In this paper, we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons. Drawing on the work of Bompadre et al. [1] and Hormann and Sukumar [2], we adopt a relative entropy measure for signed (positive or negative) shape functions, with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon. We maximize t...

We consider the problem of reconstructing frames from a video which has been compressed using the video compressive sensing (VCS) method. In VCS data, each frame comes from first subsampling the original video data in space and then averaging the subsampled sequence in time. This results in a large linear system of equations whose inversion is ill-posed. We introduce a convex regularizer to inv...

A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. A c0-saturated Banach space with an unconditional basis which has a quotient space isomorphic to l2 is constructed. A Banach space E is c0-saturated if every closed infinite dimensional subspace of E contains an isomorph of c0. In [2] and [3], it was asked whether all quotient spaces ...

A topological algebra is a complex associative algebra which is also a Hausdorff topological vector space such that the multiplication is separately continuous. A locally convex algebra is a topological algebra whose topology is determined by a family of seminorms. A complete metrizable locally convex algebra is called a B0-algebra. The topology of a B0-algebra A may be given by a countable fam...

Let A be a commutative unital R-algebra and let ρ be a seminorm on A which satisfies ρ(ab) ≤ ρ(a)ρ(b). We apply T. Jacobi’s representation theorem [10] to determine the closure of a ∑A-module S of A in the topology induced by ρ, for any integer d ≥ 1. We show that this closure is exactly the set of all elements a ∈ A such that α(a) ≥ 0 for every ρ-continuous R-algebra homomorphism α ∶ AÐ→ R wit...

The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. Introduction The Farkas Lemma, also known as the Farkas–Minkowski Lemma, plays a key role in linear programming and the relevant areas of optimization (cp. [1]). Some rather simple proof of the lemma is given in [2]. Using the technique of Boolean valued analys...

Minimizing the l0-seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the l0seminorm with the l1-norm is a commonly used approach to compute an approximate solution of the original l0-minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Pro...

We study a variant of the Whitney extension problem (1934) for the space Ck,ω(Rn). We identify Ck,ω(Rn) with a space of Lipschitzmappings from Rn into the space Pk × Rn of polynomial fields on Rn equipped with a certain metric. This identification allows us to reformulate the Whitney problem for Ck,ω(Rn) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets o...

We consider the problem of finding for a given N -tuple of polynomials (real or complex) the closest N -tuple that has a common divisor of degree at least d. Extended weighted Euclidean seminorm of the coefficients is used as a measure of closeness. Two equivalent representations of the problem are considered: (i) direct parameterization over the common divisors and quotients (image representat...

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projection technique fails to minimize any measure of...