Extended rank reduction formulas containing Wedderburn and Abaffy–Broyden–Spedicato rank reducing processes

The Wedderburn rank reduction formula and the Abaffy–Broyden–Spedicato (ABS) algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as Gram–Schmidt, conjugate direction and Lanczos methods. We present a rank reduction formula for transforming the rows and columns of A, extending the Wedderburn rank reduction formula and the ABS approach. By repeatedly applying the formula to reduce the rank, an extended rank reducing process is derived. The biconjugation process associated with the Wedderburn rank reduction process and the scaled extended ABS class of algorithms are shown to be in our proposed rank reducing process, while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations. The process provides a general finite iterative approach for constructing factorizations of A   and A^T$A^T$ under a common framework of a general decomposition V^TAP=Ω$V^{T}AP=\Omega$. We also show that the biconjugation process associated with the Wedderburn rank reduction process can be derived from the scaled ABS class of algorithms applied to A   or A^T$A^T$. Finally, we provide a list of some well-known reduction procedures as special cases of our extended rank reducing process. The approach is general enough to produce various structured decompositions as well.     

Patterson–Sullivan distributions in higher rank

For a compact locally symmetric space X _Γ of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an h-pseudo-differential calculus on X _Γ , we define and study lifted quantum limits as weak*-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson–Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.     

Torsion-free abelian α-irreducible groups of finite rank

If F is a free abelian group of finite rank and α is an endomorphism or an automorphism of its divisible hull, then the α‐ hull is determined, i.e. the minimal torsion-free abelian group with this endomorphism a. Torsion-free abelian groups of finite rank are called α-irreducible if their divisible hull is α-irreducible for an automorphism a. A complete classification is given for α-irreducible groups and this result is applied to groups of rank 2.     

Theoretical analysis on convergence behavior of rank filters

This paper systematically studies the convergence behavior of rank filters. The problem of convergence behavior of rank filters has been solved completely for bounded sequences. Moreover, some properties of its limiting sequences and recurrent sequences are obtained.     

The rank reduction procedure of Egerváry

Here we give a survey of results concerning the rank reduction algorithm developed by Egerváry between 1953 and 1958 in a sequence of papers.     

The Brill–Noether rank of a tropical curve

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill–Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill–Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill–Noether rank of a tropical curve with the dimension of the Brill–Noether locus of an algebraic curve.     

Totally geodesic submanifolds of maximal rank in symmetric spaces

We gave a complete list of totally geodesic submanifolds of maximal rank in symmetric spaces of noncompact type. The compact cases can be obtained by the duality.     

Projective maps of rank ≥2 are strongly projective

We investigate the structure of projective maps between manifolds with linear connections, showing in particular that a projective map on a connected manifold that attains a rank ≥2 at some point is strongly projective     

An example on strong shift equivalence of positive integral matrices

The following matrices are considered $$A_k = \left( {\begin{array}{*{20}c} k \\ 1 \\ \end{array} \begin{array}{*{20}c} 2 \\ k \\ \end{array} } \right), B_k \left( {\begin{array}{*{20}c} {k - 1} \\ 1 \\ \end{array} \begin{array}{*{20}c} 1 \\ {k + 1} \\ \end{array} } \right),k \in \mathbb{N},$$ which are strong shift equivalent in the sense ofWilliams [7]. In case \(k + \sqrt 2 \) is a prime number of the algebraic field \(\mathbb{Q}(\sqrt 2 )\) matrices are defined which determine the possible choices of rank two matrices connectingA _k andB _k in the sense of strong shift equivalence. A complete list of all these matrices is given.     

The quasiconvex hull for the five-gradient problem

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying ${Du \in K}$ . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each ${F \in {\rm int}\,K^{rc}}$ there exists a Lipschitz mapping u satisfying $$Du \in K\quad{\rm and}\quad u(x) = Fx\,{\rm for}\,x\,\in\,{\partial} \Omega \,.$$ Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.     

The degenerate two well problem for piecewise affine maps

The two well problem consists in finding maps u which satisfy some boundary conditions and whose gradient Du assumes values in the two wells ${\mathbb{S}_{A}, \mathbb{S}_{B}}$ . Here ${\mathbb{S}_{A}}$ (similarly ${\mathbb{S}_{B}}$ ) is the well generated by a 2 × 2 matrix A, i.e., ${\mathbb{S}_{A}}$ is the set of matrices of the form RA, where R is a rotation. We study specifically the case when at least one of the two matrices A, B is singular and we characterize piecewise affine maps u satisfying almost everywhere the differential inclusion ${Du(x) \in \mathbb{S}_{A} \cup \mathbb{S}_{B}}$ . In particular we describe the lamination and angle properties, which turn out to be different from those of the nonsingular case described in detail in [15]. We also show that the two well problem can be solved in some cases involving singular matrices, in strict contrast to the nonsingular (and not orthogonal) case.     

Global behavior of a higher order difference equation

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation $$x_{n + 1} = \frac{{A + Bx_{n - 2k - 1} }} {{C + D\prod\limits_{i = 1}^k {x_{n - 2i} } }}, n = 0,1,2, \ldots ,$$ where A, B are nonnegative real numbers, C,D > 0 and l, k are nonnegative integers such that l ≤ k.     

Zaks’ Lemma for Coherent Rings

Let A be a left and right coherent ring and C _A (resp., $C_{A^{\mathrm{op}}}$ ) a minimal cogenerator for right (resp., left) A-modules. We show that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}$ whenever flat dim C _A ?<?∞ and $\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ , and that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ if and only if the finitely presented right A-modules have bounded Gorenstein dimension.     

Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

We study the problem $$ \left\{\begin{array}{ll} {-\varepsilon^{2}\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u) = f (x, u)} \quad\; {\rm in} \; \Omega,\\ {u = 0} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {\rm on} \; \partial{\Omega}, \end{array} \right.$$ where Ω is a smooth bounded domain in ${\mathbb{R}^{N},N > 2,}$ and show it possesses nontrivial solutions for small values of ε provided f is a nonnegative continuous function which has a positive zero. The multiplicity result is based on degree theory together with a new Liouville type theorem for ${-{M}^+_{\lambda,\Lambda}(D^{2}u) = f(u)}$ in ${\mathbb{R}^{N}}$ for nonnegative nonlinearities with zeros.     

On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes

We prove that a fork-convex family $ \mathbb{W} $ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the setH of nonnegative random variables majorized by the values of elements of $ \mathbb{W} $ at fixed instants of time is bounded in probability. A securities market model with arbitrarily many main risky assets, specified by the set $ \mathbb{W}\left( \mathbb{S} \right) $ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family S, satisfies the assumptions of this theorem.     

Normal Matrices and an Extension of Malyshev's Formula

Let A be a complex matrix of order n with n ≥ 3. We associate with A the 3n × 3n matrix $Q\left( {\gamma } \right) = \left( \begin{gathered} A \gamma _1 I_n \gamma _3 I_n \\ 0 A \gamma _2 I_n \\ 0 0 A \\ \end{gathered} \right)$ where $\gamma _1 ,\gamma _2 ,\gamma _3 $ are scalar parameters and γ=(γ₁,γ₂,γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ) in the decreasing order. We prove that, for a normal matrix A, its 2-norm distance from the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to $\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in \mathbb{C}} \sigma _{3n - 2} (Q\left( \gamma \right)).$ This fact is a refinement (for normal matrices) of Malyshev's formula for the 2-norm distance from an arbitrary n × n matrix A to the set of n × n matrices with a multiple zero eigenvalue.