Rectangular arrays

In this paper we studied m×n arrays with row sums $\text{nr}\text{(n,m)}$ and column sums $\text{mr}\text{(n,m)}$ where (n,m) denotes the greatest common divisor of m and n. We were able to show that the function H_m,n(r), which enumerates m×n arrays with row sums and column sums $\text{nr}\text{(m,n)}$ and $\text{mr}\text{(n,m)}$ respectively, is a polynomial in r of degree (m?1)(n?1). We found simple formulas to evaluate these polynomials for negative values, ?r, and we show that certain small negative integers are roots of these polynomials. When we considered the generating function G_m,n(y) = Σ_r?0H_m,n(r)y^r, it was found to be rational of degree less than zero. The denominator of G_m,n(y) is of the form (1?y)^(m?1)(n?1)+3, and the coefficients of the numerator are non-negative integers which enjoy a certain symmetric relation.     

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

The orthogonality property of the Lommel polynomials and a twofold infinity of relations between Rayleigh''s σ-sums

Making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials, it is proven that the known infinite continued fraction development of the ratio of Bessel functions J_v?1(z)/J_v(z) gives rise to an orthogonality property of the Lommel polynomials $\text{{R}{}_{\mathrm{m,v}}\text{(}\text{1}\text{z}\text{)|m?}\text{N}\text{}}$ when v is real and positive. The corresponding weight function which appears to be non-negative in the interval of definition, is obtained by the application of two successive integral transforms. It consists of an infinite series of Dirac δ-functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides. For any positive v, the Lommel polynomials form a system of so-called orthogonal polynomials of a discrete variable. The orthogonality property may also be conveniently expressed by means of a Stieltjes integral. One of its corollaries is a twofold infinity of linear relations between the sums σ_v^(r) defined by σ_v^(r)=Σ_n=1^+∞1/j_v,n^2r, with $\text{v+1Σ}\text{R}{}_0{}^{\mathrm{+}}\text{, rΣ}\text{N}{}_0\text{,}\text{in which}\text{j}{}_{\mathrm{v,n}}\text{represents the}\text{n}\text{th positive zero of}\text{J}{}_{\mathrm{v}}\text{(z)}$.Another by-product consists of a complement to a theorem of Hurwitz concerning the nature and the position of the zeros of the Lommel polynomials written as g_m,v(z) in the modified notation of the mentioned author. From this study also result two interesting approximations of j_v,1 applicable for v?]?1, +1].     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

On the existence of finite central groupoids of all possible ranks. I

The problem of determining the number of finite central groupoids (an algebraic system satisfying the identity (x · y) · (y ? z) = y) is equivalent to the problem of determining the number of solutions of the matrix equation A² = J, where A is a 0, 1 matrix and J is a matrix of 1's.The existence of solutions of A² = J of all ranks r, where $\text{n ? r ? [}\text{(n}{}^2\text{+ 1)}\text{2}\text{]}$, and A is n² × n², is proven. Since these are the only possible values, the question of existence solutions of all possible ranks is completely answered. The techniques and proofs are of a constructive nature.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Generalized Bell numbers and zeros of successive derivatives of an entire function

Six different formulations equivalent to the statement that, for n ? 2, the sum ∑_{k = 1}ⁿ (?1)^kS(n, k) ≠ 0, where the S(n, k) are Stirling numbers of the second kind, are shown to hold. Using number-theoretic methods, a sufficient condition for the above statement to be true for a set of positive integers n having density 1 is then obtained. It remains open whether it is true for all n > 2. The equivalent statements then yield information on the irreducibility of the polynomials ∑_{k = 1}ⁿS(n, k)t^{k = 1} over the rationals, the nonreal zeros for successive derivatives $\text{(}\text{d}\text{dz}\text{)}{}^{\mathrm{n}}\text{exp}\text{(e}{}^{\mathrm{iz}}\text{)}$, a gap theorem for the nonzero coefficients of exp(?e^z), and the continuous solution of the differential-difference equation $\text{?(x) = 1, 0 ? x < 1, ?′(x) = ?¦x¦?(x ? 1), 1 ? x < ∞}$, where ∥ denotes the greatest integer function.     

