By Marston Morse

ISBN-10: 0691080771

ISBN-13: 9780691080772

This publication builds upon the progressive discovery made in 1974 that after one passes from functionality f to a functionality J of paths becoming a member of issues A1?A1 the connectivities R1 of the area of f might be changed through connectivities R1 over Q, universal to the pathwise elements of a uncomplicated Frechet area of periods of an identical curves becoming a member of A1 to A1. The connectivities R1, termed "Frechet numbers," are proved self reliant of the alternative of A1 ? A1, and of a alternative of Mn through any differential manifold homeomorphic to Mn.

Originally released in 1976.

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N. 19), evaluated when o- 0 = (e, 0, ••. , 0), is a matrix whose nondiagonal elements are zero and whose diagonal reduces to the n-tuple (O, W-preintegrand is nonsingular in the Conditions on preintegrands F. F l/ e, ••. , l/ e). Theorem 6. 2 follows. 1. Our major conditions on the preintegrands depend upon whether we are concerned with the "local" or "global" theory. Theorems, limited to the neighborhood of a pair (u, r) in the domain of F to the neighborhood of an extremal of J, or properties of an extremal g, or are regarded as local.

Restriction Condition. 5)' II. F(u,r) Equiva·lence Condition. BMn If and the associated W-preintegrands, then (3. 3 u f U for at u and v t V, whenever is contravariantly ¢(u) ~elated = W(v) and the vector to the vector a at r v. \i ($,V), etc. in The w-i:n:tegrals JF, JG' etc. will now be defined. By hypotheses preintegrands associated with the presentations respectively. ($ , V) (3. 6) in u, (3. 6) (¢,U) and G are and Corresponding to a piecewise regular curve, [a,b] .... U, t .... / F(u(t), u(t)) dt • ,,.

In The w-i:n:tegrals JF, JG' etc. will now be defined. By hypotheses preintegrands associated with the presentations respectively. ($ , V) (3. 6) in u, (3. 6) (¢,U) and G are and Corresponding to a piecewise regular curve, [a,b] .... U, t .... / F(u(t), u(t)) dt • ,,. a II Corresponding to a piecewise regular curve (3. 7) in V (3 • 7) y I t .... v(t) [a,b] .... lbG(v(t), v(t)) dt. II The following lemma is a consequence of the compatability condition (3. 5) ". 1. Let be equivalent presentations are respectively·associated.