x

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This paper deals with the strongly coupled parabolic system u_t= v^mΔu,v_t= u_nΔv,(x,t)∈Ω×(0,T) subject to nonlinear boundary conditions ∂u/∂η=u^αv^p,∂v/∂η=u^qv^β,(x,t)∈∂Ω×(0,T),where ΩR^N is a bounded domain,m,n are positive constants and α,β,p,q are nonnegative constants.Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established.It is proved that the positive solution of the above problem exists globally if and only if α<1,β<1 and (m+p)(n+q)≤(1-α)(1-β).

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One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

Jin Ho KWAK, Ju Mok OH

Acta Mathematica Sinica. 2006, 22(5): 1305-1320. https://doi.org/10.1007/s10114-005-0752-9

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A graph G is one-regular if its automorphism group Aut(G) acts transitively and semiregularly on the arc set.A Cayley graph Cay(Γ,S) is normal if Γ is a normal subgroup of the full automorphism group of Cay(Γ,S).Xu,M.Y.,Xu,J.(Southeast Asian Bulletin of Math.,25,355-363 (2001)) classified one-regular Cayley graphs of valency at most 4 on finite abelian groups.Marušič,D.,Pisanski,T.(Croat.Chemica Acta,73,969-981 (2000)) classified cubic one-regular Cayley graphs on a dihedral group,and all of such graphs turn out to be normal.In this paper,we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut(G) are cyclic.A classification of the same kind of graphs of valency 6 is also discussed.

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Shadowing and Inverse Shadowing for C¹ Endomorphisms

Yu Jun ZHU, Jin Lian ZHANG, Lian Fa HE

Acta Mathematica Sinica. 2006, 22(5): 1321-1328. https://doi.org/10.1007/s10114-005-0739-6

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In this paper,we consider the shadowing and the inverse shadowing properties for C¹ endomorphisms.We show that near a hyperbolic set a C¹ endomorphism has the shadowing property,and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods.Moreover,each of these shadowing properties is also "uniform" with respect to C¹ perturbation.

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The Exceptional Set in the Two Prime Squares and a Prime Problem

Ming Qiang WANG, Xian Meng MENG

Acta Mathematica Sinica. 2006, 22(5): 1329-1342. https://doi.org/10.1007/s10114-005-0701-7

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In this paper we prove that,with at most $O{\left({N^{{\frac{5} {{12}}+\in }} } \right)}$ exceptions,all positive odd integers n≤N with n≡0 or 1(mod 3) can be written as a sum of a prime and two squares of primes.

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The Frattini Subalgebra of Restricted Lie Superalgebras

Liang Yun CHEN, Dao Ji MENG, Yong Zheng ZHANG

Acta Mathematica Sinica. 2006, 22(5): 1343-1356. https://doi.org/10.1007/s10114-005-0670-x

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In the present paper,we study the Frattini subalgebra of a restricted Lie superalgebra (L,[p]).We show first that if L=A₁⊕A₂⊕…⊕A_nAn,then ø_p(L)=ø_p(A₁)+ø_p(A₂)+…+ø_p(A_n),where each A_i is a p-ideal of L.We then obtain two results: F(L)=ø(L)=J(L)=L⁽¹⁾ if and only if L is nilpotent;F_p(L) and F(L) are nilpotent ideals of L if L is solvable.In addition,necessary and sufficient conditions are found for ø_p-free restricted Lie superalgebras.Finally,we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.

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Convergence to Diffusion Waves for Nonlinear Evolution Equations with Ellipticity and Damping, and with Different End States

Chang Jiang ZHU, Zhi Yong ZHANG, Hui YIN

Acta Mathematica Sinica. 2006, 22(5): 1357-1370. https://doi.org/10.1007/s10114-005-0728-9

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In this paper,we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

with initial data

where α and ν are positive constants such that α<1,ν<4α(1-α).Under the assumption that

is sufficiently small,we show the global existence of the solutions to Cauchy problem (E) and (Ⅰ) if the initial data is a small perturbation.And the decay rates of the solutions with exponential rates also are obtained.The analysis is based on the energy method.

