Acknowledgment to the paper: “Another proof that a chain of non-empty -closed subspaces has a non-empty intersection’
J. Mioduszewski (1971)
Colloquium Mathematicae
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J. Mioduszewski (1971)
Colloquium Mathematicae
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Hermann König, Vitali Milman (2014)
Studia Mathematica
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Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ). Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form Tf = ⎧ , f’ ≥ 0, ⎨ ⎩ , f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable...
Serge Bouc (2015)
Journal of the European Mathematical Society
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Let be a prime number. This paper introduces the Roquette category of finite -groups, which is an additive tensor category containing all finite -groups among its objects. In , every finite -group admits a canonical direct summand , called the edge of . Moreover splits uniquely as a direct sum of edges of Roquette -groups, and the tensor structure of can be described in terms of such edges. The main motivation for considering this category is that the additive functors...
Justyna Kosakowska, Daniel Simson (2008)
Colloquium Mathematicae
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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor that restricts to a representation equivalence , where is a coradical square complete hereditary...
Tymoteusz Chojecki (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Suppose that is a stationary Markov chain and is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if converge in law to a normal random variable, as . For a stationary Markov chain with the spectral gap the theorem holds for all such that is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables for which the CLT holds...
Brian P. Shea, Galin L. Jones (2014)
Annales de l'I.H.P. Probabilités et statistiques
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We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let be a class of functions on the parameter space and consider estimating elements of under quadratic loss. If the formal Bayes estimator of every function in is admissible, then the prior is strongly admissible with respect to . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...
Christine Vespa (2008)
Fundamenta Mathematicae
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We continue the study of the category of functors , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in : and . As an application of these two decompositions, we give a complete description of the polynomial...
Vincent Lafforgue, Sergey Lysenko (2011)
Bulletin de la Société Mathématique de France
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We prove that the global geometric theta-lifting functor for the dual pair is compatible with the Whittaker functors, where is one of the pairs , or . That is, the composition of the theta-lifting functor from to with the Whittaker functor for is isomorphic to the Whittaker functor for .
Claire Amiot (2007)
Bulletin de la Société Mathématique de France
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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form where is a disjoint union of simply-laced Dynkin diagrams and a weakly admissible group of automorphisms of . Then we prove that for ‘most’ groups , the category is standard, ...
Hartmut von Trotha
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CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................
Marvin W. Grossman
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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...
Jacek Dębecki (2016)
Czechoslovak Mathematical Journal
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We give a classification of all linear natural operators transforming -vectors (i.e., skew-symmetric tensor fields of type ) on -dimensional manifolds to tensor fields of type on , where is a Weil bundle, under the condition that , and . The main result of the paper states that, roughly speaking, each linear natural operator lifting -vectors to tensor fields of type on is a sum of operators obtained by permuting the indices of the tensor products of linear natural...
Shiquan Ruan, Li Wang, Haicheng Zhang (2024)
Czechoslovak Mathematical Journal
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For any positive integer , let be a linearly oriented quiver of type with vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories and , where and are the two extriangulated categories corresponding to the representation category of and the morphism category of projective representations of , respectively. As a...
Jan Kurek, Włodzimierz M. Mikulski (2017)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the category of -dimensional manifolds and local diffeomorphisms and let be the tangent functor on . Let be the category of real vector spaces and linear maps and let be the category of -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors admitting -natural operators transforming classical linear connections on -dimensional manifolds into almost complex structures on .
Iain G. Gordon, Ivan Losev (2014)
Journal of the European Mathematical Society
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We study equivalences for category of the rational Cherednik algebras of type : a highest weight equivalence between and for and an action of on an explicit non-empty Zariski open set of parameters ; a derived equivalence between and whenever and have integral difference; a highest weight equivalence between and a parabolic category for the general linear group, under a non-rationality assumption on the parameter . As a consequence, we confirm special cases...
Mohamed Amouch, Hamza Lakrimi (2024)
Mathematica Bohemica
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Let be a Banach space, the algebra of bounded linear operators on and an admissible Banach ideal of . For , let and denote the left and right multiplication defined by and , respectively. In this paper, we study the transmission of some concepts related to recurrent operators between , and their elementary operators and . In particular, we give necessary and sufficient conditions for and to be sequentially recurrent. Furthermore, we prove that is recurrent...
Jiří M. Tomáš (2017)
Czechoslovak Mathematical Journal
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Let be an -dimensional manifold and a Weil algebra of height . We prove that any -covelocity , is determined by its values over arbitrary regular and under the first jet projection linearly independent elements of . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result without coordinate computations, which improves and generalizes the partial...
Saeed Nasseh, Sean Sather-Wagstaff (2015)
Czechoslovak Mathematical Journal
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We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring . Our focus is on homological properties of contracting endomorphisms of , e.g., the Frobenius endomorphism when contains a field of positive characteristic. For instance, in this case, when is -finite and is a semidualizing -complex, we prove that the following conditions are equivalent: (i) is a dualizing -complex; (ii) for some ; (iii) and is derived...
Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)
Colloquium Mathematicae
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For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....
Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)
Journal of the European Mathematical Society
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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class of morphisms in a locally presentable category of structures, the orthogonal class of objects...
M. Kuczma
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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. solutions in *................................................ 74. Change of variables..............................................................................
Jian He, Jing He, Panyue Zhou (2024)
Czechoslovak Mathematical Journal
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M. Herschend, Y. Liu, H. Nakaoka introduced -exangulated categories, which are a simultaneous generalization of -exact categories and -angulated categories. This paper consists of two results on -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an -exangulated category.
Wei-Feng Xuan, Wei-Xue Shi (2016)
Mathematica Bohemica
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A topological space has a rank 2-diagonal if there exists a diagonal sequence on of rank , that is, there is a countable family of open covers of such that for each , . We say that a space satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. We mainly prove that if is a DCCC normal space with a rank 2-diagonal, then the cardinality of is at most . Moreover, we prove that if is a first...