5,350 research outputs found
Theta series and generalized special cycles on Hermitian locally symmetric manifolds
We study generalized special cycles on Hermitian locally symmetric spaces
associated to the groups ,
and . These cycles are (covered
by) locally symmetric spaces associated to subgroups of which are of the
same type. Using oscillator representation and a construction which essentially
comes from the thesis of Greg Anderson, we show that Poincar\'e duals of these
generalized special cycles can be viewed as Fourier coefficients of a theta
series. This gives new cases of theta lifts from the cohomology of Hermtian
locally symmetric manifolds associated to to vector valued automorphic
forms associated to the groups , or
which forms a reductive dual pair with in the sense of
Howe
Convexity properties of the condition number II
In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the
condition metric in the space of maximal rank matrices. Here, we show that this
condition metric induces a Lipschitz-Riemann structure on that space. After
investigating geodesics in such a nonsmooth structure, we show that the inverse
of the smallest singular value of a matrix is a log-convex function along
geodesics (Theorem 1).
We also show that a similar result holds for the solution variety of linear
systems (Theorem 31).
Some of our intermediate results, such as Theorem 12, on the second covariant
derivative or Hessian of a function with symmetries on a manifold, and Theorem
29 on piecewise self-convex functions, are of independent interest.
Those results were motivated by our investigations on the com- plexity of
path-following algorithms for solving polynomial systems.Comment: Revised versio
Rank of Stably Dissipative Graphs
For the class of stably dissipative Lotka-Volterra systems we prove that the
rank of its defining matrix, which is the dimension of the associated invariant
foliation, is completely determined by the system's graph
Structure of almost diagonal matrices
Classical and recent results on almost diagonal matrices are presented.
These results measure the absolute and the relative distance
between diagonal elements and the appropriate eigenvalues or singular values, and in case of multiple eigenvalues or singular values, reveal special structure in matrices. Simple MATLAB programs serve to illustrate how good the theoretical estimates are
Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem
The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue
problem arising from discretized Bethe-Salpeter equation in the context of
computing exciton energies and states. A computational challenge is that at
least half of the eigenvalues and the associated eigenvectors are desired in
practice. We establish the equivalence between Bethe-Salpeter eigenvalue
problems and real Hamiltonian eigenvalue problems. Based on theoretical
analysis, structure preserving algorithms for a class of Bethe-Salpeter
eigenvalue problems are proposed. We also show that for this class of problems
all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated.
In order to solve large scale problems of practical interest, we discuss
parallel implementations of our algorithms targeting distributed memory
systems. Several numerical examples are presented to demonstrate the efficiency
and accuracy of our algorithms
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