Chaotic Attractor Explorer with Streamlit (ChAES)

Following up on the following post, I have added a Streamlit web application to visualize various chaotic attractors (link).

It is still a very rudimentary app. I hope to be able to add more features in the future.

Googol, Google and Isaac Asimov

I have been a huge fan of Isaac Asimov's fiction since I was a youngster. I have read all his Robot series and Foundation series books and many of his short stories. I was fortunate enough to hear him speak at our university when I was an undergraduate student. In one of his (many) non-fiction books he wrote that 

"For instance, in a book entitled Mathematics and the Imagination (published in 1940) the authors, Edward Kasner and James Newman, introduced a number called the `googol,' which is good and large and which was promptly taken up by writers of books and articles on popular mathematics. Personally, I think it is an awful name, but the young child of one of the authors invented it, and what could a proud father do? Thus, we are afflicted forever with that baby-talk number."

Asimov passed away in 1992, 6 years before Google was founded and before the internet is woven into the fabric of our lives, ranging from communication to entertainment to commerce. As is well known, Google chose the name as a play-on-words of the word googol. I wonder what Asimov would have thought of the fact that a word derived from the "baby-talk" word is the name of one of the largest technology companies on the planet and is also used to denote the act of finding out information via the vast information source that is the World Wide Web, technological developments that I am sure he would have love to see.

Harmonic mean of integers

The harmonic mean of a set of $n$ numbers $x_i$ is defined as $\frac{n}{\sum_{i=1}^n x_i^{-1}}$. While investigating the number of subsets of $\{1,...,n\}$ such that the harmonic mean is an integer (OEIS sequence A339453), I formulated and proved the following result, which states that for $n >1$ positive integers whose maximum is a prime power that is attained by a single element, their harmonic mean is not an integer:

Theorem: Let $x_i$ be a finite set of positive integers such that $x_j = \max_i x_i = p^k$ for some prime $p$ and positive integer $k$ and all other numbers $x_i$ are strictly less than $p^k$, then the harmonic mean of $\{x_i\}$ is not an integer.

Proof: Let $M$ be the least common multiple of $\{x_i\}$. Assume that $x_i$ are sorted in nondecreasing order. Thus $x_n = p^k$ and $x_i < p^k$ for all $i<n$ . Then $M = Wp^k$ where $p$ does not divide $W$. Let $Q_i = M/x_i$ and $Q = \sum_i Q_i$. This implies that $Q_n = W$ and $p$ divides $Q_i$ for $i <n$.

The harmonic mean $H$ can then be written as $nM/Q$. Since $p$ does not divide $W$, this implies that $p$ does not divide $Q$. Suppose $H$ is an integer. Then this implies that $Q$ divides $nM/p^k = nW$.

As $x_i < x_n$ for $i < n$, this implies that $Q_i > W$ for $i < n$, i.e. $Q > nW$, and this contradicts the fact that $Q$ divides $nW$ and thus $H$ is not an integer.

Visualizing chaotic attractors via 3D-printed models

 It has been 30 years since I first studied chaotic circuits and dynamical systems. At that time, we visualize the beautiful 3-D (and higher dimensional) chaotic attractors using a Silicon Graphics workstation. I was delighted to read in the December 2020 issue of the Notices of the AMS the article "Modeling Dynamical Systems for 3D Printing" by Stephen K. Lucas, Evelyn Sander and Laura Taalman. The ubiquity of inexpensive 3D printers makes it much easier to create 3D models of chaotic attractors. The authors provided excellent Mathematica and Matlab programs that make it easy to create STL files that can be sent to 3D printers to print these models. With python being able to be run on platforms ranging from PC to smartphones to Raspberry Pi's, I thought it would be nice to have a Python version of these programs. I ported the Matlab program to Python and utilized Python's object oriented features to make it easy to use them for other chaotic dynamical systems. 

The Python port is available here: https://github.com/postvakje/3d-printing-of-chaotic-attractor

The file dynamical_systems.py can be edited to add additional dynamical systems by providing the parameters, initial and ending simulation time, initial conditions and system equations. 

The data files that are output can be used with an script on OpenSCAD to create STL files.

The special place of the decimal number system

Modern society has been using the decimal number system for a long time, i.e. a number $n$ is expressed in base $10$ as $n = \sum_i b_i10^i$ where $0\leq b_i < 10$ are integers denoting the decimal digits of $n$.

Although there have been other cultures and civilizations who have used other number systems, most notably the Roman numerals and the Mayan base-20 number system, it was believed the decimal system was used by many due to us having 10 fingers. From a mathematical point of view, the base of the number system is arbitrary. For digital computers, using base 2 is more appropriate as it is easier to build components representing and processing 2-valued logic, or equivalently the binary digit, or the bit. It is interesting to note that one of the first (I said one of the first as there is a dispute whether the ABC computer is the first digital computer) digital computer, ENIAC, uses a decimal system and requires 10 vacuum tubes to represent a single decimal digit, each tube representing each of the numerals 0, 1, ..., 9.

Thus it came as a surprise to me that there is something inherently special about the decimal system. In 1964 Gustav Lochs proved the following theorem.

Lochs' theorem (1964): Let $m$ be the number of terms of a continued fraction expansion needed to determine the first $n$ decimal digits of a real number $x$. Then for almost all $x$, $\lim_{n\rightarrow \infty} \frac{m}{n} = \frac{ln(10)ln(64)}{\pi^2} \approx 0.970$.

What this tells us is that each coefficient of the continued fraction expansion contain slightly more information than each decimal digit. Had we use a base-11 numbering system, it would have been the opposite, each base-11 digit would contain more information than each additional continued fraction coefficient.

 

A 2D walk generated by primes

Consider starting at (0,0) on the plane and as we enumerate the primes p, except for 2 and 5, we take a step in the E, N, W, S direction depending on whether the last decimal digit of p is 1, 3, 7, 9 respectively.

The resulting 2D walk looks quite interesting. The first 200,000 steps looks like this:

The Python code used to generate this plot can be found here.

Using a different correspondence of last digit to directions, we get different, but qualitatively similar plots.

For instance (1->W, 3->E, 7->S, 9->N) gives us:

Overall, they seem more clustered and dense that a 2D random walk.

Happy casual π day!

Today is also known as π approximation day. Today is July 22, which is written as 22/7 in many countries.

22/7 is a rational number with repeating digits in decimal 3.14285714285714... and is one of the earliest approximation (besides the number 3) to π = 3.141592653589...  and has been used since antiquity. 22/7 approximates π to 2 decimal digits. 22/7 is also the second convergent in the simple continued fraction expansion of π. The convergents in this continued fraction are the best rational approximation of π, in the sense that if n/d with n and d coprime is a convergent, then no other rational number m/c with c ≤ d is closer to π.  It is interesting to note some other convergents have also been used as approximation to π. For instance the fourth convergent 355/113 = 3.141592920... matches up with π to 6 decimal places and was discovered by astronomer Zu Chongzhi (祖沖之) in the fifth century A.D.

I talked about these 2 approximations in my blog post from Pi Day 2017. For countries where the date format is YY/MM, we have to wait 2 years before we have a casual π or π approximation month during July 2022!

80's, 90's and today

I have been listening to a music channel called "80's, 90's and today" and it occurs to me that while there is only 1 decade between 80's and 90's, there are now 3 decades between 90's and today. When I was growing up, music from the 60's are considered oldies, perhaps the oldies today should include 80's and 90's music. Maybe the music channel should change its title, unless the implication is that "today" consists of music of the last 2+ decades (the 00's, the 10's and the start of the 20's) rather than grouped by a single decade, as was done before.

Prepending and appending a repunit with the same number

Consider a repunit R(n) of n digits in decimal, i,e, the number consisting of n 1's which is equal to $(10^n-1)/9$. If we append and prepend R(n) with the same number k, which we will denote as  k|R(n)|k with | denoting concatenation of digits, will k|R(n)|k be prime? For instance, for k = 324 and n = 1111, $k|R(n)|k = 3241111324 = 2^2\times 11^2 \times 281\times 23831$. It turns out that if n is a power of 2, then k needs to have at least n digits for k|R(n)|k to be prime. In particular, we have the following result:

Theorem: If $2^r$ is the largest power of 2 that divides n and k has at most $2^r-1$ digits, then k|R(n)|k is not prime. This is true even if some of the digits of k are leading zeros.

Proof: If k has m digits, then $k|R(n)|k = k(10^{n+m}+1) + R(n)10^m$ which is a multiple of gcd$(R(n),10^{n+m}+1)$.

Since $10^{2^r} - 1 = (10^{2^{r-1} -1})\times (10^{2^{r-1} + 1})$ and 9 does not divide $10^n+1$, by induction it is easy to show that $10^{2^w} + 1$ is a divisor of  $R(2^r)$ for $1 \leq w < r$. Since $R(2^r)$ is a divisor of $R(2^r s)$, $10^{2^w} + 1$ is also a divisor of $R(2^r s)$.

For $1 \leq m < 2^r$, let $t$ be the 2-adic valuation (or 2-adic order) of m, i.e. t is the largest integer such that $2^t$ divides $m$. This means that  $0 \leq t  < r$.

Then $10^{2^r s+m}+1 = 10^{2^t q}+1 = (10^{2^t})^q + 1$ for some odd number $q$.

Since the sum of two odd powers $a^q+b^q$ is divisible by $a+b$, this implies that $10^{2^r s+m}+1$ is divisible by $10^{2^t}+1$.

This implies that for $n = 2^r s$ and for all $1 \leq m < 2^r$, gcd$(R(n),10^{n+m}+1) \geq 10^{2^t}+1 > 1$, i.e. $k|R(n)|k$ is not prime. QED

For example, for n = 32, the smallest $k$ such that $k|R(n)|k$ is prime is k = 10000000000000000000000000000077, a number with 32 digits. The smallest such $k$ for different $n$'s can be found in OEIS sequence A263182 (see also OEIS sequence A329121).

Product of a number and its reversal in decimal

The number 10119126106147652568 is the smallest number such that it is the product of a number and its reversal in decimal notation in exactly 5 ways. 

10119126106147652568 = 8848263411 * 1143628488 = 8044687521 * 1257864408 = 4884561702 * 2071654884 = 4440958722 * 2278590444 = 4082378742 * 2478732804.

Multiply this number by 10 and it is the smallest number such that it is the product of a number and its reversal in decimal notation in exactly 10 ways. 

101191261061476525680 = 88482634110 * 1143628488 = 80446875210 * 1257864408 = 48845617020 * 2071654884 = 44409587220 * 2278590444 = 40823787420 * 2478732804 = 24787328040 * 4082378742 = 22785904440 * 4440958722 = 20716548840 * 4884561702 = 12578644080 * 8044687521 = 11436284880 * 8848263411.