Nous considérons dans cette note une classe d′équations différentielles fonctionnelles linéaires dont les coefficients sont analytiques; des résultats sur l′ordre des solutions et leur comportement asymptotique seront obtenus.

Il s'agit d'une généralisation à des équations du deuxiéme ordre d'une propriété d'équations d'ordre un [2]: la démonstration du théoréme principal demeure valable dans ce dernier cas.

A compactification αX of the space X is called an n -point compactification if the remainder αX — X consists of exactly n points. K. D. Magill [5] showed that if Y has an n-point compactification and if f:X→ f(x) = Y is a compact continuous mapping of the space X onto Y, then X also has an n-point compactification.

Let A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. Ml,..., Mh) be the maximal ideal of A (resp. of Ā). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces span the vector space . If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

Dans [8], nous avons étudié l'ordre de grandeur des sommes de la forme , pour ou k, où g(n) et f(n) appartiennent respectivement à une classe de fonctions multiplicatives et additives.

Let R∞ be the set of all the finite sequences of real numbers. The author shows that there are uncountably many distinct topologies on the set R∞, each of which coincides with the usual topology when restricted to an Rn. However, under each of these topologies, R∞ is always of the same homotopy type. A generalization to some other spaces is mentioned.

Suppose throughout that {pn} is a sequence of non-negative numbers with p0>0, that

and that {sn} is a sequence of real numbers.

The invariant operator range lattices of a wide class of uniformly closed algebras (including C*-algebras) are stable under weak closures. There is an algebra whose invariant operator range lattice contains properly the corresponding lattice of its norm closure. An operator range transitive algebra is operator range n -transitive for all n. A normal operator is algebraic if and only if each of its invariant operator ranges is the range of some operator commuting with it.

In considering a problem on certain summable (C, k) trigonometric series, R. D. James [13] used a symmetric pk+2- integral defined earlier to recapture the coefficients of the series from the sum function. James' formulas for the coefficients are more complicated than the usual Euler-Fourier form since the pk + 2 - integral is of order k + 2. It is shown that a generalized integral of order one for each non-negative integer k can be suitably defined to reduce James' formulas to the usual form.

Hecke, [3], has shown for x a real Dirichlet character modulo q, the associated Dirichlet L-function L(s, x) has infinitely many zeroes on the line

Here, using a method of Polya, [5], we show that both the real and imaginary parts of a function associated to L(s, x) through the functional equation, have infinitely many zeroes on any line

It is shown that techniques of Browder and Ishikawa for approximating fixed points of nonexpansive mappings extend in a more restricted sense to the locally nonexpansive case.

In [8] and subsequent papers, Jônsson (et al) developed a lattice identity which reflects precisely Desargues Law in projective geometry in that a projective geometry satisfies Desargues Law if and only if the geometry, qua lattice, satisfies this identity. This identity, appropriately called the Arguesian law, has become exceedingly important in recent investigations in the variety of modular lattices (see for example [2], [3], [9], and [12]). In this note, we supply two possible lattice identities for the Pappus' Law of projective geometry.

In [9] de la Torre proved that if is a finite measure space and T is a linear operator on a real for some fixed p, 1 < p < ∞ , such that ||T||P ≤ 1 and simultaneously ||T||∞ ≤ l, and also such that there exists with Th = h and h≠0 a.e., then the dominated ergodic theorem holds for T, i.e. for every we have

de la Torre proved his result, by showing that the operator S, defined by Sf = (sgn h) - T(f • sgn h) for is positive, and by applying Akcoglu's theorem [1] to S.

It is shown that dentability of the unit ball of a conjugate Banach space X* does not imply smoothability of the unit ball of X, answering a question raised by Kemp. A property called strong smoothability is introduced and is shown to be dual to dentability. The results are used to provide new proofs of the facts that X is an Asplund space whenever it has an equivalent Fréchet differentiable norm, or whenever X* has the Radon-Nikodym Property.

We give a classification result for "extreme-monic" polynomials in several variables having measure 1. The result implies a recent several-variable generalization, by D. W. Boyd, of Kronecker's classical theorem (that all zeros of a monic integral polynomial, with non-zero constant term and measure 1, are roots of unity).

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.

In this note we collect some observations on the outer measures ψf and ψf that have been introduced in [4] and which describe the total variation of the function f. These properties have direct applications to the study of the derivative and the relative derivative. For definitions and notation the reader is referred to [4].

I would like to express my thanks to the Canadian Mathematical Society for inviting me to present this lecture. I would also like to express my appreciation to C.J. Smyth for numerous helpful conversations during his visit this year at the University of British Columbia.

This paper follows reasonably closely the outline of the lecture presented in Ottawa. More details are given here though and a number of proofs which would not be otherwise accessible have been added as Appendices. The attentive reader will soon realize the appropriateness of the title.

Let L be a lattice, and let P and Q be partially ordered sets. We say that L is generated by P if there is an isotone mapping from P into L with its image generating L. P contains Q if there is a subset Q’ of P which, with the partial ordering inherited from P, gives an isomorphic copy of Q. For an integer n > 0, the lattice of partitions of an n-element set will be denoted by II(n); it is well-known that II(rc) is simple and complemented (cf. P. Crawley-R. P. Dilworth [1; p. 96]).

The inversion and the characterization of the convolution transform is derived via the concept of unimodality introduced by Khintchine (1938). This method yields simple and intuitively appealing proofs.