It is very well known that any two bases of a finitary matroid (see [2] for definitions) have the same cardinality. As Dlab has shown in [1], the same does not hold for arbitrary transitive exchange spaces; indeed, since the examples Dlab constructs in [1] are matroids, it does not even hold for arbitrary matroids. Nevertheless with the aid of the generalized continuum hypothesis (G. C.H.) we are able to prove the result for B-matroids.

Let E be a metric space. A mapping T of the space E into itself is said to be a contraction if there exist s a number k, with 0 ≤ k < 1 such that

for any two points x, y ∈ E. Every contraction mapping is continuous.

In [2, page 3831], Varopoulos proves that for any ∈ > 0 there exists a function on a neighbourhood of 0. Indeed, if 0 < ∈ < π/2, then f(x) defined to be equal to 1 - e when -eix, linear on [∈, 2π - ∈] and of period 2π, is an example of such a function.

The above result can be used to give a direct proof of the following result without reference to the L2 theory [1, Theorem 2.6.4].

Nous voulons montrer que le théorème de Bernstein est une conséquence simple de la formule de Taylor et d' un théoréme de Helly. Rappelons une définition: une fonction réelle f définie sur (0, ∞) est dite complètement monotone si f est indéfiniment derivable, si f ainsi que toutes ses dérivées paires sont positive set si toutes les dérivées impaires sont négatives.

In this note, a certain generalization of the Cauchy distribution is obtained, using the result of Malik [2].

The generalized gamma distribution having the density

(1)

is introduced by Stacy [1], who studied some of its properties. As remarked by Stacy [1], the familiar gamma, chi, chi-squared, exponential and Weibull distributions are special cases of (1), as are the distributions of certain functions of a normal variable - viz.

Nous voulons donner une démonstration directe du fait que la dérivée de la primitive d' une fonction f(x) intégrable au sens de Perron existe et est égale à f(x) presque partout. Nous utiliserons essentiellement le "Iemme du soleil levant" ou plutôt un de ses corollaires immédiats.

Let be an entire function, but not a polynomial. As usual let,

1