65,771 research outputs found
New vacuum solutions of conformal Weyl gravity
The Bach equation, i.e., the vacuum field equation following from the
Lagrangian L=C_{ijkl}C^{ijkl}, will be completely solved for the case that the
metric is conformally related to the cartesian product of two 2-spaces; this
covers the spherically and the plane symmetric space-times as special subcases.
Contrary to other approaches, we make a covariant 2+2-decomposition of the
field equation, and so we are able to apply results from 2-dimensional gravity.
Finally, some cosmological solutions will be presented and discussed.Comment: 15 pages, LaTeX, no figures, submitted to J. Math. Phy
On the resonances and eigenvalues for a 1D half-crystal with localised impurity
We consider the Schr\"odinger operator on the half-line with a periodic
potential plus a compactly supported potential . For generic , its
essential spectrum has an infinite sequence of open gaps. We determine the
asymptotics of the resonance counting function and show that, for sufficiently
high energy, each non-degenerate gap contains exactly one eigenvalue or
antibound state, giving asymptotics for their positions. Conversely, for any
potential and for any sequences (\s_n)_{1}^\iy, \s_n\in \{0,1\}, and
(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0, there exists a potential such that
\vk_n is the length of the -th gap, , and has exactly \s_n
eigenvalues and 1-\s_n antibound state in each high-energy gap. Moreover, we
show that between any two eigenvalues in a gap, there is an odd number of
antibound states, and hence deduce an asymptotic lower bound on the number of
antibound states in an adiabatic limit.Comment: 25 page
Approximate Approximations from scattered data
The aim of this paper is to extend the approximate quasi-interpolation on a
uniform grid by dilated shifts of a smooth and rapidly decaying function on a
uniform grid to scattered data quasi-interpolation. It is shown that high order
approximation of smooth functions up to some prescribed accuracy is possible,
if the basis functions, which are centered at the scattered nodes, are
multiplied by suitable polynomials such that their sum is an approximate
partition of unity. For Gaussian functions we propose a method to construct the
approximate partition of unity and describe the application of the new
quasi-interpolation approach to the cubature of multi-dimensional integral
operators.Comment: 29 pages, 17 figure
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