We study properties of pseudo-Riemannian metrics corresponding to Monge-Amp`ere structures on four-dimensional $T^*M$. We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Pl"ucker embedding equation. We also focus on pullbacks of the pseudo-metrics on two-dimensional $M$ and describe the corresponding Hessian structures.
{"title":"Pseudo-Riemannian and Hessian geometry related to Monge-Ampère structures","authors":"Radek Such'anek, Stanislav Hronek","doi":"10.5817/AM2022-5-329","DOIUrl":"https://doi.org/10.5817/AM2022-5-329","url":null,"abstract":"We study properties of pseudo-Riemannian metrics corresponding to Monge-Amp`ere structures on four-dimensional $T^*M$. We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Pl\"ucker embedding equation. We also focus on pullbacks of the pseudo-metrics on two-dimensional $M$ and describe the corresponding Hessian structures.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"131 5 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81147534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Let M ( X, A ) ( M ∗ ( X, A )) be the f -ring of all (bounded) real-mea-surable functions on a T -measurable space ( X, A ), let M K ( X, A ) be the family of all f ∈ M ( X, A ) such that coz( f ) is compact, and let M ∞ ( X, A ) be all f ∈ M ( X, A ) that { x ∈ X : | f ( x ) | ≥ 1 n } is compact for any n ∈ N . We introduce realcompact subrings of M ( X, A ), we show that M ∗ ( X, A ) is a realcompact subring of M ( X, A ), and also M ( X, A ) is a realcompact if and only if ( X, A ) is a compact measurable space. For every nonzero real Riesz map ϕ : M ( X, A ) → R , we prove that there is an element x 0 ∈ X such that ϕ ( f ) = f ( x 0 ) for every f ∈ M ( X, A ) if ( X, A ) is a compact measurable space. We confirm that M ∞ ( X, A ) is equal to the intersection of all free maximal ideals of M ∗ ( X, A ), and also M K ( X, A ) is equal to the intersection of all free ideals of M ( X, A ) (or M ∗ ( X, A )). We show that M ∞ ( X, A ) and M K ( X, A ) do not have free ideal.
． 让M (X) (M∗(X))的f型环(界)real-mea-surable函数T可测空间(X),让K M (X)的家庭∈f M (X),因为(f)紧凑,让米∞(X)是所有∈f M (X)){∈X: f (X) | |≥1 n}紧凑n∈n。引入M (X, A)的实紧子空间，证明M * (X, A)是M (X, A)的实紧子空间，且M (X, A)是实紧的当且仅当(X, A)是紧可测空间。对于每一个非零实Riesz映射φ: M (X, A)→R，证明了如果(X, A)是紧可测空间，对于每一个f∈M (X, A)，存在一个元素X 0∈X使得φ (f) = f (X 0)。我们证实M∞(X, A)等于M * (X, A)的所有自由极大理想的交，并且M K (X, A)等于M (X, A)(或M * (X, A))的所有自由理想的交。证明了M∞(X, A)和mk (X, A)不存在自由理想。
{"title":"Some properties of algebras of real-valued measurable functions","authors":"A. Estaji, Ahmad Mahmoudi Darghadam","doi":"10.5817/am2023-5-383","DOIUrl":"https://doi.org/10.5817/am2023-5-383","url":null,"abstract":". Let M ( X, A ) ( M ∗ ( X, A )) be the f -ring of all (bounded) real-mea-surable functions on a T -measurable space ( X, A ), let M K ( X, A ) be the family of all f ∈ M ( X, A ) such that coz( f ) is compact, and let M ∞ ( X, A ) be all f ∈ M ( X, A ) that { x ∈ X : | f ( x ) | ≥ 1 n } is compact for any n ∈ N . We introduce realcompact subrings of M ( X, A ), we show that M ∗ ( X, A ) is a realcompact subring of M ( X, A ), and also M ( X, A ) is a realcompact if and only if ( X, A ) is a compact measurable space. For every nonzero real Riesz map ϕ : M ( X, A ) → R , we prove that there is an element x 0 ∈ X such that ϕ ( f ) = f ( x 0 ) for every f ∈ M ( X, A ) if ( X, A ) is a compact measurable space. We confirm that M ∞ ( X, A ) is equal to the intersection of all free maximal ideals of M ∗ ( X, A ), and also M K ( X, A ) is equal to the intersection of all free ideals of M ( X, A ) (or M ∗ ( X, A )). We show that M ∞ ( X, A ) and M K ( X, A ) do not have free ideal.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"66 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88780151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].
{"title":"Equivalence of ill-posed dynamical systems","authors":"T. Suda","doi":"10.5817/am2023-1-133","DOIUrl":"https://doi.org/10.5817/am2023-1-133","url":null,"abstract":". The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"17 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80885356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove a Frankel type theorem for CR submanifolds of Sasakian manifolds, under suitable hypotheses on the index of the scalar Levi forms determined by normal directions. From this theorem we derive some topological information about CR submanifolds of Sasakian space forms.
