For a continuously differentiable function g defined on an interval [α,β], define ||g|| to be the uniform norm of g, i.e. ||g||=sup ∊[α,β]|g(x)|. Define ||g||1, by ||g||1=max||g||, ||g'||}. We call the norm ||.||x the function-derivative norm.

We recall that a Dirac delta function δ(x) in the real number system is the idealization of a function that vanishes outside a "short" interval and satisfies It is conceived as a function δ for which δ(0)=+ ∞, δ(t)=0 if t≠0, and This function should possess the "sifting property" for any continuous function f. Even though certain sequences of functions are used, via a limit operation, to approximate a Dirac delta function (for details, see [3] and [4]), no function in has these properties.

In his paper [2] O. N. Golovin introduced the notion of a neutral polyverbal operation on groups, of which Moran's verbal operations [4] and Gruenberg's and Šmelkin's operations [3,5] are special cases. The free and direct multiplication we shall call here trivial operations. It is known [5,4] that Mal'cev's postulate does not hold for nontrivial verbal operations but the postulate of associativity does.

In [3], Montgomery proved that if h is a pointwise periodic homeomorphism of a connected manifold without boundary onto itself, then h is periodic. Kaul generalized this result in [2] by showing that if X is a connected metrizable manifold without boundary and if (X, T)is a transformation group with T countable such that T is pointwise periodic, then T is periodic.

Moser and Abramson [4] proved: given m≥2 and 0≤kl, k2, …, kp<m, the number of p-combinations

satisfying

is

1

([x] denotes the greatest integer ≤x).