45,755 research outputs found
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
On Asymptotic Reducibility in SL(3,Z)
Recently we showed that Hessenberg matrices are proper to represent conjugacy
classes in SL(n,Z). In this paper we focus on the reducibility properties in
the set of Hessenberg matrices of SL(3,Z). We investigate the first interesting
open case here: the case of matrices having one real and two complex conjugate
eigenvalues.Comment: 24 pages, 7 figure
An Algorithmic Test for Diagonalizability of Finite-Dimensional PT-Invariant Systems
A non-Hermitean operator does not necessarily have a complete set of
eigenstates, contrary to a Hermitean one. An algorithm is presented which
allows one to decide whether the eigenstates of a given PT-invariant operator
on a finite-dimensional space are complete or not. In other words, the
algorithm checks whether a given PT-symmetric matrix is diagonalizable. The
procedure neither requires to calculate any single eigenvalue nor any numerical
approximation.Comment: 13 pages, 1 figur
Fusion rules in conformal field theory
Several aspects of fusion rings and fusion rule algebras, and of their
manifestations in twodimensional (conformal) field theory, are described:
diagonalization and the connection with modular invariance; the presentation in
terms of quotients of polynomial rings; fusion graphs; various strategies that
allow for a partial classification; and the role of the fusion rules in the
conformal bootstrap programme.Comment: 68 pages, LaTeX. changed contents of footnote no.
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
Random matrix ensembles for -symmetric systems
Recently much effort has been made towards the introduction of non-Hermitian
random matrix models respecting -symmetry. Here we show that there is a
one-to-one correspondence between complex -symmetric matrices and
split-complex and split-quaternionic versions of Hermitian matrices. We
introduce two new random matrix ensembles of (a) Gaussian split-complex
Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary
sizes. They are related to the split signature versions of the complex and the
quaternionic numbers, respectively. We conjecture that these ensembles
represent universality classes for -symmetric matrices. For the case of
matrices we derive analytic expressions for the joint probability
distributions of the eigenvalues, the one-level densities and the level
spacings in the case of real eigenvalues.Comment: 9 pages, 3 figures, typos corrected, small changes, accepted for
publication in Journal of Physics
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
Supersymmetry algebra cohomology I: Definition and general structure
The paper concerns standard supersymmetry algebras in diverse dimensions,
involving bosonic translational generators and fermionic supersymmetry
generators. A cohomology related to these supersymmetry algebras, termed
supersymmetry algebra cohomology, and corresponding "primitive elements" are
defined by means of a BRST-type coboundary operator. A method to systematically
compute this cohomology is outlined and illustrated by simple examples.Comment: v5: matches published version; 3 refs., section 5.5 and
remarks/comments in sections 1, 2.8, 3 and 7 added; minor editorial
improvements and change of titl
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