134 research outputs found
Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities
We present an algorithm for computing a holonomic system for a definite
integral of a holonomic function over a domain defined by polynomial
inequalities. If the integrand satisfies a holonomic difference-differential
system including parameters, then a holonomic difference-differential system
for the integral can also be computed. In the algorithm, holonomic
distributions (generalized functions in the sense of L. Schwartz) are
inevitably involved even if the integrand is a usual function.Comment: corrected typos; Sections 5 and 6 were slightly revised with results
unchange
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-
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
We show that the n-fold integrals of the magnetic susceptibility
of the Ising model, as well as various other n-fold integrals of the "Ising
class", or n-fold integrals from enumerative combinatorics, like lattice Green
functions, are actually diagonals of rational functions. As a consequence, the
power series expansions of these solutions of linear differential equations
"Derived From Geometry" are globally bounded, which means that, after just one
rescaling of the expansion variable, they can be cast into series expansions
with integer coefficients. Besides, in a more enumerative combinatorics
context, we show that generating functions whose coefficients are expressed in
terms of nested sums of products of binomial terms can also be shown to be
diagonals of rational functions. We give a large set of results illustrating
the fact that the unique analytical solution of Calabi-Yau ODEs, and more
generally of MUM ODEs, is, almost always, diagonal of rational functions. We
revisit Christol's conjecture that globally bounded series of G-operators are
necessarily diagonals of rational functions. We provide a large set of examples
of globally bounded series, or series with integer coefficients, associated
with modular forms, or Hadamard product of modular forms, or associated with
Calabi-Yau ODEs, underlying the concept of modularity. We finally address the
question of the relations between the notion of integrality (series with
integer coefficients, or, more generally, globally bounded series) and the
modularity (in particular integrality of the Taylor coefficients of mirror
map), introducing new representations of Yukawa couplings.Comment: 100 page
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
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