The solution to the boundary problem

where r is the distance of point (x, y) from the origin, and h is a given function the arc length s along the unit circle r = 1, is not necessarily unique, Boggio (1), Weinstein (2), Stoker (3), Martin (4). Indeed if h is a positive integer m is known that the only solutions regular analytic for r≦1 are

where r, θ denote polar coordinates and A, B are arbitrary constants.

The idea of a pregroup was introduced by Stallings and provides an axiomatic setting for a well-known argument, due to van der Waerden, used to prove normal form theorems. Details are provided in [7], Section 3.

Certain functions of infinite matrices are known to exist.† This gives rise to the following questions:

1. Whether the power series of matrices

has a zero in the field ‡ of infinite matrices, and

2. If f(A) exists for a certain infinite matrix A, is there an infinite matrix B such that

In other words, is there a matrix period for f(A)?

In this paper theorems concerning zeros and periodicity of functions of semi block infinite matrices § (defined below) are established.

When the series

is convergent, then Abel's Theorem shows that

Series of the form

are of frequent occurrence in the Analytical Theory of Heat and in other branches of Mathematical Physics, and the conditions to which f(t) is subject usually require that as t tends to zero the function f(t) should tend to the value s.

We calculate the box-dimension for a class of nowhere differentiable curves defined by Markov attractors of certain iterated function systems of affine maps.

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.

Through four generally placed lines in space of four dimensions there passes a doubly infinite system of quadric primals, but through five lines there pass in general no quadrics. It therefore follows that there must exist some special relationship between five lines in order that they may be generators of a quadric. This problem has been discussed by Richmond,1 who gives a condition which is in a restricted sense an extension of Pascal's Theorem. The five lines being taken in order certain points may be obtained which lie in a space. In Section I we state Richmond's criterion and show that it is sufficient as well as necessary. Section II is concerned with the twelve spaces which arise if all the different possible orders of the lines are considered. They cut by pairs in six planes whose configuration is developed. In Section III other lines connected with the configuration are introduced. It is shown that by taking crossers of the lines of our original figure in a certain manner five further generators are obtained, and that the same entire configuration of generators arises whether we begin with the five original or the five final lines. Furthermore, though the twelve spaces analogous to Pascal lines obtained from the final five are new, yet the six planes, their intersections by pairs, and the configuration dependent from them, are the same as those constructed from the original five.

Edmund Taylor Whittaker was born at Birkdale, Lancashire, on 24th October 1873 and belonged to a family which had been known for several generations in the district where the Rabble forms the boundary between Lancashire and Yorkshire. He entered Manchester Grammar School in 1885 and spent two years in the Classical side before turning over to the Mathematical side. In December 1891 he won a scholarship to Trinity College, Cambridge, and, after an undergraduate career of unusual distinction, was bracketed with J. H. Grace as Second Wrangler in the Tripos of 1895 ; Bromwich was Senior Wrangler. In 1896 Whittaker was elected a Fellow of Trinity College and in 1897 was first Smith's prizeman, his essay being entitled “On the reduction of the theory of multiform functions to uniform functions.”

If G is a group, let

m(G)

be the least cardinal such that G may be generated by m(G) elements, and let

m,(G)

To establish our notation N will always denote a (left) near-ring without any type of multiplicative identity (unless the contrary is stated) satisfying On = 0 for each n ∈ N where 0 is the additive identity of N. A group M, written additively, which admits N as a set of right multipliers is a (right) N-module if a ∈ M, n1, n2 ∈ N implies a(n1 + n2) = an1 + an2 and a(n1n2) = (an1)n2. When N has a two-sided identity, 1, we suppose that a 1 = a for each a ∈ M. A subgroup X of M is an N-subgroup of M if it is an N-module; X is a submodule of M if it is a normal subgroup of M and a ∈ M, x ∈ X. n ∈ N implies (a + x)n – an ∈ X. We denote by SL(M) the set of N-subgroups and by L(M) the set of submodules of M. Since N may be regarded as an N-module we can talk about N-subgroups and submodules of N although we usually call the submodules of N right ideals of N. Other definitions can be found in (6).

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

where cka, b, 0k, and .

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomials

and Pincherle's polynomials, arising from the expansions

are, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

A particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.

The alternant with powers of variables for its elements

has often been investigated and its properties are well known. In the present note an “alternant” with factorial elements,

will be discussed. Here a, b, c, …, z, α, β, γ, …, ω are integers such that

and

with the convention that a(0) = 1, O(0) = 1, and that, where negative indices occur (in the reductions of §2), the factorial is to be regarded as zero.