The following well-known theorem has been proved by Iwasawa(1) and can also be deduced without much difficulty from earlier work of Levi(2), Magnus and Witt.

The number of Seifert circuits in a diagram of a link is well known 9 to be an upper bound for the braid index of the link. The -breadth of the so-called P-polynomial 3 of the link is known 5, 2 to give a lower bound. In this paper we consider a large class of links diagrams, including all diagrams where the interior of every Seifert circuit is empty. We show that either these bounds coincide, or else the upper bound is not sharp, and we obtain a very simple criterion for distinguishing these cases.

The usual proof of the adiabatic invariance of the quantum integrals of a multiply periodic dynamical system rests on the assumption that the Hamiltonian equations remain valid for the adiabatically changing system, with the same Hamiltonian function H, or at any rate with a new Hamiltonian function differing from the old one only by terms that vanish with ȧ, (a) being the varying parameter.

This paper deals with the propagation and decay of nth-order displacement waves in linear hyperelastic rods having multiple wave speeds. The vector decay equation is integrated to find the explicit behaviour of all waves which can propagate in the rod. Examples are treated using a special form of the constitutive relations and results are given for rods of various geometries and wave multiplicities.

We work throughout in the p.l. category (see (8)) consisting of polyhedra and p.l. (piecewise linear) maps. We are concerned with the following problem which is of importance in the theory of p.l. transversality.

1. G. I. Taylor has lately* pointed out how his vorticity-transport theory of turbulent motion may be extended to three dimensions. Of the typical term expressing the effect of turbulence on the mean motion he remarks: “In general it is so complicated that it is of little practical use, but in certain special cases considerable simplifications may occur.” In the special cases which he himself discussed the mean velocity was in the direction of the axis of x, and its magnitude was a function of y only, (x, y, z) being rectangular Cartesian coordinates.

In 1982, Glyn Harman [2] proved that for almost all n, the interval [n, n+n(1/10)+ε] contains a prime number. By this we mean that the set of n[les ]N for which the interval does not contain a prime has measure o(N) as n→+∞. It follows from Huxley's work [6] that if θ>1/6 then there will almost always be asymptotically nθ(log n)−1 primes in the interval [n, n+nθ]. In 1983, Glyn Harman [3] pointed that for almost all n, the interval [n, n+n(1/12)+ε] contains a prime number, and meantime Heath-Brown gave the outline of this result in [5]. The exponent was reduced to 1/13 by Jia [10], 2/27 by Li [12] and 1/14 by Jia [11], and meantime N. Watt [16] got the same result. In this paper we shall prove the following result.

THEOREM. For almost all n, the interval

formula here

contains a prime number.

We study the existence of finite absolutely continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

For a commutative ring R, we define the notions of deformed Picard algebroids and deformed twisted differential operators on a smooth, separated, locally of finite type R-scheme and prove these are in a natural bijection. We then define the pullback of a sheaf of twisted differential operators that reduces to the classical definition when R = ℂ. Finally, for modules over twisted differential operators, we prove a theorem for the descent under a locally trivial torsor.

The aim of this note is to prove

Theorem 1. Let n ≥ 3, and let p1, p2,…, Pn be primes in ℕ: = {z ∈ ℤ:z > 0}, each congruent to 1 (mod 4), which satisfy both of the following conditions:

(i) every unit in ℚ(√(p1p2)) has norm + 1;

(ii) the graph γ = γ(p1, p2, …, pn) associated with p1, p2, …, pn is odd (in the sense of [1]).

Introduction. Many automatic computers employ magnetic drums or ultrasonic delay lines as their main store. A part of the time taken by a computation on such a machine is spent in waiting for words to become available from the store. A number of methods and devices have been used or suggested to reduce this waste of time. This paper attempts to evaluate these methods by determining on the one hand the amount of waiting time eliminated and on the other hand the design or programming complications introduced. No quantitative method for determining the amount of eliminated waiting time was known. Available estimates were based on the feeling of the programmer and varied accordingly. It was therefore necessary in the first place to develop a process by which the amount of waiting time eliminated by any given method could be determined. This process is set out while being applied to the methods discussed. The methods themselves are not new, except that closer analysis has shown that the method adopted with the ACE pilot model and known as ‘optimum programming’ is in fact not a single method but a combination of several methods with different features. Also, it appeared that, contrary to an opinion expressed elsewhere (2), there is a simple way of staggering locations in the store even with words of two lengths.

The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

Let X be a compact totally ordered space made into a semigroup by the multiplication xy=max{x, y}. Suppose that there is a continuous regular Borel measure μ on X with supp μ=X. Then the space L1(μ) of μ-integrable functions becomes a Banach algebra when provided with convolution as multiplication. The second dual L1(μ)** therefore has two Arens multiplications, each making it a Banach algebra. We shall always consider L1(μ)** to have the first of these: if F, G∈L1(μ)** and F=w*−limi ϕi, G=w*−limj ψj, where (ϕi), (ψj) are bounded nets in L1(μ), then

formula here

Given a Lie group, it is often useful to have a parametrization of the set of its lattices. In Euclidean space ℝn, for example, each lattice corresponds to a basis, and any lattice is equivalent to the standard integer lattice under an automorphism in GL(n, ℝ). In the nilpotent case, the lattices of the Heisenberg groups are classified, up to automorphisms, by certain sequences of positive integers with divisibility conditions (see [1]). In this paper we will study the set of lattices in a class of simply connected, solvable, but not nilpotent groups G. The construction of G depends on a diagonal n×n matrix Δ with distinct non-zero eigenvalues, of trace 0; we will write

formula here

An ideal I in a ring R is primal if whenever J1, …, Jn is a finite set of ideals of R with J1 … Jn = {0} then Ji ⊆ I for at least one i ∈ {1, …, n}. If the ring is commutative then is is easily shown that each primal ideal contains a prime ideal. In this paper it is shown that in a separable semi-simple Banach algebra each primal ideal contains a prime ideal, and that the space of minimal prime ideals is compact and extremally disconnected in the hull-kernel topology. An example is given of an inseparable C-algebra with a primal ideal which does not contain a prime ideal.

We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao [CFSZ20] on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

Let a finite group Τ be represented as an irreducible group of order N of linear substitutions on n variables,

The variables may be chosen so that the substitutions of the group leave invariant the Hermitian form

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

We give a hyperkähler analogue of the classification of compact homogeneous Kähler manifolds. Namely we show that, for a compact semisimple Lie group G, the complete G-invariant hyperkähler manifolds which are also locally Gc-homogeneous with respect to one of the complex structures are precisely the hyperkähler metrics of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits of Gc. As a consequence the only complete hyperkähler metrics which are of cohomogeneity one with respect to a triholomorphic action of compact semisimple Lie group are the flat metric on ℍn and the Calabi metric on T*[Copf]Pn.