Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform W_\eps(t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k), as \eps\ll1, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k) but the limit satisfies then the usual heat equation

Existence of solutions for a higher order non-local equation appearing in crack dynamics

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect)

Singular Cucker-Smale Dynamics

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation

Thermal conductivity in harmonic lattices with random collisions

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.Comment: Review paper, to appear in the Springer Lecture Notes in Physics volume "Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer" (S. Lepri ed.

Genetic contributions to visuospatial cognition in Williams syndrome: insights from two contrasting partial deletion patients

Background Williams syndrome (WS) is a rare neurodevelopmental disorder arising from a hemizygotic deletion of approximately 27 genes on chromosome 7, at locus 7q11.23. WS is characterised by an uneven cognitive profile, with serious deficits in visuospatial tasks in comparison to relatively proficient performance in some other cognitive domains such as language and face processing. Individuals with partial genetic deletions within the WS critical region (WSCR) have provided insights into the contribution of specific genes to this complex phenotype. However, the combinatorial effects of different genes remain elusive. Methods We report on visuospatial cognition in two individuals with contrasting partial deletions in the WSCR: one female (HR), aged 11 years 9 months, with haploinsufficiency for 24 of the WS genes (up to GTF2IRD1), and one male (JB), aged 14 years 2 months, with the three most telomeric genes within the WSCR deleted, or partially deleted. Results Our in-depth phenotyping of the visuospatial domain from table-top psychometric, and small- and large-scale experimental tasks reveal a profile in HR in line with typically developing controls, albeit with some atypical features. These data are contrasted with patient JB’s atypical profile of strengths and weaknesses across the visuospatial domain, as well as with more substantial visuospatial deficits in individuals with the full WS deletion. Conclusions Our findings point to the contribution of specific genes to spatial processing difficulties associated with WS, highlighting the multifaceted nature of spatial cognition and the divergent effects of genetic deletions within the WSCR on different components of visuospatial ability. The importance of general transcription factors at the telomeric end of the WSCR, and their combinatorial effects on the WS visuospatial phenotype are also discussed

Implementation of drift velocities and currents in SOLEDGE2D-EIRENE

International audienceIn order to improve cross-field transport description, drifts and currents have been implemented in SOLEDGE2D-EIRENE. The derivation of an equation for the electric potential is recalled. The resolution of current equation is tested in a simple slab case. WEST divertor simulations in forward-B and reverse-B fields are also discussed. A significant increase of ExB shear is observed in the forward-B configuration that could explain a favorable L-H transition in this case

HLA typing: Conventional techniques v. next-generation sequencing

Background. The large number of population-specific polymorphisms present in the HLA complex in the South African (SA) population reduces the probability of finding an adequate HLA-matched donor for individuals in need of an unrelated haematopoietic stem cell transplantation (HSCT). Next-generation sequencing (NGS) has numerous advantages compared with conventional typing techniques.Objective. To evaluate whether NGS can provide any additional value over conventional techniques in the SA context for the purpose of HSCT and cord blood banking.Methods. HLA genotyping was performed using NGS on 20 samples that had previously been HLA typed by conventional methods to evaluate whether NGS might provide any additional value over conventional HLA determination techniques.Results. NGS of routinely sequenced loci and exons yielded accurate genotypes for 98.5% of the five loci of interest, compared with 98% when additional exons were included.Conclusion. The study shows that the additional value of NGS over conventional techniques is limited, and unless done on a large scale to reduce cost may not be appropriate in SA at this stage in the context of HSCT and cord blood banking

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page

Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems

The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems