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Fractal negations.

Gaspar Mayor Forteza, Tomasa Calvo Sánchez (1994)

Mathware and Soft Computing

From the concept of attractor of a family of contractive affine transformations in the Euclidean plane R2, we study the fractality property of the De Rham function and other singular functions wich derive from it. In particular, we show as fractals the strong negations called k-negations.

Investigation of smooth functions and analytic sets using fractal dimensions

Emma D'Aniello (2004)

Bollettino dell'Unione Matematica Italiana

We start from the following problem: given a function f : 0 , 1 0 , 1 what can be said about the set of points in the range where level sets are «big» according to an opportune definition. This yields the necessity of an analysis of the structure of level sets of C n functions. We investigate the analogous problem for C n , a functions. These are in a certain way intermediate between C n and C n + 1 functions. The results involve a mixture of Real Analysis, Geometric Measure Theory and Classical Descriptive Set Theory.

Linear distortion of Hausdorff dimension and Cantor's function.

Oleksiy Dovgoshey, Vladimir Ryazanov, Olli Martio, Matti Vuorinen (2006)

Collectanea Mathematica

Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for...

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 the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that g ' ( x ) = 2 x for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

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