Displaying similar documents to “On the relation between the Borel sum and the classical solution of the Cauchy problem for certain partial differential equations”

On Borel summable solutions of the multidimensional heat equation

Sławomir Michalik (2012)

Annales Polonici Mathematici

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We give a new characterisation of Borel summability of formal power series solutions to the n-dimensional heat equation in terms of holomorphic properties of the integral means of the Cauchy data. We also derive the Borel sum for the summable formal solutions.

Coordinatewise decomposition of group-valued Borel functions

Benjamin D. Miller (2007)

Fundamenta Mathematicae

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Answering a question of Kłopotowski, Nadkarni, Sarbadhikari, and Srivastava, we characterize the Borel sets S ⊆ X × Y with the property that every Borel function f: S → ℂ is of the form f(x,y) = u(x) + v(y), where u: X → ℂ and v: Y → ℂ are Borel.

Normal numbers and the Borel hierarchy

Verónica Becher, Pablo Ariel Heiber, Theodore A. Slaman (2014)

Fundamenta Mathematicae

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We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.

Borel-Wadge degrees

Alessandro Andretta, Donald A. Martin (2003)

Fundamenta Mathematicae

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Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.

Recent developments in the theory of Borel reducibility

Greg Hjorth, Alexander S. Kechris (2001)

Fundamenta Mathematicae

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Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible...

Turning Borel sets into clopen sets effectively

Vassilios Gregoriades (2012)

Fundamenta Mathematicae

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We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be chosen in a hyperarithmetical way and using this we obtain some uniformity results.

Preservation of the Borel class under open-LC functions

Alexey Ostrovsky (2011)

Fundamenta Mathematicae

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Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.

A classification of ordinals up to Borel isomorphism

Su Gao, Steve Jackson, Vincent Kieftenbeld (2008)

Fundamenta Mathematicae

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We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.

Extensions of Borel Measurable Maps and Ranges of Borel Bimeasurable Maps

Petr Holický (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps f and show that some control on the relative classes of preimages and images of Borel sets under f enables one to get a bound on the absolute class of the range of f. This seems to be of some interest even within separable spaces. ...