From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2031 Path: news.gmane.org!not-for-mail From: grandis@dima.unige.it (Marco Grandis) Newsgroups: gmane.science.mathematics.categories Subject: preprint: "Ordinary and directed combinatorial homotopy,..." Date: Fri, 28 Sep 2001 16:11:22 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241018296 1814 80.91.229.2 (29 Apr 2009 15:18:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:18:16 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist@mta.ca Mon Oct 01 18:40:03 2001 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Oct 2001 18:40:03 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 3.33 #2) id 15oAWY-0005jV-00 for categories-list@mta.ca; Mon, 01 Oct 2001 18:22:50 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 63 Xref: news.gmane.org gmane.science.mathematics.categories:2031 Archived-At: The following paper is available in ps and ps.gz. It is about my talk at: 'Conference on Algebraic Topological Methods in Computer Science' Stanford, July 30 - August 3, 2001. 'Ordinary and directed combinatorial homotopy, applied to image analysis and concurrency' Marco Grandis Abstract. Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes. An image, represented by a metric space X, can be explored at a variable resolution e > 0, by equipping it with a structure t_eX of simplicial complex depending on e; this complex can be further analysed by homotopy groups \pi^e_n(X) = \pi_n(t_eX) and homology groups H^e_n(X) = H_n(t_eX). Loosely speaking, these objects detect singularities which can be captured by an n-dimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. Similarly, a directed image, represented by an 'asymmetric metric space', produces a family of directed simplicial complexes f_eX and can be explored by the fundamental n-category of the latter. The same directed tools can be applied to mathematical models of concurrent automata, like Chu-spaces. AVAILABLE AT: ftp://www.dima.unige.it/Home/grandis/public/Cmb.App.ps ftp://www.dima.unige.it/Home/grandis/public/Cmb.App.ps.gz (18p, 260K) ***** With best regards Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/ ftp://www.dima.unige.it/Home/grandis/public/ 1-Oct-2001 18:40:07 -0300,1032;000000000000-0000002b