The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a function

regular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

The Euler characteristic of an even dimensional submanifold in a space of constant curvature is given in terms of Weyl's curvature invariants. A derivation of Chern's kinematic formula in non-Euclidean space is completed. As an application of above results Weyl's tube formula about an odd-dimensional submanifold in a space of constant curvature is obtained.

The greater part of this paper consists of generalisations of well-known theorems regarding perpendiculars to the sides of a triangle, or other base-lines, the perpendiculars being replaced by isoclinals.

Much has been written in recent years on the foundations of geometry, chiefly in Germany and Italy, and the relations of the various Non-Euclidean geometries to the Euclidean system are now more generally known among mathematicians. But most of these writings involve a knowledge of more advanced mathematics, while it has been found difficult to represent even the simplest Non-Euclidean geometry—that of Bolyai-Lobatschewsky—in an elementary manner.

The generation of acoustic disturbances in a fluid of semi-infinite extent by the motion of a circular piston surrounded by a plane rigid baffle has been studied quite extensively (see (1), (2), (3), (4) and further references given in these papers). Attention has been devoted mainly to the case in which the piston executes a harmonic oscillation of small amplitude, and only comparatively recently has Oberhettinger (2) demonstrated how the time-harmonic solution can be used to solve the more general problem in which the normal velocity of the piston is an arbitrary function of time. The purpose of the present paper is two-fold. Firstly, we point out that for arbitrary normal motion of the piston the " baffled piston problem " can be solved directly, and in a particularly simple manner, by means of a technique involving integral transforms which has been applied by Mitra (5) and Eason (6) to the study of shear wave propagation in an elastic half-space. Secondly, we give a more detailed account than appears to be available in the literature of the structure of the sound pulse generated by the arbitrary normal motion of a baffled piston.

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

Menelaus's theorem is

If a transversal DEF cut the sides of triangle ABC at the points D B F, then

and conversely.

where (Am m) represents a square of m rows and m columns of a's; (Bm r) represents a rectangle of m rows and r columns of b's; (Cr m) represents a rectangle of r rows and m columns of c's;, and (Orr) represents a square of r rows and r columns of zeros.

Let be a linear differential expression involving n independent variables xi the coefficients Aik Bi, and C being functions of the independent variables but not involving the dependent variable u. Associated with F(u) is the adjoint expression

Consider the ordinary linear matrix differential system

ψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.

Some of the properties of the figure which, on account of its shape, the Greeks named the Shoemaker's Knife (ἄρβηλοσ) are given in the Lemmas attributed to Archimedes; others occur in the fourth book of Pappus's Mathematical Collection. The Lemmas (which are not extant in Greek, but have been translated from the Arabic) are generally considered to be spurious; it is, however, regarded as possible, if not probable, that the theorems among them relating to the Arbelos may be due to Archimedes. Whether they are or not, the figure and the principal proposition respecting it which Pappus gives are said by him to be “ancient.” It may be added that the Arbelos does not seem to have attracted much notice from geometers, few of them having treated of it, and fewer still having added to the properties known to the ancients. (See Steiner's Gesammelte Werke, Vol. I., pp. 47–76, and The Lady's and Gentleman's Diary for 1842 and 1845).

In the general equation of a curve that passes through the origin,

u1 ≡ a1x + b1y = 0 is the equation to the tangent at the origin; for, y q g when x and y are very small, we may neglect all the terms in comparison with those involving the first powers of x and y. If neither a1 nor b1 vanishes, we may, without loss of generality, write the equation in the form

Let υλ be a vector (i.e. a vector field) in an affinely connected space Ln, ∇k the symbol of covariant differentiation, and r the rank of the matrix ‖∇k υλ‖ then there exist two sets of n − r independent vectors and which satisfy respectively the equations

We will assume throughout this paper that polynomials are nonconstant. Let P be any complex polynomial and let p denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and multiplication is defined by fg = f ο P ο g for all f,g∈p. The near-ring p is referred to as a laminated near-ring and P is referred to as the laminating element or laminator. In [1] the problem was posed of determining Aut p the automorphism group of p. It was shown that exactly three infinite groups occur as automorphism groups of the laminated near-rings p and for each of the three groups those polynomials P were characterized such that Aut p is isomorphic to that particular group. The infinite groups turn out to be GL(2), the full linear group of all 2×2 nonsingular real matrices and two of its subgroups.

This note is concerned with “twisted“ analogues of the LP-structures (i.e.local-product structures) and grids of (4). To obtain these twisted structures, we modify the concepts of LP-structure and grid by removing the ordering of the local foliations involved in the definitions. The effect of this change is that global foliations need no longer exist, since the locally denned foils may now fit together to form self-intersecting immersed manifolds.