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S h o r t C o m m u n i c a t i o n s

This section consists of self-contained 25-100 line short communications which are prepublications of results, the details of which are to be published in either aequationes mathematicae or other journals with referee systems or equivalent systems of pre-reviewing papers. Aequationes mathematicae will endeavour to publish these short communications in the shortest possible time after the underlying papers have been accepted for publication. Unless indicated otherwise details of results described in this section will appear in subsequent issues of aequationes mathematicae.

On the Functional Equation F ( u ) = F(I - u) + F ( u / 1 - u), u ~ [0, ½] *)

DONALD B. SMALL

In this paper the author studies the functional equation

F ( u ) = F ( 1 - u) + F (1)

for functions defined over the closed unit interval, [0, 1]. Equation (1) arises in connection with the measure preserving transformation

a for 0~<a~<½ T ( a ) = 1 - a

l - a _ for ½ < a ~ < l . a

Let/~ be a Borel measure on [0, 1] and l e t f b e its cumulative distribution function.

THEOREM 1. The transformation T preserves the Borel measure # if and only if f (x) = f (x/(1 + x)) + f ( 1 ) - f (1/(1 + x)).

L e t f satisfy the functional equation f ( x ) = f ( x / ( 1 + x ) ) + f ( l ) - f ( 1 / ( 1 +x)) for 0~<x<~l. If F:I-0, 1] ~ [ 0 , 1] is defined by the equa t ion f ( x )=V(x )+f ( l ) and u is substituted for x/(1 +x) , equation (1) is obtained.

The set of solutions of (1) is not empty as shown by

THEOREM 2. l f h is any function defined on [-0, ½], there exists a unique extension F ofh to [0, 1] which satisfies (1).

In the last section of this paper, (1) is treated in a formal manner. The transforma-

*) Received July 25, 1969.

116 Short Communications AEQ, MATH.

tion u= 1/(1 + e-t0) employed in (I) yields the functional equation

( 1 ) ( 1 ) F (e ' ° ) = F 1+7:76 - F ~ e i 0 , 0 ~ < 0 < i o o . (2)

It is assumed that (2) has a solution F for which g(O)%fF(e i°) can be expanded in a

Fourier sine series of the form G(0)= ~ a k sink0. It is shown that the condition k = l

which (2) imposes on the t th coefficient, at, is that

a,= ~ a k [ ( k t ) + ( - - 1 ) ' + * ( k + t - 1 ) ] k=, k - 1 "

It is further shown that these coefficients obey a 'Pascal Triangle' type relationship based on the following theorem.

THEOREM 3. / f

a,= ~ ak [ ( ~ ) + ( - 1 ) ' + * ( k + t - 1 ) l k=a \ k - 1 '

then

j = O

On Maps x-+ x" and the Isotopy-Isomorphy Property of Moufang Loops*)

ORIN CHAIN and HALA O. PFLUGFELDER

A loop L which satisfies the identical relation

(xy) ( ~ ) = (x(yz)) x ~,

where x a is the image of x under some single valued map 2 of L into itself, is called an M-loop. While this seems to be a generalization of the concept of a Moufang loop in actuality the class of M-loops and that of Moufang loops are the same. Nevertheless, considering this class from the point of view of M-loops enables one to shed some light on the question of when a Moufang loop is isomorphic to all its loop isotopes.

I f 2 is a power mapping of the form 2:x-+ x" then the identical relation listed above is called the M.-law. If, in a loop L, the M.-law holds for some integer n> 1, and if k is the smallest such integer, then L is called an Mk loop. If no such law holds, but Lis Moufang, L is called strictly Moufang.

It is shown that if k ~ 1 (Mod 3) then every Mk-loop is isomorphic to each of its

*)Received February 9, 1970.