Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

In this paper, we will investigate discrete coactions on Hilbert C*-modules. In particular, we obtain a one-to-one correspondence between Hilbert C*-modules with discrete coactions and Hilbert C*-modules over the crossed products of the original C*-algebras which satisfies some nice properties (see 3·6 and 3·7). Then we will give some applications of this correspondence in the last three sections.

In Section 2, we give some results about discrete coactions on Hilbert C*-modules which mainly correspond to those about discrete coactions on C*-algebras (see [7]).

A theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

An S-system over a monoid S is a set A together with a function ø: A × S → A such that (a, 1) ø = a and (a, st) ø = ((a, s) ø, t) ø for all a ∈ A and for all s, t ∈ S. So an S-system is a set on which S acts unitarily on the right. For (a, s) ø we write simply as: the notions of S-homomorphism, S-subsystem etc. are defined in the obvious manner. Many papers have been written characterizing monoids by properties of their S-systems; here we are concerned with injectivity and a related concept, absolute purity. Further details of the terms we use are given in Section 2.

A generalized triangle group is a group with a presentation

GT(l, m, n; w) = 〈a, b | al = bm = wn = 1〉;

where l, m, n [ges ] 2, and w is a cyclically reduced word in the free product on {a, b}. In the setting of one-relator products of cyclic groups, Fine et al. [3] calculated their Euler characteristics for n [ges ] 3 under certain conditions on the defining word w.

It is pointed out that there is some confusion concerning the meaning of the sign of a mutual inductance. The question is examined from first principles and it is shown that since the sign of a mutual inductance depends upon conventions as to sign of current directions, these, as well as the sign of the mutual inductance, must be given in order to specify what is really required—the relation between the directions of winding of the primary and secondary coils. It is also shown that in many cases it is possible to replace mutual inductance by a quantity which is numerically equal to the mutual inductance but which is independent of current directions. The two methods applied to a Wheatstone network yield consistent results.

For t > 0, let

where xn+r = xr ≥ 0 and xr+1 + xr+2 > 0 for each r.

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such that

Random equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

In Eagleson and Weber [2] a central limit theorem for weakly exchangeable arrays is given as a consequence of a reverse martingale central limit theorem. As noted in their remarks, a direct application of this is a central limit theorem for the classical U-statistics. Here we give a corollary to the functional law of the iterated logarithm of Scott and Huggins [4] and use this to obtain laws of the iterated logarithm for weakly exchangeable arrays and hence for U-statistics under a finite (2 + δ)th moment condition.

Throughout this paper we let A = (aij) be a non-zero n × n matrix-we study real matrices only–with row sums R1,…Rn and eigenvalues λ1,…,λn, ordered λ1≥…≥λn if they are real. We denote E = (1,…,1)T and su A = ΣiΣjaij = ETAE. If A is symmetric, a simple property of the Rayleigh quotient is that

satisfies

and

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that

(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we write

where λ > 1, then the equation

defines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such that

The problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

This paper follows (4) in a quantitative study of derivations. The object is to investigate the properties of the spectrum of an inner derivation. In particular, we shall be interested in obtaining a general description of the spectrum of a derivation on a W*-algebra.

Let H(t, θ) be the hyperplane in Rn (n ≥ 2) which is perpendicular to the unit vector θ, and distant t from the origin; that is H(t, θ) = {x ε Rn: x.θ = t}. (Note that H(t, θ) and H(−t, − θ) are the same hyperplane.) If f(x) εℒ1(Rn), we will denote the integral of f with respect to (n − 1)-dimensional Lebesgue measure over H(t, θ) by F(t, θ), termed the projection or sectional integral of f over H(t,θ). By Fubini's theorem, F(t, θ) exists for almost all t for any θ. Throughout this paper we will assume that f(x) has support in X, a compact convex subset of Rn. In Section 2 we examine some of the topologies that may be defined on functions on X in terms of the F(t, θ), and in the remainder of the paper we examine the extremal problem suggested by Croft (4), that of maximizing the integral of f over the set X with the constraint that the F(t, θ) are uniformly bounded above. We examine in particular how the extremal values depend on the convex set X. In the final section the extremal problem is related to a generalization of Bang's plank theorem and the theory of capacities, and several conjectures are proposed.

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

In this paper we consider complex projective surfaces V, defined by an equation of the form fn–1 (x, y, z) w + fn (x, y, z) = 0, where fi is homogeneous of degree i, and relate the geometry of the intersections of the piane projective curves fn–1 = 0 and fn = 0 with the singularities of V. The results we obtain clarify and generalize some of those presented by Bruce and Wall (3).

Let P(x) denote a polynomial with degree n and integer coefficients. By the height h of P we mean the maximum of the absolute values of the coefficients.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

In 1973 Ronnie Lee introduced the notion of semicharacteristic classes, which are invariants of the bordism group ℜ*(Bπ) of closed manifolds equipped with a free action of a finite group π. In this paper we relativize his theory. Associated to a homomorphism G → π of finite groups, there is the relative bordism group ℜ*(BG → Bπ), which is the bordism group of compact manifolds M with a free π-action, so that the action on ∂M is induced from a free G-action, i.e. ∂M = π xGN for some manifold N with a free G-action. The invariants defined here are invariants of this relative group.

For stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.