45 research outputs found
Optimal Discrete Riesz Energy and Discrepancy
The Riesz -energy of an -point configuration in the Euclidean space
is defined as the sum of reciprocal -powers of all mutual
distances in this system. In the limit the Riesz -potential
( the Euclidean distance) governing the point interaction is replaced with
the logarithmic potential . In particular, we present a conjecture
for the leading term of the asymptotic expansion of the optimal
\IL_2-discrepancy with respect to spherical caps on the unit sphere in
which follows from Stolarsky's invariance principle [Proc.
Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two
terms of the asymptotic expansion of the optimal Riesz -energy of points
as .Comment: 8 page
A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere
A positive unit point charge approaching from infinity a perfectly spherical
isolated conductor carrying a total charge of +1 will eventually cause a
negatively charged spherical cap to appear. The determination of the smallest
distance ( is the dimension of the unit sphere) from the point
charge to the sphere where still all of the sphere is positively charged is
known as Gonchar's problem. Using classical potential theory for the harmonic
case, we show that is equal to the largest positive zero of a
certain sequence of monic polynomials of degree with integer
coefficients which we call Gonchar polynomials. Rather surprisingly,
is the Golden ratio and the lesser known Plastic number. But Gonchar
polynomials have other interesting properties. We discuss their factorizations,
investigate their zeros and present some challenging conjectures.Comment: 12 pages, 6 figures, 1 tabl
The support of the limit distribution of optimal Riesz energy points on sets of revolution in
Let A be a compact set in the right-half plane and the set in
obtained by rotating A about the vertical axis. We investigate
the support of the limit distribution of minimal energy point charges on
that interact according to the Riesz potential 1/r^{s}, 0<s<1,
where r is the Euclidean distance between points. Potential theory yields that
this limit distribution coincides with the equilibrium measure on
which is supported on the outer boundary of . We show that there are
sets of revolution such that the support of the equilibrium measure
on is {\bf not} the complete outer boundary, in contrast to the
Coulomb case s=1. However, the support of the limit distribution on the set of
revolution as R goes to infinity, is the full outer boundary for
certain sets A, in contrast to the logarithmic case (s=0)
The Riesz energy of the -th roots of unity: an asymptotic expansion for large
We derive the complete asymptotic expansion in terms of powers of for the
Riesz -energy of equally spaced points on the unit circle as . For , such points form optimal energy -point
configurations with respect to the Riesz potential , , where
is the Euclidean distance between points.
By analytic continuation we deduce the expansion for all complex values of
.
The Riemann zeta function plays an essential role in this asymptotic
expansion.Comment: Added reference to paper by Cohn and Kumar which shows optimality of
roots of unity for -2<s<-
Point sets on the sphere with small spherical cap discrepancy
In this paper we study the geometric discrepancy of explicit constructions of
uniformly distributed points on the two-dimensional unit sphere. We show that
the spherical cap discrepancy of random point sets, of spherical digital nets
and of spherical Fibonacci lattices converges with order . Such point
sets are therefore useful for numerical integration and other computational
simulations. The proof uses an area-preserving Lambert map. A detailed analysis
of the level curves and sets of the pre-images of spherical caps under this map
is given
Quasi-Monte Carlo rules for numerical integration over the unit sphere
We study numerical integration on the unit sphere using equal weight quadrature rules, where the weights are such
that constant functions are integrated exactly.
The quadrature points are constructed by lifting a -net given in the
unit square to the sphere by means of an area
preserving map. A similar approach has previously been suggested by Cui and
Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2].
We prove three results. The first one is that the construction is (almost)
optimal with respect to discrepancies based on spherical rectangles. Further we
prove that the point set is asymptotically uniformly distributed on
. And finally, we prove an upper bound on the spherical cap
-discrepancy of order (where denotes the
number of points). This slightly improves upon the bound on the spherical cap
-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm.
Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the
-nets lifted to the sphere have spherical cap
-discrepancy converging with the optimal order of
Detection of crenosoma spp., angiostrongylus vasorum and aelurostrongylus abstrusus in gastropods in Eastern Austria
Canine and feline cardiorespiratory parasites are of utmost relevance in veterinary medicine. Key epizootiological information on major pet metastrongyloids, i.e., Angiostrongylus vasorum and Crenosoma vulpis infecting dogs, and Aelurostrongylus abstrusus and Troglostrongylus brevior infecting cats, is missing from Austria. This study investigated their occurrence in 1320 gastropods collected in the Austrian provinces of Styria, Burgenland, Lower Austria, and in metropolitan Vienna. Metastrongyloid larvae were microscopically detected in 25 samples, and sequence analysis confirmed the presence of metastrongyloids in nine samples, i.e., A. vasorum in one slug (Arion vulgaris) (0.07%), C. vulpis in five slugs (one Limax maximus and four A. vulgaris) (0.4%), A. abstrusus in two A. vulgaris (0.17%), and the hedgehog lungworm Crenosoma striatum was detected in one A. vulgaris. The present study confirms the enzooticity of major cardiorespiratory nematodes in Austria and that canine and feline populations are at risk of infection