On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other

Cannabis through the looking glass: chemo- and enantio-selective separation of phytocannabinoids by enantioselective ultra high performance supercritical fluid chromatography

By using the Inverted Chirality Columns Approach (ICCA) we have developed an enantioselective UHPSFC method to determine the enantiomeric excess (ee) of (-)-Δ(9)-THC in medicinal marijuana (Bedrocan®). The ee was high (99.73%), but the concentration of the (+)-enantiomer (0.135%) was not negligible, and it is worth a systematic evaluation of bioactivity

Relational reasoning via probabilistic coupling

Probabilistic coupling is a powerful tool for analyzing pairs of probabilistic processes. Roughly, coupling two processes requires finding an appropriate witness process that models both processes in the same probability space. Couplings are powerful tools proving properties about the relation between two processes, include reasoning about convergence of distributions and stochastic dominance---a probabilistic version of a monotonicity property. While the mathematical definition of coupling looks rather complex and cumbersome to manipulate, we show that the relational program logic pRHL---the logic underlying the EasyCrypt cryptographic proof assistant---already internalizes a generalization of probabilistic coupling. With this insight, constructing couplings is no harder than constructing logical proofs. We demonstrate how to express and verify classic examples of couplings in pRHL, and we mechanically verify several couplings in EasyCrypt

Continuity of Optimal Control Costs and its application to Weak KAM Theory

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry-Mather problems corresponding to these systems.Comment: 23 pages, 1 figures, added explanations in the proofs of the main theorem and the exampl

The Arbuscular Mycorrhizal Fungus Glomus viscosum Improves the Tolerance to Verticillium Wilt in Artichoke by Modulating the Antioxidant Defense Systems

Verticillium wilt, caused by the fungal pathogen Verticillium dahliae, is the most severe disease that threatens artichoke (Cynara scolymus L.) plants. Arbuscular mycorrhizal fungi (AMF) may represent a useful biological control strategy against this pathogen attack, replacing chemical compounds that, up to now, have been not very effective. In this study, we evaluated the effect of the AMF Glomus viscosum Nicolson in enhancing the plant tolerance towards the pathogen V. dahliae. The role of the ascorbate-glutathione (ASC-GSH) cycle and other antioxidant systems involved in the complex network of the pathogen-fungi-plant interaction have been investigated. The results obtained showed that the AMF G. viscosum is able to enhance the defense antioxidant systems in artichoke plants affected by V. dahliae, alleviating the oxidative stress symptoms. AMF-inoculated plants exhibited significant increases in ascorbate peroxidase (APX), monodehydroascorbate reductase (MDHAR), and superoxide dismutase (SOD) activities, a higher content of ascorbate (ASC) and glutathione (GSH), and a decrease in the levels of lipid peroxidation and hydrogen peroxide (H2O2). Hence, G. viscosum may represent an effective strategy for mitigating V. dahliae pathogenicity in artichokes, enhancing the plant defense systems, and improving the nutritional values and benefit to human health

On the speed of approach to equilibrium for a collisionless gas

We investigate the speed of approach to Maxwellian equilibrium for a collisionless gas enclosed in a vessel whose wall are kept at a uniform, constant temperature, assuming diffuse reflection of gas molecules on the vessel wall. We establish lower bounds for potential decay rates assuming uniform $L^p$ bounds on the initial distribution function. We also obtain a decay estimate in the spherically symmetric case. We discuss with particular care the influence of low-speed particles on thermalization by the wall.Comment: 22 pages, 1 figure; submitted to Kinetic and Related Model

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

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (-4\le \gm<0), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|\gm|. For the usual $L^1$-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with -1\le \gm<0, there are algebraic upper and lower bounds on the rate of $L^1$-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates.Comment: This version contains a strengthened theorem 3, on rate of convergence, considerably relaxing the hypotheses on the initial data, and introducing a new method for avoiding use of poitwise lower bounds in applications of entropy production to convergence problem