If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.

Let L be a Lie algebra over a field κ of any characteristic, and consider the lattice ℒ(L) of all subalgebras of L. In this paper we prove that if L and M are lattice isomorphic Lie algebras, over a field of any characteristic, and L′ and M′ are nilpotent, then the difference between the orders of solvability of L and M differs by at most one.

Exact tests, based on factorization of Wilks' Λ were derived by Williams (1952, 1955) and Bartlett (1951), for testing the goodness of fit of a single hypothetical discriminant function, in the case of several groups. An analytical derivation of the distributions associated with these tests was given by the author (1964 a), after expressing the test statistics in canonical forms. Wilks' Λ to cover the case of s (s > 1) hypothetical discriminant functions. Radcliffe, in his paper, states that an analytical derivation of the distributions, as well as of the independence of the factors of Λ is desirable. This is done in the present paper, by expressing the test criteria in simpler forms and using matrix transformations in the multivariate Beta distribution (Kshirsager 1961). This is a straightforward extension of the author's paper (1964 a), which dealt with only one hypothetical discriminant function.

The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

Consider the simultaneous real differential equations , where the dot indicates differentiation with respect to time, and a, b, c, d are functions about which the only thing that we know is that they are uniformly bounded, .

A topological semiring is a system (S, +, ⋅) where S is a Hausdorff space, (S, +) and (S, ⋅) are topological semigroups (i.e., + and ⋅ are continuous associative binary operations on S) and the distributive laws , hold for all x, y, z in S. The operations + and ⋅ are called addition and multiplication respectively.

A topological semiring is a system (S, +, ⋅) where (S, +) and (S, ⋅) are topological semigroups and the distributive laws , hold for all x, y, z in S; + and ⋅ are called addition and multiplication respectively.

In this paper we study open mappings of the sphere, Sn, onto itself. In particular, sufficient conditions are given that such a mapping be a homeomorphism. For the cases n≦ 2 many of the results could be obtained from the work of G. T. Whyburn [7], [8], and [10]. For the cases n ≦ 3 the useful results of A. V. Cernavskii, [1], [4], proved to be sufficient. An application is made concerning a finite to one open mapping of one n cell onto itself. It is interesting to note that for n ≦ 2 that we could use similar proofs to show that certain quasi-monotone mappings of Sn onto Sn are necessarily monotone mappings.

A set of polynomials p0(z), p1(z), … is said to form a basic set if every polynomial can be expressed in one and only one way as a finite linear combination of them.

It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems. It is shown in this paper that a complex symmetric matrix can be diagonalised by a (complex) orthogonal transformation, when and only when each eigenspace of the matrix has an orthonormal basis; this implies that no eigenvectors of zero Euclidean length need be included in the basis. If the matrix cannot be diagonalised, then it has at least one invariant subspace which consists entirely of vectors of zero Euclidean length.

Let f(z) be a meromorphic function and write Here N(r, a) and T(r, f) have their usual meanings (see [4], [5]) and 0 ≧ |a| ≧ ∞. If δ(a, f) > 0 then a is said to be an exceptional (or deficient) value in the sense of Nevanlinna (N.e.v.), and if Δ(a, f) > 0 then a is said to be an exceptional value in the sense of Varliron (V.e.v.). The Weierstrass p(z) function has no exceptional value N or V. Functions of zero order can have atmost one N.e.v. [4, p. 114], but may have more than one V.e.v. (see [6], [8]). In this note we consider functions satisfying some regularity conditions and having one and only one exceptional value V.

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

The representations of V4 (= C2 × C2) over characteristic 2 are put down in matrix form in sect. 2 of [1]. As such representations are of particular interest to finite group theorists, we present the following “geometric” descriptions of them which give immediate insight into their structure. Indeed, without such pictures it is difficult to see how they can be handled. Finally the relative Grothendieck algebra (relative to a copy of C2 in V4) falls out immediately from these diagrams. These diagrams have already helped towards the more general calculation of such algebras [2].

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

The Rademacher functions are defined by . The Walsh functions are then given by for , where the integers ni are uniquely determined by ni+1 < ni.

Throughout this paper g is a finite group and f is a complete local principal ideal domain of characteristic p where p divides |g|. The notations of [5] are adopted; moreover we shall denote the isomorphism-class of an f g-representation module ℳ by M, the class of ℳx by Mx and the class of ℳR by MR for suitable groups K and R.

The present paper extends some earlier work [1] on heat flow in a composite system, in which a central, high-temperature region loses heat to a surrounding medium. An example of the type of situation in mind is the intrusion of igneous rock into a mass of cooler sedimentary material. As an idealization, spherical symmetry is assumed and the outer region is taken to be infinite in extent. In the earlier work, the central region and the surrounding medium were each taken to be at a constant temperature initially. The present paper gives solutions for a number of alternative situations. The temperature in the outer region is still taken to be constant initially but the temperature in the central region is represented by functions of type rn or (1/r) sin kr, where r is the distance from the centre of the system. As the system of equations for the temperature is linear, and any continuous function can be approximated arbitrarily closely by a polynomial or a Fourier series, the solutions given here can be superposed to give a solution for any continuous initial temperature distribution in the central region.

In an electro-chemical investigation, an anode in the form of an aluminium cylinder was arranged so that one circular plane end was exposed to a dilute acid electrolyte. A measured electric current was then passed and aluminium from the anode passed into solution. Employing Faraday' laws, the mass of aluminium which was expected to pass into solution as a result of the passage of the known electric charge, was calculated. It was found that, when the current density over the exposed surface of the anode was high (e.g. 100 amp/sq.cm.), the experimental and calculated results were almost in agreement. However, for small current densities (e.g. 0.1 amp/sq.cm.), it was found that the actual loss of aluminium from the anode was about 14 % greater than that predicted by Faraday's laws.

Conditions sufficient to guarantee that a generalized free product of two residually finite groups A and B is again residually finite have been given by Baumslag [1]. We here show the same conditions guarantee that a certain permutational product of A and B is also residually finite.

Let {Xn} be a sequence fo independent and identically distributed random variables such that 0 <μ = εXn ≦ + ∞ and write Sn = X1+X2+ … +Xn. Letv ≧ 0 be an integer and let M(x) be a non-decreasing function of x ≧ 0 such that M(x)/x is non-increasing and M(0) > 0. Then if ε|X1νM(|X1|) < ∞ and μ < ∞ it follows that ε|Sn|νM(|Sn|) ~ (nμ)vM(nμ) as n → ∞. If μ = ∞ (ν = 0) then εM(|Sn|) = 0(n). A variety of results stem from this main theorem (Theorem 2), concerning a closure property of probability generating functions and a random walk result (Theorem 1) connected with queues.