Let ? be a simple undirected graph on finite vertex set and let A its adjacency matrix. Then is singular if singular. The problem of characterizing graphs easy to state but very difficult resolve in any generality. In this paper we investigate the singularity for which dihedral group acts transitively vertices as automorphisms.

We study the behavior of geodesies passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed....

We construct a new class of plane-symmetric solutions possessing a curvature singularity which is null and weak, like the spacetime singularity at the Cauchy horizon of spinning (or charged) black holes. We then analyse the stability of this singularity using a rigorous non-perturbative method. We find that within the framework of (linearly-polarized) plane-symmetric spacetimes this type of nul...

Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in OS,O , the curves on the surface X = BlI (S) obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these curves are obtained. New formulas for their multiplicity, intersection numbers and order of singularity at the singularities of X are given. The semigroup of bra...

We study the homogeneous Dirichlet problem for some elliptic equations with a first order term b(u,Du) which is quadratic in the gradient variable and singular in the u variable at a positive point. Moreover, the gradient term we consider, changes its sign at the singularity. Dealing with an appropriate concept of solution that gives sense to the equation at the singularity, we prove existence ...

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singul...

In the present paper MAPLE subroutines computing Milnor and Tyurina numbers of an isolated algebraic hypersurface singularity are presented and described in detail. They represents examples, and perhaps the first ones, of a MAPLE implementation of local monomial ordering. As an application, the last section is devoted to writing down equations of algebraic stratifications of Kuranishi spaces of...

Previously, we introduced a duality transformation for Euler G– Frobenius algebras. Using this transformation, we prove that the simple A,D,E singularities and Pham singularities of coprime powers are mirror self– dual where the mirror duality is implemented by orbifolding with respect to the symmetry group generated by the grading operator and dualizing. We furthermore calculate orbifolds and ...

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singul...