We consider the existence of positive solutions of the nonlinear two point boundary value problem u′′ + λf(u) = 0, u(−1) = u(1) = 0, where f(u) = u(u − a)(u− b)(u− c)(1−u), 0 < a < b < c < 1, as the parameter λ varies through positive values. Every solution u(x) is an even function, and when it exists, it is uniquely identified by α = u(0). We study how the number of solutions changes when the parameter varies, i.e. we will be focusing on the locations of bifurcation points. The authors P. Korman, Y. Li and T. Ouyang ( ”Computing the location and the direction of bifurcation”, Mathematical Research Letters, accepted ), prove that a necessary and sufficient condition for α to be a bifurcation point is G(α) ≡ F (α) ∫ α 0 f(α)− f(τ) [F (α)− F (τ)]3/2 dτ − 2 = 0, where F (α) = ∫ α 0 f(u) du. We will prove that G(α) has vertical asymptotes at α = b, α = 1 and at any point α ∈ (0, 1) for which ∫ α 0 f(u) du = 0. We will use the asymptotic behavior of G to estimate intervals where G(α) 6= 0, that is, intervals where there is no bifurcation point.
考虑了非线性两点边值问题u ' + λf(u) = 0, u(−1)= u(1) = 0，其中f(u) = u(u−a)(u−b)(u−c)(1−u)， 0 < a < b < c < 1，且参数λ随正数值变化时正解的存在性。u(x)的每一个解都是偶函数，当它存在时，它被α = u(0)唯一标识。我们研究当参数变化时解的数目是如何变化的，即我们将关注分岔点的位置。作者P. Korman, Y. Li和T. Ouyang(“计算分岔的位置和方向”，《数学研究通讯》，已接受)证明了α是分岔点的一个充分必要条件是G(α)≡F (α)∫α 0 F (α)−F (τ) [F (α)−F (τ)]3/2 dτ−2 = 0，其中F (α) =∫α 0 F (u) du。我们将证明G(α)在α = b， α = 1以及在任意点α∈(0,1)且∫α 0 f(u) du = 0处具有垂直渐近线。我们将利用G的渐近性质来估计G(α) 6= 0的区间，即不存在分岔点的区间。
{"title":"Bifurcation theory for a class of second order differential equations","authors":"Alvaro Correa, Yi A. Li","doi":"10.17077/ETD.QMLJMX3C","DOIUrl":"https://doi.org/10.17077/ETD.QMLJMX3C","url":null,"abstract":"We consider the existence of positive solutions of the nonlinear two point boundary value problem u′′ + λf(u) = 0, u(−1) = u(1) = 0, where f(u) = u(u − a)(u− b)(u− c)(1−u), 0 < a < b < c < 1, as the parameter λ varies through positive values. Every solution u(x) is an even function, and when it exists, it is uniquely identified by α = u(0). We study how the number of solutions changes when the parameter varies, i.e. we will be focusing on the locations of bifurcation points. The authors P. Korman, Y. Li and T. Ouyang ( ”Computing the location and the direction of bifurcation”, Mathematical Research Letters, accepted ), prove that a necessary and sufficient condition for α to be a bifurcation point is G(α) ≡ F (α) ∫ α 0 f(α)− f(τ) [F (α)− F (τ)]3/2 dτ − 2 = 0, where F (α) = ∫ α 0 f(u) du. We will prove that G(α) has vertical asymptotes at α = b, α = 1 and at any point α ∈ (0, 1) for which ∫ α 0 f(u) du = 0. We will use the asymptotic behavior of G to estimate intervals where G(α) 6= 0, that is, intervals where there is no bifurcation point.","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68075371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relative homological algebra in categories of representations of infinite quivers","authors":"S. E. Domínguez, Ozdemir Salahattin","doi":"10.5072/ZENODO.26602","DOIUrl":"https://doi.org/10.5072/ZENODO.26602","url":null,"abstract":"","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70788096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-01-01DOI: 10.1093/acprof:oso/9780198702498.003.0009
G. Johnson, B. Kim
This paper explores the differential (or derivational) calculus associated with the disentangled operators arising from Feynman's operational calculi (FOCi) for noncommuting operators. (We will use the continuous case of the approach to FOCi initiated by Jefferies and Johnson in 2000.) The central part of this paper deals with a first order infinitesimal calculus for a function of n noncommuting operators. Here the first derivatives (or differentials) are replaced by the first order derivational derivatives. The derivational derivatives of the first and higher order have been useful in, for example, operator algebras, noncommutative geometry and Maslov's discrete form of Feynman's operational calculus. In the last section of this paper we will develop some special cases of higher order expansions.
{"title":"Derivational derivatives and Feynman's operational calculi","authors":"G. Johnson, B. Kim","doi":"10.1093/acprof:oso/9780198702498.003.0009","DOIUrl":"https://doi.org/10.1093/acprof:oso/9780198702498.003.0009","url":null,"abstract":"This paper explores the differential (or derivational) calculus associated with the disentangled operators arising from Feynman's operational calculi (FOCi) for noncommuting operators. (We will use the continuous case of the approach to FOCi initiated by Jefferies and Johnson in 2000.) The central part of this paper deals with a first order infinitesimal calculus for a function of n noncommuting operators. Here the first derivatives (or differentials) are replaced by the first order derivational derivatives. The derivational derivatives of the first and higher order have been useful in, for example, operator algebras, noncommutative geometry and Maslov's discrete form of Feynman's operational calculus. In the last section of this paper we will develop some special cases of higher order expansions.","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60644922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-01-01DOI: 10.1007/978-3-319-90902-8_8
M. D. J. López, S. Macías
{"title":"Induced maps on n-fold hyperspaces","authors":"M. D. J. López, S. Macías","doi":"10.1007/978-3-319-90902-8_8","DOIUrl":"https://doi.org/10.1007/978-3-319-90902-8_8","url":null,"abstract":"","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51031289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Series of Positive Terms","authors":"K. Grosse-Erdmann, G. Bennett","doi":"10.4064/BC64-0-3","DOIUrl":"https://doi.org/10.4064/BC64-0-3","url":null,"abstract":"","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2004-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70694859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-01-28DOI: 10.31274/RTD-180813-13219
J. Peters, R. Zerr
We obtain a characterization in terms of dynamical systems of those r-discrete groupoids for which the groupoid C*-algebra is approximately finite-dimensional (AF). These ideas are then used to compute the K-theory for AF algebras by utilizing the actions of these partial homeomorphisms, and these K-theoretic calculations are applied to some specific examples of AF algebras. Finally, we show that, for a certain class of dimension groups, a groupoid can be obtained directly from the dimension group's structure whose associated C*-algebra has its dimension group isomorphic to the original dimension group.
{"title":"Partial Dynamical Systems and AF C*-algebras","authors":"J. Peters, R. Zerr","doi":"10.31274/RTD-180813-13219","DOIUrl":"https://doi.org/10.31274/RTD-180813-13219","url":null,"abstract":"We obtain a characterization in terms of dynamical systems of those r-discrete groupoids for which the groupoid C*-algebra is approximately finite-dimensional (AF). These ideas are then used to compute the K-theory for AF algebras by utilizing the actions of these partial homeomorphisms, and these K-theoretic calculations are applied to some specific examples of AF algebras. Finally, we show that, for a certain class of dimension groups, a groupoid can be obtained directly from the dimension group's structure whose associated C*-algebra has its dimension group isomorphic to the original dimension group.","PeriodicalId":50398,"journal":{"name":"Houston Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2003-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69350538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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