In a finite projective plane S2, q of order q, a (k, n)-arc is a set of k points, of which some n, but no n + 1 are collinear. Barlotti (1) proved that if (n, q) = 1 then

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).

Algorithms are developed for the numerical solution of systems of first-order ordinary differential equations, the solutions of which have widely different rates of variation. The iterative procedures described use a step length of integration proportional to the rate of change of the required slowly varying solution in a region of integration, where either the transient components of the complete solution have become negligible compared with the chosen working accuracy or in a region where rapidly increasing components of the solution are theoretically possible but are made absent by the initial conditions. Several numerical examples are given to demonstrate the algorithms.

The problem of completing the mathematical structures abstracted from the time ordering of the events in the space-times of general relativity is motivated and solved. The new construction is compared with that of Geroch, Kronheimer and Penrose (5).

The importance of the spectral theory of operators in Functional Analysis cannot be over emphasized. Spectral theory is in its best form when one considers normal operators in Hilbert spaces. However, for te dimensional spaces one has the reduction to Jordan's Canonical form. In an attempt to generalize this reduction for arbitrary Banach spaces, Dunford introduced the concept of spectral operators. Considerable work has been done in recent times in the study of spectral theory in Banach spaces, almost all of which stems from the pioneering work of Dunford.

The problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.

The possibility of constructing a quantum field theory by means of fields on Euclidean space is based on works by Schwinger and Symanzik (1). Probabilistic methods were used and Nelson has shown (2) that from so-called ‘Markoff fields’ one can construct Wightman fields. This idea turned out to be unusually fruitful as it made available the statistical mechanic's techniques for consideration of quantumfield theory problems. See for example (7). However as in the Minkowski space approach, the only two-dimensional space-time nontrivial models have been proved to fulfil all Wightman axioms. Since the problem for higher dimension theories is still open and extremely difficult, it is useful to have at least some criterion which allows us to eliminate certain procedures as leading to trivial theories. One such criterion existing in the Minkowski space approach is described by a notion of Borchers class (5).

An expression is obtained for the amplitude of the waves radiated to infinity by a line source of sinusoidally-varying strength situated in water over a beach of angle α (0 < α ≤ π). It is shown that, for certain positions of the source, this amplitude is zero. Equations for the loci of these positions, and approximations to their solutions in particular cases, are derived.

The notion of orthogonality is axiomatically defined on a poset. Various notions of orthocomplementation are distinguished and conditions are given in order to induce an orthocomplementation from an orthogonality and vice versa. Subsequently ⊥-modular ⊥-poset are defined and the set of morphisms between two posets with orthogonality is briefly discussed. Given the notions of additive monoid and of positive semi-ring, an orthogonality relation is introduced on the set of idempotent elements of a positive semi-ring. Finally, the obtained results are applied to the set of idempotent and absorbent endomorphisms of an additive monoid.

This paper derives some sufficient conditions for the global validity of projective relativity theory. The basic object of study is the infinitesimal bundle, which is a generalization of a principal circle bundle. A method ofconstructing infinitesimal bundles is developed. The main result is that ifspacetime is non-compact and simply connected, and has a finite homology basis, then projective relativity is globally valid.

A double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.

The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.

In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

The de Sitter group, Spin (4, 1), is a simply connected, semi-simple, ten-dimensional Lie group which can be contracted to the inhomogeneous Lorentz group. Physical systems with the de Sitter group as asymmetry group should resemble those of the inhomogeneous Lorentz group and may provide an alternative to these systems of special-relativistic physics. Details of physics in de Sitter space from the group theoretical view-point are given in (3). The de Sitter group is also known to be a symmetry group for the bound states of the hydrogen atom, and recent work has shown how this group acts on the corresponding classical system, the Kepler problem. See (8).

In this note, we state general conditions which imply that the numerical range of a function mapping a set S into a normed linear space Y is the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

Nina M. Roy has proved (3) that every separable G-space is square, thus exposing an error in my paper (1), in which I proposed a construction of non-square G-spaces which purports to include separable examples. The mistake lies in Lemma 5, whose proof confuses the weak-* topology with the bounded weak-* topology. This note is to salvage the construction to the extent possible, which is in the nonseparable case. The context and notations of (1) are used without further comment.

An isotropic elastic material containing a random distribution of identical spherical particles of another elastic material is considered. The bonding between the spheres and the matrix is imperfect, so that slip may occur at interfaces when stress is applied to the medium. The shear stresses at the interface is assumed to be proportional to the amount of slip. The velocity and attenuation of the average harmonic elastic waves propagating through such a medium are calculated. The results are valid to the lowest order in frequency for wave lengths long compared with the radius of the sphere. The dynamic elastic moduli are obtained from these results and are compared with available results for welded contact. The variations in the P and S wave velocities for propagation across earthquake faults is discussed.

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.

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.

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j where

for each x ∈ E. The partial summation operator Sn, defined by

is a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined by

and 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.