Related papers: Level sets of signed Takagi functions

This paper studies loss functions for finite sets. For a given finite set $S$, we give sum-of-square type loss functions of minimum degree. When $S$ is the vertex set of a standard simplex, we show such loss functions have no spurious…

Euler totient function $\phi(n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying…

In some CFT models of simple current type, which are used to describe string theory on orbifolds and (adjoint) cosets of Lie groups, there arise fixed points of the simple current group. In these cases, the standard procedure to associate…

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional…

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational…

We consider iterations of integer-valued functions $\phi$, which have no fixed points in the domain of positive integers. We define a local function $\phi_n$, which is a sub-function of $\phi$ being restricted to the subdomain $\{0, ..., n…

Given a continuous function $f:[a,b]\to\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide…

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion…

The Takagi function is a classical example of a continuous nowhere differentiable function. In this paper we prove that it is nowhere approximately derivable.

Level-based and share-based loss functions are asymptotically equivalent if, in the limit, their averages converge almost surely to a constant ratio. These loss functions take a target value and its realization as arguments and are often…

We define a switch function to be a function from an interval to $\{1,-1\}$ with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given $n$ real-valued functions, $f_1,…

We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a…

The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic…

We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation,…

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN…

In this paper, we prove that for some Generalized Takagi Classes, in particular for the Takagi-Van der Waerden Class, the functions are nowhere differentiable if, and only if, the sequence of weights does not belong to $c_0$.

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating…

We consider a probabilistic approach to compute the Wiener--Young $\Phi$-variation of fractal functions in the Takagi class. Here, the $\Phi$-variation is understood as a generalization of the quadratic variation or, more generally, the…

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term…

This paper considers a parametrized family of generalized Takagi functions f_p with parameter p. Tabor and Tabor [J. Math. Anal. Appl. 356 (2009), 729-737] recently proved that for p in [1,2], f_p is (1,p)-midconvex. We give a simpler proof…