The propagation of plane waves in a viscoelastic body representing a parallel union of the Kelvin and Maxwell bodies placed in a magneto-thermal field is investigated. It is shown that the longitudinal component of the wave is in general coupled with a transverse component and the wave travels in two families. In particular if the primary magnetic field is either parallel or perpendicular to the direction of wave propagation, the three components of the wave travel unlinked, with either the longitudinal component or the transverse components unaffected by the presence of the electromagnetic field. If the electrical conductivity of the solid is infinite the effect of the primary magnetic field is to increase the values of the material constants. The effect of wave propagation on magnetic permeability is equivalent to an anisotropic rescaling of the primary magnetic field. Some of the results obtained in the earlier works are obtained as particular cases of the more general results derived here.

Let A = (ank) denote an infinite matrix. Then A is said to be of type M if the conditions

imply αn = 0(n = 0,1,2,…).

In a previous paper the author has examined the various types of non-singular surfaces of sectional genus four; in the present work the same method is applied to non-singular surfaces of sectional genus five. The examination of this case completes the classification of non-singular surfaces in higher space as far as those of the seventh order; for a septimic surface of sectional genus six, necessarily normal in S4, must lie on a quadric, and its characters may be determined from this fact.

In this paper, we study distance one surgeries between lens spaces L(p, 1) with p ≥ 5 prime and lens spaces L(n, 1) for $$n \in \mathbb{Z}$$ and band surgeries from T (2, p) to T (2, n). In particular, we prove that L(n, 1) is obtained by a distance one surgery from L(5, 1) only if n=±1, 4, ±5, 6 or ±9, and L(n, 1) is obtained by a distance one surgery from L(7, 1) if and only if n=±1, 3, 6, 7, 8 or 11.

It is shown that for k ≥ 5 in every k-chromatic graph there is a set of k distinct vertices V1, …, Vk with the property that for i, j = 1, …, k and i ≠ j the graph contains the union of a set of 4 paths connecting Vi and Vj no two of which have any edge or any vertex besides Vi and Vj in common and k − 1 paths connecting Vi and Vj no two of which have any edge in common.

A quadratic complex Q is the set of lines in 3-dimensional projective space 3 given by a single non-trivial quadratic equation in the Plücker coordinates. The lines of the complex which pass through a fixed point of 3 are, in general, the generators of a quadric cone; this cone degenerates for the points of a sub variety K of 3. Thus one can associate with Q a fibration with base 3 - K and fibre isomorphic to 1, and ask whether this fibration is associated with an algebraic vector bundle of rank 2. When the base field is and Q is non-singular, the answer is negative; this was proved some years ago by Narasimhan and Ramanan ((8), proposition 8·1), and has the consequence that there is no universal family of stable vector bundles of rank 2 and degree 0 over a curve of genus 2.

Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

For any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

When a floating body performs free simple harmonic vibrations in a vertical plane it will create a field of pulsating current flow in the liquid, and the total kinetic energy of the whole moving system will correspond to that of the floating body with an enhanced mass. The total apparent mass of the body is often called the “virtual mass” and the apparent increment the “added mass,” and these terms will be used subsequently. We are concerned in this paper with finding the “added mass” of rectangular and triangular prisms floating in water. The authors' concern with this problem is related to their investigation of the natural frequency of lateral vibration of a ship's hull, and they have contributed already two papers to these Proceedings which should be associated with the present one. In their ultimate problem the body is not moving as a whole in a vertical plane but is flexing about two or more nodal points, but in the present problem the floating body is supposed rigid and all parts of it are moving vertically with the same velocity. It is believed the present investigations are applicable to the lateral vibrations of ships' hulls, but whether or not this is so the investigation is complete in itself: for example it may be used to calculate the natural frequency of a punt or pontoon which has, say, been relieved suddenly of a dead load.

