Displaying similar documents to “On the measurability of sets of pairs of straight lines in the simply isotropic space”

Mean value densities for temperatures

N. Suzuki, N. A. Watson (2003)

Colloquium Mathematicae

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A positive measurable function K on a domain D in n + 1 is called a mean value density for temperatures if u ( 0 , 0 ) = D K ( x , t ) u ( x , t ) d x d t for all temperatures u on D̅. We construct such a density for some domains. The existence of a bounded density and a density which is bounded away from zero on D is also discussed.

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

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Let H Z 2 ω . For n ≥ 2, we prove that if Selivanovski measurable functions from 2 ω to Z give as preimages of H all Σₙ¹ subsets of 2 ω , then so do continuous injections.

How many clouds cover the plane?

James H. Schmerl (2003)

Fundamenta Mathematicae

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The plane can be covered by n + 2 clouds iff 2 .

Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande (2006)

Colloquium Mathematicae

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A function f: ℝⁿ → ℝ satisfies the condition Q i ( x ) (resp. Q s ( x ) , Q o ( x ) ) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and | ( 1 / μ ( U I ) ) U I f ( t ) d t - f ( x ) | < r . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer k 2 and every set A of words over k satisfying lim sup n | A [ k ] n | / k n > 0 there exist a word c over k and a sequence ( w n ) of left variable words over k such that the set c { c w 0 ( a 0 ) . . . w n ( a n ) : n and a 0 , . . . , a n [ k ] } is contained in A . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into d . The paper deals with Y-weak cluster points ϕ̅ of the sequence ϕ ( · , z j ( · ) ) in X, where z j : Ω m is measurable for j ∈ ℕ and ϕ : Ω × m d is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set A ϕ , the integral I ( ϕ , ν x ) : = m ϕ ( x , λ ) d ν x ( λ ) exists for x Ω A ϕ and ϕ ̅ ( x ) = I ( ϕ , ν x ) on Ω A ϕ , where ν = ν x x Ω is a measurable-dependent family of Radon probability measures on m .

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

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Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have l i m k 1 / | R k | R k χ A = χ A ( x ) for any sequence R k of sets in containing x whose diameters tend to 0. The geometric maximal operator M associated to is defined on L¹(ℝⁿ) by M f ( x ) = s u p x R 1 / | R | R | f | . The halo function ϕ of is defined on (1,∞) by ϕ ( u ) = s u p 1 / | A | | x : M χ A ( x ) > 1 / u | : 0 < | A | < and on [0,1] by ϕ(u) = u. It is shown...

On C * -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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