Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension O of a two-dimensional local regular ring (R,m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X=BlJ(R) has some point Q whose local ring OX,Q is analytically isomorphic to O.     

探寻等价无穷小关系的几种方法

介绍利用导数建立等价无穷小关系的三个方法,并举例说明．     

Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture

Summary We show that Walter Neumann's strengthened form of Hanna Neumann's conjecture on the best possible upper bound for the rank of the intersection of two subgroups of a free group is equivalent to a conjecture on the best possible upper bound for the number of edges in a bipartite graph with a certain weak symmetry condition. We illustrate the usefulness of this equivalence by deriving relatively easily certain previously known results.Oblatum 30-VIII-1993     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes

The equivalence of ergodicity and weak mixing for general infinitely divisible processes is proven. Using this result and [9], simple conditions for ergodicity of infinitely divisible processes are derived. The notion of codifference for infinitely divisible processes is investigated, it plays the crucial role in the proofs but it may be also of independent interest.     

ANALYSIS OF STRONG SINGULARITIES FOR NONLINEAR HYPERBOLIC EQUATIONS

In this paper we review our some results about the strongly singular (discontinuous,measure, or delta function) problems for nonlinear hyperbolic equations.     

On the Frobenius manifolds for cusp singularities

We show that the Frobenius manifold associated to the pair of a cusp singularity and its canonical primitive form is isomorphic to the one constructed from the Gromov–Witten theory for an orbifold projective line with at most three orbifold points. We also calculate the intersection form of the Frobenius manifold.     

A note on resolution of rational and hypersurface singularities

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.

     

一类奇异p-Laplace方程无穷多解的存在性

讨论了一类具有奇异系数的p-Laplace问题-Δpu-μ|u|u|x|p=u|x|tu+λuq-2u,x∈Ω,u=0,x∈Ω无穷多解的存在性,其中N≥3,Ω是RN中一有界光滑区域,0∈Ω,Δpu=-div(|▽u|p-2▽u),0≤μ<μ=(N-p)ppp,10,1相似文献   

On the resolution of 3-dimensional terminal singularities

We show that there is at most one nonrational exceptional divisor with discrepancy 1 over a three-dimensional terminal point of type cD. If such a divisor exists, then it is birationally isomorphic to the surface ¹ × C, where C is a hyperelliptic (for g(C) > 1) curve.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 127–140.Original Russian Text Copyright © 2005 by D. A. Stepanov.This revised version was published online in April 2005 with a corrected issue number.     

Simple singularities and topology of symplectically filling 4-manifold

Topological restrictions of symplectically filling 4-manifolds of links around simple singularities are studied by using the Seiberg-Witten monopole equations. In particular, the intersection form of minimal symplectically filling 4-manifolds of the singularity of type E ₈ is determined. Moreover, for the case of simply elliptic singularities, similar restrictions are obtained. In the proof, a vanishing theorem of the Seiberg-Witten invariant is discussed. Received: June 9, 1998.     

A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities

We apply the Tian-Todorov method, proving the Bogomolov
smoothness theorem (for deformations of compact Kähler manifolds) to deformations of the regular part of a Stein space with a finite number of isolated singular points. By the argument based on the Hodge structure on a strongly pseudo-convex Kähler domain or on a punctured Kähler space, we obtain an unobstructed subspace of the infinitesimal deformation space.

     

Universal Abelian covers of rational surface singularities and multi-index filtrations

In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series.     

Foliations of isonergy surfaces and singularities of curves

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.      

集值映射的Nash平衡点集的通有稳定性

本文讨论了集值映射的Nash平衡点的存在及平衡点集的通有稳定性,得到大多数的集值映射的Nash平衡点集是稳定的。     

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

In this paper, the stress analysis of moderately thick functionally graded sector plate is developed for studying the singularities in vicinity of the vertex and effects of boundary layer. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations. These equations are solved analytically and the effects of boundary layer function on stress components are shown. Also, the singularities of shear force, moment resultants and boundary layer function are discussed for both salient (α?180)$(\alpha ?180)$ and re-entrant (α>180)$(\alpha >180)$ sectorial plates. In order to verify the accuracy of the results, the governing equations are also solved using differential quadrature method (DQM). By comparing the results of exact method with DQM, a good agreement can be seen.     

Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arborescence of order n > 1. This is a particular case of Sumner's conjecture: every tournament of order 2n − 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n − 5)/2 contains every oriented tree of order n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244–256, 2000     

Smale's mean value conjecture and the coefficients of univalent functions

We study Smale's mean value conjecture and its connection with the second coefficients of univalent functions. We improve the bound on Smale's constant given by Beardon, Minda and Ng.