/mg/ = /math/ general: Completions Edition

Im studying about Illuminati.

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>>9036358

Studying basic tensor calculus (aka index shuffling for physicists) and trying to into abstract algebra.

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How do you train you brain (besides solving multiple exercises)? Do you do anything specific? I think solving puzzles in general helps.

>>9036475
>What about his videos on algebraic topology?
They're good for the most part, same with the diff geo parts.

How do I get into differential geometry and topology ?

>>9036358
Reading Abstract Algebra by Dummit and Foote. I haven't been too dedicated, only on chapter 5 since I started this summer (busy with programming things)

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Are semi math related jokes allowed? If so pic related is worth a look for a laugh.

I'm currently learning quantum field theory. I'm not very far, just coming to grips with interacting fields and Feynman diagrams. Been using Mandl/Shaw and Lahiri/Pal. Going to read some Zee and Peskin/Schroeder. Zee especially looks very interesting since he starts off with path integrals.

>>9037148
The important thing is to make sure you're trying difficult problems and content. Solving a bunch of exercises is pointless if they're too easy. Actually solving problems isn't necessary either. As long as you're thinking and making an attempt you'll learn. Pick a topic, theorem, or problem that interests you and try to learn/prove/solve it.

>>9037160
Start with Spivak's calculus on manifolds if you don't know that stuff yet. I lightly recommend Hicks, it's pretty old with awful typesetting but gave me good intuition. Lee's smooth manifolds is a much more modern and highly recommend text so give that a try.

>>9037160
You'll want a background in (multivariable) analysis and point set topology. If you don't already feel comfortable with that, do that first.

For Differential Topology, I'd recommend:
>Differential Topology by Guillemin and Pollack
>Topology from the Differentiable Viewpoint by Milnor

For Differential Geometry, I'd recommend
>Introduction to Smooth Manifolds by Lee
>Differential Geometry of Curves and Surfaces by Do Carmo

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>>9037160
Heinrich Guggenheimer.

>>9037178
This is shit

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Another good textbook. So long anon.

>>9037415
Howard anton's kill this in one shot

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How do I into algebraic geometry?
t-brainlet with only an undergrad courses.

>>9037416
Found the engineer.

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>>9037431

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Just started this one recently, I rather like it so far. Short, and too the point.

>>9036358
I'm studying introduction to differential equations.

I got an 85 on my first midterm. really bummed about it because I made a few stupid mistakes.

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>>9037415
Why do people like this book? I don't really get it. I've given it a try many times but it's just hard to read, even now after knowing plenty of abstract algebra. His treatment is painfully concrete, this is most apparent in how he treats the determinant in the first chapter by just giving a formula, no motivation at all. Hoffman and Kunze has served me much better, although his treatment of determinant still isn't enlightening. Evan Chen's napkin has a section on determinant and trace which is extremely nice, though you have to know a bit of algebra.

>>9037437
That is a good book. Though later on it goes a little crazy with the characteristic polynomial for everything. I'm not convinced getting trace and determinant from it is really all that sensible. You could take a look at Hoffman and Kunze as well, it's a bit harder than Axler but not that much.

>>9037431
Milne has the notes you want
http://www.jmilne.org/math/CourseNotes/
Here's some more books with AG background material and basic AG
http://dmat.cfm.cl/library/ac.pdf
https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf
Did you say you want more possible books to read you fuck? Well let me tell you, when I wanted to learn AG I went to mathstacks and found a pretty good set of answers that worked out for me.
https://math.stackexchange.com/questions/285201/path-to-basics-in-algebraic-geometry-from-hs-algebra-and-calculus/285355#285355
https://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne/57019#57019

>>9037457
>muh motivation
If your textbook doesn't use this definition it is shit tier:
https://math.stackexchange.com/questions/21614/is-there-a-definition-of-determinants-that-does-not-rely-on-how-they-are-calcula/21617#21617
(I'm being facetious.)

>>9037473
that's in hoffman&kunze...
multilinear forms and even the grassman ring

>>9037457
Just a formula is how determinants were historically defined in the first place. Abstract stuff like this >>9037473 is a recent development. True of most things. Are you also dissatisfied with textbooks that define the derivative by the limit of a formula?

>>9037473
>>9037479
That's also the definition in Chen's napkin I mentioned
https://usamo.files.wordpress.com/2017/02/napkin-2017-02-15.pdf
I actually really like definition. All you need to do is define a vector product which is bilinear and preserves area signature (v x w = - w x v), construct the space of r products over a vector space of dimension n, then consider a generic linear map f:V->V and define a map on the product space via mapping v x w to fv x fw. Now the dimension of the space is the binomial coefficient n!/(r!(n-r)!) so when n=r it just has dimension 1. Now the product space derived map is a linear map on a 1 dimensional space so it's just multiplication by a constant, which is what we call the determinant of f.
>>9037480
The historical development of a subject is not always the best way to go about teaching it. There's plenty of simpler motivation for the determinant besides the admittedly abstract one above. Just consider each column as a geometric vector and consider a a signed area of the shape created.
Also I am dissatisfied by defining the derivative with a limit, at least initially. A limit is a fairly complex notion compared to the simple notion of change in y over change in x that the first users of calculus used. Thompson's calculus made easy treats it using differentials which is very easy for students to grasp and merely requires algebraic manipulation and geometry. After this you can easily state limits and it will be a well motivated rigorous definition.

How should I learn homological algebra? Also, how much category theory do I need for homological?

>>9037487
The motivation is how determinants are used. If you can't develop your own intuition behind concepts then maybe math isn't for you.

>>9037494
intuition is passed along
concepts should be motivated
expecting newcomers to a topic to develop all the intuition themselves is ridiculous. that's why talks are given

>>9037494
But >>9037473 isn't using determinants at all. It's just some assumptions, a wedge product, which leads up to both an interpretation of the determinant in general vector spaces and it also gives a method of calculating them. Nowhere does it attempt to use the determinant to solve a problem.

>>9037498
I'm sorry, but you simply cannot teach intuition.

Your motivation should be the drive to understand, as fully as possible, the topic you're trying to learn. If that isn't your motivation, then you probably should be in some other field.

>>9037492
people usually learn both at the same time. grab any algebraic topology book and it will teach both

>>9037416
Either a high schooler or engineer. I read Anton is high school bro

>>9037513
not motivation as drive to keep going, you asshole
motivation as in why was this definition the right one? etc etc

you cannot teach general ability to grasp intuition, but you sure as hell can communicate the main idea of a topic and the gist behind the main results that lead to the right definitions quite concisely

is there any difference between [math]\all x,y[/math] and [math]\all x \all y [/math]

I see the first as quantifying all pairs x,y while the other quantifies for each x all values of y

but they to me seem to result in the same pairings of x and y

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>>9037457
Shilov's book is pretty comprehensive for its size and introduces notions as they are used. It also has a neat little go-between with category theory at the end. Hoffman's and Kunze's textbook is very elegant but is not how most mathematicians actually use the tools of linear algebra. I wouldn't recommend it as a first book on the subject.

Also, Shilov's book has an [math]aesthetic[/math] cover that fit that image.
>>9037431
Give this a read later on, after you have some notions of how algebraic geometry is served.

>>9037558
>I see the first as quantifying all pairs x,y
No, it's just shortened notation for [math] \forall x\forall y [/math].

>>9037591
I see, thanks.

>>9036358
So my professor in Real Analysis keeps writing the real line as [math]\phi \neq s \subseteq R[/math]

Is this notation correct? Because its annoying me and I'm not sure you can combine them like that.

>>9037604
Obviously not, the correct symbol for the empty set is [math] \emptyset [/math]. Also, it is standard to write R as [math] \mathbb{R} [/math] to make it clear that it is the real numbers we are talking about not some random ring [math] R [/math].

>>9037604
>the real line
>Is this correct?
No.

>>9037611
I'm not sure how to get the Real symbol in latex which is why I didn't put it.

Thanks for noting the difference between the empty set symbol and phi. I've been saying its phi for years and only now realized that its different.

>>9037612
In what way? Is it wrong to say that the nonempty subset of reals is the real number line? I was just thinking the notation is funky and needs to be separated into two statements.

>>9037431
Atiyah-MacDonald and then Hartshorne.

Maybe skip chapter 1 of Hartshorne and read something better for classical algebraic geometry.

>>9037620
>In what way?
"""Real""" numbers don't really exist.

>>9037620
https://tex.stackexchange.com/questions/58098/what-are-all-the-font-styles-i-can-use-in-math-mode
Also, there is nothing wrong with that notation. It's a contracted form of [math] (\emptyset \neq s) \wedge (s \subseteq \mathbb{R}) [/math].
(Since we're on the topic of autism, stop reddit spacing or fuck off.)

>>9037431
read Vakil's notes

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>>9036358
>can understand all the random shit I've talked about here
>still think Wilson-Kadanoff renormalization group is black magic
When will we have a mathematical theory of renormalization group(oid)s? I feel like all I'm doing is stumbling around a dark room when all I have to go on are examples.

>>9037668
You might be interested in what Hairer is doing then
https://www.youtube.com/watch?v=5ZTRqVeSUKI
https://arxiv.org/pdf/1303.5113.pdf

>>9023781
>>9023784
>>9023786
>>9036071
>>9036337
I got it:

first, we rewrite the curve equation as a polynomial [math] f(x,y) = x^3 - 4x - ky^3 + ky [/math]. It is clear that this polynomial is irreducible over [math] \mathbb{C} [/math] so by the genus-degree formula its genus is 1.
next consider the homogenized polynomial
[math] F(X,Y,Z) = X^3 - 4XZ^2 - kY^3 + kYZ^2 = X(X-2Z)(X+2Z) - kY(Y-Z)(Y+Z) [/math]
it is obvious that [math] F [/math] has a rational point, for example [math] (1 : \frac{1}{2} : \frac{1}{2}) [/math].
all is left to show that it has no singular points so we need to find if there are values of [math] k [/math] for which the partial derivatives
[math] \dfrac{\partial F}{\partial X} = 3X^2 - 4Z^2 [/math], [math] \dfrac{\partial F}{\partial Y} = -3kY^2 + kZ^2 [/math] and [math] \dfrac{\partial F}{\partial Z} = -8XZ + 2kYZ [/math] don't vanish.
let [math] X,Y,Z \neq 0 [/math]
from the first derivative we get [math] X = \pm \cfrac{2Z}{\sqrt{3}} [/math] from the second [math] Z = \pm Y\sqrt{3} [/math] (if [math] k \neq 0 [/math]) and from the third [math] X = \cfrac{kY}{4} [/math] which gives [math] k = \pm 8 [/math]

so the answer is [math] k \in \mathbb{R} \setminus \{ -8, 0, 8 \} [/math]

has >>9034896 posted the answer?

what i proposed

>>9036129
>>9036090

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>>9037719
I'm kinda surprised it took almost a whole week for someone to post a solution to the easiest problem. I was under the impression that there are a lot of curious high-schooler and college freshmen here, given how often people asks for textbook recommendations.

>>9036358
I was just at a university summer school to study mathematics, because I have no formal education in the subject and they wanted some proof of my abilities before they would accept me. I am some 95% sure I failed, but in that month I learned more about mathematics than I had in the rest of my entire life. I'm going to study basic mathematics over the summer, and if I really did fail, I suppose I'll move on to linear algebra and single variable calculus to fill up my free time over the next year.

What is there beyond topology
Is it the end

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>there are true statements which are not axioms which cannot be proven
Does this bother anyone else?

what level of algebra do i need to start doing proper alg topology

>>9037949
God is hiding behind them.

>>9037949
No, because the only unprovable statements are self-referential.

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>>9037966
Be careful with that word "only".

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Anybody here had any experience with fragrant sets?

>>9037977
>Anybody here had any experience with fragrant sets?

No. Things like "fragrant" are made up definitions for competitions. No one studies fragrant sets. Just look at other competition math problems in number theory and you'll see many define some concept.

>>9037966
> the only unprovable statements are self-referential
Which I might add, are largely vacuous. Mathematicians have actively tried to avoid self-referential statements since the dawn of time anyway. It's pretty interesting how it has always been non-mathematicians who made a fuss over Gödel's result.
>>9037971
Are you trying to say that there are statements that are true but unprovable that are not self-referential?

>>9037982
>No. Things like "fragrant" are made up definitions for competitions.

That would explain why I can't find anything on it~

>>9037986
>That would explain why I can't find anything on it~

Yeah, it's pretty common. I like competition number theory problems so I'm working on it. i already proved that b has to be bigger than 1. I'll post if I reach something.

>>9037983
The problem is that some statements could be "stealth-ly" self-referential in that they could depend on self-referential statements in convoluted ways.

Anyone can post that math iceberg infographic

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>>9037989
I'm not sure if I'm on the right track at all as I don't have much experience with this kind of math but, I'm guessing that a 'fragrant' set would not include prime numbers, which would mean that the elements for the above set we need to find would be located somewhere in between two prime number solutions for x^2+x+1.

I tried setting a = 0 and then kind of brute forcing through to see what I could find but I didn't get anything.
I found two sets where it would have been fragrant if not for one out of place element each time.

I really have no idea how else to go about this problem, was thinking of picking it back up after I sleep.

How do I begin to learn algebraic topology?
t-undergrad

>>9037983
>Are you trying to say that there are statements that are true but unprovable that are not self-referential?
Not that guy, but there definitely are. The whole point of Godel's theorem is that it produces unprovable statements that are just innocent number-theoretic properties, along the lines of "there is an x such that for all y there is a prime p such that F(p, y) = x". There properties also have a self-referential *interpretation*, which is why they are unprovable; but at the same time, there's a boring and innocent number-theoretic property about prime numbers that is unprovable yet true.

>>9038015
I forgot to mention but my thinking was if I could find this set of solutions that satisfy the fragrant property with a = 0 then a would equal whatever b-1 is and b would equal the cardinality of the set.

Hopefully this isnt all obvious~

>>9038025
The self-referential aspect is there whether you want it to be or not. The scare-quotes are unnecessary and misleading.
It's Gödel btw. or Goedel if you can't type an umlaut.

>>9038015
>I'm guessing that a 'fragrant' set would not include prime numbers
Is there a reason you don't think it could include exactly one prime?

>>9037977
let f(n) = n^2 + n + 1 for brevity
use polynomial gcd and modulo arithmetic to find out the following:
f(n) and f(n+1) are always coprime
f(n) and f(n+2) can only have common factor 7, and thats only if n=2 mod 7
f(n) and f(n+3) can only have common factor 3, that happens when n=1 mod 3
f(n) and f(n+4) can only have common factor 19, that's when n=7 mod 19
now go through b from 1 to 5 and see that none can work:
b=1 obvious
b=2 cant work since f(n) and f(n+1) are always coprime
b=3 same reason when you look at the middle term
b=4 and b=5 similar reasoning

for b=6 you can match the numbers like this:
i want f(a+1) and f(a+4) have common factor 3
f(a+2) and f(a+6) common factor 19
f(a+3) and f(a+5) common factor 7
so all i need is
a+1 = 1 mod 3
a+2 = 7 mod 19
a+3 = 2 mod 7

which has some solutions (chinese remainder theorem)
so the answer is 6

>>9038160
Nice.

I am not the guy who asked the question but I was already up to proving that for b=4 it was impossible. I was already considering the problem in terms of congruences so I guess if I had kept going at it for 20 more minutes I would have got it.

>>9038154
Now that you mention it I do think it could include exactly one prime, thank you.
>>9038160
>>9038176
Thanks for the interest in the question guys i'm loving the replies.

>>9038216
>Thanks for the interest in the question guys i'm loving the replies.
Where did you get it from?

>>9038218
2016 IMO questions

>>9038069
>The self-referential aspect is there whether you want it to be or not.
Yes. But so is the number-theoretic aspect. Self-reference or no, the theorem still shows that there is a clearly-meaningful true property on natural numbers that is unprovable.

>The scare-quotes are unnecessary and misleading.
What scare quotes?

>It's Gödel btw. or Goedel if you can't type an umlaut.
I know. I'm just lazy.

>>9038264
Nice. Keep making more.

>>9038025
does that have any implications on actual research?
even in the unlikely event that such unprovable statement would be of interest to number theorists, it would be only unprovable in some first-order theory for example PA
but literally no one cares about proving things in PA so the question would be eventually resolved by other means

>>9038264
can you do one of these, but with dumb girl holding some babby-tier book

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Well this is awkward

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>>9038296
>Well this is awkward
What about it is awkward?

>>9038288
The whole point of Godel's first incompleteness theorem is that you can move to a stronger theory than PA (say ZFC), but then within that stronger theory you can apply a similar construction to get a new set-theoretic statement that ZFC can't prove.

The second incompleteness theorem then gives a constructive example of such a statement, usually written as Con(ZFC), and establishes that it cannot be proved in ZFC unless ZFC is inconsistent (i.e. unless ZFC also proves 0=1). And so on.

