18,071 research outputs found
Counting invertible Schr\"odinger Operators over Finite Fields for Trees, Cycles and Complete Graphs
We count invertible Schr\"odinger operators (perturbations by diagonal
matrices of the adjacency matrix) over finite fieldsfor trees, cycles and
complete graphs.This is achieved for trees through the definition and use of
local invariants (algebraic constructions of perhapsindependent
interest).Cycles and complete graphs are treated by ad hoc methods.Comment: Final version to appear in Electronic Journal of Combinatoric
Computationally efficient recursions for top-order invariant polynomials with applications
The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.
Generalized Quantum Dynamics as Pre-Quantum Mechanics
We address the issue of when generalized quantum dynamics, which is a
classical symplectic dynamics for noncommuting operator phase space variables
based on a graded total trace Hamiltonian , reduces to Heisenberg
picture complex quantum mechanics. We begin by showing that when , with a Weyl ordered operator Hamiltonian, then the generalized
quantum dynamics operator equations of motion agree with those obtained from
in the Heisenberg picture by using canonical commutation relations. The
remainder of the paper is devoted to a study of how an effective canonical
algebra can arise, without this condition simply being imposed by fiat on the
operator initial values. We first show that for any total trace Hamiltonian
which involves no noncommutative constants, there is a conserved
anti--self--adjoint operator with a structure which is closely
related to the canonical commutator algebra. We study the canonical
transformations of generalized quantum dynamics, and show that is a
canonical invariant, as is the operator phase space volume element. The latter
result is a generalization of Liouville's theorem, and permits the application
of statistical mechanical methods to determine the canonical ensemble governing
the equilibrium distribution of operator initial values. We give arguments
based on a Ward identity analogous to the equipartition theorem of classical
statistical mechanics, suggesting that statistical ensemble averages of Weyl
ordered polynomials in the operator phase space variables correspond to the
Wightman functions of a unitary complex quantum mechanics, with a conserved
operator Hamiltonian and with the standard canonical commutation relations
obeyed by Weyl ordered operator strings. Thus there is a well--defined sense inComment: 79 pages, no figures, plain te
Global unitary fixing and matrix-valued correlations in matrix models
We consider the partition function for a matrix model with a global unitary
invariant energy function. We show that the averages over the partition
function of global unitary invariant trace polynomials of the matrix variables
are the same when calculated with any choice of a global unitary fixing, while
averages of such polynomials without a trace define matrix-valued correlation
functions, that depend on the choice of unitary fixing. The unitary fixing is
formulated within the standard Faddeev-Popov framework, in which the squared
Vandermonde determinant emerges as a factor of the complete Faddeev-Popov
determinant. We give the ghost representation for the FP determinant, and the
corresponding BRST invariance of the unitary-fixed partition function. The
formalism is relevant for deriving Ward identities obeyed by matrix-valued
correlation functions.Comment: Tex, 22 page
Introduction to Random Matrices
These notes provide an introduction to the theory of random matrices. The
central quantity studied is where is the integral
operator with kernel 1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y). Here
and is the characteristic function
of the set . In the Gaussian Unitary Ensemble (GUE) the probability that no
eigenvalues lie in is equal to . Also is a tau-function
and we present a new simplified derivation of the system of nonlinear
completely integrable equations (the 's are the independent variables)
that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case
of a single interval these equations are reducible to a Painlev{\'e} V
equation. For large we give an asymptotic formula for , which is
the probability in the GUE that exactly eigenvalues lie in an interval of
length .Comment: 44 page
Algebraic quantization of the closed bosonic string
The gauge invariant observables of the closed bosonic string are quantized
without anomalies in four space-time dimensions by constructing their quantum
algebra in a manifestly covariant approach. The quantum algebra is the kernel
of a derivation on the universal envelopping algebra of an infinite-dimensional
Lie algebra. The search for Hilbert space representations of this algebra is
separated from its construction, and postponed.Comment: 22 pages. Revised: minor changes as in the published version (CMP
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