T. Schneider [1] has shown that a transcendental function with a limited rate of growth cannot assume algebraic values at too many algebraic points. It is not clear however whether a transcendental function may assume algebraic values at all algebraic points in an open set in its domain of analyticity. We show, using a method of B. H. Neumann and R. Rado [2] that this can be the case. Indeed we construct transcendental entire functions which, together with all their derivatives, assume, at every point in every algebraic number field, values in that field.

If (X, ) is a set X with topology we shall say that is connected if (X, ) is a connected topological space. We shall investigate the existence of and the properties of maximal connected topologies.

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.

There are infinitely, but at most continuously, many varieties of groups; the precise cardinal is unknown. It is easy to see that if there is no infinite properly descending chain of varieties (equivalently, if the laws of every variety have a finite basis), then the number of varieties is countable infinity; the converse implication does not seem to have been proved. This note presents an argument which implies that if the locally finite or the locally nilpotent varieties fail to satisfy the minimum condition, then there are continuously many such varieties. Alternatively, one can conclude that if a locally finite or locally nilpotent variety has a finite basis for its laws but some subvariety of has none, then there are continuously many varieties between and . This points again to the interesting question: is every locally finite [locally nilpotent] variety contained in a suitable locally finite [locally nilpotent] variety which has a finite basis for its laws? (That is, must be locally finite [locally nilpotent] for some finite n?) For, if the answer were affirmative, it would follow that the number of locally finite [locally nilpotent] varieties is either countable or the cardinal of the continuum, depending exactly on the existence of finite bases for the laws of such varieties.

No general rule for determining the number N(n) of topologies definable for a finite set of cardinal n is known. In this note we relate N(n) to a function Ft(r1,…, rt+1) defined below which has a simple combinatorial interpretation. This relationship seems useful for the study of N (n). In particular this can be used to calculate N(n) for small values. For n 3, 4, 5, 6 we find N(3) = 29, N(4) = 355, N(5) = 7,181, N(6) = 145,807.

We consider a queueing system with k identical servers in parallel, the services being negative exponential with parameter μ. The input is a natural generalisation of the usual general recurrent input. If we denote the sequence of arrival points by {An, n ≧ 0} then the inter-arrival intervals are given by where the ƒi: are (integrable) non-negative functions and {Ui} is a sequence of identically and independently distributed random variables. In the simplest case, p = 0, this is just a general recurrent input. We write U(·) for the probability distribution function of the Un.

The problem of finding the most economical coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice, which is equivalent to a problem concerning the inhomogeneous minima of positive definite quadratic forms, has been discussed recently by Barnes and Dickson [1]. The reader is referred to [1] for a complete background on the problem. Terms and notations used will be as in that paper.

Let be a complex Banach algebra, possibly non-commutative, with identity e. By a Reynolds operator we mean here a bounded linear operator T: → satisfying the Reynolds identity for all x, y ∈ . We prove that under certain conditions the resolvent of T, R(p, T) = (pI−T)−1, has the form where s = −log(e−Te) and exp y = e+y+y2/2!+….

A number of studies [1] have concerned themselves with properties of artificial poly-peptide chains, which differ from naturally occurring poly-peptides in two ways: There is only one kind of amino-acid in each polymer chain, but that acid occurs both in its right-handed and in its left-handed (D and L) forms, with some random order of L and D constituents.

This paper is devoted to a complete investigation into a problem initiated by Davenport [4], and further studied by Kanagasabapathy [6], [7], from whom I borrow the title. The question is a hybrid of the two classical results of Hurwitz and Minkowski on indefinite binary quadratic forms.

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition

Let M1 and M2 be two sets of probability measures defined on Rn. Ameasurable R1 valued function h (l ≧1) is said to distinguish M1from M2unbiasedly if there are numbers or vectors I1 and I2 (I1≠I2) such that ∫Rnh(x)m(dx) = Ii if m is in Mi (i = 1, 2). Here we shall be concerned with the case where M1 and M2 are translation families, in that all of the elements of Mi are translates of a single measure mi. This means that if, for any t in Rn, mtt is the measure defined by , where E−t {x−t: x ∈ E}, then Mi = , where T is a subset of Rn. If M1 and M2 are of this type, we will investigate the conditions under which there does not exist a function to distinguishing M1 from M2 unbiasedly. A case of special interest arises if m2(E) = m1(BE) = m1 with B a non-degenerate n × n matrix, and particularly a nonzero multiple (scale parameter) of the identity matrix, cf. [1], [2]. For simplicity, take l = 1.

Let Q(x1, …, xn) be an indefinite quadratic form in n-variables with real coefficients, determinant D ≠ 0 and signature (r, s), r+s = n. Then it is known (e.g. see Blaney [2]) that there exist constants Γr, s depending only on r and s such for any real numbers c1, …, cn we can find integers x1, …, xn satisfying

In this paper we find an expression for ex as the limit of quotients associated with a sequence of matrices, and thence, by using the matrix approach to continued fractions ([5] 12–13, [2] and [4]), we derive the regular continued fraction expansions of e2/k and tan 1/k (where k is a positive integer).

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

An integral on a locally compact Hausdorff semigroup S is a nontrivial, positive linear function μ on the space K(S) of real-valued continuous functions on S with compact support. If S has the property: is compact whenever A is compact subset of S and s ∈ S, then the function fa defined by fa(x) = f(xa) is in K(S) whenever f ∈ K(S) and a ∈ S An integral on a locally compact semigroup S with the property (P) is said to be right invariant if μ(fa) = μ(f) for all f ∈ K(S) and a ∈ S.

If the metric of an n-dimensional space is taken in the form ds2 = u2dτ2+dσ2, where dτ2 and dσ2 are cartesian metrics of r and (n−r) dimensions, respectively, the various forms of u for flat space are quite simple. The study of accelerated motions in special relativity by various authors has led to four dimensional metrics of this form. Those in which u = ±1 at the space origin for all values of time are of particular interest. They are locally cartesian at the accelerated observer, and so the coordinates in the neighbourhood of the observer correspond directly to physical measurements. Hence, such metrics provide convenient means of describing physical conditions experienced by accelerated observers. If the τ-space contains the time direction and is of one or two dimensions, arbitrary rectilinear motions are allowed.

Recently Orrin Frink (see [2]) gave a neat internal characterization of Tychonoff or completely regular T spaces. This characterization was given in terms of the notion of a normal base for the closed sets of a space X. A normal base for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of . In a normal space the ring of closed sets is a normal base.

An algorithm for summing the series , where the coefficients an are assumed known, and the quantities pn, satisfy a linear three term recurrence relation, has been given by Clenshaw [1]. If we suppose that the pn satisfy the recurrence relation where αn and βn are, in general, functions of n, then PN may be found by constructing a sequence {bn} for n = N(−1)0, where the b satisfy the inhomogeneous recurrence relation with the conditions, The sum PN is then given by This result can be readily verified by multiplying each side of equation (1.2) by pn, summing from n = 0 to N, and making use of equations (1.1) and (1.3).

In 1962, O. Frink [2] showed that in a pseudo-complemented semilattice 〈P; ∧, *, 0〉, the closed elements form a Boolean algebra. We shall consider an extension of this result to arbitrary commutative semigroups with zero.