And the existence of Con(ZFC), or Con(T) for a general theory T, has pretty much shaped the development of modern set theory (via model theory), so I'd say it has had significant "implications on actual research".

>>9038235
>Self-reference or no
The self-reference is crucial.

>>9036358
>what are you studying
I just learned that you can solve cubic equations in planar geometry with the mathematics of paper folding
my dick is harder than neutronium. so hard that it violates all known laws of physics.

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Anyone read Byrne's Elements?

https://archive.org/details/firstsixbooksofe00eucl

Overall it's a very good piece of design. In most cases, demonstrations are either confined to a single page, or occupy two facing pages. Things get a bit silly in book V however as the graphical devices are made to stand in for algebraic variables, which introduces a little semantic confusion.

Also found this cheeky fucker (pic related).

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I took the GRE yesterday. This question on the math was the reason I didn't get a 170: "How many integers from 1 to 2000 (inclusive) are both the squares and cubes of integers?"

The answer choices ranged from 3 to like 44 or something. Please show your work.

>>9038419
The first is 1 = 1^6

Then 2^6 = 8^2 = 4^3

Then 3^6 = 27^2 = 9^3

Then 4^6 > 2000 and thus there are only 3.

Is GRE an american exam? Because this shit is literally first grade arithmetic. I can't believe american grad students can't do it. The US is a fucking joke.

>>9038419
>he literally can't calculate [math]\lfloor 2000^{\frac{1}{6}} \rfloor[/math]
Woe is you.

>>9038419
[math] 2000 = 2^4 \cdot 5^3 [/math]
[math] x^2 = y^3 [/math
[math] x = y \sqrt{y} \in \mathbb{Z} \implies \exists a \in (1, 20 \sqrt{5}) \cap \mathbb{Z},\
y = a^2 [/math]
[math] \lfloor 20 \sqrt{5} \rfloor = 44 [/math]

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>>9038458
What the fuck are you doing, retard?

>>9038465
No, what the fuck are YOU doing. Why'd you answer the babby's question?

>>9038469
I'm not doing anything.
Now I'm even more confused.

>>9038472
Good.

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>>9038478
is it?

>>9037487
Thompson has proofs with infinitessimals that have
to be completely refactored to work with limits.
E.g. derivative of sine goes something like:
[math]dy=\sin(x+dx)-\sin(x)=\sin(x)\cos(dx)+\cos(x)\sin(dx)-\sin(x)[/math]
And then by small angle approximation:
[math]dy=\sin(x)+\cos(x)\cdot dx -\sin(x)=\cos(x)\cdot dx[/math]
and then divide by [math]dx[/math]. The standard(?)
proof given by Wikipedia is completely different.

>>9038294
Why in the world would I edit pictures of 3DPDs?

>>9038524
It's actually the same idea. It just makes the small angle identities for sinx/x and (cosx-1)/x as being 1 and 0 respectively rigorous by using limits and squeeze theorem.

>>9038294
>>>/wsr/

Fuck off.

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>>9037689
This has a very Strocchi-like flavor, i.e. studying regularity via functional analysis instead of operator algebras like with second quantization. This looks very interesting and promising, though it's already quite well understood that critical theories (e.g. Heisenberg ferromagnet) can be Haldane mapped to field theories (e.g. [math]\phi^4[/math]) so I guess this is still very much in development.
Guess I gotta put "constructing a general mathematical framework for renormalization" under my bucket list alongside "constructing a general mathematical framework for AdS/CFT".

>>9038725
>Massive cowtits as a drawbak
>He doesn't want mummy's milkies like the good little faggot he is
Ask me how I know you're an algebraicist.

>>9036358

Should it bother me that I can't do any excersizes in my book?

I've read it carefully and if it were a book for a course I would have the prerequisites

>>9039049
Yes
if you can't even do the textbook problems you really don't understand shit about what you just read

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studying green chemistry and atomistics

The sum of all spaces inside each unique n-dimensional unit sphere adds up to roughly 43.

>>9037949
No. I have divine knowledge. Proofs are merely an exercise for the human mind.

>>9037949
Just use a system which can prove every true statement if it bothers you that much.

>>9037940
Category theory

>>9039270
Anonymous homosexuals.

>>9039277
Proof?

>>9038022
wildberger

>>9039287
These literal faggots: >>9038864 >>9038967

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>>9038967
Why is it so hard to find more people with taste like this? It's extremely common to see headpatting and cuddling but it's usually the guy doing it to the girl. I need more content of the opposite way which is the superior way. I want to be headpatted, cuddled and told "shhhh" and "it's ok now" by a tender girl.

>>9039206
is one of the axioms of that system "every statement is provable except for this one?"

>>9039375
>Why is it so hard to find more people with taste like this?
Because most /ss/ patricians keep quiet about their tastes, lest they be called manlets.

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>>9039411
Incorrect. How unfortunate.

Ultimately, every man seeks to fuck someone who resembles their own mother. This is a subconscious fact, and your consciousness is not aware of it because your subconscious intuitively interprets it as blasphemy, so it retains this. You then spend the entirety of your life searching for someone just like your mother: feminine, caring, loving, and who will raise kids who were once like you, and having whatever else features you like (or not having whatever features you dislike) about your mother. Facing your own inner fears is something that requires balls, it's not anyone who can do it. I'm much more of a man than you will ever be, kid.

And I'm 6'4.

>>9039437
>Ultimately, every man seeks to fuck someone who resembles their own mother.
Speak for yourself.

>>9039449
Ask me how I know you are insecure.
Read Freud.

>>9039457
>Ask me how I know you are insecure.
How?

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>>9039437
>Ultimately, every man seeks to fuck someone who resembles their own mother. This is a subconscious fact, and your consciousness is not aware of it because your subconscious intuitively interprets it as blasphemy, so it retains this. You then spend the entirety of your life searching for someone just like your mother: feminine, caring, loving, and who will raise kids who were once like you, and having whatever else features you like (or not having whatever features you dislike) about your mother.

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>>9039375
Yeah, I agree, it's very fucking patrician. I hope to one day be good enough for an thicc amazonian qt3.14 mommy gf (a man can dream).
>>9039399
What's /ss/ anon?
>>9039411
It's much more mommy issues desu, but you're also dead on the money.
>>9039457
>Read Freud.
Kek. But I also want to know how you knew he was insecure.

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>>9039449
>>9039461
>>9039465
>think they're smart & knowledgeable
>haven't read Freud
Leave this thread, and kill yourselves.

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>>9039475
>he thinks reading Freud makes you smart & knowledgeable

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>>9039478
>He's in /mg/ and he actually misconstrued a very simple statement of a necessary condition into a causal relation
It's nice to be among fellow brainlets desu.

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>>9039461
>>9039471
He bounced off what was said in a very defensive way. Like >>9039478, he read what I said not to understand but to respond. It's the kind of person who takes hasty decisions, is highly disagreeable with, probably had plenty of misunderstanding with his father during teen age days. All of this is the reflection of a troubled subconscious that avoids making decisions by picking whatever choice seems to be the easiest, the closest to reach. Ultimately, he is afraid of blame, afraid of being guilty.
>>9039475
They don't even need to read Freud. That kind of thing is usually taught in schools. Unless you are American or from some random shit hole, that is.
>>9039478
It's elementary knowledge any person should know. It's as fundamental as, say, Nietzsche.

>>9039497
>He bounced off what was said in a very defensive way.
If "speak for yourself" is somehow considered defensive, how does one respond to a statement like "Ultimately, every man seeks to fuck someone who resembles their own mother" where you claim to know the perspective of every man without being defensive?

>>9039497
Sounds like a load of bullshit to me, but ok.

>>9039501
You are doing it again.

>>9039505
>You are doing it again.
What is "it"?

>>9039507
Showing your insecurity.

>>9039508
I am not insecure.

>>9039508
It's easy to make blanket statements and just repeatedly call people insecure for not accepting them, regardless of whether the person is insecure or not. So once again,
>If "speak for yourself" is somehow considered defensive, how does one respond to a statement like "Ultimately, every man seeks to fuck someone who resembles their own mother" where you claim to know the perspective of every man without being defensive?

>>9039510
How old were you when your parents divorced?
>>9039512
You don't. It's a fact that has been proven over and over.

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>>9039514
>How old were you when your parents divorced?
Did I say something wrong?
[spoiler]Try to guess.[/spoiler]

>>9039514
>How old were you when your parents divorced?
How long ago did you binge watch The Mentalist?

>>9039514
>It's a fact that has been proven over and over.
Ah so psychosexual complexes are "proven" to universally apply, this makes more sense now.

Are you sure you read Freud?

