Pacific Institute for the Mathematical Sciences

Abstract: The pairwise comparison of RNA secondary structures is a fundamental problem, with direct application in mining databases for annotating putative noncoding RNA candidates in newly sequenced genomes. An increasing number of software tools are available for comparing RNA secondary structures, based on different models (such as ordered trees or forests, arc annotated sequences, and multilevel trees) and computational principles (edit distance, alignment). We describe here the website BRASERO that offers tools for evaluating such software tools on real and synthetic datasets.

Abstract: Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we prove that there exists an effectively computable constant c(m,n) depending only on m and n such that the following holds. Let f_1,...,f_n be polynomials with coefficients in K, which are disintegrated polynomials of degree at most m and let F be the induced coordinate-wise self-map of the n-th dimensional affine space, i.e. F(x_1,..,x_n)=(f_1(x_1),...,f_n(x_n)). Then the period of every irreducible F-periodic subvariety of the n-th dimensional affine space with non-constant projection to each coordinate axis is at most c(m,n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt's theory of polynomial decomposition.

Abstract: This study looks at the dynamics of a Covid-19 type epidemic with a dual purpose. The first objective is to propose a reliable temporal mathematical model, based on real data and integrating the course of illness. It is a daily discrete model with different delay times, and whose parameters are calibrated from the main indicators of the epidemic. The model can be broken down in two decoupled versions: a mortality-mortality version, which can be used with the data on the number of deaths, and an infection-infection version to be used when reliable estimates of infection rate are available. The model allows to describe realistically the evolution of the main markers of the epidemic. In addition, in terms of deaths and occupied ICU beds, the model is not very sensitive to the current uncertainties about IFR. The second objective is to study several original scenarios for the epidemic’s evolution, especially after the period of strict lockdown. A coherent strategy is therefore proposed to contain the outbreak and exit lockdown, without going into the risky herd immunity approach. This strategy, called zigzag strategy , is based on a classification of the interventions into four lanes, distinguished by a marker called the daily reproduction number. The model and strategy in question are flexible and easily adaptable to new developments such as mass screenings or infection surveys. They can also be used at different geographical scales (local, regional or national).

Abstract: Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree δ ≥ 2 is said to be disintegrated if it is not linearly conjugate to x or ±Tδ(x) where Tδ(x) is the Chebyshev polynomial of degree δ. Let d and n be integers greater than 1, we prove that there exists an effectively computable constant c(d, n) depending only on d and n such that the following holds. Let f1, . . . , fn ∈ K[x] be disintegrated polynomials of degree at most d and let φ = f1 × · · · × fn be the induced coordinate-wise self-map of AK . Then the period of every irreducible φ-periodic subvariety of AK with non-constant projection to each factor AK is at most c(d, n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are the Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt’s theory of polynomial decomposition.