Propagation of chaos for the Keller–Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.     

超短脉冲激光诱导周期性表面结构

激光诱导周期性表面结构（Laser-induced periodic surface structures,LIPSS）具有纳米尺度的特征结构和自重复的微观尺度的排列图案,因此,LIPSS在传感器、太阳能发电、光催化等方面具有广泛的应用前景。本文首先介绍LIPSS形成过程中超快激光与物质相互作用的复杂过程,强调瞬态光学性质和表面结构变化的作用。然后综述几种具有代表性的LIPSS形成机理,并且讨论了各自的优缺点。接着介绍了LIPSS形成过程中材料的变化,主要包括材料化学成分、晶体结构和表面微观结构的变化。最后综述了LIPSS在材料表面处理、光学和机械等方面的应用。     

Large Time Behavior for a Quasilinear Diffusion Equation with Critical Gradient Absorption

We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption

$$\begin{aligned} \partial _t u-\Delta _{p}u+|\nabla u|^{q_*}=0 \quad \hbox {in}\, (0,\infty )\times \mathbb {R}^N, \end{aligned}$$

for \(p\in (2,\infty )\) and \(q_*:=p-N/(N+1)\). We show that the asymptotic profile of compactly supported solutions is given by a source-type self-similar solution of the p-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results.

     

Genus mapping class groups are not Kähler

The goal of this note is to prove that the mapping class groups of closed orientable surfaces of genus 2 (with punctures) are not Kähler. An application to compactifications of the moduli space of genus curves (with punctures) is given.

     

Longitudinal dynamics of trains—a non-smooth approach

An efficient and straightforward algorithm for the integration of the constrained systems is developed in the present work. Some of the fundamental issues of contact dynamics are briefly reviewed. Considering Gauss?? least constraint principle, an intuitive and effective computation method of the constraint forces is arrived at. The contact problem is solved by means of the generalized matrix inverse (GI) embedded in an appropriate algorithm and the numerical integration is event-driven. The method is applied to the study of train longitudinal dynamics which is essential for comfort and running safety. Due to the complexity of the phenomena and the presence of dry friction, the investigations have been carried out by numerical methods using regularization. Based on the non-smooth dynamics approach, in the present paper, a comprehensive model is presented. Set-value friction of Coulomb??s law type is accounted for and motion equations are formulated as a differential inclusion. Application examples are presented. Simulations demonstrate the existence of specific phenomena like stick?Cslip and offset in the final equilibrium position. Computational efficiency is dramatically improved compared with previous models, while computation speed is increased by at least an order of magnitude. A wide class of problems may be solved using the present method, as is pointed out in the article.     

Topological Hilbert Nullstellensatz for Bergman spaces

We prove Bergman space analogues of a conjecture of Douglas and Paulsen related to the classification of invariant subspaces for multiplication operators in several variables.     

Melons are Branched Polymers

Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3.     

Dynamic Trapping as a Selective Route to Renewable Phthalide from Biomass-Derived Furfuryl Alcohol

A novel route for the production of the versatile chemical building block phthalide from biorenewable furfuryl alcohol and acrylate esters is presented. Two challenges that limit sustainable aromatics production via Diels–Alder (DA) aromatisation—an unfavourable equilibrium position and undesired regioselectivity when using asymmetric addends—were addressed using a dynamic kinetic trapping strategy. Activated acrylates were used to speed up the forward and reverse DA reactions, allowing for one of the four DA adducts to undergo a selective intramolecular lactonisation reaction in the presence of a weak base. The adduct is removed from the equilibrium pool, pulling the system completely to the product with a fixed, desired regiochemistry. A single 1,2-regioisomeric lactone product was formed in up to 86 % yield and the acrylate activating agent liberated for reuse. The lactone was aromatised to give phthalide in almost quantitative yield in the presence of Ac₂O and a catalytic amount of strong acid, or in 79 % using only catalytic acid.     

Conversion of Natural Narciclasine to Its C-1 and C-6 Derivatives and Their Antitumor Activity Evaluation: Some Unusual Chemistry of Narciclasine

During the search for a general, efficient route toward the synthesis of C-1 analogues of narciclasine, natural narciclasine was protected and converted to its C-1 enol derivative using a novel semi-synthetic route. Attempted conversion of this material to its triflate in order to conduct cross-coupling at C-1 resulted in a triflate at C-6 that was successfully coupled with several functionalities. Four novel compounds were fully deprotected after seven steps and subjected to evaluation for cytotoxic activity against three cancer cell lines. Only one derivative showed moderate activity compared to that of narciclasine. Spectral and physical data are provided for all new compounds.