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>>9039517
>>9039518
I am a Physics PhD who likes studying stuff as a hobby not your therapist.

>>9039524
Too afraid to guess the wrong number doc?

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>>9039524
>The subject is obviously evading my question about The Mentalist because of his deeply rooted need for social validation. Interestingly, he still commits to this practice even in an anonymous forum.
So did your father play ball with you when you were a child?

>>9039524
>I am a Physics PhD
No one cares

>Math general
>It's full of weebs
I don't know why but it cracks me up. Recommend me a good measure theory book fuccbois.

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>>9039554

>>9039554
Stein Shakarchi

Folland

>>9039554
The one by Tao seems pretty good, but I don't have experience with the subject so I'd like to hear other anons' perspective on this.

>>9039559
>>9039561
>>9039564
Can you elaborate on why you think these are good?

>>9039562
I'm always up for anime pussy.

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>>9039572
1.- It's motherfucking Tao
2.- Recommended by proffessor
3.- Started from the bottom up to pretty nice and relevant stuff
4.- Very interesting exercises

In general, it's a book designed to be worked through by hardworking students on their own.
>>9039562
Get away from my Anonymous-kun!

>>9039564
>>9039588
Tao is a brainlet though.

>>9039367
If that's your idea of a "proof" you shouldn't be posting in these threads.

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>>9039657
*Blocks your path*

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>>9039686
*recommends you John Oliver*

>>9039689
>How many layers of liberalism are you on?
>Like, maybe 5 or 6 right now, my dude
>You're like a little baby, watch this
>TRUCC
I wish I were good enough at MS paint to make this a reality

I don't want to do this. I love math but I can't bring myself away from it. I'm obsessed with it and I have to be doing it. My life revolves around it, my expectations, my pride, my love for myself - all revolves around my capability of doing math successfully and competently.

I don't want to grow up like this anons. I'm 19, and if I don't do math I beat myself up harder and harder to the point where I'm seriously contemplating suicide. I work so hard at it until I'm repulsed and it perpetuates a cycle of horrible emotional behavior.

I don't want to commit my life to math. I don't want to enslave myself to my own expectations. I want to die, I want to leave, I just want to get out, but math's all I got. I've had girlfriends, I've had sex, I've had hobbies, I've had hobbies, I've had some pretty good times in my life, but they're all temporary. I cringe when I don't do work on weekends, I cringe at the thought of socializing now, I cringe at the thought of leaving the confines of my room if not to do work.

I can't bring myself away from it. I need help. What the fuck is happening? Is there a scientific explanation? I don't want to die like this - I'll die alone. I'll never reach my expectations, and I'll never truly love myself.

Please fucking help me

>>9039735
>math: not even once

>>9039737
I'm serious. I used to think this was a good thing, that I knew what I wanted to do, but I consistently keep setting higher and higher expectations and I'll end up killing myself.

I love math, but it's literally consuming my life, and I want off this ride

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>>9039270
Does anyone know?

>>9039735
>Please fucking help me
Sorry, I don't help reddit-spacers.

>>9039735
>I've had girlfriends, I've had sex
You sound like a huge faggot.

>>9039747
Absolute savage. Stay blown the fuck out nigger.

>>9039735
>>9039745
I think I know how you feel, except instead of pushing myself harder and harder I slip away further and further. I don't even remember if I've ever asked for help, but if I did it's all lost to the fog that is my mind. Why is it that you want to excel at maths? What makes you push yourself to this limit? Answering these and more questions will help other anons help you.

>>9039750
>>I've had girlfriends, I've had sex
>You sound like a huge faggot

Well, I feel compelled to say this else one of the first replies I would've got is "lel you need get laid permavirgin fag". I'm becoming too fucking antisocial to even hangout with girls by this point, because any kind of relationship would distract me

>>9039754
>Why is it that you want to excel at maths?

I like it and want to be very good at it. I can't find any other reason tbf

>What makes you push yourself to this limit?

I need to be good, really good, and if I'm not, I'm better off dead. That's what I tell myself.

>>9039772
But why do you want to be good at it in the first place?

>>9039772
>I need to be good
You won't be, might as well off yourself right now. People who are inherently good at math don't act this way.

>>9039375
Were you molested by an older girl/woman as a child? I also have similar fetishes and I was toyed with sexually by two of my older cousins.

>>9039782
How does that make you feel? And how did it make you feel back then? Not him, I'm the guy he replied to.

>>9039772
>lel you need get laid permavirgin fag
This is math general.

Honestly you're actually in a better spot than most people believe it or not. You have identified what you want to do in life and are making steps to fulfill it. Most people just pretend to have realistic dreams and of course they never actually try and eventually fail. Your problem is that you beat yourself up too much. You're allowed to make mistakes and you're allowed to take breaks. You can enjoy doing other things and still be good at math. Mistakes are how you learn things.

But ultimately you should be having fun. The people that are the best at math are the ones who don't care about being the best. They spend everyday doing math problems and they love it. That's why they're the best, people who are doing it purely for competitive or validation(as in being good at something) just can't compete. Just because you're good at something doesn't mean you should devote your life to it. Devote it to something you enjoy instead. Hope you get yourself sorted out eventually.

>>9039779
Because I like it. It's just who I am I guess. If you're not really good at something, don't bother with it.

>>9039780
Never said I was inherently good at math. I just work really hard.

>>9039793
Thank you anon, your words have resonated with me.

>>9039794
>Never said I was inherently good at math
Which means you can just kill yourself right now. You won't be good at it.

>>9039772
>spacing
>>>/r/eddit

>>9039793
>Most people just pretend to have realistic dreams and of course they never actually try and eventually fail.
I'm one of those people. Any tips?
>>9039804
Do you really think this? And are you inherently good at maths?

>>9039790
Conflicted now and extremely confused back then.

>>9039806
stop shitting boars with this idiotic meme, go back to your shit website if you like it that much

>>9039825
It's not a meme and it's not idiotic. Kill youself you fucking plebbitor.

>>9039825
Shut up, disgusting piggot.

>>9039834
responding to random people just to sperg out about your website for idiots because you don't like the way they lay out posts is the most retarded meme you could possibly come up with, you fucking moron

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I saw an interesting undecided question posed
right here on /sci/ as follows: does this series
(the "Flint Hill" series) converge, yes or no?

>>9039841
I'm thinking "yes" but I'm not prepared to prove it.

>>9039841
the proof has to do with the Liouville theorem
about algebraic numbers, and its counterpart
(I don't know the name) about transcendentals

>>9039554
Terence Tao's book "An Introduction to Measure Theory" is a pretty good book for self studying. I particularly enjoy Tao's exposition. His textbook style really makes you reinforce the material by having exercises as you read. However, due to this kind of organization, I would say that Tao's book is a poor reference book, but I am not recommending this book to be used as a reference, but as a book to learn from. After you finish this book, you can follow it up with his "An Epsilon of Room Part 1". Also, it is legally free on his website, so you can grab it and if you don't like it, then you can find other books.

Real Analysis by Stein and Shakarchi is also good. Tao's book makes reference to this book, and there are similarities between the two, but I like Tao's organization of the topics more than SS.

Folland's book is a good reference, but it is horrible for studying the material for the first time. I would say Royden's book is better than Folland in that regard. Many people complain about Royden's fourth edition, due to its large number of errata(which is indeed true, it has 20 pages of errata), but I would say it is a good book otherwise. The amount of errata really is a shame though.

All of the books I mentioned above except for Folland start of by an example of a measure on Euclidean space, and then slowly abstract the process. Folland(as well as Rudin's Real and Complex Analysis) prefer to do everything with the utmost generality, which is an approach that I do not like when learning the material.

>>9039848
>the proof
How is there a proof if it's an open problem?

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>>9039706

>>9039871
How do you read "TRUCC"?

>>9039841
I looked at the Taylor expansion about
[math]n=n_i[/math]
where [math]n_i[/math] is the numerator of the i-th convergent of π
and the series seems to converge.

>>9039871
If you made this meme yourself, what program did you use? Also:
>In how many dimensions are you simply connected?
>Like, maybe 5 or 6 right now, my dude
>You're like a little baby. Watch this
>RICC
With Perelman on top of meme man. If someone can make it or recommend me a tool to make it myself, I'd be deeply indebted to that person, and I'd repay them with unquestioning and eternal gratitude.

>>9039841
No one knows. Stop trolling with this series. It is an unsolved problem.

>>9039893
Use GIMP. It's good enough for memes. Examples of what one can achieve in GIMP:
>>9037411
>>9037415
>>9037586
>>9038264

>>9039900
Ok anon, thanks.

