/sci/ - Science & Math

>>11070543
why did floer commit sudoku?

>>11070543
Damn, why can’t all mathematicians look like this?

Damn, why can’t all mathematicians look like this?

Don't I have to distribute the (1+1/x) individually to the big fraction with the ln in the denominator? My answer was 1/(x^2 + lnx) but it's wrong. I thought fractions multiplied straight across.

>>11070803
>Don't I have to distribute the (1+1/x) individually to the big fraction with the ln in the denominator?
(a/b)=((a*c)/(b*c))=/=c*(a/b)=(c*a)/b

>>11070803
Anon, he literally just multiplied the numerator by the numerator.

>>11070809
Yeah, but when you multiply fractions, I thought you multiply numerators and denominators together, so you would distribute the x in the denominator of 1/x to the x making it x^2 and to the lnx making it xlnx

>>11070728
most people in the planet dont have good genes

>>11070890
>but when you multiply fractions, I thought you multiply numerators and denominators together
Yes, you multiply the numerator by 1+1/x and the denominator by 1. Since you are multiplying with the fraction (1+1/x)/1.
I have an idea what you are trying to do, but I think it is rooted in severe inexperience with algebraic manipulations, keep in mind that a*(b+c) equals a*b + q*c.

>>11070543
Bro, you must tell us the edition.

>>11070543
Given any set X, a metric d on X such that fora any x,y in X if x = y then d(x,y) = 0 and if x is not equal to y then d(x,y) = 1
Can anyone give me an intuition on why balls and spheres look like they do? For example any sphere with radius 1 equals to X\{x} !!! any closed ball with radii 1 equals whole space X and with radii just 0.9999999 it immediately shrinks to a single point {x} wtf

>>11070728
hole fuck, look at his nails

>>11070922
literally a discrete set of points all of them being at the same distance from each other

>>11070934
what? I take R^2, I set that distance it's not obvious why Ball placed in 0 with radius 1 suddenly takes all plane

>>11070958
>I completely defom a set's topology but keep visualizing it the same way because it's the same set
You understood absolutely nothing, reread the entire book.

>>11070543
Aside from academic careers, what else can I do with a Mathematics degree?
I am Math major but sometimes I feel like my future is blurred. Is there a chance for me to join an industry?

>>11070958
dude if you picture R^2 you're automatically thinking the Euclidean topology. if you change the metric, it's not a "plane" anymore, it has literally no relation to a Euclidean plane whatsoever. >>11070961 is right.

>>11070958
Because distance between 0 and any other point is 1. The entire concept of distance has changed feim the usual Euclidean distance.

>>11070962
Depends on your field of interest.
Discrete math translates well to computer science jobs for example

>>11070920
no edition edition

>>11070962
Statistics/probabilities/being able to process large amount of data is a useful skills in some private research groups.
The profile isn't really one of a pure mathematician, but someone comfortable with the software/code to get & process the data AND the mathematical insight behind it.

>>11070920
Floer Homology edition

I was asked to prove that
[math]\sup \{ x \in \mathbb{Q} : x^2 < 2 \} = \sup \{ x \in \mathbb{R} : x^2 < 2 \}[/math]
(the left set will be denoted A and the right B for convenience), I used the fact that A is a subset of B to show that sup A is less than or equal to sup B and then used the density of the rational numbers to show that sup A is greater than or equal to root 2. Combining these facts I deduced the equality. Was this a good idea?

>>11071115
>then used the density of the rational numbers to show that sup A is greater than or equal to root 2
If you can do that it is trivial to show that both sups are equal to sqrt(2), which proves the equality even simpler.
But I think what you did works just as well.

>>11071115
>I used the fact that A is a subset of B to show that sup A is less than or equal to sup B
correct
>used the density of the rational numbers to show that sup A is greater than or equal to root 2
needs more elaboration to know if you're right

anyway, I highly approve that you're proving an equality by proving two inequalities. you're doing good.

>>11071126
>>11071140
I'll take what I can get. It's an elementary analysis class tought by an extreme ESL teacher, (she's super enthusiastic I just can't understand her), and admittedly this pure logic shit just isn't my natural strength

What is the most basic math book? Because the ones in the wiki are too complex for me.

>>11071250
Euclid Elements

>>11071115
Let me guess: MAT157?

Threadly reminder to work with physicists.

>>11071115
You use the fact that the rationals are a subset of the reals to get that the left hand side can’t be larger than the right hand side. Use the fact that the least upper bound of a set of real numbers is, itself, a real number, and just use the density of rational numbers to show that the right hand side can’t be larger than the left hand side. And, so, equality is the only possibiliy.

https://www.codechef.com/IEMATIS1
>IEMATIS1 - International Mathematics Olympiad
>There will be 5 levels and each level will have one question.
>The answers will be integer type and each participant will have three chances to lock an answer for a particular question. After three unsuccessful attempts their id would be blocked and they cannot continue participation.
>Level of question will be of International Mathematics Olympiad(IMO) level.

Does the set of cardinal numbers with addition and multiplication defined respectively form any sort of algebraic structure?

>>11071756
All cardinals together don't form a set, they form a proper class. They don't have any interesting algebraic structure.

>>11071756
They form a commutative (class) semiring

>>11071768
darn

>>11071772
What do you refer to with the "class" an brackets?

Do you just mean they form a class as in
>>11071768
?
I'd reason the fact that a group theory in ZF foundations is defined as a set with a binary operation is pretty much another set theory artifact.
Classes are just as good collections as sets (or do you really need to put your collection of group elements inside another set to study your group for some reason?)
Being the goto example of universal algebra, you don't even need an existential quantifier to speak about groups, pic related.
Reasoning another way, you just need product (pullback over a terminal object) and three arrows
m : G × G G, e : 1 G, inv: G G
for a group object. Don't need for G to be a set for anything, I'd say

m : G × G -> G,
e : 1 -> G,
inv : G -> G

>>11071768
Wait, semirings appear in a lot of places... a lot of the theory of associative algebras over commutative rings works over commutative semirings; they show up in logic via distributive lattices/bisemilattices; they show up in the theory of barycentric algebras; they show up in K-theory; etc, etc. Especially if you bound the cardinals under some "universal" set, so that they're a small semiring. See Glazek, "A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences" for a pretty long list of references on the topic.

>>11071790
There are plenty of foundational reasons to differentiate between small and large objects (i.e. set or proper class-sized objects) IMO. For instance, the cardinals wouldn't be a member of the category of (small) semirings, which is a valid (and often useful) perspective for studying algebraic objects.

what do you fear, /mg/ ?

>>11071819
Okay, sure, this is an issue.

Still, I don't really know how to parse
>They don't have any interesting algebraic structure.

The questions around taking N to 2^N kept some people awake. And if that's not algebraic structure, then I don't know what about cardinal arithmetic is.

>>11071886
Oh I entirely agree that there's interesting structure. I'm not >>11071768, and I mean to point out that it's a commutative semiring, and hence has some sort of interesting structure.

>>11071903
Semirings are fantastic. The concise algebraic relation
[math] x^* = 1 + x^* \cdot x [/math]
whereever the star operation exists, gives rise to variants of [math] \sum_{n=0} x^n [/math] and so on.
A lot of automata theory is also really representations of some generic ring like that.

>>11071867
That thing, algebraic geometry, it scares me.

>>11071867
Mazur manifolds.
Witten.
Ignoring techniques and results from physics that become useful in stuff I'm interested in.
Dying a virgin.
Smashing my head on accident and becoming a number theorist.
Gorillas.

>>11071930
How did you even come across Mazur manifolds and why do they scare you? The definitions seem pretty clear to me, and although I never worked with them it looks more like a combinatorical hassle than some sort of geometric trouble to deal with them. Might be totally off though.

>>11071930
I have a free day tomorrow and am currently interested in analytic (complex) functions and some non-trivial formulas related to it.
Do you have any reference, a nice read, well written, in that direction.

whoever responds will receive some questions back eventually.

>>11071947
At what level are you at approximately? If you already know some complex analysis I would recommend Huybrecht's "Introductino to Complex Geometry", nice and well digestable book. Otherwise check out some complex analysis lecture notes, there's a ton of them out there.

>>11071947
One variable or several?
>>11071940
Exotic smooth structures scare me.

>>11071955
>Exotic smooth structures
It's far removed from any geometric intuition imo, so yeah I understand your feelings.

>>11071955
Very few or very many.
The nice, well written text criterium has prominence over the choice

>>11071972
Grauert and Fritzsche's Several Complex Variables.

>>11071979
Thanks but I didn't mean to read a book tomorrow.
I was also going more in the physics and modern theory.

>>11071867
Non reflexive banach spaces that are isometric to their double dual

Can anyone hear give a quick expose to Floer Homology, in the spirit of this thread?

>>11070728
A true wizard of mathematics

>>11069983
You don't belong in mathematics, it has become painfully obvious that your experience with logic has not breached even an undergraduate level of understanding. Try to define "this" in your definition of P rigorously. You simply cannot.
>B-but what if I encode all propositions with an index set and then write the universal proposition "Prop i is false", then taking the index of this proposition and applying kleene's recursion theorem to-
Yes, congratulations. You're proving basic computability results to yourself. This proposition doesn't make sense at all - it can never refer to itself.
>I-I was just showing how your intuition can be flawed...
My intuition would never be so naive as to claim the well-definedness of a statement P = "this proposition is false." My intuition is tuned to more reasonable pursuits (e.g. axiom of choice)

Hey guys,

Is there a text that combines undergrad real, complex, and functional analysis into one textbook?

I'm finishing off my undergrad in Applied Maths at the end of 2020 (my uni doesn't offer pure maths), but I want to prepare for an M.S. in Pure Maths (at another university).

Also I can't cover three textbooks in six months, that would be too much, and that would make me deter away my coursework.

Thanks

>>11071886
i interpreted the question of interesting algebraic structure, as the usual stuff, groups rings fields, etc. Cardinal addition and multiplication are boring, it is literally the max operation, the only thing that is interesting is exponentiation.

>>11071906
I'm open for alternatives?
Please.

Pic related gave up his PhD and went into sex therapy.

>>11072033
The stein and shakarchi series is what you're looking for. Extremely high quality.

>>11072184
No anon,you misunderstood. He doesn't want a self-contained series that covers those subjects.
He wants a book-sized summary.
Because he's retarded.

I can think of books that cover real and complex or real and functional, but not all three.

>>11072184
>The stein and shakarchi series is what you're looking for.
He/she asked for a single textbook.

>>11072033
Single one, No.
Search Shilov Real Analysis and Complex, plus elementary functional analysis in libgen.is

>>11072206
>>11072217
It's a single textbook split into volumes.
In any case, the Real Analysis book has some basic functional as well, it would totally suffice.

>>11072033
Just go with rudin real and complex. It has enough functional for you.

>>11070728
Is he smart because he eats only eggs?

>>11072033
Why do you need it to be a single book? That's retarded. It's too much material for one book, and if someone were to make one it would just leave out too much stuff, and end up being useless.

Seems like you just want some way to convince yourself that you can adequately learn the material by blitzing through it at a surface level, and avoiding lots of things. Just read wikipedia summaries then.

>>11071930
>Dying a virgin.
Stop watching anime then

>>11072013
>more reasonable pursuits (e.g. axiom of choice)
Is there a genuine answer for why it's remotely reasonable? I know it sounds retarded, but I've been stressing over this for a few days now and I'm even thinking of dropping out of maths altogether because of it...

>>11073122
kek

>>11073122
the axiom of choice is obviously true, the well ordering principle is obviously false, and zorn's lemma depends on a personal taste

>>11073122
Admirable.
Also kek

>>11073122
Forget about platonism, math isn't real anyway. There's nothing natural about mathematical "objects", it's just a mental study of some man-made rules. Choice or no choice - nothing is inherently better.

>>11073218
>math isn't real anyway
On the contrary, math is very real. I don't see how ''platonism"" has any relevance in this discussion though.

>>11073218
>math isn't real
>math isn't describing real world
retard

>>11073242
>>11073232
It is real, in the sense that it's a mental construct, not something universal, independent of human mind. For example, there aren't "natural numbers" in the world. The natural numbers don't actually exist. If you say they do, then you are a platonist.
And I didn't say that math doesn't describe the real world. However, we don't know why real world behaves like our calculations would predict (read "The Unreasonable Effectiveness of Mathematics in the Natural Sciences").

