# /mg/ - Math general

What are you learning, /mg/?

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What are some applications of vector spaces which do not have a basis?

everything i should have in the last 2 months since my exams are next week

want to keep working on first semester modern algebra homework
professor assigned our second exam online last week
exam was open book, open notes. instructions were to print, complete, scan, submit in a 2hr 15min period.
finished on saturday morning (two days ago).
0/100
my 98.8% average dropped to a 64.5%.
have no fucking clue what went wrong.
there is no way I missed every single question on the exam
either I submitted the wrong pdf (I don't think I did that, I remember double-checking) or I am suspected of cheating (which I didn't do).
worst case scenario, it was a bug in our open source online homework platform
I have fought against such bugs before, it rarely ends in my favor
swear to fucking god I'll withrdraw from enrollment and join the navy if this doesn't end well. I am tired of playing the online student game as a resident student.
I have no fucking idea what I should do, short of talking to the prof during his office hours tomorrow. Should I talk to my advisor? He's respected in the department. Would he be able to help if this goes south?

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Every vector space has a basis.

On the other hand, not every module has a basis (i.e. not every module is free).

He's a subhuman shitposter. Don't reply to him.

People like you are why sci sucks.

/mg/ is not the same as /b/, my dude. I've received solid advice from Veeky Forums before.

no u

Lurk or read the archive before posting, idiot.

Theorem: Proof.
QED.

Email the professor ASAP.

congesting /mg/ with pointless arguments

What are you learning, /mg/?
not >Crucify the shitposter and repliers, /mg/

this
especially if you can remember your answers, if even one is right, it means you hit a buggu

Yes I just barely passed my calc 1 class and am barely passing calc 2

Every vector space has a basis.
ZFC + ¬AC begs to differ.

Fugg I meant ZF + ¬AC lol

to be fair, ZFC + ¬AC also proves that some vector spaces don't have bases (:

Even in ZFC most vectorspaces don't have bases unless you assume they do

One of the biggest sources of difficulty in math is bad learning habits. People get used to not actually understanding what they're doing, and then they forget what actually understanding something is like in the context of math, and then their ability to learn math at all atrophies, because learning requires you to be able to actively engage in a process of fixing what you don't get.

Semirelated quote:
Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Working in the context of ZF, the axiom of choice is provably equivalent to the statement that every vector space has a basis. Therefore, it is true in ZFC that every vector space has a basis—the assumption you mention is precisely the C in ZFC.

Therefore, it is true in ZFC that every vector space has a basis—the assumption you mention is precisely the C in ZFC.
Right, like I said, they don't have a bases unless you assume they do

Yes, you are correct.

I think you meant ZF and not ZFC then buddy

Don't feed the trolls.

thank you user, I'll think about that

I think you meant ZF and not ZFC then buddy
Choice and existence of bases are equivalent in the category of axioms (i.e. isomorphic objects), just labelled differently like $\frac{1}{2}=\frac{2}{4}$

Good luck!

ksi
dropped

PDEs

I'm glad this is the last pure math course I ever have to take ;_;

Category Theory
github.com/hmemcpy/milewski-ctfp-pdf

I'm confused user. What do you think happens in applied math?

What's a good source to learn finite calcul us?

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Numerical solutions, what else?

Seperating math into "applied", "pure", "whatever", was a bad idea imo. Applied math is mostly considered to be anything that is concerned with PDEs and of course numerics. But numerics is not just using big calculators, the theoretical aspects of it are usually more complicated than a lot of "pure" stuff since you need a lot of different concepts from other fields like analysis, (metric) topology, combinatorics, graph theory...

Unless you don't study proper math, then yes, applied math is literally using calculators.

What is categorized as pure math

Am I an applied math or pure mathematician

Seperating math into "applied", "pure", "whatever", was a bad idea imo.
You can't separate something which is not well-defined.

