Commutators of trace zero matrices over principal ideal rings

We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XY ? YX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is simpler than the earlier one.     

A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

In algebraic topology it is well known that, using the Mayer–Vietoris sequence, the homology of a space X can be studied by splitting X into subspaces A and B and computing the homology of A, B, and A∩B. A natural question is: To what extent does persistent homology benefit from a similar property? In this paper we show that persistent homology has a Mayer–Vietoris sequence that is generally not exact but only of order 2. However, we obtain a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms. This implies that persistent homological features of A and B can be found either as persistent homological features of X or of A∩B. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

A note on transitively <Emphasis Type="Italic">D</Emphasis>-spaces

In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ? such that for any non-closed subset A of X there is some V ∈ ? such that |V ∩ A| ? ω, then X is transitively D. As a corollary, if X is a sequential space and has a point-countable wcs*-network then X is transitively D, and hence if X is a Hausdorff k-space and has a point-countable k-network, then X is transitively D. We prove that if X is a countably compact sequential space and has a pointcountable wcs*-network, then X is compact. We point out that every discretely Lindelöf space is transitively D. Let (X, τ) be a space and let (X, ?) be a butterfly space over (X, τ). If (X, τ) is Fréchet and has a point-countable wcs*-network (or is a hereditarily meta-Lindelöf space), then (X, ?) is a transitively D-space.     

The real symmetric matrices of odd order with a P-set of maximum size

Suppose that A is a real symmetric matrix of order n. Denote by m_A(0) the nullity of A. For a nonempty subset α of {1, 2,..., n}, let A(α) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α. When m_A(α)(0) = m_A(0)+|α|, we call α a P-set of A. It is known that every P-set of A contains at most ?n/2? elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs G under which there is a real symmetric matrix A whose graph is G and contains a P-set of size (n ? 1)/2.     

On a recursive inverse eigenvalue problem

Let s ₁, ..., s _n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s _i is an eigenvalue of the leading principal submatrix A _i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s ₁, ..., s _n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A _i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A _i and A _{i + 1}.     

Factorization of cp-rank-3 completely positive matrices

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB^?. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.     

Remarks on discretely absolutely star-Lindelöf spaces

In this paper, we prove the following statements(1) There exists a Hausdorff Lindelöf space X such that the Alexandroff duplicate A(X) of X is not discretely absolutely star-Lindelöf.(2) If X is a regular Lindelöf space, then A(X) is discretely absolutely star-Lindelöf.(3) If X is a normal discretely star-Lindelöf space with e(X) < ω ₁, then A(X) is discretely absolutely star-Lindelöf.     

A stability result using the matrix norm to bound the permanent

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |per(A)| ≤ ||A|| ⁿ ₂ with equality iff A/||A||₂ ∈ P (where ||A||₂ is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ||A|| ⁿ ₂. In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than ||A|| ⁿ ₂ unless all but at most αn rows contain entries of modulus at least ||A||₂(1?β).     

Ultrafilter semigroups generated by direct sums

Let λ be an infinite cardinal and for every ordinal α<λ, let A _α be a set with a distinguished element 0 _α ∈A _α . The direct sum of sets A _α , α<λ, is the subset \(X=\bigoplus_{\alpha<\lambda}A_{\alpha}\) of the Cartesian product ∏_α<λ A _α consisting of all x with finite supp?(x)={α<λ:x(α)≠0 _α }. Endow X with a topology by taking as a neighborhood base at x∈X the subsets of the form {y∈X:y(α)=x(α) for all α<γ} where γ<λ. Let Ult?(X) denote the set of all nonprincipal ultrafilters on X converging to 0∈X. There is a natural partial semigroup operation on X which induces a semigroup operation on Ult?(X). We show that if direct sums X and Y are homeomorphic, then the semigroups Ult?(X) and Ult?(Y) are isomorphic.     

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A ^[t+3]) ≤ γ S(A ^[t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A ^[t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method.     

A gradual rank increasing process for matrix completion

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign “appropriate values” to these entries. A common way to solve this problem is to compute a rank-k matrix, B _k , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B _k . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices \(B_{k}, k = 1, 2, \dots \), where B _k is a rank-k matrix that approximates A in a least squares sense. The computational effort is reduced by using B _k-1 as starting point in the computation of B _k . The ability of B _k to serve as substitute for A is measured with two objective functions: a “training” function that measures the distance between the known part of A and the corresponding part of B _k , and a “probe” function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach.     

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z?=?XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Asymptotic distribution of singular values of powers of random matrices

Let x be a complex random variable such that \( {\mathbf{E}}x = 0,\,{\mathbf{E}}{\left| x \right|^2} = 1 \), and \( {\mathbf{E}}{\left| x \right|^4} < \infty \). Let \( {x_{ij}},i,j \in \left\{ {1,2, \ldots } \right\} \), be independent copies of x. Let \( {\mathbf{X}} = \left( {{N^{ - 1/2}}{x_{ij}}} \right) \), 1≤i,j≤N, be a random matrix. Writing X ^? for the adjoint matrix of X, consider the product X ^m X ^?m with some m ∈{1,2,...}. The matrix X ^m X ^?m is Hermitian positive semidefinite. Let λ₁,λ₂,...,λ _N be eigenvalues of X ^m X ^?m (or squared singular values of the matrix X ^m ). In this paper, we find the asymptotic distribution function \( {G^{(m)}}(x) = {\lim_{N \to \infty }}{\mathbf{E}}F_N^{(m)}(x) \) of the empirical distribution function \( F_N^{(m)}(x) = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\mathbb{I}\left\{ {{\lambda_k} \leqslant x} \right\}} \), where \( \mathbb{I}\left\{ A \right\} \) stands for the indicator function of an event A. With m=1, our result turns to a well-known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457–483, 1967].     

Stratified modules over an extension algebra

Let A be a standard Koszul standardly stratified algebra and X an A-module. The paper investigates conditions which imply that the module Ext* _A (X) over the Yoneda extension algebra A* is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.