>>11073218
>>11073232
>>11073274
Spoiler, utterances like "exist" have no simple denotation and while you guys maybe think you have a sensible conversation, you actually don't. Anybody may say "numbers don't exist like cars exist" and the other may go "but that's not the kind of exist I mean". Yeah bravo, you can do this all night. You may well have agood argument why math is a thing of the mind. You may also argue that even if no conscious being were to exist, the plant with twice as many leaves gets twuc as much sun and for evolutionary purposes you end up finding th golden ratio in the angles between the leaves - and suddenly we'd have a long discussion about why "math is of the mind" must be interpreted not to take this into account.
Spoiler, human language isn't math and a small finite number of words can't grasp a vast range of possible concepts and conceptions.
You can't no nuffin in philosophy is also correct. Whatever you argue for, you always base it on simple ideas and Occam's razor notions that can't be proven.
Maybe an elephant with godlike capabilities interrupted you while you read this, but then made you forget you ever saw him and then he send you back in time to before you saw him and it all seemed to you like nothing storage happened. Well that's a possibility we couldn't ever rule out or disprove, we can only talk it off by the sheer absurd randomness and arbitrariness of the story. We can all convince ourselves of the absurdity, but that doesn't mean it didn"t and indeed it can't be proven wrong. All explanation would be based on what we think to this point (e.g. what we think is possible in the physical world's....). Platonic real, or the reality of math, or numbers, etc., this all can't be argued for or against in a closing way, because it relies being able to come up with a good notion of reality that we'd agree upon (an unending topic in philosophy) and it also relies on being able to be use logic without having to argue for logical rules.

>>11073274
You have a good point, but the human mind doesn't exist independently of the human body, and the human body doesn't exist independently of the real world.
Consider looking into this book for a perspective more informed than some asshole on the internet like me can give you: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

>>11073294
This seems to be about doing mathematics and so "where does mathematics (the activity and the capability to do it) comes from" is explored in a biological/psychological way.
But beyond the notion of math as an activity, you have question of mathematically describable effective relations in the physical world - not sure if this discussed the omnipresence of the seeming mathematical phenomenon (reasonability of events in the universe)

>>11072184
the four book series: fourier, real, complex and functional analysis'

good series but a lot to cover, but i give up on trying to cover three materials, perhaps two at a time then.

im mostly interest in their fourier analysis text, but ill pick that in my applied maths degree (thankfully).

>>11072253
thanks

i like the dover books. ill try working on the real and complex analysis one first, perhaps tomorrow to feel it out. i generally like looking at examples and solutions, if that doesn't work then i guess ill go with undergraduate analysis by lang (since it has a solutions manual).

>>11072789
im not at that level... thanks though.

>>11073294
>Consider looking into this book
I don't trust non-mathematicians to be coherent when answering the question it poses. Does the book fall into the usual traps when "philosophers" try to talk about maths?

>>11073289
>You can't no nuffin
I Know that AC is true, philosophically. I don't believe it, but I simply Know it due to certain undescribable phenomena.

>>11073366
AOC is true, power set is wrong. Remove inaccessible sets, and you're left with a universe where AoC isn't problematic. That's why AoC is a strange question.
It's not wrong to use a knive, it's just wrong to use a knive at the cinema

Is this correct? If the radius of a circle is an integer, then it's circumference is always a transcendental number. If the circumference is an integer then the radius is always a transcendental number

Let a > 0 and a ≠ 1. Knowing that d/dx a^x = a^x*ln(a) and using the derivative of inverse functions, show that d/dy loga(y) = 1 / yln(a)

Why is this so hard? Am I just a brainlet? If I try it I get 1 / a^yln(a), but I can't get the a^y away no matter what I try.

>>11073122
I have yet to hear a real argument for why choice is not reasonable. THERE ARE TONS OF THINGS IN MATHEMATICS WHICH YOU CANNOT WRITE DOWN. JUST BECAUSE YOU CANNOT WRITE THEM DOWN DOES NOT MEAN THEY DO NOT EXIST. For fuck's sake!
I mean damn, you don't even want to be able to take cartesian products!

>>11073340
>i'm not at that level
what the fuck does that mean? you're not at the level to do math? it's just more fucking math.

>>11073374
I don't have a problem with this. "Cartesian products of nonempty sets are nonempty" is a much more reasonable axiom than "the set of all subsets of a set is a set." That should already send off alarm bells since having the set of all subsets introduces contradictions. Axioms which guarantee sets of sets of sets and so on will of course end up generating things which cannot be described.
But I don't really give a shit, because I'm not a pseud who needs math written down in front of him to accept it's real.

>>11073512
This is quite possibly the stupidest question I've ever seen in this thread.
Did you forget the circumference formula from middle school? It's [math]C= 2 \pi r[/math]

It's really detrimental that math always sticks with shitty notation for 400 years
On the one hand it's good that you can print a paper by Gauss in LaTeX and you'd barely guess how old it is - but shit like the
"Reciprocity law"
[math] \left(\frac{p}{q}\right) \left(\frac{p}{q}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}[/math]
are really confusing until you look into it, even if more streamlined notation would immediately tell anyone that it's just a statement about (anti-)symmetrial properties of certain predicates/functions
[math] f(y,x) = (-1)^{\frac{x-1}{2}\frac{y-1}{2}} \, f(x,y) [/math]

Same goes for standard names of trigonometric functions and many other notations.
Stuff like\frac is generally bad, as it doesn't trasnalte well to computer written text. Same goes for dot notation over characters.

It's really detrimental that math always sticks with shitty notation for 400 years
On the one hand it's good that you can print a paper by Gauss in LaTeX and you'd barely guess how old it is - but shit like the
"Reciprocity law"

[math] \left ( \frac{p} {q} \right) \left( \frac {q} {p} \right) = (-1)^{ \frac {p-1} {2} \frac {q-1} {2} } [/math]

are really confusing until you look into it, even if more streamlined notation would immediately tell anyone that it's just a statement about (anti-)symmetrial properties of certain predicates/functions

[math] f(y,x) = (-1)^{\frac{x-1}{2}\frac{y-1}{2}} \, f(x,y) [/math]

Same goes for standard names of trigonometric functions and many other notations.
Stuff like\frac is generally bad, as it doesn't trasnalte well to computer written text. Same goes for dot notation over characters.