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Global Classical Solutions to the Equations of Two-dimensional MHD Transverse Flow

Yi Peng SHI

Acta Mathematica Sinica. 2006, 22(5): 1371-1384. https://doi.org/10.1007/s10114-005-0700-8

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A new method for constructing the symmetric form for the hyperbolic systems is introduced.Then a new symmetric form of the equations of MHD transverse flow is constructed by adding an additional equation.With this new form,we obtain the local existence of smooth solution including the case that the initial density may tend to the vacuum state at infinity.Furthermore,the uniformly a priori estimation for the classical solutions is established and the global smooth solutions for a kind of initial data are obtained.

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The Exterior Tricomi Problem for Generalized Mixed Equations with Parabolic Degeneracy

Guo Chun WEN

Acta Mathematica Sinica. 2006, 22(5): 1385-1398. https://doi.org/10.1007/s10114-005-0707-1

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This paper deals with the exterior Tricomi problem for generalized mixed equations with parabolic degeneracy.Firstly the representation of solutions of the problem for the equations is given,and then the uniqueness and existence of solutions are proved by a new method.

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Explicit Reciprocity Law for Lubin-Tate Formal Groups

Lei CAO

Acta Mathematica Sinica. 2006, 22(5): 1399-1412. https://doi.org/10.1007/s10114-005-0663-9

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In this article,using Fontaine's ΦΓ-module theory,we give a new proof of Coleman's explicit reciprocity law,which generalizes that of Artin-Hasse,Iwasawa andWiles,by giving a complete formula for the norm residue symbol on Lubin-Tate groups.The method used here is different from the classical ones and can be used to study the Iwasawa theory of crystalline representations.

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α-Transience and α-Recurrence of Right Processes

Min Zhi ZHAO, Jian Gang YING

Acta Mathematica Sinica. 2006, 22(5): 1413-1424. https://doi.org/10.1007/s10114-005-0667-5

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In this article,we mainly discuss some potential theory in the framework of right Markov processes.We introduce the concept of α-excessive function,α-recurrence and α-transience for right processes with α≤0,and give a thorough investigation.

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Global Existence of a Shock for the Supersonic Flow Past a Curved Wedge

Hui Cheng YIN

Acta Mathematica Sinica. 2006, 22(5): 1425-1432. https://doi.org/10.1007/s10114-005-0611-8

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This note is devoted to the study of the global existence of a shock wave for the supersonic flow past a curved wedge.When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value,we show that a shock attached at the wedge will exist globally.

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The Unfolding of Equivariant Bifurcation Problems with Two Types of State Variables in the Presence of Parameter Symmetry

Deng Lan CUI, Yang Cheng LI

Acta Mathematica Sinica. 2006, 22(5): 1433-1440. https://doi.org/10.1007/s10114-005-0712-4

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The unfolding of equivariant bifurcation problems with two types of state variables under the action of group K(Γ,Δ) is discussed by using DA-algebraic tools.One of the main results is the equivariant versal unfolding theorem.

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Asymptotics for the Korteweg-de Vries-Burgers Equation

Nakao HAYASHI, Pavel I. NAUMKIN

Acta Mathematica Sinica. 2006, 22(5): 1441-1456. https://doi.org/10.1007/s10114-005-0677-3

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We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation
$u_{t}+uu_{x}-u_{{xx}}+u_{{xxx}}=0,x \in {\text{R}},t >0.$
We are interested in the large time asymptotics for the case when the initial data have an arbitrary size.We prove that if the initial data u₀∈H^s (R)∩L¹ (R),where s>1/2 then there exists a unique solution u (t,x)∈C^∞ ((0,∞);H^∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation,which has asymptotics
$u{\left(t \right)}=t^{{-\frac{1} {2}}} f_{M} {\left({{\left(\cdot \right)}t^{{-\frac{1} {2}}} } \right)}+o{\left({t^{{-\frac{1} {2}}} } \right)}$
as t→∞,where f_M is the self-similar solution for the Burgers equation.Moreover if xu₀ (x)∈L¹ (R),then the asymptotics are true
$u{\left(t \right)}=t^{{-\frac{1} {2}}} f_{M} {\left({{\left(\cdot \right)}t^{{-\frac{1} {2}}} } \right)}+O{\left({t^{{-\frac{1} {2}-\gamma }} } \right)},$
where $\gamma \in {\left({0,\frac{1} {2}} \right)}.$

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Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L ² Initial Data

Shang Bin CUI, Dong Gao DENG, Shuang Ping TAO

Acta Mathematica Sinica. 2006, 22(5): 1457-1466. https://doi.org/10.1007/s10114-005-0710-6

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In this paper we study solvability of the Cauchy problem of the Kawahara equation $\partial _{t} u+au\partial _{x} u+\beta \partial ^{3}_{x} u+\gamma \partial ^{5}_{x} u=0$ with L² initial data.By working on the Bourgain space X^r,s(R²) associated with this equation,we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r(R) and -1<r≤0.This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L²(R).

