Aubry-Mather measures in the non convex setting

The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053–1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301–319], is used to construct analogues to the Aubry–Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed. Copyright © 2011 Society for Industrial and Applied Mathematic

Optimal regularity and structure of the free boundary for minimizers in cohesive zone models

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, \alpha}$, for some $\alpha \in (0, 1)$.Comment: 39 page

Rigidity for perimeter inequality under spherical symmetrisation

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950

Convergence of a semi-discretization scheme for the Hamilton--Jacobi equation: a new approach with the adjoint method

We consider a numerical scheme for the one dimensional time dependent Hamilton--Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L. C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(sqrt{h}) convergence rate in terms of the L^infty norm and O(h) in terms of the L^1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper

A global method for deterministic and stochastic homogenisation in BV

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation

Efficacy and Safety of Cannabidiol in Epilepsy: A Systematic Review and Meta-Analysis

Background: Approximately one-third of patients with epilepsy presents seizures despite adequate treatment. Hence, there is the need to search for new therapeutic options. Cannabidiol (CBD) is a major chemical component of the resin of Cannabis sativa plant, most commonly known as marijuana. The anti-seizure properties of CBD do not relate to the direct action on cannabinoid receptors, but are mediated by a multitude of mechanisms that include the agonist and antagonist effects on ionic channels, neurotransmitter transporters, and multiple 7-transmembrane receptors. In contrast to tetra-hydrocannabinol, CBD lacks psychoactive properties, does not produce euphoric or intrusive side effects, and is largely devoid of abuse liability. Objective: The aim of the study was to estimate the efficacy and safety of CBD as adjunctive treatment in patients with epilepsy using meta-analytical techniques. Methods: Randomized, placebo-controlled, single- or double-blinded add-on trials of oral CBD in patients with uncontrolled epilepsy were identified. Main outcomes included the percentage change and the proportion of patients with ≥ 50% reduction in monthly seizure frequency during the treatment period and the incidence of treatment withdrawal and adverse events (AEs). Results: Four trials involving 550 patients with Lennox–Gastaut syndrome (LGS) and Dravet syndrome (DS) were included. The pooled average difference in change in seizure frequency during the treatment period resulted 19.5 [95% confidence interval (CI) 8.1–31.0; p = 0.001] percentage points between the CBD 10 mg and placebo groups and 19.9 (95% CI 11.8–28.1; p < 0.001) percentage points between the CBD 20 mg and placebo arms, in favor of CBD. The reduction in all-types seizure frequency by at least 50% occurred in 37.2% of the patients in the CBD 20 mg group and 21.2% of the placebo-treated participants [risk ratio (RR) 1.76, 95% CI 1.07–2.88; p = 0.025]. Across the trials, drug withdrawal for any reason occurred in 11.1% and 2.6% of participants receiving CBD and placebo, respectively (RR 3.54, 95% CI 1.55–8.12; p = 0.003) [Chi squared = 2.53, degrees of freedom (df) = 3, p = 0.506; I2 = 0.0%]. The RRs to discontinue treatment were 1.45 (95% CI 0.28–7.41; p = 0.657) and 4.20 (95% CI 1.82–9.68; p = 0.001) for CBD at the doses of 10 and 20 mg/kg/day, respectively, in comparison to placebo. Treatment was discontinued due to AEs in 8.9% and 1.8% of patients in the active and control arms, respectively (RR 5.59, 95% CI 1.87–16.73; p = 0.002). The corresponding RRs for CBD at the doses of 10 and 20 mg/kg/day were 1.66 (95% CI 0.22–12.86; p = 0.626) and 6.89 (95% CI 2.28–20.80; p = 0.001). AEs occurred in 87.9% and 72.2% of patients treated with CBD and placebo (RR 1.22, 95% CI 1.11–1.33; p < 0.001). AEs significantly associated with CBD were somnolence, decreased appetite, diarrhea, and increased serum aminotransferases. Conclusions: Adjunctive CBD in patients with LGS or DS experiencing seizures uncontrolled by concomitant anti-epileptic treatment regimens is associated with a greater reduction in seizure frequency and a higher rate of AEs than placebo

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.Comment: 30 page

Rapid evolution of female-biased genes among four species of Anopheles malaria mosquitoes.

Understanding how phenotypic differences between males and females arise from the sex-biased expression of nearly identical genomes can reveal important insights into the biology and evolution of a species. Among Anopheles mosquito species, these phenotypic differences include vectorial capacity, as it is only females that blood feed and thus transmit human malaria. Here, we use RNA-seq data from multiple tissues of four vector species spanning the Anopheles phylogeny to explore the genomic and evolutionary properties of sex-biased genes. We find that, in these mosquitoes, in contrast to what has been found in many other organisms, female-biased genes are more rapidly evolving in sequence, expression, and genic turnover than male-biased genes. Our results suggest that this atypical pattern may be due to the combination of sex-specific life history challenges encountered by females, such as blood feeding. Furthermore, female propensity to mate only once in nature in male swarms likely diminishes sexual selection of post-reproductive traits related to sperm competition among males. We also develop a comparative framework to systematically explore tissue- and sex-specific splicing to document its conservation throughout the genus and identify a set of candidate genes for future functional analyses of sex-specific isoform usage. Finally, our data reveal that the deficit of male-biased genes on the X Chromosomes in Anopheles is a conserved feature in this genus and can be directly attributed to chromosome-wide transcriptional regulation that de-masculinizes the X in male reproductive tissues

Deep venous thrombosis and abortion: an unusual clinical manifestation of severe form of pectus excavatum

Pectus excavatum is a chest wall malformation with a strong psychological and aesthetic impact. Rarely, pectus excavatum patients can show respiratory or cardiac symptoms occurring mainly during physical exertion. We report a case of a 34-year-old pregnant woman with a severe degree of pectus excavatum who developed serious cardiovascular disease resulting in spontaneous twin abortion at the twenty-first week of gestation. Cardiovascular disease was resolved after open surgical correction of pectus excavatum. This case shows how a tardive diagnosis and a delayed surgical approach for pectus excavatum can lead to severe consequences

Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana