Upper bounds for permanents of (1, − 1)-matrices

Let Ω _n denote the set of alln×n (1, ? 1)-matrices. In 1974 E. T. H. Wang posed the following problems: Is there a decent upper bound for |perA| whenAσΩ _n is nonsingular? We recently conjectured that the best possible bound is the permanent of the matrix with exactlyn?1 negative entries in the main diagonal, and affirmed that conjecture by the study of a large class of matrices in Ω _n . Here we prove that this conjecture also holds for another large class of (1, ?1)-matrices which are all nonsingular. We also give an upper bound for the permanents of a class of matrices in Ω _n which are not all regular.     

On the truncated assignment polytope

We investigate the convex polytope Ω_m,n(r) which is the convex hull of the m × nr-subpermutation matrices. The faces of Ω_m,n(r) are characterized, and formulae are obtained to compute their dimensions. The faces of Ω_m,n(r) are themselves convex polytopes, and we determine their facets.     

Permanental Polytopes of Doubly Stochastic Matrices

A subpolytope Γ of the polytope Ω_n of all n×n nonnegative doubly stochastic matrices is said to be a permanental polytope if the permanent function is constant on Γ. Geometrical properties of permanental polytopes are investigated. No matrix of the form 1⊕A where A is in Ω₂ is contained in any permanental polytope of Ω₃ with positive dimension. There is no 3-dimensional permanental polytope of Ω₃, and there is essentially a unique maximal 2-dimensional permanental polytope of Ω₃ (a square of side $\text{1}\text{3}$). Permanental polytopes of dimension $\text{(n}{}^2\text{?3n+4)}\text{2}$ are exhibited for each n?4.     

Some rationally looking faces of Ω n having irrational minimum permanents

Let Ω _n denote the convex polytope consisting of all n × n doubly stochasiic matrices. We determine the minimum permanents which may or may not be rational and the permanent-minimizing matrices over some rationally looking faces of Ω _n We also discuss the barycentricity of the (0, l)-matrices with which we consider the permanent-minimization problem.     

Isolated sets of diagonals,diagonal hypergraphs,and simplices of doubly stochastic matrices

We investigate a connection between isolated sets of diagonals of a matrix and simplicial faces of the polytope Ω _n of doubly stochastic matrices. We also consider the minimum permanent on simplicial faces of Ω _n .     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

Cauchy integral decomposition for harmonic forms

Let ann-dimensional differential form Ω be defined at points of aC ¹-smooth boundary π of a domainG ? ? ⁿ . Under what condition can Ω be represented as Ω = Ω₊ + Ω₊ + Ω_-, where Ω_± are forms insideG and outsideG, harmonic in the sense of Hodge? A necessary condition is that both restrictions Ω{inπ and *Ω{inπ be closed in the sense of currents. This condition, with an additional smoothness assumption, turns out to be sufficient as well. This is an analogue of the Cauchy integral decomposition of functions in the plane.     

Generalized triangulations and diagonal-free subsets of stack polyominoes

For n?3, let Ω_n be the set of line segments between vertices in a convex n-gon. For j?1, a j-crossing is a set of j distinct and mutually intersecting line segments from Ω_n such that all 2j endpoints are distinct. For k?1, let Δ_n,k be the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing. For example, Δ_n,1 has one maximal set for each triangulation of the n-gon. Dress, Koolen, and Moulton were able to prove that all maximal sets in Δ_n,k have the same number k(2n-2k-1) of line segments. We demonstrate that the number of such maximal sets is counted by a k×k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We generalize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.     

On some questions concerning permanents of (1,−1)-matrices

Let Ω _n denote the set of alln×n-(1,?1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω _n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 ^k ?1,k≧2, even for singularA∈Ω _n . Moreover, we prove that perA≠0 ifA∈Ω _n ,n=2 ^k ?1, and we derive new results concerning the divisibility of the permanent in Ω _n by powers of 2.     

The group-theoretic complexity of subsemigroups of boolean matrices

We find the group-theoretic complexity of many subsemigroups of the semigroup B_n of n × n Boolean matrices, including Hall matrices, reflexive matrices, fully indecomposable matrices, upper triangular matrices, row-rank-n matrices, and others.     

