7,968 research outputs found
Lorentz and Galilei Invariance on Lattices
We show that the algebraic aspects of Lie symmetries and generalized
symmetries in nonrelativistic and relativistic quantum mechanics can be
preserved in linear lattice theories. The mathematical tool for symmetry
preserving discretizations on regular lattices is the umbral calculus.Comment: 5 page
Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
We present a general solution-generating result within the bidifferential
calculus approach to integrable partial differential and difference equations,
based on a binary Darboux-type transformation. This is then applied to the
non-autonomous chiral model, a certain reduction of which is known to appear in
the case of the D-dimensional vacuum Einstein equations with D-2 commuting
Killing vector fields. A large class of exact solutions is obtained, and the
aforementioned reduction is implemented. This results in an alternative to the
well-known Belinski-Zakharov formalism. We recover relevant examples of
space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and
double Myers-Perry black holes, black saturn, bicycling black rings)
dilaton Weyl multiplet in 4D supergravity
We construct the dilaton Weyl multiplet for conformal supergravity in
four dimensions. Beginning from an on-shell vector multiplet coupled to the
standard Weyl multiplet, the equations of motion can be used to eliminate the
supergravity auxiliary fields, following a similar pattern as in five and six
dimensions. The resulting 24+24 component multiplet includes two gauge vectors
and a gauge two-form and provides a variant formulation of conformal
supergravity. We also show how this dilaton Weyl multiplet is contained in the
minimal 32+32 Poincare supergravity multiplet introduced by Muller in
superspace.Comment: 15 page
Bi-differential calculi and integrable models
The existence of an infinite set of conserved currents in completely
integrable classical models, including chiral and Toda models as well as the KP
and self-dual Yang-Mills equations, is traced back to a simple construction of
an infinite chain of closed (respectively, covariantly constant) 1-forms in a
(gauged) bi-differential calculus. The latter consists of a differential
algebra on which two differential maps act. In a gauged bi-differential
calculus these maps are extended to flat covariant derivatives.Comment: 24 pages, 2 figures, uses amssymb.sty and diagrams.sty, substantial
extensions of examples (relative to first version
Differential Calculi on Associative Algebras and Integrable Systems
After an introduction to some aspects of bidifferential calculus on
associative algebras, we focus on the notion of a "symmetry" of a generalized
zero curvature equation and derive Backlund and (forward, backward and binary)
Darboux transformations from it. We also recall a matrix version of the binary
Darboux transformation and, inspired by the so-called Cauchy matrix approach,
present an infinite system of equations solved by it. Finally, we sketch recent
work on a deformation of the matrix binary Darboux transformation in
bidifferential calculus, leading to a treatment of integrable equations with
sources.Comment: 19 pages, to appear in "Algebraic Structures and Applications", S.
Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics,
202
Noncommutative Geometry of Lattice and Staggered Fermions
Differential structure of a d-dimensional lattice, which is essentially a
noncommutative exterior algebra, is defined using reductions in first order and
second order of universal differential calculus in the context of
noncommutative geometry (NCG) developed by Dimakis et al. This differential
structure can be realized adopting a Dirac-Connes operator proposed by us
recently within Connes' NCG. With matrix representations being specified, our
Dirac-Connes operator corresponds to staggered Dirac operator, in the case that
dimension of the lattice equals to 1, 2 and 4.Comment: Latex; 13 pages; no figures. References added. Accepted by Phys.
Lett.
An Overview of Variational Integrators
The purpose of this paper is to survey some recent advances in variational
integrators for both finite dimensional mechanical systems as well as continuum
mechanics. These advances include the general development of discrete
mechanics, applications to dissipative systems, collisions, spacetime integration algorithms,
AVI’s (Asynchronous Variational Integrators), as well as reduction for
discrete mechanical systems. To keep the article within the set limits, we will only
treat each topic briefly and will not attempt to develop any particular topic in
any depth. We hope, nonetheless, that this paper serves as a useful guide to the
literature as well as to future directions and open problems in the subject
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