Affine and combinatorial binary m-spaces

Let V(n) denote the n-dimensional vector space over the 2-element field. Let a(m, r) (respectively, c(m, r)) denote the smallest positive integer such that if n ? a(m, r) (respectively, n ? c(m, r)), and V(n) is arbitrarily partitioned into r classes C_i, 1 ? i ? r, then some class C_i must contain an m-dimensional affine (respectively, combinatorial) subspace of V(n). Upper bounds for the functions a(m, r) and c(m, r) are investigated, as are upper bounds for the corresponding “density functions” $\text{a}\text{(m, ?)}$ and $\text{c}\text{(m, ?)}$.     

Finitely generated submodules of differentiable functions II

Let [$\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of [$\text{E}$(Ω)]^p are of the form im(F) where F: [$\text{E}$(Ω)]^q → [$\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in [$\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.     

On a problem of Chvátal and Erdös on hypergraphs containing no generalized simplex

Let X be an n-element set and $\text{T}$ a family of k-subsets of X. Let r be an integer, k > r ? 2. Suppose that $\text{T}$ does not contain r + 1 members having empty intersection such that any r of them intersect non-trivially. Chvátal and Erdös conjectured that for (r + 1) k ? rn we have |$\text{F}$|$\text{?}\text{n?1}\text{k?1}$. In this paper we first prove that This conjecture holds asymptotically (Theory 1). In Theorems 4 and 5 we prove it for r = 2, K ? 5, n > n_o(k); k ? 3r, n > n_o(k, r), respectively.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Topological cliques of random graphs

Given a graph G, denote by tcl(G) the largest integer r for which G contains a TK^r, a toplogical complete r-graph. We show that for every ? > 0 almost every graph G of order n satisfies

$\text{(2?ε)n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{?}\text{tlc}\text{(G)?(2+ε)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

     

On maximal intersecting families of finite sets

Let r be a positive integer. A finite family $\text{H}$ of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ? $\text{H}$, |A| ? r there exists a member E ? $\text{H}$ such that $\text{A ∩ E = ?}$. For instance, a finite projective plane of order r ? 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove $\text{N(r) ?}\text{3}\text{4}\text{r}{}^2$ whenever a projective plane of order $\text{r}\text{2}$ exists. This disproves the known conjecture N(r) ? r² ? r + 1.     

A radical diophantine equation

All solutions in positive integers x, yz of the diophantine equation $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>m</mtext>}}\text{+ y}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{= z}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}$ are determined, where m, n, r are given positive integers. The proof makes use of a simple criterion for the irreducibility of the polynomial xⁿ ? a over the rationals, where a is a positive rational.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

A note on the functional law of iterated logarithm for maxima of Gaussian sequences

Let {X_n, n ≥ 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let M_n = max{X_t, i ≤ n} and H_n(t) = (M_[nt] ? b_n)a_n^?1 be the maximum resp. the properly normalised maximum process, where $\text{c}{}_{\mathrm{n}}\text{= (2}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{a}{}_{\mathrm{n}}\text{=}\text{(}\text{log log}\text{n)}\text{c}{}_{\mathrm{n}}$ and $\text{b}{}_{\mathrm{n}}\text{= c}{}_{\mathrm{n}}\text{?}\text{1}\text{2}\text{(}\text{log}\text{(4π}\text{log}\text{n))}\text{c}{}_{\mathrm{n}}$. We characterize the almost sure limit functions of (H_n)_n≥3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, ∞), if r(n) (log n)^3?Δ = O(1) for all Δ > 0 or if r(n) (log n)^2?Δ = O(1) for all Δ > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence.