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...

Previously described algorithms for calculating the Laurent expansion of the inverse of a matrix-valued analytic function become impractical already for singularity orders as low as around p = 6, since they require over O(28) matrix multiplications and correspondingly large amounts of memory. In place of using mathematically exact recursions, we show that, for floating point calculations, a rat...

A new method is presented for controlling the trajectories of straight-line Cartesian paths near the kinematic singularities of simple manipulators. Conventional approachs to this problem, which typically employ a pseudo-inverse of the manipulator Jacobian, result in path deviations and have diiculty controlling joint accelerations. The more global approach described here uses a \potential func...

Abstract Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p , sampled according to one of the following two models: either every entry is independently taken be probability (the ‘Bernoulli’ model) or each row uniformly from set all length- n vectors exactly pn ones ‘combinatorial’ model). We give simple proofs (essentially best-possible) fact that in both models, if $\min(p,1-p)\ge...

This paper introduces q-series for a curve singularity (C, 0) in the affine space (Cn, 0) via subspace arrangements. These q-series are certain multivariable generating functions whose coefficients are the characteristic polynomials of subspace arrangements associated with the singularity (C, 0) at various orders; the q is the variable in the characteristic polynomials of the subspace arrangeme...

Hallin and Ley (2012) investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) Fisher degeneracy problem, showing that it can be more or less severe, inducing n (“simple singularity”), n (“double singularity”), or n (“triple singularity”) consistency ra...

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to...

There exists a bijection between the configuration space of a linear pentapod and all points (u,v,w, px, py, pz)∈R located on the singular quadric Γ : u2+v2+w2 = 1, where (u,v,w) determines the orientation of the linear platform and (px, py, pz) its position. Then the set of all singular robot configurations is obtained by intersecting Γ with a cubic hypersurface Σ in R6, which is only quadrati...