CASC2017

   

   
    Tutorials

 

CASC 2018 will again feature two tutorials on topics related to the CASC universe. They will take place on the first day of the conference. They will be presented by leading experts in the respective fields and are targeted at graduate students and non-specialists. Participation is open to all registered participants of CASC 2018 without any additional fees.

 

Nathalie Verdière

Le Havre, France

 

Applications of computer algebra in the identifiability study and the parameter estimation. Application in neurosciences: 

Identifiability analysis adresses the question whether it is possible to estimate the model parameters for a given choice of measurement data and experimental input. This study should precede the parameter estimation in ideal case. Indeed, if the input-output model is not identifiable, this will lead to the lack of unique global extremum regarding the optimization problem corresponding to parameter estimation. Computer algebra offers an efficient way to do such a study as a numerical method to obtain a first initial guess of the unknown parameters without any knowledge of them. Indeed, it permits to obtain input-output polynomials depending only on observable (i.e. known) state variables and the unknown parameters that the experimenter wants to estimate.

The aim of the course will consist in presenting i) the algebra tools to obtain these input-output polynomials, ii) the way to use them to do an identifiability study, iii) the parameter estimation method, iv) the basic elements to understand the biological mechanism leading to the electrical behavior of a neuron v) the way to use computer algebra on classical neuronal models to identify and estimate the parameters.


 

Mark Moreno Maza

London, Ontario, Canada

 

Computing limits with the RegularChains and PowerSeries libraries: from rational functions to topological closures.

While computer algebra systems can perform highly sophisticated algebraic tasks, they are much less equipped for solving problems from mathematical analysis in a symbolic manner. Elementary problems in analysis, like the manipulation of Taylor series and the calculation of limits of univariate functions are supported, with some limitations, in general-purpose computer algebra systems such as Maple and Mathematica. However, limits of multivariate functions and more advanced notions of limits, like topological closures, are almost absent from such systems. For instance, and quite surprisingly, Maple is not capable of computing limits of rational functions in more than two variables.

Many fundamental concepts in mathematics are defined in terms of limits and it is highly desirable for computer algebra to implement those concepts. However, limits are, by essence, hard to compute, or even not computable in an algorithmic fashion, say by doing finitely many rational operations on polynomials or matrices.

In this tutorial, we shall see how various types of limits can be computed by means of algebraic calculations. Examples will cover the Zariski closure of a constructible set, the tangent cone of an algebraic set at one of its singular points, and the limit of a real multivariate rational function at one of its poles.

The tutorial will include a presentation of the underlying mathematical concepts and algorithms as well as an extended software demonstration powered by the RegularChains and PowerSeries libraries.