Time dependent orbifolds with spacelike or null singularities have recently been studied as simple models of cosmological singularities. We show that their apparent simplicity is an illusion: the introduction of a single particle causes the spacetime to collapse to a strong curvature singularity (a Big Crunch), even in regions arbitrarily far from the particle.

A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka–Volterra type that result translates into a simple condition on the coefficients.

The recent demonstration of current singularity formation by Low et al. assumes that potential fields will remain potential under simple expansion or compression (Low 2006, 2007; Janse & Low 2009). An explicit counterexample to their key assumption is constructed. Our findings suggest that their results may need to be reconsidered.

The existence of sufficiently many finite-order (in the sense of Nevanlinna) mero-morphic solutions of a difference equation appears to be a good indicator of integra-bility. This criterion is used to single out dP II from a natural class of second-order difference equations. The proof given uses an estimate related to singularity confinement. The Painlevé property has been used as a detector o...

We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is −2. In the particular case of subregular nilpotent orbits we show that the structure may be computed by means of ...

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl g...

the existence is a categorical concept according to kant’s philosophy. kant, going to be a subjectivist philosopher, negates the existence as a perfection and concludes the separation between the mind and extra-mental world. but the externality is not an origin which we deny it easily, even if it is faced with multiple problems such as the issue of simplicity and multiplicity and also being his...

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...

In 1985 Goode and Wainwright devised the concept of an isotropic singularity. Since that time, numerous authors have explored the interesting consequences, in mathematical cosmology, of assuming the existence of this type of singularity. In this paper, we collate all examples of cosmological models which are known to admit an isotropic singularity, and make a number of observations regarding th...