We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

Fix an odd prime p. Let $\mathcal{D}_n$ denote a non-abelian extension of a number field K such that $K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $ and whose Galois group has the form $ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $ where g > 0 and $0 \lt n'\leq n$. Given a modular Galois representation $\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$ which is p-ordinary and also p-distinguished, we shall write $\mathcal{H}(\overline{\rho})$ for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant

\begin{equation} \lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of } \text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big) \end{equation}

$f\in\mathcal{H}(\overline{\rho})$

$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$

$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$

$\mathcal{H}(\overline{\rho})$

$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$

For example, if $\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$ has the structure of a p-adic Lie extension then our formulae include the cases where: either (i) $\mathcal{D}_{\infty}/K$ is a g-fold false Tate tower, or (ii) $\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$ has dimension ≤ 3 and is a pro-p-group.

We construct characteristic classes for the change of rings spectral sequences. These have their values in appropriate Ext groups and provide descriptions of the first non-zero differentials in the spectral sequences. We use these classes to do some calculations in the cohomology spectral sequence of an extension of a finite cyclic group by a finite cyclic group.

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:

Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.

In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.

The radioactivity of the naturally and artificially produced lutecium isotope has been investigated on a number of occasions (l, 2). A 4 hr. activity was observed when lutecium was activated by slow neutrons and later a 6-day period was observed by Hevesi and Levi (3). Heyden and Wefelmeier (4) discovered the natural radioactivity of lutecium, while Libby (5) made measurements indicating that this natural activity consisted of a negative electron spectrum of maximum energy 0·215 MeV. Mattauch and others (6) showed from mass spectrograph evidence that in addition to the main isotope of lutecium of mass 175 there was an isotope of mass 176 with an abundance of 2·5%. The second isotope had been reported previously by Gollnow (7), but not identified. Since Hf176 and Yb176 are known to be stable, whereas Hf175 and Yb175 do not exist in nature, the radioactivity of natural lutecium must be assigned to Lu176. Flammersfeld and Mattauch (8) measured the energy of the natural spectrum by the absorption method and found it to be 0·4 MeV. They observed also a γ-radiation (1–2 quanta per β-particle) of 0·26 MeV. In addition, they investigated the radiations resulting from slow neutron bombardment of lutecium. They estimate the shorter period, previously reported by Hevesi and Levi, to be 3·4 hr. and to consist of a β-spectrum of maximum energy 1·150 MeV. They estimate the longer period to be 6·6 days and to consist of β-spectrum of maximum energy 0·440 MeV. In view of the closeness of this latter energy to 0·4 MeV., Flammersfeld and Mattauch allocate the 6·6 days' activity to an isomeric state of Lu176. The short period is allocated by these authors to Lu177.

1. Let Γn be the representation group or spin group (4, 9) of Sn. Then the irreducible representations of Γn are of two distinct types. These are (a) ordinary representations, which are the irreducible representations of the symmetric group and (b) spin or projective representations. Corresponding to every partition (λ) = (λ1, λ2, …, λm) of n with λ1 > λ2 > … > λm > 0 there is an irreducible spin representation 〈λ〉 of Γn.

We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.

In this paper we study certain aspects of the problem of analytic continuation of the scattering operator and Eisenstein integral which we introduced in [7, 8, 10], for Kleinian groups Γ with exponent of convergence δ(Γ) ≥ 1. The corresponding problem for groups with δ(Γ) < 1 was examined and solved in [9].

Every smooth, closed, orientable manifold of dimension 3 contains a fibred knot, i.e. an imbedded circle K such that M – K is the total space of a fibre bundle over S1 whose fibre F is standard near K. This means the boundary of the closure of F is K so that K is null-homologous in M. A natural problem suggested by Rolfsen (see (2), problem 3·13) is to determine which elements of π1M3 may be represented by fibred knots. We deal with this by proving the

Theorem. The collection of all elements of π1 M3 which can be represented by fibred knots is exactly the commutator subgroup [π1 M3, π1 M3].

