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.

It is shown that Einstein's 1929 Unified Field Theory and that of Levi-Civita are incompatible.

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〉.

An elementary, purely algebraic derivation of the most general Lorentz transformations without spacial rotation and of the Thomas precession is given.

If Φ is a real number with |Φ| ≧ 1, then a semiregular continuet fraction development of Φ is denoted by where the ai are integers such that |ai| ≧ 2. The expansions arise geo-. metrically by considering the sequence of divided cells of two-dimensional grids (see [1]), and are described by the following algorithm: for all n ≧ 0, taking Φ = Φ.0 Hence where in this case the square brackets are used to signify the integer-part function. It follows that each irrational Φ has uncountably many such expansions, none of which has a constantly equal to 2 (or -2) for large n.

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.

In a recent paper [1] Adati and Miyazawa studied conformally recurrent spaces, that is, Riemannian spaces defined by where is the conformal curvature tensor: λi is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor gij.

Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.

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.

K. W. Chang generalizing a result of Lazer [3], proved in [4] the following THEOREM 1. Suppose that f:I → R+ = (0, + ∞), I = ([t0, + ∞), t0 0, is a non-decreasing function whose derivatives of orders ≧ 3 exist are continuous on ([t0, + ∞). Moreover, limt→+∞f(t) = +∞ and for some α, 1≧ α ≧ 2, and F =f-1/α then every solution x(t) of the equation tends to zero as t - + ∞.

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 this short note we shall apply the theory of semi-groups of operators, (cf: Hille and Phillips, [2]), to the problem of representing solutions of certain differential equations with non-constant coefficients. When the coefficients are constant, this representation reduces to the usual Laplace transform solution of the relevant equation.

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.

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

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.

In a paper by Polimeni [3] the concept of a c-group was introduced. A group is called a c-group if and only if every subnormal subgroup is characteristic. His paper claims to characterize finite soluble c-groups, which we will call fsc-groups. There are some errors in this paper; see the forthcoming review by K. W. Gruenberg in Mathematical Reviews.

For any point x, we use the symbol x to denote {(x, x)}. By a triform, we mean a triple (X, Fx, S[X]) where X is a nonempty set, Fx is a family of subsets of X, [X] is a family of subsets of X x X and the following conditions are satisfied: .

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.