4,053 research outputs found
Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures
The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian
integrable systems with every untwisted affine Lie algebra. We compute the
complete set of invariants of the related bihamiltonian structures with respect
to the group of Miura type transformations.Comment: 73 pages, no figure
Moment ideals of local Dirac mixtures
In this paper we study ideals arising from moments of local Dirac measures
and their mixtures. We provide generators for the case of first order local
Diracs and explain how to obtain the moment ideal of the Pareto distribution
from them. We then use elimination theory and Prony's method for parameter
estimation of finite mixtures. Our results are showcased with applications in
signal processing and statistics. We highlight the natural connections to
algebraic statistics, combinatorics and applications in analysis throughout the
paper.Comment: 26 pages, 3 figure
The Geometry of Supermanifolds and New Supersymmetric Actions
We construct the Hodge dual for supermanifolds by means of the Grassmannian
Fourier transform of superforms. In the case of supermanifolds it is known that
the superforms are not sufficient to construct a consistent integration theory
and that the integral forms are needed. They are distribution-like forms which
can be integrated on supermanifolds as a top form can be integrated on a
conventional manifold. In our construction of the Hodge dual of superforms they
arise naturally. The compatibility between Hodge duality and supersymmetry is
exploited and applied to several examples. We define the irreducible
representations of supersymmetry in terms of integral and superforms in a new
way which can be easily generalised to several models in different dimensions.
The construction of supersymmetric actions based on the Hodge duality is
presented and new supersymmetric actions with higher derivative terms are
found. These terms are required by the invertibility of the Hodge operator.Comment: LateX2e, 51 pages. Corrected some further misprint
Covariant techniques for computation of the heat kernel
The heat kernel associated with an elliptic second-order partial differential
operator of Laplace type acting on smooth sections of a vector bundle over a
Riemannian manifold, is studied. A general manifestly covariant method for
computation of the coefficients of the heat kernel asymptotic expansion is
developed. The technique enables one to compute explicitly the diagonal values
of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De
Witt-Seeley coefficients, as well as their derivatives. The elaborated
technique is applicable for a manifold of arbitrary dimension and for a generic
Riemannian metric of arbitrary signature. It is very algorithmic, and well
suited to automated computation. The fourth heat kernel coefficient is computed
explicitly for the first time. The general structure of the heat kernel
coefficients is investigated in detail. On the one hand, the leading derivative
terms in all heat kernel coefficients are computed. On the other hand, the
generating functions in closed covariant form for the covariantly constant
terms and some low-derivative terms in the heat kernel coefficients are
constructed by means of purely algebraic methods. This gives, in particular,
the whole sequence of heat kernel coefficients for an arbitrary locally
symmetric space.Comment: 31 pages, LaTeX, no figures, Invited Lecture at the University of
Iowa, Iowa City, April, 199
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