Let P be the parabolic differential operator

where E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form

1

has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.

Let M, M′ be C∞ Riemann manifolds such that

(1.0) M is compact;

(1.1) M′ is complete and its sectional curvatures are non-positive.

In terms of local coordinates x = (x1, … , xn) on M and y = (y1, … , ym) on M′, let the respective Riemann elements of arc-length be

and Γijk, Γ′αβγ be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M′) or its coordinates in some local coordinate system.

Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying

1. (i) ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and

2. (ii) ϕ0 = e, the identity map on S.

   A semi-group {ϕt} of maps in S is said to be

3. (iii) of class (C0) if ϕt(x) → x as t → 0 for each x in S,

4. (iv) separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and

5. (v) doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.

In a recent paper (2), Helson and Kahane consider the problem of the existence of real numbers x such that the sequence (λn x) (when reduced modulo 1) is not summable by a given regular Toeplitz method, where (λn) is a lacunary sequence of positive real numbers. Thus, as an example, they show the existence of uncountably many x such that the sequence (θn x) does not have a distribution function modulo 1, where θ is some fixed number > 1.

A family Ω of valuations of the field K is said to be of finite character if only a finite number of valuations are non-zero at any non-zero element of K. If w ∈ Ω has ring and maximal ideal , then A = ∩w∈Ω is said to be defined by Ω and ∩ A is a prime ideal called the centre of w on A and denoted by Z(w). If = Az(w), then w is said to be an essential valuation for A. A domain defined by a family of finite character in which every valuation is essential is called a ring of Krull type.

If X is a complex Banach space and B(X) denotes the space of bounded linear operators on X, then the class of meromorphic operators consists of those T in B(X) such that the non-zero points of σ(T) are poles of the resolvent Rλ(T). If we also require that each non-zero eigenvalue of T have finite multiplicity, members of the class ⊆ so defined have been called operators of Riesz type. and have been studied in (2, 6, 7) and (1,4) respectively.

This paper forms a continuation of (1), extending the concept of a meromorphic operator to not necessarily bounded, closed linear operators in complex Banach space. Let T denote such an operator with range and domain in Banach space X. We shall study the class of such operators T where λ = 0 and λ = ∞ are the only allowable points of accumulation of σ(T) and every isolated point of σ(T) is a pole of Rλ(T). We shall write (0, ∞) to represent the class of such operators. If λ = 0 (λ = ∞) is the only allowable point of accumulation of σ(T), we shall write (0) ((∞)) to denote the corresponding class of operators.

Given a probability measure space (Ω, , P) consider the following packing problem. What is the maximum number, b(K, Λ), of sets which may be chosen from so that each set has measure K and no two sets have intersection of measure larger than Λ < K?

In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the following minimal paving problem. In a non-atomic a-finite measure space (Ω, , μ) what is the measure, V(b, K, Λ), of the smallest set which is the union of exactly b subsets of measure K such that no subsets have intersection of measure larger than Λ?

Following (2) we say that a measure μ on a ring is semifinite if

Clearly every σ-finite measure is semifinite, but the converse fails.

In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.

This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.

In this paper we shall study properties of a locally convex space (l.c.s.) which guarantee that it is a direct product of normed linear spaces or Banach spaces. The conditions will be given both as properties of the original space itself and as properties of the dual, and will take the form of a completeness condition and the existence of sub-basic sets of pseudo-norms with certain properties (a set of pseudo-norms is basic if the set of unit balls of its members is a base of neighbourhoods of 0.

G will denote a finite (or infinite) group of order n. In a normal multiplication table (n.m.t.) of G (7, 8, 9, 12) all entries in one diagonal are equal to e (the identity of G), and if the entry on the intersection of the ith column and jth row is gi,j ∈ G, then

The following is a n.m.t. of Z6 = {0, 1, 2, 3, 4, 5}:

Remark. The cyclic groups Zn will always be written in additive notation. The table is uniquely defined by every one of its columns, in particular by the first.

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

A set of points M of a graph G is a point cover if each line of G is incident with at least one point of M. A minimum cover (abbreviated m.c.) for G is a point cover with a minimum number of points. The point covering number α(G) is the number of points in any minimum cover of G. Let [V1, V2, … , Vr], r > 1 be a partition of V(G), the set of points of G. Let Gi be the subgraph of G spanned by Vi for i = 1, 2, … , r.

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

Let

1.1

denote the nth partial sum of the Fourier Legendre series of a function ƒ(x). The references available to us, except (5), prove only that Sn(ƒ, x) converges uniformly to ƒ(x) in [— 1, 1] if ƒ(x) has a continuous second derivative on [—1, 1]. Very recently Suetin (5) has shown by employing a theorem of A. F. Timan (7) (which is a stronger form of Jackson's theorem) that Sn(ƒ, x) converges uniformly to ƒ(x) ƒ(x) belongs to a Lipschitz class of order greater than 1/2 in [—1, 1]. More generally he has proved the following theorem.

In this paper properties of symmetric coordinate spaces and symmetric bases are investigated. Since a space which possesses a basis is essentially a space of sequences (12, p. 207), the interrelation of these two concepts naturally suggests itself.

Section 2 is a summary of the terminology and methods employed, which fall into four categories: (1) set theoretical properties of coordinate spaces such as symmetry and dual spaces; (2) the notion of FK and BK space (12, p. 202; 13); (3) the theory of the Schauder basis in F-space applied to the case when (see § 2) is a basis for a coordinate space; (4) the concept of a sequential norm, which the author introduced in (7) to illustrate the underlying unity of the first three ideas.

In order to deal with certain problems in the theory of interpolation spaces, it is convenient to consider operators of the following form:

Let k be a non-negative measurable function on the half-line R+, and let ƒ be a measurable function on R+ with

1

Then the operator T is defined by

2

with the domain of T, D(T), consisting of all ƒ which satisfy (1).

Consider a set of n elements α1, … , αn (denoted by S) and the set of all possible multiplication tables on these elements. The total number of such tables is clearly and each table can be represented by a square matrix [μij] where μij is the product αiαj (μij ∈ S, i = 1, … , n; j = 1, … , n). The triple (αi, αj, αk) is said to be associative if the following equation is satisfied:

1.1

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.

The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.