Pub Date : 2023-11-30DOI: 10.2989/16073606.2023.2276751
K. Dhifaoui, M. Mabrouk
Let be a complex Hilbert space and denote by the algebra of all linear bounded operators on . For any , denote by V0(T) its maximal numerical range. We prove that if is a surjective map such tha...
{"title":"Maps preserving maximal numerical range of operator products","authors":"K. Dhifaoui, M. Mabrouk","doi":"10.2989/16073606.2023.2276751","DOIUrl":"https://doi.org/10.2989/16073606.2023.2276751","url":null,"abstract":"Let be a complex Hilbert space and denote by the algebra of all linear bounded operators on . For any , denote by V0(T) its maximal numerical range. We prove that if is a surjective map such tha...","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"7 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536640","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 : 2023-11-30DOI: 10.2989/16073606.2023.2277841
Elgiz Bairamov, Kenan Tas, Ekin Uğurlu
In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduc...
本文研究了一类奇异左定哈密顿系统，并构造了该系统的特征矩阵理论。利用这一理论的结果，我们介绍……
{"title":"On the characteristic functions and Dirchlet-integrable solutions of singular left-definite Hamiltonian systems","authors":"Elgiz Bairamov, Kenan Tas, Ekin Uğurlu","doi":"10.2989/16073606.2023.2277841","DOIUrl":"https://doi.org/10.2989/16073606.2023.2277841","url":null,"abstract":"In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduc...","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"84 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536646","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}
By using canonical transformation technique, we convert the nonlinear weakly canonical fourth-order delay differential equations into strongly canonical equations. Then we apply oscillation results ...
利用正则变换技术，将非线性弱正则四阶时滞微分方程转化为强正则方程。然后我们应用振荡结果…
{"title":"Oscillation results for nonlinear weakly canonical fourth-order delay differential equations via canonical transform","authors":"Natarajan Prabaharan, Ethiraju Thandapani, Ercan Tunç","doi":"10.2989/16073606.2023.2281567","DOIUrl":"https://doi.org/10.2989/16073606.2023.2281567","url":null,"abstract":"By using canonical transformation technique, we convert the nonlinear weakly canonical fourth-order delay differential equations into strongly canonical equations. Then we apply oscillation results ...","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536648","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247726
A.V. Arhangel’skii
AbstractThis paper has features of mixed nature. It contains new results, in particular, on semitopological groups. We also pose some problems, introduce new definitions and describe in details certain techniques we need below, providing the proofs of not so well-known theorems for the sake of completeness. Hence, this article can be treated also as a kind of a short survey.Mathematics Subject Classification (2020): 54A2554D2054D4054E18Key words: Semitopological grouphomogeneous spacep-spaceLindelöf p-spaces-spaceremaindercountably compactσ-compactgroup number
{"title":"Some concepts of topology and questions in topological algebra","authors":"A.V. Arhangel’skii","doi":"10.2989/16073606.2023.2247726","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247726","url":null,"abstract":"AbstractThis paper has features of mixed nature. It contains new results, in particular, on semitopological groups. We also pose some problems, introduce new definitions and describe in details certain techniques we need below, providing the proofs of not so well-known theorems for the sake of completeness. Hence, this article can be treated also as a kind of a short survey.Mathematics Subject Classification (2020): 54A2554D2054D4054E18Key words: Semitopological grouphomogeneous spacep-spaceLindelöf p-spaces-spaceremaindercountably compactσ-compactgroup number","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"215 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325936","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247730
Hans-E. Porst
AbstractThe Feistel-Toffoli construction of a bijective Boolean function out of an arbitrary one, a fundamental tool in reversible computing and in cryptography, has recently been analyzed (see [12]) to be a special instance of the construction of a monoid homomorphism from the X -fold cartesian power of a monoid M into the endomorphism monoid of the free M -set over the set X . It is the purpose of this note to show that this construction itself is in fact a genuine monoidal one. The generalization of the Feistel-Toffoli construction to internal categories in arbitrary finitely complete categories of [12] then becomes a special instance of this monoidal description.Mathematics Subject Classification (2020): 18M0518D4068Q09Key words: Convolution monoids and Hopf monoids in monoidal categoriesinternal categoryKleisli categoryspansFeistel schemeToffoli gate
{"title":"The monoidal nature of the Feistel-Toffoli construction","authors":"Hans-E. Porst","doi":"10.2989/16073606.2023.2247730","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247730","url":null,"abstract":"AbstractThe Feistel-Toffoli construction of a bijective Boolean function out of an arbitrary one, a fundamental tool in reversible computing and in cryptography, has recently been analyzed (see [12]) to be a special instance of the construction of a monoid homomorphism from the X -fold cartesian power of a monoid M into the endomorphism monoid of the free M -set over the set X . It is the purpose of this note to show that this construction itself is in fact a genuine monoidal one. The generalization of the Feistel-Toffoli construction to internal categories in arbitrary finitely complete categories of [12] then becomes a special instance of this monoidal description.Mathematics Subject Classification (2020): 18M0518D4068Q09Key words: Convolution monoids and Hopf monoids in monoidal categoriesinternal categoryKleisli categoryspansFeistel schemeToffoli gate","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271916","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247739
David Holgate, Minani Iragi
AbstractIn a category with a proper ()-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous orders induced by pointed and co-pointed endofunctors. We also lift the topogenous orders along an -fibration. This permits one to obtain the lifting of interior and neighbourhood operators along an -fibration and includes the lifting of closure operators found in the literature. A number of examples presented at the end of the paper demonstrates our results.Mathematics Subject Classification (2020): 18A0518F6054A1554B30Key words: Closure operatorinterior operatorcategorical topogenous orderheredity(co)pointed endofunctors-fibrationsstrict, co-strictinitial and final morphisms
{"title":"Topogenous orders and related families of morphisms","authors":"David Holgate, Minani Iragi","doi":"10.2989/16073606.2023.2247739","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247739","url":null,"abstract":"AbstractIn a category with a proper ()-factorization system, we study the notions of strict, co-strict, initial and final morphisms with respect to a topogenous order. Besides showing that they allow simultaneous study of four classes of morphisms obtained separately with respect to closure, interior and neighbourhood operators, the initial and final morphisms lead us to the study of topogenous orders induced by pointed and co-pointed endofunctors. We also lift the topogenous orders along an -fibration. This permits one to obtain the lifting of interior and neighbourhood operators along an -fibration and includes the lifting of closure operators found in the literature. A number of examples presented at the end of the paper demonstrates our results.Mathematics Subject Classification (2020): 18A0518F6054A1554B30Key words: Closure operatorinterior operatorcategorical topogenous orderheredity(co)pointed endofunctors-fibrationsstrict, co-strictinitial and final morphisms","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"214 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325938","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}
AbstractLet p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces.Mathematics Subject Classification (2020): 54B3054E3554F45Key words: FunctordistancemonoidHausdorff distancefinite supportdimension
{"title":"The <i> ℓ <sup>p</sup> </i> -metrization of functors with finite supports","authors":"Taras Banakh, Viktoria Brydun, Lesia Karchevska, Mykhailo Zarichnyi","doi":"10.2989/16073606.2023.2247240","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247240","url":null,"abstract":"AbstractLet p ∈ [1, ∞] and F : Set → Set be a functor with finite supports in the category Set of sets. Given a non-empty metric space (X, dX), we introduce the distance on the functor-space FX as the largest distance such that for every n ∈ ℕ and a ∈ Fn the map Xn → FX, f → Ff(a), is non-expanding with respect to the ℓp-metric on Xn. We prove that the distance is a pseudometric if and only if the functor F preserves singletons; is a metric if F preserves singletons and one of the following conditions holds: (1) the metric space (X, dX) is Lipschitz disconnected, (2) p = 1, (3) the functor F has finite degree, (4) F preserves supports. We prove that for