Soit {pk(n)}w(n)k=1 la suite croissante des facteurs premiers distincts d'un entier n. Nous donnons, lorsque k → ∞, une approximation uniforme de la loi de répartition limite de la fonction arithmétique n [map ] pk(n), précisant ainsi un résultat classique d'Erdős. Deux applications en sont déduites, relatives à la médiane de cette loi et à celle de la fonction “ nombre de facteurs premiers ”.

We give a simple proof that a measure on a locally compact abelian group G is quasi-translation invariant with continuous translation densities if and only if it is equivalent to Haar measure on G and has a continuous positive density.

In this paper comparisons are made between the class of continuous functions of generalized bounded variation and the class of continuous functions with graphs having σ-finite length i.e. linear measure. An investigationof the differentiability and approximate differentiability of such functions discloses the fact that the latter class is considerably more extensivethan the former one. The following definitions will be needed:

(1) A function f is said to be of bounded variation (VB) on a set E if

where the supremum is taken over all sequences {[ai, bi]} of non-overlapping intervals with endpoints in E.

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Starting from an arbitrary algebra in place of the algebra of smooth functions on a differentiable manifold M, we construct an algebra which generalizes the algebra of polynomial functions on T*M. We define a Poisson bracket on this algebra and show that it can be obtained from a natural one-form which generalizes the fundamental one-form on T*M.

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

1. Given a series

we will throughout write {sn} for the sequence of its partial sums; that is to say,

We will write p, q to denote the sequences {pn}, {qn}; we will write throughout

and will take P−1 = p−1 = 0.

A simple sufficient condition is given for a stochastic process x(t), 0 ≤ t ≤ 1, to have the following property: There is an integer m ≥ 2 such that for any non-degenerate subinterval J ⊂ [0, 1], there exist m disjoint subintervals I1, …, Im ⊂ J such that the intersection of the images of I1,…, Im under the mapping by x(·) has positive Lebesgue measure, almost surely. There is also a version for vector random fields; and the main result is shown to apply to large classes of processes.

The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.

By a well-known theorem of Kirchhoff the strain energy of an elastic solid is less in the equilibrium position than in any other position satisfying the same boundary conditions, and under the same body forces. The theorem contradicts the fact that elastic instability can occur, since two or more positions of equilibrium can then exist, and both cannot have the smaller strain energy. Numerous writers (Bryan (1), Southwell (2), Dean (3)) have explained the apparent discrepancy as due to the neglect of second-order terms in the elastic equations. This is correct so far as it goes, but it does not explain why the usual discussions of elastic instability give the right answers. The elastic constants vary somewhat with stress and in any case will be different according as a stressed or an unstressed state is taken as the standard. If any second-order terms should be included, we might expect that this variation would make a contribution comparable with those hitherto considered. Further, several theories of elasticity now exist that differ in the higher terms (Dean (3), Seth (8), Murnaghan (6)), and it may be asked whether they should lead to the same estimates of the critical loads.

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x by

We denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the type

We write

The object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

Let be the self-dual CCR algebra over a triplet (K, r, Γ), where K is assumed to be a Hilbert space, and let {τt}t ∈ R be a Bogoliubov automorphism group of induced by a strongly continuous one-parameter unitary group on K. In this paper, we introduce some continuity for linear functionals on and, under this continuity, we study existence, uniqueness, and non-uniqueness of KMS states for .

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.

Let $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . Suppose $(\textsf{A},\textsf{B})$ is a co-t-structure with coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended coheart $\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{S}* \Sigma \textsf{S}$ , which is an extriangulated category. We show that there is a bijection between co-t-structures $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ in $\textsf{T}$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ and complete cotorsion pairs in the extended coheart $\textsf{C}$ . In the case that $\textsf{T}$ is Hom-finite, $\textbf{k}$ -linear and Krull–Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\textsf{C}$ and functorially finite torsion classes in $\textsf{mod}\, \textsf{S}$ .