Sign patterns that require almost unique rank

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive (respectively, negative, zero) entry of B by + (respectively, −, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B:sgn(B)=A}. The minimum rank mr(A) (maximum rank MR(A)) of a sign pattern matrix A is the minimum (maximum) of the ranks of the real matrices in Q(A). Several results concerning sign patterns A that require almost unique rank, that is to say, the sign patterns A such that MR(A) = mr(A) + 1, are established and are extended to sign patterns A for which the spread is d=MR(A)-mr(A). A complete characterization of the sign patterns that require almost unique rank is obtained.     

MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear b...     

The determinant of the sum of two normal matrices with prescribed eigenvalues

Given complex numbers α₁,...,α_n, β₁,...,β_n, what can we say about the determinant of A+B, where A (B) is an n×n normal matrix with eigenvalues α₁,...,α_n (β₁,...,β_n)? Some partial answers are offered to this question.     

General preservers of quasi-commutativity on self-adjoint operators

Let H be a separable Hilbert space and Bsa(H) the set of all bounded linear self-adjoint operators. We say that A,B∈Bsa(H) quasi-commute if there exists a nonzero ξ∈C such that AB=ξBA. Bijective maps on Bsa(H) which preserve quasi-commutativity in both directions are classified.     

On a restricted cross-intersection problem

Suppose A and B are families of subsets of an n-element set and L is a set of s numbers. We say that the pair (A,B) is L-cross-intersecting if |A∩B|∈L for every A∈A and B∈B. Among such pairs (A,B) we write PL(n) for the maximum possible value of |A||B|. In this paper we find an exact bound for PL(n) when n is sufficiently large, improving earlier work of Sgall. We also determine P{2}(n) and P{1,2}(n) exactly, which respectively confirm special cases of a conjecture of Ahlswede, Cai and Zhang and a conjecture of Sgall.     

Generalizations of the reverse order law with related results

This paper presents conditions which are necessary and sufficient for (AB>)⁺ = B⁺A^ω for all normalized generalized inverses A^ω of the complex matrix A. Corresponding conditions are stated which are equivalent to the situation where (AB)⁺ = B_ωA⁺ is satisfied by each weak generalized inverse B_ω of B. The results are applied to theorems by Baskett and Katz and by Schwerdtfeger.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

Implicitly-weighted total least squares

In a total least squares (TLS) problem, we estimate an optimal set of model parameters X, so that (A-ΔA)X=B-ΔB, where A is the model matrix, B is the observed data, and ΔA and ΔB are corresponding corrections. When B is a single vector, Rao (1997) and Paige and Strakoš (2002) suggested formulating standard least squares problems, for which ΔA=0, and data least squares problems, for which ΔB=0, as weighted and scaled TLS problems. In this work we define an implicitly-weighted TLS formulation (ITLS) that reparameterizes these formulations to make computation easier. We derive asymptotic properties of the estimates as the number of rows in the problem approaches infinity, handling the rank-deficient case as well. We discuss the role of the ratio between the variances of errors in A and B in choosing an appropriate parameter in ITLS. We also propose methods for computing the family of solutions efficiently and for choosing the appropriate solution if the ratio of variances is unknown. We provide experimental results on the usefulness of the ITLS family of solutions.     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

Independence for partition regular equations

A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies Ax=0. We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerden's theorem and Schur's theorem, may naturally be interpreted as statements about partition regularity. Those matrices which are partition regular over the positive integers were completely characterised by Rado in 1933.Given matrices A and B, we say that A Rado-dominates B if any set which is PR for A is also PR for B. One trivial way for this to happen is if every solution to Ax=0 actually contains a solution to By=0. Bergelson, Hindman and Leader conjectured that this is the only way in which one matrix can Rado-dominate another. In this paper, we prove this conjecture for the first interesting case, namely for 1×3 matrices. We also show that, surprisingly, the conjecture is not true in general.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

Birational maps between generalized Severi-Brauer varieties

A conjecture of Amitsur states that two Severi-Brauer varieties V(A) and V(B) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi-Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi-Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi-Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.     

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

Intervals and tests for weighted sum of percentiles θ(p) and θ(1−p) for symmetrical populations

Available is a random sample from a distribution that is continuous, symmetrical, and reasonably well-behaved. Approximate one-sided and two-sided confidence intervals (provide tests) are developed forAθ(p)+Bθ(1?p), where θ(p) is the population 100p percentile, 1/2≦p<1, andA, B can be any positive or negative numbers. The interquantile and other ranges are special cases. Asymptotically, a confidence coefficient value is precisely determined. The statistics used are weighted sums (with weightsA andB) of two percentage points of the sample. Many comparisons of population percentiles can be made through suitable choice ofp, A, andB.     

Euclidean ramsey theorems. I

The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B. For a?A denote by R(a) the set {b?B | (a, b) ?R}. R is called r-Ramsey if for any r-part partition of B there is some a?A with R(a) in one part. We investigate questions of whether or not certain R are r-Ramsey where B is a Euclidean space and R is defined geometrically.     

On the growth of solutions of second order linear differential equations with extremal coefficients

In this paper, we consider the differential equation f + A(z)f + B(z)f = 0, where A and B ≡ 0 are entire functions. Assume that A is extremal for Yang's inequality, then we will give some conditions on B which can guarantee that every non-trivial solution f of the equation is of infinite order.     

Imbedding conditions for λ-matrices

We say that A(λ) is λ-imbeddable in B(λ) whenever B(λ) is equivalent to a λ-matrix having A(λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A(λ) to be λ-imbeddable in B(λ). The solution is given in terms of the invariant polynomials of A(λ) and B(λ). We also solve an analogous problem when A(λ) and B(λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B, over a field F, with prescribed similarity invariant polynomials and a prescribed principal submatrix A.     

Long cycles in bipartite graphs

Let G be a 2-connected bipartite graph with bipartition (A, B), where |A| ≥ |B|. It is shown that if each vertex of A has degree at least k, and each vertex of B has degree at least l, then G contains a cycle of length at least 2min(|B|, k + l ? 1, 2k ? 2). Then this result is used to determine the minimum number of edges required in a bipartite graph to ensure a cycle of length at least 2m, for any integer m ≥ 2.