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Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which l i m i n f n → ∞ | f n ( x ) - f n ( y ) | = 0 and l i m s u p n → ∞ | f n ( x ) - f n ( y ) | > 0 . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

Determinant criteria of solvability

V. S. Kalnitsky (2013)

Banach Center Publications

In the paper [3] the determinant criterion of solvability for the Kuczma equation [4] was given. This criterion appeared in the natural way as barycenter of some mass system. It turned out that determinants do appear in many different situations as solvability criteria. The present article is aimed to review the mostly classical results in the theory of functional equations from this point of view. We begin with classical results of the linear functional equations and the determinant equations solved...

Développements asymptotiques q -Gevrey et séries G q -sommables

Changgui Zhang (1999)

Annales de l'institut Fourier

Nous donnons une version q -analogue de l’asymptotique Gevrey et de la sommabilité de Borel, dues respectivement à G. Watson et E. Borel et systématiquement développées depuis une quinzaine d’années par J.-P. Ramis, Y. Sibuya, etc. Le but de ces auteurs était l’étude des équations différentielles dans le champ complexe. De même notre but est l’étude des équations aux q -différences dans le champ complexe, dans la ligne de G.D. Birkhoff et W.J. Trjitzinsky.Plus précisément, nous introduisons une nouvelle...

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...

Difference of Function on Vector Space over F

Kenichi Arai, Ken Wakabayashi, Hiroyuki Okazaki (2014)

Formalized Mathematics

In [11], the definitions of forward difference, backward difference, and central difference as difference operations for functions on R were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems [3], [10]. Differential cryptanalysis [4] that undertakes a general purpose attack against...

Difference operators from interpolating moving least squares and their deviation from optimality

Thomas Sonar (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the classical Interpolating Moving Least Squares (IMLS) interpolant as defined by Lancaster and Šalkauskas [Math. Comp. 37 (1981) 141–158] and compute the first and second derivative of this interpolant at the nodes of a given grid with the help of a basic lemma on Shepard interpolants. We compare the difference formulae with those defining optimal finite difference methods and discuss their deviation from optimality.

Difference operators from interpolating moving least squares and their deviation from optimality

Thomas Sonar (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the classical Interpolating Moving Least Squares (IMLS) interpolant as defined by Lancaster and Šalkauskas [Math. Comp.37 (1981) 141–158] and compute the first and second derivative of this interpolant at the nodes of a given grid with the help of a basic lemma on Shepard interpolants. We compare the difference formulae with those defining optimal finite difference methods and discuss their deviation from optimality.

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