>>9039893
Please pay me in anime images. One cute image will be enough.
>>9039900

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>>9039912
I hope you find this image as /comfy/ as I do.

>>9039893
I used paint.net and I originally intended to completely reverse engineer the original, but couldn't figure out how to do the 4th panel and decided to wing it

it took me the better part of an hour (11 times longer than any shitposter should ever dedicate to anything) so I probably won't make any more

>>9039811
>Do you really think this? And are you inherently good at maths?

I'm a CS major. That should speak for itself.

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>>9039960

>>9039974
>implying he's not implying he's shit at maths
i could believe it. ~80% of CS majors in my country are shit at math and are just in it for money. maybe he's one of the few among that 80% which is self aware.

>>9039811
>Do you really think this? And are you inherently good at maths?
Yes to both.

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>>9039897
no U

>>9040003
Care to elaborate? Perhaps enlighten us to your ways? How you fly by uni with As without attending lectures/reading the texts because you grasp concepts intuitively? You're a LARPing fag on /sci/ bro, """inherently""" good at math just means you work hard and understand topics.

You're just projecting insecurities onto the other anon

Also, being a CS major is a joke and they're notoriously bad at math>>9039960

>>9040264
>Care to elaborate?
Sure, if you rewrite your post without reddit spaces.
>being a CS major
I didn't claim to be a C* major.

>>9039811
>Any tips?
Suicide.

>>9039960
I have taken some CS courses they were all complete jokes, I honestly don't know why you people are taken seriously.

At one point in the second! semester the professor spend solid 20 minutes explaining some trivial induction to show that some search algorithm is in a certain "O(something), seriously infuriating.
Especially no one of the CS student seemed to grasp it and the professor continued explaining.

If you have a brain and bother to put in the time to remember what a constant virtual abstract constructor is then CS is a joke.

A 4D chessboard has 8^4 tiles. A king can move to at most 3^4-1 other tiles. A queen can do any number of king moves in the same direction at a time. How many white queens does it take to checkmate a black king in the worst case? Ignore the 50 moves rule.

>>9040428
4.

>>9040394
>constant virtual abstract constructor
That is what they call ""pr*gramming"", not CS.

>>9040486
But you also learn how to pr*gram in CS...

Else none of these retards would be able to get a job.

>>9040489
>But you also learn how to pr*gram in CS...
Which is absolutely retarded. I don't know why anyone would waste time "learning" to pr*gram while "studying" "CS".

>>9039735
cringe

>>9040493
That is exactly why CS is such a joke it is a some math plus programming.
But programming is not hard to learn if you have a brain, most of it is just memorization.

Forget about CS and find the ratio between the area of a square inscribed in a circle and an equilateral triangle circumscribed about that circle.
You have 5 minutes. Any additional minute you take past that point halves your score.

>>9040506
>CS
>pr*gramming
No. Most of CS is a joke for completely different reasons.

Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.

>>9040556
B- bait.

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I just realized something. Since for any Field [math](\mathbb{K},+,\times)[/math] with corresponding identities [math]0,1[/math]
[math]a\times 0 = 0[/math] follows essentially from distributivity, one cannot extend the notion of a field to an
algebraic object that is distributive in a symmetric fashion consistently.

By that I mean that taking a set [math]\mathbb{K}[/math] such that [math]\mathbb{K}^+ = \mathbb{K}\backslash\{1\}[/math], [math]\mathbb{K}^\times = \mathbb{K}\backslash\{0\}[/math]
form abelian groups [math]\left((\mathbb{K}^+,+),(\mathbb{K}^\times,\times)\right )[/math] and enforce
[eqn]a\times(b+c) = (a\times b)+(a\times c)\\
a+(b\times c) = (a+b)\times(a+c)[/eqn]
as well as associativity you'll find that [math]1 + a = 1[/math] and [math]0 \times a = 0[/math] being well-defined,
but expressions like [math] 0\times 0[/math] becoming ill-defined for any case where [math]\mathbb{K}\ne\{1,0\}[/math]
(I did not check that one, however).

>>9040582
That's wrong... [math] (\mathbb{Q}^{+},+) [/math] is not a group.

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>>9040592
>[math]\mathbb{Q}^{+}[/math]
I think you're mixing up notations. re-read how I defined [math]\mathbb{K}^{+}[/math] and notice that I am not talking about subsets of [math]\mathbb{R}[/math] at all.

>>9040592
yes it is

>>9040599
>>9040604
Are you saying your [math] \mathbb{K} [/math] unity cannot be the sum of any two elements in [math] \mathbb{K} [/math] ? Why?
>>9040604
What is [math] \frac{1}{2} + \frac{1}{2} [/math] in [math] \mathbb{Q}^{+} [/math] ?

>>9040613
identity*

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>>9040582
Suppose [math]0\neq a\neq1[/math]. Then [math]1+a, -a\in\mathbb{K}^+[/math], but then [math]1=(1+a)+(-a)\in\mathbb{K}^+[/math].

>>9040620
[math]a\neq-1[/math], sorry

>>9040620
>>9040582
How in the world can you add 1 when it's not an element of the purported additive group? This whole construction is batshit retarded.
>>9040592
It's not even a magma.
Addition is not closed in Q\{1}.

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>>9040630
IF [math]a\neq0[/math], then [1+a\neq1[/math], so [math]1+a\in\mathbb{K}^+[/math]. Similarly, if [math]a\neq-1[/math], then [math]-a\in\mathbb{K}^+[/math]. If we then suppose we have a group, we get [math](1+a)+(-a)\in\mathbb{K}^+[/math].

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>>9040599
>math]\mathbb{R}[/math]
No such thing.

>>9040640
1+a doesnt exist.[math] + : \mathbb{K}^{+} \rightarrow \mathbb{K}^{+} [/math] 1 [math] \notin \mathbb{K}^{+} [/math].
Please tell me you're just baiting me at this point.

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>>9040645
Is 1+a 1? If not, then it exists. My butt instinct is telling me that you aren't baiting, and I find that sensation quite terrifying. Please check the definition of that impossible additive group, and then you will see 1+a indeed is there, for every non-zero element a.

>>9040645
I meant K+ times K+.
Whatever. Addition with 1 is not defined.

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The way I realized this btw, assuming I did not fuck up, is rather straightforward but ugly.
If nothing else is stated, assume that [math]a,b,c[/math] are chosen in a way that neither 0 nor 1
appear in the equations implicitly.

First, you consider [math]a+(b\times 1) = a+b[/math], apply distributivity and find after
some substitution that [math]c\times (a+1) =c[/math].
Heck, if one wants to make no assumptions and case studies this expression is good enough.
Next, consider [math](1+1)\times(b+c)[/math]. Applying distributivity yields
[math](1+1)\times(b+c) = b+c+b+c[/math]
Yet, [math](a\times 1)+(b\times 1) = a+b[/math] but
[eqn](a\times 1)+(b\times 1) = a+b \\
a+b= ((a\times 1)+b)\times((a\times 1)+1)\\
a+b= (a+b)\times((a+1)\times(1+1))\\
a+b= ((a+b)\times(a+1))\times(1+1)\\
a+b= (a+b)\times(1+1)=a+b+a+b[/eqn]

I'll reply in a minute that took a while to type.

First off, here is the quick-and-dirty proof for everyone interested >>9040652

>>9040620
>>9040630
1+a is nontrivial to evaluate since you can only get it via analyzing it's algebraic properties

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>>9040650
You're retarded.
>>9040652
You're also retarded. This is "proof that 1 = 0" tier bullshit.

>>9040660
>""""""""proof""""""""
>>>/b/

>>9040582
>>9040652
Friendly reminder that this imbecile is a student (if not researcher) at UofBC. Isn't affirmative action great?

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>>9040660
Do you not understand how your construction blows ass because you end up having the prohibited element in your "group"?

>>9040662
Please tell me more. Show my mistake. You claim that element doesn't exist there, but don't want to give a proper explanation on why it does not exist.

These two posters are summer incarnate.

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>>9040662
no it isn't. It's in complete analogy to the proof that 0*1 = 0, which follows from (a+0)*b = a+b. Since 0 is not in K*, this is the only way to prove it. What I've shown is that you break something if both identities are treated symmetrically, since distributivity makes the unity of one group the absorbing element of the other. This yields inconsistencies.

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>>9040672
>>9040671
are you people retarded? Of course I can extend binary operations of groups to elements they are not defined on within the constraints of the group. How else do you define 0*a otherwise? * is not defined for 0 in the group (K*,*) but can be trivially expanded due to distributivity because the definitions needed for 0*_ are uniquely induced by distributivity.

