Connectivity of a general class of inhomogeneous random digraphs

We study a family of directed random graphs whose arcs are sampled independently of each other, and are present in the graph with a probability that depends on the attributes of the vertices involved. In particular, this family of models includes as special cases the directed versions of the Erd?s‐Rényi model, graphs with given expected degrees, the generalized random graph, and the Poissonian random graph. We establish a phase transition for the existence of a giant strongly connected component and provide some other basic properties, including the limiting joint distribution of the degrees and the mean number of arcs. In particular, we show that by choosing the joint distribution of the vertex attributes according to a multivariate regularly varying distribution, one can obtain scale‐free graphs with arbitrary in‐degree/out‐degree dependence.     

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.     

关于最小二乘QR分解算法(LSQR)的一个注记

本文从最小多项式出发,通过寻找包含奇异线性系统Ax=b最小范数解的一个解空间,获得了一个更简单的求解广义逆的计算公式.并从理论上对最小二乘QR分解算法（LSQR）收敛性进行了简单分析,分析表明LSQR的收敛性与矩阵A的非零奇异值密切相关,并用A的非零奇异值以及所寻找到的最小范数解空间将最小范数解线性表出.     

Hyperplane sections of determinantal varieties over finite fields and linear codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated.     

Canonical symbolic dynamics for one-dimensional generalized solenoids

We define canonical subshift of finite type covers for Williams' one-dimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.

     

Ample fields as a basis for possibilistic processes

Ample fields play an important role in possibility theory. These fields of subsets of a universe, which are additionally closed under arbitrary unions, act as the natural domains for possibility measures. A set provided with an ample field is then called an ample space. In this paper we generalise Wang's notions of product ample field and product ample space. We make a topological study of ample spaces and their products, and introduce ample subspaces, extensions and one-point extensions of ample spaces. In this way, a first step towards a mathematical theory of possibilistic processes is made.     

Statistical learning control of uncertain systems: theory and algorithms

It has recently become clear that many control problems are too difficult to admit analytic solutions. New results have also emerged to show that the computational complexity of some “solved” control problems is prohibitive. Many of these control problems can be reduced to decidability problems or to optimization questions. Even though such questions may be too difficult to answer analytically, or may not be answered exactly given a reasonable amount of computational resources, researchers have shown that we can “approximately” answer these questions “most of the time”, and have “high confidence” in the correctness of the answers.