Let $V$ be a complete nonsingular projective surface defined over an algebraic number field $k$, such that the Picard variety of $V$ is trivial and Pic($\bar{V}$) is torsion-free. Since our main interest is in necessary conditions for $V(k)$ not to be empty, we shall further assume that $V(k_v)$ is non-empty for every completion $k_v$ of $k$. We do not assume that $\bar{V}$ is rational, and indeed the case which primarily interests us is when $V$ is a K3 surface. Our objective is to describe effective ways of computing the Brauer–Manin obstructions to the existence of points on $V$ defined over $k$. Write \[ \hbox{Br}^0 (V) = \{\hbox{Ker}(\hbox{Br}(V)\longrightarrow\hbox{Br}(\bar{V}))\}/\{{\rm Im}({\rm Br}(k)\longrightarrow\hbox{Br}(V))\};\] then it is known that $\hbox{Br}^0(V)$ is isomorphic to ${\rm H}^1(k, \hbox{Pic}(\bar{V}))$, though to the best of our knowledge there is in general no known algorithm for computing either $\hbox{Br}^0(V)$ or this isomorphism. It is generally believed that $\hbox{Br}^0(V)$ contains all the information about the Brauer group Br$(V)$ which is useful in this context. Most of this paper is concerned with computing groups isomorphic to $\hbox{Br}^0(V)$, and with describing in terms of these groups the Brauer–Manin obstructions coming from elements of $\hbox{Br}^0(V)$.

Two of the strongest emission lines in the spectra of the planetary nebulae were identified by Boyce, Menzel and Payne (1) as due to forbidden transitions in the ground configuration of doubly ionized neon. The transitions are of wave-length 3967 and of wave-length 3869 in the configuration Ne III 1s22s22p4. These lines are excited by inelastic collisions of electrons with the neon ions. In order to investigate these transitions in detail a knowledge of the collision cross-sections is required. As a first step in this direction, wave functions have been calculated for the 3P state by the method of the self-consistent field, including the effect of exchange. In later papers these wave functions will be used in the calculation of collision cross-sections, in a study of the departures from the Landé rule in the triplets of the p2 and p4 configurations and, it is hoped, in the calculation of line strengths for certain permitted lines of astrophysical importance in the spectrum of Ne II.

This paper is concerned with a particular class of acceleration waves in heat-conducting elastic materials, namely those waves for which the acceleration amplitude is a proper vector of both the isothermal and isentropic acoustical tensors. Acceleration waves with this characteristic property are found to fall naturally into two distinct sub-classes, and close parallels are shown to exist between the members of the sub-classes and waves for which the acceleration amplitude is either longitudinal or transverse to the direction of propagation. Particular attention is given here to waves of the quasi-longitudinal type, the quasi-transverse variety having been studied in some detail elsewhere.

Signature invariants of odd dimensional links from irregular covers and non-abelian covers of complements are obtained by using the technique of Casson and Gordon. We show that the invariants vanish for slice links and can be considered as invariants under Fm-link concordance. We illustrate examples of links that are not slice but behave as slice links for any invariants from abelian covers.

For a fixed finite group G, the numbers Ng of equivalence classes of orientation-preserving actions of G on closed orientable surfaces Σg of genus g can be encoded by a generating function [sum ]Ngzg. When equivalence is determined by the isomorphism class of the quotient orbifold Σg/G, we show that the generating function is rational. When equivalence is topological conjugacy, we examine the cases where G is abelian and show that the generating function is again rational in the cases where G is cyclic.

1. It is well known that, if two n × n matrices A, B commute, then there is a non-singular matrix P such that P−1AP, P−2BP are both triangular (i.e. have all their subdiagonal elements zero). This result has been generalized, and, in particular, has been shown to hold even if the commutator K = AB − BA is not zero, provided that K is properly nilpotent in the polynomial ring generated by A, B.

We give a framework for combining n monads on the same category via distributive laws satisfying Yang–Baxter equations, extending the classical result of Beck which combines two monads via one distributive law. We show that this corresponds to iterating n-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict n-category monad on n-dimensional globular sets; we first construct for each i a monad for composition along bounding i-cells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary Yang–Baxter equations.

