Évènements

Surveying natural populations is challenging due to their scattered distribution across a territory. To create spatially balanced samples, surveys typically divide the territory into a spatial grid and either use the grid nodes to form the sample or select points within the grid cells. Sampling the cells adds an additional stage, as currently employed by the French National Forest Inventory (NFI) for annual estimates. However, little attention has been given to accounting for this stage. Double sampling for stratification is a general method that helps reduce the size of a field sample, which is particularly costly. To improve sampling efficiency, we propose a new framework called two-stage two-phase sampling, incorporating a two-stage sampling design in the first phase.

The Horvitz-Thompson estimator is used to estimate the total value. In the first stage, cells are sampled using spatially systematic sampling, and in the second stage, points within these cells are sampled uniformly. The classification of first-phase points into strata is performed through photo-interpretation. In the second phase, points are sampled using spatially systematic sampling over the first-phase sample, based on varying sampling intensities across the strata. To calculate the variance estimator, the global first-phase sample is modeled as uniform sampling, and the global second-phase sample is modeled as stratified simple random sampling. Our results indicate that the expansion estimator remains unbiased and the variance estimators are moderately conservative for the sampling design used by the French NFI.

Additionally, the forest is undergoing rapid changes due to various disturbances, which can be large-scale, such as windthrow or fire, or small-scale, like bark beetle infestations. Our project focuses on large-scale disturbances. Estimating the area affected by such disturbances, known as the area of interest, is interesting for foresters. To address this, we are considering the intensification method, which increases sampling intensity in the area of interest. This method requires higher sampling intensity in specific zones compared to others, resulting in different sampling intensities across regions. A two-stage two-phase sampling framework is particularly useful for managing these varying sampling rates during the second phase, as disturbance information only becomes available at this phase.

Je proposerai une excursion aux « Fondements mathématiques » (dans le sens de l’intitulé d’une unité de notre L1 que j’étais amené à enseigner à Nancy pendant ces dernières années) : depuis le 19e siècle, la théorie des fondements des nombres et de l’analyse réels, et celle de la théorie des ensembles, se sont nourries mutuellement (Dedekind, Cantor,…). Au 20e siècle, cette interaction a pris un nouveau tournant : du coté théorie des ensembles, l’univers de von Neumann permet de sortir indemne de la « crise des fondements » ; du coté de la théorie des nombres, John Horton Conway proposa, dans son livre « On Numbers and Games » (connu sous le sigle ONAG), une nouvelle approche qui permet de voir les nombres réels dans un cadre beaucoup plus vaste de « tous les nombres » (« All Numbers Great and Small »). Le terme « nombres surréels », crée par Donald Knuth dans son livre Surreal numbers – how two ex-students turned on to pure mathematics and found total happiness (qui est paru même avant ONAG), est un peu malheureux car il suggère une analogie avec le courant d’art de même nom, ce qui est trompeur. Dans cet exposé, je tenterai de vous expliquer que ces nombres sont aussi réels que tout objet mathématique vivant dans l’univers mathématique, et pour lequel l’univers de von Neumann fournit un modèle. Il s’agit d’un travail en cours, loin d’être terminé.

(Joint Work with Michele Serra.)

In his paper  » Automorphisms of fields of formal power series » (Bull. Am. Math. Soc. 50, 1944) Otto Schilling described the automorphism group of k((t)), the field of Laurent series with coefficients in a ground field k and exponents in the group of integers. In our paper « The automorphism group of a valued field of generalised formal power series » (J. Algebra 605, 2022) we generalise his results to the case when the exponents lie in an arbitrary abelian group. In particular, our results apply to a variety of such fields, e.g. to the field of Puiseux series, of multivariate rational functions, of multivariate Laurent series, or to the field of surreal numbers.
The talk will be self contained talk and geared towards a general audience.