4,639 research outputs found
Tree expansion in time-dependent perturbation theory
The computational complexity of time-dependent perturbation theory is
well-known to be largely combinatorial whatever the chosen expansion method and
family of parameters (combinatorial sequences, Goldstone and other Feynman-type
diagrams...). We show that a very efficient perturbative expansion, both for
theoretical and numerical purposes, can be obtained through an original
parametrization by trees and generalized iterated integrals. We emphasize above
all the simplicity and naturality of the new approach that links perturbation
theory with classical and recent results in enumerative and algebraic
combinatorics. These tools are applied to the adiabatic approximation and the
effective Hamiltonian. We prove perturbatively and non-perturbatively the
convergence of Morita's generalization of the Gell-Mann and Low wavefunction.
We show that summing all the terms associated to the same tree leads to an
utter simplification where the sum is simpler than any of its terms. Finally,
we recover the time-independent equation for the wave operator and we give an
explicit non-recursive expression for the term corresponding to an arbitrary
tree.Comment: 22 pages, 2 figure
Transitive factorizations of permutations and geometry
We give an account of our work on transitive factorizations of permutations.
The work has had impact upon other areas of mathematics such as the enumeration
of graph embeddings, random matrices, branched covers, and the moduli spaces of
curves. Aspects of these seemingly unrelated areas are seen to be related in a
unifying view from the perspective of algebraic combinatorics. At several
points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th
birthda
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
Determinants of (generalised) Catalan numbers
We show that recent determinant evaluations involving Catalan numbers and
generalisations thereof have most convenient explanations by combining the
Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a
simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This
approach leads also naturally to extensions and generalisations.Comment: AmS-TeX, 16 pages; minor correction
Computations involving differential operators and their actions on functions
The algorithms derived by Grossmann and Larson (1989) are further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear dynamical systems. These algorithms are extended in two different directions: the algorithms are generalized so that they apply to differential operators on groups and the data structures and algorithms are developed to compute symbolically the action of differential operators on functions. Both of these generalizations are needed for applications
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