Continuous Non-differentiable Function Research Articles

Variation of topological pressure and dimension: from polynomials to complex Hénon maps

AbstractWe study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

Level sets of the Takagi function: local level sets

The Takagi function τ: [0,1] → [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : τ(x) = y} of the Takagi function τ(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a “generic” full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a “generic” full Lebesgue measure set of abscissas x, the level set L(τ(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly \({\frac{3}{2}}\).

A Novel Approach to Processing Fractal Signals Using the Yang-Fourier Transforms

In the present paper, local fractional continuous non-differentiable functions in fractal space are studied, and the signals in fractal-time space are reflectively investigated using the Yang-Fourier transforms based on the local fractional calculus. Two illustrative examples are given to elaborate the signal process and reliable results.

Level sets of the Takagi function: generic level sets

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that for a full Lebesgue measure set of ordinates y, these level sets are finite sets, but whose expected number of points is infinite. Complementing this, it shows that the set of ordinates y whose level set has positive Hausdorff dimension is itself a set of full Hausdorff dimension 1 (but Lebesgue measure zero). Finally it shows that the level sets have a nontrivial Hausdorff dimension spectrum. The results are obtained using a notion of "local level set" introduced in a previous paper, along with a singular measure parameterizing such sets.

On the critical case of Okamoto’s continuous non-differentiable functions

In a recent paper in this Proceedings, H. Okamoto presented a parameterized family of continuous functions which contains Bourbaki’s and Perkins’s nowhere differentiable functions as well as the Cantor-Lebesgue singular function. He showed that the function changes it’s differentiability from ‘differentiable almost everywhere’ to ‘non-differentiable almost everywhere’ at a certain parameter value. However, differentiability of the function at the critical parameter value remained unknown. For this problem, we prove that the function is non-differentiable almost everywhere at the critical case.

Level sets of the Takagi function: Hausdorff dimension

The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity of τ(x) up to affine shifts.

Scale calculus and the Schrödinger equation

This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,ε) such that f(t,ε)→f(t) when ε goes to zero. This led to several new notions as representations: fractal functions and ε-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schrödinger equation via the classical Newton’s equation of dynamics using the scale operator. Under special assumptions we recover the classical Schrödinger equation and we discuss the relevance of these assumptions.

Automorphic forms and differentiability properties

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.

FUNCTIONS THAT HAVE NO FIRST ORDER DERIVATIVE MIGHT HAVE FRACTIONAL DERIVATIVES OF ALL ORDERS LESS THAN ONE

A question of classical mathematical analysis the existence of a continuous non-differentiable function is treated here in a detailed setting. We consider this question within the framework of fractional derivatives and show that there exist continuous functions f(x) which nowhere have an ordinary first order derivative, but have continuous fractional derivatives of any order U < l. We begin with notation in Section 1 because there exist different forms of fractional integration and differentiation which do not necessarily coincide with each other. We use Riemann-Liouville, Liouville, Marchaud and Weyl fractional differentiation. In the introductory Section 2 we specify the setting of the problem under consideration and give some comments. Section 3 contains statements of

An iterative approach to VLSI circuit lifetime estimation

Abstract In this paper, the problem of VLSI circuit lifetime estimation is discussed. It is shown that lifetime determination is equivalent to finding the smallest zero of a continuous non-differentiable function for a specific circuit realization. An iterative multiple cubic spline approximation algorithm is proposed to determine the lifetime. Application in VLSI circuit analysis is given, in which the degradation resulting from the hot electron effects are considered. The results show that our proposed approach is efficient.

A quantitative condensation of singularities on arbitrary sets

This paper deals with quantitative extensions of the classical condensation principle of Banach and Steinhaus to arbitrary (not necessarily countable) families of sequences of operators. Some applications concerned with the sharpness of approximation processes, with (Weierstrass) continuous nondifferentiable functions as well as with the classical counterexample of Marcinkiewicz on the divergence of Lagrange interpolation polynomials, illustrate this unifying approach to various condensations of singularities in analysis.

Random walks with self-similar clusters

We construct a random walk on a lattice having a hierarchy of self-similar clusters built into the distribution function of allowed jumps. The random walk is a discrete analog of a Lévy flight and coincides with the Lévy flight in the continum limit. The Fourier transform of the jump distribution function is the continuous nondifferentiable function of Weierstrass. We show that, for cluster formation, it is necessary that the mean-squared displacement per jump be infinite and that the random walk be transient. We interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker. The difference in dimensions is related to the fractal (Hausdorff-Besicovitch) dimension of the self-similar clusters.

Table of a Weierstrass Continuous Non-Differentiable Function

Table of a Weierstrass Continuous Non-Differentiable Function

Table of a Weierstrass continuous non-differentiable function

Table of a Weierstrass continuous non-differentiable function

Some Remarks on the Construction of Continuous Non-Differentiable Functions

Some Remarks on the Construction of Continuous Non-Differentiable Functions

Some Remarks on the Construction of Continuous Non-Differentiable Functions

Proceedings of the London Mathematical SocietyVolume s2-50, Issue 1 p. 463-481 Articles Some Remarks on the Construction of Continuous Non-Differentiable Functions F. A. Behrend, F. A. BehrendSearch for more papers by this author F. A. Behrend, F. A. BehrendSearch for more papers by this author First published: 1948 https://doi.org/10.1112/plms/s2-50.6.463Citations: 4AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-50, Issue11948Pages 463-481 RelatedInformation

A note on certain continuous non-differentiable functions

A note on certain continuous non-differentiable functions

An Elementary Example of a Continuous Non-Differentiable Function

An Elementary Example of a Continuous Non-Differentiable Function

An Elementary Example of a Continuous Non-Differentiable Function

(1927). An Elementary Example of a Continuous Non-Differentiable Function. The American Mathematical Monthly: Vol. 34, No. 9, pp. 476-478.