>mfw i'm taking calculus after taking diff geometry and topology
all the more humiliating if i don't ace exams.

>>11073831
So i guess i was correct, thanks for being a douche

>>11073122
I am in a state of crisis right now because of this. I hope some anon replies satisfactorily when I wake up.

Give a man a mask and he’ll tell you the truth.

>>11074003
>why do most people think the action of choice is true
There are quite a few ways of thinking about it:
1 - You can eenie meenie uncountable amounts of things.
2 - It's common knowledge that even without choice, you can still choose based on some conditions, i.e. Russell's left boot analogy. So instead of thinking of randomly choosing, you can imagine that there are sufficiently many predicates to make such choices over. Intuitively speaking, there should be as many as predicates as there are sets. So if I have uncountable boots, I can try to get just the brown ones, just the black ones, the ones with a smudge exactly here, the ones that were either made by Kanye or are yellow, etc, and at least one of those is in the cartesian prouct.

>>11073122
>>11074003
>why it's remotely reasonable

We're stuck with AoC just for historical reasons. It has momentum in functional analysis and other frameworks and is thus hard to get rid.

When Cantor was young, in the 1870's or so, he worked on analysis and invented transfinite ordinals (N extended to w's) to make statements about compositions of trigonometric functions. This lead him to define set theory.
Around the same time, Frege came up with his predicate logic (quantifiers in a fully formal logic) and in the next 30 years, people would realize they can squeeze all of math into a formal set theory.
Then they realized that taking over intuitions from sets of naturals to a general notion of "set" is sketchy. For decades to come, people tried to wrap their head around the various notions of impredicativity. Some could be removed, some we still use today, some are not present in constructive set theory.

Here's a soft looking but mean example. If [math] n, S [/math] are resp. variables standing in for some natural number and som subset of natural numbers, consider the selection of natural numbers defined by
[math] T := \{ n \, | \, \forall (S \subset {\mathbb N} \}. P(n,S) \} [/math]
That's allowed to think about but already problematic. To inspect what the elements of T are, you must go through all subsets S of N, but T is among those, so you get in a cycle. But that was a soft digression to highlight the kind of issues..

The Axiom of Choice was written down by Zermelo for practical purposes around 1900 for some desirable ordering properties of sets, but it wasn't the focus of anyone back then.
ZF fixed the fatal impredicativity issues in set theory (Russel issues) by Fraenkels Axiom of Replacement, around 1920 and that's the one making the set theory universe V what is it (a transfinite sequence of applying the power set to N).
When Gödel explored his constructive ordering L inside V in 1940, Choice was already 40 years old and the standard was set.

>>11073883
>Stuff like\frac is generally bad, as it doesn't trasnalte well to computer written text.
But it is much clearer to read by a human.
The way programming languages require you to write down fractions is horrific for readability.

>Same goes for dot notation over characters.
What does that mean?

In generally I consider most math notation quite good and have seen very few suggestions that would, leaving adoption hurdles aside, be actually an improvement.

>>11074122
>The way programming languages require you to write down fractions is horrific for readability.
I think you miss my point. Multiplication is not too bad either, so division doesn't have to be as well. Write a#b for [math]\frac{a}{b}[/math]. It's all just conventions and we're stucking to thousand year old ones. (Not saying it's possible to change it now, given that everybody is used to it.)

What does that mean?
[math] \dot{x} (t)[/math]

I know it's good and efficient for humans. But I think by now I wrote down more math in text than paper. Those peoples habits won't change in the other direction in the future. I presume.

>improvement
Well writing function application on the side you're writing to would be a good start. If you write from left to right like we do, then xfg instead of gfx would help everybody. I know it looks weird once you're used to something, but changing state of things by appending is much easier to parse than adding more prefixes.

Similarly, base 12 is better than base 10 is many way. It's nor an accident that time (60) and angles (360) are multiples of 12. It's because humans are good in the range up to maybe 20, and the number 12=2*2*3 has more prime factors than 10=2*5.
One hour having 60 minues means that
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 minutes
are all sensible divisions of an hour - a large amount only because the numbers were choosen to have many divisors. 60 really is only adding that next prime divisor (5) that 12 doesn't have yet.

Okay the last one isn't actually a notation thing, but goes along the same lines.

>>11074161
>Write a#b for abab
But that IS the programming way, almso always denoted by "/", there is even the ÷ symbol which is recognized as what you want # to be.
But that notation is PURE GARBAGE and also every mathematician makes the conscious choice of \frac over ÷ because \frac is easier to read and more natural to write.

The issue you do not understand is that a•b•c isn't ambiguous, a÷b÷c is, so you have to constantly bracket everything or have to rely on conventions which aren't visually there.

>What does that mean?
That is physicist notation, which spilled over into mathematics \partial_t x(t) us what a mathematician in pure mathematics would write.
The \partial_x is here treated as an operator.

>Well writing function application on the side you're writing to would be a good start.
Irrelevant. There is no real benefit gained from changing the direction.

>Similarly, base 12
Not a mathematical issue, it's a global standard with very little relevance in mathematics.

Just a question. Do you have any formal training in mathematics? None of your complaints are sensible and they portray a severe lack of understanding of the notation, for example you not knowing that there is recognized symbol for division.

>>11073831
How do you know pi is transcendental?

>>11074248
I'd converse with you if you weren't just one of those internet people who are hostile for no apparent reason

>>11074350
I am not more hostile then what your ideas deserve.
You obviously lack the mathematical education to comment on these issues as, again, you appear to not know that your proposal exist but is consciously avoided by pretty much any mathematicians.

My hostility is founded in you making extremely bad arguments and wasting my time with them.
Try saying something interesting or worth considering.
Aside from that, you are on 4chan, not Reddit, so get used to it.