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Application of Hilbert’s Projective Metric on Symmetric Cones

Khalid KOUFANY

Acta Mathematica Sinica. 2006, 22(5): 1467-1472. https://doi.org/10.1007/s10114-005-0755-6

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Let Ω be a symmetric cone.In this note,we introduce Hilbert's projective metric on Ω in terms of Jordan algebras and we apply it to prove that,given a linear invertible transformation g such that g(Ω)=Ω and a real number p,|p|>1,there exists a unique element x∈Ω satisfying g(x)=x^p.

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Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

Zong Xia LIANG

Acta Mathematica Sinica. 2006, 22(5): 1473-1480. https://doi.org/10.1007/s10114-005-0816-x

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In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral
$X_{t} {\left(w \right)}=\eta {\left(w \right)}+{\int_0^t {a_{s} X_{s} {\left(w \right)}d} }{\text{\bf W}}_{s}+{\int_0^t {b_{s} X_{s} {\left(w \right)}ds,t \in {\left[ {0,1} \right]}} },$
where (W_s) is the canonical Wiener process defined on the standard Wiener space (W,H,μ),a is non-smooth and adapted,but η and b may be anticipating to the filtration generated by (W_s).The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense,in the existing literature.The idea is to approach the non-smooth diffusion coefficient a by smooth ones.

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SD-splittings for Some Bordered 3-Manifolds

Feng Chun LEI, Yang LI

Acta Mathematica Sinica. 2006, 22(5): 1481-1484. https://doi.org/10.1007/s10114-005-0732-0

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In the present paper,we characterize the bordered 3-manifolds with genus 1 D-splittings and SD-splittings.We also describe a minimal genus SD-splitting for a handlebody and surface×I.

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Inverse Scattering for a Schrödinger Operator with a Repulsive Potential

Fran cois NICOLEAU

Acta Mathematica Sinica. 2006, 22(5): 1485-1492. https://doi.org/10.1007/s10114-005-0767-2

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We consider a pair of Hamiltonians (H,H₀) on L²(Rⁿ),where H₀=p²-x² is a Schrödinger operator with a repulsive potential,and H=H₀+V (x).We show that,under suitable assumptions on the decay of the electric potential,V is uniquely determined by the high energy limit of the scattering operator.

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Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

Veli B. SHAKHMUROV

Acta Mathematica Sinica. 2006, 22(5): 1493-1508. https://doi.org/10.1007/s10114-005-0764-5

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This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E₀,E.Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E₀ and E.In particular,the most regular class of interpolation spaces E_α between E₀,E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued L_p,γ spaces is proved.These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal L_p,γ regularity and R-positivity uniformly with respect to these parameters.

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Hörmander’s Multipliers for the Weighted Herz-type Hardy Spaces

Ming Yi LEE(Ming-Yi LEE), Chin Cheng LIN(Chin-Cheng LIN), Yu Tien LIN(Yu-Tien LIN)

Acta Mathematica Sinica. 2006, 22(5): 1509-1518. https://doi.org/10.1007/s10114-005-0801-4

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In this article,we apply the molecular characterization of the weighted Hardy space developed by the first two authors to show the boundedness of Hörmander multiplier on the weighted Herz-type Hardy spaces H

(|x|t;|x|t) and HK₂^α,p (|x|t;|x|t).

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Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

M. KADAKAL, O. Sh. MUKHTAROV

Acta Mathematica Sinica. 2006, 22(5): 1519-1528. https://doi.org/10.1007/s10114-005-0751-x

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In this paper,discontinuous Sturm-Liouville problems,which contain eigenvalue parameters both in the equation and in one of the boundary conditions,are investigated.By using an operator-theoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions.We modify some techniques of [Fulton,C.T.,Proc.Roy.Soc.Edin.77 (A),293-308 (1977)],[Walter,J.,Math.Z.,133,301-312 (1973)] and [Titchmarsh,E.C.,Eigenfunctions Expansion Associated with Second Order Differential Equations I,2nd edn.,Oxford Univ.Pres,London,1962],then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.

