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.     

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 new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

Let A be an n × n complex matrix, and write A = H + iK, where i² = ?1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f(x, y, z) = det(zI ? xH ? yK). Suppose f(x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U^?1AU is block diagonal. We prove that if f(x, y, z) has a linear factor of multiplicity greater than n?3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

On a problem of Oppenheim concerning “factorisatio numerorum”

Let f(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is “highly factorable” if f(m)<f(n) for all m < n. We prove that f(n)=n·L(n)^?1+0(1) for n highly factorable, where $\text{L(n)=}\text{exp}\text{{}\text{log}\text{n}\text{logloglog}\text{n}\text{loglog}\text{n}\text{}}$. This result corrects the 1926 paper of Oppenheim where it is asserted that f(n)=n·L(n)^?2+0(1). Some results on the multiplicative structure of highly factorable numbers are proved and a table of them up to 10⁹ is provided. Of independent interest, a new lower bound is established for the function Ψ(x, y), the number of n≤x free of prime factors exceeding y.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Calculation of the Shannon information

Let ($\text{Ω, β, μ}{}_{\mathrm{X}}$) and $\text{(?, F, μ}{}_{\mathrm{N}}\text{)}$ be probability spaces, with $\text{f: Ω × ? ? ?}\text{a}\text{β × F|F}$ measurable map. Define μ_XY on β × $\text{F}$ by μ_XY(A) = μ_X ? μ_N{(x, y): (x, f(x, y)) ?A}, and let μ_Y = (μ_X ? μ_N)^of^?1. An expression is determined for computing the Shannon information in the measure μ_XY. This expression is used to compute the information for the non-linear additive Gaussian channel, and can be used to solve the channel capacity problem.     

A generalization of the fast LUP matrix decomposition algorithm and applications

We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(m^α?1n) time, where the complexity of matrix multiplication is O(m^α). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(m^α?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations $\text{A}\text{x}\text{=}\text{b}$ (where $\text{b}$ is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, $\text{A}{}^1$, of A (i.e., $\text{AA}{}^1\text{A = A}$). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.     

Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) $\text{x′ =}\text{A}\text{?}\text{(θ + ωt)x,}\text{where}\text{A}\text{?}\text{(θ)}\text{is an}\text{(n × n)}$ matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ? C^N(T^k), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix $\text{A}\text{?}\text{(θ + ωt)}$ in (1) is the Jacobian matrix of a nonlinear vector field $\text{?(x)}$ evaluated along a quasi-periodic solution x = φ(t) of (2) $\text{x′ = ?(x)}$. We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of $\text{Ω = H(φ)}$, the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system.     

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.     

C1-algebras with finite duals

A forest is a finite partially ordered set F such that for x, y, z?F with x ? z, y ? z one has x ? y or y ? x. In this paper we give a complete characterization of all separable C¹-algebras $\text{A}$ with a finite dual $\text{A}\text{?}$, for which Prim $\text{A}$ is a forest with inclusion as partial order. These results are extended to certain separable C¹-algebras $\text{A}$ with a countable dual$\text{A}\text{?}$. As an example these results are used to characterize completely all separable C¹-algebras $\text{A}$ with a three point dual.     

Equivalence of measures for some class of Gaussian random fields

We consider two Gaussian measures P₁ and P₂ on (C(G), $\text{B}$) with zero expectations and covariance functions R₁(x, y) and R₂(x, y) respectively, where R_ν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A^(ν) of order $\text{2m, m ≥ [}\text{d}\text{2}\text{] + 1}$, on a bounded domain G in $\text{R}$^d (ν = 1, 2). It is shown that if the order of A⁽²⁾ ? A⁽¹⁾ is at most $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are equivalent, while if the order is greater than $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are not always equivalent.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

A general approach to optimal control of a regression experiment

Let $\text{θ}\text{θ}\text{?}$ = ($\text{θ}\text{θ}\text{?}$₁, $\text{θ}\text{θ}\text{?}$₂, …, $\text{θ}\text{θ}\text{?}$_n)′ be the least-squares estimator of θ = (θ₁, θ₂, …, θ_n)′ by the realization of the process y(t) = Σ_{k = 1}ⁿθ_kf_k(t) + ξ(t) on the interval T = [a, b] with f = (f₁, f₂, …, f_n)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and D_s-optimality are used as criteria for choosing f in X. A-, D-, and D_s-optimal models can be constructed explicitly by means of r.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.