Tropical Surface Singularities

In this paper, we study tropicalizations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional geometric type, show that they can generically have only finitely many singular points, and describe all possible locations of singular points. More precisely, we show that singular points must be either vertices, or generalized midpoints and barycenters of certain faces of singular tropical surfaces, and, in some case, there may be additional metric restrictions to faces of singular tropical surfaces.     

退化矩阵Liouville,Beta,Dirichlet分布

该文引进和讨论了退化矩阵Liouville分布，由此导出退化矩阵Beta分布、退化矩阵Dirichlet分布．推广了文献［1］关于退化Wishart分布和秩为1的退化矩阵Beta分布的结果。     

Virtual singular braids and links

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.     

Spectral properties of disjointly strictly singular operators

Spectral properties of strictly singular and disjointly strictly singular operators on Banach lattices are studied. We show that even in the case of positive operators, the whole spectral theory of strictly singular operators cannot be extended to disjointly strictly singular operators. However, several spectral properties of disjointly strictly singular operators are given.     

SINGULAR INTEGRAL OPERATORS ANDSINGULAR QUADRATURE OPERATORSASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS

1IntroductionSingUlarlintegralequations(SIEs)withCauchytypekernelsoftheformappearfrequelltlyinproblemsOfthetheoriesofelasticity.Heretheinputfunctionsa)b,f,l,aretheH5lder-continuousfunctionsfortheirvariables,Aisagivenconstant,anditisrequiredtofindthesolutionWintheclassho[1,2].Theclassicaltheoryoftheseequationsisrathercomplete[1,2].Inthepasttwentyyearsagreatdealofinteresthasarisenintheirnumericalsolution.VariouscollocationmethodsforSIEshaveappeared,forwhichsomereferencescanbefoundinthesurv…     

Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension.The set of singular points consists of some singular lines and some isolated singular points.It is proved that near a singular line or a singular point,each weak solution can be decomposed into two parts,a singular part and a regular part.The singular parts are some finite sum of particular solutions to some simpler equations,and the regular parts are bounded in some norms,which are slightly weaker than that in the Sobolev space H~2.     

Singular set of a Levi-flat hypersurface is Levi-flat

We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.     

在有限部分具有四个奇点二次系统的无穷远奇点

为了研究具有四个奇点二次系统的结构,本文研究了这类系统的无穷远奇点。 一般来说,二次系统的无穷远奇点是比较复杂的。但是在有限范围内具有四个奇点的次系统,它的无穷远奇点则具有某些特殊性质。为了方便,下面以E_2~4表示这种系统。     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

矩阵奇异值的极值性质

讨论了矩阵奇异值的问题,利用奇异值分解定理给出了奇异值的极值性质,并用其证明了矩阵论中关于奇异值的一些经典结论.     

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed.     

Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.      

The generalized center problem of degenerate resonant singular point

In this paper, generalized center condition and integrability of degenerate resonant singular point for a class of complex polynomial differential system were studied. The method was based on a homeomorphic transformation of the degenerate singular point into elementary singular point, which allows us to compute the generalized singular point quantities and determine the generalized center condition for the origin. In the end, we obtained the necessary and sufficient conditions of generalized complex center of degenerate resonant singular point.     

Riemannian Geometry of Conical Singular Sets

In this paper, we study a class of singular Riemannian manifolds. The singular set itself is a smooth manifold with a cone-like neighborhood. By imposing a reasonable convergence condition on the metric, we can determine the local geometrical structure near the singular set. In general, the curvature near the singular set is unbounded. We prove that a bounded curvature assumption would have a strong implication on the geometrical and topological structures near the singular set. We also establish the Gauss–Bonnet–Chern formula, which can be applied to the study of singular Eistein 4-manifolds.     

快变量空间中的同宿轨道分支

本文考虑奇摄动问题的位于快变量空间中的奇异同宿轨道的保存和周期轨道分支问题.文中关于奇异同宿轨道保存的结论推广了一些已知的结果,而周期轨道产生于奇异同宿轨道的分支则提供了一种新的分支类型.     

On singular formal deformations

We discuss the notion of singular formal deformation in algebraic setup. Such deformations show up in both finite and infinite dimensional structures. It turns out that there is a stronger version of singular deformation—called essentially singular—which arises from a singular curve in the base of the versal deformation.     

非可换的奇异l-群

通过建立奇异元以及奇异l-群的刻划，研究了一般非可换的奇异l-群的性质及相关的结构．     

Interlacing inequalities for generalized singular values

A generalization of the concept of singular value of a complex matrix introduced by J. F. Maitre and N. Huu Vinh is considered. We give the relation between the singular values of two matrices and the singular values of their sum, and we find Cauchy interlacing-type inequalities for some of these singular values.