#starting with algebra

[vee1021]

Ugh, I have to do make-up work for school eeeewwww

[currentlyinflames]

WDOFHWJNFN THE PJO EP TITLES ARE OUT AND THEY'RE EVERYTHING I EVER WANTED AND MORE

They actually kept the og chapter titles and I'm so happy!!

[wheatormeat]

Maybe it's cause I'm surrounded by art people all the time but I feel like math as a school subject gets a bad wrap. Like now that I'm not forced to take classes for it, I just start doing it on my own for fun. It gets my brain stimulated. I'm doing calculus like it's a word puzzle; it's a stress reliever and it's rewarding. It's encountering a problem I already know all the steps to even if I've semi-forgotten them. And there's minimal consequences if I get the answer wrong. I think it's fun to go to town with a ballpoint pen on some dirty scrap paper, just letting the scribbly numbers flow.

[bubbloquacious]

Okay so to get the additive group of integers we just take the free (abelian) group on one generator. Perfectly natural. But given this group, how do we get the multiplication operation that makes it into the ring of integers, without just defining it to be what we already know the answer should be? Actually, we can leverage the fact that the underlying group is free on one generator.

So if you have two abelian groups A,B, then the set of group homorphisms A -> B can be equipped with the structure of an abelian group. If the values of homorphisms f and g at a group element a are f(a) and g(a), then the value of f + g at a is f(a) + g(a). Note that for this sum function to be a homomorphism in general, you do need B to be abelian. This abelian group structure is natural in the sense that Hom(A ⊗ B,C) is isomorphic in a natural way to Hom(A,Hom(B,C)) for all abelian groups A,B,C, where ⊗ denotes the tensor product of abelian groups. In jargon, this says that these constructions make the category of abelian groups into a monoidal closed category.

In particular, the set End(A) = Hom(A,A) of endomorphisms of A is itself an abelian group. What's more, we get an entirely new operation on End(A) for free: function composition! For f,g: A -> A, define f ∘ g to map a onto f(g(a)). Because the elements of End(A) are group homorphisms, we can derive a few identities that relate its addition to composition. If f,g,h are endomorphisms, then for all a in A we have [f ∘ (g + h)](a) = f(g(a) + h(a)) = f(g(a)) + f(h(a)) = [(f ∘ g) + (f ∘ h)](a), so f ∘ (g + h) = (f ∘ g) + (f ∘ h). In other words, composition distributes over addition on the left. We can similarly show that it distributes on the right. Because composition is associative and the identity function A -> A is always a homomorphism, we find that we have equipped End(A) with the structure of a unital ring.

Here's the punchline: because ℤ is the free group on one generator, a group homomorphism out of ℤ is completely determined by where it maps the generator 1, and every choice of image of 1 gives you a homomorphism. This means that we can identify the elements of ℤ with those of End(ℤ) bijectively; a non-negative number n corresponds to the endomorphism [n]: ℤ -> ℤ that maps k onto k added to itself n times, and a negative number n gives the endomorphism [n] that maps k onto -k added together -n times. Going from endomorphisms to integers is even simpler: evaluate the endomorphism at 1. Note that because (f + g)(1) = f(1) + g(1), this bijection is actually an isomorphism of abelian groups

This means that we can transfer the multiplication (i.e. composition) on End(ℤ) to ℤ. What's this ring structure on ℤ? Well if you have the endomorphism that maps 1 onto 2, and you then compose it with the one that maps 1 onto 3, then the resulting endomorphism maps 1 onto 2 added together 3 times, which among other names is known as 6. The multiplication is exactly the standard multiplication on ℤ!

A lot of things had to line up for this to work. For instance, the pointwise sum of endomorphisms needs to be itself an endomorphism. This is why we can't play the same game again; the free commutative ring on one generator is the integer polynomial ring ℤ[X], and indeed the set of ring endomorphisms ℤ[X] -> ℤ[X] correspond naturally to elements of ℤ[X], but because the pointwise product of ring endomorphisms does not generally respect addition, the pointwise operations do not equip End(ℤ[X]) with a ring structure (and in fact, no ring structure on Hom(R,S) can make the category of commutative rings monoidal closed for the tensor product of rings (this is because the monoidal unit is initial)). We can relax the rules slightly, though.

