Some theorems
@CihanPostsThms
Posting some theorems, and occasionally other stuff. By @bahran_cihan
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NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.
[Davis 1955]. Given a lattice (L, ≤), TFAE: (1) Every monotone f:L→L has a fixed point. (2) L is complete.
This position with white to move is impossible to reach within the current laws of chess. Source: anselan.com/tutorial.html
It follows that the topological version of the Whitney embedding theorem holds for compact (connected) n-manifolds: they all embed in ℝ²ⁿ. The non-compact ones are also believed to embed as such, but this does not seem to be written anywhere.
For a field k and a (finite) set X of variables, let k((X)) be the field of fractions of the formal power series ring k[[X]]. Theorem. There is a chain C of subfields within the field extension k((x,y)) → k((x))((y)) with |C| = |ℝ|.
Every position of Rubik's cube can be solved in at most 20 half-turn moves. Moreover there are positions for which 19 moves is not enough.
I can attest that @EricRWeinstein is a brilliant mathematical physicist. I recently did a deep dive into his Geometric Unity theory and I think it has the potential to unify all forces of nature in a beautiful & novel way; explain dark matter; and give a new explanation of dark…
A set X is called amorphous if X is infinite and whenever X = Y ⊔ Z, either Y or Z is finite. Theorem. ZF has models in which there are amorphous sets.
Given any year in the commonly used Gregorian calendar, • the last day of February, • April 4 (4/4), • June 6 (6/6), • August 8 (8/8), • October 10 (10/10), • December 12 (12/12) all occur on the same day of the week. For 2025 this so-called Doomsday is Friday.
In ZF the following are equivalent: (1) ℕ is a Lindelöf space. (2) ℚ is a Lindelöf space. (3) ℝ is a Lindelöf space. (4) The axiom of countable choice holds for subsets of ℝ.
Let M̅_g,n be the Deligne–Mumford–Knudsen compactification of the moduli space M_g,n of curves of genus g with n marked points. [Canning–Larson–Payne 2023]. Given g ≥ 1, n ≥ 0, TFAE: (1) H¹¹(M̅_g,n; ℚ) ≠ 0. (2) g=1 and n≥11.
22⋅arctan(24478/873121) + 17⋅arctan(685601/69049993) equals 𝜋/4.
>17 year old trans girl has article written about her mathematical contributions >68 year old creep sexologist (redundant adjective) complains about the article not sexualising her trans women truly are the women of women
To those interested in AGP: The article linked in the following post is about a trans woman, Hannah Cairo, who is a math prodigy. Although she is out as transgender (which has been disappeared from this article), and is definitely later-onset, AGP is of course not mentioned.
[Papanikolaou 1997]: If the Euler–Mascheroni constant 𝛾 is rational, then its denominator exceeds 10²⁴⁴⁶⁶³.
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