Certain interesting distributions are characterized by two or more symmetry properties, and it is a challenge to link these properties in an effective way. Dynkin's advice Discuss the simplest case first is always the best, and here we follow it (and avoid technical jargon) by discussing in detail just one concrete example, which concerns a discounted version of the arc-sine law. Theorem 1 B gives the symmetry properties for this case, and Theorem 1 A shows how they may be exploited to obtain an asymptotic result which we find rather surprising. Section 5 points to one of the generalizations which we shall examine in a sequel, and describes the natural formulation of the strange symmetry property which we first meet at (16b). (The sequel will be in many ways very different from this paper, but will share with it the fact that its results are not those which we originally conjectured.) In Sections 6 and 7 we consider some of the bizarre numerical analysis associated with the current problem.

1. Introduction. The optimum design of plane reinforced concrete slabs frequently requires consideration of deflected shapes for the slab such that through every point there is at least one principal curvature (κ1) direction on the deflected surface which is a constant curvature direction. The most interesting cases are those where only one principal direction has this property.

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:

Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,

and its analogue in algebraic K-theory:

Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be finitary if it consists of elements of finite rank. We study the situation when L acts irreducibly on the infinite-dimensional vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I. Moreover I is simple and L/I is solvable.

The object of this note is to direct attention to an entirely false inference which was drawn from the analysis given in my previous paper with the same title.

A coupled cell network is an inflation of if the dynamics of is embedded in as a quotient network. We give necessary and sufficient conditions for the existence of a strongly connected inflation of a strongly connected network. We provide a simple algorithm for the construction of a strongly connected inflation as a sequence of simple inflations.

The theory of the intersection-complex Ch · Ck, due originally to Veblen, Weyl, Alexander and Lefschetz, has recently been extended by Flexner to cover intersections on a “topological n-manifold” (locally Cartesian n-space) in the case h + k = n. In his theory the whole of Lefschetz's work for ordinary simplicial manifolds is presupposed, including the rather difficult approximations. The object of the present paper is to provide a combinatory theory of the general intersection, (h + k ≥ n), which would serve equally well as a basis for Flexner's work, and avoids the wasteful process of using two independent approximations.

It is shown that the white light fringe obtained with Brewster's fringes or with ‘scatter’ fringes has a breadth which decreases with the reflexion coefficient R of the film surfaces. The decrease of intensity from the centre of the fringe is given by the usual Fabry-Perot formula except that R is replaced by R2.

Thirty years ago, in a paper on continued fractions, Stieltjes published a definition of the integral which bears his name. His replacement of the variable of integration x by a more general “base function” φ(x)—a change which throws so much light upon other theories of integration—received at first little attention, but has later sprung into greater prominence; so much so that Professor Hildebrandt, in summarizing these various theories in a paper to the American Mathematical Society, makes the statement that “it [the Stieltjes Integral] seems destined to play the central rôle in the integrational and summational processes of the future.” Yet even now the integral and the allied theory of differentiation with respect to a function have been subjected to little detailed analysis, and the possibilities of extension have been only touched upon. It is the object of this present paper to establish certain results which are of some value in themselves and which prepare the way for an attack upon the integral.

The method for the solution of problems of surface waves on water with which this note is concerned can be sufficiently illustrated by a particular example given by Lamb (1). In Lamb's example it is supposed that a steady stream in which the x, y components of the velocity are U, 0, where U is constant, is slightly disturbed by the application of a constant pressure ρPU2 to a band of finite width 2a of the surface of the stream; ρ denotes the density of the water and P a non-dimensional constant. In the solution Lamb, using an artifice due originally to Rayleigh and widely used since, assumes that ‘the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity’, and states that this ‘law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion’.

In this paper it is shown how all the regular polytopes (including the Kepler-Poinsot polyhedra and Hess's analogous star polytopes in four dimensions) arise as rational solutions of trigonometrical equations.

In a recent paper (1) it has been established that a spherically symmetric space-time of class one cannot describe an electromagnetic distribution of energy. In this investigation the inconsistency of class one conditions with Rainich algebraic conditions was shown in case of spherical symmetry. In the present paper it is shown that the result is true for space-time of class one, in general.

Two spatial embeddings of a graph are said to be delta edge-homotopic if they are transformed into each other by self delta moves and ambient isotopies. In this paper we classify theta curves up to delta edge-homotopy in terms of the third coefficient of the Conway polynomial of an associated 2-component link. In particular, we show that every boundary theta curve is delta edge-homotopically trivial, and two cobordant theta curves are delta edge-homotopic.

Let $f$ be a real function, continuous over ${\mathbb R}^d$, $d\geq1$, and let $\Gamma(f,D)\,{\hbox{\sun}}\,\{(x,f(x))|x\in D\}$ be its graph over a domain $D\subseteq {\mathbb R}^d$ and $\Gamma{f}$ its graph over ${\mathbb R}^d$. The question naturally arises about which (and how) smoothness properties of $f$ are related to the Hausdorff dimension of $\Gamma(f)$. It is a well-known result that if $f$ is locally $\alpha$-Hölder, which we denote by $f\in\Fspaceloc{C}{\alpha}$, for some $\alpha>0$, then $\mathrm{dim}_{\mathcal{H}}(\Gamma(f))\leq \max(d,d+1-\alpha)$. Of course, in this inequality, one can replace $\alpha$ by the Höolder exponent of $f$, i.e. by $\sup\{s:f\in\Fspaceloc{C}{s}\}$. The question of the sharpness of the obtained bound yields different outputs depending on $f$: one can show that the equality is satisfied over some classes of functions (see [5] and [1, chapter 8]) but it is not difficult to find examples for which the inequality is strict. One should also mention that in the simple case of the well-known Weierstrass function, neither the equality nor a smaller upper bound could be proved (see [11] for related results).

Let $F$ be a perfect field and $M(F)$ the non-associative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then ${\rm Aut}(M(F))$ is equal to $G_2(F) \rtimes {\rm Aut}(F)$. In particular, every automorphism of $M(F)$ is induced by a semilinear automorphism of $O(F)$. The proof combines results and methods from geometrical loop theory, groups of Lie type and composition algebras; the result being an identification of the automorphism group of a Moufang loop with a subgroup of the automorphism group of the associated group with triality.

Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation

$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$

The initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.

It is well known that a spherically symmetric space-time is, in general, of class two. A necessary and sufficient condition for a spherically symmetric space-time to be of class one has been obtained in terms of the Riemann curvature tensor. By means of a transformation property of s.s. space-time, three distinct cases are shown to exist. The incompatibility of class one spherically symmetric space-times with Rainich algebraic conditions is established in these three cases. It is concluded that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions.

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

Let S+ (resp. S−) denote the class of fundamental groups of closed orientable (resp. non-orientable) 2-manifolds of genus ≥ 2, and let surface = S+ ∪ S−. In the list of problems raised at the 1977 Durham Conference on Homological Group Theory occurs the following([7], p. 391, (G. 3)).