90,633 research outputs found
Swiss journal of geosciences (1888-2006) / Carte des gorges de l'Areuse
No full textpar Mce Borel et Aug. Dubois ; coloriée géologiquement par H. Schardt et Aug. Dubois, 1899-1901Accompagne une étude géologique des gorges de l'Areuse1 CARTE (1 f.) de 70 x 47 cm, pliée 24 x 12 cmBeilage zu: Bull. Soc. Neuch. Sc. nat., T. XXX, [1903]Exlibrisstempel: "Geolog. Institut von Polytechnikum und Universität in Zürich" Exemplar der ETH-BIBIn: Eclogae geologicae helvetiae. - Basel. - Vol. 7(1903), pl. 1
Arsenin -- Kunugui Theorem And Weak Forms Of Borel Bimeasurability
No full text. Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then M is also Luzin, that f \Gamma1 (y) is a K oe set for all, except for countably many, y 2 M , and that the Borel classes of the sets f(F ), F closed in L, are bounded by a countable cardinal. It gives a counterpart to the classical theorem of Arsenin and Kunugui that enables to state it in the form of an equivalence. As a particular case we get a theorem of Taimanov saying that the image of a Luzin space by a closed continuous mapping is a Luzin space. The method is based on a straightforward construction that gives a Hurewicz theorem and on the use of the Jankov -- von Neumann selection theorem. A classical theorem of Novikov says that a Borel measurable mapping f : L ! Y of a Borel subset L of a Polish space X to a Polish space Y is Borel bimeasurable (i.e. f(B) is Borel for every Borel subset B of L) if ..
F. Borel, Les Sanza
No full textKabana Kongo. F. Borel, Les Sanza. In: L'Homme, 1988, tome 28 n°108. Les Animaux : domestication et représentation. pp. 182-183
Absolute and non-absolute F-Borel spaces
No full textWe investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results.
Borel structures on the set of Borel mappings
No full textAbstractLet Y and Z be two Borel spaces. By B(Y,Z) we denote the set of all Borel maps of Y into Z. In Aumann (1961) [2] and Rao (1971) [10] the authors tried to generalize the results of R. Arens and J. Dugundji (see Arens and Dugundji (1951) [1]) for Borel spaces. Unfortunately as R.J. Aumann observed in Aumann (1961) [2], the results of Arens and Dugundji (1951) [1] are not true for Borel spaces, since for some of the simplest Borel spaces for example it is impossible to defined a Borel structure on the set B(Y,Z) such that the map e:B(Y,Z)×Y→Z with e(f,y)=f(y) for every f∈B(Y,Z) and y∈Y is Borel. Even if we consider the discrete structure on B(Y,Z), then e will not in general be Borel. It is for this reason that in Aumann (1961) [2] and Rao (1971) [10] the authors studied subsets F of B(Y,Z) and Borel structures on F such that the restriction of the map e on F×Y is Borel.In this paper we study the above problem and try to generalize the results presented in Arens and Dugundji (1951) [1] for Borel spaces. More precisely in Section 1 the paper preliminaries are given. In Sections 2 and 3 we give and study Borel A-splitting and A-admissible structures on B(Y,Z), where A is an arbitrary family of Borel spaces, and prove that there exists at most one Borel structure on B(Y,Z) which is both Borel splitting and admissible. When this structure exists, it coincides with the greatest Borel splitting structure, which always exists. We also present and study some special Borel structures on B(Y,Z). In Section 4 some remarks for Borel structures on B(Y,Z) are stated. In Section 5 we define and study some relations between the Borel structures of the set B(Y,Z) and the Borel structures of the set BZ(Y) consisting of all subsets f−1(B) of Y, where f∈B(Y,Z) and B is an element of the Borel structure of Z, concerning the notions of Borel A-splitting and Borel A-admissible Borel structures. Finally, some open questions for Borel structures on the set of Borel mappings are posed
The Classification of Hyperfinite Borel Equivalence Relations
No full textLet X be a standard Borel space and E a Borel equivalence relation on X. We call E hyperfinite if there is a Borel automorphism T of X such that xEy ⇔ ∃ n Є ℤ(T^nx = y).
For Borel equivalence relations E,F on X, Y resp. we write
E ⊑ F ⇔ 3 ƒ : X → Y(ƒ Borel, injective with E = ƒ^(-1)[F])
E ≈ F ⇔ E ⊑ F and F ⊑ E E ≅ F ⇔ ∃ ƒ :X → Y(ƒ a Borel isomorphism with E= ƒ^(-1)[F]) A Borel equivalence relation E on X is called smooth if there is a Borel map ƒ: X → Y (Y
some standard Borel space) with xEy ⇔ ƒ(x) = ƒ(y)
Representation theory, Borel cross-sections, and minimal measures
No full textLet E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:Cb(E) → L∞(μ) with Γ(f∘Π) = f for every f ∈ Cb(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.
Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L∞(μ) with Γ(f∘Π) = f for every f in Cb(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.Ph. D
Borel and Baire Sets in Bishop Spaces
No full textWe study the Borel sets Borel(F) and the Baire sets Baire(F) generated by a Bishop topology F on a set X. These are inductively defined sets of F-complemented subsets of X. Because of the constructive definition of Borel(F), and in contrast to classical topology, we show that Baire(F) = Borel(F). We define the uniform version of an F-complemented subset of X and we show the Urysohn lemma for them. We work within Bishop's system BISH * of informal constructive mathematics that includes inductive definitions with rules of countably many premises
On Borel summability and analytic functionals
No full textAbstract. We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function f is of exponential type if and only if the formal power series ∞ n=0 f (n) (0)z n is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon
On Borel summability and analytic functionals
No full textWe show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function is of exponential type if and only if the formal power series is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon
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