193 research outputs found
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
Differential Calculi on Commutative Algebras
A differential calculus on an associative algebra A is an algebraic analogue
of the calculus of differential forms on a smooth manifold. It supplies A with
a structure on which dynamics and field theory can be formulated to some extent
in very much the same way we are used to from the geometrical arena underlying
classical physical theories and models. In previous work, certain differential
calculi on a commutative algebra exhibited relations with lattice structures,
stochastics, and parametrized quantum theories. This motivated the present
systematic investigation of differential calculi on commutative and associative
algebras. Various results about their structure are obtained. In particular, it
is shown that there is a correspondence between first order differential
calculi on such an algebra and commutative and associative products in the
space of 1-forms. An example of such a product is provided by the Ito calculus
of stochastic differentials.
For the case where the algebra A is freely generated by `coordinates' x^i,
i=1,...,n, we study calculi for which the differentials dx^i constitute a basis
of the space of 1-forms (as a left A-module). These may be regarded as
`deformations' of the ordinary differential calculus on R^n. For n < 4 a
classification of all (orbits under the general linear group of) such calculi
with `constant structure functions' is presented. We analyse whether these
calculi are reducible (i.e., a skew tensor product of lower-dimensional
calculi) or whether they are the extension (as defined in this article) of a
one dimension lower calculus. Furthermore, generalizations to arbitrary n are
obtained for all these calculi.Comment: 33 pages, LaTeX. Revision: A remark about a quasilattice and Penrose
tiling was incorrect in the first version of the paper (p. 14
Soliton equations and the zero curvature condition in noncommutative geometry
Familiar nonlinear and in particular soliton equations arise as zero
curvature conditions for GL(1,R) connections with noncommutative differential
calculi. The Burgers equation is formulated in this way and the Cole-Hopf
transformation for it attains the interpretation of a transformation of the
connection to a pure gauge in this mathematical framework. The KdV, modified
KdV equation and the Miura transformation are obtained jointly in a similar
setting and a rather straightforward generalization leads to the KP and a
modified KP equation.
Furthermore, a differential calculus associated with the Boussinesq equation
is derived from the KP calculus.Comment: Latex, 10 page
The INECO experience: Main outcomes and lessons learned from participatory Case Study processes
Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations
By considering a lattice model of extended phase space, and using techniques
of noncommutative differential geometry, we are led to: (a) the conception of
vector fields as generators of motion and transition probability distributions
on the lattice; (b) the emergence of the time direction on the basis of the
encoding of probabilities in the lattice structure; (c) the general
prescription for the observables' evolution in analogy with classical dynamics.
We show that, in the limit of a continuous description, these results lead to
the time evolution of observables in terms of (the adjoint of) generalized
Fokker-Planck equations having: (1) a diffusion coefficient given by the limit
of the correlation matrix of the lattice coordinates with respect to the
probability distribution associated with the generator of motion; (2) a drift
term given by the microscopic average of the dynamical equations in the present
context. These results are applied to 1D and 2D problems. Specifically, we
derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in
velocity space, thus indicating the way random walk models are incorporated in
the present context; (II) Kramers' equation, by further assuming that, motion
is deterministic in coordinate spaceComment: LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfi
A Better Understanding of the Performance of Rate-1/2 Binary Turbo Codes that Use Odd-Even Interleavers
The effects of the odd-even constraint - as an interleaver design criterion -
on the performance of rate-1/2 binary turbo codes are revisited. According to
the current understanding, its adoption is favored because it makes the
information bits be uniformly protected, each one by its own parity bit. In
this paper, we provide instances that contradict this point of view suggesting
for a different explanation of the constraint's behavior, in terms of distance
spectrum
Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula
One of the key ingredients of A. Connes' noncommutative geometry is a
generalized Dirac operator which induces a metric(Connes' distance) on the
state space. We generalize such a Dirac operator devised by A. Dimakis et al,
whose Connes' distance recovers the linear distance on a 1D lattice, into 2D
lattice. This Dirac operator being "naturally" defined has the so-called "local
eigenvalue property" and induces Euclidean distance on this 2D lattice. This
kind of Dirac operator can be generalized into any higher dimensional lattices.Comment: Latex 11pages, no figure
Noncommutative Geometry of Finite Groups
A finite set can be supplied with a group structure which can then be used to
select (classes of) differential calculi on it via the notions of left-, right-
and bicovariance. A corresponding framework has been developed by Woronowicz,
more generally for Hopf algebras including quantum groups. A differential
calculus is regarded as the most basic structure needed for the introduction of
further geometric notions like linear connections and, moreover, for the
formulation of field theories and dynamics on finite sets. Associated with each
bicovariant first order differential calculus on a finite group is a braid
operator which plays an important role for the construction of distinguished
geometric structures. For a covariant calculus, there are notions of invariance
for linear connections and tensors. All these concepts are explored for finite
groups and illustrated with examples. Some results are formulated more
generally for arbitrary associative (Hopf) algebras. In particular, the problem
of extension of a connection on a bimodule (over an associative algebra) to
tensor products is investigated, leading to the class of `extensible
connections'. It is shown that invariance properties of an extensible
connection on a bimodule over a Hopf algebra are carried over to the extension.
Furthermore, an invariance property of a connection is also shared by a `dual
connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late
Electromagnetism and Gauge Theory on the Permutation Group
Using noncommutative geometry we do U(1) gauge theory on the permutation
group . Unlike usual lattice gauge theories the use of a nonAbelian group
here as spacetime corresponds to a background Riemannian curvature. In this
background we solve spin 0, 1/2 and spin 1 equations of motion, including the
spin 1 or `photon' case in the presence of sources, i.e. a theory of classical
electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs
from the U(1) Maxwell theory in noncommutative geometry), including the moduli
spaces of flat connections. We show that the Yang-Mills action has a simple
form in terms of Wilson loops in the permutation group, and we discuss aspects
of the quantum theory.Comment: 28 pages, LaTex as revised March 2001 -- expanded remarks in last
section on the quantum theory, but no sig. change
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
- …