The prototype for applications of group theory to physics, and to mathematical physics, is the quantum theory of angular momentum [1] ; the use of such techniques is now almost universal, and familiarly (through somewhat imprecisely) known as “Racah algebra”. To categorize, group theoretically, those characteristics which underlay this applicability to physical problems, Wigner [30] isolated two significant conditions, and designated groups possessing these properties as simply reducible.

The two conditions for simple reducibility are:

(a)Every element is equivalent to its reciprocal, i.e., all classes are ambivalent.

(b) The Kronecker (or “direct“) product of any two irreducible representations of the group contains no representation more than once.

Computation of the velocity of a given motion depends on measurement of nearby position changes only. Computation of acceleration, on the other hand, depends on measurement of nearby changes in velocity. But since velocity vectors are attached to positions so that even nearby ones are not a priori comparable, acceleration is not computable until a rule for comparison of vectors along a curve is given. Such a rule-parallel translation or linear connexion - exists automatically in Euclidean spaces. For motions in more general manifolds, for example (semi-) Riemannian ones, parallel translation is a less obvious consequence of the metric properties.

Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system

over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard [3] has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.

The study of metrization has led to the development of a number of new topological spaces, called generalized metric spaces, within the past fifteen years. For a survey of results in metrization theory involving many of these spaces, the reader is referred to [13]. Quite a few of these generalized metric spaces have been studied extensively, somewhat independently of their role in metrization theorems. Specifically, we refer here to characterizations of these spaces by various workers as images of metric spaces. Results in this area have been obtained by Alexander [2], Arhangel'skii [3], Burke [5], Heath [10], Michael [15], Nagata [16], and the author [1], to mention a few. Later we will recall specifically some of these results.

The recent study of operational statistics (see [2; 4; 5; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16]) describes a generalized sample space which represents the set of all possible outcomes of a collection of coherently related operations (experiments). This approach to probability generalizes the classical notion of a sample space due to A. N. Kolmogorov [8], and it gives the concept somewhat wider applicability. For instance in [4] and [14], D. J. Foulis and C. H. Randall set out the start of a program wherein a generalized sample space (hereafter called a GSS) and its affiliated partially ordered set of generalized propositions could be a framework within which a genuinely operational interpretation of the so called “logic” of quantum mechanical systems may be found.

A Steiner triple system (briefly STS) is a pair (S, t) where S is a set and t is a collection of 3-subsets of S (called triples) such that every 2-subset of S is contained in exactly one triple of t. The number |S| is called the order of the STS (S, t). It is well-known that there is an STS of order v if and only if v = 1 or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all v it is understood that v = 1 or 3 (mod 6).

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

Oscillation criteria will be obtained for the linear elliptic partial differential equation

in an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by

where each α (i), i = 1, … , n, is a non-negative integer.

On étudie certains sous-espaces de l'algèbre de Banach Bp(G) des multiplicateurs de AP(G). On montre entre autres que l'espace des coefficients d'une représentation de G sur un espace Lp n'est pas nécessairement fermé dans Bp(G) à moins que p soit égal à deux.

Dans [4] C. S. Herz soulève la question suivante:

Soit G = SL(2, R) le groupe des matrices réelles d'order deux, de déterminant un; soit H = SL(2, Z) le sous-groupe discret de G, formé des matrices à coefficients entiers. Existe-t-il une constante absolue c > 0 telle que pour toute function f de At(H) à support fini, on puisse trouver une function de AP(G), à support compact dont la restriction à H coïncide a vec f et

Cardan motion in Euclidean geometry may be defined as the motion of a plane Г1 with respect to a coinciding plane Г such that two points A1, A2 of Г1 move along two orthogonal lines ai, a2 of Г1 The properties of this classical motion are well-known: the path of a point of Г1 is in general an ellipse with its center at the intersection o of a1 and a2; there are ∞ l points of Г1 (their locus being the circle C1 with A1, A2 = 2d as diameter) the paths of which are line segments. The moving polhode is the circle Ci, the fixed polhode is the circle (o; 2d). We investigate here Cardan motion-defined in the same way-in the elliptic plane.

At the beginning of the chapter on induced representations in the treatise of Curtis and Reiner [8] on representation theory, they write “Most of the results have not yet found suitable generalization to rings with minimum condition or finite dimensional algebras, …”. The purpose of this paper is to indicate how some of the more basic theorems concerning induced representations can, in fact, be generalized to rings and algebras. In most cases we can do this by bringing together known results, so that in this sense this paper does not contain substantially new results.

In a lecture in 1966 (see [6]), the second author considered briefly those first order properties which hold for a lattice if and only if they hold for the lattice of all ideals of . The best known examples of such properties are those given by identities. The well-known connection between the modular identity ϵ and the five-element nonmodular lattice transforms the above result for ϵ into the following statement: is a sublattice of a lattice if and only if is a sublattice of .

This paper will continue certain investigations into the geometric nature of the well-known traveling salesman problem: that of determining the extreme Hamiltonian circuits (H-circuits) of a graph.

A finite set of generators for the isotopy classes of selfhomeomorphisms of closed surfaces was given by Lickorish in three papers [2; 3; 4]. In [5] the group of isotopy classes for a particular, well-known cube with holes was presented. There the structure was “tight” enough to allow the computation of the relators as well as the generators. In this paper we give a finite set of generators for the group of isotopy classes of self-homeomorphisms on the solid double torus, the cube with two handles. Let us remark that the group of isotopy classes for the solid torus is well-known.

Consider the wth order scalar ordinary differential equation

with pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:

DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decomposition

where M1 and M2 are subspaces of X such that

(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;

(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

It is known that minimally thin and semithin sets in R2 are preserved by conformai mappings (see [3]) but it is not known whether or not analogous results hold true for ordinary thin and semithin sets respectively. In an unpublished work, Brelot has shown that ordinary thin sets at the origin are preserved by mappings of the form f(z) = z∝ (∝ > 0) where one always considers the principal branch of the mapping. We shall prove this result along with an analogous one for ordinary semithin sets and will see that the implications established by Jackson (see [4, Theorem 4]), for ordinary and minimally thin sets and by Brelot (see [2, p. 152]) for semithin sets in a half plane hold true for any wedge shaped region with vertex at the origin.

Let H denote the set of all entire functions of a single complex variable equipped with the topology of convergence uniform on all compact subsets of C, the set of complex numbers. Then an entire function f is mean periodic if the subspace spanned by f and its complex translates is not dense in H. It was shown by Schwartz [13, p. 922] in 1947, to whom this definition is due, that any such function is the limit in H of a certain sequence of exponential polynomials.

In [5] Pall denned a partitioning of a quadratic space over a field of characteristic not 2 to be a collection of disjoint (except for ﹛0﹜ ) maximal totally isotropic subspaces whose union formed the set of isotropic vectors. Clearly isometric quadratic spaces simultaneously do or do not have partitionings. Pall exhibited the existence of partitionings for the spaces associated with

over formally real fields for n = 1, 2, 4, 8 and over Z/(p), p prime, for n = 1, 2. Using the latter, he was able to find a new proof for Jacobi's formula for the number of representations of a positive integer as the sum of four integral squares.

In 1950, R. C. James [7] exhibited a nonreflexive Banach space with a basis that is of finite codimension in its second dual. This space is the first example of a quasi-reflexive space. General results on quasi-reflexive spaces have been obtained by P. Civin and B. Yood [3], and quasi-reflexive spaces with bases have been studied by D. Dean, B. L. Lin, and I. Singer [4 ; 12].