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 As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.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...

Integrating factor

Jan Mařík (1994)

Mathematica Bohemica

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The problem of integrating factor for ordinary differential equations is investigated. Conditions are given which guarantee that each solution of ∂ 1 F ( x , y ) + y ' ∂ 2 F ( x , y ) = 0 is also a solution of M ( x , y ) + y ' N ( x , y ) = 0 where ∂ 1 F = μ M and ∂ 2 F = μ N .

Curves with finite turn

Jakub Duda (2008)

Czechoslovak Mathematical Journal

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In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence...