Categories of rigid/plastic response are investigated along arbitrary paths of homogeneous deformation. Successive yield surfaces are presumed to be convex and self-similar relative to certain canonical measures of stress and finite strain. After a general analysis attention is focussed on path-sensitive response in configurations where the canonical rate of hardening vanishes. The analysis is subsequently extended to other types of eigenstate, especially those associated with actual or prospective loading devices. The results are illustrated in the context of formability tests on sheet metal.

In this paper, we are concerned with the boundedness of all the solutions of the two-dimensional differential system $$ \left.\begin{array}{@{}l@{}}x'\,{=}\,{-}|y|^{p-2} y+f(t)\\ y'\,{=}\,|x|^{q-2} x+g(t) \end{array}\right\} $$ where $1\,{<}\,p$, $1\,{<}\,q\,{<}\,2$ and $5pq\,{>}\,3p+4q$, $pq/(p+q)\,{<}\,1$, $f, g$ are $C^5$ 1-periodic functions.

It is concluded that the “overshoot phenomenon” in superconductivity described by F. B. Silsbee was due to his method of mounting the specimen.

Methods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.

Bagnera(1) shows that when a certain primitive finite group of order 576, G576, is represented as a group of collineations in three-dimensional space, it contains twenty-four harmonic inversions with respect to points and planes, and he shows how to calculate the coordinates of the points and planes of the inversions. The purpose of this note is to discuss the configuration of these points, which we shall call vertices, to show that the whole group is generated by the twenty-four harmonic inversions, termed projections after Prof. Baker (2) and Dr J. A. Todd (4), and to enumerate the conjugate sets of the group.

In this paper wave-maker theory including the effect of surface tension is determined for three-dimensional motion of water in a semi-infinite rectangular channel with outgoing surface wave modes allowed for at infinity; the motion is generated by a harmonically oscillating vertical plane wave-maker at the end of the channel and the cases of both infinite and finite constant depth are treated. The solution of the boundary-value problem for the velocity potential is more complicated in the presence of surface tension due mainly to the additional effect of the channel walls at which the normal free surface slopes are prescribed—as also is the slope at the wave-maker—to ensure uniqueness. The simpler three-dimensional solution for a semi-infinite region—obtained long ago by Sir Thomas Havelock in the absence of surface tension for the case of infinite depth—is also noted.

A Kolmogorov-type inequality is derived from a Jackson-type inequality and a Bernstein-type inequality. Several new results follow as corollaries.

Let F:${\mathbb R}$n→${\mathbb R}$n be a polynomial local diffeomorphism and let SF denote the set of not proper points of F. The Jelonek's real Jacobian Conjecture states that if codim(SF) ≥ 2, then F is bijective. In this work we prove a weak version of such Conjecture, but for more general maps than polynomial, namely: the semi-algebraic maps.

For any category A and group G there is a category AG of A-objects with G-action. An object of AG is an A-object A together with a group morphism, θA:G → Isom (A, A), of G into the group of invertible A-morphisms A → A. A morphism of AG is an A-morphism f: A → B which commutes with the G-action; that is, for each g∈G, the following diagram commutes.

Let a(n) denote the number of non-isomorphic Abelian groups of order n. It is well-known that

for a natural number k we define

and

A spray is a second-order differential equation field on the slit tangent bundleof a differentiable manifold, which is homogeneous of degree 1 in the fibrecoordinates in an appropriate sense; two sprays which are projectivelyequivalent have the same base-integral curves up to reparametrization. We showhow, when the base manifold is two-dimensional, to construct from any projectiveequivalence class of sprays a conformal class of metrics on a four-dimensionalmanifold, of signature (2, 2); the Weyl conformal curvature of these metrics issimply related to the projective curvature of the sprays, and the geodesics ofthe sprays determine null geodesics of the metrics. The metrics in question havepreviously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a differentmethod involving the exploitation of a close analogy between the Cartan geometryof second-order ordinary differential equations and of three-dimensionalCauchy–Riemann structures. From this perspective the derived metricsare seen to be analoguous to those defined by Fefferman in the CR theory, andare therefore said to be of Fefferman type. Our version of the constructionreveals that the Fefferman-type metrics are derivable from the scalar tripleproduct, both directly in the flat case (which we discuss in some detail) and bya simple extension in general. There is accordingly in our formulation a verysimple expression for a representative metric of the class in suitablecoordinates.

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.

In this paper we describe solutions of the equation: F(A(z)) = G(B(z)), where A,B are polynomials and F,G are continuous functions on the Riemann sphere.

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

Every isogeny class over an algebraically closed field contains a principally polarized abelian variety ([10, corollary 1 to theorem 4 in section 23]). Howe ([3]; see also [4]) gave examples of isogeny classes of abelian varieties over finite fields with no principal polarizations (but not with the degrees of all the polarizations divisible by a given non-zero integer, as in Theorem 1·1 below). In [17] we obtained, for all odd primes [lscr ], isogeny classes of abelian varieties in positive characteristic, all of whose polarizations have degree divisible by [lscr ]2. We gave results in the more general context of invertible sheaves; see also Theorems 6·1 and 5·2 below. Our results gave the first examples for which all the polarizations of the abelian varieties in an isogeny class have degree divisible by a given prime. Inspired by our results in [17], Howe [5] recently obtained, for all odd primes [lscr ], examples of isogeny classes of abelian varieties over fields of arbitrary characteristic different from [lscr ] (including number fields), all of whose polarizations have degree divisible by [lscr ]2.

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

We introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.

In a previous paper solutions have been derived for the deflexion of a circular annular plate under various combinations of boundary conditions and subject to certain line loadings spread along the circumference of a concentric circle. These solutions, which were used in the analysis of circular ring plates and sectorial plates acted upon by a concentrated load or a concentrated couple at any point, are utilized in this paper to obtain solutions appropriate to lateral loadings of the type distributed over the entire circular ring plate. Solutions corresponding to the case φ(r) = rm are then explicitly found, and the principle of superposition can be applied to get the solution when the load on the annulus can be expanded into a Laurent-Fourierseries in the polar coordinates r, θ. The method of images is used to deduce solutions for an isotropic plate bounded by two arcs of concentric circles and two radii, and loaded by the normal pressure p = p0rm when the plate is simply supported along the straight edges and is subject to various conditions along the circular edges.