Small Ball Estimates for Quasi-Norms

This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood–Offord type estimates for quasi-norms. This generalizes results which were previously obtained in Friedland and Sodin (C R Math Acad Sci Paris 345(9):513–518, 2007), and Rudelson and Vershynin (Commun Pure Appl Math 62(12):1707–1739, 2009).     

The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

In this paper we discuss the notion of singular vector tuples of a complex-valued \(d\) -mode tensor of dimension \(m_1\times \cdots \times m_d\) . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., \(m_1=\cdots =m_d\) . We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank- \((r_1,\ldots ,r_d)\) approximation for \(d\) -mode tensors.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

The maximal orders of finite subgroups in

We give a relatively simple proof that the orthogonal group over the integers is the unique finite subgroup (up to a conjugation) in of the maximal order for .

     

Phase coherence of the electrons in δ-doped gaAs

The phase coherence length of the electrons is determined by magnetoconductivity for a wide range of sheet concentrations of Si-donors and free electrons. For diffusive carrier motion a comparison of the experimental and theoretical results for weak fields shows that the weak localisation is limited by inelastic electron-electron-scattering as expected for disturbed systems. For higher magnetic fields the role of mutual electron-electron interaction is also significant.     

Inverse eigenvalue problems

The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.     

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λ _d (p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤ ^d , where p∈[0,1] is the dimer density. We give upper and lower bounds for λ _d (p) in terms of expressions involving λ _d−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤ ^d is bounded above by λ _d (p). We compute the first three terms in the formal asymptotic expansion of λ _d (p) in powers of  \frac1d\frac{1}{d}. We prove that the lower asymptotic matching conjecture is satisfied for λ _d (p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.     

On the eigenvalues of non-negative Jacobi matrices

The spectrum σ of a non-negative Jacobi matrix J is characterized. If J is also required to be irreducible, further conditions on σ are needed, some of which are explored.