This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...

this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...

This paper is devoted to the proof of the unique solvability ofthe inverse problems for second-order differential operators withregular singularities. It is shown that the potential functioncan be determined from spectral data, also we prove a uniquenesstheorem in the inverse problem.

In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.

In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as ...

In Arnold’s classification list of singularities (Arnold [1].) we find interesting singularities to be studied. Though we find singularities of any dimension in Arnold’s list, we consider singularities of dimension two in particular. Among them there is a class called exceptional singularities or triangle singularities. This class consists of fourteen singularities. It is known that they are cl...

Three types of singularity that can arise in fluid dynamical problems will be distinguished and discussed. These are: (i) singularities driven by boundary motion in conjunction with viscosity (e.g. corner singularities, or the Euler-disc finite-time singularity); (ii) free-surface (cusp) singularities associated with surface-tension and viscosity; (iii) interior point singularities of vorticity...

Rational singularities are a class of singularities which have been heavily studied since their introduction in the 1960s. Roughly speaking, an algebraic variety has rational singularities if its structure sheaf has the same cohomology as the structure sheaf of a resolution of singularities. Rational singularities enjoy many useful properties, in particular they are both normal and Cohen-Macaul...