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Prochains exposés

SPOT 55. Lundi 1 Octobre 2018. Lieu : N7, salle des thèses

14h – Salma Kuhlmann (Univ Constance, Allemagne) - Positive polynomials and moment problems

Hilbert’s 17th problem asked whether a real polynomial p(x1,..,xn)
which takes non-negative values as a function on R^n is a finite sum of
squares (SOS) of real rational functions q(x1,..,xn)/r(x1,..,xn).
A complete positive answer was provided by Artin and Schreier (1927),
giving birth to real algebraic geometry. The question when the (SOS)
representation is denominator free is however of particular interest for
applications. In his pioneering 1888 paper, Hilbert gave a general
answer (in terms of degree and number of variables). Subsequent general
results, such as Krivine’s Positivstellensatz, pertain to a relative
situation, where one considers polynomials non-negative on a basic
closed semi-algebraic set K and SOSs weighted with inequalities
defining K. Stronger results hold when K is compact; the Archimedean
Positivstellensatz of Putinar and Jacobi-Prestel is a fundamental tool
in theory and applications. By the classical Riesz-Haviland theorem
(1930s), the problem of characterizing positive polynomials on a given
closed subset K of R^n is dual to the finite dimensional moment problem
(i.e. that of representing a linear functional on the polynomial
algebra R[x1,..,xn] as integration with respect to a Borel measure).
An algebraic approach was taken in a series of papers by Ghasemi-
Kuhlmann-Marshall (2013-2016) who study the moment problem on a
general not necessarily finitely generated commutative unital real
algebra, a context adapted to infinite dimensional moment problems. In
this talk I will survey (with examples) various Positivstellensaetze and
their corresponding moment problem interpretations.

 

15h – Jean-Claude Yakoubsohn (Univ Paul Sabatier Toulouse) – Certified and Fast Computation of SVD

This work is in progress with the collaboration  Joris Van Der Hoeven of the Laboratory LIX.  We define and study a new  method to approximate locally the SVD of a general  matrix: this method  has  a local quadratic convergence.  This solves the problem of the fast certified approximation of the multiple singular values or clusters of singular values.  We also give result of complexity of this approximation using such a type of homotopy method.

 


Comité local d’organisation

Cf un glossaire expliquant ces sigles et affiliations du système universitaire toulousain.


 Fréquence et structure

Une séance par mois environ, avec deux conférenciers chaque fois (deux conférences de type différent : une orientée fondements et une orientée applications, un conférencier de l’environnement toulousain et un conférencier extérieur, un conférencier du milieu académique et un conférencier du milieu de l’industrie et des services, etc.).

Horaire habituel : le lundi après-midi de 14h à 16h.


Lieu

Sauf indication contraire, à l’amphithéâtre des (SPOT-) thèses à l’ENSEEIHT (N7), 2 rue Charles Camichel, 31000 Toulouse  (métro B, François Verdier). Attention, présentez-vous au poste de garde afin d’accéder au site.