Who says we need the multiplication (or addition, for that matter) on End(ℤ[X])? We still have the bijection ℤ[X] ↔ End(ℤ[X]), so we can just give ℤ[X] the composition operation by transfering along the correspondence anyway. If p and q are polynomials in ℤ[X], then p ∘ q is the polynomial you get by substituting q for every instance of X in p. By construction, this satisfies (p + q) ∘ r = (p ∘ r) + (q ∘ r) and (p × q) ∘ r = (p ∘ r) × (q ∘ r), but we no longer have left-distributivity. Furthermore, composition is associative and the monomial X serves as its unit element. The resulting structure is an example of a composition ring!

The composition rings, like the commutative unital rings, and the abelian groups, form an equational class of algebraic structures, so they too have free objects. For sanity's sake, let's restrict ourselves to composition rings whose multiplication is commutative and unital, and whose composition is unital as well. Let C be the free composition ring with these restrictions on one generator. The elements of this ring will look like polynomials with integers coefficients, but with expressions in terms of X and a new indeterminate g (thought of as an 'unexpandable' polynomial), with various possible arrangements of multiplication, summation, and composition. It's a weird complicated object!

But again, the set of composition ring endomorphisms C -> C (that is, ring endomorphisms which respect composition) will have a bijective correspondence with elements of C, and we can transfer the composition operation to C. This gets us a fourth operation on C, which is associative with unit element g, and which distributes on the right over addition, multiplication, and composition.

This continues: every time you have a new equational class of algebraic structures with two extra operations (one binary operation for the new composition and one constant, i.e. a nullary operation, for the new unit element), and a new distributivity identity for every previous operation, as well as a unit identity and an associativity identity. We thus have an increasing countably infinite tower of algebraic structures.

Actually, taking the union of all of these equational classes still gives you an equational class, with countably infinitely many operations. This too has a free object on one generator, which has an endomorphism algebra, which is an object of a larger equational class of algebras, and so on. In this way, starting from any equational class, we construct a transfinite tower of algebraic structures indexed by the ordinal numbers with a truly senseless amount of associative unital operations, each of which distributes on the right over every previous operation.

[lipshits-continuous]

Pretty neat that for a topological group G with identity element e, we have that π₁(G,e) is abelian

[coyotetatertot]

Promotional for Tate's company in my interp of A Better World AU.

FULL TEXT BENEATH THE CUT‼️‼️

God, I love exploring what he can do if he hadn't suffered through his father abandoning them and then YEARS of caretaker burnout as he tried in vain to heal his dad. What if he hadn't learned to fear his intellect and skill. What if Appalachia hadn't been cut out of him by being raised in the Bay Area. What if his abilities and cultural identity were both nurtured and encouraged by loving parents and a strong educational support system. What then. 👁️

I think he definitely still has his issues, because public figures often do lol. Fame causes so many problems. But fuck if I don't wanna let this lil scruffy genius out of his mental cage of repression, burnout, and depression. I think he's wild, enthusiastic, and has so much heart and spirit underneath all those layers of bullshit. 30 years of suffering and he is in his 30s, the divergence of the AU puts him on a radically different path from childhood and that makes him a TOTALLY new person.

On the highest peaks in the world, the strongest tethers aren't your rope, but the emotional ties which unite your climbing team and keep you connected to those waiting for you back home. Whether it's by blood or by choice, Tater Higgs McGucket understands the importance of family. Son of revolutionary inventor and co-founder of the Institute of Oddology Fiddleford Hadron McGucket, Tate describes his father as his closest friend, collaborator, and mentor. In collaboration with family friend and other co-founder of the Institute Stanford ("Ford") Pines, the three first designed their renowned supplemental oxygen delivery system after an expedition studying anomalies in the Himalayas.

"Our investigation took us to Camp 1 of Manaslu," Tate described in an exclusive interview with Mountaineering Monthly last week, "And I was shocked by the amount of traffic. This was some of the roughest terrain on the planet, but we saw more people out there than on some of my hiking trips back home in Oregon. . . Ford was our interpreter, and after talking with the locals, we realized that there were all these companies selling tickets to the top — with sherpas puttin' themselves on the line just to ferry tourists to the summit."

The influx of inexperienced climbers has had disastrous consequences, as Tate witnessed firsthand. "A lot of these people, they're physically and mentally capable of makin' that kinda climb, but maybe they don't follow best practice. You can summit without any oxygen, if ya stop and acclimatize along the way. But that takes a while, so it can be really temptin' to ignore your body and throw an oxygen bandaid at the problem. But then you're puttin' yourself in an emergency situation if it fails. While we were there, one of those climbers ran out, and a sherpa had to run more oxygen up there. I told him there was a storm a-comin', but he went up anyway. And we ended up losin' 'em both."

Tate's growing twang was underscored by a nervous bouncing of his leg, and he took a moment to collect himself before resuming the interview.

"Dad and I had a look at these open circuit breathing apparatuses. While they were reliable, we saw they were plum wasteful. Knew we could make somethin' better. There's a growin' culture of risk-takin' 'round them mountains. And maybe we cain't stop the industry that's causin' these problems, but we can at least make it safer for them climbers. 'Cuz at the end of the day, regardless of what ya think about these people? With an accident like that, there’s people left behind that're a-hurtin' somethin' fierce. Partners, friends, kids without parents. I mean, just the thought of losin' my dad like that is enough to break my heart — but that's reality, for both the families of that climber and the sherpa who died tryin' to save him. . . Naw, I reckon we can do better."

That was how the youngest McGucket, who had become a household name in the 1990s for his work in designing personal computers with his father's company, first ventured into the world of alpinism. But what he hadn't expected was to fall in love during the process.

"I always needed nature," he explained, "I get overstimulated awfully easy, and so I go out there to clear my head. Been hikin' and fishin' since I was a kid. . . And so, after workin' with climbers to test this equipment — I saw a lot of them eight-thousanders up close, right? And one day, I just knew I had to see it from the top."

But having become familiar with the dangers involved, Tate knew that preparing himself for such a climb would be no easy task.

Luckily, he found a trainer in Ford's twin brother, Stanley Pines.

“Stanley is a stand-up guy. Real old school. Throws a hell of a punch, catches a hell of a catfish.” Tate said of his mentor, “He’s a fighter. So I knew I needed him, because all it takes is one slip up or act of god for these expeditions to turn life-or-death. And he’s been great. Neither of us knew much about rock climbin’ or mountaineering before all this. But we’ve learned together. And having summited a few eight-thousanders now, I can tell ya, I wouldn’t be here without his help.”

Also aiding in his expeditions were his prototype real-time weather and vital monitoring systems, which have since become standard issue in all McGucket brand protective wear. But Tate is most proud of his high-frequency beacon system, which allows climbers to communicate with their partners and first responders — even from inside perilous crevasses.

"The danger of avalanche or serac collapse is real. There are times when your life just ain’t in your own hands. Our systems allow climbers to communicate when they’re entering or exiting a perilous area, and can send out an SOS. They’re also constantly pinging, so in the event somethin’ does happen, they’ll help your climbing partners or first responders find you.”

But high altitudes aren’t the only place you’ll find the twin peaks of McGucket Mountaineering. Tate’s inventions have seen heavy use by first responders of all stripes, from firefighters to wilderness search and rescue — and he has recently signed a contract to manufacture respirators for medical use.

"At the end of the day, it’s all about making it home safely.” Tate concluded, “You gotta prioritize what matters most. You can do incredible things in this world, but none of it matters if you can’t share them with the people who love you.”

[proof-by-intimidation]

alright, Lurie's higher topos theory hits so hard. Can't wait to get into it seriously

[the-jedi-ninja]

When you realise that Sally was sitting in the rain in the first episode because she was probably trying to feel closer to Poseidon, or at least like he was with her in that moment.

