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Dieudonné-type theorems for lattice group-valued k -triangular set functions

Antonio Boccuto, Xenofon Dimitriou (2019)

Kybernetika

Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for k -triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems.

Difference functions of periodic measurable functions

Tamás Keleti (1998)

Fundamenta Mathematicae

We investigate some problems of the following type: For which sets H is it true that if f is in a given class ℱ of periodic functions and the difference functions Δ h f ( x ) = f ( x + h ) - f ( x ) are in a given smaller class G for every h ∈ H then f itself must be in G? Denoting the class of counter-example sets by ℌ(ℱ,G), that is, ℌ ( ℱ , G ) = H ⊂ ℝ / ℤ : ( ∃ f ∈ ℱ G ) ( ∀ h ∈ H ) Δ h f ∈ G , we try to characterize ℌ(ℱ,G) for some interesting classes of functions ℱ ⊃ G. We study classes of measurable functions on the circle group 𝕋 = ℝ / ℤ that are invariant for changes on null-sets (e.g. measurable...

Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem

Bruno Franchi, Marco Marchi, Raul Paolo Serapioni (2014)

Analysis and Geometry in Metric Spaces

A Carnot group G is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. Here ‘natural’ is meant to stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensional intrinsic Lipschitz graphs are sets with locally finite G-perimeter....

Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

Julian Ławrynowicz, Tatsuro Ogata, Osamu Suzuki (2009)

Banach Center Publications

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III)....

Differential inclusions and multivalued integrals

Kinga Cichoń, Mieczysław Cichoń, Bianca Satco (2013)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented...

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