36,638 research outputs found
Eigenfunction expansions for a fundamental solution of Laplace's equation on in parabolic and elliptic cylinder coordinates
A fundamental solution of Laplace's equation in three dimensions is expanded
in harmonic functions that are separated in parabolic or elliptic cylinder
coordinates. There are two expansions in each case which reduce to expansions
of the Bessel functions or , , in
parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that
is a fundamental solution and is the Riemann function of
partial differential equations on the Euclidean plane
The inner kernel theorem for a certain Segal algebra
The Segal algebra is well defined for arbitrary locally
compact Abelian Hausdorff (LCA) groups . Despite the fact that it is a
Banach space it is possible to derive a kernel theorem similar to the Schwartz
kernel theorem, of course without making use of the Schwartz kernel theorem.
First we characterize the bounded linear operators from
to by distributions in . We call this the "outer kernel theorem". The "inner kernel theorem" is
concerned with the characterization of those linear operators which have
kernels in the subspace , the main subject of
this manuscript. We provide a description of such operators as regularizing
operators in our context, mapping into test functions
in , in a -to norm continuous manner. The
presentation provides a detailed functional analytic treatment of the situation
and applies to the case of general LCA groups, without recurrence to the use of
so-called Wilson bases, which have been used for the case of elementary LCA
groups. The approach is then used in order to describe natural laws of
composition which imitate the composition of linear mappings via matrix
multiplications, now in a continuous setting. We use here that in a suitable
(weak) form these operators approximate general operators. We also provide an
explanation and mathematical justification used by engineers explaining in
which sense pure frequencies "integrate" to a Dirac delta distribution
Drawing graphs with vertices and edges in convex position
A graph has strong convex dimension , if it admits a straight-line drawing
in the plane such that its vertices are in convex position and the midpoints of
its edges are also in convex position. Halman, Onn, and Rothblum conjectured
that graphs of strong convex dimension are planar and therefore have at
most edges. We prove that all such graphs have at most edges
while on the other hand we present a class of non-planar graphs of strong
convex dimension . We also give lower bounds on the maximum number of edges
a graph of strong convex dimension can have and discuss variants of this
graph class. We apply our results to questions about large convexly independent
sets in Minkowski sums of planar point sets, that have been of interest in
recent years.Comment: 15 pages, 12 figures, improved expositio
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