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。     

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.     

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.     

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.     

Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

Singular random matrix decompositions: distributions

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.     

Isomonodromic confluence of singular points

We consider the problem of confluence of singular points under isomonodromic deformations of linear systems. We prove that a system with irregular singular points is a result of isomonodromic confluence of singular points with minimal Poincaré ranks, i.e., of singular points whose Poincaré rank does not decrease under gauge transformations.     

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

General stationary iterative methods with a singular matrix M for solving range‐Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range‐Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n,K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8,K) and show that every locally singular hyperplane of DQ(8,K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.     

Sensitivity analysis for the generalized singular value decomposition

In this paper, we discuss the sensitivity of multiple nonzero finite generalized singular values and the corresponding generalized singular matrix set of a real matrix pair analytically dependent on several parameters. From our results, the partial derivatives of multiple nonzero singular values and their left and right singular vector matrices are obtained.Copyright © 2012 John Wiley & Sons, Ltd.     

Balanced split sets and Hamilton�CJacobi equations

We study the singular set of solutions to Hamilton?CJacobi equations with a Hamiltonian independent of u. In a previous paper, we proved that the singular set is what we called a balanced split locus. In this paper, we find and classify all balanced split loci, identifying the cases where the only balanced split locus is the singular locus, and the cases where this does not hold. This clarifies the relationship between viscosity solutions and the classical approach of characteristics, providing equations for the singular set. Along the way, we prove more structure results about the singular sets.     

Solvability,controllability, and observability of singular systems

The problems of solvability, controllability, and observability for the singular systemKx(t)=Ax(t)+Bu(t) are studied, whereK is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable. The classical theories of matrix pencils are first related to the solvability of singular systems. Then, the concepts of reachability, controllability, and observability of regular systems are extended to singular systems. Finally, the set of reachable states is described. The proposed matrix conditions for testing the controllability and observability of singular systems are simple and always feasible.     

A singular function with a non-zero finite derivative

This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

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.      

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

Global structure for a class of dynamical systems

In this paper, we shall consider the global structure of positive bounded systems on the plane which have m singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have at least m−1 connecting orbits; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set. Each of other orbits tends to a singular point as t→+∞, and approaches to the infinity as t→−∞.     

A finite element method dealing the singular points with a cut-off function

We consider an elliptic partial differential equation with input function inH ^-1. We assume that the type of singular part of solution is known and the solution is expressed as a sum of regular and singular term. We use a cut-off function to isolate the singular part and pose a variational problem to find the regular part of the solution. Then we have the solution by adding the regular solution and singular part. Error analysis and some example with computation will be given.