Unless you don't study proper math
I'm an engineer

K, then you can probably relax a little in your applied math courses.
Really brings my neuronal schlongers into swinging motion. Also yes.
What is categorized as pure math
Usually everything that is not PDEs, numerics, stochastics, statistics. It's stupid though.

My rule is that if the professor uses the word "theorem," it's pure math.

applied math
No such thing.

math
No such thing.

This is a legal game of Connect Four. After some experimentation, I found a sequence of play (encoded as the natural number 2167127614743451375325622663157315723654) which ends with the second player (yellow) winning.

What's slightly interesting about this is that it illustrates how a /legal/ (although certainly not typical in "normal" play!) game may conclude with multiple "fours", obtained all at once, by the winning player. It also calls attention to the central importance of the game's center column, and especially the board's central two cells. Consider that in principle, either such cell may participate in one of thirteen game-winning four-in-a-rows; here, the winning move simultaneously generates eleven distinct fours in the same blow. The result is the more interesting in that upon inspection the board-state given happens to show the conclusion of a legal game (which may be reached by multiple distinct games, of course).

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I've stopped using khanacademy as a learning tool and I've found myself at a faster pace by using just a textbook, Paul's Online Math Notes, Professor Leonard and profrobob's youtube tutorials.

How long would it take for me to get to calculus? I am currently on Algebra II factoring.

Does a functor mapping spectra to (to the equivalence classes of) their Postnikov towers serve any purpose?

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Math simply, platonically, Is. Xie doesn't exist to entertain you.

That's retarded.

you guys are fucking lucky.
must be why people here talk about killing themselves so often huh

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must be why people here talk about killing themselves so often huh
That's just one of the female regulars

Applied math is math being used for real-world applications. Pure math is math for its own sake. The latter often grows out of the former, for example geometry came about because of a practical need in construction and land measurement, but when Euclid wrote his elements the primary concern was the mathematical objects and structures themselves.

It's so tempting~

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pure maths is for smart people
applied maths is where brainlets are redirected by default when they have less than a 4.0

What are you learning, /mg/?

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Not a single thing in mathematics implies intelligence. I wouldn't be surprised if the regulars here were mostly people who just memorize theorems and definitions for exams and then forget them, never understanding the stuff. This would explain why these threads are full of failures posting about categorical stuff and thinking they are cool, but really their posts are just barrages of buzzwords without any real content. An example of such a post would be as a reply to a legit question. These failures come together to these threads to post the same stuff every time, and some unlucky and innocent people fall for their traps every time, or are the other losers and start a shitpost fest by replying to them just for the sake of shitposting. Smartness isn't just having a high IQ, but also about being able to use it for something. These people may have almost 120 points as their scores, but for all mankind they have contributed nothing. This should itself prove you don't need to be smart to study math. On the other hand, your submissive attitude makes me think it is better for you to simply stop trying or even breathing, as you would only get trampled by people with stronger wills.

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Not a single thing in mathematics implies intelligence. I wouldn't be surprised if the regulars here were mostly people who just memorize theorems and definitions for exams and then forget them, never understanding the stuff. This would explain why these threads are full of failures posting about categorical stuff and thinking they are cool, but really their posts are just barrages of buzzwords without any real content. An example of such a post would be as a reply to a legit question. These failures come together to these threads to post the same stuff every time, and some unlucky and innocent people fall for their traps every time, or are the other losers and start a shitpost fest by replying to them just for the sake of shitposting. Smartness isn't just having a high IQ, but also about being able to use it for something. These people may have almost 120 points as their scores, but for all mankind they have contributed nothing. This should itself prove you don't need to be smart to study math. On the other hand, your submissive attitude makes me think it is better for you to simply stop trying or even breathing, as you would only get trampled by people with stronger wills.
cringe

then why do math and physics graduates have higher average IQ than other fields?

lang basic math

sometimes, it's better to ask anons on Veeky Forums, then friends and family. Actually Veeky Forums is a great place to ask irl questions, bc, literally there are thousands of people, who definitely know more than your family,who are much smarter, who work and study in many many different professions, who can unselfishly give you a good irl advice, and answer probably all of your questions.

