10,575 research outputs found
Density results for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets
We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin
spaces on rough sets. Our main results, stated in the simplest Sobolev space
setting, are that: (i) for an open set ,
is dense in whenever has zero Lebesgue
measure and is "thick" (in the sense of Triebel); and (ii) for a
-set (), is dense in whenever for some . For (ii), we provide
concrete examples, for any , where density fails when
and are on opposite sides of . The results (i) and (ii)
are related in a number of ways, including via their connection to the question
of whether for a
given closed set and . They also
both arise naturally in the study of boundary integral equation formulations of
acoustic wave scattering by fractal screens. We additionally provide analogous
results in the more general setting of Besov and Triebel--Lizorkin spaces.Comment: 38 pages, 6 figure
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Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation
A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined
A Lefschetz type coincidence theorem
A Lefschetz-type coincidence theorem for two maps f,g:X->Y from an arbitrary
topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence
index is equal to the Lefschetz number. It follows that if L(f,g) is not equal
to zero then there is an x in X such that f(x)=g(x). In particular, the theorem
contains some well-known coincidence results for (i) X,Y manifolds and (ii) f
with acyclic fibers.Comment: The final version, 23 pages, to appear in Fund. Mat
Open problems in Banach spaces and measure theory
We collect several open questions in Banach spaces, mostly related to measure
theoretic aspects of the theory. The problems are divided into five categories:
miscellaneous problems in Banach spaces (non-separable spaces,
compactness in Banach spaces, -null sequences in dual spaces),
measurability in Banach spaces (Baire and Borel -algebras, measurable
selectors), vector integration (Riemann, Pettis and McShane integrals), vector
measures (range and associated spaces) and Lebesgue-Bochner spaces
(topological and structural properties, scalar convergence)
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