8,939 research outputs found
Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory
No full textBishop\u27s measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space , and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of -integrable functions , rely on quantification over proper classes (from the constructive point of view). In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions , as the completion of an integration space, is included
A Univalent Formalization of Constructive Affine Schemes
No full textWe present a formalization of constructive affine schemes in the Cubical Agda proof assistant. This development is not only fully constructive and predicative, it also makes crucial use of univalence. By now schemes have been formalized in various proof assistants. However, most existing formalizations follow the inherently non-constructive approach of Hartshorne\u27s classic Algebraic Geometry textbook, for which the construction of the so-called structure sheaf is rather straightforwardly formalizable and works the same with or without univalence. We follow an alternative approach that uses a point-free description of the constructive counterpart of the Zariski spectrum called the Zariski lattice and proceeds by defining the structure sheaf on formal basic opens and then lift it to the whole lattice. This general strategy is used in a plethora of textbooks, but formalizing it has proved tricky. The main result of this paper is that with the help of the univalence principle we can make this lift from basis strategy formal and obtain a fully formalized account of constructive affine schemes.22 pages; title changed, introduction and conclusion restructured, typos corrected, references adde
The Functor of Points Approach to Schemes in Cubical Agda
No full textWe present a formalization of quasi-compact and quasi-separated schemes (qcqs-schemes) in the Cubical Agda proof assistant. We follow Grothendieck’s functor of points approach, which defines schemes, the quintessential notion of modern algebraic geometry, as certain well-behaved functors from commutative rings to sets. This approach is often regarded as conceptually simpler than the standard approach of defining schemes as locally ringed spaces, but to our knowledge it has not yet been adopted in formalizations of algebraic geometry. We build upon a previous formalization of the so-called Zariski lattice associated to a commutative ring in order to define the notion of compact open subfunctor. This allows for a concise definition of qcqs-schemes, streamlining the usual presentation as e.g. given in the standard textbook of Demazure and Gabriel. It also lets us obtain a fully constructive proof that compact open subfunctors of affine schemes are qcqs-schemes
Cs10Ta29.27O78
No full textSingle crystals of caesium tantalate(V), Cs10Ta29.27O78, were obtained as a serendipitous product in a welded tantalum ampoule by a blank reaction of CsBr and bismuth subnitrate [Bi5O(OH)9(NO3)4] with the container material. The crystal structure of the title compound is made up of a three-dimensional framework constituted by two types of layers, viz. (Ta6O15)n and (Ta3O9)n, parallel to (001), which are linked together by TaO6 octahedra (3m. symmetry) along [001]. This framework has cavities where three independent Cs+ ions (3m. and 6m2 symmetry, respectively) are located. The compound reveals a Ta deficiency at one trigonal prismatic coordinated site (6m2 symmetry). The composition of the title compound was verified by energy-dispersive X-ray analysis of single crystals
Univalent Foundations of Constructive Algebraic Geometry
No full textWe investigate two constructive approaches to defining quasi-compactand quasi-separated schemes (qcqs-schemes). First, we introduce qcqs-schemes as locally ringed lattices, refining an approach of Coquand, Lombardi and Schuster using ringed lattices. We then consider qcqs-schemes as functors from rings to sets, building on a formalized definition by Zeuner and Hutzler. We work in Homotopy TypeTheory and Univalent Foundations, but reason informally. The main result is a constructive and univalent proof that the two definitions coincide, giving an equivalence between the respective categories of qcqs-schemes.arXiv version available at: https://arxiv.org/abs/2407.17362</p
Univalent Foundations of Constructive Algebraic Geometry
No full textWe investigate two constructive approaches to defining quasi-compactand quasi-separated schemes (qcqs-schemes). First, we introduce qcqs-schemes as locally ringed lattices, refining an approach of Coquand, Lombardi and Schuster using ringed lattices. We then consider qcqs-schemes as functors from rings to sets, building on a formalized definition by Zeuner and Hutzler. We work in Homotopy TypeTheory and Univalent Foundations, but reason informally. The main result is a constructive and univalent proof that the two definitions coincide, giving an equivalence between the respective categories of qcqs-schemes.arXiv version available at: https://arxiv.org/abs/2407.17362</p
Univalent Foundations of Constructive Algebraic Geometry [Elektronisk resurs]
No full textWe investigate two constructive approaches to defining quasi-compactand quasi-separated schemes (qcqs-schemes). First, we introduce qcqs-schemes as locally ringed lattices, refining an approach of Coquand, Lombardi and Schuster using ringed lattices. We then consider qcqs-schemes as functors from rings to sets, building on a formalized definition by Zeuner and Hutzler. We work in Homotopy TypeTheory and Univalent Foundations, but reason informally. The main result is a constructive and univalent proof that the two definitions coincide, giving an equivalence between the respective categories of qcqs-schemes.</p
Enseñanza de la escritura de Max Aub: comprensión y memoria
No full textEste texto analiza a obra testimonial de Max Aub sobre su experiencia en los campos de concentración en Francia desde una perspectiva de discursos comparados. Para destacar las estrategias de la escritura del autor recuperables por otros proyectos discursivos que persigan la sensibilización y la denuncia a través del cruce entre la comunicación y la éticaThis text analyses the testimonial work of Max Aub about his experience in the French concentration camps in France from comparative discourses approach. It emphasizes the writing strategies used by the author useful for other awareness and denounce discourses through the dialogue among communication and ethic
Max Brooks literary reading flier
No full text2012 Bismarck State College Visiting Writers Series and ArtsQuest present: Max Brooks. April 25, 7:30 p.m.; Belle Mehus Auditorium. Max Brooks is the author of World War Z: An Oral History of the Zombie War and the graphic novel The Zombie Survival Guide: Recorded Attacks
Max Frisch's novel: Stiller. A study
No full text The attempt is made in the following study to present an interpretation of the novel "Stiller" by the Swiss author, Max Frisch, by tracing through the novel the dominant themes of the graven-image or 'Bildnis' and that of the problem of freedom with reference to the novel's main character. ThesisMaster of Arts (MA
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