Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in characteristic 3

We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic three.     

Log Structures on Generalized Semi-Stable Varieties

In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structure on this family of morphisms.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Decomposition of Log Crepant Birational Morphisms between Log Terminal Surfaces

We prove that every log crepant birational morphism between log terminal surfaces can be decomposed into log flopping type divisorial contraction morphisms and log blowdowns. Repeating these two kinds of contractions, we reach a minimal log minimal surface from any log minimal surface.     

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.

     

On initial value problems related to p-Laplacian and pseudo-Laplacian

Summary We establish some qualitative properties in the sense of weak singularity and super singularity for a certain system of two nonlinear differential equations related to the radially symmetric solution of p-Laplacian and pseudo-Laplacian problems. For the transformed system of differential equations we carry out the classification in the sense of weak singularity, singularity and super singularity. The choice of initial values at the point of singularity for correct settings of Cauchy problem is also considered.     

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.     

一类三次系统的细奇点

本文研究了具有一个高阶奇点的系统（１,２）,通过在实系统中引入奇点量的概念使焦点量和鞍点量得到统一,并且计算了系统（１．２）的各阶奇点量,证明了这样的系统存在珐多三阶细奇点,而且同时存在细奇点的最大个数是四。当系统是四个细奇点时,它们可能是四个一阶细奇点或四个三阶细奇点。     

Vector bundles on log terminal varieties

Let be an -dimensional variety and an ample vector bundle on of rank . We give a complete classification of pairs , with log terminal and such that is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

     

On the splash singularity for the free-surface of a Navier–Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for d-dimensional flows, $d=2$ or 3, the free-surface of a viscous water wave, modeled by the incompressible Navier–Stokes equations with moving free-boundary, has a finite-time splash singularity for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.     

Singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel

In a recent paper [3], Cao and Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend the result of Cao and Xu and to establish the singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. The iterated singularity preserving Galerkin method is also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

An evolution compressible Stokes system in a polygon

An evolution compressible Stokes system is studied in a polygon. A lowest order corner singularity by the Laplacian operator is sorted out and an increased regularity is established in a suitable Banach space. In the lowest level, for each time the order of the corner singularity of the system is the same as that of the Laplacian. In addition, a singular function for pressure is defined and shows that the corner singularities originated from the corners of polygon may propagate along the directed streamlines.     

Investigation of the stress singularity of a magnetoelectroelastic bonded antiplane wedge

The order of the stress singularity of a magnetoelectroelastic bonded antiplane wedge is analyzed by complex potential function and eigenfunction expansion method. Contrary to the familiar problem of elastic anisotropic bonded wedges which always produce real values for the order of singularity, the results of the magnetoelectroelastic bonded wedges may be real or complex. Numerical results are presented for problems with different boundary conditions. In particular, special behaviors of the order of the stress singularity for some degenerate composite materials and for some special wedge angles are noted.     

基于小波变换的信号奇异性分析

本文研究得出了用小波变换进行奇异性检测时的小波基的一些选取原则和检测算法，并应用实例予以验证，从而证明了这些原则的有效性，为小波变换在奇异性检测中的实践运用奠定了一定的理论基础．     

The evolution compressible Navier-Stokes system on polygonal domains

The evolution compressible Navier-Stokes system is considered on polygonal domains. It is shown that the lowest order of the corner singularity of the system is the same as that of the heat equation. In a suitable Banach space the velocity is split into singular and regular parts and the coefficient of the singularity is expressed by convolution of some two functions in the time variable. By a formula of the pressure we observe propagation of the corner singularity along the characteristic lines emanating from the corners and also unboundedness of derivatives of pressure there. An increased regularity for the remainder part is established.     

Two concepts of singularity for matrices over semirings

The two common concepts of singularity for matrices over semirings are being studied since the 1970’s and arise from natural generalizations of the determinant and linear dependence. They were introduced in the context of schedule algebras by Gondran and Minoux, who proved later that the concepts discussed are equivalent over any selective invertible semiring. We present an approach that uses a generalization of power series arithmetic and, in particular, allows to derive a short proof for the theorem of Gondran and Minoux. Our main result is a complete concise characterization of semirings over which the two concepts of singularity are equivalent.     

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.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point).      

A note on a singular integral equation arising in water wave scattering

A quick method of solution of a singular integral equationof the first kind involving both logarithmic singularity aswell as Cauchy-type singularity is explained.