#starting with algebra
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vee1021 · 10 months ago
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Ugh, I have to do make-up work for school eeeewwww
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writingraccoon · 2 months ago
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Genuinely I think we as a society need to figure out why 90% of the population is unbelievably bad at math and hates it like why have we demonized truly one of the most important subjects in school
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onewantwo · 4 months ago
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imagine getting outscored in the drivers’ standings by a high schooler
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kusnechik · 5 months ago
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Guess I'll dump these all at once since they're from the same genre anyways
Hc that when he can afford it, Hyde just keeps wearing Jekyll's shirt because he's too lazy to change
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hana-bobo-finch · 3 months ago
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i love making low effort shitposts instead of properly explaining my oc’s lore 🥰🥰
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asterberry · 1 month ago
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Riddle HCs
Sharing a couple headcanons i have because Riddle is my favorite twst character and has been since day 1
• He has naturally wavy hair but straightens it
• Riddle has very faint freckles on his cheeks and across his nose, they're just barely visible when you get close to his face
• His favorite drink is a shirley temple, he first tried one at the mostro lounge when Cater forced
• Has a small collection of stickers in a sticker book, the collection first started when Ace gave him a hedgehog sticker (Floyd gave him his second sticker, a goldfish one, because he got jealous and, despite Riddle protestinf, he ended up keeping it)
• Kept a journal when he was a kid that included his time spent Che'nya and Trey, it was hidden under his bed and his mother never found it, but he stopped writing in it after his mother learned about him sneaking out (he started a new one after his overblot)
• Riddle sometimes watches educational children's shows (like Wordgirl, Super Why, Little Einsteins, etc. They exist in twst simply for my angsty headcanons) and movies because he never got to as a child and they bring him comfort
• He went through a very short phase of wanting a lip piercing, simply because he heard of someone who had one (either a character in a book or maybe his mom telling him about a patient) and thought it would be cool (sometimes he still wants one)
• He'd never admit it aloud but when Riddle orders a strawberry tart from the Mostro Lounge he always hopes Floyd makes it since his taste the best (this is my only slight FloRid headcanon im including. Sue me i love them)
• Riddle loves flower meanings and researched them a bunch when he was younger. He always puts in a lot of thought into flower bouquets
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lipshits-continuous · 7 months ago
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Got this for Christmas!
(this is the one with all those cool pieces of art!)
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currentlyinflames · 2 years ago
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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!!
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wheatormeat · 1 year ago
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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.
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bubbloquacious · 11 months ago
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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.
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proof-by-intimidation · 8 months ago
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alright, Lurie's higher topos theory hits so hard. Can't wait to get into it seriously
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paradox-crows · 2 months ago
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Found footage of me doing 2 days of intensive studying in the space of 6 hours:
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coyotetatertot · 10 months ago
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Promotional for Tate's company in my interp of A Better World AU.
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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.”
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justablah56 · 3 months ago
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sighs I need to move somewhere where I could make a living off of just exclusively tutoring algebra <- guy who remembered a free math course thing it knows about and has been doing algebra for fun for the last couple hours
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chaos-bringer-13 · 1 year ago
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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
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lipshits-continuous · 1 year ago
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Pretty neat that for a topological group G with identity element e, we have that π₁(G,e) is abelian
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