Non-Autonomous Evolution Equations on Nonsmooth Domair

The solution of a stationary boundary value problem on a domain with conical points has singularities near these points. Here we first consider existence results in appropriate weighted Sobolev spaces in order to incorporate the singularities. We secondly use these results to prove existence, uniqueness and regularity of solutions of non-autonomous second order evolution equations on such domains.     

Simple Cohen–Macaulay Codimension 2 Singularities

In this article, we provide a complete list of simple isolated Cohen–Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.     

Equisingular Deformations of Sandwiched Singularities

We relate the equisingular deformation theory of plane curve singularities and sandwiched surface singularities. We show the existence of a smooth map between the two corresponding deformation functors and study the kernel of this map. In particular we show that the map is an isomorphism when a certain invariant is large enough.     

A Characterization of Quasi-Homogeneous Purely Elliptic Complete Intersection Singularities

It is well-known that quasi-homogeneity is characterized by equality of the Milnor and Tjurina numbers for isolated complex analytic hypersurface singularities and for certain low-dimensional singularities. In this paper we prove that this characterization extends to isolated purely elliptic complete intersection singularities, with bounds on neither the embedding codimension nor the dimension of the singularity.     

Singularities of the radon transform of a class of piecewise smooth functions in <Emphasis Type="Italic">R</Emphasis><Superscript>2</Superscript>

A class of piecewise smooth functions in R2 is considered.The propagation law of the Radon transform of the function is derived.The singularities inversion formula of the Radon transform is derived from the propagation law.The examples of singularities and singularities inversion of the Radon transform are given.     

Seshadri constants and the spaces of curves with prescribed singularities and positions

We will give optimal bounds for Seshadri constants of an ample line bundle at multiple points on a complex projective surface X.We also present a solution to the long-studied classical problem on the existence of curves on X with given topological singularities at r arbitrary points p1,...,pr.Namely,we obtain a universal lower bound on the degree of curves for the existence.It is independent of the position of the singular points.     

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Equivariant resolution of points of indeterminacy

We prove an equivariant form of Hironaka's theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro's equivariant Chow lemma.

     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Deformation of isolated hypersurface singularities and their moduli algebras

By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence. Research supported by NNSF     

An accurate semi-analytic finite difference scheme for two-dimensional elliptic problems with singularities

A high-order semi-analytic finite difference scheme is presented to overcome degradation of numerical performance when applied to two-dimensional elliptic problems containing singular points. The scheme, called Least-Square Singular Finite Difference Scheme (L-S SFDS), applies an explicit functional representation of the exact solution in the vicinity of the singularities, and a conventional finite difference scheme on the remaining domain. It is shown that the L-S SFDS is “pollution” free, i.e., no degradation in the convergence rate occurs because of the singularities, and the coefficients of the asymptotic solution in the vicinity of the singularities are computed as a by-product with a very high accuracy. Numerical examples for the Laplace and Poisson equations over domains containing re-entrant corners or abrupt changes in the boundary conditions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 281–296, 1998     

The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in L_p-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.     

Formal completions and idempotent completions of triangulated categories of singularities

The main goal of this paper is to prove that the idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason's theorem on dense subcategories and a relation to the negative K-theory.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

Spectral Bounds on Orbifold Isotropy

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.This research was partially supported by NSF grant DMS 0072534.     

Simple singularities of parametrized plane curves in positive characteristic

Let K be an algebraically closed field of characteristic p>0. The aim of the article is to give a classification of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of singularities which are not simple such that almost all singularities deform to one of those and show that remaining singularities are simple.     

SINGULAR INTEGRAL EQUATIONS ON THE REAL AXIS WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.     

Bifurcations in parametric nonlinear programming

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem.This research was supported by the Air force Office of Scientific Research through grant number AFOSR-88-0059.     

THEORY AND COMPUTATION OF SECONDARY BIFURCATIONS NEAR A DOUBLE EIGENVALUE

This paper deals with secondary bifurcations near a double eigenvalue of a nonlinear equation with two parameters. Utilizing symmetries (or more generally, equivariances ) and introducing two new parameters, we give some extended systems so that the double singular points, secondary bifurcation points and initial secondary branches respectively become their regular solutions. The methods in this paper not only give more general conditions of secondary bifurcation but also avoid the adjacent singularities of existing extended systems for computing the simple bifurcation points on non-trivial solution branches A numerical example is presented, showing the effectiveness of our methods.     

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.