By a group A is meant throughout an additively written abelian group. A is said to have linear topology if there is a system of subgroups Ui (i∈I) of A such that, for a ∈ A, the cosets a+Ui(i∈I) form a fundamental system of neighborhoods of a. The group operations are continuous in any linear topology; the topologies are always assumed to be Hausdorff, that is, ∩iUi = 0. A linearly compact group is a group A with a linear topology such that if aj+Aj (j∈I) is a system of cosets modulo closed subgroups Aj with the finite intersection property (i.e. any finite number of aj+Aj have a non-void intersection), then the intersection ∩j(aj+Aj) of all of them is not empty.

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.

M. H. Stone raised the problem ([1] Problem 70) of characterising the class of distributive pseudo-complemented lattices ℒ = 〈L; ∨, ∧, 0, 1〉 in which a* ∨ a** = 1 holds identically. Several solutions to this problem have now been offered — the first being by G. Grätzer and E. T. Schmidt [6], who gave this class of lattices the name Stone lattices. Later solutions were given by J. Varlet [11], O. Frink [4] and G. Grätzer [5]; see also G. Bruns [2].

In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ 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 ℒ.

Some recent papers have revived interest in some questions concerning the motion of a simple pendulum which is oscillating with small angular amplitude under gravity, when the length of the pendulum changes with time in some prescribed manner.

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.

In studying the compression of loops of fabric between flat parallel plates, the problem arises of the compression of bends. A ‘bend’ is illustrated in Figs. 1 and 2. Two bends are joined to form a loop. The material is postulated to yield in bending such that the bending moment where B is a bending rigidity, KI the current impressed curvature, and KR the ‘remanent’ curvature. This remanent curvature is the free, or natural curvature in the material caused by its past and present deformation. We study the case when the remanent curvature that is, the remanent curvature is proportional to the greatest previously impressed curvature. We call equations (1) and (2) the ‘Remanent Curvature Hypothesis’.

In [1], Aull and Thron introduce several separation axioms between T0 and T1. In particular, they define a space X to be TD if for each x ∈ X, {x}' is a closed set.

A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

Ricci identities in a Finsler space have been given by C. I. Ispas [1], H. Rund [2], R. S. Mishra and H. D. Pande [3] and others. Here we shall prove some identities using the principle of mathematical induction. Considering a second order contravariant tensor depending on the element of support , we have the following theorems.

Extensions of semigroups have been studied from two points of view; ideal extensions and Schreier extension. In this paper another type of extension is considered for the class of inverse semigroups. The main result (Theorem 2) is stated in the form of the classical treatment of Schreier extensions (see e.g.[7]). The motivation for the definition of idempotentseparating extension comes primarily from G. B. Preston's concept of a normal set of subsets of a semigroup [6]. The characterization of such extensions is applied to give another description of bisimple inverse ω-semigroups, which were first described by N. R. Reilly [8]. The main tool used in the proof of Theorem 2 is Preston's characterization of congruences on an inverse semigroup [5]. For the standard terminology used, the reader is referred to [1].

A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.

As the applications of category theory increase, we find ourselves wanting to imitate in general categories much that was at first done only in abelian categories. In particular it becomes necessary to deal with epimorphisms and monomorphisms, with various canonical factorizations of arbitrary morphisms, and with the relations of these things to such limit operations as equalizers and pull-backs.

In a recent paper [4], H. Sharp, Jr., has discussed the problem of finding formulae for the following naturally defined integers: the numbers t(n), tc(n), t0(n), tc0(n), and ts(n) of all homeomorphism classes of finite topological spaces with n elements, which are respectively (i) arbitrary, (ii) connected, (iii) T0, (iv) connected and T0, (v) symmetric. Here, a finite topological space X is called symmetric provided the following relation ≦ is symmetric: x ≦ y if and only if x ∈ Uv, the intersection of all open sets containing y.

In this paper the Toeplitz determinant of order s ≧ 1 generated by the rational function , with , and , is evaluated exactly for all values of s ≧ m, as , where in with and , thus proving Szegö's formula for the function fm, n(z).

By forming the rational approximation of the generating function the formula is then extended to enabling the evaluation of the limit of Toeplitz determinants generated by certain classes of complex valued functions.

Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

Let be a given series with its partial sums {Sn} and {Pn} a sequence of real or complex parameters. Write. The transformation given by defines the Nörlund means of {Sn} generated by {Pn}. The series Σann is said to be absolutely summable (N, pn) or summable ∣N, pn∣, if {tn} is of bounded variation, i.e., Σ|tn—tn−1| converges.

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffer's stroke operation and. Byrne's brief formulations of Boolean algebras [1], Sholander's characterization of distributive lattices [7] and Sorkin's famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a−1; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab−1. [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.

In [1], the following theorem is proved:THEOREM. If f ∈C2π, α is a positive integer, and then Where , .