Search results for "Linear Algebra."

From quantale algebroids to topological spaces: Fixed- and variable-basis approaches

2010

Using the category of quantale algebroids the paper considers a generalization of the classical Papert-Papert-Isbell adjunction between the categories of topological spaces and locales to partial algebraic structures. It also provides a single framework in which to treat the concepts of quasi, standard and stratified fuzzy topology.

Existence of Unconditional Bases in Spaces of Polynomials and Holomorphic Functions

2002

Our main result shows that every Montel Kothe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an unconditional basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the bornological topology τδ associated to τ0 has an unconditional basis (for coechelon spaces τδ equals τ0). The main idea is to extend the concept of the Gordon-Lewis property from Banach to Frechet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of unconditional basis in spaces of m-th (symmetric) tenso…

VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE

2003

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Kothe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions ‖f‖X : Ω → [0,∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the pvariation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE(X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X))∗ = VE′ (X ∗), that VE(X) can be identified with space of cone absolutely summing operators fr…

An example concerning the zero set of the Jacobian

2006

AbstractLet f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L1(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L1(Ω) and Jf vanishes in a set of positive measure.

Stochastic factorizations, sandwiched simplices and the topology of the space of explanations

2003

We study the space of stochastic factorizations of a stochastic matrix V, motivated by the statistical problem of hidden random variables. We show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra, and use this geometrical model to gain some insight into its structure and topology. We prove theorems describing its homotopy type, and, in the case where the rank of V is 2, we give a complete description, including bounds on the number of connected components, and examples in which these bounds are attained. We attempt to make the notions of topology accessible and relevant to statisticians.

Thin Bases of Order Two

2001

AbstractA set A⊆N0 is called a basis of order two if A+A≔{a+a′∣a, a′∈A}=N0. If n∈N then A(n) denotes the number of a∈A with 1⩽a⩽n. In this paper bases A, B, C of order two are given such thatlimA(n)n=253,limB(n)n=72andlimC(n)n=101653.

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

2013

Let J(n) be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G-gradings on J(n) where G is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n - 1, where n is the dimension of the vector space V defining J(n). We prove that in this case the algebra J(n) is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.

A note on the closed graph theorem

1977

Timed Sets, Functional Complexity, and Computability

2012

AbstractThe construction of various categories of “timed sets” is described in which the timing of maps is considered modulo a “complexity order”. The properties of these categories are developed: under appropriate conditions they form discrete, distributive restriction categories with an iteration. They provide a categorical basis for modeling functional complexity classes and allow the development of computability within these settings. Indeed, by considering “program objects” and the functions they compute, one can obtain models of computability – i.e. Turing categories – in which the total maps belong to specific complexity classes. Two examples of this are introduced in some detail whi…

Locally Supported Wavelets on the Sphere

1998

We construct explicitly wavelets on the sphere that provide a locally supported and stable basis for the Sobolev spaces H2,0 ⩽ s < 1. We get at hand at fast wavelet transform with almost optimal complexity. This basis can be easily implemented in numerical schemes. We apply the wavelet transform to singularity detection and data compression. This contribution summarizes the results of [1].