1,124 research outputs found
Highest weight irreducible representations of the Lie superalgebra
Two classes of irreducible highest weight modules of the general linear Lie
superalgebra are constructed. Within each module a basis is
introduced and the transformation relations of the basis under the action of
the algebra generators are written down.Comment: 24 pages, TeX; Journ. Math. Phys. (to be published
Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation
We consider a class of real random polynomials, indexed by an integer d, of
large degree n and focus on the number of real roots of such random
polynomials. The probability that such polynomials have no real root in the
interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the
exponent associated to the decay of the persistence probability for the
diffusion equation with random initial conditions in space dimension d. For n
even, the probability that such polynomials have no root on the full real axis
decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a
physical realization of real random polynomials. We further show that the
probability that such polynomials have exactly k real roots in [0,1] has an
unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x)
is a universal large deviation function.Comment: 4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev.
Let
Wavefunctions and counting formulas for quasiholes of clustered quantum Hall states on a sphere
The quasiholes of the Read-Rezayi clustered quantum Hall states are
considered, for any number of particles and quasiholes on a sphere, and for any
degree k of clustering. A set of trial wavefunctions, that are zero-energy
eigenstates of a k+1-body interaction, and so are symmetric polynomials that
vanish when any k+1 particle coordinates are equal, is obtained explicitly and
proved to be both complete and linearly independent. Formulas for the number of
states are obtained, without the use of (but in agreement with) conformal field
theory, and extended to give the number of states for each angular momentum. An
interesting recursive structure emerges in the states that relates those for k
to those for k-1. It is pointed out that the same numbers of zero-energy states
can be proved to occur in certain one-dimensional models that have recently
been obtained as limits of the two-dimensional k+1-body interaction
Hamiltonians, using results from the combinatorial literature.Comment: 9 pages. v2: minor corrections; additional references; note added on
connection with one-dimensional Hamiltonians of recent interes
Singular vectors by Fusions in affine su(2)
Explicit expressions for the singular vectors in the highest weight
representations of are obtained using the fusion formalism of
conformal field theory.Comment: 7 page
Classification of linearly compact simple Nambu-Poisson algebras
We introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A, by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8, 1\u2013145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) n-Nambu-Poisson algebras over an algebraically closed field of characteristic zero
Open Systems Viewed Through Their Conservative Extensions
A typical linear open system is often defined as a component of a larger
conservative one. For instance, a dielectric medium, defined by its frequency
dependent electric permittivity and magnetic permeability is a part of a
conservative system which includes the matter with all its atomic complexity. A
finite slab of a lattice array of coupled oscillators modelling a solid is
another example. Assuming that such an open system is all one wants to observe,
we ask how big a part of the original conservative system (possibly very
complex) is relevant to the observations, or, in other words, how big a part of
it is coupled to the open system? We study here the structure of the system
coupling and its coupled and decoupled components, showing, in particular, that
it is only the system's unique minimal extension that is relevant to its
dynamics, and this extension often is tiny part of the original conservative
system. We also give a scenario explaining why certain degrees of freedom of a
solid do not contribute to its specific heat.Comment: 51 page
Designed Interaction Potentials via Inverse Methods for Self-Assembly
We formulate statistical-mechanical inverse methods in order to determine
optimized interparticle interactions that spontaneously produce target
many-particle configurations. Motivated by advances that give experimentalists
greater and greater control over colloidal interaction potentials, we propose
and discuss two computational algorithms that search for optimal potentials for
self-assembly of a given target configuration. The first optimizes the
potential near the ground state and the second near the melting point. We begin
by applying these techniques to assembling open structures in two dimensions
(square and honeycomb lattices) using only circularly symmetric pair
interaction potentials ; we demonstrate that the algorithms do indeed cause
self-assembly of the target lattice. Our approach is distinguished from
previous work in that we consider (i) lattice sums, (ii) mechanical stability
(phonon spectra), and (iii) annealed Monte Carlo simulations. We also devise
circularly symmetric potentials that yield chain-like structures as well as
systems of clusters.Comment: 28 pages, 23 figure
Fusion and singular vectors in A1{(1)} highest weight cyclic modules
We show how the interplay between the fusion formalism of conformal field
theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for
the singular vectors in the highest weight representations of A1{(1)}.Comment: 42 page
Highest weight representations of the quantum algebra U_h(gl_\infty)
A class of highest weight irreducible representations of the quantum algebra
U_h(gl_\infty) is constructed. Within each module a basis is introduced and the
transformation relations of the basis under the action of the Chevalley
generators are explicitly written.Comment: 7 pages, PlainTe
Complex-valued fractional derivatives on time scales
We introduce a notion of fractional (noninteger order) derivative on an
arbitrary nonempty closed subset of the real numbers (on a time scale). Main
properties of the new operator are proved and several illustrative examples
given.Comment: This is a preprint of a paper whose final and definite form will
appear in Springer Proceedings in Mathematics & Statistics, ISSN: 2194-1009.
Accepted for publication 06/Nov/201
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