Irrelevant. That is not needed to get a degree.

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Someone is jealous.

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Jealous for what?

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lang basic math
Lang is a meme.

it's a good meme

What's the geometric intuition for projective transformations?

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jealous of that hot 2D body

Your entire life is a meme, yet you are alive somehow. Why haven't you killed yourself user?

gpa
measure of intelligence

Die whore

I talked to him today. He gave everyone a zero on the exam because he hadn't finished grading them(!). Apparently, I was the fifth person to ask him about it since he posted the grades yesterday morning. You thought he would have posted an announcement or something on the class website?

Can someone please tell me why would someone in it's right mind prefer to define the tangent space in terms if derivations? Just so you can justify abuse of notation? I thought mathematicians weren't like this.

I cannot think of a good explanation, but you might want to look into some "mathematics for open gl" book. This might be a good source of intuition.

Give me some simple but tricky problems.

U just got troled!

Currently I'm trying to understand the classification of semisimple Lie algebras and their representation theory. Humphreys presents this pretty well, in my opinion.

Because it works for any locally ringed space. i.e. Manifolds, Analytic Spaces, Varieties, Schemes

PDEs
Eh PDEs can definitely be pure math.

Here's one that I came up with for myself. It's rather easy but you need a slight idea:
Let $n$ be and even, positive integer and let
$\pi: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ be bijective.
Show that there are $x,y \in \mathbb{Z}/n\mathbb{Z}$ with $x \neq y$ and
$x - y = \pi(x) - \pi(y)$.

Then why it's also used in more practical textbooks designed for physicists? Lie brackets make no sense with that formalism.or maybe I'm just a brainlet idk.

Lie brackets in that formalism are essentially just the commutator. Nothing really confusing there.

The derivation definition is also often more convenient for actually computing tangent spaces.

How does one go about actually computing concrete tensor products in practice?
I mostly understand how they're defined and what they do but I struggle to figure out on my own even very simple examples of them because there doesn't seem to be any obvious way of "solving" for it

how irresponsible of him, hopefully you end up doing well on the exam in that case

Put them in a basis and distribute the sum, but besides quantum entanglement and information, it's really not something done in practice I believe

Mathematical Statistics and Real Analysis. I've finished chapters 1-7 of baby rudin, and gonna start doing chapters 9 and 11 next. I've heard it's shit though. What's a good book to learn about the Lebesgue integeral/measure theory?

Also no one here knows any good stats books, huh

Not long. I would finish your algebra 2 stuff and then just dive right in with calculus. I did pre-calculus before calculus and I honestly dont know if it was necessary at all. Just make sure you learn some trigonometry as you do algebra 2.

Don't be scared of calculus, the operations are actually pretty easy. You'll be fine if you do well on the algebra 2 material.

tensor products of what

There are no good stats books. Statistics is a dung heap of a subject kept alive by it's practicality.

As for measure theory, there's a book by Robert Ash called Probability and Measure Theory that I really like.
I think it's a really good idea to teach measure and probability simultaneously, it's the easiest way to see that rigorous integration isn't just pointless autism but is actually good for something. Most other places where it's worthwhile to ditch Riemann are difficult to show to undergrads.

Crammed through 200 pages of linear algebra textbook without sleep

I just want to die

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and tomorrow you will remember 15% of it

studying only because I have a test on it tomorrow

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Statistics is a dung heap of a subject kept alive by it's practicality.
lmao, yeah its a weird subject. Somehow nothing is rigorous even in "mathematical" statistics. Thanks for the book rec, will check it out.