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:

where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Then

provided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

The expression

for the number of linearly independent line complexes of degree n in space of r dimensions was derived by Sisam as a particular case in the determination of the number of linearly independent hyperconnexes of given degrees. The purpose of this note is to generalize Sisam's result by determining the number of linearly independent algebraic forms of a special type to which we give the name k-connex.

Singularity theory is concerned with the study of smooth mappings between smooth manifolds. Given two such manifolds X and Y and a pair of smooth mappings f1,f2: X→Y we say that f1 and f2 are -equivalent if there are diffeomorphisms α: X→X and β: Y→Y with βof1oα = f2. Clearly -equivalence is an equivalence relation, and one aims to classify smooth mappings f: X→Y up to this equivalence.

A complete lattice, L, is constructively completely distributive, (CCD) (L), if the sup map defined on down-closed subobjects has a left adjoint. It was known that in Boolean toposes (CCD) (L) is equivalent to (CCD) (Lop). We show here that the latter property for all L (sufficiently, for Ω.) characterizes Boolean toposes.

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.

Investigations of reactions of the type

are of interest for their own sake, and also for the fact that, if is a radioactive nucleus, as it usually is, being a stable isobar, activation of a few different A nuclei in the same fast neutron flux will yield some information about the neutron spectrum. This follows from the fact that these (np) reactions (and the similar (nα) reactions) are nearly all endothermic, showing a threshold of neutron energy below which no activation takes place. Thus knowledge of the excitation functions of a few such reactions would enable the above-mentioned crude neutron spectroscopy to be realized. The detectors (A nuclei) would have a standard shape and size, and the induced β-activity would be measured with a standard counter in fixed geometrical conditions. Though exact spectroscopy would not be possible, any change in the relative activations of the detectors would be a quite sensitive indicator of a change in the incident neutron spectrum.

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 < EX ≡ μ < ∞ and finite positive variance 0 < var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .

We work in the DIFF category. A knot K = (Sn+2, Sn) is a ribbon knot if Sn bounds an immersed disc Dn+1 →Sn+2 with no triple points and such that the components of the singular set are n-discs whose boundary (n – l)-spheres either lie on Sn or are disjoint from Sn. Pushing Dn+1 into Dn+3 produces a ribbon disc pair D = (Dn+3, Dn+1), with the ribbon knot (Sn+2, Sn) on its boundary. The double of a ribbon (n+1)-disc pair is an (n + l)-ribbon knot. Every (n+l)-ribbon knot is obtained in this manner.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. It is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. (These special situations are eliminated in a companion paper by the first author.) An example coming from the exceptional group E6(q) shows that the kernel of the action of Gx on the ball of radius 2 around x can be non-trivial.

We consider some elastostatic and related potential problems in which an unbounded uniform medium contains a plane elliptic disc as an inhomogeneity. Various appropriate conditions on the disc and at infinity are specified. These problems have many physical applications. In the elastostatic case the disc can be a crack or rigid inclusion, for example. The potential problems include the classical one of finding the electrostatic field around an elliptic disc charged to a given potential.

1. Let G be a group, ZG its integral group ring and Δ = ΔG the augmentation ideal ZG By an augmentation quotient of G we mean any one of the ZG-modules

where n, r ≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotients Qn(G) = Qn,1(G) and

(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determine Qn(G) and Pn(G) for finite G it is sufficient to assume that G is a p-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelian p-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: if G is a finite group then there exist natural numbers n0 and π such that Qn(G) ≅ Qn+π (G) for all n ≥ n0; if Gω is the nilpotent residual of G and G/Gω has class c then π divides l.c.m. {1, 2, …, c}. There do not appear to be any examples in the literature of this periodic behaviour for c > 1. One of goals here is to present such examples. These examples will be from the class of finite p-groups in which the lower central series is an Np-series.

Recently, the first author and Y. Wang proved that (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.

We prove a decomposition theorem similar to the well-known result of Voiculescu's: namely that completely positive maps A → (B)/B factor through a given homomorphism ι: A →(B)/B when the homomorphism ι has a certain infiniteness property.

The algebra B is only assumed to be separable and nonunital; in particular, it is not assumed to be stable. The C*-algebra A is assumed to be separable, unital and nuclear.