mathematics

Statistical Tests to determine whether a bit-stream can be considered "random"

If we were to flip a coin 10 times, we would expect to see roughly 5 heads and 5 tails. Let's assign 00 to heads and 11 to tails. Therefore, we might see a sequence like this...

cross-posted from: https://sh.itjust.works/post/48633930

Normally, we use a place-value system. This uses exponentials and multiplication.

1234
^^^^
||||
|||└ 4 * 10^0 = 4
||└ 3 * 10^1 = 30
|└ 2 * 10^2 = 200
1 * 10^3 = 1000

1000 + 200 + 30 + 4 = 1234

More generally, let d be the value of the digit, and n be the digit's position. So the value of the digit is d * 10^n^ if you're using base 10; or d * B^n^ where B is the base.

1234
^^^^
||||
|||└ d = 4, n = 0
||└ d = 3, n = 1
|└ d = 2, n = 2
d = 1, n = 3

---

What I came up with was a base system that was polynomial, and a system that was purely exponential, no multiplication.

In the polynomial system, each digit is d^n^. We will start n at 1.

polynomial:
1234
^^^^
||||
|||└ 4^1 = 4 in Place-Value Decimal (PVD)
||└ 3^2 = 9 PVD
|└ 2^3 = 8 PVD
1^4 = 1 PVD

1234 poly = 1 + 8 + 9 + 4 PVD = 22 PVD

This runs into some weird stuff, for example:

• Small digits in high positions can have a lower magnitude than large digits in low positions
• 1 in any place will always equal 1
• Numbers with differing digits being equal!

202 poly = 31 poly
PVD: 2^3 + 2^1 = 3^2 + 1^1
8 + 2 = 9 + 1 = 10

---

In the purely exponential system, each digit is n^d^. This is a bit more similar to place value, and it is kind of like a mixed-base system.

1234
^^^^
||||
|||└ 1^4 = 1
||└ 2^3 = 8
|└ 3^2 = 9
4^1 = 4

1234 exp = 4 + 8 + 9 + 1 PVD = 22 PVD

However it still runs into some of the same problems as the polynomial one.

• Small digits in high positions can have a lower magnitude than large digits in low positions (especially if the digit is 1)
• The digit in the ones place will always equal 1
• Numbers with differing digits can still be equal

200 exp = 31 exp
PVD: 3^2 = 2^3 + 1^1
9 = 8 + 1

---

So there you have it. Is it useful? Probably not. Is it interesting? Of course!

https://arxiv.org/abs/2410.20392

cross-posted from: https://sopuli.xyz/post/22688165

Random thought on magic squares:

If I view the smallest possible non-trivial magic square

2 7 6
9 5 1
4 3 8

since its rows and diagnoals sum up to 2+5+8 = 2+7+6 = 4+5+6 = 2+9+4 = … = 15

in the article as a 3x3 Matrix, its determinant is Δ = -360 . Its inverse:

-37/360 19/180 23/360
17/90 1/45 -13/90
-7/360 -11/180 53/360

note how this is a magic square, rows and diagonals sum up to 1/15.

https://matrix.reshish.com/inverse.php

Now if you are really bored (I can not do this): proof that for any non trivial magic squares the inverse …

• exists (i.e. every non-trivial magic square has an inverse)
• is a magic square.

Extracting Fernando Gouvêa's article on Spivak's Calculus book. I, like many others, used this book as my first introduction to "real" math, and it still has an important place in my heart. But the rest of the memorial is definitely worth reading. I had no idea that Spivak was gay, and it is good to see that representation.

Michael Spivak’s Calculus

By Fernando Q. Gouvêa

Let me begin with some autobiography. I was introduced to calculus during my first year at the University of São Paulo, in Brazil. At that time, we didn’t use a textbook; instead, we used references and small booklets of notes, prepared by one of the teachers. The references were a variety of books that our professors felt could be consulted with profit. Among them was Spivak’s Calculus. I didn’t buy a copy at first, because I was fairly happy with the notes and with the books I already had. But the name stuck in my head.

A couple of months into the course, I was strolling through the aisles of my favorite bookstore, and saw a big gray calculus textbook. I think it was the first edition. It was definitely in soft cover, perhaps an edition for sale in third world markets. I recognized the author’s name, bought the book, took it home, and set to reading it.

It was love at first sight. I’ll admit that I wasn’t doing all—or even most—of the problems. (There are lots of very hard problems; instructors will be glad that there is an answer book.) But Spivak’s account of what calculus is all about, his careful but precise account of the theoretical underpinnings of the material, his chapter on the “hard theorems” (the ones that require an understanding of the completeness of the reals), his pictures, even his asides…This was calculus as an intellectual adventure, deep, compelling, and beautiful. I read the whole book, making my way far beyond what we had covered in class by that point. I even read, with some small level of understanding, the sections on complex power series and the explanation of Dedekind cuts.

It will give you a measure of Spivak’s impact on me to note that I took quite seriously the annotated suggested reading list given in the back. I sought out many of them, read a good chunk of the ones I bought, and understood a few of the ones I read.

I never forgot some bits of the book. Spivak’s comment that “mathematicians like to pretend that they can’t even add, but most of them can when they have to” (on page 179) stuck in my mind. I’ve used it in class many times. His chapter on integration in finite terms is another I remembered, and especially problem 7 in that chapter (described as “Potpourri. No holds barred.”). I remember “the world’s sneakiest substitution” $t=\tan (x/2)$. One of the problems was marked as “An exercise to convince you that this substitution should be used only as a last resort.” The next problem had the note: “a last resort.”

The treatment of power series also stuck with me; as a result, I’ve been a fan of series ever since. I’m grateful that the edition I read didn’t include the section on Kepler’s laws, added in the third edition, because I don’t think I could have followed it. The reading list, alas, was never updated.

A calculus textbook that does everything honestly, even if gently enough so that good first-year students can follow it, is not for everyone. Many people would want to see more applications or more history. In addition, there is no multivariable calculus, which is a pity; I’d have loved to see what Spivak could do with that at this level. (His Calculus on Manifolds is, of course, a classic, but it is so terse as to be impenetrable for most students.)

When I use it in class, students always find the book very hard to read and understand. But they also understand that they are getting the real thing, that they are being treated as intellectual adults. Spivak’s achievement was to produce a textbook that has the potential to lead talented students into an appreciation for both the subject and the methods of mathematics. There must be many mathematicians today who cut their teeth on this book; may there be many more.

A link from @highergeometer@mathstodon.xyz to a 1972 talk by Grothendieck (including a recording; I don't think I'd ever actually heard his voice before!). It's interesting how complaints that seem very modern have existed for so long.

I’ve been taking every opportunity to meet scientists, whether in public discussions like this one or in private, and raise these questions. In particular: “Why do we do scientific research?” [...] The extraordinary thing is to see how incapable my colleagues are of answering this question. In fact, for most of them, the question is simply so strange, so extraordinary, that they refuse even to contemplate it. In any case, they are extremely reluctant to give any kind of answer.

I’ve come to realize that in fact this satisfaction that scientists are supposed to derive from exercising their cherished profession, is a pleasure… which is not a pleasure for everyone! [...] Once you’ve got your job, it’s an imperative to move up the ladder. Once you’ve moved up the ladder, even assuming you’ve made it to the top, it’s an imperative to be considered in the running. You’re expected to produce. Scientific production—like any other kind of production in the ambient civilization—is considered an imperative in itself. The remarkable thing about all this is that in the end, the content of research becomes a second thought. It’s all about producing a certain number of “papers”. In extreme cases, a scientist’s productivity is measured by the number of pages they publish. Under these conditions, for a large number of scientists—certainly for the overwhelming majority, with the real exception of a few who are fortunate enough to have an exceptional gift or to be in a social position and disposition that enables them to free themselves from these feelings of constraint—for most, scientific research is a real constraint that kills the pleasure one can have in carrying it out.

Calegari's Notices essay mentioned in the article.