Nowhere differential functions

A. Continuous, Nowhere Differentiable Functions and Deterministic Fractals... 

We consider a few nowhere differentiable functions. Some of them have such amazing properties that they have been given the names of the great mathematicians who invented them Bolzano, Cantor, Peano, Weierstrass, Koch, Van der Waerden, Sierpinski, and others. 

About 1830, Bolzano showed that continuous, nowhere differentiable functions exist [16, 17]. 

The manuscript of Bolzano s was discovered only in 1920. So that the example of a nowhere differentiable function found by Weierstrass in 1871 was deemed the first example of such a function. Nowadays many examples of nowhere differentiable functions exist. Let us consider a nowhere differentiable Bolzano function paying tribute to Bolzano as the first scientist who formulated a nowhere differentiable function (Fig. 1). 

Thus, the Bolzano function B(x) is defined on the whole segment [0, 1]. It can be rigorously proved that B(x) is a nowhere differentiable function [16, 17]. 

The Bolzano construction for obtaining a nowhere differentiable function can be simplified and made more graphic [18]. 

Next, we divide each part into four parts again and construct 16 isosceles triangles. The graph so obtained, y = fl (x), is added to the previous construction. Continuing this process, we obtain a nowhere differentiable function (Van der Waerden function) [18]. 

Another example of a nowhere differentiable function is the Weierstrass function, defined by [19]... 

Along with the discovery of nowhere differentiable functions, geometric shapes were created and it was difficult to say whether they were lines, surfaces, or volumes. 

Wiener s process (i.e., Brownian motion) and Kolmogorov s turbulence (i.e., a nonsmooth vector field) may be cited as examples of phenomena which can be described by continuous, nowhere differentiable functions (fractal functions). 

Our rapid overview of mathematical monsters would not be complete without a brief reference to an interesting family of continuous, nowhere differentiable functions. They are occasionally referred to as Weierstrass-like but, for historical reasons, it seems more appropriate to call them Bolzano-Weierstrass-like . 

Indeed, Bernard Bolzano [16] appears to have been the first (in a manuscript written around 1830 but published only in 1930) to provide an example of a continuous, nowhere differentiable function. A few years later, in 1872, Karl Weierstrass [17] showed that the function... 

Over the years, many examples of continuous, nowhere differentiable functions have been published see Edgar [11 (pp. 7, 341)]. One of them, the so-called Weierstrass-Mandelbrot function, assumes a particular significance in environmental science because it constitutes the theoretical basis of the first article that used fractal geometry in connection with soils data. Burrough [19] used the Weierstrass-Mandelbrot function to describe the often erratic-looking spatial variation of soil properties along transects. 

Observations made by scientists over the years provided additional momentum in the same direction. The physical process whose study was perhaps most influential in stimulating interest in nowhere-differentiable functions is Brownian motion, described in Section 2.2.5. Besides Brownian motion, several other physical processes contributed to foster interest in nowhere-differentiable functions. A few years after Wiener s work, Dedebant and Wehrld [51] studied a number of scale issues arising in meteorology and concluded that many meteorological processes, observed at a very small scale, are like the continuous, nondifferentiable functions that seemed forever to belong to the realm of speculative mathematicians [51 (p. 83)]. 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

Richardson, in his 1926 investigation of turbulence, observed that the velocity field of the atmospheric wind is so erratic that it probably cannot be described by an analytic function [79]. He suggested a Weierstrass function as a candidate to represent the velocity field, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-field data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion ... 

Along with functions discontinuous at all points, continuous functions having no derivatives at any point were discovered—that is, functions which are almost everywhere continuous but nowhere differentiable. 

In order to justify our introduction of the Peano functions (which as will be shown below are nowhere differentiable) and to better our understanding of the... 

The second approximation to the curve x = ip (f) can be constructed in a similar way again dividing each segment < t < into nine equal parts. This approximation is shown in Fig. 10b, and it already indicates that the function x = (pit) may be nowhere differentiable. Now we will prove that it is really so. 

The third part of the spectral function Oj =Oj(p) is the pathological portion of the spectral function 0(p). It is continuous but almost nowhere differentiable as the trajectories of a classical Brownian particle or of a random walker in the limit of vanishing step width (Ax 0) and time... 

Diffusion model used in this work For the purpose of this work the evolution of the particle position is described in terms of the stochastic differential equation. The major problem with using this technique is that the particle velocity cannot be described (since the diffusion sample paths are nowhere differentiable). The central limit theorem does however provide the reassurance that the diffusion equation accurately describes evolution of the transition density. For all chemical systems investigated as part of this work, the transient period occurs on a timescale of tens of picoseconds or possibly even longer. This value is much bigger than typical values of the velocity autocorrelation function, making the use of the diffusion equation justifiable. 

In this section we will prove that the Lieb functional is differentiable on the set of E-V-densities and nowhere else. The functional derivative at a given E-V-density is equal to — v where v is the external potential that generates the E-V-density at which we take the derivative. To prove existence of the derivative we use the geometric idea that if a derivative of a functional G[n] in a point n0 exists, then there is a unique tangent line that touches the graph of G in a point (n0, G[ 0 ). To discuss this in more detail we have to define what we mean with a tangent. The discussion is simplified by the fact that we are dealing with convex functionals. If G B — TZ is a differentiable and convex functional from a normed linear space B to the real numbers then from the convexity property it follows that for n0,nj 5 and 0 < A < 1 that... 

As a next step we will show that this implies that the functional FL is Gateaux differentiable at every E-V-density and nowhere else. For the application of the next theorem it is desirable to extend the domain of FL to all of L1 DL3. We follow Lieb 1 and define... 

Theorem 10. The functional FL is Gateaux differentiable for every E-V-density in the set S and nowhere else. Moreover, the functional derivative at an E-V-density is equal to — v where v is the potential that generates this density. 

We have seen that the Lieb functional Fh is differentiable at the set of E-V-densities in S and nowhere else. For this reason it is desirable to know a bit more about these densities. The question therefore is which densities are ensemble v-representable In this section we will prove a useful result which will enable us to put the Kohn-Sham approach on a rigorous basis. 

We see that this is simply the Lieb functional with the two-particle interaction omitted. All the properties of the functional Fh carry directly over to Th. The reason is that all these properties were derived on the basis of the variational principle in which we only required that 7 + IV is an operator that is bounded from below. This is, however, still true if we omit the Coulomb repulsion W. We therefore conclude that Th is a convex lower semicontinuous functional which is differentiable for any density n that is ensemble v-representable for the noninteracting system and nowhere else. We refer to such densities as noninteracting E-V-densities and denote the set of all noninteracting E-V-densities by >0. Let us collect all the results for 7) in a single theorem ... 

Since both Th and FL are defined on the set S the exchange-correlation functional E,lc is also defined on that set. Since F and 7] are differentiable, respectively, on the sets B and B0 and nowhere else, the functional Exc is differentiable on B D B0 and nowhere else. The derivative of equation (243) on that set is given by... 

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