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System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

Xie Ping DING

Acta Mathematica Sinica. 2006, 22(5): 1529-1538. https://doi.org/10.1007/s10114-005-0671-9

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A new class of locally finite continuous topological spaces (for short,locally FC-spaces) and a class of system of generalized vector quasi-equilibrium problems are introduced.By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author,a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces.By applying our equilibrium existence theorem of generalized game,some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces.These theorems improve,unify and generalize many known results in the literatures.

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On the Semivalues and the Least Square Values Average Per Capita Formulas and Relationships

Irinel DRAGAN

Acta Mathematica Sinica. 2006, 22(5): 1539-1548. https://doi.org/10.1007/s10114-005-0818-8

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In this paper,it is shown that both the Semivalues and the Least Square Values of cooperative transferable utilities games can be expressed in terms of n² averages of values of the characteristic function of the game,by means of what we call the Average per capita formulas.Moreover,like the case of the Shapley value earlier considered,the terms of the formulas can be computed in parallel,and an algorithm is derived.From these results,it follows that each of the two values mentioned above are Shapley values of games easily obtained from the given game,and this fact gives another computational opportunity,as soon as the computation of the Shapley value is efficiently done.

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Inductive Rings and Systems of Diophantine Equations

Rong Fang BIE, Shi Qiang WANG

Acta Mathematica Sinica. 2006, 22(5): 1549-1556. https://doi.org/10.1007/s10114-005-0834-8

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In this paper,by using model-theoretic methods,it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers.These inductive rings are not fields,and every element of them is a sum of 4 cubes and a sum of 3 squares.Also some of them satisfy the Goldbach conjecture and some others don't.

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Characterization of Totally Geodesic Totally Real 3-dimensional Submanifolds in the 6-sphere

Miroslava ANTIĆ, Mirjana DJORIĆ, Luc VRANCKEN

Acta Mathematica Sinica. 2006, 22(5): 1557-1564. https://doi.org/10.1007/s10114-005-0798-8

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In this paper,we study Lagrangian submanifolds of the nearly Kaehler 6-sphere.We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold.Our pinching result improves a previous result obtained by H.Li.

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M-Cross-Validation in Local Median Estimation

Ying YANG

Acta Mathematica Sinica. 2006, 22(5): 1565-1582. https://doi.org/10.1007/s10114-005-0855-3

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M-cross-validation criterion is proposed for selecting a smoothing parameter in a nonparametric median regression model in which a uniform weak convergency rate for the M-cross-validated local median estimate,and the upper and lower bounds of the smoothing parameter selected by the proposed criterion are established.The main contribution of this study shows a drastic difference from those encountered in the classical L_2-,L_1- cross-validation technique,which leads only to the consistency in the sense of the average.Obviously,our results are novel and nontrivial from the point of view of mathematics and statistics,which provides insight and possibility for practitioners substituting maximum deviation for average deviation to evaluate the performance of the data-driven technique.

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The Total Embedding Distributions of Cacti and Necklaces

Yi Chao CHEN, Yan Pei LIU, Yan Pei LIU, Tao WANG

Acta Mathematica Sinica. 2006, 22(5): 1583-1590. https://doi.org/10.1007/s10114-005-0856-2

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Total embedding distributions have been known for only a few classes of graphs.In this paper the total embedding distributions of the cacti and the necklaces are obtained.Furthermore we obtain the total embedding distributions of all graphs with maximum genus 1 by using the method of this paper.

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Obstruction Theory and Coincidences in Positive Codimension

Daciberg GONC ALVES, Jerzy JEZIERSKI, Peter WONG

Acta Mathematica Sinica. 2006, 22(5): 1591-1602. https://doi.org/10.1007/s10114-005-0797-9

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Let f,g:X→Y be two maps between closed manifolds with dim X≥dim Y=n≥3.We study the primary obstruction o_n(f,g) to deforming f and g to be coincidence free on the n-th skeleton of X.We give examples for which obstructions to deforming f and g to be coincidence free are detected by o_n(f,g).

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15 October 2006, Volume 22 Issue 5

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15 October 2006, Volume 22 Issue 5