This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.

Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no essential map between Eilenberg-MacLane spaces of different dimensions has a kernel. In this paper we define homotopy pull-backs and push-outs, which do exist and which behave like pull-backs and push-outs, and we give some of their properties. Applications may be found in [3; 5; 6 and 14].

In §1, we define a differential-integral operator, which for positive real indices is commonly known as the Liouville-Riemann generalized integral. For positive integer indices, we obtain an iterated integral. For negative real indices we obtain the Riemann-Holmgren (5; 9) generalized derivative, which for negative integer indices gives the ordinary derivative of order corresponding to the negative of such an integer. Following M. Riesz (10) we extend these ideas to include complex indices.

An n X n real matrix A = (aij) is called stable (resp. semistable) if each of its eigenvalues has negative (resp. nonpositive) real part. These notions are important because of their close connection with the stability of motion, which can be described especially simply for a system

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:

No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

Introduction. Soit G un groupe de Lie connexe et soit B un sous-groupe fermé de G. Supposons que la variété homogène G/B ou G opère transitivement à gauche soit munie d'une structure complexe invariante par les opérations de G, et qu'il existe sur G/B un volume invariant par G. Ce volume étant unique à un facteur constant près, la forme hermitienne de G/B qui est canoniquement associée au volume et à la structure complexe se trouve entièrement déterminée par la structure complexe de G/B et le groupe transitif G (5).

It is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

A quasi-Poisson manifold is a $G$-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ${{\Lambda }^{3}}\mathfrak{g}$ associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity

Let Md and be Riemannian manifolds. We shall say that an isometric immersion ϕ: Md —> is isotropic provided that all its normal curvature vectors have the same length. The class of such immersions is closed under compositions and Cartesian products. Umbilic immersions (e.g. Sd ⊂ Rd+1) are isotropic, but the converse does not hold. If M and are Kähler manifolds of constant holomorphic curvature, then any Kähler immersion of M in is automatically isotropic (Lemma 6). We shall find the smallest co-dimension for which there exist non-trivial immersions of this type, and obtain similar results in the real constant-curvature case.

The two ternary quadratic forms

are said to be reciprocal (to each other) since their coefficients form inverse matrices :

Dans un précédent article [6, §4] j'ai essayé de traiter le problème suivant, qui fait l'objet du présent travail: on donne 2n abscisses, marquées 1 , 2 , … , 2n, de gauche à droite, sur un axe horizontal. On les joint deux à deux par n arcs convexes, tracés au-dessus de l'axe, de manière que chaque abscisse soit l'origine ou l'extrémité d'un seul arc, l'origine étant à gauche et l'extrémité à droite. On obtient ainsi p2n = 1.3.5. … (2n — 1) configurations et on demande le nombre de celles qui ont p points doubles.

Let A, B be two square matrices with complex coefficients, of respective orders n and m, where n ≥ m. We shall say that B is imbeddable in A if there exists a unitary matrix U of order n such that U*AU contains B a s a principal submatrix. In other words, B is said to be imbeddable in A if there exists a matrix V of type n × m such that V*V = IW (= the identity matrix of order m) and V*AV = B.

The basic problem of linear programming is to minimize (or maximize) a linear function of a finite number of variables constrained by a finite number of linear inequalities. From a mathematical point of view the subject may be regarded as being divided into two areas. One is primarily analytical and deals with certain questions of duality and consistency. The other is algorithmic and is concerned with computational questions and methods.

This paper is a result of an investigation into general methods of determining the irreducible characters of GL(n, q), the group of all non-singular linear substitutions with marks in GF(q), and of the related groups, SL(n, q), PGL(n, q), PSL(n, q), the corresponding group of determinant unity, projective group, projective group of determinant unity, respectively.

The notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but which are not expansive homeomorphisms. For example, the shift maps of Knaster's indecomposable chainable continua are continuum-wise expansive homeomorphisms, but they are not expansive homeomorphisms. Also, there is a continuum-wise expansive homeomorphism on the pseudoarc. We study several properties of continuum-wise expansive homeomorphisms. Many theorems concerning expansive homeomorphisms will be generalized to the case of continuum-wise expansive homeomorphisms.

1. Introduction. If M is any finite set we define a chain on M as a mapping f of M into the set of ordinary integers. If a ∈ M then f(a) is the coefficient of a in the chain f. The set of all a ∈ M such that f(a) ≠ 0 is the domain |f| of f. If |f| is null, that is if f(a) = 0 for all a, then f is the zero chain on M. If M is null it is convenient to say that there is just one chain, a zero chain, on M.

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.

If f(x) is denned on [0, 1], then its corresponding Bernstein polynomial

approaches f(x) uniformly on [0, 1], if f(x) is continuous on [0, 1]. If f(x) is bounded on [0, 1], then at every point x where the second derivative exists (Voronowskaja [7], see also [5])