Tensor product of vectors could be made in the next way. You have a vector $a = (a_1, a_2) \in A$ and $b = (b_1,b_2,b_3) \in B$ . Vector $c \in C = A \otimes B$ will be $c = (a_1 b_1, a_1 b_2, a_1 b_3, a_2 b_1, a_2 b_2, a_2 b_3)$. I dunno what are you afraid of.

Tensor product of vectors could be made in the next way. You have a vector $a = (a_1, a_2) \in A$ and $b = (b_1,b_2,b_3) \in B$. Vector $c \in C = A \otimes B$ will be $c = (a_1 b_1, a_1 b_2, a_1 b_3, a_2 b_1, a_2 b_2, a_2 b_3)$. I dunno what are you afraid of.

I did. I felt fucking destroyed yesterday.
He's sharp when it comes to statistics and operations research related things. I get the feeling that he isn't so sharp with other things, mostly technology. He does most of his lecturing with this digital whiteboard and he fucks up the most basic shit (screenshots and copying-pasting images in Windows 10). I'm sure giving us all zeros was a quick-fix to a "bug" he ran into while grading. Who the hell knows.
Anything is better than a 0/100 after this shit, and it was only worth 20% of our final grade. Can't wait for this class to end desu. Operations research without spreadsheets is just fucking tedious.

Im currently majoring in Mathematics and I feel like im not hitting the potential that I could with amphetamines. My smartest friends use adderall, vyvanese etc. but I used to use meth as a young teen so I feel that I could relapse if I do this. Anyone here use Kratom as a performance-enhancer for mathematics?

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GTFO degenerate junkie.

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0.5 correlation is non-negligible

My smartest friends use adderall, vyvanese etc

I have a feeling that they are not very smart

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I need some form of strictly noncommutative algebra (i.e. a*b != b*a for all distinct (a,b)) for an proof attempt at representing combinatorical objects algebratically

The only noncommutative algebra I know of that can be calculated easily are matrices but they don't suit my needs. Are there any other such algebratic structures?

differential operators

$\sum_k f_k(x) \dfrac{d^k}{dx^k}$

How do amphetamines compare to nootropics?

Amphetamines are nootropics

As in nootropics like racetams, ampakines, etc

jacobian, hessian matrices, that kind of stuff. I'm a 2nd year math undergraduate in Paris-VI and I've an exam tomorrow

i thought the french liked abstract nonsense

is pi a group or ring homo? or is it a set function?

any group modulo the center

How does one go about actually computing concrete tensor products in practice?
That's more of an engineering question.

This would explain why these threads are full of failures posting about categorical stuff and thinking they are cool, but really their posts are just barrages of buzzwords without any real content.
What are you referring to?

Use them! They help a lot! Also there are no sideffects at all.

Is is common to think of the phase space for a system with 2n degrees of freedom as a complex vector space of degree n?

Refer to Veeky Forumscatalog#s=eg%2F.

Many things wrong. Phase space is in general a manifold and it doesn't have to be a vector space. Also, complex manifolds have many strong properties you really don't ask for a physical system. You can embedded a manifold in some R^2n and then identify it as C^(n) but it's not needed.

physical system
Refer to boards such as /toy/ and the engineering thread at Veeky Forumscatalog#s=en%2F.

Hamiltonian mechanics is math, you fool.

Not really. Maybe you should post in threads which actually discuss the things you are interested in? I suggest /toy/catalog#s=physics%2F and Veeky Forumscatalog#s=en%2F.

What's a good Android calculator app?

Right now I'm delving deep into the murky depths of econometrics, specifically classical panel data model tests and the linear algebra related to their proofs

Fucking hausman, I want his brain to tell me it's secrets

Veeky Forumscatalog#s=sqt%2F

I feel triggered after I got 77% on my combinatorics test today

It's a weird feeling knowing that I'm neither a complete brainlet nor a genius

How do I deal with this feel

Something for /math/ to know about: a decade ago, a newly minted Ph.D. died a gruesome and rather romantic death in rural Nebraska, which has become the subject of a memoir and a documentary film.

