On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $nin mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.     

Infinite groups with large balls of torsion elements and small entropy

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. Received: 3 August 2005     

Distributed caching over heterogeneous mobile networks

Sharing content over a mobile network through opportunistic contacts has recently received considerable attention. In proposed scenarios, users store content they download in a local cache and share it with other users they meet, e.g., via Bluetooth or WiFi. The storage capacity of mobile devices is typically limited; therefore, identifying which content a user should store in her cache is a fundamental problem in the operation of any such content distribution system. In this work we propose Psephos, a novel mechanism for determining the caching policy of each mobile user. Psephos is fully distributed: users compute their own policies individually, in the absence of a central authority. Moreover, it is designed for a heterogeneous environment, in which demand for content, access to resources, and mobility characteristics may vary across different users. Most importantly, the caching policies computed by our mechanism are optimal: we show that Psephos maximizes the system’s social welfare. To the best of our knowledge, our work is the first to address caching with heterogeneity in a fully distributed manner.     

An analytic approach to the ergodic theory of a stochastic variational inequality

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.     

Stochastic EM algorithms for parametric and semiparametric mixture models for right-censored lifetime data

Mixture models in reliability bring a useful compromise between parametric and nonparametric models, when several failure modes are suspected. The classical methods for estimation in mixture models rarely handle the additional difficulty coming from the fact that lifetime data are often censored, in a deterministic or random way. We present in this paper several iterative methods based on EM and Stochastic EM methodologies, that allow us to estimate parametric or semiparametric mixture models for randomly right censored lifetime data, provided they are identifiable. We consider different levels of completion for the (incomplete) observed data, and provide genuine or EM-like algorithms for several situations. In particular, we show that simulating the missing data coming from the mixture allows to plug a standard R package for survival data analysis in an EM algorithm’s M-step. Moreover, in censored semiparametric situations, a stochastic step is the only practical solution allowing computation of nonparametric estimates of the unknown survival function. The effectiveness of the new proposed algorithms are demonstrated in simulation studies and an actual dataset example from aeronautic industry.     

Erratum to: Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education -     

Metamorphosis of images in reproducing kernel Hilbert spaces

Metamorphosis is a method for diffeomorphic matching of shapes, with many potential applications for anatomical shape comparison in medical imagery, a problem which is central to the field of computational anatomy. An important tool for the practical application of metamorphosis is a numerical method based on shooting from the initial momentum, as this would enable the use of statistical methods based on this momentum, as well as the estimation of templates from hyper-templates using morphing. In this paper we introduce a shooting method, in the particular case of morphing images that lie in a reproducing kernel Hilbert space (RKHS). We derive the relevant shooting equations from a Lagrangian frame of reference, present the details of the numerical approach, and illustrate the method through morphing of some simple images.     

Maximal solutions of semilinear elliptic equations with locally integrable forcing term

We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝ ^N with compact boundary, assuming thatf ∈ (L _loc ¹ (Ω))₊ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC _1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition, we discuss the question of uniqueness of large solutions. This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274.     

The role of the Ellingsrud-Strømme construction in the classification of instantons

We review a construction of Ellingsrud-Strømme relating instantons of charge n on the ordinary projective space and theta-characteristics on a plane curve of degree n with some extra-structure.