1,161 research outputs found
Galilean contractions of -algebras
Infinite-dimensional Galilean conformal algebras can be constructed by
contracting pairs of symmetry algebras in conformal field theory, such as
-algebras. Known examples include contractions of pairs of the Virasoro
algebra, its superconformal extension, or the algebra. Here, we
introduce a contraction prescription of the corresponding operator-product
algebras, or equivalently, a prescription for contracting tensor products of
vertex algebras. With this, we work out the Galilean conformal algebras arising
from contractions of and superconformal algebras as well as of the
-algebras , , , and . The latter results provide
evidence for the existence of a whole new class of -algebras which we call
Galilean -algebras. We also apply the contraction prescription to affine Lie
algebras and find that the ensuing Galilean affine algebras admit a Sugawara
construction. The corresponding central charge is level-independent and given
by twice the dimension of the underlying finite-dimensional Lie algebra.
Finally, applications of our results to the characterisation of structure
constants in -algebras are proposed.Comment: 45 pages, v2: minor changes, references adde
Picard curves over Q with good reduction away from 3
Inspired by methods of N. P. Smart, we describe an algorithm to determine all
Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A
correspondence between the isomorphism classes of such curves and certain
quintic binary forms possessing a rational linear factor is established. An
exhaustive list of integral models is determined, and an application to a
question of Ihara is discussed.Comment: 27 pages; A previous lemma was incorrect and has been removed;
Corrected computation has identified 18 new such curves (63 in total
Elements of a Theory of Simulation
Unlike computation or the numerical analysis of differential equations,
simulation does not have a well established conceptual and mathematical
foundation. Simulation is an arguable unique union of modeling and computation.
However, simulation also qualifies as a separate species of system
representation with its own motivations, characteristics, and implications.
This work outlines how simulation can be rooted in mathematics and shows which
properties some of the elements of such a mathematical framework has. The
properties of simulation are described and analyzed in terms of properties of
dynamical systems. It is shown how and why a simulation produces emergent
behavior and why the analysis of the dynamics of the system being simulated
always is an analysis of emergent phenomena. A notion of a universal simulator
and the definition of simulatability is proposed. This allows a description of
conditions under which simulations can distribute update functions over system
components, thereby determining simulatability. The connection between the
notion of simulatability and the notion of computability is defined and the
concepts are distinguished. The basis of practical detection methods for
determining effectively non-simulatable systems in practice is presented. The
conceptual framework is illustrated through examples from molecular
self-assembly end engineering.Comment: Also available via http://studguppy.tsasa.lanl.gov/research_team/
Keywords: simulatability, computability, dynamics, emergence, system
representation, universal simulato
Class number formulas via 2-isogenies of elliptic curves
A classical result of Dirichlet shows that certain elementary character sums
compute class numbers of quadratic imaginary number fields. We obtain analogous
relations between class numbers and a weighted character sum associated to a
2-isogeny of elliptic curves.Comment: 19 pages; To appear in the Bulletin of the London Mathematical
Societ
Higher-order Galilean contractions
A Galilean contraction is a way to construct Galilean conformal algebras from
a pair of infinite-dimensional conformal algebras, or equivalently, a method
for contracting tensor products of vertex algebras. Here, we present a
generalisation of the Galilean contraction prescription to allow for inputs of
any finite number of conformal algebras, resulting in new classes of
higher-order Galilean conformal algebras. We provide several detailed examples,
including infinite hierarchies of higher-order Galilean Virasoro algebras,
affine Kac-Moody algebras and the associated Sugawara constructions, and
algebras.Comment: 15 page
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