Generalized barycentric coordinates are widely used to represent a point inside a polygon as an affine combination of the polygon’s vertices, and it is desirable to have coordinates that are non-negative, smooth, and locally supported. Unfortunately, the existing coordinate functions that satisfy all these properties do not have a simple analytic expression, making them expensive to evaluate an...

It is known that interval Newton methods can verify existence and uniqueness of solutions of a nonlinear system of equations near points where the Jacobi matrix of the system is not ill-conditioned. Recently, we have shown how to verify existence and uniqueness, up to multiplicity, for solutions at which the Jacobi matrix is singular. We do this by efficient computation of the topological index...

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r+ where g, r ∈ C(R) and the function r+ is defined by r+(y) = { r(y), r(y) ≥ 0 0, r(y) < 0 , ∀y ∈ R . G...

In this paper, we study the support recovery guarantees of underdetermined sparse regression using the `1-norm as a regularizer and a non-smooth loss function for data fidelity. More precisely, we focus in detail on the cases of `1 and `∞ losses, and contrast them with the usual `2 loss. While these losses are routinely used to account for either sparse (`1 loss) or uniform (`∞ loss) noise mode...

Several wav elets from well known biorthogonal families are shown to be unbounded on every interval. One, in fact, is shown to be infinite at each dyadic rational. Not withstanding these facts, we show how to compute exact values for these wav elets at many points and thus achieve exact pictures for these functions.

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics. Mathematics Subject Classification (2000): Primary: 46F30; secondary: 46T30, 53B20.