Several authors have recently considered the problem of establishing sufficient criteria to guarantee the oscillation or non-oscillation of all solutions of a second order elliptic equation or system. We mention in particular the papers of C. A. Swanson, [15; 16], K. Kreith [9], Kreith and Travis [10], Noussair and Swanson [13], Allegretto and Swanson [3], Allegretto and Erbe [2] and the references therein.

Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,

and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].

Soit f: [0,1] → [0,1] une fonction continue. C[0,1] est l'espace (de Banach) des fonctions continues à valeurs complexes définies sur [0, 1], muni de la norme définie par le suprémum. L'opérateur de composition [f] se définit très naturellement sur cet espace:

A transversal of a family of non-empty sets is a 1-1 map

such that φ(v) ∊ Fv (v ∊ I) . A number of problems in combinatorial mathematics reduce to the question of whether or not a certain family of sets has a transversal. An up-to-date account of this theory is to be found in the book by Mirsky [9]. The best known result of this kind is the following theorem.

In his paper [2] O. N. Golovin introduced the notion of a neutral polyverbal operation on groups, of which Moran's verbal operation [6], and Gruenberg's and Šmel'kin's operations [3 ; 7] are special cases. (Bronštein [1] proved, more generally, that every regular operation for which MacLane's postulate (see [2]) holds and which is invariant under addition of trivial factors, is a neutral polyverbal operation.)

In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.

Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent:

(i) every derivation of A is inner;

(ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit.

In this paper we derive a generation procedure for the simple (3-valent) 3-polytopes with cyclically 5-connected graphs. (A graph is called cyclically n-connected if it cannot be broken into two components, each containing a cycle, by the removal of fewer than n edges.) We define three new types of face splitting and we show, in Theorems 16 and 17, that the simple 3-polytopes with cyclically 5-connected graphs are exactly the polytopes obtained from the dodecahedron by these face splittings.

We let G denote an infinite compact group, and Ĝ its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and denotes its dual space under the pairing 〈f, ϕ〉 , (f ∊ A (G), ϕ ∊ ). Further, note is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A (G) on is defined by the following: for f ∊ A (G), ϕ ∊ , f · ϕ ∊ by

Let us associate to each open set U ⊂ Rn+1 the space HU of real functions f which are twice continuously differentiable in x1 . . . xn and once continuously differentiable in xn+1 and which satisfy the heat equation: Δf = ∂f/∂xn+1 where Then we have what in the axiomatic of Bauer is called a strong harmonic space [2, p. 61]. We will call functions of HUharmonic in U.

If is a variety of groups that can be denned by n-variable laws and (m) is the variety all of whose m-generator groups are in then there corresponds the chain: (1) ≧ (2) ≧ . . . ≧ (n) = . In this paper such chains are investigated to determine which of the inclusions are proper for certain varieties . In particular the inclusions are shown to be all proper for the varieties where is the variety of nilpotent-of-class-c groups, is the abelian variety and is the variety of centre-bymetabelian groups. For the inclusions are likewise proper but for the corresponding chain is:

It has been shown (Mallory and Sion [6]) that the problem of finding "limit" measures for inverse systems of measure spaces (Xi, μi)i∊I can be successfully attacked by establishing the existence of a “limit” measure on the product space , then considering the restriction to the inverse limit set .

In this paper we use a similar point of view to establish conditions under which a system of Radón measures has a “limit” measure which is also Radón.

Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case.

Let (X, , μ) be a complete, σ-finite measure space. Let {Tt} be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For Tt positive, it was shown independently in [2] and [5] that

1.1

almost everywhere on X, for any f ∊ L1. The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.

Let be a category with nice factorization-properties. If a functor G: —> which has a left-adjoint behaves nice with respect to factorizations then it can be shown quite easily that G behaves well in many other respects, especially that it lifts nice properties from into .

Let G be a group. Denote by ZG the group ring of G over the integers and by Δ = Δ(G) the augmentation ideal of ZG, that is, the kernel of the augmentation map ϵ : ZG → Z defined by . Then Δ is a free abelian group with a free basis . A filtration of Δ is a sequence

The desire to study constructive properties of given mathematical structures goes back many years; we can perhaps mention L. Kronecker and B. L. van der Waerden, two pioneers in this field. With the development of recursion theory it was possible to make precise the notion of "effectively carrying out" the operations in a given algebraic structure. Thus, A. Frölich and J. C. Shepherdson [7] and M. O . Rabin [13] studied computable algebraic structures, i.e. structures whose operations can be viewed as recursive number theoretic relations. A. Robinson [18] and E. W. Madison [11] used the concepts of computable and arithmetically definable structures in order to establish the existence of what can be called non-standard analogues (in a sense that will be specified later) of certain subfields of R and C, the standard models for the theories of real closed and algebraically closed fields respectively.

The work of various authors (e.g. Frink [3] and Markushevitch [7]) suggests the possibility of studying complete biorthogonal sequences in Banach spaces as a generalization of orthogonal families of continuous functions. But except for the case where the complete biorthogonal sequence is a Schauder basis such studies have not led to a very rich theory. The main reason for this is that an arbitrary complete biorthogonal sequence is not likely to have many helpful properties. For instance, in every separable Banach space X one can find a complete biorthogonal sequence {ei, Ei} which is not one-summable.

In this paper we have established the existence of an arithmetical difference system by a constructive method. Our arithmetical difference systems are a generalization of cyclic difference sets.

This note is concerned specifically with links of two (disjoint) n-spheres in an (n + 2)-manifold M, i.e. embeddings L:Sn + Sn —> M. The links L0 and L1 are isotopic if they are t he ends of a continuous family Li:Sn + Sn —> M (0 ≦ t ≦ 1) of links. They are ambient isotopic or equivalent if there is a continuous family of self-homeomorphisms ht: M—> M (0 ≦ t ≦ 1) such that h0 = identity and h1◯L0 = L1. Ambient isotopic links are isotopic, but not conversely. For example, an isotopy can tie and untie little knots (as in Figure 3) in the components of a link, thus changing the original link into one which is inequivalent to the original.

This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.

In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(np) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general

This communication is in essence a note tying together the structure theory of Bourbaki and the categorical structure theory exemplified in the works of Sonner, and Hedrlin, Pultr, and Trnkova, to name a few. The work of Bourbaki [1] appeared in 1957, and quietly succumbed to the explosion of category theory. Sonner [6] used the nomenclature of Bourbaki to classify certain types of functors, while Hedrlin et al use the functor approach, studying structures by way of the associated forgetful functors. More discussion of this approach appears in section 3.