Search results for "15b"
showing 10 items of 11 documents
Adjacency matrices of random digraphs: singularity and anti-concentration
2017
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, …
New spaces of matrices with operator entries
2019
In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as $H^\infty(\mathbb{D})$ and $A(\mathbb{D})$ when dealing with upper t…
Circular law for sparse random regular digraphs
2020
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studyi…
Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations
2017
We study the spectrum of large a bi-diagonal Toeplitz matrix subject to a Gaussian random perturbation with a small coupling constant. We obtain a precise asymptotic description of the average density of eigenvalues in the interior of the convex hull of the range symbol.
Structure of eigenvectors of random regular digraphs
2018
Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have cardin…
CCDC 1977492: Experimental Crystal Structure Determination
2020
Related Article: Jingyu Zhang, Jing Li, Jas S. Ward, Khai-Nghi Truong, Kari Rissanen, Markus Albrecht|2020|J.Org.Chem.|85|12160|doi:10.1021/acs.joc.0c01373
CCDC 795379: Experimental Crystal Structure Determination
2011
Related Article: J.Barjau, G.Schnakenburg, S.R.Waldvogel|2011|Angew.Chem.,Int.Ed.|50|1415|doi:10.1002/anie.201006637
CCDC 1961567: Experimental Crystal Structure Determination
2020
Related Article: Kibrom Gebreheiwot Bedane, Lukas Brieger, Carsten Strohmann, Ean-Jeong Seo, Thomas Efferth, Michael Spiteller|2020|J.Nat.Prod.|83|2122|doi:10.1021/acs.jnatprod.0c00060
CCDC 866803: Experimental Crystal Structure Determination
2012
Related Article: A.V.B.Djoumessi, L.P.Sandjo, J.C.Liermann, D.Schollmeyer, V.Kuete, V.Rincheval, A.M.Berhanu, S.O.Yeboah, P.Wafo, B.T.Ngadjui, T.Opatz|2012|Tetrahedron|68|4621|doi:10.1016/j.tet.2012.04.027
The rank of random regular digraphs of constant degree
2018
Abstract Let d be a (large) integer. Given n ≥ 2 d , let A n be the adjacency matrix of a random directed d -regular graph on n vertices, with the uniform distribution. We show that the rank of A n is at least n − 1 with probability going to one as n grows to infinity. The proof combines the well known method of simple switchings and a recent result of the authors on delocalization of eigenvectors of A n .