The Division Relation: Congruence Conditions and Axiomatisability

We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y ₁,…, y _n ) involving all of the listed variables, and elements c ₁,…, c _n such that t ^A (a, c ₁,…, c _n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.     

Principal minors and diagonal similarity of matrices

Let A, B be n × n matrices with entries in a field F. Our purpose is to show the following theorem: Suppose n⩾4, A is irreducible, and for every partition of {1,2,…,n} into subsets α, β with ¦α¦⩾2, ¦β¦⩾2 either rank A[α¦β]⩾2 or rank A[β¦α]⩾2. If A and B have equal corresponding principal minors, of all orders, then B or B^t is diagonally similar to A.     

Isomorphism and elementary equivalence of multilinear maps

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures:

1. the (n+2)-tuples (A ₁…,A _n+1,f),where n≥2 is an integerA ₁…,A _n+1 are disjoint finitely generated abelian groups and f A ₁×…×A _n →A _n+1: is a n-linear map;

2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A ⁿ →B is a n-linear map;

3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A ⁿ →A is a n-linear map.

For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

On the convex hull of the multiform numerical range

Let A ₁,…,A_m be nxn hermitian matrices. Definine

W(A ₁,…,A_m )={(xA₁x ^?,…xA_mx ^?):x?C ⁿ ,xx ^?=1}. We will show that every point in the convex hull of W(A ₁,…,A_m ) can be represented as a convex combination of not more than k(m,n) points in W(A ₁,…,A_m ) where k(m,n)=min{n,[√m]+δ _n ² _m+1}.     

Failure of cancellation conditions for additive linear orders

Coordinate independence assumptions, also known as cancellation conditions, play a central role in the representational theory of measurement for an ordering relation ≻ on a finite Cartesian product set A₁ × A₂ × ··· × A_m. A sequence of increasingly complex cancellation conditions is known to be sufficient for additive representability in the form (a₁, a₂, ⋖ a_m) ≻ (b₁, b₂, ⋖ b_m) ⇔ ∑_i v(a_i) > &sum_i v(b_i). A longstanding open problem is to determine the simplest subset of cancellation conditions as a function of the size of A₁ × ··· × A_m that is violated by every order ≻ that is not additively representable. This article proves a lower bound on minimum subset sufficiency when all A_i are binary. We conjecture that this lower bound, which is very near to a known upper bound, is the exact minimum. The binary-factors version of the problem is reformulated under a first-order independence assumption by a map from ≻ on {0,1}^m into a subset L of {1,0,−1}^m that is referred to as an additive linear order. The lower bound is then established by examples of additive linear orders on {1,0,−1}^m that exhibit worst-case failures of cancellation. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:353–365, 1997     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

Wave equations with time-dependent spatial operators of higher order

We study the initial-boundary value problem for ?_t²u(t,x)+A(t)u(t,x)+B(t)?_tu(t,x)=f(t,x) on [0,T]×Ω(Ω??ⁿ) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Eigenvalue Inequalities and Schubert Calculus

Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A₁,…, A_r ? ¢^n×n with the spectrum of the sum A₁ + … + A_r. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices.     

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s _t  ∈ A), where ?⁺ ? t → s _t  ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Doubly stochastic matrices which have certain diagonals with constant sums

Let A be an n×n doubly stochastic matrix and suppose that 1?m?n?1. Let τ₁,…,τ_m be m mutually disjoint zero diagonals in A, and suppose that every diagonal of A disjoint from τ₁,…,τ_m has a constant sum. Then aall entries of A off the m zero diagonals have the value (n?m)^?1. This verifies a conjecture of E.T. Wang.     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

On the existence of unitarily invariant norm under some conditions

Let r ₁, …, r _m be positive real numbers and A ₁, …, A _m be n × n matrices with complex entries. In this article, we present a necessary and sufficient condition for the existence of a unitarily invariant norm ‖·‖, such that ‖A _i ‖ = r _i , for i = 1, …, m. Then we identify the greatest unitarily invariant norm which satisfies this condition. Using this, we get an approximation of unitarily invariant norms. Although the minimum unitarily invariant norm which satisfies this condition does not exist in general, we find conditions over A _i s and r _i s which are sufficient for the existence of such a norm. Finally, we get a characterization of unitarily invariant norms.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

r-Numbers completion problems of partial upper canonical form I

A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {r_j}_j=1^p as the r-numbers sequence if r₁ = rank(B) and r_j = rank[B AB … A^j−1 B] − rank[B AB … A^j−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion A_c of A, such that the pair (A^c, B) has an r-numbers sequence prescribed under some restrictions.     

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)