{"title":"A Frankel type theorem for CR submanifolds of Sasakian manifolds","authors":"D. Di Pinto, A. Lotta","doi":"10.5817/am2023-5-369","DOIUrl":"https://doi.org/10.5817/am2023-5-369","url":null,"abstract":". We prove a Frankel type theorem for CR submanifolds of Sasakian manifolds, under suitable hypotheses on the index of the scalar Levi forms determined by normal directions. From this theorem we derive some topological information about CR submanifolds of Sasakian space forms.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"54 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84732770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus, due to significance of such materials in the design and construction of buildings, semi-heuristic computational models must be taken into consideration. These models generate mathematical problems, whose solvability is not transparent frequently, which limits the credibility of all results of ad hoc designed numerical simulations. In this short paper such phenomena are demonstrated on a simple model problem, covering both micro-and macro-cracking, with references to needful generalizations and more realistic computational settings.
{"title":"Numerical approaches to the modelling of quasi-brittle crack propagation","authors":"J. Vala","doi":"10.5817/am2023-3-295","DOIUrl":"https://doi.org/10.5817/am2023-3-295","url":null,"abstract":". Computational analysis of quasi-brittle fracture in cement-based and similar composites, supplied by various types of rod, fibre, etc. reinforcement, is crucial for the prediction of their load bearing ability and durability, but rather difficult because of the risk of initiation of zones of microscopic defects, followed by formation and propagation of a large number of macroscopic cracks. A reasonable and complete deterministic description of relevant physical processes is rarely available. Thus, due to significance of such materials in the design and construction of buildings, semi-heuristic computational models must be taken into consideration. These models generate mathematical problems, whose solvability is not transparent frequently, which limits the credibility of all results of ad hoc designed numerical simulations. In this short paper such phenomena are demonstrated on a simple model problem, covering both micro-and macro-cracking, with references to needful generalizations and more realistic computational settings.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"12 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89822327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This paper is devoted to the investigation on the stability for two characteristic functions f 1 ( z ) = z 2 + pe − zτ + q and f 2 ( z ) = z 2 + pze − zτ + q , where p and q are real numbers and τ > 0. The obtained theorems describe the explicit stability dependence on the changing delay τ . Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.
{"title":"Delay-dependent stability conditions for fundamental characteristic functions","authors":"H. Matsunaga","doi":"10.5817/am2023-1-77","DOIUrl":"https://doi.org/10.5817/am2023-1-77","url":null,"abstract":". This paper is devoted to the investigation on the stability for two characteristic functions f 1 ( z ) = z 2 + pe − zτ + q and f 2 ( z ) = z 2 + pze − zτ + q , where p and q are real numbers and τ > 0. The obtained theorems describe the explicit stability dependence on the changing delay τ . Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"7 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84410249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This paper is concerned with blow-up of solutions to a two-species chemotaxis-competition model with production from only one species. In previous papers there are a lot of studies on boundedness for a two-species chemotaxis-competition model with productions from both two species. On the other hand, finite-time blow-up was recently obtained under smallness conditions for competitive effects. Now, in the biological view, the production term seems to promote blow-up phenomena; this implies that the lack of the production term makes the solution likely to be bounded. Thus, it is expected that there exists a solution of the system with single production such that the species which does not produce the chemical substance remains bounded, whereas the other species blows up. The purpose of this paper is to prove that this conjecture is true.
{"title":"Finite-time blow-up in a two-species chemotaxis-competition model with single production","authors":"M. Mizukami, Yuya Tanaka","doi":"10.5817/am2023-2-215","DOIUrl":"https://doi.org/10.5817/am2023-2-215","url":null,"abstract":". This paper is concerned with blow-up of solutions to a two-species chemotaxis-competition model with production from only one species. In previous papers there are a lot of studies on boundedness for a two-species chemotaxis-competition model with productions from both two species. On the other hand, finite-time blow-up was recently obtained under smallness conditions for competitive effects. Now, in the biological view, the production term seems to promote blow-up phenomena; this implies that the lack of the production term makes the solution likely to be bounded. Thus, it is expected that there exists a solution of the system with single production such that the species which does not produce the chemical substance remains bounded, whereas the other species blows up. The purpose of this paper is to prove that this conjecture is true.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"159 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81727322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Riccati matrix differential equation and the discrete order preserving property","authors":"Viera Štoudková Růžičková","doi":"10.5817/am2023-1-125","DOIUrl":"https://doi.org/10.5817/am2023-1-125","url":null,"abstract":"","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"PP 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84299192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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