>>9040688
You have not shown that extension dumb-ass. Addition by the multiplicative identity is undefined in your construction.
>>9040673
>This yields inconsistencies.
That's what I'm saying. The construction is nonsensical.

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>>9040688
You have [math]1\inK^+[/math]. Your whole construction is retarded.

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>>9040700
Not my Latex day today.

>>9040699
Yes, which sets it apart from a regular fields. That was my whole point to begin with. That you cannot make an algebraic object analogous to a field that treats both operators symmetrically. That I find an interesting conclusion to draw, since it means that the seemingly arbitrary break in symmetry induced by distributivity cannot be fixed on algebraic objects while it can in other contexts (there are structures that provide double-distributivity after all). This has nothing to do with the groups, it is a purely distributivity-induced phenomenon.
>>9040700
you fundamentally missed the point I was making, I did no such thing.

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>>9040715
One final time:
If [math]-1\neq a\neq0[/math], then [math](1+a), -a\in\mathbb{K}^+[/math]. Assuming it is a group, it must then contain [math](1+a)+(-a)[/math], but [math](1+a)+(-a)=1+(a-a)=1[/math]. Maybe you just forget about fields and go play with limits and stuff.

>>9040715
You have problems with the construction way WAY before any distributivity compatibility considerations kick-in. Refer to my original counter example: >>9040592 >>9040613
Let [math] a, b \in (K, +, \times) [/math] with [math] a+b = 1 [/math]. What is [math] a+b [/math] in [math] K^{+} [/math]?
If you start with a field and remove its [math] 1 [/math] addition is not necessarily well defined. All the symbol manipulation you do is irrelevant gibberish.
The only field this construction works with is [math] (\mathbb{Z}_2, +, *) [/math], because the additive group reduces to the trivial group after you remove [math] 1 [/math].

>>9040722
I never demanded the existence of -1. It does not exist in my setting. 1 is NOT in K+ like 0 is NOT in K*. It can't be, which is the trivial part of the whole ordeal. the fact that you can fix that but not 1+1 or 0*0 is the interesting thing. Those two operations are the problem. Everything else can be trivially fixed.

>>9040293
I don't like walls of texts and I've never even gone on the site you immense Cunt. And yes you did, see the post again.

Why the fuck did you censor CS?

>>9040739
I never said that K,+,* form a field in another context, or that + and * are operations that are even capable of forming a field. I was looking for a field-analogue which needs not to arise from a field. I thought that was fucking obvious the moment I set the definition.

>>9040493
Why does this nigga keep censoring words?

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>>9040740
>I never demanded the existence of -1
It is also there, because you start with a field, unless your field only has 2 elements. That is probably the only case in which you would actually get a group out of that retarded shit you posted. As long as you have more than 2 elements and you only remove the unit element, you will have the problem I have pointed out. This should be obvious, but it doesn't seem to be for you.

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>>9040753
>It is also there, because you start with a field
I didn't, you disabled fuck. I did not think I had to make myself so clear that I can't recycle the same symbol [math]\mathbb{K}[/math] to highlight the similarity without people thinking it was supposed to mean the same thing. [math]\mathbb{K}[/math] only belonged to a field in the very, very first line. Afterwards it is taken to be an arbitrary set which satisfies the initial conditions I demanded. Do I have to make myself any clearer or do you finally get it?

>>9040745
It does not matter. Say you start with some set [math] K [/math] with more than two elements, and define 2 distinct binary operations [math] + : K \times K \rightarrow K [/math] and [math] * : K \times K \rightarrow K [/math] such that [math] (K, +, 1_{+}) [/math] and [math] (K, *, 1_{*}) [/math] are groups.
What guarantees there do not exist any [math] a, b \in K [/math] with [math] + a\ b = 1_{*} [/math] or any [math] c, d \in K [/math] with [math] * c\ d = 1_{+} [/math] ?

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>>9040759
Be more explicit, fuckface. Do you even have an example of such sets other than a set of two elements?

>>9040763
In fact, you can prove the contrary: there always exist [math] a, b \in K [/math] such that [math] +a\ b = 1_{*} [/math] and [math] c, d \in K [/math] such that [math] *c\ d = 1_{+} [/math].

>>9040743
>And yes you did, see the post again.
That wasn't me. I though even you could infer that from the context, but I'll state it explicitly - I'm not a C* major.

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>>9040763
You mean besides the definition that clearly states that +: K+ -> K+ forms a group (K+,+) and *: K* -> K* forms a group (K*,*)? It's all there. The group statements explicitly say that 1 and 0 are not included, see above. 1+x and 0*x need to be defined afterwards. You will find this consistent with the standard definition of a field that never explicitly defines multiplication with 0 as there is only one definition that arises consistently from distributivity.

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>>9040772
>you can prove the contrary
Go ahead.

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>>9040769
I can't since I've shown that no such set can exist. But for that I had to assume it's existence and then find a contradiction within that. In retrospect I admit I was being sloppy. Sorry for that. I did however explicitly say it was meant to be an arbitrary set with the given properties so I assumed it was clear that I would not talk about the same thing since it otherwise could not exist.
>>9040772
see >>9040775

>>9040775
>besides the definition
Forget the field-like construction. Show me a model of your two groups.

>>9040786
I did not care to construct one. {0} and {1} of course work. But I did not see any immediate value in thinking of groups that satisfy that condition beyond that. Since I was more interested in seeing a potential field-like play out I studied that first and realized if such groups exist, they do not connect interestingly like that.

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>>9040785
The K being both a field and that set confused me, np. It does work for sets of 2 elements. You will then get a pair of trivial groups, and this would give a trivial gorilloid, but the gorilloid sturcture would be justified iff there would be a non-trivial example. Sry for being mean.

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Would anyone have and recommended reading for financial applications of, for example calculus. I've just flipped my undergrad major from Major-Finance Minor-Mathematics, to Major-Mathematics Minor-Finance, and the I'd like to have the right groundwork for building my knowledge base. Maybe something that someone would benefit from learning fairly early on, to hedge against bad information I may encounter thought out my academic career.

>>9040779
[math] + 1_{*} 1_{*}^{'} = 1_{+} = +x\ x^{'} \implies +x\ x^{'}\ 1_{*}^{'} = 1_{*} [/math]

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>>9040801
It's fine man, a heated argument is better than not getting to talk about it at all. Thank you for having been patient with me.

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>>9040815

>>9040799
>{0} and {1} of course work
I already know that. Show me a non-trivial model of your two groups.

>>9040812
Cheeky. I like it.

>>9040819
Like I said, didn't construct one since I first assumed them to exist and found it would not matter if. Shame, kinda. But if you find a proof that such a pair can't exist it would be immensely interesting.
Sadly I gotta go, hope the thread is still up in a few hrs.

>>9040824
> I first assumed them to exist
But what if they don't exist? If assuming stuff was all it took dragons would be flying in the sky as a type this.

Let [math] \{0\},\{1\} \not\subset S \neq \emptyset [/math], and [math]K=S\cup\{0\}\cup\{1\} [/math].
Define [math] K^{+} = S\cup\{0\} [/math] and [math] K^{*} = S\cup\{1\} [/math].
Let [math] + \subseteq K^{+}\times K^{+}\times K^{+} [/math] with the following properties:

[math] \forall (x,y)\in K^{+}\times K^{+}\ \exists z\in K^{+} : (x,y,z)\in + [/math]
[math] \forall (x,y)\in K^{+}\times K^{+} : ((x,y,v)\in +) \wedge ((x,y,w)\in +)\implies v=w [/math]
[math] \forall x,y,z,u,v,w,p \in K^{+} : ((x,y,u)\in +)\wedge ((y,z,v)\in +)\wedge ((x,v,w)\in +)\wedge ((u,z,p)\in +) \implies w = p [/math]
[math] \forall x,y \in K^{+}: (0,x,y) \in + \implies x = y [/math]
[math] \forall x,y \in K^{+}: (x,0,y) \in + \implies x = y [/math]
[math] \forall x\in K^{+}\ \exists y \in K^{+} : (x,y,0)\in +[/math]

Then [math] (K^{+}, +, 0) [/math] is a group.
Similarly, [math] (K^{*}, *, 1) [/math] is also a group for some [math] * \subseteq K^{*} \times K^{*} \times K^{*} [/math].
This completes the construction of a model of the pre-gorillon structure.

>>9040833
The groups would have been merely means to the end of providing the algebraic structure. But it's a good question.

>>9041134
yeah one should be in principle able to brute force it into existence like that.

