Elementary differential calculus on discrete and hybrid structures

Howard A. Blair, David W. Jakel, Robert J. Irwin, Angel Rivera

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationLogical Foundations of Computer Science - International Symposium, LFCS 2007, Proceedings
Pages41-53
Number of pages13
StatePublished - Oct 29 2007
EventInternational Symposium on Logical Foundations of Computer Science, LFCS 2007 - New York, NY, United States
Duration: Jun 4 2007Jun 7 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4514 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

OtherInternational Symposium on Logical Foundations of Computer Science, LFCS 2007
CountryUnited States
CityNew York, NY
Period6/4/076/7/07

Keywords

  • Convergence space
  • Differential
  • Discrete structure
  • Hybrid structure

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    Blair, H. A., Jakel, D. W., Irwin, R. J., & Rivera, A. (2007). Elementary differential calculus on discrete and hybrid structures. In Logical Foundations of Computer Science - International Symposium, LFCS 2007, Proceedings (pp. 41-53). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4514 LNCS).