Let f(ζ) be a power series of the form

where lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the form

where lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.

Let X be a complex normed space, with dual space X′, and T a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x∊X, f∊ X′, ∥x∥ = ∥f∥ = f(x) = 1}. Let ⃒V(T)⃒ denote sup {⃒λ⃒: λ∊ V(T)}. Our purpose is to prove the following theorem.

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:

(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.

(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by |G|p, and |A(G)|p, where A(G) is the automorphism group of G. It was shown by Herstein and Adney [8] that |A(G)|p ≧ p whenever |G|p= ≧P2. Later Scott [16] showed that A(G)p≧P2. For the special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup of G is abelian, and gave an example where |G|p= p4 and |A(G)|P =p2. We are able to show in Theorem 4.5 that, if |G|p = ≧ p5, then |A(G)| = ≧ p3.

The equations. We shall solve the equation

giving the solution in two forms, and give a new solution of

originally solved by Carlemann.

The latter will be extended to the case where the limits are a and b with a < x <b.

An n-tournament is a set of n labelled points, each pair A, B of which is joined either by the oriented line AB or by the oriented line BA. There are N = n(n –1)/2 such pairs and so Fn different n-tournaments, where Fn = 2N. A tournament is reducible if the points can be separated into two non-empty subsets and ℬ, such that every line joining a point in to a point in ℬ is directed towards the point in ℬ. Rado [3] showed that an irreducible tournamentis strongly connected; i.e. for every ordered pair of points A, B, there is a sequence of correctly oriented lines AC1, C1C2, …, ChB in the tournament, and conversely that a strongly connected tournament is irreducible.

Let Pn be the class of normalised polynomials of the form

of degree n which are univalent in U = {|z| < 1}. In this note we discuss the coefficients of polynomials in Pn and in some of its subclasses.

In this paper, we investigate various “arithmetical” functions associated with the factorisation of polynomials in GF[q, X1, …, Xk], where k ≥ 1 and GF[q]is the finite field of order q. We shall assume throughout that all polynomials discussed are non-zero and have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. The constant polynomial will be denoted by 1. With this normalisation, GF[q, X1, …, Xk] becomes a unique factorisation domain. When k = 1, normalisation is achieved by considering only monic polynomials. By the degree of a polynomial A(X1, …, Xk) will be understood the ordered set (m1, …, mk), where m1 is the degree of A(X1, …, Xk) in X1,(i = 1, …, k).

In [1, p. 130] the following partition theorem was deduced from a general theorem concerning the limit of a recurrent sequence.

It is well known that, provided that the indexing set I is not too large, the product

of a family of bornological locally convex topological vector spaces Eαis bornological. Products of bornological spaces were first studied by Mackey [3]. He reduced the problem to the study of R1, showing that this space is bornological if and only if I satisfies a certain condition, related to a problem in measure theory posed by Ulam [5]. We shall therefore call it the Mackey-Ulam condition on I. A similar study of the spaces R1 is to be found in the paper [4] by Simons; see also [1, Ch. IV, §6, exercise 3].

The purpose of this paper is to generalize a result of K. Iseki [1]. In his note, Iseki proves that, in a normal space S, for every countable discrete collection ℋ = {H1, H2, …} of sets from S, there exists a countable collection = {U1, U2, …} of mutually disjoint open sets from S such that Hi ⊂ Ui for every i.

Einleitung. Bei der Betrachtung automorpher Formen zu diskreten Untergruppen Г von SL2(R)mit nicht ganzzahligem Gewicht bzgl. des Automorphiefaktors J(A, z)=, hat man einen Zweig L(A, z) des Logarithmus von J(A, z)zu fixieren.

Let Y be a subspace of a topological space X. Then S(X, Y) denotes the semigroup, under composition, of all continuous selfmaps of X which also carry Y into Y. In the special case Y = X, the simpler notation S(X)is used. We have devoted several recent papers ([4], [7] and [8]) to the problem of determining when S(Z) and S(X, Y)are isomorphic and, more generally, when S(Z) is a homomorphic image of S(X, Y). In this paper, we investigate the analogous problem for certain semigroups of functions on spaces which were introduced in [5]. These include semigroups of closed functions which are treated in further detail.

In [1, p. 97], Bruck and Bose ask the question ”Has every (right) Veblen-Wedderburn system finite dimension over its left operator skew-field?” It is the purpose of this note to show that, in general, this question has a negative answer.

This paper is concerned with three basic transforms

The first of these has been studied by Widder [1], who points out that f(t) can be interpreted as the temperature u(0, t) on the time axis, where u(x, t) is the solution of the heat equation withsymmetric initial temperature u(x, 0) = g(|x|). The second has also been studied by Widder [2], where it is pointed out that f(t) can be interpreted as the value of the harmonic function u(x, t) on the t-axis arising from the boundary data u(x, 0) = g(|x|).

The Banach space lp(p≥1) is the space of all infinite sequences x = (x1, x2, x3, …) of real or complex numbers such that is convergent, with the norm defined by

The unit sphere S of lp is the set of all points x ∈ lp with ∥x∥ ≤ 1 and the sphere of radius a ≥ 0 centred at y ∈ lp is denoted by Sa(y), so that

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the form

The purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations:

(i) In case the perturbing term is linear in y, i.e. f(t, y) = B(t)y, the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6),

The hypothesis given in Trench's theorem is sufficient to imply (*) but not (6). A similar comment applies to Remark 5.

Since all the semigroups considered in this paper are commutative, we shall use the terms “semigroup” and “group” where we actually mean “commutative semigroup” and “commutative group”. Some basic results from the theory of semigroups are required and will be used without explicit mention; these results may be found in [1, § 4.3]. We shall denote the additive semigroups of integers, positive integers, negative integers, positive rationals by Z, Z+, Z-, Q+ respectively.