Abstract:
We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected value of this spectrum and obtain some almost sure convergence results thanks to classical properties of Crump-Mode-Jagers (CMJ) processes counted by random characteristics. Then, by using multitype CMJ-processes, we get asymptotic properties about the number of alleles that have undergone a fixed number of mutations with respect to the ancestral allele of the population.

Abstract:
In the present work, we consider spectrally positive L\'evy processes $(X_t,t\geq0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting 0. This way we obtain a new conditioning of L\'evy processes to stay positive. The (honest) law $\pfl$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For L\'evy processes with infinite variation paths, this martingale is $(\int\tilde\rt(\mathrm{d}z)e^{\alpha z}+I_t)\2{t\leq T_0}$ for some $\alpha$ and where $(I_t,t\geq0)$ is the past infimum process of $X$, where $(\tilde\rt,t\geq0)$ is the so-called \emph{exploration process} defined in Duquesne, 2002, and where $T_0$ is the hitting time of 0 for $X$. Under $\pfl$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\pfl$ as $x\to0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)critical \emph{splitting tree}. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

Abstract:
We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P_1,P_2,...) of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter theta/b.

Abstract:
The standardised precipitation index (SPI) is an index that allows monitoring the intensity and spatial extension of droughts at different time scales (3, 6, 12 and 24 months). The SPI is linked to the probability occurrence of dry or wet events. The SPI allows monitoring operationally any location with a 30-year time series. It is also used here to do a retrospective analysis of the spatial extension and intensity of droughts in South Africa since 1921. According to this index, the 8 most severe droughts at the 6-month time scale for the summer rainfall region of South Africa happened in 1926, 1933, 1945, 1949, 1952, 1970, 1983 and 1992. There is considerable decadal variability and an 18 to 20 year cycle is only found in the number of dry districts. The total number of wet and dry districts per decade seems to have increased since the 1960s. Drought lasting 3 years is not uncommon for each of the 8 South African rainfall regions defined by the South African Weather Service. Combining the retrospective analysis with real time monitoring could be extremely beneficial in the development of response, mitigation strategies and awareness plans. WaterSA Vol.29(4) 2003: 489-500

Abstract:
We show that the standard generating set of a Coxeter group is of minimal cardinality provided that the non-diagonal entries of the Coxeter matrix are sufficiently large.

Abstract:
The structure of the stratum corneum contributes to the barrier function of the epidermis. Skin barrier recovery is of utmost importance after epidermal tissue damage. The aim of this study was to describe, at the cellular level, the structural effects resulting from topical application of a hand-cream onto normal skin and to investigate the potential repair mechanisms induced by the emollient on altered tissue. Transmission electron microscopy (TEM) was used to compare the architectures of the horny layers from: 1) ex-vivo cultured human skin; 2) skin treated by topical application of a hand-cream emulsion; 3) explants exposed to sodium lauryl sulfate (SLS); 4) SLS-treated explants that underwent subsequent topical application of the emollient emulsion. These TEM assessments allowed identifying the structural changes occurring in the stratum corneum of skin explants exposed to SLS and/or treated with an emollient. Results strongly suggest that both, SLS-induced damage and emollient-driven repair process take place in the stratum corneum, at the cellular level. One can envisage that the observed restructuring effects after topical application of the skin-care product are likely to ameliorate or restore the barrier function of the stratum corneum. In this, the properties of the emollient go beyond the cosmetic feel.

Abstract:
We review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time . We give closed-form formulae for the expected frequency spectrum at and prove a pathwise convergence to an explicit limit, as , of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models. 1. Introduction We consider a general branching model, where particles have i.i.d. (not necessarily exponential) life lengths and give birth at a constant rate during their lives to independent copies of themselves. The genealogical tree thus produced is called splitting tree [1–3]. The process that counts the number of the alive particles through time is a Crump-Mode-Jagers process (or general branching process) [4] which is binary (births occur singly) and homogeneous (constant birth rate). We enrich this genealogical model with mutations. In Model I, each child is a clone of her mother with probability and a mutant with probability . In Model II, independently of other particles, each particle undergoes mutations during her life at constant rate (and births are always clonal). For both models, we are working under the infinitely many alleles model; that is, a mutation yields a type, also called allele, which was never encountered before. Moreover, mutations are supposed to be neutral; that is, they do not modify the way particles die and reproduce. For any type and any time , we call family the set of all particles that share this type at time . Branching processes (and especially birth and death processes) with mutations have many applications in biology. In carcinogenesis [5–10], they can model the evolution of cancerous cells. In [11], Kendall modeled carcinogenesis by a birth and death process where mutations occur during life according to an inhomogeneous Poisson process. In [8, 10], cancerous cells are modeled by a multitype branching process where a cell is of type if it has undergone mutations and where the more a cell has undergone mutations, the faster it

Abstract:
In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.

Abstract:
Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices --especially of high-dimensional ones-- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.