Showing papers on "Geometry and topology published in 1971"

Computer Determination of the Geometry and Topology of Stream Networks

Abstract: Previously proposed binary string procedures for describing stream network topology do not greatly assist geomorphic and engineering studies that require network geometry information. A variation of these procedures is described that can determine the network geometry as well as its topology. Two years' experience has shown that this extension of the, binary string methods, which includes geometric properties, is an efficient and practical method of studying real stream networks.

On Two Classical Theorems of Algebraic Topology

Abstract: For theorems which have stimulated so much further research, beginning with the work of H. Hopf and continuing to the present, and whose content is so clear and easy to state, they are surprisingly difficult to prove, even in the simplest cases-the unit disk and the ordinary 2-sphere. Proofs are customarily given in standard courses in algebraic topology, but only after a fairly extensive theory is developed. In a brief, but very readable and elegant book [3 ], J. Milnor gave relatively simple proofs, based in part on a very original approach due to M. Hirsch [4]. Since this book goes into many generalizations of these theorems, it introduces and uses some techniques of differential topology which we wish to avoid, for example Sard's theorem. In this paper it is our purpose to make use of the fact that several advanced calculus texts discuss both manifolds and exterior differential forms in the presentation of multiple integrals and Stokes' theorem, and thus to rewrite the standard classical proofs using these techniques to avoid overt use of algebraic topology. In fact the present treatment goes very little beyond material to be found in the texts of either Fleming [6] or Devinatz [7] or in the more specialized book of Flanders [5], and the proofs of these theorems become, then, a somewhat delicate exercise in advanced calculus.

Geometry and extension and representation theorems for ordered linear spaces

Abstract: .................................................... V