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Register a new userCrouzeixs Conjecture and related problems
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Added by
Felix Schwenninger
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we establish several results related to Crouzeixs conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.
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This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with very recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finite-dimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.
A Latin square of order $n$ is an $n times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order $ntimes n$ contains a transversal of order $n-1$. In this paper we prove the existence of a transversal of order $n-O(log{n}/log{log{n}})$, improving the celebrated bound of $n-O(log^2n)$ by Hatami and Shor. Our approach (different from that of Hatami-Shor) is quite general and gives several other applications as well. We obtain a new lower bound on a 40 year old conjecture of Brouwer on the maximum matching in Steiner triple systems, showing that every such system of order $n$ is guaranteed to have a matching of size $n/3-O(log{n}/log{log{n}})$. This substantially improves the current best result of Alon, Kim and Spencer which has the error term of order $n^{1/2+o(1)}$. Finally, we also show that $O(nlog{n}/log{log{n}})$ many symbols in Latin arrays suffice to guarantee a full transversal, improving on previously known bound of $n^{2-varepsilon}$. The proofs combine in a novel way the semirandom method together with the robust expansion properties of edge coloured pseudorandom graphs to show the existence of a rainbow matching covering all but $O(log n/log{log{n}})$ vertices. All previous results, based on the semi-random method, left uncovered at least $Omega(n^{alpha})$ (for some constant $alpha$) vertices.
A matrix $Ainmathbb{C}^{ntimes n}$ is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every $Ain mathbb{C}^{ntimes n}$ is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E.B. Davies: for each $deltain (0,1)$, every matrix $Ain mathbb{C}^{ntimes n}$ is at least $delta|A|$-close to one whose eigenvectors have condition number at worst $c_n/delta$, for some constants $c_n$ dependent only on $n$. Our proof uses tools from random matrix theory to show that the pseudospectrum of $A$ can be regularized with the addition of a complex Gaussian perturbation. Along the way, we explain how a variant of a theorem of Sniady implies a conjecture of Sankar, Spielman and Teng on the optimal constant for smoothed analysis of condition numbers.
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The purpose of this paper is concerned with the approximate solution of split equality problems. We introduce two types of algorithms and a new self-adaptive stepsize without prior knowledge of operator norms. The corresponding strong convergence theorems are obtained under mild conditions. Finally, some numerical experiments demonstrate the efficiency of our results and compare them with the existing results.
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