any Lipschitz map f : (X, dX) → (Y, dY) between metric spaces the map is Lipschitz with Lipschitz constant Lip(Ff) ≤ Lip(f). If the functor F is finitary, has finite degree (and preserves supports), then F preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula dim FpX ≤ deg(F) dim X. Using injective envelopes, we introduce a modification of the distance and prove that the functor Dist → Dist, , in the category Dist of distance spaces preserves Lipschitz maps and isometries between metric spaces.Mathematics Subject Classification (2020): 54B3054E3554F45Key words: FunctordistancemonoidHausdorff distancefinite supportdimension","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"45 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271716","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247729
Maria Manuel Clementino, Fernando Lucatelli Nunes
AbstractLet Ord be the category of (pre)ordered sets. Unlike Ord/X , whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category Ord//X . In this paper we show that the forgetful functor Ord//X → Ord is topological if and only if X is complete. Moreover, under suitable hypothesis, Ord//X is complete and cartesian closed if and only if X is. We end by analysing descent in this category. Namely, when X is complete, we show that, for a morphism in Ord//X , being pointwise effective for descent in Ord is sufficient, while being effective for descent in Ord is necessary, to be effective for descent in Ord//X .Mathematics Subject Classification (2020): 06A0718A2518A3018N1018D2018E50Key words: Effective descent morphismslax comma 2-categoriescomma categoriesexponentiabilitycartesian closed categoriestopological functorsenriched categoriesOrd-enriched categories
{"title":"Lax comma categories of ordered sets","authors":"Maria Manuel Clementino, Fernando Lucatelli Nunes","doi":"10.2989/16073606.2023.2247729","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247729","url":null,"abstract":"AbstractLet Ord be the category of (pre)ordered sets. Unlike Ord/X , whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category Ord//X . In this paper we show that the forgetful functor Ord//X → Ord is topological if and only if X is complete. Moreover, under suitable hypothesis, Ord//X is complete and cartesian closed if and only if X is. We end by analysing descent in this category. Namely, when X is complete, we show that, for a morphism in Ord//X , being pointwise effective for descent in Ord is sufficient, while being effective for descent in Ord is necessary, to be effective for descent in Ord//X .Mathematics Subject Classification (2020): 06A0718A2518A3018N1018D2018E50Key words: Effective descent morphismslax comma 2-categoriescomma categoriesexponentiabilitycartesian closed categoriestopological functorsenriched categoriesOrd-enriched categories","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"44 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271718","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247720
Jiří Adámek, Miroslav Hušek, Jiří Rosický, Walter Tholen
AbstractQuillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest?This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.Mathematics Subject Classification (2020): 18F6054B3054D10Key words: Finitely presentable objectfinitely generated objectfinitely small objectdirected colimitHausdorff spaceT0-spaceT1-spacecompact space
{"title":"Smallness in topology","authors":"Jiří Adámek, Miroslav Hušek, Jiří Rosický, Walter Tholen","doi":"10.2989/16073606.2023.2247720","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247720","url":null,"abstract":"AbstractQuillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest?This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.Mathematics Subject Classification (2020): 18F6054B3054D10Key words: Finitely presentable objectfinitely generated objectfinitely small objectdirected colimitHausdorff spaceT0-spaceT1-spacecompact space","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"50 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135271912","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 : 2023-11-01DOI: 10.2989/16073606.2023.2247795
David Holgate
{"title":"Horst Herrlich – A deep and enduring contribution to South African mathematics","authors":"David Holgate","doi":"10.2989/16073606.2023.2247795","DOIUrl":"https://doi.org/10.2989/16073606.2023.2247795","url":null,"abstract":"","PeriodicalId":49652,"journal":{"name":"Quaestiones Mathematicae","volume":"60 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135272068","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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