Explicit enumeration of triangulations with multiple boundaries

We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte

Percolating paths through random points :

We prove consistency of four different approaches to formalizing the idea of minimum average edge-length in a path linking some infinite subset of points of a Poisson process. The approaches are (i) shortest path from origin through some $m$ distinct points; (ii) shortest average edge-length in paths across the diagonal of a large cube; (iii) shortest path through some specified proportion $\delta$ of points in a large cube; (iv) translation-invariant measures on paths in $\Reals^d$ which contain a proportion $\delta$ of the Poisson points. We develop basic properties of a normalized average length function $c(\delta)$ and pose challenging open problemComment: 28 page

On one property of distances in the infinite random quadrangulation

We show that the Schaeffer's tree for an infinite quadrangulation only changes locally when changing the root of the quadrangulation. This follows from one property of distances in the infinite uniform random quadrangulation

Stochastic Models for Phylogenetic Trees on Higher-order Taxa

Simple stochastic models for phylogenetic trees on species have been well studied. But much paleontology data concerns time series or trees on higher-order taxa, and any broad picture of relationships between extant groups requires use of higher-order taxa. A coherent model for trees on (say) genera should involve both a species-level model and a model for the classification scheme by which species are assigned to genera. We present a general framework for such models, and describe three alternate classification schemes. Combining with the species-level model of Aldous-Popovic (2005), one gets models for higher-order trees, and we initiate analytic study of such models. In particular we derive formulas for the lifetime of genera, for the distribution of number of species per genus, and for the offspring structure of the tree on genera.Comment: 41 pages. Minor revision

Birth and death processes on certain random trees: Classification and stationary laws

The main substance of the paper concerns the growth rate and the classification (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process of parameter $\lambda$ and leaves can be deleted at a rate $\mu$. The main results lay the stress on the famous number $e$. A complete classification of the process is given in terms of the intensity factor $\rho=\lambda/\mu$: it is ergodic if $\rho\leq e^{-1}$, and transient if $\rho>e^{-1}$. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Some basic stationary laws are computed, e.g. the number of vertices and the height. Various bounds, limit laws and ergodic-like theorems are obtained, both for the transient and ergodic regimes. In particular, when the system is transient, the height of the tree grows linearly as the time $t\to\infty$, at a rate which is explicitly computed. Some of the results are extended to the so-called multiclass model

Growth Rate and Ergodicity Conditions for a Class of Random Trees

Projet MEVALThis paper gives the growth rate and the ergodicity conditions for a simple class of random trees. New edges appear according to a Poisson process, and leaves can be deleted at a rate $\mu$. The main results lay the stress on the famous number $e$. In the case of a pure birth process, i.e. $\mu=0$, the height of the tree at time $t$ grows linearly at the rate $e$, in mean and almost surely as $t\to\infty$. When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor $\rho=\lambda/\mu\,$: it is ergodic if $\rho\leq e^{-1}$, and transient if $\rho>e^{-1}$. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Bounds are obtained for the transient regime

Reinforcement Learning based Curriculum Optimization for Neural Machine Translation

We consider the problem of making efficient use of heterogeneous training data in neural machine translation (NMT). Specifically, given a training dataset with a sentence-level feature such as noise, we seek an optimal curriculum, or order for presenting examples to the system during training. Our curriculum framework allows examples to appear an arbitrary number of times, and thus generalizes data weighting, filtering, and fine-tuning schemes. Rather than relying on prior knowledge to design a curriculum, we use reinforcement learning to learn one automatically, jointly with the NMT system, in the course of a single training run. We show that this approach can beat uniform and filtering baselines on Paracrawl and WMT English-to-French datasets by up to +3.4 BLEU, and match the performance of a hand-designed, state-of-the-art curriculum.Comment: NAACL 2019 short paper. Reviewer comments not yet addresse

Birth and Death Processes on Certain Random Trees : Classification and Stationary Laws

Projet MEVALThe main substance of the paper concerns the growth rate and the classificatio- n (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process, and leaves can be deleted at a rate . The main results lay the stress on the famous number e. In the case of a pure birth process, i.e. =0, the height of the tree at time t grows linearly at the rate e, in mean and almost surely as t. When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor =/¸: it is ergodic if e^-1, and transient if >e^-1. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jump. Bounds are obtained for the transient regime. Some basic stationary laws are computed, e.g. the number of nodes and the height. An extension to the so-called multiclass case is presented, with a more complex classification