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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.

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

David Cimasoni (2012)

Journal of the European Mathematical Society

Let ∑ be a flat surface of genus g with cone type singularities. Given a bipartite graph Γ isoradially embedded in ∑ , we define discrete analogs of the 2 2 g Dirac operators on ∑ . These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair Γ ⊂ ∑ for these discrete Dirac operators to be Kasteleyn matrices of the graph Γ . As a consequence, if these conditions are met, the partition function of the dimer...

Distribution of zeros and shared values of difference operators

Jilong Zhang, Zongsheng Gao, Sheng Li (2011)

Annales Polonici Mathematici

We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that Δ c k f ≠ 0 for n ≥ 2, and a is a small function with respect to f, then f ⁿ Δ c k f equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

Équations aux différences associées à des groupes, fonctions représentatives.

Nicolas Marteau (2004)

Annales de l’institut Fourier

Inspiré par un travail de J.-P. Bézivin et F. Gramain sur les systèmes d’équations aux différences, on caractérise les sous-groupes H d’un groupe de Lie réel (resp. complexe) G , pour lesquels toute fonction f : G → ℂ continue (resp. entière) telle que l’ensemble des H -translatées engendrent un ℂ -espace vectoriel de dimension finie, engendrent aussi un ℂ -espace vectoriel de dimension finie par G - translation. On fait le lien avec les systèmes d’équations aux différences à coefficients constants.

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