Maximum and minimum principles for certain ordinary differential equations of order 2m are derived in a unified manner from the theory of complementary variational principles for multiple operator equations. The minimum principle is known in the literature, but the maximum principle appears to be new.

In a previous paper ((l)) the authors gave the solution for the two-dimensional circular inclusion problem in a medium containing a circular cavity. This paper seeks to solve the more general problem of a similar inclusion when the cavity is replaced by an inhomogeneity which could be of a different elastic material. The solution consists in finding three sets of suitable complex potential functions ø(z) and ψ(z) for three regions: the inhomogeneity, the inclusion and the rest of the material. The solution depends upon the evaluation of the complex potentials for a material containing the inhomogeneity when on the former a finite force is acting at some fixed point. It may be noted that two sets of ø(z) and ψ(z) have to be found in this case: one for the inhomogeneity and the other for the rest of the material. This may be taken as an auxiliary problem.

Throughout we let ($T,{\frak m},k$) denote a commutative Noetherian local ring with maximal ideal ${\frak m}$ and residue field $k$. We let $I\,{\subseteq}\,T$ be an ideal generated by a regular sequence of length $c$ and set $R\,{\colonequal}\,T/I$. In the important paper [1], Avramov addresses the following question. Given a finitely generated $R$-module $M$, when does $M$ have finite projective dimension over a ring of the form $T/J$, where $J$ is generated by part (or all) of a set of minimal generators for $I$? He gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over $R$. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about $M$ (orits syzygies) coming from $T$, in particular, information gleaned from various Fitting ideals defined over $T$. The following theorem from Section 3 is one of our main results. We use $\Fitt_T(M)$ to denote the Fitting ideal of $M$.

Let $G$ be a universal Chevalley group defined over an algebraically closed field $F$ of arbitrary characteristic. In this paper we investigate the restrictions to parabolic subgroups of $G$ of the irreducible $FG$-modules corresponding to minuscule highest weights via a combinatorial scheme of “chamber systems” indexed over certain cosets in the Weyl group.

The object of this note is to prove the following generalization (from Riemann to Lebesgue integrable functions) of a theorem already proved by the authors (1). The notation and terminology of the earlier paper are used.

We are interested in the dynamics that can arise close to rotational group orbits after forced symmetry-breaking to discrete symmetries. In particular we ask how simple or complicated the dynamics induced by symmetric linear vector fields can be. We look in detail at related tetrahedral and dihedral examples, and there we find precise conditions for a linear field to exhibit homoclinic orbits.

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

A knot in a thickened surface $F^2 \times \mathbb{R}$ is called a global knot if its projection to $F^2$ is transversal to a vector field on this surface, which has at most critical points of index −1. Global knots generalize closed braids. We introduce new knot invariants of finite type which are trivial for all global knots. These invariants are a very effective tool for showing that a given knot in $F^2 \times \mathbb{R}$ is not isotopic to any global knot.

Let $G$ be a group, $g$ an element of $G$ and $n$ a natural number, $n\geqslant 2$. In [8] B.H. Neumann considered the equation $r_0(t)\deq t^ng=1$ and proved its solvablility in an overgroup. A natural generalisation of this equation is the equation $r(t)=1$, where $r(t)$ is defined as follows: let $W\in G\ast\langle t\rangle$ be a cyclically reduced word of length at least 2 which is not a proper power in $G\ast\langle t\rangle$. Define ${(\ast)}\hspace{4.8cm}r(t)\deq r_0(W)=W^ng$. In this paper we show that for $n\geqslant 3$ the equation $r(t)=1$ is solvable in an overgroup with some structural properties, under certain conditions on $g$ and $W$. It is conjectured in [3, p. 131] that this equation is solvable over an arbitrary group, without conditions on $W$ or $g$.

In this paper we continue the study of elliptic curves defined over a quadratic field with good reduction at primes not dividing 2 begun in [9] (referred to as I). We extend the results of I to show that such a curve must have a point of order 2 also defined over when d = −7, −3, −2, −1, 2, 3 or 5 and list all such curves over .

Let A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that

$$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$

$$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equation

Here the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

Let G be a connected split reductive group defined over a finite field [ ]q, and G([ ]q) the group of [ ]q-rational points of G. For each maximal torus T of G defined over [ ]q and a complex linear character θ of T([ ]q), let RGT(θ) be the generalized representation of G([ ]q) defined in [DL]. It can be seen that the conjugacy classes in the Weyl group W of G are in one-to-one correspondence with the conjugacy classes of maximal tori defined over [ ]q in G ([C1, 3·3·3]). Let c be the Coxeter conjugacy class of W, and let Tc be the corresponding maximal torus. Then by [DL] we know that πθ = (−1)nRGTc(θ) (where n is the semisimple rank of G and θ is a character in ‘general position’) is an irreducible cuspidal representation of G([ ]q). The results of this paper generalize the pattern about the dimensions of cuspidal representations of GL(n, [ ]q) as an alternating sum of the dimensions of certain irreducible representations of GL(n, [ ]q) appearing in the space of functions on the flag variety of GL(n, [ ]q) as shown in the table below.

An investigation of the random or stochastic integral equations of the form

and

is presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.

A recent result of the first two authors has bridged the gap between the Timantype pointwise estimate and the norm estimate for polynomial approximation in C[– 1, 1]. Here we show that, when considering the second order modulus of smoothness, the aforementioned result holds with polynomials that preserve monotonicity, convexity and the values of the function at the end points of the interval [–1, 1]. This yields a generalization of results by Telyakovskii, Gopengauz, DeVore, Yu and the third author. Moreover, the construction here does not depend on the construction for the non-constrained approximation by polynomials.

Conditions are examined under which plane shook waves of constant strength can propagate through a gas in a given homentropic motion. Since the entropy change across the shock is taken to be constant homentropic flow exists behind the shock also. The motion behind the shock is determined by solving a Cauchy problem with data given on the back of the shock. The theory is illustrated by two examples, one of which generalizes a result obtained by Copson.

Notation and results. The main results of this paper are (a) a new proof of the canonical Schoenflies theorem and (b) a non-compact version of the canonical Schoen-flies theorem.

The result (a) is already known(5); the existing proof is based on the proof by Brown of the ordinary Schoenflies theorem, whereas ours is based on that of Mazur and Morse.

The theory of a previous paper is used to discuss Eddington's work in fundamental physics.

1. Introduction. A previous paper (2), referred to here as I, described a theory depending on an unorthodox view of the relation of mathematics to physics. This theory was begun from the study of the work of Eddington which we take to be summarized in Fundamental Theory (5). In the present paper is shown the extent to which, and the way in which, the two theories are connected.

A method is presented whereby all locally defined conserved currents of the Klein-Gordon field are found. The mathematical background to the method includes a generalization of the Poincaré lemma of the calculus of exterior differential forms. It is found that the only conserved currents are essentially a countably infinite set of functions, bilinear in the field, together with a single current in the case where the mass is zero. The usual energy-momentum tensor is included amongst these functions. The method does not depend on the use of any canonical formulation of the field theory.