The aim of this note is to give a sharp lower bound for rational approximations to ζ(2) = π2/6 by using a specific Beukers' integral. Indeed, we will show that π2 has an irrationality measure less than 6.3489, which improves the earlier result 7.325 announced by D. V. Chudnovsky and G. V. Chudnovsky.

Let pn(x) be a real algebraic polynomial of degree n, and consider the Lp norms on I = [−1, 1]. A classical result of A. A. Markoff states that if ‖pn‖. ∞ ≤ 1, then ‖P′n‖∞ ≤ n2. A generalization of Markoff's problem, first suggested by P. Turán, is to find upper bounds for ‖pn(J)‖p if ∣pn(x)∣≤ ψ(x)x ∈ I. Here ψ(x) is a given function, a curved majorant. In this paper we study extremal properties of ‖p′n‖2 and ‖p″n‖2 if pn(x) has the parabolic majorant ∣p(x)∣≤ 1 − x2, x ∈ I. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ‖(1 − x2)

A module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.

As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

In the paper we study the asymptotic behaviour of the number of trees with n vertices and diameter k = k(n), where k/√n → ∞ as n → ∞ but k = o(n).

Theorems 1 and 2 are known results concerning Lp–Lq estimates for certain operators wherein the point (1/p, 1/q) lies on the line of duality 1/p + 1/q = 1. In Theorems 1′ and 2′ we show that with mild additional hypotheses it is possible to prove Lp-Lq estimates for indices (1/p, 1/q) off the line of duality. Applications to Bochner-Riesz means of negative order and uniform Sobolev inequalities are given.

We generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

Let G be a group acting on a ring R. We study the problem of determining necessary and sufficient conditions in order that the skew group ring RG be von Neumann regular. Complete characterizations are given in some particular situations, including the case where all idempotents of R are central. For a regular ring R admitting a G-invariant pseudo-rank function N, with G finite, we obtain a necessary condition for RG being regular in terms of the induced action of G on the N-completion of R.

We study polynomials over an integral domain R which, for infinitely many prime ideals P, induce a permutation of R/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field of R. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

Bo Ju Jiang introduced an invariant lying in the braid group which is the best lower bound of the number of fixed points in a homotopy class of a given pair of self maps of a surface. Here we modify this construction to get a lower bound of the number of coincidence points of a pair of maps between two closed surfaces.

This paper provides an asymptotic estimate for the expected number of K-level crossings of the random trigonometric polynomial g1 cos x + g2 cos 2x+ … + gn cos nx where gj (j = 1, 2, …, n) are dependent normally distributed random variables with mean zero and variance one. The two cases of ρjr, the correlation coeffiecient between the j-th and r-th coefficients, being either (i) constant, or (ii) ρ∣j−r∣ρ, j ≠ r, 0 < ρ < 1, are considered. It is shown that the previous result for ρjr = 0 still remains valid for both cases.

The notion of pointwise cleavability is introduced. We clarify those results concerning cleavability which can be or can not be generalized to the case of pointwise cleavability.

The importance of compactness in this theory is shown. Among other things we prove that t, ts, πx, the property to be Fréchet-Urysohn, radiality, biradiality, bisequentiality and so on are preserved by pointwise cleavability on the class of compact Hausdorff spaces.

This article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.

A recursive resampling method is discussed in this paper. Let X1, X2,…, Xn, be i.i.d. random variables with distribution function F and construct the empirical distribution function Fn. A new sample Xn+1 is drawn from Fn and the new empirical distribution function 1 in the wide sense, is computed from X1, X2,…, Xn, Xn+1. Then Xn+2 is drawn from 1 and 2 is obtained. In this way, Xn+m and m are found. It will be proved that m converges to a random variable almost surely as m goes to infinity and the limiting distribution is a compound beta distribution. In comparison with the usual non-recursive bootstrap, the main advantage of this procedure is a reduction in unconditional variance.

Certain omissions in the recently introduced dual for fractional minimax programming problem ‘minimize max y ∈ Y f(x, y) / h(x, y), subject to g(x) ≤ 0’, are indicated and two modified duals for this problem are presented. Various fractional programming and generalized fractional programming duals are shown to be special cases of this study.

A well-known result of Zygmund states that if f ∈ L (log+L) ½ on the circle group T and E is a Hadamard set of integers, then . In this paper we investigate similar results for the classes on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual Γ. These results are obtained as special cases of more general results concerning a new class of lacunary sets Sαβ, 0 < α ≤ β, where a subset E of Γ is an Sα β set if . We also prove partial results on the distinctness of the Sαβ sets in the index β.

A balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

In this paper conditions of M-symmetry, strong, semimodularity and θ-modularity for the congruence lattice L (S) of a regular ω-semigroup S are studied. They are proved to be equivalent to modularity. Moreover it is proved that the kernel relation is a congruence on L(S) if and only if L(S) is modular, generalizing an analogous result stated by Petrich for bisimple ω-semigroups.

We give an alternative short proof of a recent theorem of J. A. Hillman and P.A. Linnell that an elementary amenable group with finite Hirsch number has, modulo its locally finite radical, a soluble normal subgroup with index and derived length bounded only in terms of the Hirsch number of the group.

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings.