Difference between revisions of "Van der Waerden function"
From specialfunctionswiki
| Line 2: | Line 2: | ||
$$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s \left(10^k x \right)}{10^k}.$$ | $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s \left(10^k x \right)}{10^k}.$$ | ||
Note: to calculate $s(x)$ you may use $s(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right)$, where $\lfloor \cdot \rfloor$ denotes the [[floor]] function and $\lceil \cdot \rceil$ denotes the [[ceiling]] function. | Note: to calculate $s(x)$ you may use $s(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right)$, where $\lfloor \cdot \rfloor$ denotes the [[floor]] function and $\lceil \cdot \rceil$ denotes the [[ceiling]] function. | ||
| + | |||
| + | |||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Vanderwaerdenplot.png|Plot of the van der Waerden function. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
Revision as of 17:45, 22 January 2016
Define $s(x)=\inf_{n \in \mathbb{Z}} |x-n|$ (i.e. the distance from $x$ to the set of integers $\mathbb{Z}$). The van der Waerden function is defined by the formula $$V(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s \left(10^k x \right)}{10^k}.$$ Note: to calculate $s(x)$ you may use $s(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right)$, where $\lfloor \cdot \rfloor$ denotes the floor function and $\lceil \cdot \rceil$ denotes the ceiling function.
Plot of the van der Waerden function.
Properties
Theorem: The van der Waerden function is continuous.
Proof: █
Theorem: The van der Waerden function is nowhere differentiable on $\mathbb{R}$.
Proof: █
