Computability, noncomputability and undecidability of maximal intervals of IVPs
Autor(a) principal: | |
---|---|
Data de Publicação: | 2009 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.1/1018 |
Resumo: | Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initial-value problem dx dt = f(t, x) x(t0) = x0, where E is an open subset of Rm+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. |
id |
RCAP_66d7d54fd3fb3e6ca45f1a659c66cbb6 |
---|---|
oai_identifier_str |
oai:sapientia.ualg.pt:10400.1/1018 |
network_acronym_str |
RCAP |
network_name_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository_id_str |
7160 |
spelling |
Computability, noncomputability and undecidability of maximal intervals of IVPsLet (α, β) ⊆ R denote the maximal interval of existence of solution for the initial-value problem dx dt = f(t, x) x(t0) = x0, where E is an open subset of Rm+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.SapientiaGraça, DanielZhong, NingBuescu, Jorge2012-04-13T08:45:30Z20092009-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.1/1018engAUT: DGR01772;info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-24T10:11:56Zoai:sapientia.ualg.pt:10400.1/1018Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:55:16.763988Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
title |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
spellingShingle |
Computability, noncomputability and undecidability of maximal intervals of IVPs Graça, Daniel |
title_short |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
title_full |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
title_fullStr |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
title_full_unstemmed |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
title_sort |
Computability, noncomputability and undecidability of maximal intervals of IVPs |
author |
Graça, Daniel |
author_facet |
Graça, Daniel Zhong, Ning Buescu, Jorge |
author_role |
author |
author2 |
Zhong, Ning Buescu, Jorge |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Sapientia |
dc.contributor.author.fl_str_mv |
Graça, Daniel Zhong, Ning Buescu, Jorge |
description |
Let (α, β) ⊆ R denote the maximal interval of existence of solution for the initial-value problem dx dt = f(t, x) x(t0) = x0, where E is an open subset of Rm+1, f is continuous in E and (t0, x0) ∈ E. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f and (t0, x0) whose maximal interval of existence (α, β) is noncomputable. The fact that f may be taken to be analytic shows that this is not a lack of regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that (α, β) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable. |
publishDate |
2009 |
dc.date.none.fl_str_mv |
2009 2009-01-01T00:00:00Z 2012-04-13T08:45:30Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10400.1/1018 |
url |
http://hdl.handle.net/10400.1/1018 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
AUT: DGR01772; |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
|
_version_ |
1799133154980659200 |