Global regularity of the\overline \partial-neumann problem on an annulus between two pseudoconvex manifolds which satisfy property (P)problem on an annulus between two pseudoconvex manifolds which satisfy property (P)

LetX be a complex manifold of dimensionn≥3. Let Ω₁, Ω₂ be two open pseudoconvex submanifolds with smooth boundary such that Ω₁ ? Ω₂ ?X . Let Ω = Ω₂ \ $\overline \Omega_1 $ . Assume thatbΩ₁ andbΩ₁ satisfy Catlin's condition (P). Then the compactness estimate for (p, q)-forms with 0<q<n?1 holds for the $\overline \partial$ -Neumann problem on Ω. This result implies that given a $\overline \partial$ -closed (p, q)-form α with 0<q<n?1, which isC ^∞ on $\overline \Omega$ and which is cohomologous to zero on Ω, the canonical solutionu of the equation $\overline \partial$ u=α is smooth on $\overline \Omega$ .     

Generalized doubly stochastic and permutation matrices over a ring

Let R be a ring with unity. A combinatorial argument is used to show that the R-module Δ_n(R) of all n × n matrices over R with constant row and column sums has a basis consisting of permutation matrices. This is used to characterize orthogonal matrices which are linear combinations of permutation matrices. It is shown that all bases of Δ_n(R) consisting of permutation matrices have the same cardinality, and other properties of bases of Δ_n(R) are investigated.     

Asymptotic expansions of the distributions of the latent roots in MANOVA and the canonical correlations

Asymptotic expansions are given for the density function of the normalized latent roots of S₁S₂^?1 for large n under the assumption of $\text{Ω = O(n)}$, where S₁ and S₂ are independent noncentral and central Wishart matrices having the $\text{W}{}_{\mathrm{p}}\text{(b, Σ; Ω)}$ and W_p(n, Σ) distributions, respectively. The expansions are obtained by using a perturbation method. Asymptotic expansions are also obtained for the density function of the normalized canonical correlations when some of the population canonical correlations are zero.     

Products of irreducible matrices

In this paper we characterize the subsemigroup of B_n (B_n is the multiplicative semigroup of n × n Boolean matrices) generated by all the irreducible matrices, and hence give a necessary and sufficient condition for a Boolean matrix A to be a product of irreducible Boolean matrices. We also give a necessary and sufficient condition for an n × n nonnegative matrix to be a product of nonnegative irreducible matrices.     

The weights of Hadamard matrices

Let $\text{Ω}{}_{\mathrm{n}}$ denote the set of all n × n Hadamard matrices. For $\text{H ∈ Ω}{}_{\mathrm{n}}$, define the weight of H to be w(H) = number of 1's in H, and $\text{w(n) =}\text{max}\text{{w(H); H ∈ Ω}{}_{\mathrm{n}}\text{}}$. In this paper, we derive upper and lower bounds for w(n).     

Functions with derivatives given by polynomials in the function itself or a related function

We construct the set of holomorphic functions S ₁ = {f: Ω_f ? ? → ?} whose members have n-th order derivatives which are given by a polynomial of degree n+1 in the function itself. We also construct the set of holomorphic functions S ₂ = {g: Ω_g ? ? → ?} whose members have n-th order derivatives which are given by the product of the function itself with a polynomial of degree n in an element of S ₁. The union S ₁ ∪ S ₂ contains all the hyperbolic and trigonometric functions. We study the properties of the polynomials involved and derive explicit expressions for them. As particular results, we obtain explicit and elegant formulas for the n-th order derivatives of the hyperbolic functions tanh, sech, coth and csch and the trigonometric functions tan, sec, cot and csc.     

Classical verigin problem as a limit case of verigin problem with surface tension at free boundary

In this psper we consider Verigin problem with surface tension st free     

Orbits of Nilpotent Matrices

The action under conjugation of invertible lower triangular n×n matrices (over an infinite field) on lower triangular nilpotent matrices, N_n, divides N_n into orbits. We show that for n?6 the number of orbits is infinite.