360 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
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
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
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
Connectivity and equilibrium in random games
We study how the structure of the interaction graph of a game affects the
existence of pure Nash equilibria. In particular, for a fixed interaction
graph, we are interested in whether there are pure Nash equilibria arising when
random utility tables are assigned to the players. We provide conditions for
the structure of the graph under which equilibria are likely to exist and
complementary conditions which make the existence of equilibria highly
unlikely. Our results have immediate implications for many deterministic graphs
and generalize known results for random games on the complete graph. In
particular, our results imply that the probability that bounded degree graphs
have pure Nash equilibria is exponentially small in the size of the graph and
yield a simple algorithm that finds small nonexistence certificates for a large
family of graphs. Then we show that in any strongly connected graph of n
vertices with expansion the distribution of the number
of equilibria approaches the Poisson distribution with parameter 1,
asymptotically as .Comment: Published in at http://dx.doi.org/10.1214/10-AAP715 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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