МыслиШизика

lol

Да блять! Отложка

Не хочу много букв но как получится.

Я много думаю про эту новость с решеним ИИ задачи тысячелетия, и в один момент стало даже очень грустно, от мысли, куда может скатиться работа математика. Но поговорив с коллегами, посмотрев пару роликов, я понял куда большую проблему от всего этого — потеря новых задач.

Решая задачи, математики находят новые ветки развития, связывают то, что уже было известно, видят куда можно пойти дальше. У ИИ нет такой цели, он просто решает то что дали. Ищет контрпримеры, закономерности и т.п. Если мы будем действовать таким подходом к каждой задаче, то можем столкнуться с тем, что не будет прогресса. Зачем придумывать новые обходы если ИИ может перебрать огромный пласт примеров, который человеку не перебрать, и найти ответ.

Придумал.

А у вас было тако,что вы решили сделать запросик для Fable про гипотезу, которую не могли решить ~90 лет и получить корректный ответ?

Если этот пост вышел значит отложка слетела :(

So we get:
⭐⭐, ⭐⭐⭐.

These are exactly the Cauchy--Riemann equations, which describe when a mapping locally preserves angles and transforms small squares into rotated and scaled squares.

Let
⭐⭐⭐⭐⭐, ⭐⭐⭐⭐⭐⭐⭐⭐
be a function.

Consider a small square:
⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐, where ⭐⭐⭐.

We want to find under what condition the mapping⭐sends this infinitesimal square to another square (up to rotation and scaling).

Take the vectors:
⭐⭐⭐⭐,⭐⭐⭐⭐.

For small ⭐⭐, we use linear approximation:
⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐,

⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐.

For the image to be a square, two conditions must hold:

1) Orthogonality:
⭐⭐⭐⭐⭐⭐.

2) Equal length:
⭐⭐⭐⭐⭐⭐.

Let us denote:
⭐⭐, ⭐⭐, ⭐⭐, ⭐⭐.

Then the conditions become:
⭐⭐⭐⭐,

⭐⭐⭐⭐⭐.

This means that the vectors ⭐⭐⭐and ⭐⭐⭐are orthogonal and have the same length.

Therefore, one vector is obtained from the other by a rotation by⭐. Hence:
⭐⭐, ⭐⭐.

Lately, I’ve been spending time learning about interactive proofs using LEAN. I wanted to write a small post about it, but I felt that using only text would be a bad idea. In LEAN, proofs are built with tactics, and it is much better to see how each tactic works instead of only reading about it.

To solve this, I decided to learn Manim and use animations for explanations.

You can see a template of the future video.

Channel photo updated

This is a 10 Turkish lira banknote featuring Cahit Arf.

Printed next to his portrait is the Arf invariant:
Arf(q) = Σ q(aᵢ)q(bᵢ) ∈ ℤ₂

Here, (q) is a quadratic form over ( \mathbb{F}_2 ), and ((a_i, b_i)) is a symplectic basis.
The remarkable fact: this single bit (0 or 1) completely classifies non-degenerate quadratic forms over ( \mathbb{F}_2 ) up to equivalence.

Abstract algebra, compressed into a banknote.

Today I am 4! Maybe I'll make it to 5!

One day you have a some problem and you know what your friend Cauchy (of course, all my Russians friends have a friend with name Cauchy) can help you to solve your problem. You invite them to your room and you together trying to solve a problem that you have. After few hours your friend Cauchy says — "If I have *this* instrument, I could solve the problem". You remember that three days ago, your friend Jacob told you that he had bought *this* instrument. You invite a Jacob to your room and you all trying to solve your (seems to hard) problem. It seemed like the problem had almost been solved, but then Jacobi says, ‘We've solved your problem, but it will only work if Zermelo solves his problem.’ You call Zermelo to find out how his problem is progressing, and he says — "Sorry, bro, my problem can't be solved in ZFC, try another one."

It explains how mathematics works for me and why I love it 😌!

Mathematics has grown a lot over the last thousand years, but the connections between different topics have stayed the same.
I think Geometry and Algebra are a great combination. The idea of using a coordinate system to solve geometric problems is very beautiful.

The second thing I found beautiful was the connection between number theory and analysis. Number theory seems to be something discrete, countable and simple, since we most often work with natural numbers.
Analysis, on the other hand, uses the language of infinitesimals and functions, which doesn't seem to match number theory at first.

To show how number theory and analysis are connected, I will give you three simple proofs of statements about prime numbers.

Music that help me solving a problems📝

I would like to write a post about Lobachevsky, because I was in Kazan, but I realized that I had no idea what to consider there. That's why I wanted to come back and tell you about such a wonderful mathematician as Galois and his theory made before his death in a duel. But even then I gave up 😩

Since I've been studying number theory lately, a good example would be to talk about the problems that occur there, but I haven't gotten around to it.
So I came up with the following. I want to write a list of topics/tasks that I have to describe in a month (or maybe faster) I'll start this month.

January

- What interesting in number theory?
- Favorite mathematicians 🤩
- Why I love math?
- The problem that I am currently trying to solve.

#January