Steven Paul Haataja (pronounced Ha-deee-JAH), algebraist and longtime teacher, had spent his life in Minnesota, South Dakota, and finally Nebraska, where he earned his Ph.D at Lincoln, a place where he'd also taught in the past. Shortly after completing his doctorate in 2006, Haataja accepted a post in very rural Chadron, Nebraska and struck up relations with the locals - and a few months later, went missing. Still a few more months later, his body was found burnt to a crisp, and tied to a tree, off in a back country behind the local college. Since nothing ever happens in Chadron, the death became a frame story for local author and eccentric Poe Ballantine, and has become a sort of small town mystery. Ballantine's book (and later doco) of the same name, "Love and Terror on the howling plains of nowhere", chronicle all of this. Technically, the case remains open, and although the circumstances suggest foul play on its face, Haataja had also attempted suicide in the recent past, and is supposed to have gotten some coal and liquor on the night of his disappearance.

I have managed to discover two publications of Haataja's (an early one seems to have regrettably misspelled his name as Stephen), and his third, the dissertation. the two relevant algebrai items, C* algebras and all that:

t. a family friend retired to Chadron in recent years which is how I came to be aware of this story

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You could just make multiplication work like a free group. It's somewhat unsatisfactory, but it should work.

la fac

Why are there so many frogs here?

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Help me with an easy proof, /math/?
For G a transitive group on (1,2,...n) let Ki be the subgroup that leaves elements 1 to i fixed. Prove G = Sn if and only if Ki != Kj for all pairs i,j such that i != j and i < n-1.
the only if is easy but the only way I figured out how to prove the if is through construction generating first (n-1, n) and then going down the ladder. It was really involved compared to the other proofs in this book, which is at a pretty low level, so I think I'm missing something obvious. The section deals with normal cosets, lagrange's theorem on groups, and the subgroup generated by an arbitrary subset of a group. the previous problem was a proof that for Hi the stabilizer of i in a transitive subgroup of Sn, |G| = n * |Hi|

It's pronounced HAA-ta-ya, where Haa is pronounced like Haar but without the r, ta is like in tapir, and ya is like in ya'll niggas don't even smoke crack. This is how amerimutts twist even the names of their ancestors. Really makes me want to puke.

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B.C.-born professor awarded 'Nobel Prize' of mathematics
A Canadian mathematician has been awarded the Abel Prize — often referred to as the Nobel prize of mathematics, for a theory 50 years in the making.
Robert Langlands, 81, who was born in New Westminster, B.C., was awarded the prize for developing what the Abel Prize citation describes as a "grand unified theory of mathematics."

Isn't the Fields medal often referred to as the Nobel prize of mathematics?

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How do i prove the collatz conjecture?

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By inspection.

its trivial
so trivial nobody can be bothered to write it down

exhaustion

big
frog
brains

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Hey user this is neat!

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discrete math is literally shit

Engineering has a tendency to be literal shit.

true that. especially if studying on such a shitty uni, like i (unfortunately) do

Is there a legal game where more than eleven different wins are achieved? 13 seems a bit of a stretch, but maybe you can get 12, can you not?

Got some good introductions to set theory and some question sets?

Are question sets related to Freyd's solution sets?

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Probably not...

Nah, don't think you can have more than 11 wins on a legal game, at least I think that is the case. The horizontal and vertical lines are already full and the stones fall so yozu can't have mroe than one win vertically.

What may be interesting is the total number of wins possible on an n/n+1 board of 4 connect.
Or the total number of wins possible on an n/n+1 board of m-connect. (n,m are natural numbers of course)

The first person who replied is not the one who made the initial post FWIW, I am.

The game's available material (in this instance, of course) is itself a limiting factor. In order to actually have an arrangement with thirteen simultaneous four (in-a-rows), (leaving aside game legality!), the winning player would need to have laid down 22 checkers (imagine the pic being filled up in the last two spots with yellow checkers). But each player only has 21 checkers, and plays in turn either until a win or until the board is filled up with no win, producing a draw.

