We present a Lie algebraic framework for vortex sheets as singular 2-forms with support of codimension 1, i.e. singular elements of a completion of the dual to the Lie algebra of divergence-free vector fields. This framework allows one to define the Poisson and symplectic structures on the space of vortex sheets, which interpolate between the corresponding structures on filaments and smooth vor...

We introduce a constructive model for algebraic tori based on reduced divisors on singular curves. By using a singular hyperelliptic model, this provides an alternative representation, and computational model, for groups of rational points on an algebraic tori [4]. We obtain a represention of elements on certain tori of dimension r in compact representation using r +1 elements. By embedding the...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

In this paper, we propose a simplified pollution adaptive mesh generation algorithm using singular elements. The algorithm based on the element pollution error indicator concentrate on boundary nodes. The automatic mesh generation method is fo llowed by either a node-relocation or a node-insertion method. The boundary node relocation phase is introduced to reduce pollution error estimates witho...

A weakly singular form of the hypersingular boundary integral equation (BIE) (traction equation) for 3-D elastic wave problems is developed in this paper. All integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements, regular quadrature and collocation procedures used for conventional BIEs are sufficient for the discretization of the or...

Let A be a C*-algebra with identity and suppose A has real rank 0. Suppose a complex-valued function is holomorphic and bounded on the intersection of the open unit ball of A and the identity component of the set of invertible elements of A. We show that then the function has a holomorphic extension to the entire open unit ball of A. Further, we show that this does not hold when A = C(S), where...

Similar to the well known Schur Horn theorem that characterizes the relationship between the diagonal entries and the eigenvalues of a Hermitian matrix the Sing Thompson theorem characterizes the relationship between the diagonal en tries and the singular values of an arbitrary matrix It is noted in this paper that based on the induction principle such a matrix can be constructed numerically by...

There are two problems need to be dealt with for Non-negative Matrix Factorization (NMF): choose a suitable rank of the factorization and provide a good initialization method for NMF algorithms. This paper aims to solve these two problems using Singular Value Decomposition (SVD). At first we extract the number of main components as the rank, actually this method is inspired from [1, 2]. Second,...

We consider the product ofM quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary Ensemble with M = 1. In this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix sizeN and t...

It is shown that some Schrödinger operators, with nonprimitive substitution potentials, have pure singular continuous Cantor spectrum with null Lebesgue measure for all elements in the respective hulls.