By Orlicz L.

The current quantity of Commentationes Mathematicae is devoted to its Editor, Professor Wladyslaw Orlicz, at the celebration of his 75-th birthday.Professor Wladyslaw Orlicz, an exceptional mathematician, was once as a member of the recognized Lwow university one of the founders of contemporary practical research. The articles accumulated during this quantity, a lot of them by way of Professor Orlicz's former scholars, have been encouraged via the information raised within the paintings of Professor Orlicz. it really is, we think, the simplest image of recognize that his acquaintances all over the world carry for Professor Orlicz.

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Proof. 5), the identity + dyf(M(x)) dX'W{x') ο ldxf(u(x)) 0 ο dxu(x)-] 0 = Sxg(u(x)) + dyg(u(x)) ο ôxu(x) ο ο is obtained, where χ' = f (u(x)). Because of the obvious relation ο 3,/(«(*)) + 5 , / ( « ( * ) ) 0 ο ÔMx) = W) 0 ο 1 u(x) 1 and one similar to it for the mapping g, the identity can be rewritten as (dx,uXx'))°(DfoU(x)) 1 On = Dgou(x). 4) is rewritten as ν ° Df = Dg and its 1 composition is taken with u(x\ there 1 1 results (ι; ο u(x)) ο (Df ο u(x)) = 1 1 1 1 1 Dg ο u(x). 6) for k = 1. 6) is established, for general /c, by induction.

This fact will be studied completely in Chapter V. ( b ) Let ξ · d = xdx + ydy be an operator of dilatation and φ = χ — y. 1) is satisfied, although φ is not an 1 invariant of the group G ^ ) . The universal invariant of this group is J = y/x and with its aid the manifold φ = 0 is defined by the equation J = 1 everywhere, except at the special point χ = y = 0. 2 (c) Let ξ · d = dy be an operator of translation and φ = y . 1) holds. ;), 33 THE C O N T I N U A T I O N THEORY §4. which are x' = x, y' = y + a.