Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.Comment: LaTeX; 40 pages; review pape

Growth of the Brownian forest

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton--Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams' decomposition for Brownian motion with drift.Comment: Published at http://dx.doi.org/10.1214/009117905000000422 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dual random fragmentation and coagulation and an application to the genealogy of Yule processes

The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.Comment: 14 page

Influence of homology and node-age on the growth of protein-protein interaction networks

Proteins participating in a protein-protein interaction network can be grouped into homology classes following their common ancestry. Proteins added to the network correspond to genes added to the classes, so that the dynamics of the two objects are intrinsically linked. Here, we first introduce a statistical model describing the joint growth of the network and the partitioning of nodes into classes, which is studied through a combined mean-field and simulation approach. We then employ this unified framework to address the specific issue of the age dependence of protein interactions, through the definition of three different node wiring/divergence schemes. Comparison with empirical data indicates that an age-dependent divergence move is necessary in order to reproduce the basic topological observables together with the age correlation between interacting nodes visible in empirical data. We also discuss the possibility of nontrivial joint partition/topology observables.Comment: 14 pages, 7 figures [accepted for publication in PRE

Goethite on Mars - A laboratory study of physically and chemically bound water in ferric oxides

Thermogravimetric study of physically and chemically bound water in ferric oxides of limonite with application to goethite on Mar

Abrupt change in climate and climate models

First, we review the evidence that abrupt climate changes have occurred in the past and then demonstrate that climate models have developing capacity to simulate many of these changes. In particular, the processes by which changes in the ocean circulation drive abrupt changes appear to be captured by climate models to a degree that is encouraging. The evidence that past changes in the ocean have driven abrupt change in terrestrial systems is also convincing, but these processes are only just beginning to be included in climate models. Second, we explore the likelihood that climate models can capture those abrupt changes in climate that may occur in the future due to the enhanced greenhouse effect. We note that existing evidence indicates that a major collapse of the thermohaline circulation seems unlikely in the 21st century, although very recent evidence suggests that a weakening may already be underway. We have confidence that current climate models can capture a weakening, but a collapse in the 21st century of the thermohaline circulation is not projected by climate models. Worrying evidence of instability in terrestrial carbon, from observations and modelling studies, is beginning to accumulate. Current climate models used by the Intergovernmental Panel on Climate Change for the 4th Assessment Report do not include these terrestrial carbon processes. We therefore can not make statements with any confidence regarding these changes. At present, the scale of the terrestrial carbon feedback is believed to be small enough that it does not significantly affect projections of warming during the first half of the 21st century. However, the uncertainties in how biological systems will respond to warming are sufficiently large to undermine confidence in this belief and point us to areas requiring significant additional work

The Action of Sulphuric Acid on Mercury.

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Random trees with superexponential branching weights

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case when $w_n$ grows faster than exponentially with $n$. In this case the measures on trees of finite size $N$ converge weakly as $N$ tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form $w_n=((n-1)!)^\alpha$ with $\alpha >0$ we obtain more refined results about the approach to the infinite volume limit.Comment: 19 page