Information percolation and cutoff for the random‐cluster model

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.     

Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation

The present paper is concerned with the asymptotic behaviors of radially symmetric solutions for the multi-dimensional Burgers equation on the exterior domain in ${\mathbb{R}}^n,n\ge 3$, where the boundary and far field conditions are prescribed. We show that in some case where the corresponding 1-D Riemann problem for the non-viscous part admits a shock wave, the solution tends toward a linear superposition of stationary and rarefaction waves as time goes to infinity, and also show the decay rate estimates. Furthermore, we improve the results on the asymptotic stability of the stationary waves which are treated in the previous papers [2], [3]. Finally, for the case of $n=3$, we give the complete classification of the asymptotic behaviors, which includes even a linear superposition of stationary and viscous shock waves.     

Monte Carlo simulation of electric conductivity for pure and doping fullerene (C60)

In this paper, we have used Monte Carlo (MC) method to simulate and study the temperature and doping effects on the electric conductivity of fullerene (C60). The results show that the band gap has reduced by the doping and the charge carrier transport is facilitated from valence band to conduction band by the temperature where is touched a 300 K. In this case, the conductivity reached a value of $4\times {10}^{-7}\text{S}{\text{cm}}^{-1}$. The electric conductivity of C60 can increase by the triphenylmethane dye crystal violet (CV) alkali metal to reach $4\times {10}^{-3}\text{S}{\text{cm}}^{-1}$ at 303 K. Our results of MC simulation have a good agreement with those extracted from literature [10], [33].     

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable family of lattices ${({\mathrm{\Gamma }}_j)}_{j\in J}$ in a locally compact abelian group G, there exist functions ${(g_j)}_{j\in J}$ such that the resulting generalized shift-invariant system ${(g_j(??\gamma ))}_{j\in J,\gamma \in {\mathrm{\Gamma }}_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Regularity results of the thin obstacle problem for the p(x)-Laplacian

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.     

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.     

A new daphnane diterpenoid from Excoecaria venenata with inhibitory effect on human leukaemia HL-60 cells

A new daphnane diterpenoid, venenatin (1), along with six known compounds, was isolated from the aerial parts of Excoecaria venenata. The structure of 1 was elucidated on the basis of extensive spectroscopic analysis. Compound 1 exhibited growth inhibitory effect on human leukaemia HL-60 cell line with IC₅₀ value of 28.10
60;μM.     

Fused-Pentagon Isomers of C60 Fullerene Isolated as Chloro and Trifluoromethyl Derivatives

The carbon cage of buckminsterfullerene I_h-C₆₀, which obeys the Isolated-Pentagon Rule (IPR), can be transformed to non-IPR cages in the course of high-temperature chlorination of C₆₀ or C₆₀Cl₃₀ with SbCl₅. The non-IPR chloro derivatives were isolated chromatographically (HPLC) and characterized crystallographically as ¹⁸⁰⁹C₆₀Cl₁₆, ¹⁸¹⁰C₆₀Cl₂₄, and ¹⁸⁰⁵C₆₀Cl₂₄, which contain, respectively two, four, and four pairs of fused pentagons in the carbon cage. High-temperature trifluoromethylation of the chlorination products with CF₃I afforded a non-IPR CF₃ derivative, ¹⁸⁰⁷C₆₀(CF₃)₁₂, which contains four pairs of fused pentagons in the carbon cage. Addition patterns of non-IPR chloro and CF₃ derivatives were compared and discussed in terms of the formation of stabilizing local substructures on fullerene cages. A detailed scheme of the experimentally confirmed non-IPR C₆₀ isomers obtained by Stone–Wales cage transformations is presented.     

上海市力学学会60年

上海市力学学会成立于1959年9月20日，到今年正好是一个甲子．60个春秋真实写照了上海市力学学会的建立、成长、发展与壮大的过程．60年来，在上海市科协的领导和中国力学学会的指导下，在各团体成员单位的大力支持下，上海市力学学会一代代力学工作者薪火相传、开拓耕耘，历届理事会及下属工作委员会和专业委员会团结奋进、不断创新，广大会员踊跃参与，为上海力学大力发展、力学人才精心培养、力学知识科学普及、力学服务国家发展战略和区域经济建设中做出了重要贡献，学会持续发展壮大．在庆祝力学学会成立六十周年之际，让我们共同抒怀上海力学六十年薪火相传、回顾近10年重要发展，展望上海力学新时代．