Assume $P$ is a family of primes, and let $()_P$ represent the $P$-localization functor. If $1\,{\to}\,N\stackrel{\iota}{\to} G\stackrel{\epsilon}{\to} Q\,{\to}\,1$ is an exact sequence of groups with $N$ finite, we prove that the sequence $N_P\stackrel{\iota_P}{\to} G_P\stackrel{\epsilon_P}{\to} Q_P\,{\to}\,1$ is exact. Moreover, we provide an explicit description of $\mbox{Ker}\ \iota_P$ when $Q$ belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all $P$-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-nilpotent groups.

In the year 1663 John Wallis, Savilian Professor of Geometry in the University of Oxford, delivered a lecture in which he claimed that he had proved Euclid's Postulate of Parallels according to the strictest laws of demonstration after Euclid's manner.

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution for

but not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed that

as p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimate

It is the purpose of this paper to show that this same estimate also holds for quartic polynomials.

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

For C*-algebras A1, A2 the map (I1, I2) → ker(qI1 ⊗ qI2) from Id′(A1) × Id′(A2) into Id′(A1 ⊗minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1 ⊗minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1 ⊗minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1 ⊗minA2 with certain other properties are presented.

A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$ . We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$ . In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$ . These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$ .

The rate of sputtering of magnesium in oxygen was observed with an apparatus which permitted the field immediately in front of the target to be reversed. Change in the sign of the field did not affect the rate of sputtering which remained abnormally low. It is therefore concluded that, while the low rate is in some way due to the presence of an oxide film, it is not due to charged particles being returned to the cathode by the electric field, as had been suggested by other workers.

A random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the ‘division polynomials’ associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.

Ever since the solution of the -Neumann problem by Kohn [4], refinements, extensions and estimates of the solution have been made by a lot of people up to the present day. Kohn discovered that, where the -Neumann problem was solvable, the space of harmonic forms is finite dimensional and it followed from Hörmander's solution [3] that on bounded pseudoconvex domains in ℂn the appropriate spaces of harmonic forms are zero dimensional.

The Feynman integral is evaluated for classical waves using a Hamiltonian approach to the summation over histories.

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.

The Statistical Mechanics which has been developed in accordance with the requirements of the new Quantum Theory is concerned with distribution laws over energy values only—over, that is, the characteristics of Schrödinger's equation. To obtain a space distribution law, even for the Classical limit, some use must be made of the characteristic functions. A formula has been suggested by Fowler, but it has not been shown that this formula gives the Classical law for gases at ordinary temperatures and pressures. In this paper we shall show that this is so, but before doing so we shall sketch the analogous method of obtaining the law, on the Classical theory.

We study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$, where $L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$.

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.