>>9041134
To simplify the notation we write [math] x + y = z [/math] whenever [math] (x,y,z) \in + [/math], [math] xy = z [/math] whenever [math] (x,y,z) \in * [/math] and [math] -x [/math] for the [math] y [/math] that satisfies the last property of [math] + [/math] (uniqueness of [math] y [/math] is left as an exercise to the reader), respectively [math] x^{-1) [/math] in the case of [math] * [/math].

Question: how does one extend [math] + [/math] to [math] 1 [/math] and [math] * [/math] to [math] 0 [/math] such that they are compatible with the group structures [math] (K^{+},+,0) [/math] and [math] (K^{*},*,1) [/math], respectively?

Since all inverses in a group are unique, we can see that [math] 1 + 1 = 0 [/math] and [math] 0*0 = 1 [/math] are the only possible extensions, whereby [math] 1 + x = y \implies 1 + y = x [/math] and [math] 0x = y \implies 0y = x [/math] (and similar relations for the commuted cases).
But, [math] 1+x = y \implies 1+x+(-y) = 0 \implies x+(-y) = 1 [/math] which is absurd, as [math] x+(-y) [/math] already has a value in [math] K^{+} [/math]. Similarly [math] 0x = y \implies xy^{-1} = 0 [/math] is again not possible in [math] K^{*} [/math].

So [math]\forall x\in S : 1 + x = x + 1 \in \{ 0,1 \} [/math] and [math]\forall x\in S : 0x = x0 \in \{ 0,1 \} [/math] are the only possible values left.
But [math] 1 + x = 1 \implies x = 0 [/math] and [math] 1 + x = 0 \implies x = 1 [/math] both of which are absurd. Similarly, [math] 0x [/math] is impossible to define.

This proves that any group operation is impossible to extend to a single new element in a way that is compatible with the preexisting structure (of course, this is a known fact).

In short: [math] 1 + a [/math] doesn't fucking exist.
The gorillon is impossible to define for non-empty [math] S [/math], i.e. the trivial gorillon [math] \{ 0,1 \} [/math] is the only gorillon there is. Distributivity of operations doesn't even begin to matter.

/bump

>>9041581
uh, no: suppose you don't want the extension of [math] + [/math] to [math] 1 [/math] to be compatible with the group structure but you want a monoid [math] (K, +) [/math] instead. Then it's possible, for example the trivial extension [math] \forall a \in K\ a + 1 = 1 [/math] works.

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>>9041652
yes, that is what I found. But that falls apart quickly if you assume too much.
>>9041581
Distributivity matters once you assume + and * to be well-defined, but not invertible outside their original range. In that case you have to oppose further restrictions on the operations to make the connection unique. Since you however generally do not want to mess with the group structure, you demand 0 and 1 to be absorbing elements or something equivalent (as mentioned, distributivity implies the "inner" operation's unity to be an absorbing element). The issue is that trying to then fix everything you find that

1) you must drop both distributivities or
2) you must drop a group property
none of which I wanted to pursue further.

For example, one could set [math]0*a 0 a*0 = 0[/math], [math]1+a=a+1=1[/math] for all [math]K[/math] including the units.
Thus we can resolve all standard problems carried over by inversion (all equations containing 1+ or 0* follow special rules such as standard multiplication of the actual real number 0), but immediately run into issues if we only allow even one distributivity law, say [math]a*(b+c) = a*b + a*c [/math], because [math]a = a*(1+1) = a*1 + a*1 = a + a[/math].

So what could work is to build essentially a half-gorillion which is not distributive in either direction. I'm currently thinking of an instance.

>>9041713
[math]a*0 = 0*a = 0 [/math]*

>>9041652
>the trivial extension ∀a∈K a+1=1∀a∈K a+1=1 works.
it's the only one of two that works
a + 1 = b =/= 1 => 1 = -a + b impossible, -a + b already exists
a + 1 = 0 => 1 = -a impossible

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>>9041713
If we have no distributivity at all we can in fact write down many structures that work easily.
Take the cyclic group [math]C_2[/math] twice and let [math]K=\{0,1,a\}[/math]. Then [math]a=a^{-1}=-a[/math],
[math]0*a=0*0 = 0*1 =0[/math] and [math]1+a=1+1 = 1+0 = 1[/math].
This is essentially consistent, but boring, because you can't relate + and * in any way. This makes the notion of a field extremely unique, in a way. Two groups in which one has an extra element can produce a consistent algebra. Having two "interlocking" groups only yields trivial algebras. The next question would be whether you can make consistent algebras with more than 2 groups/binary operations that have more than one unique element in one set. However, that is far from my field of expertise and the notation becomes tedious.

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The death of the female fields medalist prompted me to do something I'd been meaning to do for some time: collect basic statistics on the history of the prize.

As of today, a supermajority of all medalists are still alive. Fully and exactly 3/4, or 42 of 56 medalists are still breathing, while 1/4 or 14 of 56 have left this vale of tears.

This can be explained by a number of factors. First, the Fields Medal is a relatively young prize, having been given out for less than a century, and only in its "continuous" format since 1950, following the war (many top-tier international prizes and competitions were interrupted in some way by the war, such as the olympics, the nobel prizes, and the world cup, which just happens to occur on the same years as the fields medal is awarded, and has been contested for about as long, in about as many instances: ~20 now).

Second, mathematicians tend to live dull and comfy lives. Third, of course, is that this short-lived prize is only awarded to younger people - math is a young man's game, and then you are free to keep going or putter around for the next several decades of life, if you are so lucky (and most are, in this case). The female medalists' death is all the more striking when seen in this order, against all the older men who are still alive. At a glance, I'm guessing that the female has also had the shortest lifespan of any medalist.

One wonders whether Ramanujan would have been so lionized had he lived a bit longer, or if the prize had begun issue a bit earlier.

Let [math]E\subset \mathbb{R}.[/math] Prove that for every [math]\alpha \in (0, 1),[/math] there exists an interval [math]I[/math] such that [math]m^*(E \cap I) \geq \alpha m^*(I),[/math] where [math]m^*[/math] is Lebesgue outer measure.

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>>9041983
Choose the interval to be a degenerate interval (a singleton).

>>9041987
I'm pretty sure there exists non trivial intervals that satisfy the property.

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>>9042014
Irrelevant when the assignment is to prove there exists an interval. It's like showing Euclidean spaces are manifolds: you can choose a ball nbd around a point and show it is homeomorphic to the whole space etc, or you can just choose the whole space as this nbd. IF you wanna get done faster, choose the one with less work.

>>9041983
>>9041987
Oops, that should be an open interval.

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>>9041983
>Let [math]E\subset \mathbb{R}[/math]
>[math]\mathbb{R}[/math]
But there is no such thing.

I was just able to solve a competition level problem and I feel like latexing the proof. Enjoy

(Putnam 1989) How many primes amongst the positive integers written as usual in base-ten are such that their digits are alternating 1's and 0's beginning and ending in 1?

All of these integers will be of the form [math] \sum_{k=0}^{n} 100^k [/math] with n>0 and by the finite geometric series formula, this is equal to [math] \frac{100^{n+1} - 1}{99} [/math].

But notice that [math] 100^{n+1} - 1 = (10^{n+1})^2 - 1 = (10^{n+1} - 1)(10^{n+1} + 1) [/math]

And therefore the integer is of the form [math] \frac{(10^{n+1} - 1)(10^{n+1} + 1)}{99} [/math]

But notice that because of the diference formula, [math]10^{n+1} - 1 = 9(10^n + 10^{n-1} + ... + 10 + 1) [/math]

And thus the integer is of the form [math] \frac{(10^n + 10^{n-1} + ... + 10 + 1)(10^{n+1} + 1)}{11} [/math].

Now notice that if n is even them [math]10^{n+1} + 1 \equiv -1 + 1 \equiv 0 \mod 11 [/math] and therefore 11 will divide that side of the fraction.

So for the number to be prime, with n even, it must be that [math] \frac{10^{n+1} + 1}{11} = 1 [/math]. This is only true when n=0, but we assume n>0 and thus this will never happen.

Now notice that if n is odd, then 11 divides the other part of the fraction as modulo 11, an alternating series of the form -1 + 1 -1 +1 - ... with an even number of terms will emerge, which will be equal to 0 modulo 11.

So for the number to be prime now it must be that [math] \frac{10^n + 10^{n-1} + ... + 1}{11} = 1[/math] and notice that this only occurs when [math] n=1 [/math].

So n=1 is the only remaining possibility for this number to be prime. Computing the corresponding number yields 101 which is indeed prime. Therefore, 101 is the only prime of this form.

