#application of algebra

[math1089]

The Power of Algebra in Finding the Missing Digit – Part 2

One cannot escape the feeling that these mathematical formulas have an independent existence and anintelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get moreout of them than was originally put into them.Heinrich Hertz Algebra, often perceived as a daunting challenge, possesses incredible potential to solve complex real-world problems. In this…

[writingraccoon]

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

[flying-fangirls]

How many Fausts do you think Noel had in the Prison Pits?

We know Arthur had one Faust during his three months in the pits (1:3). If we assume that this ratio applies to the rest of the pits, then at a rate of approximately 1 Faust per every 3 months, there would be 4 Fausts per year (1/3 x 12 = 4). We also know that Noel was in there for ten years, and so we must multiply our yearly total to find the decade total (10 x 4 = 40).

Therefore, we can assume that Noel killed and ate at least 40 Fausts during his time in the Prison Pits.

[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.

[uranglaisteach]

My dumbass getting a degree in engineering getting absolutely burnt out by math but then looking into PhD programs and seeing they want me to take MORE MATH THAN MY DEGREE REQUIRES 💔💔💔 give me a break I’m begging

[beaniebabs]

i don't wanna brag or anything but i finished my first college semester with 3 A's B)

[brandboosterx]

In this video, we delve into the art of solving logarithmic equations with different bases, demystifying the process for you step by step. Whether you're a student brushing up on logarithms or someone facing more complex problems, we've got you covered. Understanding how to work with different bases is crucial when faced with logarithmic equations. We break down the techniques, providing clear explanations using frequently used words to ensure that you grasp the concepts effortlessly. No more getting stuck on those seemingly perplexing logarithmic problems!

[supernovaa-remnant]

abt to face the horrors (my calc lecture)

[9nid]

🧠 Master Graphical Applications of Quadratic Equations – Part 2 | JEE Series Part-15 | By 9nid

🎯 Introduction Quadratic Equations are one of the most tested and important topics in JEE Mathematics, and understanding their graphical behavior can make even the toughest questions feel easy! In Part-15 of the JEE Series by @9nid, we dive deeper into the Graphical Application of Quadratic Equations (Part-2) — focused on visualizing inequalities and interpreting expressions using…

[edusquaremaths]

🧠 Master Graphical Applications of Quadratic Equations – Part 2 | JEE Series Part-15 | By 9nid

🎯 Introduction Quadratic Equations are one of the most tested and important topics in JEE Mathematics, and understanding their graphical behavior can make even the toughest questions feel easy! In Part-15 of the JEE Series by @9nid, we dive deeper into the Graphical Application of Quadratic Equations (Part-2) — focused on visualizing inequalities and interpreting expressions using…

[perfectdiction-ary]

“Well, algebra is a tool, like the plow or the hammer, and a good tool to those who know how to use it.”

Round the Moon - Jules Verne

[math-journal]

Application Problem

[then-ponder]

Mathematicians will see this and be like: “is there a surface that doesn’t have the property in which four points A,B,C,D with the relationship that |AB| = |CD| and |AD| = |BC| aren’t tangent to the surface at any orientation around a given Origin.

https://youtube.com/shorts/FArZQkQ3Ah8

[associativeglassdesert]

gonna throw a curveball here and say fav algebraic topology least fav differential topology

Math enthusiasts of tumblr. What math subjects have you studied and which ones were your favorite? Which ones were your least favorite? Which ones were the hardest?

[supernovaa-remnant]

astronomy is my passion but calc... calc is my enemy

[as-if-and-only-if]

I'm interested in forming a sort of...math & physics reading group network. well, with some very important modifications to the concept of "reading group".

for example, right now, I'd like to learn algebraic geometry, qft, and/or refresh myself on representation theory with someone—maybe just one or two people—meaning that traditionally, we'd pick a text for one of these topics, discuss the material (asynchronously or synchronously?), exchange exercises, etc.

but currently (being Between Institutions), my best bet is posting on tumblr. and that's a pretty good bet, tbh! there are a lot of us here!

though, wouldn't it be great if there were a way to coordinate groups like this across institutions? you make a post proposing a group, specifying your goals and constraints...

even that would be a boon. but I think the concept of a "reading group" itself could be changed in interesting ways. this is what I'm really interested in.

there are variations among reading groups themselves already. sometimes you have directed reading groups, where someone already knows the material and "leads" it; some people are looking for more or less people involved; and there are probably things to explore for making sure that reading groups stick through it instead of falling apart when some motivation flags. default meeting times help with this, for example.

there are many experiments to be done! I think lessons for group-making can be taken from a maybe-surprising source: theater. there are a lot of things that make groups which put on shows more robust and rewarding than reading groups. a sense of building to something; many factors that create informal group cohesion (e.g. such a structure should make sure it creates more-informal "cafe" time in addition to more-formal "practice" time, just as rehearsal in physical spaces facilitates that casual sort of interaction on its periphery); ways to get into the right headspace during discussions (just as warm-ups do in theater; the engagement with this material is an event); clear goals (e.g. "understand ___"); successive shared accomplishments...to that end I wonder if it makes sense to form math troupes, which do successive reading groups together, drawn from its members.

it might be useful to envision some sort of public-facing artifact created as the culmination of this learning, whether a presentation, or an article, or some novel application or research...the crucial question is: how do we choose a goal that we find meaning in?

one idea, for example, is to have a collection parallel reading groups learning different things, and end by presenting to each other! that way we know what we learn will be meaningful to others, too, from the beginning. in general, I think it's important to feel that our own development of insight and understanding can be meaningful to others and to the group. it's nice to participate; it's nice to be able to offer something that is valued. what form can this take? how can you set up the interactions such that everyone has a part to play, and so this meaningfulness is tangled up in participation in the group?

I've also got a couple of ideas for "activities" that let us engage, re-engage, and play with the concepts we're learning with each other, beyond the text itself. how can we give ourselves the opportunity to toss around the concepts we're learning? I believe that the fun ultimately comes from the understanding itself, and therefore that any group exercise which lets us effectively play with the ideas will be fun.

it's a lot to ask people to come up with structure like that themselves, but using a pre-existing structure is not so difficult! sort of like how it's hard to make a TTRPG itself, so simply saying "go off and roleplay" isn't that helpful, but it's easy to use the structure of an existing one to run a game.

you might say, well, the existing form of a reading group is fine. okay! existing reading group structures can be low-stakes, relaxed, and accessible...but they can also fall apart easily (especially when not tied to an institution, in my experience), and you have to get lucky to find a truly rewarding one. I find reimagining our mechanisms of learning pretty exciting, and I think the space of ways to learn math with each other is underexplored at this level (emphasis on the with each other). there's a lot of potential!

anyway! reply or tag with "!" if this is something you could maybe be interested if done well? or if you're at least curious! I'm just taking a temperature. :)