TY - GEN

T1 - Elementary differential calculus on discrete and hybrid structures

AU - Blair, Howard A.

AU - Jakel, David W.

AU - Irwin, Robert J.

AU - Rivera, Angel

PY - 2007/10/29

Y1 - 2007/10/29

N2 - We set up differential calculi in the Cartesian-closed category CONV of convergence spaces. The central idea is to uniformly define the 3-place relation_is a differential of_at_for each pair of convergence spaces X, Y in the category, where the first and second arguments are elements of Hom(X, Y) and the third argument is an element of X, in such a way as to (1) obtain the chain rule, (2) have the relation be in agreement with standard definitions from real and complex analysis, and (3) depend only on the convergence structures native to the spaces X and Y. All topological spaces and all reflexive directed graphs (i.e. discrete structures) are included in CONV. Accordingly, ramified hybridizations of discrete and continuous spaces occur in CONV. Moreover, the convergence structure within each space local to each point, individually, can be discrete, continuous, or hybrid.

AB - We set up differential calculi in the Cartesian-closed category CONV of convergence spaces. The central idea is to uniformly define the 3-place relation_is a differential of_at_for each pair of convergence spaces X, Y in the category, where the first and second arguments are elements of Hom(X, Y) and the third argument is an element of X, in such a way as to (1) obtain the chain rule, (2) have the relation be in agreement with standard definitions from real and complex analysis, and (3) depend only on the convergence structures native to the spaces X and Y. All topological spaces and all reflexive directed graphs (i.e. discrete structures) are included in CONV. Accordingly, ramified hybridizations of discrete and continuous spaces occur in CONV. Moreover, the convergence structure within each space local to each point, individually, can be discrete, continuous, or hybrid.

KW - Convergence space

KW - Differential

KW - Discrete structure

KW - Hybrid structure

UR - http://www.scopus.com/inward/record.url?scp=35448932350&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35448932350&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:35448932350

SN - 3540727329

SN - 9783540727323

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 41

EP - 53

BT - Logical Foundations of Computer Science - International Symposium, LFCS 2007, Proceedings

T2 - International Symposium on Logical Foundations of Computer Science, LFCS 2007

Y2 - 4 June 2007 through 7 June 2007

ER -