1,896 research outputs found
Eventually rational and m-sparse points of linear ordinary differential operators with polynomial coefficients
AbstractLet L(y)=0 be a linear homogeneous ordinary differential equation with polynomial coefficients. One of the general problems connected with such an equation is to find all points a (ordinary or singular) and all formal power series ∑n=0∞cn(x−a)n which satisfy L(y)=0 and whose coefficient cn — considered as a function of n — has some ‘nice’ properties: for example, cn has an explicit representation in terms of n, or the sequence (c0,c1,…) has many zero elements, and so on. It is possible that such properties appear only eventually (i.e., only for large enough n). We consider two particular cases: 1.(c0,c1,…) is an eventually rational sequence, i.e., cn=R(n) for all large enough n, where R(n) is a rational function of n;2.(c0,c1,…) is an eventually m-sparse sequence, where m⩾2, i.e., there exists an integer N such that(cn≠0)⇒(n≡N(modm))for all large enough n.Note that those two problems were previously solved only ‘for all n’ rather than ‘for n large enough’, although similar problems connected with polynomial and hypergeometric sequences of coefficients have been solved completely
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Symbolic-numeric interface: A review
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
The Magnus expansion and some of its applications
Approximate resolution of linear systems of differential equations with
varying coefficients is a recurrent problem shared by a number of scientific
and engineering areas, ranging from Quantum Mechanics to Control Theory. When
formulated in operator or matrix form, the Magnus expansion furnishes an
elegant setting to built up approximate exponential representations of the
solution of the system. It provides a power series expansion for the
corresponding exponent and is sometimes referred to as Time-Dependent
Exponential Perturbation Theory. Every Magnus approximant corresponds in
Perturbation Theory to a partial re-summation of infinite terms with the
important additional property of preserving at any order certain symmetries of
the exact solution. The goal of this review is threefold. First, to collect a
number of developments scattered through half a century of scientific
literature on Magnus expansion. They concern the methods for the generation of
terms in the expansion, estimates of the radius of convergence of the series,
generalizations and related non-perturbative expansions. Second, to provide a
bridge with its implementation as generator of especial purpose numerical
integration methods, a field of intense activity during the last decade. Third,
to illustrate with examples the kind of results one can expect from Magnus
expansion in comparison with those from both perturbative schemes and standard
numerical integrators. We buttress this issue with a revision of the wide range
of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its
applications to several physical problem
Resultant-based Elimination in Ore Algebra
We consider resultant-based methods for elimination of indeterminates of Ore
polynomial systems in Ore algebra. We start with defining the concept of
resultant for bivariate Ore polynomials then compute it by the Dieudonne
determinant of the polynomial coefficients. Additionally, we apply
noncommutative versions of evaluation and interpolation techniques to the
computation process to improve the efficiency of the method. The implementation
of the algorithms will be performed in Maple to evaluate the performance of the
approaches.Comment: An updated (and shorter) version published in the SYNASC '21
proceedings (IEEE CS) with the title "Resultant-based Elimination for Skew
Polynomials
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