Diophantine inequalities with mixed powers,II

It is shown that if λ₁, …, λ₅ are non-zero real numbers, not all of the same sign, and at least one of the ratios $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ (1 ≤ j ≤ 3) is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^2\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^3$ for integer values of x₁, …, x₅ are everywhere dense on the real line. Similar results are proved for the polynomials $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_1{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ … + λ}{}_6\text{x}{}_6{}^3$ and $\text{λ}{}_1\text{x}{}_1{}^2\text{+ λ}{}_2\text{x}{}_2{}^2\text{+ λ}{}_3\text{x}{}_3{}^3\text{+ λ}{}_4\text{x}{}_4{}^3\text{+ λ}{}_5\text{x}{}_5{}^4\text{+ λ}{}_6\text{x}{}_6{}^4$.     

A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

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 $\text{M(y)1H}$.     

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 $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

A vector exchange property of submodular systems

Generalizing the multiple basis exchange property for matroids, the following theorem is proved: If x and y are vectors of a submodular system in $\text{R}{}^{\mathrm{E}}$ and $\text{x}{}_1\text{,x}{}_2\text{?}\text{R}{}^{\mathrm{E}}$ such that x = x₁ + x₂, then there are $\text{y}{}_1\text{,y}{}_2\text{?}\text{R}{}^{\mathrm{E}}$ such that y = y₁ + y₂ and both x₁ + y₁ and x₂ + y₂ belong to the submodular system.An integral analogue holds for the integral submodular systems and a non-negative analogue for polymatroids.     

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: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

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.     

Hilbert space representations of general discrete time stochastic processes

We show that under mild conditions the joint densities Px₁,…,x_n) of the general discrete time stochastic process X_n on $\text{pH}$ can be computed via

$\text{Px}{}_1\text{,…,x}{}_{\mathrm{n}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{) = 6?T(x}{}_1\text{)…T(x}{}_{\mathrm{n}}\text{)6}{}^2$

where ? is in a Hilbert space $\text{pH}$, and T (x), x ? $\text{pH}$ are linear operators on $\text{pH}$. We then show how the Central Limit Theorem can easily be derived from such representations.     

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.     

The existence,uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system $\text{?x}\text{?t}\text{= Δx + F(x,y),}\text{?y}\text{?t}\text{= G(x,y)}$ is investigated, where x and y are scalar functions of time (t ? 0), and n space variables $\text{(ξ}{}_1\text{,…, ξ}{}_{\mathrm{n}}\text{), Δx ≡ ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{x}\text{?ξ}{}_{\mathrm{i}}{}^2$, and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution $\text{(x(r),y(r)),}\text{where}\text{r = (ξ}{}_1{}^2\text{+ … + ξ}{}_{\mathrm{n}}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which is bounded for r ? 0 and satisfies x(0) >x₀, y(0) > y₀, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.     

Minkowski's conjecture for n = 5; a theorem of Skubenko

A theorem of Skubenko asserts that if L is a lattice in R₅, then there exist positive real numbers λ₁,…, λ₅ such that L has five linearly independent points on the boundary of the ellipsoid $\text{Σ}{}_{\mathrm{i=1}}{}^5\text{λ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^2\text{≤ 1}$ and none other than the origin in its interior. A different proof of this result is given for the case when L has homogeneous minimum different from zero.     

Sturm—Liouville problems with several parameters

We consider the regular linear Sturm-Liouville problem (second-order linear ordinary differential equation with boundary conditions at two points x = 0 and x = 1, those conditions being separated and homogeneous) with several real parameters λ₁,…,λ_N. Solutions to this problem correspond to eigenvaluesλ = (λ₁,…,λ_N) forming sets $\text{R}$^N determined by the number of zeroes in (0, 1) of solutions. We describe properties of these sets including: boundedness, and when unbounded, asymptotic directions. Using these properties some results are given for the system of N Sturm-Liouville problems which share only the parameters λ. Sharp results are given for the system of two problems sharing two parameters. The eigensurfaces for a single problem are closely related to the cone $\text{K={λ R}{}^{\mathrm{N}}\text{:λ}{}_1\text{a}{}_1\text{(x)+…+λ}{}_{\mathrm{N}}\text{a}{}_{\mathrm{N}}\text{(x)?0 for all x in [0,1]}}$, particularly in questions of boundedness. The cone K and related objects are discussed, and a result is given which relates cones with two oscillation conditions known as “Right-Definiteness” and “Left-Definiteness.”     

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.     

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.     

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.     

An example of bifurcation to homoclinic orbits

Consider the equation $\text{x}\text{?}\text{? x + x}{}^2\text{= ?λ}{}_1\text{x + λ}{}_2\text{?(t)}\text{where}\text{?(t + 1) = ?(t)}\text{and}\text{λ = (λ}{}_1\text{, λ}{}_2\text{)}$ is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations.     

Starting in maximum-polynomial-degree Nordsieck-Gear methods

The intent of this paper is to show that the Nordsieck-Gear methods with maximum polynomial degree k+1, first described in [1], admit of matched starting methods which are exact for all polynomials of degree ?k+1. In general, it is shown that these starting methods yield starting errors of the required order, O(h^k+2), for all initial-value problems

$\text{y}{}^{\mathrm{(P)}}\text{(x)=f(x,y,y}{}^{\mathrm{(1)}}\text{,y}{}^{\mathrm{(2)}}\text{,…,y}{}^{\mathrm{(p?1)}}\text{),}$

$\text{y(0)=y}{}_0\text{, y}{}^{\mathrm{i}}\text{(0)=y}{}_0{}^{\mathrm{(i)}}\text{, i=1,2,…,p?1,}$

where f is k+1 times continuously differentiable in a neighborhood of the graph of the exact solution (x,?(x)), x?[0,X]. Two theorems are proved. The first is the constructive existence of an algorithm which requires (k-p+1)(k-p+2)/2 evaluations of the function f to obtain approximations of the method's required higher-order scaled derivatives at the origin:

$\text{h}{}^{\mathrm{p+1}}\text{y}\text{?}{}^{\mathrm{(p+1)}}\text{(0)}\text{(p+1)!}\text{,…,}\text{h}{}^{\mathrm{k}}\text{y}\text{?}{}^{\mathrm{(k)}}\text{(0)}\text{k!}\text{,}$

each of which is accurate to O(h^k+2). The second, less general theorem, shows that when f is a polynomial in x, y, and its higher order derivatives y⁽¹⁾,y⁽²⁾,…,y(^p?1), an algorithm can be constructed for obtaining the higher-order scaled derivatives exactly. These results lay to rest once and for all any heuristic arguments against varying corrector minus predictor coefficients for preserving maximal order (polynomial degree) because starting values are inexact. Furthermore, and perhaps most importantly, the maximum-polynomial-degree Nordsieck-Gear methods are shown to have a unique property of zero starting error for an important class of ordinary differential equations.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

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.     

A determinant of symmetric forms

Let Δ(α + β) = |H_λ2?r+1| where H_r is the complete symmetric function in (α₁ + β₁), (α₂ + β₂), …, (α_n + β_n). It is proved that Δ(α + β) ? Δ(α) + Δ(β). This inequality is generalised for certain symmetric functions defined by Littlewood. Let . Then we prove that Ω(α + β) ? Ω(α) + Ω(β). Here λ₁, λ₂, λ₃, …, λ_n is a partition such that λ_n > λ_n?1 > ··· > λ₂ > λ₁.