170 research outputs found
Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten
A determinant evaluation is proven, a special case of which establishes a
conjecture of Bombieri, Hunt, and van der Poorten (Experimental Math\. {\bf 4}
(1995), 87--96) that arose in the study of Thue's method of approximating
algebraic numbers.Comment: AMSTe
Directed-loop Monte Carlo simulations of vertex models
We show how the directed-loop Monte Carlo algorithm can be applied to study
vertex models. The algorithm is employed to calculate the arrow polarization in
the six-vertex model with the domain wall boundary conditions (DWBC). The model
exhibits spatially separated ordered and ``disordered'' regions. We show how
the boundary between these regions depends on parameters of the model. We give
some predictions on the behavior of the polarization in the thermodynamic limit
and discuss the relation to the Arctic Circle theorem.Comment: Extended version with autocorrelations and more figures. Added 2
reference
Spanning forest polynomials and the transcendental weight of Feynman graphs
We give combinatorial criteria for predicting the transcendental weight of
Feynman integrals of certain graphs in theory. By studying spanning
forest polynomials, we obtain operations on graphs which are weight-preserving,
and a list of subgraphs which induce a drop in the transcendental weight.Comment: 30 page
Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations
We consider the numerical discretization of the time-domain Maxwell's
equations with an energy-conserving discontinuous Galerkin finite element
formulation. This particular formulation allows for higher order approximations
of the electric and magnetic field. Special emphasis is placed on an efficient
implementation which is achieved by taking advantage of recurrence properties
and the tensor-product structure of the chosen shape functions. These
recurrences have been derived symbolically with computer algebra methods
reminiscent of the holonomic systems approach.Comment: 16 pages, 1 figure, 1 table; Springer Wien, ISBN 978-3-7091-0793-
In Praise of an Elementary Identity of Euler
We survey the applications of an elementary identity used by Euler in one of
his proofs of the Pentagonal Number Theorem. Using a suitably reformulated
version of this identity that we call Euler's Telescoping Lemma, we give
alternate proofs of all the key summation theorems for terminating
Hypergeometric Series and Basic Hypergeometric Series, including the
terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz
sum, and their -analogues. We also give a proof of Jackson's -analog of
Dougall's sum, the sum of a terminating, balanced, very-well-poised
sum. Our proofs are conceptually the same as those obtained by the WZ method,
but done without using a computer. We survey identities for Generalized
Hypergeometric Series given by Macdonald, and prove several identities for
-analogs of Fibonacci numbers and polynomials and Pell numbers that have
appeared in combinatorial contexts. Some of these identities appear to be new.Comment: Published versio
A double bounded key identity for Goellnitz's (big) partition theorem
Given integers i,j,k,L,M, we establish a new double bounded q-series identity
from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon
for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the
identity yields a strong refinement of Goellnitz's theorem with a bound on the
parts given by L. This is the first time a bounded version of Goellnitz's (big)
theorem has been proved. This leads to new bounded versions of Jacobi's triple
product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on
Symbolic Computation
Logarithmic and complex constant term identities
In recent work on the representation theory of vertex algebras related to the
Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic
analogues of (special cases of) the famous Dyson and Morris constant term
identities. In this paper we show how the identities of Adamovic and Milas
arise naturally by differentiating as-yet-conjectural complex analogues of the
constant term identities of Dyson and Morris. We also discuss the existence of
complex and logarithmic constant term identities for arbitrary root systems,
and in particular prove complex and logarithmic constant term identities for
the root system G_2.Comment: 26 page
On two-point boundary correlations in the six-vertex model with DWBC
The six-vertex model with domain wall boundary conditions (DWBC) on an N x N
square lattice is considered. The two-point correlation function describing the
probability of having two vertices in a given state at opposite (top and
bottom) boundaries of the lattice is calculated. It is shown that this
two-point boundary correlator is expressible in a very simple way in terms of
the one-point boundary correlators of the model on N x N and (N-1) x (N-1)
lattices. In alternating sign matrix (ASM) language this result implies that
the doubly refined x-enumerations of ASMs are just appropriate combinations of
the singly refined ones.Comment: v2: a reference added, typos correcte
Enumeration of quarter-turn symmetric alternating-sign matrices of odd order
It was shown by Kuperberg that the partition function of the square-ice model
related to the quarter-turn symmetric alternating-sign matrices of even order
is the product of two similar factors. We propose a square-ice model whose
states are in bijection with the quarter-turn symmetric alternating-sign
matrices of odd order, and show that the partition function of this model can
be also written in a similar way. This allows to prove, in particular, the
conjectures by Robbins related to the enumeration of the quarter-turn symmetric
alternating-sign matrices.Comment: 11 pages, 13 figures; minor correction
Dynamical Belyi maps
We study the dynamical properties of a large class of rational maps with
exactly three ramification points. By constructing families of such maps, we
obtain infinitely many conservative maps of degree ; this answers a question
of Silverman. Rather precise results on the reduction of these maps yield
strong information on the rational dynamics.Comment: 21 page
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