We show that, for a finite group G and a prime p, the following facts are equivalent: (i) the p-localization homomorphism l: G —> Gp induces p-localization on integral homology; (ii) the higher homotopy groups of the Bousfield-Kan Zp-completion of a K(G, 1) vanish; (iii) the group G is p-nilpotent.

A sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.

Let α be a positive irrational real number, and let fα (n) = [(n + l)α] — [nα] — [α],n > 1, where [x] denotes the greatest integer not exceeding x. It is shown that the sequence fα has a certain 'substitution property' if and only if α is the root of a quadratic equation over the rationals.

We define an arithmetical function, f(n), which gives a lower bound for the rank of elliptic curves, y2 = x3 + nx, n square-free. Thus, if f(n) is unbounded for square-free values of n, then there are elliptic curves of arbitrarily large rank. We show that f(n) is unbounded as n ranges over all integers.

Let V be a vector space of dimension n over a field k of characteristic p. Let G ⊆ Gl(V) be a finite group with p-Sylow subgroup P. G and P act on the symmetric algebra R of V. Denote the respective rings of invariants by RG and Rp. We show that if Rp is Cohen-Macaulay (CM) so also is RG, generalizing a result of M. Hochster and J. A. Eagon. If P is normal in G and G is generated by P and pseudo-reflections, we show that if RG is CM so also is Rp. However, in general, RG may even be polynomial with Rp not CM. Finally, we give a procedure for determining a set of generators for RG given a set of generators for Rp.

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

Mahler's measure of a monic polynomial is equal to the product of modules of its roots which lie outside the unit circle. By classical theorem of Kronecker it is strictly greater than 1 for any polynomial that is not a product of cyclotomic factors. In this case a number of lower bounds of the measure, depending either on the degree of the polynomial or on the number of its non-zero coefficients, has been found. Here is given an improvement of the bound of the latter type previously found by the author, A. Schinzel and W. Lawton.

A distributional solution to the Hilbert problem in dimension > 1 is given.

We present examples showing that certain results on the invariance of link polynomials under generalized mutation are the best possible. They show, moreover, that this generalized mutation cannot be effected by a sequence of ordinary mutations. One of the examples also shows that the reduced Jones polynomial can be a more sensitive invariant than the Jones polynomial itself.

The authors previously observed that the space of completely bounded maps between two operator spaces can be realized as an operator space. In particular, with the appropriate matricial norms the dual of an operator space V is completely isometric to a linear space of operators. This approach to duality enables one to formulate new analogues of Banach space concepts and results. In particular, there is an operator space version ⊗μ of the Banach space projective tensor product , which satisfies the expected functorial properties. As is the case for Banach spaces, given an operator space V, the functor W |—> V ⊗μ W preserves inclusions if and only if is an injective operator space.

Let T be a non-singular, conservative, ergodic automorphism of a Lebesgue space. We study a kind of weighted cocycles called H-cocycles. We introduce the notions of H-superrecurrence and H-supertransience. We use skew products to give necessary and sufficient conditions for H-superrecurrence.

Computations of the Iwasawa λ -invariant for imaginary quadratic fields showed a discrepancy in the proportion of even and odd traces of certain integers from these imaginary quadratic fields. This paper shows that such a discrepancy is in some sense to be expected and that the proportion of even and odd traces of principal generators of powers of prime ideals in imaginary quadratic fields is related to the 3-primary component of the class group.

Let A(D) be the usual disc algebra, and let A+ be the subalgebra of A(D) consisting in those f € A(D) which have an absolutely convergent series of Taylor coefficients. Set m1 = {f € A(D) | f(1) = 0}, m = m1 ∩ A+, let G be the function and K = A+ ∩ I1.

We show that the quotient algebras m1/I1 and m/K are similar in the sense of J. Esterle.

J. Hayden [2] proved that, if a finite abelian group is a Cartesian group satisfying a certain "homogeneity condition", then it must be an elementary abelian group. His proof required the character theory of finite abelian groups. In this note we present a shorter, elementary proof of his result.

Let T be an ergodic automorphism with rational discrete spectrum and ϕ a ℤ2-cocyle for T. We show that the resulting two-point extension of T is cohomologous to a Morse cocycle if ϕ is approximated with speed o(1/n).

On the other hand, we show by example that this is in general false when the speed of approximation is O(1/n).

We consider the density in C[a, b] of generalized polynomials of the form The main point of this note is that total positivity of K(x, t) has little relationship to density: There is a symmetric, analytic, totally positive (in fact ETP (∞)) kernel K for which these generalized polynomials are not dense.

Suppose that f, g € L∞ [0,1 ] have discontinuities of the first kind only. Using the measure, max{ ∥f — h∥p, ∥g — h∥p}, of simultaneous Lp approximation, we show that the best simultaneous approximations f and g by nondecreasing functions converge uniformly as p → ∞. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. Assuming only that f and g are approximately continuous, we show that their simultaneous best monotone Lp approximation is continuous.

A Banach space X is said to have the Dunford-Pettis Property if every weakly compact linear operator T: X —> Y, where Y is any Banach space, is completely continuous (that is, T maps weakly convergent sequences to strongly convergent ones). In this paper, we prove that if A is a nest algebra of operators on a separable, infinite dimensional Hilbert space, then A fails to have the Dunford-Pettis Property. We also investigate a certain algebra associated to A, analogous to a construction used by Bourgain and others in connection with the Dunford-Pettis Property for function algebras. We show that this algebra must lie between A and the quasi-triangular algebra A + K and we give examples to show that either extreme or something in between is possible. Finally, we consider the algebra of analytic Toeplitz operators and give a result for the corresponding associated algebra which is analogous to a result of Cima, Jansen, and Yale for H∞.

We prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

For each nonnegative integer r,

is represented by a multiple series which is expressed in terms of rational numbers and the special values of the zeta function Thus, the set serves as a kind of basis for expressing all of the values