>>11074003
basically anything you can prove with the axiom of choice you can prove without it, the only diffrence is that proof without the axiom of choice will be more constructive and thus harder and also harder because the initial intuition that rested on the axiom of choice of why you thought a statement is true is now no longer viable and you need to think of another completely different intuition. And also you won't be able to prove the same theorems that were true with AC for all objects but only for ones that you have previously constructed but that in my sense is even better.
;tldr people still keep axiom of choice in the set of axioms because they want to prove things with swiping generality and don't get bothered with set-theoretic details
math without the axiom of choice is a hard mode but essentially is better but you don't have to go that way, just try to notice when you're using it and think for a couple seconds isn't there any way to avoid it. That will make you proof writing stronger

>>11074364
>then
lol

>>11074383
Yes, sorry that I am not a native speaker.

Anything interesting to say? Except that we should use a stupid notation consciously avoided by pretty much any mathematician.

>>11074377
>basically anything you can prove with the axiom of choice you can prove without it
Existence of a basis?

RIP
http://www.math.harvard.edu/history/tate/index.html

>>11074395
F

>>11074003
>>11074377
I forgot to mention, there is also another axiom that people (set-theorists) keep to not get bothered with (set-theoretic) details, it's call the axiom of regularity. Like the axiom of choice it is completely optional and you won't even encounter it anywhere in normal mathematics

>>11074401
>Like the axiom of choice it is completely optional and you won't even encounter it anywhere in normal mathematics
But you DO encounter AC, or it's consequences quite often.
Hahn Banach, If I recall correctly, requires it (or Zorn) for example.

>>11074386
This (>>11074383) was not me. Replying wtih >lol is just as unnecessary. I'm not a native speaker either.

Yes, I do have some math notation, I have a PhD in chemaical enigneering and I'm not reddit, I'm on /sci/ since 2006.
I'm saying two dimensional notation like \frac and \dot make issues and post-fixing notations for functions make for a nicer writing style and this somehow triggers you and makes you feel the need to get insulting. You're just angry.

>>11074393
well, you can certainly prove an existence of a basis for finite-dimensional vector spaces and use some weaker, more resonable forms of choice to prove an existence of a basis for some infinite-dimensional spaces

>>11074386
This (>>11074383) was not me. Replying wtih >lol is just as unnecessary. I'm not a native speaker either.

Yes, I do have some math education, I have a PhD in chemaical enigneering and I'm not reddit, I'm on /sci/ since 2006.
I'm saying two dimensional notation like \frac and \dot leads to issues when we all use computers every day and post-fixing notations for functions make for a nicer writing style and this somehow triggers you and makes you feel the need to get insulting. You're just angry.

>>11074428
>I'm saying two dimensional notation like \frac and \dot make issues
What is the alternative?
÷ is avoided for a very good reason as a every mathematician knows about it, but nobody uses it.
Again denoting time derivatives is physicist notation but I also find it "okay" mathematical notation.
The two dimensionality is only an issue if you are writing on fixed height documents, like pure text or on a typewriter, neither when using LaTeX not when using paper is it an issue and you gave no argument why it is bad or why mathematicians shouldn't continue to refuse your alternative.

>post-fixing notations for functions make for a nicer writing style
No, it simply makes no difference.

>You're just angry.
Yes, I told you already that your post annoyed me as it contained only very bad arguments and examples.

>>11074433
So you can't prove everything without it...

>>11074452
The second thing you said was
>The issue you do not understand is that a•b•c isn't ambiguous, a÷b÷c is

Do you say to people you talk to in collage
>None of your complaints are sensible and they portray a severe lack of understanding of the notation

This is just asshole behavior and you want to vent something.

We read from left to right so writing
x f A
for what's commonly called A f x would make sense.
Going from a line
f x
to a line
A f x
is also more confusing than going from a line
x f
to a line
x f A
It's a nicer reading flow.

It's clear that having to use brackets for one dimensional notation such as (3/5) is annoying, but that doesn't mean keeping on with old two dimensional notations is a good thing. It's only natural it won't happen because people are unwilling to go through a confusing transitory phase. I added this caveat from the start.

Stop using the word stupid if people didn't attack you. You not liking an argument isn't a reason to sperg out. People will dislike you if you're a person with such an attitude.

>>11074474
>Do you say to people you talk to in collage
No, of course not. Why would I?

>and you want to vent something.
For the third time. Yes, your post annoyed me.

>It's a nicer reading flow.
You make a very bad case for your argument here.
The argument I would kinda see is that x A as an operator from X to Y starts with an object from X and ends with an object from Y.
But that is a very minor thing, even though, yes, there is some extremely slight room for improvement there.

>but that doesn't mean keeping on with old two dimensional notations is a good thing.
You give no alternative and no reason why it is bad.
And I have seen no alternative, this is actually a huge issue in programming languages where most things involving fractions are unreadable.

The 2D fractions are DEFINITELY preferable to any common 1D alternative or any alternative that I have seen. Your proposal was the worst one I have seen as you tried to reintroduce an existing and rejected idea with a different symbol.

>It's only natural it won't happen because people are unwilling to go through a confusing transitory phase.
You actually have to argue why it is bad and what is a better solution BEFORE any transition phase, as your proposal has been rejected by pretty much every mathematician, although an alternative is actually commonly understood and would need no transition phase.

>Stop using the word stupid if people didn't attack you. You not liking an argument isn't a reason to sperg out. People will dislike you if you're a person with such an attitude
Why should I care that you feel hurt by me calling your stupid ideas stupid?

>>11074495
>Why should I care that you feel hurt by me calling your stupid ideas stupid?
Because we're all human.

>>11074501
>Because we're all human.
Yes, I am human too and as such I felt offended and hurt by your second post and expressed my frustrations in harsh terms.

If "being human" to you means portraying a soulless husk devoid of emotion, fine, but I won't do that, because I am not empty.

>>11070906
>I think it is rooted in severe inexperience with algebraic manipulations

sh-shutup, t-take that back

>>11070543
What are the best Dover books on mathematics? No, I do not intend to read them, just purchase them.

