Polypotentials on a Riemannian manifold

Polyharmonic functions are considered on open sets in a Riemannian manifold R and their potential-theoretic properties are studied using the notion of complete m-potentials. Also one obtains here some characterizations of domains in R on which such complete m-potentials exist.     

An inhomogeneous multispecies TASEP on a ring

We reinterpret and generalize conjectures of Lam and Williams as statements about the stationary distribution of a multispecies exclusion process on the ring. The central objects in our study are the multiline queues of Ferrari and Martin. We make some progress on some of the conjectures in different directions. First, we prove Lam and Williams' conjectures in two special cases by generalizing the rates of the Ferrari–Martin transitions. Secondly, we define a new process on multiline queues, which have a certain minimality property. This gives another proof for one of the special cases; namely arbitrary jump rates for three species.     

Induced saturation of P6

A graph $G$ is called $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but removing any edge from $G$ creates an induced copy of $H$ and adding any edge of $G^c$ to $G$ creates an induced copy of $H$. Martin and Smith studied a related problem, and proved that there does not exist a $P_4$-induced-saturated graph, where $P_4$ is the path on 4 vertices. Axenovich and Csikós gave examples of families of graphs $H$ for which $H$-induced-saturated graph $G$ exists, and asked if there exists a $P_n$-induced-saturated graph when $n\ge 5$. Our aim in this short note is to show that there exists a $P_6$-induced-saturated graph.     

Random subtrees and unimodal sequences in graphs

For a complete graph $K_n$ and a nonnegative integer k, we study the probability that a random subtree of $K_n$ has exactly $n-k$ vertices and show that it approaches a limiting value of $\frac{e^{-k-e^{-1}}}{k!}$ as n tends to infinity. We also consider the (conditional) probability that a random subtree of $K_n$ contains a given edge, and more generally, a fixed subtree. In particular, if e and f are adjacent edges of $K_n$, Chin, Gordon, MacPhee and Vincent [J. Graph Theory 89 (2018), 413-438] conjectured that $P[e\subseteq T|f\subseteq T]\le P[e\subseteq T]$. We prove this conjecture and further prove that $P[e\subseteq T|f\subseteq T]$ tends to three-quarters of $P[e\subseteq T]$ as $n\to \infty$. Finally, several classes of graphs are given, such as star plus an edge, lollipop graph and glasses graph, whose subtree polynomials are unimodal.     

完全活性空间组态相互作用能量的拟合和外推

完全活性空间组态相互作用计算与完全活性空间中的活性电子数和活性轨道数有关,但完全活性空间组态相互作用的能量不是活性电子数和活性轨道数的单调递减函数,因此活性轨道数和活性电子数不能用来外推完全活性空间组态相互作用的能量。为此,我们定义了一个新的变量:活性空间中的最大未占满轨道数。我们对一系列单重态、双重态和三重态分子进行了完全活性空间组态相互作用的计算,并利用活性空间中的活性电子数和最大未占满轨道数这两个变量,对这些基态能量进行了拟合和外推,拟合的均方根误差都在10~(-6)数量级。外推能量的精度优于MP4,对小分子体系,其精度高于CCSD。外推的完全的组态相互作用(FCI)能量值和实际计算的FCI值也很接近。另外,我们还利用外推能量来优化双原子分子的平衡键长,并计算谐振频率,其精度优于CASSCF。     

Stable local cohomology

For a module having a complete injective resolution, we define a stable version of local cohomology. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this functor behaves like the usual local cohomology functor. Our main result is that when there is only one nonzero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.     

Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity

The physics of polycrystalline materials is described via microscopic processes such as grain boundary migration, grain growth, grain rotation, polygonization (the bending and breaking of crystallites) and evolution of dislocation density. The importance of taking these processes into account lies in their influence on the macroscopic mechanical behaviour of the material. Constitutive equations to describe such phenomena have been proposed in the literature. The main result of this paper is to give a general and thermodynamically consistent approach for such constitutive equations. The framework of the Theory of Mixtures with Continuous Diversity (TMCD) is used. The inclusion of both orientation and grain-size distributions is presented in this paper for the first time. Their introduction requires the formulation of a new and general constitutive theory that is, therefore, given. The method of Lagrange multipliers used in the context of the entropy principle (Liu, Arch. Rat. Mech. Anal. 46, 131–148 (1972)) provides the restrictions of the second law of thermodynamics on the constitutive equations. The success of this work is that all the main results present in the literature can be incorporated in this framework.     

三种红层岩石常规三轴压缩下的强度与变形特性研究

利用MTS815Teststar程控伺服岩石力学试验系统研究了川东地区一红层边坡中的砂岩、粉砂岩和泥岩围压为0³MPa的应力-应变全过程曲线,建立了峰值强度、峰值强度前弹性模量以及峰值强度后的弹性模量和围压的关系。将低围压下红层的全应力-应变曲线概化成5个阶段,分别为压密段、弹性段、屈服段、应变软化段和塑性流动阶段。试验结果得出,红层弹性模量随围压的增加而提高且变化明显,砂岩和粉砂岩在此围压内为脆性破坏,泥岩为塑性破坏的规律。     

Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

This article studies the zero divisor graph for the ring of Gaussian integers modulo n, Γ (? _n [i]). For each positive integer n, the number of vertices, the diameter, the girth and the case when the dominating number is 1 or 2 is found.

Complete characterizations, in terms of n, are given of the cases in which Γ (? _n [i]) is complete, complete bipartite, planar, regular or Eulerian.     

Modules with Reducible Complexity,II

We continue studying the class of modules having reducible complexity over a local ring. In particular, a method is provided for computing an upper bound of the complexity of such a module, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.