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On Multi-Dimensional Random Walk Models Approximating Symmetric Space-Fractional Diffusion Processes

Umarov, Sabir, Gorenflo, Rudolf (2005)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 47B06, 47G30, 60G50, 60G52, 60G60.In this paper the multi-dimensional analog of the Gillis-Weiss random walk model is studied. The convergence of this random walk to a fractional diffusion process governed by a symmetric operator defined as a hypersingular integral or the inverse of the Riesz potential in the sense of distributions is proved.* Supported by German Academic Exchange Service (DAAD).

On non-absolutely convergent integrals

Štefan Schwabik (1996)

Mathematica Bohemica

The influence of Jan Marik in the field of non absolute integration is described in the plane of Czech mathematics. A short historical account on the development of integration theory in the Czech region is presented in this connection together with the recent Riemann sum approach to the general Perron integral.

On nowhere weakly symmetric functions and functions with two-element range

Krzysztof Ciesielski, Kandasamy Muthuvel, Andrzej Nowik (2001)

Fundamenta Mathematicae

A function f: ℝ → {0,1} is weakly symmetric (resp. weakly symmetrically continuous) at x ∈ ℝ provided there is a sequence hₙ → 0 such that f(x+hₙ) = f(x-hₙ) = f(x) (resp. f(x+hₙ) = f(x-hₙ)) for every n. We characterize the sets S(f) of all points at which f fails to be weakly symmetrically continuous and show that f must be weakly symmetric at some x ∈ ℝ∖S(f). In particular, there is no f: ℝ → {0,1} which is nowhere weakly symmetric. It is also shown that if at each point x we...

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz Natkaniec, Piotr Szuca (2010)

Colloquium Mathematicae

We prove that if f: → is Darboux and has a point of prime period different from 2 i , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

On principal iteration semigroups in the case of multiplier zero

Dorota Krassowska, Marek Zdun (2013)

Open Mathematics

We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero and establish connections among them. The common characteristic property of each definition is conjugating of an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas and satisfy either Böttcher’s or Schröder’s functional equation.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 , = φ: ℝ → ℝ|φ is continuous, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Rajkovic, Predrag, Marinkovic, Sladjana, Stankovic, Miomir (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of integration, we consider the fractional q–derivative of Caputo type. Especially, its applications to q-exponential functions allow us to introduce q–analogues of the Mittag–Leffler function. Vice versa, those functions can be used for defining generalized operators in fractional q–calculus.

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