Abelian fibrations on S[n]

Let $\text{S}\text{→}\text{φ}\text{P}{}^1$ be an elliptic fibration on a K3 surface S. Then the composition $\text{S}{}^{\mathrm{[n]}}\text{→}\text{π}\text{S}{}^{\mathrm{(n)}}\text{→}\text{sym}{}^{\mathrm{n}}\text{φ}\text{P}{}^{\mathrm{n}}$ gives an Abelian fibration on S^[n]. Let E be the exceptional divisor of π, then symⁿφ°π(E) is of dimension n?1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

Recurrence relations based on minimization and maximization

Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and g(n) = δ_nI. (2) α + β > 1, min(α, β) > 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and (a) g(n) = δ_n1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases $\text{M(n) = Ω(n}{}^{\mathrm{<mtext>1\; +\; </mtext><mtext>1</mtext><mtext>\gamma </mtext>}}\text{)}$, where γ is defined by α^?γ + β^?γ = 1, and that $\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{M(n)}\text{n}{}^{\mathrm{1\; +\; \gamma }}$ does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even.     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

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.     

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.     

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rⁿ let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in $\text{L}{}^2\text{(Ω))}$. The existence of n commuting self-adjoint operators $\text{H}{}_1\text{,…, H}{}_{\mathrm{n}}\text{in}\text{L}{}^2\text{(Ω)}$ such that each H_j is a restriction of $\text{?i β}\text{βx}{}_{\mathrm{j}}$ (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rⁿ such that the restrictions to Ω of the functions exp i ∑ λ_jx_j form a total orthogonal family in $\text{L}{}^2\text{(Ω)}$. If it is required, in addition, that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$, then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rⁿ. The measurable sets Ω ?Rⁿ (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rⁿ as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rⁿ by some subgroup Γ (essentially the dual of Λ).     

La “conjecture des cylindres”

The “cylinder conjecture” is to suppose that, if K is a gauge, the critical constants of C(K) = K ×] ? 1, +1 [? Rⁿ⁺¹ and of its basis K ? Rⁿ are equal. The connection with packing constants is studied. The concept of Za(ssenhaus)-packing is introduced. ⊕_i=1^hG + (i ? 1)a (G a lattice) is a linear h-lattice, $\text{ζ}{}_{\mathrm{h}}\text{′(K),}\text{ζ}\text{′}{}_{\mathrm{h}}\text{(K)}$, $\text{η}{}_{\mathrm{h}}\text{′(K),}\text{η}\text{′}{}_{\mathrm{h}}\text{(K)}$ the maximum density for translates of K by a linear h-lattice if the translates form a Za-packing for ζ, a packing for η, and if this packing is strict for ^. For K a bounded central star body, it is possible to find H with ζ₁(C(K)) ≤ 2 ζ_H′(K). H is precised for K a gauge and for K = Bⁿ. It is proved by Woods' methods that $\text{η}{}_1\text{(C(B}{}^4\text{)) =}\text{sup}{}_{\mathrm{i=1,\; 3,\; 4,\; 6,}}\text{7}\text{η}{}_{\mathrm{i}}\text{(B}{}^4\text{)}$; a result of Cleaver is used.     

On manifolds supporting quasi-Anosov diffeomorphisms

Let M be an n-dimensional manifold supporting a quasi-Anosov diffeomorphism. If n=3 then either $\text{M=}\text{T}{}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of $\text{Z}{}^6$. If n=4 then Π₁(M) contains a copy of $\text{Z}{}^4$, provided that the diffeomorphism is not Anosov. To cite this article: J. Rodriguez Hertz et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323.     

Critical exponents for the Pucci's extremal operatorsLes exposants critiques pour l'opérateur extrémal de Pucci

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

(1)$\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}\text{(D}{}^2\text{u)+u}{}^{\mathrm{p}}\text{=0,u?0}\text{in}\text{R}{}^{\mathrm{N}}\text{.}$

Here N?3, p>1 and $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{+}}$ denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents $\text{1<p}{}^{\mathrm{s}}{}_{\mathrm{+}}\text{<p}{}^1{}_{\mathrm{+}}\text{<p}{}^{\mathrm{p}}{}_{\mathrm{+}}$, that satisfy: (i) If $\text{1<p<p}{}^1{}_{\mathrm{+}}$ then there is no nontrivial solution of ($\text{1}$). (ii) If $\text{p=p}{}^1{}_{\mathrm{+}}$ then there is a unique fast decaying solution of ($\text{1}$). (iii) If $\text{p}{}^1\text{<p?p}{}^{\mathrm{p}}{}_{\mathrm{+}}$ then there is a unique pseudo-slow decaying solution to ($\text{1}$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($\text{1}$). Similar results are obtained for the operator $\text{M}{}_{\mathrm{<mtext>\lambda ,</mtext><mtext>\Lambda </mtext>}}{}^{\mathrm{?}}$. To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

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.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Some remarks on the Steiner problem

Let Σ be a set of n points in the plane. The minimal network for Σ is the tree of shortest total length L_M(Σ) whose vertices are exactly the points of S. The Steiner minimal network for Σ is the tree of shortest possible total length L_S(Σ) when the vertices are allowed to be any set Σ′ ? Σ. Clearly L_S(Σ) ? L_M(Σ), since the minimization in L_S is over a larger set. It has long been conjectured that, conversely, $\text{L}{}_{\mathrm{<mtext>S</mtext>}}\text{(Σ) ? (}\text{3}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{2}\text{) L}{}_{\mathrm{<mtext>M</mtext>}}\text{(Σ)}$, but this has previously been proved only if n = 3. In this paper, among other results, this is proved for n = 4. Unfortunately the proof is sufficiently complicated that immediate generalization to arbitrary n, no matter how desirable, is unlikely.     

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.     

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.     

On positional games

Let {A_i} be a family of sets and let S = ∩_iA_i. By a positional game we shall mean a game played by two players on {A_i}. The players alternately pick elements of S and that player wins who fist has all the elements of one of the A_i. This paper deals with almost disjoint hypergraphs only, i.e., |A_i∪A_j| ? 1 if i ≠ j. Let $\text{M}{}^1\text{(n)}$ be the smallest integer for which there is an almost disjoint n-uniform hypergraph $\text{|T| = M}{}^1\text{(n)}$, so that the first player has a winning strategy. It is shown that $\text{lim}{}_{\mathrm{n}}\text{[M}{}^1\text{(n)]}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{= 4}$, which was conjectured by Erdös. The same method is applied to prove a conjecture of Hales and Jewett on r-dimensional tick-tack-toe if r is large enough. Finally we prove that for an arbitrary almost disjoint n-uniform hypergraph the second player has such a strategy that the first player unable to win in his mth move if m < (2 ? ?)ⁿ.     

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 → ∞.