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.

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.

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.

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

1