>>11070922
you are using a really contrived metric in which the distance between any pair of distinct points is 1... of course the spheres are going to look weird

>>any closed ball with radii 1 equals whole space X and with radii just 0.9999999 it immediately shrinks to a single point {x} wtf

There are no points that have distance 0.9999.. from each other. I think you are trying to imagine X to look like Euclidean space, when it just doesn't

>>11074904
Reformulation: What are YOUR favorite cheap mathematics books?

>purchasing books
*cough* libgen.is *cough*

>>11070803
It's the same thing do whatever you like.

>>11074904
>>11074982
Counterexamples in Topology by Seebach and Steen.

>>11074904
purchasing dover books to not read is pointless, they're cheap ugly shit
collect AMS publications if you want to buy pretty things to make yourself feel smart

>>11075391
this is a cool book

>>11071643
Begins in 25 mins

>>11075398
it's so, so much fun

>>11075414

>>11075469
well? obviously just lagrange multipliers. lmao.

>>11075469
>>11075509
alternatively, symmetry clearly shows that all values must be the same. cause if one is smaller than the others then its cubic term is much smaller than the other cubic terms, which makes the reciprocals bad. this symmetry is common with lagrange multipliers. so it's just a = b = c = 1, and the sum is 3/sqrt(2), and the answer is 13.

>>11075469
>>11075515
i'm sorry, how is this anywhere near an imo level problem? am i missing something obvious?

>>11075509
>>11075519
While it's true that the question is shit and very easy (especially since it's just asking for you to punch in some integers, so you can just use a symmetry heuristic and literally get the answer in your head without writing anything down), it's not really accurate to say that because an inequality can be bashed to pieces with calculus that it's not IMO-worthy.

Constrained inequalities appear regularly on most olympiads (I remember them being particularly common on USAMOs) and they're all pretty much just completely trivialized by Lagrange multipliers. The people writing them know this, and they know a substantial chunk of the students know the technique, but they don't care much because it's both a written and unwritten rule that the contest is intended to be elementary. If a question has a challenging but nice elementary solution, then it's fine. It doesn't matter if you can break it with calculus and 5 pages of hideous meaningless algebra.
It's also a pretty risky strategy to employ, because it's easy to make an error somewhere when Lagrange multipliers get complicated, and graders have no respect for kids trying to cheese questions with shitty calculus solutions so they will ream your ass if you mess up at all. If you forget to check an irrelevant edge case with basic algebra, you might lose 1-2 points. If you do this with calculus you might _get_ 1-2 points on the question.

For those two reasons it's basically a last-resort thing for when there's 20 minutes left and you're flat out of ideas.

>>11075563
it's just taking a fucking derivative, jesus christ. these are high schoolers. it's not complicated machinery.
sure, using the residue theorem on a high school olympiad might qualify. lagrange multipliers are part of a standard high school curriculum.

>>11075668

>lagrange multipliers are part of a standard high school curriculum.

I never saw them until calc 3 and never understood them (really) until after DE. Where is a HS student supposed to encounter Hessians or understand gradients?

>>11074829
>sh-shutup, t-take that back
Anon, I would feel bad lying to you, but you should really consider putting some effort into learning how these manipulations work.

>>11074003
The axiom of choice is useful since it is equivalent to every vector space having a basis, which is certainly useful in some contexts.
>>11070543
Haven't been on /mg/ in a while, how're things going with you anons? Been typing up a bunch of stuff. Honestly, I'm still not sure how one judges the value of their own work. I look at some papers over a decade old with barely any citations, yet they look like articles of at least decent quality. Can you imagine spending months and months on something only for no one to give a fuck? I mean, besides your thesis.

>>11075849
it the problem with the current publishing system.
too many papers with too little new stuff in them.

Does anyone know why in the multiplicative group G {p, p^2,p^3,p^4,p^5,p^6=1} (6th roots of unity) the only generators of the cyclic group are p and p^5.

>>11075891
coprimality

>>11075707
you don't need to understand them to use them. and what in the world were you doing in high school if not calculus? lmao.

>>11075849
well, there are plenty of foundationmorons right now who keep talking about choice and LEM, and you're encouraging them.
it's not as bad as >>>/lit/ guy though.

>>11075891
because if you try the other ones they don't give you everything
why else? there's only a few things to try. come on now.

Nice try, butthurt samefag >>11075984

[math]integral_(-1)^1 \cos(0.5 \pi x) \cos(1.5 \pi x) dx = -1.41358×10^{-16}[/math]

Should this be zero? I believe the functions should be orthogonal because of self-adjointness of Sturm-Liouville problems.

All of the references assume orthogonality holds for [math]\cos(n\pi x) \cos(m \pi x)[/math] for n, m integers. Is n-1/2, m-1/2 the only special case for which this also holds? 5AM and too sleep deprived to work this out. pls help

>>11075668
>high schoolers
>lagrange multipliers
where the fuck do you live

>>11076086
Obviously he's being sensationalistic for (You)s. Somewhat related:

>Calculus courses in the USA have been transformed from strong mathematical crucibles, in which approximation and geometrical proofs were part and parcel of the subject, into much less rigorous courses taken by all or most incoming freshman science majors. When Rudin wrote this book, calculus courses included epsilon-delta limit arguments and inequalities on the real line alongside related rates, solving differential equations and calculating volumes and areas using standard integral formulas. Looking at the books of the past — such as Lipman Bers’ Calculus and Edwin E. Moise’s Calculus — it’s easy to see why Rudin was the book of choice for analysis courses. It was reasonable to expect that students who did well in such calculus courses would have more then sufficient background to be able to tackle Rudin, despite the effort it would require of even good students.

>Today’s students don’t stand a chance — most are simply overwhelmed due to lack of preparation. It’s as simple as that. Unless they’ve had the good fortune and talent to be guided through high school to a good honors calculus course as freshmen — such as those based on Spivak’s Calculus — reading this book is going to be a real struggle, to say nothing of the exercises.

>>11076154
My calculus courses were taught from Stewart. I could have taken honors freshman courses that were more rigorous, but back then I lacked confidence and assumed I wouldn't be able to handle it.
When it came time to take a course on the level of Rudin I made sure to start studying the text in advance, solving lots of practice problems. I was a tryhard, with something to prove, and I ended up outperforming many people who had taken the honors classes.
Maybe they weren't no-lifeing as hard as me, maybe I'm just smarter, I dunno. But I can say that I didn't know what a supremum was prior to reading Rudin, which is fucking laughable.
There's just too much fucking demand for degrees now. Higher education is ruined by this flood of brainlets. Fuck overpopulation, and especially fuck the stinking third world.

