Diffeomorphic random sampling using optimal information transport

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge--Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.Comment: 8 pages, 3 figure

Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact

Bosonic sector of the two-dimensional Hubbard model studied within a two-pole approximation

The charge and spin dynamics of the two-dimensional Hubbard model in the paramagnetic phase is first studied by means of the two-pole approximation within the framework of the Composite Operator Method. The fully self-consistent scheme requires: no decoupling, the fulfillment of both Pauli principle and hydrodynamics constraints, the simultaneous solution of fermionic and bosonic sectors and a very rich momentum dependence of the response functions. The temperature and momentum dependencies, as well as the dependency on the Coulomb repulsion strength and the filling, of the calculated charge and spin susceptibilities and correlation functions are in very good agreement with the numerical calculations present in the literature

Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in L^1(\R \times \R,\d \x \otimes \d\v) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

A fast and cost-effective approach to develop and map EST-SSR markers: oak as a case study

Background: Expressed Sequence Tags (ESTs) are a source of simple sequence repeats (SSRs) that can be used to develop molecular markers for genetic studies. The availability of ESTs for Quercus robur and Quercus petraea provided a unique opportunity to develop microsatellite markers to accelerate research aimed at studying adaptation of these long-lived species to their environment. As a first step toward the construction of a SSR-based linkage map of oak for quantitative trait locus (QTL) mapping, we describe the mining and survey of EST-SSRs as well as a fast and cost-effective approach (bin mapping) to assign these markers to an approximate map position. We also compared the level of polymorphism between genomic and EST-derived SSRs and address the transferability of EST-SSRs in Castanea sativa (chestnut). Results: A catalogue of 103,000 Sanger ESTs was assembled into 28,024 unigenes from which 18.6% presented one or more SSR motifs. More than 42% of these SSRs corresponded to trinucleotides. Primer pairs were designed for 748 putative unigenes. Overall 37.7% (283) were found to amplify a single polymorphic locus in a reference fullsib pedigree of Quercus robur. The usefulness of these loci for establishing a genetic map was assessed using a bin mapping approach. Bin maps were constructed for the male and female parental tree for which framework linkage maps based on AFLP markers were available. The bin set consisting of 14 highly informative offspring selected based on the number and position of crossover sites. The female and male maps comprised 44 and 37 bins, with an average bin length of 16.5 cM and 20.99 cM, respectively. A total of 256 EST-SSRs were assigned to bins and their map position was further validated by linkage mapping. EST-SSRs were found to be less polymorphic than genomic SSRs, but their transferability rate to chestnut, a phylogenetically related species to oak, was higher. Conclusion: We have generated a bin map for oak comprising 256 EST-SSRs. This resource constitutes a first step toward the establishment of a gene-based map for this genus that will facilitate the dissection of QTLs affecting complex traits of ecological importance

Celebrating Cercignani's conjecture for the Boltzmann equation

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation. 24 pages. V2: correction of some typos and one ref. adde

Річард Смоллі і знамениті «десять вересневих днів»

У вересні цього року виповнюється 27 років, як було відкрито фулерен — нову сфероподібну форму вуглецю. Ця подія буквально приголомшила вчених, які на той час вважали, що про елементарний вуглець їм відомо практично все. Історія відкриття цієї речовини досить незвичайна. Ще в 1971 р. можливість існування молекули фулерену була передбачена японським ученим Е. Осавою (E. Osawa), за два роки радянські хіміки-теоретики Д.А. Бочвар і О.Г. Гальперн квантово-хімічними розрахунками підтвердили стабільність молекули С60, і лише у 1985 р. Р. Смоллі, Р. Керл та Г. Крото експериментально отримали кластери із 60 атомів вуглецю в стійкій формі, яку вони пояснили структурою молекули у вигляді футбольного м’яча. Натхненнику цього відкриття, видатному вченому, нобелівському лауреату, активному популяризатору нанотехнологій Річарду Смоллі присвячено цей матеріал.В сентябре этого года исполняется 27 лет с момента открытия фуллерена — новой сферообразной формы углерода. Это событие буквально потрясло ученых, которые в то время считали, что об элементарном углероде им известно практически все. История открытия этого вещества довольно необычна. Еще в 1971 г. возможность су ществования молекулы фуллерена была предсказана японским ученым Е. Осавой (E. Osawa), через два года советские химики-теоретики Д.А. Бочвар и Е.Г. Гальперн с помощью квантово-химических расчетов подтвердили стабильность молекулы С60, и только в 1985 г. Р. Смолли, Р. Керл и Г. Крото экспериментально получили кластеры из 60 атомов углерода в устойчивой форме, которую они объяснили структурой молекулы в виде футбольного мяча. Вдохновителю этого открытия, выдающемуся ученому, нобелевскому лауреату, активному популяризатору нанотехнологий Ричарду Смолли посвящен этот материал.27 years since the discovery of fullerene, the new form of carbon, is observed in September of this year. This event has literally shocked scientists, who believed at that time that they know almost everything about the elementary carbon. History of this discovery is rather unusual. Long ago, in 1971 the possibility of the existence of a fullerene molecule was predicted by an young Japanese scientist E. Osawa. Then two Soviet chemists and theorists D.A. Bochvar and E.G. Hal pern confirm the stability of the C60 molecule using quantum chemical calculations, and in 1985 at last R. Smalley, R. Curl and H. Kroto experimentally obtained clusters of 60 carbon atoms in a sustainable form. They explained the structure of this molecule as the structure of a soccer ball. This material is devoted to the inspirer of this discovery, an outstanding scientist, Nobel laureate, active popularizer of nanotechnology — Richard Smalley

A Study of the Antiferromagnetic Phase in the Hubbard Model by means of the Composite Operator Method

We have investigated the antiferromagnetic phase of the 2D, the 3D and the extended Hubbard models on a bipartite cubic lattice by means of the Composite Operator Method within a two-pole approximation. This approach yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of both the model and the algebra. The complete phase diagram, as regards the antiferromagnetic and the paramagnetic phases, has been drawn. We firstly reported, within a pole approximation, three kinds of transitions at half-filling: Mott-Hubbard, Mott-Heisenberg and Heisenberg. We have also found a metal-insulator transition, driven by doping, within the antiferromagnetic phase. This latter is restricted to a very small region near half filling and has, in contrast to what has been found by similar approaches, a finite critical Coulomb interaction as lower bound at half filling. Finally, it is worth noting that our antiferromagnetic gap has two independent components: one due to the antiferromagnetic correlations and another coming from the Mott-Hubbard mechanism.Comment: 20 pages, 37 figures, RevTeX, submitted to Phys. Rev.

From log Sobolev to Talagrand: a quick proof

We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent develop- ment [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincar\ue9 inequality