A simple example of the continuous function without derivative

Takagi, Tatsuya

doi:10.1007/978-4-431-54995-6_3

Cited by 107 publications

(130 citation statements)

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“…Figure (7) shows a recursive computation of T q (y), for q = 1/3. This example is very similar to the Takagi function [13] T (y) = y + 1 2 T (2y), 0 ≤ y < 1 2 ,…”

Section: Stationary State Under Flux Boundary Conditionsmentioning

confidence: 74%

Entropy production in a persistent random walk

Gilbert

Dorfman

2000

Physica A: Statistical Mechanics and its Applications

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We consider a one-dimensional persisent random walk viewed as a deterministic process with a form of time reversal symmetry. Particle reservoirs placed at both ends of the system induce a density current which drives the system out of equilibrium. The phase space distribution is singular in the stationary state and has a cumulative form expressed in terms of generalized Takagi functions. The entropy production rate is computed using the coarse-graining formalism of Gaspard, Gilbert and Dorfman. In the continuum limit, we show that the value of the entropy production rate is independent of the coarse-graining and agrees with the phenomenological entropy production rate of irreversible thermodynamics.

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“…Figure (7) shows a recursive computation of T q (y), for q = 1/3. This example is very similar to the Takagi function [13] T (y) = y + 1 2 T (2y), 0 ≤ y < 1 2 ,…”

Section: Stationary State Under Flux Boundary Conditionsmentioning

confidence: 74%

Entropy production in a persistent random walk

Gilbert

Dorfman

2000

Physica A: Statistical Mechanics and its Applications

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“…Obviously, it is also subadditive, even and 1-periodic. Note that h is the classical example of a nowhere differentiable function due to Takagi [13] (and rediscovered by van der Waerden [14]). …”

Section: Nowhere Differentiable Continuous Subadditive Even Periodic mentioning

confidence: 99%

“…In section 5 we show that nowhere differentiable continuous periodic function of Takagi [13] (rediscovered by van der Waerden [14]) is subadditive.…”

Section: Introductionmentioning

confidence: 99%

Subadditive periodic functions

Matkowski¹

2011

OpMath

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Abstract. Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function f0 : [0, 1) → R is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.

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“…It turned out that a year before the Japanese mathematician Takagi Teiji published an article with almost the identical title "Simple example of continuous function without derivatives" [2], where he described the socalled "blancmange" curve (currently referred to as Takagi-Landsberg curve), which was the nearest relative of the Koch snowflake. Then the family of curves under consideration was expanded in 1933 by Hildebrandt [3] and in 1957 by de Rham [4].…”

Section: Introductionmentioning

confidence: 99%

Family of Generalized Triadic Koch Fractals: Dimentions and Fourier Images

Arzamastseva¹,

Evtikhov²,

Lisovsky³

et al. 2016

RENSIT

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Abstract. Fourier images of generalized triadic Koch fractals (curves and snowflakes) with variable vertex angle of generator were obtained by digital methods. The comparison of different methods of fractal dimensions determination using Fraunhofer diffraction patterns was made. Analysis of the size ratio dependence of central (fractal) and peripheral (lattice) parts of diffraction pattern both upon vertex angle at a fixed value of prefractal generation number and upon prefractal generation number at a fixed value of vertex angle was made. The features of Koch curves and Koch snowflakes Fourier images are discussed.

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A simple example of the continuous function without derivative

Cited by 107 publications

References 0 publications

Entropy production in a persistent random walk

Entropy production in a persistent random walk

Subadditive periodic functions

Family of Generalized Triadic Koch Fractals: Dimentions and Fourier Images

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