Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem $\text{T}{}_{\mathrm{m}}\text{x}{}_{\mathrm{m}}\text{=}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}\text{x}{}_{\mathrm{m}}$, 0 ≠ x_m?H_m, m = 1 … k. (1) Here T_m and V_mn are self-adjoint operators on Hilbert spaces H_m. The index is based on the number of negative eigenvalues of T_m ? ∑_{n = 1}^kλ_nV_mn and on the sign of the determinant δ₀ with (m, n)th entry (x_m, V_mnx_m). We assume that certain cofactors of δ₀ are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ₀ is positive.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Generalizations of two inequalities involving hermitian forms

Let λ₁ and λ_N be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(h_ij), and x=(x₁,x₂,…,x_N) with (x,x)=1. Then, it is known that (1) λ₁?(x,Hx)?λ_N and (2) if, in addition, H is positive definite, $\text{(λ}{}_1\text{+λ}{}_{\mathrm{N}}\text{)}{}^2\text{4λ}{}_1\text{λ}{}_{\mathrm{N}}\text{?(x,Hx)(x,H}{}^{\mathrm{?1}}\text{x)?1}$. Assuming that y=(y₁,y₂,…, y_N) and |y_i|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H^?1 are, respectively, replaced by the Hadamard products $\text{M(y)1H}$ and $\text{M(y)1H}{}^{\mathrm{?1}}$, where M(y) is a matrix defined by $\text{M(y)=(δ}{}_{\mathrm{ij}}\text{+(1?δ}{}_{\mathrm{ij}}\text{)}\text{y}{}_{\mathrm{i}}\text{y}{}_{\mathrm{j}}$. Subsequently, these results are extended to improve the spectral bounds of $\text{M(y)1H}$.     

Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

Matrix extensions of Liouville-Dirichlet-type integrals

The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x₁,…,x_k: x_i?0, i=1,…,k, s?∑^k₁x_i?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes $\text{Ω={V}{}_1\text{,…,V}{}_{\mathrm{k}}\text{: V}{}_{\mathrm{i}}\text{>0, i=1,…,k, 0?∑V}{}_{\mathrm{i}}\text{?t}}$, where now each V_i is a p×p symmetric matrix and A?B means that A?B is positive semidefinite.     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Comparison cones for multiparameter eigenvalue problems

We consider the multiparameter eigenvalue problem (T_r + ∑_{s = 1}^kλ_sV_rs) x_r = 0, x_r ≠ 0, 1 ? r ? k, where T_r and V_rs are self-adjoint linear operators on Hilbert spaces H_r, the V_rs being bounded. The problem may be posed in either ⊕_{r = 1}^kH_r or ⊕_{r = 1}^kH_r and we develop variational approaches for both settings. We explore the rôles played in both settings by $\text{C ={λ ∈ R}{}^{\mathrm{k}}\text{|}\text{∑}\text{k}\text{s=1}\text{λ}{}_{\mathrm{s}}\text{(V}{}^{\mathrm{rs}}\text{x}{}_{\mathrm{r}}\text{,x}{}_{\mathrm{r}}\text{? 0}$ for some nonozero and related cones in $\text{R}$^k. We also compare certain geometrical conditions on C with analytical definiteness conditions already in the literature.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

Uniqueness and continuous dependence results for solutions of singular differential equations

Solutions of Cauchy problems for the singular equations $\text{u}{}_{\mathrm{tt}}\text{+ (}\text{Ψ(t)}\text{t}\text{) u}{}_{\mathrm{t}}\text{= Mu}$ (in a Hilbert space setting) and $\text{u}{}_{\mathrm{t}}\text{+ Δu +}\text{∑}\text{m}\text{i=1}\text{((}\text{k}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)(}\text{?}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{)) + g(t)u=0}\text{in}\text{ω × |0,T)}$, ω={(x₁,…,x_M)ε$\text{R}$^m: 0 < x_i < c_i for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Temporal exponential decay for the Stark effect in atoms

Let $\text{H}{}_1\text{= ?∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{(Δ}{}_{\mathrm{i}}\text{+ V(}\text{x}{}_{\mathrm{i}}\text{)) + ∑}{}_{\mathrm{1\; ?\; i\; <j\; ?\; N}}\text{¦}\text{x}{}_{\mathrm{i}}\text{?}\text{x}{}_{\mathrm{j}}\text{¦}{}^{\mathrm{?1}}\text{, V(}\text{x}{}_{\mathrm{i}}\text{) = N ∝ ¦}\text{x}\text{?}\text{y}\text{¦}{}^{\mathrm{?1}}\text{?(}\text{y}\text{)d}\text{y}$, with ? a normalized Gaussian. Suppose $\text{E}$ ≠ 0 and that $\text{H = H}{}_1\text{+}\text{E}\text{· (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{)}$ has no eigenfunctions in L²($\text{R}$^3N. If H₁ψ = μψ with μ < infσ_ess(H₁), then (ψ, e^?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.     

The probability that k positive integers are relatively r-prime

If r, k are positive integers, then $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ denotes the number of k-tuples of positive integers (x₁, x₂, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r = 1. An explicit formula for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}$ is derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(n)}\text{n}{}^{\mathrm{k}}\text{=}\text{1}\text{ζ(rk)}$.If S = {p₁, p₂, …, p_a} is a finite set of primes, then 〈S〉 = {p₁^a₁p₂^a₂…p_s^a_s; p_i ∈ S and a_i ≥ 0 for all i} and $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ denotes the number of k-tuples (x₁, x₃, …, x_k) with 1 ≤ x_i ≤ n and (x₁, x₂, …, x_k)_r ∈ 〈S〉. Asymptotic formulas for $\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}$ are derived and it is shown that $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Abstract multiparameter theory,I

In this paper we continue our investigation of multiparameter spectral theory. Let H₁,…, H_k be separable Hilbert spaces and H = ?_{r = 1}^kH_r, be their tensor product. In each space H_r we have densely defined self-adjoint operators T_r and continuous Hermitian operators V_rs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ₁,…, λ_n) ?R^k and eigenvectors $\text{? = ?}{}_1\text{? ··· ? ?}{}_{\mathrm{k}}\text{? H}$ such that $\text{T}{}_{\mathrm{r}}\text{?}{}_{\mathrm{r}}\text{+ ∑}{}_{\mathrm{s\; =\; 1}}{}^{\mathrm{k}}\text{λ}{}_{\mathrm{s}}\text{V}{}_{\mathrm{rs}}\text{?}{}_{\mathrm{r}}\text{= 0}$. We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations.