Nice to see math being ranked high (heh) and above guns even if I suck at it :P

https://archive.ph/kIgcr

This paper discovered the continuous math equivalent of the digital NAND gate. It turns out that a single binary operation paired with the constant 1 can generate every single standard elementary function. That operation is defined as eml(x,y)=exp(x)-ln(y). You can reconstruct constants like pi and the imaginary unit alongside basic addition and complex calculus tools using nothing but this one function.

The implications for machine learning and symbolic regression are massive. Normally when artificial intelligence tries to discover mathematical formulas from data it has to search through a chaotic space of different operators and syntax rules. Because the EML operator turns every mathematical expression into a uniform binary tree of identical nodes the search space becomes perfectly regular. You can basically treat a mathematical formula like a neural network circuit. The paper shows that when you train these EML trees using standard gradient optimizers like Adam the weights can actually snap to exact closed-form symbolic expressions instead of just giving fuzzy numerical approximations.

This finding could change how we design analog circuits and specialized computing hardware. If you only need a single instruction to execute any complex mathematical function you could build physical hardware or single instruction stack machines optimized purely for the EML operation. The fact that this was discovered by computationally stripping down a calculator rather than through purely theoretical derivation highlights how much structural beauty is still hiding in basic math.

cross-posted from: https://lemmy.world/post/45146291

To saturate the inequality, we need something wigglier. The heavyweight champions for polynomial wiggliness are the Chebyshev polynomials , which are motivated and described at length in this previous post.

https://epoch.ai/frontiermath/open-problems

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!