[juvia-is-beast]

Hey Y'all I rarely post stuff of my own but I'm in need of help. I've always had a hard time asking for help but I've become desperate. I just moved out of my toxic Indian household at 27 with no savings because my mom would take all of my SSI checks and huge chunks of my paycheck that I worked for. I'm having a lot of health issues and chronic pain flare ups. I work at a college but I've been put in an administrative role until I can get my health in order. I don't have as many hours but now I can actually see doctors. My family was preventing me from getting help, from seeing doctors or getting my meds on time. I have PTSD and I can't drive because of it. I need financial help. I've been approved for SNAP but I'm waiting on my EBT card. I'll be going to a food bank tomorrow to get some food but that still leaves me with no funds to buy medical marijuana. I've been experiencing nerve pain the last 2 months and hormonal migraines for a straight month. I live in FL and it's super heavily taxed and I'm having trouble getting my muscle relaxers re-prescribed because I need a specialist to prescribe them for my insurance to keep paying for them. I have no other proper pain management rn. Please help me. I know the global climate is at its worst right now and I feel guilty making this post with everything going on in Palestine but that doesn't change the fact that I NEED HELP.

If you can help in any way please, even suggestions on what to do better with e-begging would be greatly appreciated.

My Cash@pp: $ButtPirate27

I can also tutor you online in Algebra if you need a math tutor I can help with Pre-Calc and Trig too but I'm far too rusty on Calculus to tutor but I would gladly tutor for any financial help.

If you want more info on my situation I don't mind sharing. I've been on Tumblr for 11 years and barely ever posted about my own life. I know that there are definitely people here willing to help but there have also been a fair share of scammers so I understand the hesitation. Here's my cat Ares, something cute to look at. I want to get him a cat tree too and a bin to make a housed litter box for him.

[chaos-bringer-13]

First of all, I need to know if sewing is actually a whole subject in USA. Is it just... sewing? Not a stupidly long topic in shop class?

Second, you have sewing for boys too? That's so freaking cool. In Russia boys and girls are usually divided in two groups for shop class, and that's such bullshit.

Third... Why haven't I seen anyone use the fact that Tucker excels at not just computers but also sewing? That sounds like one of those facts that fandom picks up and runs with, and yet I haven't seen anyone mention it. Imagine Tucker sewing gothic clothes for Sam because she wants something unique, something she can't just buy. Imagine ghost king AU where Tucker helps Danny with his royal clothes. Imagine Tucker making a hero outfit for himself, combining tech and clothes, because Danny has his ghost look and Sam can make an outfit with plants, but Tucker has to make one. I HAVE IDEAS

[bronzetomatoes]

Tried to search for this video and youtube gave me a hotline

[shalom-iamcominghome]

Little celebratory thing: it feels like I'm getting a bit better at alef bet! Even though the vowel markings can feel redundant, they're still something I've been able to remember and recognize individually!

[inquisitor-julia]

me the entire summer: i just don't feel like drawing

me the second i have math hwk i don't want to do: if I don't draw Wren Lavellan stargazing right this second I'm going to scream

[subjxctseventeen]

just need everyone to know that i absolutely crushed the math puzzle in night springs episode 2

i'm getting such a good grade in video games (which is normal to want and possible to achieve)

[genuflectx]

Maaan I love the optimistic advice "keep practicing and you'll get better at art" as much as the next artist, but it always rubs me the wrong way when that evolves into "just keep practicing and you WILL 100% succeed and CAN get into the industry."

It changes from good general advice to implying you're just doing something wrong if still haven't "made it" yet. Not in the industry? Well, you just haven't worked hard enough, obviously, as if there aren't plenty of other factors that play into "succeeding” in a highly competitive industry like art.

Don’t let advice that’s supposed to be encouraging turn into something discouraging 😭

[ringneckedpheasant]

hyperfixating on an oc I know almost nothing about in a setting I know only slightly more than nothing about & that I only have 2 pieces of art for is fucking agonizing. it’s just me & my binder full of notes on graph paper & my pinterest board 😔