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.     

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.     

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.     

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.     

Maximal and extremal singular graphs

A graph G is singular if the nullspace of its adjacency matrix is nontrivial. Such a graph contains induced subgraphs called singular configurations of nullity 1. We present two algorithms. One is for the construction of a maximal singular nontrivial graph G containing an induced subgraph, which is a singular configuration with the support of a vector in its nullspace as in that of G. The second is for the construction of a nut graph, a graph of nullity one whose null vector has no zero entries. An extremal singular graph of a given order, with the maximal nullity and support, has a nut graph as a maximal singular configuration.     

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.     

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.     

Solvability of a singular boundary value problem in a domain of the Lobachevskii plane

Under study is some singular elliptic boundary value problem in a domain of the Lobachevskii plane with an isolated boundary point. We introduce the new function spaces that coincide with the spaces of Sobolev-Nikol’skii-Besov type outside the singular point, and the concept of σ-trace at the singular point. The main result is a proof that the singular boundary value problem is uniquely solvable.     

Sufficiency of Lakshmibai–Sandhya Singularity Conditions for Schubert Varieties

We establish one direction of a conjecture by Lakshmibai and Sandhya which describes combinatorially the singular locus of a Schubert variety. We prove that the conjectured singular locus is contained in the singular locus.     

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.     

Robust finite-time H∞ control of singular stochastic systems via static output feedback

This paper addresses the problem of robust finite-time stabilization of singular stochastic systems via static output feedback. Firstly, sufficient conditions of singular stochastic finite-time boundedness on static output feedback are obtained for the family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Then the results are extended to singular stochastic H_∞ finite-time boundedness for the class of singular stochastic systems. Designed algorithm for static output feedback controller is provided to guarantee that the underlying closed-loop singular stochastic system is singular stochastic H_∞ finite-time boundedness in terms of strict linear matrix equalities with a fixed parameter. Finally, an illustrative example is presented to show the validity of the developed methodology.     

奇异脉冲系统的奇异H_∞控制

主要研究奇异脉冲系统的奇异H∞控制问题.当系统不满足正则条件时,给出奇异脉冲系统的奇异H∞控制问题可解的充分条件,控制律使得闭环系统在保证内稳定的条件下达到干扰衰减.     

Isochronicity Problem of Higher-Order Singular Point for Polynomial Differential Systems

Due to the difficulty, the isochronicity problems with respect to higher-order singular point (or degenerate singular point) of polynomial differential systems are far from being solved. The calculation of period constants is an effective way to find necessary conditions for isochronicity. In this paper, by means of a homeomorphic transformation, higher-order singular point is transferred into the origin. At the same time, a new recursive algorithm to compute period constants at the origin of the transformed system is deduced which is easy to realize with the computer algebraic system such as MATHEMATICA or MAPLE. Finally, to illustrate the effectiveness of our algorithm, the pseudo-isochronous center conditions of higher-order singular point for a class of septic system are investigated. Our work is new in terms of research about the isochronicity problem of higher-order singular point and consists of the existing results related to the origin as a special case when it is an elementary singular point.     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

On the equivalence of singular point quantities and the integrability of a fine critical singular point

In this paper, the equivalence of singular point quantities and the integrability of a fine critical singular point are discussed. For an extensive class of complex autonomous differential systems, the theoretical basis for the methods of using formal series and integrating factors to calculate the singular point quantities for the origin (a fine critical singular point) and judge its integrability is obtained. Two recursion formulas, which are much simpler than focal values for computation because the operation is rational, for computing singular point quantities are introduced.     

基于谱变换的奇异二阶系统解耦研究

针对奇异二阶系统的解耦问题,提出了一种基于谱变换的解耦方法.首先分析了质量矩阵非奇异的解耦条件,提出一种非奇异二阶同谱对角系统的构造解耦方法,然后引入谱变换将首系数转换为非奇异的,利用非奇异的构造方法来构造同谱对角系统,最后对解耦系统进行还原,从而实现奇异二阶系统的解耦.数值试验证明该方法确实有效.     

Singular mean-field control games

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

On necessary and sufficient conditions for near-optimal singular stochastic controls

In this paper we discuss the necessary and sufficient conditions for near-optimal singular stochastic controls for the systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland’s variational principle and some delicate estimates of the state and adjoint processes. It is well known that optimal singular controls may fail to exist even in simple cases. This justifies the use of near-optimal singular controls, which exist under minimal conditions and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to choose suitable ones, that are convenient for implementation. This result is a generalization of Zhou’s stochastic maximum principle for near-optimality to singular control problem.