In this note we continue the study, initiated in [1], of the class S*(α) of functions

(1.1)

that are analytic and univalent in the unit disc U and satisfy the condition

(1.2)

S*(1) is the frequently studied class of univalent star-like functions. For each α, S*(α) is a subclass of the class K(α) of close-to-convex functions of order α introduced by Pommerenke [4]. Properties of the class S*(α) proved useful in studying the coefficient behaviour of bounded univalent functions that are analytic and map U onto a convex domain [1]. In this note we investigate the problem of determining

but we are able to give only a partial solution.

In this paper, we develop techniques for solving ternary Diophantine equations of the shape $A{{x}^{n}}+B{{y}^{n}}=C{{z}^{2}}$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A,\,B\,\text{and}\,\text{C}$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

Nous savons que dans le champ gravitationnel de la relativité (i.e. défini par les équations Rμv = 0) les ondes dites gravitationnelles se propagent avec la vitesse de la lumière. Nous allons maintenant voir, dans le présent travail que si, dans le même continuum, on impose aux potentiels gμv, une oscillation de fréquence assez élevée, il en résulte une onde du type maxwellien.

The continuous real functions on a topological space X are partially ordered in a natural way by putting ƒ ≦ g if and only if ƒ(x)≦ g(x) for all x in X. With respect to this partial ordering these functions constitute a lattice, the lattice operations ∪ and ∩ being defined by the relations (ƒ ∪ g) (x) = max (ƒ(x), g(x)) ƒ ∩ g )(x) = min (ƒ(x), g(x)). The lattice character of any partially ordered system merely expresses the existence of least upper and greatest lower bounds for any finite set of elements in the system. Many partially ordered systems enjoy much stronger boundedness properties than these: for example, every subset with an upper bound may have a least upper bound, as in the case of the real number system.

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).

In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each of the first order. The method of Kirchhoff provides certain constants of the motion. If we describe the positions of the vortices by their point-traces on a plane perpendicular to them, the following facts follow from the theory of Kirchhoff:

(1.1) The mean centre of the system is fixed.

A recent survey article by J. Fournier and J. Stewart (Bull.AMS 13 (1985), 1-21) explains how amalgams of Lp with lq (as function spaces over any locally compact abelian group G) can be used as an effective tool for the treatment of various problems in harmonic analysis. The present article may be seen as a complement to this survey, indicating further advantages that arise if one works with generalized amalgams (introduced in 1980 under the name of Wiener-type spaces by the author [10]). The main difference between amalgams and these more general spaces is the fact that they allow a more precise description of the local behavior of functions (or distributions) by rather arbitrary norms and that the conditions on the global behavior (of the quantity obtained using that chosen local norm) is described in a way that includes both growth and integrability conditions (not only lq-summability).

In this paper we study unit preserving *-endomorphisms of and type II1 factors. A *-endomorphism α which has the property that the intersection of the ranges of αn for n = 1 , 2 , … consists solely of multiples of the unit are called shifts. In Section 2 it is shown that shifts of can be characterized up to outer conjugacy by an index n = ∞ 1, 2 , …. In Section 3 shifts of R the hyperfinite II1 factor are studied. An outer conjugacy invariant of a shift of R is the Jones index [R: α(R)]. In Section 3 a class of shifts of index 2 are studied. These are called binary shifts. It is shown that there are uncountably many binary shifts which are pairwise non conjugate and among the binary shifts there are at least a countable infinity of shifts which are pairwise not outer conjugate.

1. Introduction. In his well-known book on graphs [1] König proposed the following problem: “When can a given abstract group be represented as the group of the automorphisms of a (finite) graph, and if possible how can the graph be constructed?”

To understand this problem well let us begin with the precise definition of a graph and its group (of automorphisms).

A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.

In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.

1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].

C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K0(X)), construit à l'aide des -modules localement libres sur un schéma X[14]. La somme directe et le produit tensoriel de modules font de K0(X) un anneau, et les opérations de puissances extérieures lui fournissent une structure supplémentaire, que Grothendieck appelle λ-anneau. Un λ-anneau est muni d'une filtration décroissante, la γ-filtration, et un des principaux résultats de Grothendieck est que, si X est lisse sur un corps, le groupe

est isomorphe, à la torsion près, au groupe de Chow CH1(X) des cycles de codimension i sur X, modulo l'équivalence linéaire.

Dans un article antérieur (3), j'ai donné quelques théorèmes généraux sur les représentations unitaires des groupes de Lie nilpotents simplement connexes. On va, dans le présent article, construire explicitement les représentations unitaires irréductibles et la formule de Plancherel des groupes de Lie nilpotents simplement connexes de dimension ⩽ 5.

In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

A pro-p-group G is said to be a Demushkin group if

1. (1) dimFp H1(G, Z/pZ) < ∞,

2. (2) dimFp H2(G, Z/pZ) = 1,

3. (3) the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.

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].

For a non-negative integer s and a finite simplicial complex K, let βS(K) denote the s-dimensional Betti number of K and let fs(K) denote the number of s-simplices of K. Our theorem, like Poincaré's, applies to combinatorial manifolds M, but it concerns the numbers fs(M) instead of the numbers βS(M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices of n-dimensional convex poly topes which have a given number i of (n — 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system of i linear inequalities in n variables, and was the original motivation for our study.

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

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.