Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.     

In-chain functionalization through the combination of ring opening copolymerization and oxime “Click” reaction towards X-ray opaque polylactide copolymers

X-ray imaging functionalization of biodegradable polyesters is a great demand and challenge in biomedical applications.In this work,a strategy of in-chain functionalization through the combination of ring opening copolymerization and oxime "Click" postfunctionalization was developed towards X-ray opaque polylactide copolymers.A functionalized cyclic carbonate was first synthesized and used as comonomer of polylactide copolymers,which were subjected to postfunctionalization of oxime "Click" reaction towards iodinated polylactide copolymers.The chemical structure and physical properties of the target products were traced and confirmed.In vitro cytotoxicity evaluation with 3T3-Swiss albino by Alamar blue demonstrated a low cytotoxicity.The X-ray radiopacity was analyzed by Micro-CT and quantified by Hounsfield Units value,which could be tailorable by the feedstock.It is a promising X-ray visible implantable biomaterial in biomedical applications.     

MALDI MSI Reveals the Spatial Distribution of Protein Markers in Tracheobronchial Lymph Nodes and Lung of Pigs after Respiratory Infection

Respiratory infections are a real threat for humans, and therefore the pig model is of interest for studies. As one of a case for studies, Actinobacillus pleuropneumoniae (APP) caused infections and still worries many pig breeders around the world. To better understand the influence of pathogenic effect of APP on a respiratory system—lungs and tracheobronchial lymph nodes (TBLN), we aimed to employ matrix-assisted laser desorption/ionization time-of-flight mass spectrometry imaging (MALDI-TOF MSI). In this study, six pigs were intranasally infected by APP and two were used as non-infected control, and 48 cryosections have been obtained. MALDI-TOF MSI and immunohistochemistry (IHC) were used to study spatial distribution of infectious markers, especially interleukins, in cryosections of porcine tissues of lungs (necrotic area, marginal zone) and tracheobronchial lymph nodes (TBLN) from pigs infected by APP. CD163, interleukin 1β (IL-1β) and a protegrin-4 precursor were successfully detected based on their tryptic fragments. CD163 and IL-1β were confirmed also by IHC. The protegrin-4 precursor was identified by MALDI-TOF/TOF directly on the tissue cryosections. CD163, IL-1β and protegrin-4 precursor were all significantly (p < 0.001) more expressed in necrotic areas of lungs infected by APP than in marginal zone, TBLN and in control lungs.     

Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.     

基于平面度的燃料电池电化学性能模拟

 固体氧化物燃料电池的翘曲会影响电极-盖板界面的接触情况，从而影响电化学性能，对相关制造工艺提出了很大的挑战．为了分析燃料电池平面度对放电过程的影响，揭示其潜在的风险，我们建立了两个基于有限元法的仿真模型，对考虑平面度缺陷的燃料电池封装和放电进行分析．在对固体氧化物燃料电池进行平面度测量的基础上，首先建立了具有真实燃料电池翘曲特性的几何模型，分析封装过程中接触压力的分布情况．然后将接触电阻的仿真结果导入到三维多物理场耦合模型中，模拟具有平面度缺陷的燃料电池电化学性能．计算结果展示了燃料电池两侧封装过程中接触压力的分布情况．通过对比有接触电阻和无接触电阻的燃料电池电流密度，分析了电池与盖板的接触对放电过程的影响．结果表明，燃料电池的凹陷面较难达到满意的接触状态，需要比凸起面更大的封装压力．燃料电池表面接触电阻的变化将引起电流传导路径的变化，产生局部高电流或低电流．这项工作强调了在燃料电池中保持均匀分布的接触电阻的重要性，为今后的优化工作奠定了基础．     

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.     

Nonnegative solutions to reaction–diffusion system with cross-diffusion and nonstandard growth conditions

We establish the existence of nonnegative weak solutions to nonlinear reaction–diffusion system with cross-diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.     

安培环路定理的拓扑学描述

近年来拓扑学在量子力学中得到了广泛的运用.本文将安培环路定理积分式重新表达为一矢量场在轮胎参数面上的第一类陈数积分.数值模拟展示了该积分值为一整数即第一陈数,其代表矢量场的整体性质:当经历连续变换时,矢量场的局部数值发生改变但整体积分值即陈数仍保持不变;若陈数发生改变,则表明矢量场变换的连续性条件发生破坏,矢量场出现奇点.进一步通过高斯映射将该矢量场从参数轮胎面映射到单位球面上,并给出了第一陈数的直观几何意义.理论和数值结果揭示了安培环路定理的拓扑学本质,表明拓扑概念在经典物理学中也会有广泛应用.     

Can Anti-Aufbau DFT Calculations Estimate Singlet Excited State Aromaticity? Correspondence on “Dibenzoarsepins: Planarization of 8π-Electron System in the Lowest Singlet Excited State”

The simple anti-aufbau DFT approach for estimating singlet excited state aromaticity suggested in a recent Communication published in this journal is shown to produce incorrect results because it targets a linear combination of the singlet and triplet configurations involving the HOMO and LUMO rather than the first singlet excited state. If the S₁ state of a molecule is dominated by the HOMO→LUMO excitation, a comparably simple but theoretically consistent and qualitatively correct approximation to the S₁ wavefunction can be achieved by performing a small “two electrons in two orbitals” CASSCF(2,2) calculation which can be followed by the evaluation of magnetic aromaticity criteria such as NICS.