A paper on the solution of connect four. It's fairly autistic and more CS-y than math-y, but of interest.

informatik.uni-trier.de/~fernau/DSL0607/Masterthesis-Viergewinnt.pdf

Pic related is a heatmap of the board I whipped up quick. Each number represents the possible fours in which a given cell may participate. As you can see, center spots are at a premium.

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No. In general given a coordinate space $M$ the phase space is $M\times T^M[math], where your momenta are treated as vector fields on [math]M$. Hence the smooth structure as well as the differential structure is already fixed. You can only do it if you in addition also have a complex structure on M.

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halp with this? It's for self study so I'm never going to get told how to do it properly

nice TeX brainlet

on all levels but physical, you want to be a Calabi-Yau manifold.

Have you done any math today? I didn't know enough to actually understand a paper on arxiv, but it introduced quandles to me. I learned something new.

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But coalgebras and Hopf algebras seem pretty dank.

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You're cute :3

It works whenever $\pi$ is a bijection of sets. You don't need it to be a group homomorphism.

what do you know about group actions? My first instinct would be to try using the following theorem: A transitive left action on a set A is G-equivariant to a left action on G/H where H = stab(a) for any a in A

i havent cracked it but after a couple computations i think its actually stronger: for any x, there exists a y such that x-y=pi(x)-pi(y)

This doesn't seem to be right: on $\mathbb{Z}/4\mathbb{Z}$ take the cycle (2 3 4). Then for x = 0 there is no other y that fulfils the condition.

(1 2 3), I meant.

trying to learn some basic homotopy theory out of hatcher

will i ever stop forgetting that my maps have to be maps of pairs/triples? always makes me feel like a brainlet when it hits me

Or just Kahler

Yes, once you do enough of the proofs.

I grew up in Chadron. I was around when Haataja went missing. I know most of the people Ballantine interviewed in his book/documentary. I even studied math at Chadron State College, so I know the people and the program pretty well. If you have any burning questions, please ask. I may be able to answer them.
Ballantine probably wrote that pronunciation key. I don't know what the fuck he was getting at, no one in Chadron pronounces it Ha-deee-JAH.

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I wouldn't be surprised if the regulars here were mostly people who just memorize theorems and definitions for exams and then forget them, never understanding the stuff. This would explain why these threads are full of failures posting about categorical stuff and thinking they are cool, but really their posts are just barrages of buzzwords without any real content.
I agree agree with this part. But please don't post pictures of my girlfriend without her consent.

journalists

Why do you agree with it?

I haven't seen many insightful posts here. It's mostly circle jerking.

The last few months have been pretty shit, but I've seen plenty of "insightful" or at least helpful posts in the past which were made precisely by those 1-2 "regulars" who that post seems to disparage.

regulars

absolutely nothing, it's from an introductory text. I don't know what equivariant means, but I'll look it up I guess. My approach was this horrible proof by induction where I demonstrated that you can transform each element in Kn not in K(n-1) to the transposition (a,n) for each a in [1,n], to give you an idea of the level it's at. I just had this nagging feeling that there should be something about e order of the group being (n-1)! that I don't see

circle-up-to-homeomorphism jerking or circle-up-to-homotopy-equivalence jerking?

Holy fucking shit
I finally finished my linear algebra test
Went through 300 pages of material in 2 days

linear algebra test
What are you, some kind of first-year brainlet?

3rd year course
I regret taking this everyday

pic related
it's so incredibly boring I literally fell asleep in the only few lectures I attended

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It's your own fault for taking engineering courses.

1) Are you aware of any other mathematical works whatsoever having Haataja as an author, apart from the two I linked, and the dissertation? If so, please give a pointer.