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>>9042202
>pr*mes
Didn't even bother reading your garbage.

>>9042207
What is wrong with prime numbers?

>>9042222
The word "pr*mes" appearing in the very beginning of that "proof" already means it's fucking garbage.

>>9042235
The word prime does not appear until the end of the proof. It appears at the beginning in the statement of the problem, not the proof. And why do you censor the word prime?

>>9042238
>why do you censor the word prime
Because pr*mes are uninteresting garbage, just like the rest of your problem.

>>9042242
>t. brainlet who realized he wasn't good enough for number theory so now he is stuck in some arbitrary field proving weird theorems nobody likes or cares about in the hope that 300 years from now a REAL mathematician uses one of his theorems as a lemma in a proof about a statement in number theory: the queen of mathematics

Feels bad to have failed in life. Sometimes I wake up in the morning and think about what would happen if I did not end up being good enough for number theory and had to just study a lesser field. Just thinking about it makes me want to kill myself.

But anyways, even a brainlet like you should appreciate how a non-trivial statement about prime numbers can be proven almost entirely with algebra. That's pretty cool isn't it?

>>9042251
Who are you quoting?

>>9042253
No one, I am saying that if you do not appreciate prime numbers then you are a brainlet.

>>9042251
>non-trivial statement about prime numbers
No such thing. The entirety of number theory is trivial.

>>9042257
>No such thing. The entirety of number theory is trivial.
Sheeeiiiiiiiiiiiiiiiiiiiiiiiiit

Prove that there are infinitely many twin trimes then big boy.

>>9042256
>No one
Why did you quote this text then? ">t. brainlet who realized he wasn't good enough for number theory so now he is stuck in some arbitrary field proving weird theorems nobody likes or cares about in the hope that 300 years from now a REAL mathematician uses one of his theorems as a lemma in a proof about a statement in number theory: the queen of mathematics
"
>I am saying that if you do not appreciate prime numbers then you are a brainlet
Are you implying it is somehow possible to be a brainlet and a non-brainlet?

>>9042260
>Sheeeiiiiiiiiiiiiiiiiiiiiiiiiit
What sort of retarded dialect of English is this? Are you "Black" by any chance?

>>9042260
>Prove that there are infinitely many...
Assuming your system is consistent, that's impossible.

>>9042263
>Why did you quote this text then?
Greentext is not just for quoting, newfag.

>Are you implying it is somehow possible to be a brainlet and a non-brainlet?

[math] nonbrainlet \equiv \neg brainlet [/math]

So yeah.

>What sort of retarded dialect of English is this? Are you "Black" by any chance?

I'm not black but I have used the internet for more than 2 months unlike you so I guess I know more memes.

>>9042270
>Greentext
What is "greentext"?
>So yeah.
You do sound like one indeed.
>I'm not black
I find that very hard to believe.
>memes
What are these "memes"?

>>9042202
>number theory
>But notice that
>But notice that
>Now notice that if
>Now notice that if
>So for the number to be prime
>So for the number to be prime
I genuinely hope you're still in high school.

>>9042285
Sophomore university actually.

I agree that my writing is kinda lazy, I repeat too many words but seriously, what is wrong with number theory? Number theory is the queen of mathematics.

>>9042289
>Sophomore university actually.
I feel sorry for you. I'm not even kidding.
>""queen of mathematics""
Undefined.

>>9042292
>I feel sorry for you. I'm not even kidding.
I would feel bad if I had you in high regard but given that you are a brainlet that cannot appreciate the most important field of mathematics then this makes me happy because if a brainlet thinks I'm a brainlet then that means I am not a brainlet.

>>9042295
18+ please.

>>9042300
My age is a prime number so that means I am not 18.

>>9042295
Lurk for at least 2 years before posting.

>>9042260
Oh, look at him, he swears like a big boy!

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Fuck yeah, my ban expired just in time to post a problem that will trigger brainlets:

Let [math] P=A_1A_2\cdots A_k [/math] be a convex polygon in the plane. The vertices [math] A_1, A_2, \ldots, A_k [/math] have integer coordinates and lie on a circle. Let [math] S [/math] be the area of [math] P [/math]. An odd positive integer [math] n [/math] is given such that the squares of the side lengths of [math] P [/math] are integers divisible by [math] n [/math].
Prove that [math] 2S [/math] is an integer divisible by [math] n [/math].
>>9042202
Nice.

>>9040513
[math] \cfrac{2}{3\sqrt{3}} [/math]: you find the area of the [math] \Delta [/math] in terms of its median which is of length [math] 3r [/math] and the area of the [math] \square [/math] in terms of its diagonal which is of length [math] 2r [/math] ([math] r [/math] is the radius of the circle).

>>9040809
>polluting mathematics with finance of all things
Couldn't you at least minor in less disgusting shit, like physics?
Commerce kills Art.

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>>9037178
>not studying psychoanalytic knot theory

>>9042318
Who are you and why did they ban you? Also, sweet fucking Christ does your problem look like a fucking bitch to solve, and the worst part is that there's probably one weird trick required and then it's straightforward. Am I right?

>>9042318
>Anonymous 07/17/17(Mon)07:07:07
11000 / 7 = 1571.(428571)
>71
11000 = 11*5*5*5*2*2*2
(11 - 5*2) + 2 + 2 + 2 = 7
>7
5 + 2 = 7
>7
2*2*2 - (11 - 5*2) = 7
>7

2SPOOKY4ME

>>9042318
since coordinates are integers, 2S is also an integer
only question is divisibility by n;
try k=3 first: just plug stuff into heron's formula and the result is obvious since if you multiply the terms in the formula you are left with only squares of side lengths
similar formula exists for quadrilateral
so just use induction and trigonometry to deal with all k

>>9042202
impressive.

>>9042318
>>9042708
Wouldn't it be easier to do it by Gauss' shoelace formula?

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When exactly are the next Fields nominations decided and who's going to win the medal next year?

>>9042882
In August, next year.
>who's going to win the medal
I am 100% certain that either of Sophie Morel or Maryna Viazovska will get a medal. The IMU is dying to give the medal to another woman and in their case there exist excuses.
I think from now on a diversity quota will be implemented and the Fields medal will steadily lose its prestige.

I'm learning discrete mathematics at the moment, I'm enjoying it. But it's probably the most formal math education I've received thus far and I'm bad at articulating myself. For example;

"Rewrite the statement:
Everybody trusts somebody in formal language. "

Let, X = people, and
Y = a person
P(x, y) = ∀ x, ∃ y such that x trusts y.

This just doesn't seem as formal as the examples in my text.

>>9042938
The predicate is "x trusts y", so "everybody trusts somebody in formal language" should be [math] \forall x \exists y P(x,y) [/math]. Think of predicates as functions which output true or false for some given input.

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>>9042938
>"Rewrite the statement:
>Everybody trusts somebody in formal language. "

>>9042946
In that case, [math] P : X \times X \rightarrow \{ \text{true}, \text{false} \} [/math], where [math] X [/math] is the set of people and, [math] x,y \in X [/math] are persons.
You could of course, tentatively, think about a predicate that takes a statement about people as an input. The problem is that not all statements about people are well-formed (i.e. can be formalised in a rigorous manner) or decidable (i.e. the truth of those statements is undefined). Predicates should always have a definite value for a given input.
So if you want to have a predicate that has formulas which include quantifiers over some variables and such, you have to be extra careful with what types of formulae are admissible and whatnot.
For [math] \mathbb{T} [/math] being statements about people (formulae using elements from [math] X [/math], quantifiers and predicates which take persons as inputs), a predicate [math] P : \mathbb{T} \rightarrow \{ \text{true}, \text{false} \} [/math] might not be well defined. For most interesting universes of statements, any such predicate usually isn't well defined, not in a way that models anything meaningful in any case.
>>9042995
kek

>>9042882
http://owl-sowa.blogspot.ro/2014/08/and-who-actually-got-fields-medals.html
read the whole thread, the fields medal is a vile affair
good on perelman for refusing that poisoned chalice

>>9042995
>>9043028

This makes more sense.

I'm doing the practice questions in my textbook but I have no answers.

Let P(x) be the predicate “-10 < x < 10” with domain Z (set of all positive integers. Find the truth set of p(x).

Surely the truth set is just = {1,2,3,4,5,6,7,8,9}

>>9043094
[math] \mathbb{Z} [/math] includes negative integers, so maybe you meant [math] \mathbb{N}^{*} [/math], in which case that set is the right one.

>>9042917
>Maryna Viazovska
I wouldn't be particularly mad if she won, solving (at least a special case) one of the biggest open conjectures is no small feat.