Let $F$ be a free group, freely generated by a non-empty set $X$ and let $R$ be a cyclically reduced word in $F$. Let $X_0$ be the subset of elements of $X$ which occur in $R$ or in $R^{-1}$. Let $G$ be the quotient of $F$ by the normal closure of $R$. By the classical Dehn–Magnus Freiheitssatz, if $Y$ is a subset of $X$ which does not contain $X_0$, then the subgroup of $G$ generated by the image of $Y$ in $G$ is freely generated by it.

Free products of groups can be viewed as generalisations of free groups, by replacing the infinite cyclic groups generated by individual elements of $X$, by arbitrary groups. It is known that in special cases the corresponding version of the Freiheitssatz holds true for one-relator quotients of free products of two groups (components), for example if the relator satisfies a small cancellation condition.

In this work we extend Magnus' Freiheitssatz to an arbitrary number of components and where the subset corresponding to $Y$ above contains arbitrary number of components, provided that the relator satisfies the small cancellation condition C(8) and some further conditions. As an application we imitate, under certain conditions, the result of Magnus stating that one-relator groups have the structure of HNN extensions.

The method of proof includes a combinatorial analysis of van Kampen diagrams, relying on the structure theorem for small cancellation diagrams.

In a long series of papers published some years ago both here and elsewhere, Dr L. J. Slater has derived a large number of valuable results concerned with the theory of generalized hypergeometric functions.

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GL-projection on a pre-symmetric space represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system. Two neutral GL-projections S and T on the pre-symmetric space A* are said to be L-orthogonal if for all elements x in SA* and y in TA*,By studying the algebraic properties of the dual space A of A*, which is a JBW*-triple, it is shown that, provided that the orthogonal neutral GL-projections S and T satisfy a certain geometrical condition, there exists a smallest neutral GL-projection S∨T majorizing both S and T, and that S, T and S∨T form a compatible family.

Let be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

For α∈ (0,1) and x∈ [0,π, we define S0(x,α) = 1 and where Also, we set, for α∈ (0,1), where it is agreed that infØ =∞. It is shown in this paper that and 8≤ N(α)<∞ if α0<α<α*, where α*:=0.33542. . . is the unique solution α∈ (0,1) of the equation while α0= 0.308443. . . is the unique solution α∈ (0,1) of the equation In the case α*<α<1, this extends a result of Brown, Wang and Wilson (1993).

We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson–Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson–Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.

Magyar has shown that if B ⊂ ℤd has positive upper density (d ⩾ 5), then the set of squared distances {||b1 − b2||2 : b1, b2 ∈ B} contains an infinitely long arithmetic progression, whose period depends only on the upper density of B. We extend this result by showing that B contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of B and the number of vertices of the sought tree). In particular, B contains all chains of elements with gaps in some given arithmetic progression (which depends on the length of the sought chain). This is a discrete analogue of a result obtained recently by Bennett, Iosevich and Taylor on chains with prescribed gaps in sets of large Haussdorf dimension. Our techniques are Ergodic theoretic and may be of independent interest to Ergodic theorists. In particular, we obtain Ergodic theoretic analogues of recent optimal spherical distribution results of Lyall and Magyar which, via Furstenberg's correspondence principle, recover their combinatorial results.

There are several ways of shortening a curve on a surface in order to establish the existence of closed geodesics. These methods are often useful in the study of much more general spaces, e.g. locally compact length spaces (see [1] chapter 10). Recently, J. Hass and P. Scott in [6] have introduced a new curve flow, the polygonal flow, which takes into consideration the complexity of the resulting geodesic. They construct the polygonal flow on surfaces so that the number of self-intersection points of a curve does not increase throughout the flow. In this paper we are concerned with ideal polyhedra with finitely many cells of dimension 2. These spaces consist of finite ideal hyperbolic triangles glued together by isometries along their sides.

The aim of this note is to obtain some formulae for generalized Rice polynomials as defined by (2)

Self Δ-equivalence is an equivalence relation for links, which is stronger than link-homotopy defined by J. W. Milnor. It was shown that any boundary link is link-homotopic to a trivial link by L. Cervantes and R. A. Fenn and by D. Dimovski independently. In this paper we will show that any boundary link is self Δ-equivalent to a trivial link.