An open subgroup U of a topological group G is always closed, since U is the complement of the open set . An arbitrary closed subgroup C of G is almost never open, unless G belongs to a small family of exceptional groups. In fact, if G is a locally compact abelian group in which every non-trivial subgroup is open, then G is the additive group δp of p-adic integers or the additive group Ωp of p-adic rationale (cf. Robertson and Schreiber[5[, proposition 7). The fact that δp has interesting properties as a topological group has many roots. One is that its character group is the Prüfer group ℤp∞, which makes it unique inside the category of compact abelian groups. But even within the bigger class of not necessarily abelian compact groups the p-adic group δp is distinguished: it is the only one all of whose non-trivial subgroups are isomorphic (cf. Morris and Oates-Williams[2[), and it is also the only one all of whose non-trivial closed subgroups have finite index (cf. Morris, Oates-Williams and Thompson [3[).

As in [2], we define a geometry Γ = (B2, …, Br; *) to be an ordered sequence of r pairwise disjoint non-empty sets Bi together with a symmetric incidence relation * on their union B = B1 ∪ … ∪ Br such that if F is any maximal set of pairwise incident elements (i.e. a maximal flag), then |F ∩ Bi| = 1 for i = 1,…, r. The number r is called the rank of Γ. The geometry Γ is called connected if the r-partite graph (Γ, *) is connected.

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let

where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let

where

and

where

The heavy elastic structure of a ship possesses many natural periods of lateral vibration, and if any periodic disturbing force within the hull happens to synchronise approximately with any of these natural periods, considerable vibration may result. Experience has shown that the damping of the hull structure is small, and when the rate of revolution of the propeller coincides with a natural period of the ship, a vibration of large amplitude is set up. No doubt reciprocating machinery is competent to produce more violent vibration than turbine machinery, but the inevitable lack of balance of the propeller itself is sufficient to vibrate seriously a turbine-driven ship.

For a continuous function f(x) on the reals or on the circle T (continuous and 2π periodic) we say that f(x) belongs to the generalized Lipschitz class, denoted by f ∈ Lip* α, if

where and Δhf(x) = f(x + ½h)−f(x−½h). For a convolution approximation process given by

where

we shall investigate equivalence relations between the asymptotic behaviour of (d/dx)rAn(f, x) and f ∈ Lip* α.

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the region

contains a pair of generating points of Λ; and that, unless the lattice Λ is of the special form

where both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,

inscribed in the region (1). With simple geometrical arguments, he showed that

(i) Пλ always contains a point of Λ, other than 0,

(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says that

for all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

Let r, s be two fixed integers greater than 1. A positive integer will be called r-free if it is not divisible by the rth power of any prime.

In a series of papers ((l)–(5)) Evelyn and Linfoot considered the problem of determining an asymptotic formula for the number Qr, s(n) of representations of a large positive integer n as the sum of s r-free integers; for s ≥ 4 their results were subsequently sharpened by Barham and Estermann(6).

Let A and E be C*-algebras, let A ⊗ B denote the minimal C*-tensor product, and let ε A *. The right slice map R: A ⊗ B → B is the unique bounded linear mapping with the property that R (a ⊗ b) = (a)b (a ε A, b ε B)(10). A triple (A, B, D), where D is a C*-subalgebra of B, is said to have the slice map property if whenever x ε A ⊗ B and R(x) D for all ε A* then x ε A ⊗ D). It is known that (A, B, D) has the slice map property whenever A is nuclear (11,13), but it appears to be still unknown whether the nuclearity of B will suffice (unless some extra condition is placed on D (l)).

Using a technique developed by Lighthill, surface waves are studied when a concentrated pressure point oscillating with a constant frequency moves along OX in a rotating frame O X Y Z on the free surface of rotating liquid bounded below by a horizontal plane. The effect of rotation is to split the coincident gravity modes of the non-rotating case and further to produce a new system of wavelets (countably infinite) which do not have any counterpart in the non-rotating case.

Computer simulation of landscapes and skylines has recently attracted a great deal of interest: see [6, 7]. Specification of a ‘landscape’ requires a function f: D → ℝ on a subset D of ℝ2, selected so that the apparent irregularity and randomness of the surface {(t,f(t)): t ∈ D} corresponds to what might be observed in nature. It is natural to look to random fields (that is, stochastic processes in two variables), and in particular to Gaussian fields, for functions with such properties. Even when an appropriate random field has been selected, determination of a typical sample function is far from easy [7].