A new form of trigonometric orthogonality and Gaussian-type quadrature

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x_i), i = 1(1)2n+1, gives a trigonometric sum of the n^th order, S_2n+1(x = a₀ + ∑_jⁿ = 1^(a_jcos jx + b_jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S_2r+1(x) is one that satisfies

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{2r+1}}\text{(x)S}{}_{\mathrm{2r\prime +1}}\text{(x)dx=0, r′<r}$

and two other arbitrarily imposable conditions needed to make S_2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S_2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S_4r?1(x). We have

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{4r?1(x)}}\text{dx=∑}{}_{\mathrm{j=1}}{}^{\mathrm{2r}}\text{A}{}_{\mathrm{j}}\text{S}{}_{\mathrm{4r?1}}\text{(x}{}_{\mathrm{j}}\text{)}$

where the nodes x_j, j = 1(1)2r, are the zeros of the orthogonal S_2r+1(x). It is proven that A_j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S_2r+1(x), x_jandA_j, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.     

Subproper splitting for rectangular matrices

In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?C^{m × n}_r. The splitting A = M ? N is called subproper if R(A) ? R(M) and $\text{R(A}{}^1\text{) ?R(M}{}^1\text{)}$. Consider the iteration $\text{x}{}_{\mathrm{i}}\text{= M}{}^2\text{Nx}{}_{\mathrm{i}}{}_{\mathrm{?1}}\text{+ M}{}^2\text{b}$. We characterize the convergence of this scheme to a solution of the linear system. When A?R^m×ⁿ_r, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.     

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

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 $\text{lim}{}_{\mathrm{n\to \infty }}\text{T}{}_{\mathrm{k}}{}^{\mathrm{r}}\text{(S, n)}\text{n}{}^{\mathrm{k}}\text{=}\text{(p}{}_1\text{… p}{}_{\mathrm{a}}\text{)}{}^{\mathrm{rk}}\text{ζ(rk)(p}{}_1{}^{\mathrm{rk}}\text{? 1) … (p}{}_{\mathrm{s}}{}^{\mathrm{rk}}\text{? 1)}$.     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

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.     

An asymptotic evaluation of the cycle index of a symmetric group

Let $\text{Z(S}{}_{\mathrm{n}}\text{;?(x))}$ denote the polynomial obtained from the cycle index of the symmetric group Z(S_n) by replacing each variable s_i by f(x¹). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=x^k + a_k+1x^k+1 + a_k+2x^k+2+? with every a₁?0. Finally, let 0<x<1 and let x??. We prove that

This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of $\text{1}\text{n!e}$.In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented.     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

Explicit expressions for Bezoutians

The following commutator identity is proved:

$\text{[u(S}{}^1\text{), v(S)] = [v}{}_1\text{(S}{}^1\text{), u}{}_1\text{(S)]}$

. Here S is the n by n matrix of the truncated shift operator S = (Γ_i,i+1), i = 0, 1,…, n ? 1, and u, v are two polynomials of degree not exceeding n. The reciprocal polynomial f;₁ of a polynomial f; of degree ?n is defined by $\text{f}{}_1\text{(z) = z}{}^{\mathrm{n}}\text{f(}\text{1}\text{z}\text{)}$. The commutator identity is closely related to some properties of the Bezoutian matrix of a pair of polynomials; it is used to obtain the Bezoutian matrix in the form of a simple expression in terms of S and S¹. To demonstrate the advantage of this expression, we show how it can be used to obtain simple proofs of some interesting corollaries.     

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.     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

Connected matroids with the smallest whitney numbers

In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if _X(G) = Σμ(0, x)λ^{r ? r(x)} = Σ (?1)^{r ? k}W_kλ^k is the characteristic polynomial of G, then

$\text{w}{}_{\mathrm{k}}\text{?}\text{r}\text{k}\text{+n}\text{r?1}\text{k}$

Thus γ(G) ? 2^{r ? 1} (n+2), where γ(G) = Σw_k. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then

$\text{w}{}_{\mathrm{i}}\text{?}\text{r}\text{i}\text{+ n}\text{r}\text{i+1}\text{for i>1; w}{}_1\text{?r+n}\text{r}\text{2}\text{? 1;}$

|μ| ? (r? 1)n; and γ ? (2^{r ? 1} ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, $\text{r(G) = r(}\text{G}\text{?}\text{) = 4}$, and $\text{n(G) = n(}\text{G}\text{?}\text{) = 3}$ then $\text{(w}{}_1\text{(G))}{}^4{}_{\mathrm{t-G}}\text{? (w}{}_1\text{(}\text{G}\text{?}\text{)) = (8, 20, 18, 7, 1)}$. Further, if β is the Crapo invariant,

$\text{β(G)=}\text{d}{}_{\mathrm{X}}\text{(G)}\text{dλ}\text{(1)}\text{,}$

then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Matrix identities of the fast fourier transform

Let A_n(ω) be the nxn matrix A_n(ω)=(a_ij with a_ij=ω^ij, 0?i,j?n?1, ωⁿ=1. For n=rs we show

$\text{A}{}_{\mathrm{n}}\text{(w)P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{P}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{∵}\text{0}\text{s?1}\text{A}{}_{\mathrm{r}}\text{(w}{}^{\mathrm{s}}\text{)}\text{P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{{T}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(w)}}\text{∵}\text{0}\text{r?1}\text{A}{}_{\mathrm{s}}\text{(w}{}^{\mathrm{r}}\text{)}$

=(A_r?I_s)T^s_r(I_r?A_s). When r and s are relatively prime this identity implies a wide class of identities of the form PA_n(ω)Q^T=A_r(ω^αs)?A_s(ω^βr). The matrices P^s_r, P^r_s, P, and Q are permutation matrices corresponding to the “data shuffling” required in a computer implementation of the FFT, and T^s_r is a diagonal matrix whose nonzeros are called “twiddle factors.” We establish these identities and discuss their algorithmic significance.     

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.     

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.     

Author index

A matrix T=(t_ik) is introduced, the coefficients of which are defined by $\text{k}{}_{\mathrm{ik}}\text{:= (}\text{i}{}^{\mathrm{k}}\text{(ik)!)}\text{Σ}{}_{\mathrm{x?S}}{}_{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{(x)}{}^{\mathrm{k}}\text{, i, k?N={1, 2, 3,…,}}$, where a_i(x) denotes the s the number of i cycles in the element x of the symmetric group S_n. It is shown that these numbers are natural numbers, that they are easy to evaluate, and that they serve very well in order to formulate an infinite number of characterizations of multiply transitive subgroups of symmetric groups in terms of the cycle structure of their elements.     

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.     

Sobolev inequalities with remainder terms

The usual Sobolev inequality in $\text{R}$ⁿ, n ? 3, asserts that , with S_n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? $\text{R}$ⁿ. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with $\text{p =}\text{2}{}^1\text{2}$ and with $\text{q =}\text{n}\text{(n ? 1)}$. (ii) If ? ≠ 0 on ?Ω, then with $\text{q =}\text{2(n ? 1)}\text{(n ? 2)}$. Some further results and open problems in this area are also presented.