Diff for "days24/abstracts"

This page contains the abstracts for the talks given at Sage days 24.

Mohamed S. Boudellioua --- On the simplification of systems of linear multidimensional equations

Linear multidimensional equations arise in the treatment of systems of partial differential equations, delay- differential equations, multidimensional discrete recursive equations, etc. The purpose of this talk is to present a constructive result on the simplification of a linear multidimensional system to an equivalent system which contains fewer equations and unknowns. In particular the case when the reduced system consists of only one equation is considered. It is shown that the transformation of zero-coprime equivalence forms the basis of such simplification. This transformation has been studied by a number of authors and has been shown to play an important role in the theory of multidimensional linear systems.

Stefan Böttner --- Mixed Transcendental and Algebraic Extensions for the Risch-Norman Algorithm

The problem of integration in finite terms for elementary functions has been solved since 1969 with the invention of the Risch algorithm. However, ever since then the sine and cosine functions have been rewritten in terms of other functions, originally using complex exponentials. Later, for the Risch-Norman algorithm, an alternative has been proposed where they are rewritten in terms of a tangent of half the angle.

We discuss extensions to the Risch-Norman algorithm that admit functions satisfying systems of differential equations (and thus also functions satisfying a differential equation of higher order). We further improve the method to allow algebraic relations to exist among the functions, paying particular attention to new logarithms that may appear and need to be predicted. This results in a heuristic but quite powerful algorithm that is able to deal with a large class of special functions and a variety of algebraic functions. In particular, it is able to work with the sine and cosine functions directly without the need to rewrite them in terms of other functions.

Frédéric Chyzak --- DDMF (Dynamic Dictionary of Mathematical Functions) and its DynaMoW

We present the prototype of a new system for displaying dynamic mathematical contents on the web (DynaMoW), together with an application based on it, our interactive web-based encyclopedia of mathematical functions (DDMF, http://ddmf.msr-inria.inria.fr). As part of DynaMoW, we developed an extension of the Ocaml language that is based on quotations and antiquotations to embed fragments of computer-algebra and mathematics-presentation languages. This extension controls the simultaneous interactions between a user and one or several computer-algebra systems, as well as the generation of mathematical documents. Our encyclopedia DDMF focuses on so-called differentiably finite functions, and can in principle be augmented with any such function. For each mathematical function, the current version (v1.5) algorithmically computes then displays: its potential symmetries; Taylor and Chebyshev series expansions; more generally, asymptotic expansions given in closed form or through definitions by recurrence; calculations of guaranteed, arbitrary-precision numerical approximations; real plots; its Laplace transform. Upon request by the user, more terms in series expansions or more digits in numerical approximations can be computed incrementally. For some of the properties, human-readable proofs are also automatically generated and displayed. (DynaMoW is joint work in progress with Alexis Darrasse; DDMF is joint work in progress with Alexandre Benoit, Alexis Darrasse, Stefan Gerhold, Marc Mezzarobba, and Bruno Salvy.)

Simon King --- Completeness criteria for modular group cohomology

The modular cohomology of a finite group is a graded commutative algebra over a finite field. Using projective resolutions and the stable element method, the algebraic structure can be "approximated" to arbitrary degree. Since the modular cohomology has a finite presentation, it is isomorphic to its degree-n approximation, if n is big enough.

Jon Carlson was the first to give a completeness criterion, that tells when n is big enough. He used it for the first modular cohomology computation for all groups of order 64. More recent completeness criteria are due to Dave Benson, Peter Symonds, David Green and myself. I implemented them in Sage, obtaining the first modular cohomology computation for all groups of order 128 and for various interesting non-prime-power groups, including the Higman-Sims group and the third Conway group.

Veronika Pillwein --- CAD and Special Functions Inequalities

Cylindrical algebraic decomposition (CAD) is a widely known tool to handle (possibly quantified) systems of polynomial equations and inequalities. As Stefan Gerhold and Manuel Kauers discovered, CAD can also be applied for proving special functions inequalities that go beyond the scope of the original area of applications. It is their approach that primarily caught my interest in CAD and in this talk I want to briefly introduce CAD, the Gerhold/Kauers-method and to present a non-trival application of their method to show the positivity of a sum over certain Gegenbauer polynomials.

Nico Temme --- Special Functions and Computer Algebra

The following points will be discussed.

An overview of basic numerical methods to compute special functions, such as series expansions, recurrence relations, continued fractions, and numerical quadrature.

Examples of certain certain asymptotic forms of special functions, which forms are missing and/or would be welcomed.

Examples where Maple and Mathematica produce wrong or too difficult answers in special functions evaluations.

A few other topics in connection with special functions and computer algebra, such as methods based on Zeilberger's summation method.