>>11075849
>The axiom of choice is useful since it is equivalent to every vector space having a basis, which is certainly useful in some contexts.

This is a tautology, since the context only arise by the axiom.

In the context where I want to consider a world where all cars are red, the postulate that all cards are red is useful. Say your mother likes red cars. Your "context" is that you are "too lazy" to say
>If my mother sees a car, and the car is red, then she likes it.
and instead, in your framework, you say
>If my mother sees a car then she likes it
(which is true in your framework because all cars are red, per axiom)
Instead of saying, for some proposition P of vector spaces with property Q
>Consider a vector space V with property Q, if it has a basis B, then P(V) is true
you use the axiom of chose to be able to say
>Consider a vector space V with property Q, then P(V) is true

Same goes for proves. Instead of having to restrict yourself to vector spaces where one actually can work with a basis, one just postulates that a basis always "exists" (even if it can provably never be written down) and then ends up proving more non-constructive theorems about it.
It would be easy to just prove theorems about vector spaces which have a basis and end up with hands on results.

This is not done, out of convenience and habit.
This is not a very strong rant about AoC - use whatever axioms you want. But the "it's useful" argument as formulated above has little content, given that the habit led to mathematicans work in "those contexts".

I can shill for any axiom that has non-constructive flavor and make it mainstream, and after 50 years there sure will be many "contexts" in academia where people wouldn't want to bother and rewrite all the statements that have my shill axiom as implicit assumption worked into every second proof.

>>11075849
>The axiom of choice is useful since it is equivalent to every vector space having a basis, which is certainly useful in some contexts.
This is a tautology, since the context only arise by the axiom.

In the context where I want to consider a world where all cars are red, the postulate that all cards are red is useful. Say your mother likes red cars. Your "context" is that you are "too lazy" to say
>If my mother sees a car, and the car is red, then she likes it.
and instead, in your framework, you say
>If my mother sees a car then she likes it
(which is true in your framework because all cars are red, per axiom)
Instead of saying, for some proposition P of vector spaces with property Q
>Consider a vector space V with property Q, if it has a basis B, then P(V) is true
you use the axiom of choice to be able to say
>Consider a vector space V with property Q, then P(V) is true

Same goes for proofs. Instead of having to restrict yourself to vector spaces where one actually can work with a basis, one just postulates that a basis always "exists" (even if it can provably never be written down) and then ends up proving more non-constructive theorems about it.
It would be easy to just prove theorems about vector spaces which have a basis and end up with hands on results.
This is not done, out of convenience and habit.
This is not a very strong rant about AoC - use whatever axioms you want. But the "it's useful" argument as formulated above has little content, given that the habit led to mathematicians work in "those contexts".

I can shill for any axiom that has non-constructive flavor and make it mainstream, and after 50 years there sure will be many "contexts" in academia where people wouldn't want to bother and rewrite all the statements that have my shill axiom as implicit assumption worked into every second proof.

>>11076284
To give a mathematical example, the axiom
>every group is commutative
of course is "helpful" in many context: E.g. when you actually work in a different field and stumble upon a group which actually happens to be commutative, and you try to fish for some theorems you can use.
Of course group theorists should be able to study commutative groups if they wish to - but it's a restriction of the subject they should be aware of.

The same thing happens with the axiom of choice (equivalent to well-ordering of all sets, which is a more hands on formulation). Many set theorists happened to willingly restrict themselves to the models where all sets are automatically "list like" in the sense that they can be well-ordered disregarding the cardinality.
At least set theorists are aware of the axiomatic aspect of it and still study more generic models of ZF and work out whether or not choice holds in those models.

But the functional analysis use choice by habit and aren't even aware of the convenience cut they got themselves in. They aren't aware that they are group theorists who got restricted to commutative groups by using foundations from another field.

>Manin disagrees with the Baconian story, that Hilbert set the agenda for the mathematics of the twentieth century when he presented his famous list of twenty-three unsolved problems to the In-ternational Congress of Mathematicians in Paris in 1900. According to Manin, Hilbert’s problems were a distraction from the central themes of mathematics. Manin sees the important advances in mathematics coming from programs, not from problems. Problems are usually solved by apply-ing old ideas in new ways. Programs of research are the nurseries where new ideas are born. He sees the Bourbaki program, rewriting the whole of mathematics in a more abstract language, as the source of many of the new ideas of the twentieth century. He sees the Langlands program, unifying number theory with geometry, as a promising source of new ideas for the twenty-first. People who solve famous unsolved problems may win big prizes, but people who start new programs are the real pioneers.

>>11070962
You can literally do any job with a math degree

Minus any job that requires specific education like engineering, Medicine, and Law.

If you don't have a masters, you won't go into any math specific job (probably). Studies and surveys show that the vast majority of undergraduates from all backgrounds go onto careers where they don't use their degree.

In fact, you will probably work alongside people with liberal arts degrees.

Jobs in programming require no thought - you sit at a desk and edit some code for somebody else. Or you just combine a program with another already made program.

What are the matrix representations of the object [math] \sqrt {2} [/math], i.e. what are matrices such that their square is 2 times the identity matrix.
I remember that this works for all [math] \sqrt {n} [/math], but I forgot how.

>>11074395
>one of the biggest legends dies
>only 2 replies
the absolute state of /mg/

>>11076702
maybe you should stop welcoming normalfag tourists onto your site

>>11076699
>I forgot how.
Think about it.

>>11076702
Big whoop. Every 48 hours, we lost
>500 due to medical errors
>300 due to flu
>250 due to suicide
>200 to car accidents
>40 to homicide

>>11072096
Wait what? Qiaochu gave up his phd?

How do you deal with the Collatz conjecture being still unsolved?

>>11076702
Old men die. I mean it's sad if people die, but it happens every day.
But yes, let's honor his contributions. Good guy.

