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Merely In The Name Of Self-Outward Violence

Roos ∗ They/Them ∗ Fifth Year Math Student ∗ Main is @bubbliterally but I talk more here ∗ An outlet for my mathematical ramblings >:^) ∗ Icon and banner are by @tamberella

orteil42

(non-french/dutch bilinguals don’t interact)
aux pays-bas une tasse de café c'est copieux

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a-set-of-tight-measures

I have found a cool Wikipedia article

List of named matrices - Wikipedia

en.m.wikipedia.org

oaticedeggshells

this is so cool!

reminds me of this, relations between rings (i think all commutative with identity), ED is Euclidean Domain, PID is principal ideal domain, UFD is unique factorization domain and ID is integral domain

and this, some stuff with Lie groups and how we classify them by Dynkin diagrams

math fun fact: if R[X] is a field then so is R

A Wikipedia screenshot. It features a drawing of a small fish near the tentacles of an off-screen Portuguese man o' war, as well as a block of text. The man-of-war fish, Nomeus gronovii, is a driftfish native to the Atlantic, Pacific and Indian Oceans. It is notable for its ability to live within the deadly tentacles of the Portuguese man o' war, upon whose tentacles and gonads it feeds. Rather than using mucus to prevent nematocysts from firing, as is seen in some of the clownfish sheltering among sea anemones, the fish appears to use highly agile swimming to physically avoid tentacles.[59][60]

Holy fuck

Source: Wikipedia

brightgreendandelions

the Elongated square gyrobicupola! gotta be one of my favorite genders

math

A geometric diagram exhibiting a scissors congruence from a figure consisting of the two squares of the right sides of a right triangle to the square of the hypotenuse.

How do you guys like this very convoluted proof of the Pythagorean theorem I just made

math for my thesis btw hehe one good thing about this one is that it is a direct scissors congruence using only translations though there's definitely simpler diagrams that have that same property :p this one i got by applying zylev's algorithm to the common visual proof where you add 4 triangles to make a square with side length (a+b)

cure-typhoon

rose lalonde

hs VERY important

I think I’ve finally come up with the right definition of a coverage for abstract scissors congruence purposes :))

A screenshot of a mathematical document. [text copied from the pdf cause it's really long, apologies if the ID is not very clear] Let F be a small category. If a is an arrow of F, let dom(a) or doma and cod(a) or coda denote the domain and codomain of a respectively. Unless otherwise specified, if we denote an arrow by some lower-case letter then we will denote its domain by the corresponding capital letter. For example, if a∗i is some arrow, then A∗i = dom(a∗i ). We write A ∈ Obj(F) to mean that A is an object of F, and a ∈ Arr(F) to mean that a is an arrow of F. For A,B ∈ Obj(F), let F(A,B) denote the set of arrows from A to B. Let a, b ∈ Arr(F) be such that coda = codb. We say that b factors through a, denoted b ≤ a, if there issomec:B→Asuchthatb=a◦c. Fora∈Arr(F),let↓adenotetheset{b∈Arr(F):b≤a}. For A ∈ Obj(F) let IA denote the set {b ∈ Arr(F) : codb = A}, and notice that IA = ↓idA. LetA∈Obj(F). AsetofarrowsintoAisasubsetB⊂IA. AsieveonAisasetofarrowsS into A such that if b ∈ S, then ↓b ⊂ S. That is, S is closed under precomposition. If S is a sieve on A and b ∈ IA is any arrow, then the pullback of S along b is the sieve b∗(S) on B defined as b∗(S)={c∈Arr(F):b◦c∈S}. Notethatifb∈S,thenb∗(S)=IB. IfSisasieveonA,and b ∈ IA and c ∈ IB are arrows, then (b ◦ c)∗(S) = c∗(b∗(S)). Any intersection of sieves on A is a sieve on A. It follows that there is a closure operator on the collection of all sets of arrows into A that maps a set of arrows onto the unique smallest sieve containing it. If B is any set of arrows into A then we denote this sieve by ↓B, which we call the sieve generated by B. It consists of all arrows into A that factor through some b ∈ B. We call IA the maximal sieve on A. The empty set ∅ ⊂ IA of arrows is also a sieve on A, which we call the empty sieve. They are indeed the top and bottom elements of the complete lattice of sieves on A. A coverage on F is a function J that assigns to every A ∈ Obj(F) a set J(A) of sieves on A. The elements S ∈ J(A) are called covering sieves on A. We require J to satisfy the following properties for all A ∈ Obj(F): ∗ (S1: Existence of trivial covers) IA ∈ J(A). ∗ (S2: Pullback-stability) If S ∈ J(A) and b ∈ IA, then b∗(S) ∈ J(B). ∗(S3: Composability)IfB⊂IA issuchthat↓B∈J(A)andforallb∈Bwehavesome Sb ∈J(B),then{b◦c:b∈B,c∈Sb}∈J(A). A small category F equipped with a coverage J on F is called a site. If F is a site, then the objects A ∈ Obj(F) are called figures of F and the morphisms a ∈ Arr(F) are called transformations. Transformations that are isomorphisms we call congruences.

math note that there are several definitions of coverages of various strengths this one should be weaker than a grothendieck coverage/topology in particular i do not assume the existence of any pullbacks whatsoever but i'll have to see whether it actually allows me to define scissors congruences lol optimistic though!!