Displaying similar documents to “Curves in Banach spaces which allow a C 1 , BV parametrization or a parametrization with finite convexity”

On vector functions of bounded convexity

Libor Veselý, Luděk Zajíček (2008)

Mathematica Bohemica

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Let X be a normed linear space. We investigate properties of vector functions F : [ a , b ] → X of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity K a b F is equal to the variation of F + ' on [ a , b ) . As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.

A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

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P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ℝ n propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for n = 2 , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) ψ ( x ) = ( x , y 1 ( x ) - y 2 ( x ) ) , x ∈ [ 0 , α ] , where y 1 , y 2 are convex and Lipschitz on [ 0 , α ] . In other words: singularities propagate along arcs with finite turn. ...

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

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Let [ a , b ] be an interval in ℝ and let F be a real valued function defined at the endpoints of [ a , b ] and with a certain number of discontinuities within [ a , b ] . Assuming F to be differentiable on a set [ a , b ] ∖ E to the derivative f , where E is a subset of [ a , b ] at whose points F can take values ± ∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that 𝒦ℋ -vt ∫ a b f = F ( b ) - F ( a ) , where 𝒦ℋ -vt denotes the total value of the integral. The paper ends with a few examples that illustrate the...