It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + 1 , . . . , r + m — 1, each of which is a kth power residue of p. Let r(k, m, p) denote the least such r.

A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .

It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.

When, for the generalized summation of series, we use A and B methods, giving A and B sums, respectively, we say that the A method is included in the B method, A ⊂ B, if the B sum exists and is equal to the A sum whenever the latter exists. A theorem proving such a result is called an Abelian theorem. For example, there is an Abelian theorem stating that if the A and B sums are the first Cesàro mean and the Abel mean, respectively, then A ⊂ B. If A ⊂ B and B ⊂ A, we say that A and B are equivalent, A = B. For example, the nth Hölder and nth. Cesàro means are equivalent.

Cet exposé fait suite à des travaux antérieurs et notamment à mon article (4). Il a fait l'objet d'une communication au Colloque de géométrie différentielle et de topologie algébrique de Zürich en juin 1960. Dans une première partie on aborde, par les méthodes cohomologiques, le problème de la restriction du groupe structural G d'un espace fibre à un sous-groupe N, ce qui, lorsque N n'est pas normal, conduit à une "cohomologie à coefficients dans un faisceau d'espaces homogènes." Dans le second on donne une interprétation cohomologique des espaces fibres en théorie simpliciale, une fonction tordante étant interprétée comme un cocycle.

We shall consider the classification problem for space forms of (Riemannian manifolds which are covered by) real, complex, and quaternionic Grassmann manifolds. In the particular case of the real Grassmann manifold of oriented 1-dimensional subspaces of a real Euclidean space, this is the classical "spherical space form problem" of Clifford and Klein. We shall not consider space forms of the Cayley projective plane because it is easy to see that there are no non-trivial ones.

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

A decomposition of a topological space X is a partitioning of X into non-empty, disjoint sets called elements of the decomposition. An element of a decomposition is non-degenerate if it contains more than one point. Associated with each decomposition D of X is a topological space D*, called the hyperspace of the decomposition. A classical problem on decompositions of topological spaces is to find conditions under which D* is homeomorphic to X. Often decompositions arise from mappings: if g is a mapping of a space X onto a space Y, then D = {g-1(y) |y ∊ Y} is a decomposition of X.

Let J be a Dedekind ring, F its quotient field, F′ a finite separable extension of F, and J′ the integral closure of J in F′. It has been shown by Artin (1) that a necessary and sufficient condition that J′ have an integral basis over J is that a certain ideal of F (namely, √(D/Δ), where D is the discriminant of the extension and Δ is the discriminant of an arbitrary basis of the extension) should be principal. More generally, he showed that if is an ideal of F′, then a necessary and sufficient condition that have a module basis over J is that N()√(D/Δ) should be principal.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by

1.1

where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

A graph G is called complete if any two of its vertices are connected by an edge; a set of vertices of G are said to be independent if no two of them are connected by an edge. It follows from a well-known theorem of Ramsay (1) that for each pair of positive integers k, l there is an integer f(k, l), which we take to be minimal, such that every graph with f(k, l) vertices either contains a complete graph of k vertices or a set of l independent points.

Let En be the n-dimensional euclidean real space and the group of unimodular affine transformations which operates on it. It is known that the sets of linear h-spaces Lh (0 < h < n) have no invariant measure with respect to u (5). We wish now to consider sets of elements

1.1

composed by q parallel subspaces of dimensions h1, h2, … , hq which transform transitively by .

Projectors, spectral functions, carriers, and collections of these objects are some of the tools which have been used to study vector lattices. One of our objectives in this paper is to show that these various approaches are not essentially different. We do this by proving that each of the above-mentioned objects can be identified with a collection of prime ideals.

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Let C(M, Q) be the Clifford algebra of an even dimensional vector space M relative to a quadratic form Q. When Q is non-degenerate, it is well known that there exists an isomorphism of the orthogonal group O(Q) onto the group of those automorphisms of C(M, Q) which leave invariant the space M ⊂ C(M, Q). These automorphisms are inner and the group of invertible elements of C(M, Q) which define such inner automorphisms is called the Clifford group.

If instead of the group O(Q) we take the group of similitudes γ(Q) or even the group of semi-similitudes Γγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elements C+(M, Q) ⊂ C(M, Q). Each one of the automorphisms of C+(M, Q) so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism of C(M, Q), although the extension is not unique.

In 1947 Einstein and Strauss (2) proposed a unified field theory based on a four-dimensional manifold characterized by a nonsymmetric tensor gij and a non-symmetric connection , where

(1)

Using a variational principle in which gij and are independently varied, the above authors obtain the equivalent of the following field equations:

(2a)

(2b)

In these equations a comma denotes partial differentiation with respect to the co-ordinates of the manifold, Wij is the Ricci tensor formed from and the notation

for the symmetric and skew-symmetric parts of geometric objects Q is employed.

Real functions

may be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing to

when f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.

The groups in question are generated by elements A and B subject to the relations

in which α and β are fixed integers. We prove:

Theorem. Each group of the class just presented is finite when neither α nor β equals 1, and is nilpotent. Its order is a factor of 27 (α — 1) (β — l)∈8where ∈ is the greatest common divisor of α — 1 and β — 1, and its nilpotency class is at most 8.

We denote the commutator X-1Y-1XY by [X, Y], and Y-1XY by XY. If X1, X2, … , Xr are elements of some group then {X1, X2, … , Xr} will mean the subgroup which they generate.