29,503 research outputs found
Singular Higher Order Divergence-Conforming Bases of Additive Kind and Moments Method Applications to 3D Sharp-Wedge Structures
We present new subsectional, singular divergence conforming vector bases that incorporate the edge conditions for conducting wedges. The bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood. Singular bases of this kind, complete to arbitrarily high order, are described in a unified and consistent manner for curved quadrilateral and triangular elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. Our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components, adequate modelling of the divergence, and removal of spurious solutions. These singular high-order bases provide more accurate and efficient numerical solutions of surface integral problems. Several test-case problems are considered in the paper, thereby obtaining highly accurate numerical results for the current and charge density induced on 3D sharp-wedge structures. The results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problem
The inverse problem for representation functions for general linear forms
The inverse problem for representation functions takes as input a triple
(X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a
function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A
\subseteq X such that there are f(x) solutions (counted appropriately) to
L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists.
This paper represents the first systematic study of this problem for
arbitrary linear forms when X = Z, the setting which in many respects is the
most natural one. Having first settled on the "right" way to count
representations, we prove that every primitive form has a unique representation
basis, i.e.: a set A which represents the function f \equiv 1. We also prove
that a partition regular form (i.e.: one for which no non-empty subset of the
coefficients sums to zero) represents any function f for which {f^{-1}(0)} has
zero asymptotic density. These two results answer questions recently posed by
Nathanson.
The inverse problem for partition irregular forms seems to be more
complicated. The simplest example of such a form is x_1 - x_2, and for this
form we provide some partial results. Several remaining open problems are
discussed.Comment: 15 pages, no figure
Extended graphical calculus for categorified quantum sl(2)
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum
sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here
we enhance the graphical calculus introduced and developed in that paper to
include two-morphisms between divided powers one-morphisms and their
compositions. We obtain explicit diagrammatical formulas for the decomposition
of products of divided powers one-morphisms as direct sums of indecomposable
one-morphisms; the latter are in a bijection with the Lusztig canonical basis
elements. These formulas have integral coefficients and imply that one of the
main results of Lauda's paper---identification of the Grothendieck ring of his
2-category with the idempotented quantum sl(2)---also holds when the 2-category
is defined over the ring of integers rather than over a field.Comment: 72 pages, LaTeX2e with xypic and pstricks macro
Additivity, subadditivity and linearity: automatic continuity and quantifier weakening
We study the interplay between additivity (as in the Cauchy functional
equation), subadditivity and linearity. We obtain automatic continuity results
in which additive or subadditive functions, under minimal regularity
conditions, are continuous and so linear. We apply our results in the context
of quantifier weakening in the theory of regular variation completing our
programme of reducing the number of hard proofs there to zero.Comment: Companion paper to: Cauchy's functional equation and extensions:
Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and
Beurling's equation Updated to refer to other developments and their
publication detail
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