1st TOMCAT Workshop

PROGRAMME

• ​Asymptotic enumeration for partitions with forbidden parts​ (Stella Brassesco) :

  ​A family of random variables \( X(t), t \in (0,1) \) is naturally associated to the generating function \(f(t)\) of \(p(n)\), the number of partitions of an integer \(n\). From the product formula of Euler, the variables \(X(t)\) are, for each \(t\), a sum of independent random variables \(X_j(t)\) associated likewise with each factor, and thus corresponding to the part \(j\). As \(t \to 1\), a local Central Limit Theorem for \(X(t)\) conveniently normalized holds, as observed by L. Báez-Duarte [1], who deduced then the asymptotic formula of Hardy and Ramanujan for \(p(n)\). Moreover, a further analysis based on an expansion in terms of the cumulants of \(X(t)\) yields an asymptotic expansion for \(p(n)\) [2].
  The approach is well suited to analyse partitions with restrictions on its parts. In particular, we obtain the asymptotics as \(n \to \infty\) of the number of partitions of \(n\) in the case there is a finite number of missing parts, extending results in [3]. (Joint work with A. Meyroneinc)

  [1] Luis Báez-Duarte. Hardy–Ramanujan asymptotic formula for partitions and the central limit theorem. Advances in Mathematics, 125 (1997), 114–120.
  [2] Stella Brassesco, Arnaud Meyroneinc. An expansion for the number of partitions of an integer. Ramanujan Journal, 51 (2020), 563–592.
  [3] J.L. Nicolas, A. Sárkosy. On partitions without small parts. Journal de Théorie des Nombres de Bordeaux, tome 12,1 (2000), 227–254.

• Perturbation of unimodal maps with application to synchronisation (Sandro Vaienti) :

  We consider a system of two coupled unimodal maps and we study the onset of synchronisation using dimension theory and a random annealed approach.

• A panorama on piecewise contracting maps (Pierre Guiraud) :

  A map \( f : X \to X \) defined on a compact metric space is a piecewise contracting map (PCM), if there exists a collection of disjoint open subsets of \( X \), whose union is dense, and such that each restriction of \( f \) to one of this set is a contraction. In this talk, I will propose a panorama and open questions about the asymptotic dynamics of this kind of maps. We will consider general PCM, as well as interval PCM. We will be interested in the nature of their possible attractors, the complexity of their symbolic dynamics, and the statistics of their stochastic perturbations. 

• Topological dynamics of piecewise continuous maps of the interval (Alfredo Calderón) :

  ​In the context of continuous discrete-time systems, the notion of perturbation has become a fundamental tool for describing stable and/or generic dynamic properties. However, in a discontinuous context, it is a real challenge to define a concept of perturbation that allows for the description of interesting dynamic phenomena. In fact, some authors have opted to work with perturbations that are not induced by topologies, or simply describe dynamic properties based on non-topological notions of genericity.
  In this talk, we introduce a new concept of perturbation (induced by a metric) on the space of piecewise  continuous maps of the interval. We analyze the advantages and disadvantages of considering this concept of perturbation, posing some open questions from the developed theory.

• Asymptotic formulae for the number of some coloured partitions of an integer (Arnaud Meyroneinc) :

  We present asymptotic formulae for the number of \(k\)-coloured partitions of an integer \(n\), and for the number of partitions "of \(n\) with \(n\) colours". The framework is of random partitions, following Rosenbloom's original observation on a result by Hayman, and the approach by Fristedt and Báez-Duarte. The formulae are obtained from sharp estimates of the cumulants of a family of random variables that are involved in an inversion formula, and are incidentally accurate for small \(n\). The expansion for the number of \(k\)-coloured partitions is a generalization of the case \(k=1\) obtained in [1], and its first term already improves a formula obtained by Dewar and Murty [2]. The formula for the number of partitions "of \(n\) with \(n\) colours" has been conjectured ​by Kotesovec [3]. We will address in the end a relation with some symbolic dynamical systems.  (Joint work with S. Brassesco and Y. Vargas)

  [1] Stella Brassesco, Arnaud Meyroneinc. An expansion for the number of partitions of an integer. Ramanujan Journal, 51 (2020), 563–592.
  [2] M. Ram Murty. The Partition Function Revisited. The Legacy of Srinivasa Ramanujan, RMS-Lecture Notes Series No. 20, pp. 261–279 , 2013.
  [3] OEIS Foundation Inc. (2022), Entry A008485 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A008485.

• Piecewise deterministic Markov processes in pharmacodynamic models (Lisandro Fermín) :

  We propose a Piecewise-Deterministic Markov Process (PDMP) to model the drug concentration in the case of multiple intravenous-bolus (multi-IV) doses and poor patient adherence situation: the scheduled time and doses of drug administration are not respected by the patient, the drug administration considers switching regimes with random drug intake times. We study the randomness of drug concentration and derive probability results on the stochastic dynamics using the PDMP theory, focusing on two aspects of practical relevance: the variability of the concentration and the regularity of its stationary probability distribution. The main result shows as the regularity of the concentration is governed by a bifurcation parameter, which quantifies in a precise way the situations where drug intake times are too scarce concerning the elimination rate. 

Organizing Comitee

• Pierre Guiraud (UV)
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• Gerardo Honorato (UV)
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• Arnaud Meyroneinc (UV)
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