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On - associated comonotone functions

Ondrej Hutník, Jozef Pócs (2018)

Kybernetika

We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to + -associatedness of functions (as proved by Boczek and Kaluszka), but also to their -associatedness with being an arbitrary strictly...

On belated differentiation and a characterization of Henstock-Kurzweil-Ito integrable processes

Tin-Lam Toh, Tuan-Seng Chew (2005)

Mathematica Bohemica

The Henstock-Kurzweil approach, also known as the generalized Riemann approach, has been successful in giving an alternative definition to the classical Itô integral. The Riemann approach is well-known for its directness in defining integrals. In this note we will prove the Fundamental Theorem for the Henstock-Kurzweil-Itô integral, thereby providing a characterization of Henstock-Kurzweil-Itô integrable stochastic processes in terms of their primitive processes.

On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

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 also extended...

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