A combinatorial application of the Frobenius inequality on rank function to maximum set of commuting nilpotent matrices

In this note we show how the Frobenius inequality on rank function can be applied to determine the maximum cardinality of a setS of pairwise-commutative n×n nilpotent matrices with the property that no finite product of distinct elements of S equals zero.     

Outliers in the spectrum of iid matrices with bounded rank perturbations

It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier eigenvalues. We show that if the perturbation is small, then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation; but if the perturbation is large, then many more outliers can be created, and their law is governed by the zeroes of a random Laurent series with Gaussian coefficients. On the other hand, these outliers may be eliminated by enforcing a row sum condition on the final matrix.     

On the dimension of some real spaces of bounded rank matrices

Given n∈N, let X be either the set of hermitian or real n×n matrices of rank at least n-1. If n is even, we give a sharp estimate on the maximal dimension of a real vector space V⊂X∪{0}. The results are obtained, via K-theory, by studying a bundle map induced by the adjunction of matrices.     

Large spaces of symmetric matrices of bounded rank are decomposable

Let k and n be positive integers such that k≤n. Let S_n(F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n(F) is said to be a k-subspace if rank A≤k for every AεL.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n(F) is decomposable if there exists in Fⁿ a subspace W of dimension n-r such that x^tAx=0 for every xεWAεL.

We show here, under some mild assumptions on kn and F, that every k∥-subspace of S_n(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n.     

Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup T _k (X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| ∈ «3, 4» or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).     

On bilinear maps determined by rank one matrices with some applications

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map ? from M_n×M_n to M_n satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,y∈M_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.     

Profinite groups with nonabelian crowns of bounded rank and their probabilistic zeta function

It has been conjectured by Mann that the infinite sum Σ _H μ(H,G)/|G:H| ^s , where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group.     

Linear maps preserving pairs of Hermitian matrices on which the rank is additive and applications

Denote the set ofn×n complex Hermitian matrices byH _n . A pair ofn×n Hermitian matrices (A, B) is said to be rank-additive if rank (A+B)=rankA+rankB. We characterize the linear maps fromH _n into itself that preserve the set of rank-additive pairs. As applications, the linear preservers of adjoint matrix onH _n and the Jordan homomorphisms ofH _n are also given. The analogous problems on the skew Hermitian matrix space are considered.     

On the rank of certain matrices

This note is an appendix to the paper “Osculating spaces and diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier)” by M. Bolognesi and G. Pirola.     

Global optimization of a quadratic function subject to a bounded mixed integer consraint set

In this paper we consider the optimization of a quadratic function subject to a linearly bounded mixed integer constraint set. We develop two types of piecewise affine convex underestimating functions for the objective function. These are used in a branch and bound algorithm for solving the original problem. We show finite convergence to a near optimal solution for this algorithm. We illustrate the algorithm with a small numerical example. Finally we discuss some modifications of the algorithm and address the question of extending the problem to include quadratic constraints.Supported by grants from the Danish Natural Science Research Council and the Danish Research Academy.     

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle More concretely, we prove that if , then the period function of the center is monotonically increasing.

     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

On the complete set packing and set partitioning polytopes: Properties and rank 1 facets

This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary $n$-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets.     

Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

We prove that if is of bounded variation, then the uncentered maximal function is absolutely continuous, and its derivative satisfies the sharp inequality . This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.

     

Strongly stable rank and applications to matrix completion

We perform an in-depth study of strongly stable ranks of modules over a commutative ring. Here we define the strongly stable rank of a module to be the supremum of the stable ranks of its finitely generated submodules. As an application, we give non-Noetherian generalizations of known facts about outer products and matrix completions over PIRs and Dedekind domains. We construct Noetherian and non-Noetherian domains of arbitrary strongly stable rank. We also consider strongly n-generated ideals, and we characterize the rings in which every ideal is strongly 2-generated and the domains in which every ideal is strongly 3-generated.