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Displaying similar documents to “On generalized “ham sandwich” theorems”

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

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Let Λ n × m and k be a positive integer. Let f : n m be a locally bounded map such that for each ( ξ , η ) Λ , the derivatives D ξ j f ( x ) : = d j d t j f ( x + t ξ ) | t = 0 , j = 1 , 2 , k , exist and are continuous. In order to conclude that any such map f is necessarily of class C k it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ ( ξ , η ) which is linear in η = ( η 1 , η 2 , , η m ) and homogeneous of degree k in ξ = ( ξ 1 , ξ 2 , , ξ n ) . This generalizes a result of J. Boman for the case k = 1 . The statement and the proof of a theorem of Boman for the case k = is...

Hausdorff dimension of the maximal run-length in dyadic expansion

Ruibiao Zou (2011)

Czechoslovak Mathematical Journal

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For any x [ 0 , 1 ) , let x = [ ϵ 1 , ϵ 2 , , ] be its dyadic expansion. Call r n ( x ) : = max { j 1 : ϵ i + 1 = = ϵ i + j = 1 , 0 i n - j } the n -th maximal run-length function of x . P. Erdös and A. Rényi showed that lim n r n ( x ) / log 2 n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than log 2 n , is quantified by their Hausdorff dimension.

Vector integral equations with discontinuous right-hand side

Filippo Cammaroto, Paolo Cubiotti (1999)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the integral equation u ( t ) = f ( I g ( t , z ) u ( z ) d z ) , with t I = [ 0 , 1 ] , f : 𝐑 n 𝐑 n and g : I × I [ 0 , + [ . We prove an existence theorem for solutions u L ( I , 𝐑 n ) where the function f is not assumed to be continuous, extending a result previously obtained for the case n = 1 .