WEIERSTRASS-TYPE FUNCTIONS IN p-ADIC LOCAL FIELDS

The Weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in [Formula: see text]. This paper investigates the Weierstrass-type function in the [Formula: see text]-adic local field [Formula: see text] whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the [Formula: see text]-adic calculus and the dimensions of the corresponding graphs.

Mathematics

This chapter presents an overview of Bolzano’s work in mathematics and its philosophy, while presenting some interesting samples of his work. It begins with a discussion of his views on mathematical method in their historical context, followed by an exposition of some of his best work in real analysis. In particular, the chapter discusses his early work on infinite series and his analysis of continuity, beginning with the Purely Analytic Proof (1817), and extending to his construction of a continuous, nowhere differentiable function in the 1830s, called Bolzano’s function. (85 words)

BOX DIMENSIONS OF α-FRACTAL FUNCTIONS

The box dimension of the graph of non-affine, continuous, nowhere differentiable function [Formula: see text] which is a fractal analogue of a continuous function [Formula: see text] corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of [Formula: see text]-fractal function [Formula: see text] for equally spaced as well as arbitrary data sets are found.

Hausdorff dimension of level sets of generalized Takagi functions

This paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form \begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*} where φ(x) is the distance from x to the nearest integer, and for each n, ωn is a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n), j=0,1,. . .,2n − 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension of f−1(y) is $\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωn is constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph of f with lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points of f. The paper ends with a list of open problems.