Le problème est de calculer où est une mesure de probabilité sur un espace et est intégrable. Si est une suite de variables aléatoires indépendantes et distribuées selon , alors on peut approximer par
qui est dit un estimateur de Monte-Carlo.
En pratique, il peut être difficile de générer . On préférera alors introduire une mesure , avec absolument continue par rapport à , de sorte que
soit une estimée de plus commode à calculer. Cette technique, dite de l’échantillonnage préférentiel, peut aussi servir à améliorer la qualité de l’estimateur par exemple en réduisant sa variance.Read More »
Amino acids are small molecules of the form
where is a side chain called the -group. There are 20 different amino acids found in proteins, each characterized by its -group.
Peptides and proteins are chains of amino acids. Proteins are long such chains, whereas peptides and polypeptides are shorter ones. The amino acids are linked together by peptide bonds:
The following is based on a weekend project that I also presented as a short talk in an undergraduate combinatorics seminar. The project is self-contained and mostly based on independent work. Ideas and inspiration came from discussions with my teacher and from the introduction of Diaconis and Holmes (1998). Theorem 2 is from Semple and Steel (2003). Tree pictures were produced with Sagemath and Latex.
A phylogenetic tree is a rooted binary tree with labeled leaves.
These trees are used in biology to represent the evolutive history of species. The leaves are the identified species, the root is a common anscestor, and branching represents speciation.
An interesting problem is that of reconstructing the phylogenetic tree that best explains the observed biological characterics of a set of species. A naive mathematical formulation of this problem is proposed in section 4, and used to implement a tree reconstruction algorithm.
Consider the problem of estimating the volume of a tumor, given X-ray scans along orthogonal axes. It may be known that the tumor has a somewhat spherical shape. To formalise this idea, let be the tumor, the area of its surface, its volume and . From the isoperimetric inequality, we have , with equality iff is a ball. Correspondingly, we say that is a quasi-ball if . In reality, is unknown but its distribution may be determined.
We are now given the areas , and of the projections of along orthogonal axes. From the Loomis-Whitney inequality (or Cauchy-Schwarz in this case), we have the following estimate of the volume of .
Can we find such an estimate of that is close to sharp when is close to a ball?
Loomis, L. H.; Whitney, H. An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 (1949), no. 10, 961–962. http://projecteuclid.org/euclid.bams/1183514163.