2) Attached is a map giving approximate locations. In red at top is 200 1/2 Bordeaux (or thereabouts), in orange in the middle is the rough area where Math/Sci is located, and the big black dot is the approximate location where the body was found. All info taken from the film. Please let me know if any of this is factually wrong, or misrepresents locations.

3) What is your personal view on the circumstances of Dr. Haataja's death?

4) Did you happen to take any classes with Haataja?

5)

no one in Chadron pronounces it Ha-deee-JAH

If that is true, then why is it that in the film, community members Kathy Bahr (12:46), Poe Ballantine of course (31:35, 4:00 among others), DA Vance Haug (4:45), local paper publisher George Ledbetter (41:58) and grad student Steve Welch (1:00:32) ALL USE THE EXACT SAME HAA-DEE-JAH pronounciation which is clearly audible at 0:50 of the link?

map.

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Ah, I see. Then I have no problem with it.

5) here makes me worry, though.

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5. My apologies, I misinterpreted your pronunciation key. I thought the "JAH" in your key sounded like "jaw" instead of "yah."
1. I think you've found them all. I searched for his pubs in the UNL digital commons and found a partial copy of his PhD thesis (search.proquest.com/docview/305273734). He might have a master's thesis floating around too? He did finish an MS at UNL before tackling the PhD.
2. Your dots for 200 1/2 Bordeaux and the Math and Science Bldg. are definitely accurate. I am less confident in the placement of the black dot because I haven't actually visited the site of Haataja's death. The bearing looks accurate (southeast of Math and Science), but I am less confident in the distance. It's been a while since I watched the documentary, how did Ballantine describe the location of Haataja's death? Didn't he show a map?
3. When Haataja's body was first discovered, my mom was convinced it was a hate crime. She (and many others) thought Haataja was a nutty gay professor that a bunch of hicks decided drag out and kill. Once I learned that Haataja was not bound with barbed-wire at the time of his death (a rumor that was floating around Chadron at the time), I accepted the idea that he committed suicide.
4. I did not take any classes with Haataja, but I have taken classes with Cary and Vogl. I worked with Wentworth, but never took any of her classes.

Thanks for your replies. Since several posts have been spilt on the pronounciation issue already, with at least one admitted misinterpretation, I felt it was essential to get some audio in the conversation.

As to your question about the body site, a map in the film clearly shows the approx. location in relation to Chadron (a short piece SE of the college on the ranch land). It is (was?) a very small copse of trees on the land, it seems. And yes, the film shows a map and even shows Haataja's death certificate, Social Security Number and all (despite face blurrings on old photos), which is where I got the 200 1/2 Bordeaux address. To clarify with the above, this is what I assume to have been Haataja's final residence. That he was (?) apparently in the habit of walking to and from work and home for his brief time in Chadron would further validate the notion of him being able to take his final walk under his own power, his recent accident notwithstanding, if that's what happened.

Your mother's view also resonates with the initial thoughts depicted in the film. However, the film's "third act" basically gives up the charcol and liquor purchases, plus prior attempt, which is where things actually click for me. Upon viewing the film, I also came away with the impression that it is in fact a suicide, albeit a very strange one, and so I'm gratified that a local close to the story has the same impression. Haataja's depression and prior attempt, and moving out to rural Nebraska ffs (no offense) also validate this view.

As I say, I have slight connections. Dad's ex-bandmate moved in and has apparently played chess with Poe's autistic kid. The ex-bandmate gave dad the DVD, and now I have it which is how I got hip to all this. Also after me learning of all this, one time at the local grocery store a woman had a Chadron sweater and I mentioned that I knew it, without going into gory detail. She was impressed that anyone knew Chadron.

There's a small section on it in Concrete Math in case you haven't already read that

Concrete "Math"
Please don't litter. Put that garbage in a trash can.

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I hate this general.

user, I can't discern what level the book is at if you don't tell me exactly what you've learned about group actions, especially, because the theorem i posted is literally the second thing they prove in the book i looked up for reference. At least tell me what book you're using