GEOMETRY OF COMPLETE HYPERSURFACES EVOLVED BY MEAN CURVATURE FLOW

Some geometric behaviours of complete solutions to mean curvature flow before the singularities occur are studied. The author obtains the estimates of the rate of the distance between two fixed points and the derivatives of the second fundamental form. By use of a new maximum principle, some geometric properties at infinity are obtained.     

关于(f,τ) -相容Hopf代数对(B,H)

该文定义了(f,τ) -相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H -余模范畴^HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数$B$的乘法,构造出Yetter-Drinfeld范畴中^H_HYD的Hopf代数.     

Aleksandrov maximum principle and bony maximum principle for parabolic equations

The author simplifies the proof of Aleksandrov maximum principle for parabolic equations given by Krylov and obtains finer results. He further proves Bony maximum principle for parabolic equations by using the above results.     

Characterization of Solid Cones

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.     

与直径和围长有关的图的最大亏格

利用图的直径和围长来研究图的最大亏格的下界,得到了如下结果：设G是直径为d的简单图,若G的围长不小于d（其中d为不小于3的整数）,则ξ（G）≤2,即γM（G）≥1/2β（G）-1．而且,在这种意义下,所得到的界是最好的．     

Affine Structures on a Ringed Space and Schemes

The author first introduces the notion of affine structures on a ringed space and then obtains several related properties. Affine structures on a ringed space, arising from complex analytical spaces of algebraic schemes, behave like differential structures on a smooth manifold.     

Twistor Spinors and Quasi-twistor Spinors

The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.     

A Gradient Estimate for Solutions of the Heat Equation II

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.     

SINGULAR INTEGALS IN SEVERAL COMPLEX VARIABLES (IV)-THE DERIVATIVE OF CAUCHY INTEGRAL ON SPHERE

The aim of this paper is to study the boundary properties of the derivative of Cauchy integral on the sphere. Using the concept of the Hadamard principal value of singular integral of higher order introduced in [1], the author obtains the corresponding Plemelj formula expressed by Hadamard principal value.     

Singular numbers of the imbedding operators for certain classes of analytic and harmonic functions

One obtains formulas of the strong asymptotics of the s-numbers of the imbedding operators of the weighted classes of analytic and harmonic functions into the weighted spaces L₂ on the interior subdomains.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 56–66, 1984.The author expresses his gratitude to M. Z. Solomyak for his constant support.     

遗传覆盖性质和选择理论

通过引入局部下半连续集值映射的概念,建立了遗传覆盖性质与多种选择之间的一些重要联系,说明了选择构造本身与原象空间的分离性无关.     

Necessary conditions of the minimum in an impulse optimal control problem

The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

具有两个不同Ricci主曲率的局部共形平坦Riemann流形

得到了具有常m阶Schouten曲率与两个不同Schouten主曲率(或者等价地,两个不同Ricci主曲率)的完备局部共形平坦Riemann流形的分类结果.作为应用,得到了若干Schouten张量的pinching性质.     

INFINITE ELEMENT METHOD FOR PROBLEMS ON UNBOUNDED AND MULTIPLY CONNECTED DOMAINS

1 IntroductionDifferent kinds of numerical n1etl1ods llavc been apPlied to so1lle proble1l1s o11 exteriordolllai11s successf1lly, e.g. tlle bou11dary element metl1od, the absorbillg boundary condi-tion lnethod, the spectrunl 11letllod, a11d the i11fillite eleIue1lt 1nethod. Tlle il1finite elementllletllod llas beell applied to tl1e Laplace equatioll['], the Stokes equation[']['], the plane elastic-ity systeln['1, alld the Heln1ho1tz equatioll[n. We study tl1e infinite ele111ent llletl1od tbr…     

Z[(1 21~(1/2))/2]上正定幺模格的分类(英文)

本文综合利用邻格方法及Siegelmass公式对Z[(1 2 1) /2 ]上秩 4的正定幺模格实现了分类 ,得到了gen(I4 )的类数为 9,偶模格的类数为 3,并且给了代表格     

On the geometry of polynomial dynamical systems

In this paper, the author performs a global qualitative study of plane polynomial dynamical systems and suggests a new geometric approach to solving the sixteenth Hilbert problem on the maximum number and mutual location of their limit cycles in two special cases of such systems. First of all, using the geometric properties of four parameters rotating the vector field of a new canonical system constructed in the paper, the author proposes the proof of his early conjecture, which asserts that the maximum number of limit cycles of an arbitrary quadratic system is equal to 4, and their location (3: 1) is uniquely possible [4]. Then using the same geometric approach, the author solves the primary problem for the polynomial Liénard system (in this special case, it is considered as the thirteenth Smale problem), and generalizing the obtained results, the author formulates the theorem on the maximum number of limit cycles enclosing one singular point in the case of a polynomial system. Moreover, applying the Wintner-Perko termination principle for multiple limit cycles, the author develops an alternative approach to solving the sixteenth Hilbert problem, and using this approach, the author completes the global qualitative study of a general cubic Liénard system having three singular points in the finite part of the plane. In conclusion, the author discusses one more known approach to solving the sixteenth Hilbert problem. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

不同步交易模型的参数估计

对金融证券中高频数据的不同步交易模型,讨论了参数的矩估计,极大似然估计及估计的有关性质.     

Asymptotics of the spectra of variational problems on the solutions of an elliptic equation in a domain with a piecewise-smooth boundary

In a bounded domain with a piecewise-smooth boundary one considers the ratio of differential quadratic forms on the solutions of a homogeneous, strongly elliptic equation. One obtains the principal term of the asymptotics of the spectrum of the compact operator defined by this ratio.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 179–183, 1985.The author is sincerely grateful to M. Sh. Birman for guidance.     

SOME TOPOLOGICAL PROPERTIES OF BENZENOID Sn,Tn-ISOMERS WITH n-RADICAL CONNECTION

We consider a series of benzenoid isomers obtained by attaching fragments to ann-radical. Some of their topological properties, such as the number of Kekulé patterns and the maximum number of aromatic π-sextets are established. Research of this author is supported in part by the RDG grant of the Pennsylvania State University.     

Asymptotics of the spectra of variational problems on the solutions of a homogeneous elliptic equation,under the presence of a constraint on part of the boundary

One obtains the principal term of the asymptotics of the spectrum of the ratio of two differential quadratic forms, considered in a bounded domain on the solutions of a homogeneous, regularly elliptic equation, under the presence of additional conditions on part of the boundary. This condition has the form of a pseudodifferential constraint and it is imposed on the Dirichlet data of the solutions. On the leading symbol of the corresponding pseudodifferential operator on the boundary, one imposes a nonsingularity conditioa.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 84–97, 1984.In conclusion, the author expresses her gratitude to her scientific adviser M. Sh. Birman for his assistance and also to M. Z. Solomyak for his interest in the paper and for a series of useful consultations.