>>11076780
He became frustrated with the topic he said he was assigned and seemed to have burned out

https://thicketforte.com/2018/06/05/performing-desire/

>>11072096

Yes, please do tell!

>>11076780

True or False: Two categories are equivalent iff they have isomorphic skeletons.

is there a polynomial in 2 variables of degree 3 or less with no roots in the integers but infinitely many positive and negative values

>>11076822
Probably false because data lost from compositions.
Not willing to construct a counter example, tho. I think there exists a finite one, tho.

>>11076822
>>11076833
I think the question (the claim) was literally copied from Wikipedia.

>>11076827
i+X^2+Y^2

>>11076834
Oh hey, the Wiki on Skeletons says the claim is true.

https://en.wikipedia.org/wiki/Skeleton_(category_theory)

>>11076834
Oh, right.
I think I mixed up the skeleton with the nerve.

Thank you! Next dumb question: Is the category of associative (unital?) algebras over a field an abelian category? I checked Wikipedia first this time. ;)

>>11076856
Do the morphisms need to send the identity to the identity?

>>11076866
Considering unital algebras, yeah. But I'm concerned that the category isn't abelian in either case for the same reason that neither RING nor RNG are abelian.

>>11076866
Actually, that's irrelevant.
The tensor product doesn't coincide with the direct product.
>>11076872
Yup.

[math] \frac{d}{dt}e^{X(t)} = e^{X}\frac{dX(t)}{dt} + e^{X} \sum_{k = 1}^\infty \frac{(-1)^k}{(k + 1)!}(\mathrm{ad}_X)^k \frac{dX(t)}{dt} [/math]

Are there mathematicans or physicists other than Gödel and Hilbert who had some sense of fashion and style?
I mean it's a sad joke.

>>11076991
Villani.

>>11075982
>and what in the world were you doing in high school if not calculus? lmao.
>t. virgin

>>11076086
what are you on about? i finished AP Calc BC in junior year and then did multivariable the next year. isn't that standard for students who are even the least bit talented in math?

>>11076154
rudin isn't difficult at all, it's a fucking meme. the person writing this is either out of touch or a complete pseud. i did calculus with stewart, never revisited it, and trounced rudin in like a month.

>>11076780
he was a pseud, he was never gonna make it

>>11077010
yes, is that a problem here, or unusual? who gives a shit about sex. lol. what exactly is it that brings you to a mathematics general on 4chan if you submit to carnal pleasures?

>>11077041
*tips fedora*

>>11077035
How so? He was and is extremely knowledgeable in various fields, especially abstract algebra.
Nobody "makes it". There's 500,000 mathematicans in the history, and 500 of them are remembered.

>>11077073
>nobody "makes it"
500 people did, according to your post
also "abstract" algebra is not mathematics - you're outing yourself as just another sophomore undergrad.
>pic related: qiaochu

>>11072096
>Yes, I love Sex Therapists a lot young and soft sex therapists, fellow mathematics.
Finances.

[math]\sqrt n A[/math], where [math]A[/math] is an idempotent matrix.

>>11077162
fugg, meant as a reply to >>11076699

>>11076284
I really appreciate your effort of explaining these things to this very ignorant individual. Although I admit it must be quite a tough experience. He literally doesn't get 80% of what you say.
Thumbs up anyway.

t. observing constructivist

>>11077117
>He thinks abstract algebra isn't real math
Nice job for exposing yourself as an undergrad, retard

>>11077021
>isn't that the standard?
It depends a lot on the country.
Here, you can be an absolute genius, you'll still follow the usual route, while possibly studying ahead on your own.

>>11072096
Wait, I knew he had quit his PhD but didn't what he had switched to. How do you know that?

>>11070728
If you put all your eggs in one carton,the others are going to be empty

>why yes, I prefer to use synthetic methods, how did you know? In fact, I periodically reformulate my objects of study into short, tight and self-contained geometric axioms that rely only on set theory.

>>11076811
>https://thicketforte.com/2018/06/05/performing-desire/

kekeke my sides have gone into orbit

>>11077291
>Imagine wrote about have sex with prostitutes as thesis report in your blog and Facebook.

>>11077291
>>11077323
Guys, what did he switch to, I'm curious now.

>>11077345
Machine learning, search Him name in LinkedIn
>drop PhD and $500K for sex therapist.

>>11071867
Spectral sequences that don't collapse at degree 2.

>>11076991
Observe this gentleman.
https://en.m.wikipedia.org/wiki/Johan_Jensen_(mathematician)

But not dressing like a slob is really a pretty outdated Idea, not that I am not a total slob either.

>>11076856
No. You usually don’t get a morphism of rings after adding morphisms of rings

>>11077278
>that rely only on set theory.
If you use synthetic axiom, why the fuck do you need sets?

>>11077171
How do you spot me, scary

>>11077162
Pretty sure this is about representations of matrices over Z

>>11075849
>Can you imagine spending months and months on something only for no one to give a fuck?
This, coupled with the specificity and seemingly arbitrariness of research is why I'm not pursuing a PhD and will try to find an actual job in 2020.

>>11077512
We used to have some nice discussions here over the years.
I also read your web page and watched some of your vids.
You seem really talented unlike me

I have a set of 4 symbols. I want to see how many ways I can arrange these symbols into a word with 20 symbols. How do I solve this?

its not n choose k because n<k

>>11077663
it's just 4^20... pretty simple anon

>>11077671
thanks im shit at combinations and permutations
when do i use n choose k?

>>11077663
follows same logic as license plate and phone number problems.
for instance if you have license plates which are 3 letters followed by 3 numbers the possibilities would be 26 characters for first spot, 2nd and 3rd, and 10 digits for 4th, 5th and 6th.
so (26^3)*(10^3) different possible license plates of that form

>>11077680
When you can't repeat letters and words are modulo permutation.

>>11077680
you use n choose k when you're trying to figure our where in a group certain things fall, so if you had those same 4 objects but lets say you have 5 of each 5,5,5,5 and you want to know how many words you make with that
it would be 20!/5!5!5!5!
if you wanted to know how many words out of all possible words with those 4 symbols contain equal amounts of them being used (5,5,5,5) you would have this answer (20!/5!5!5!5!) over 4^20
I really like probability, I think its fun

What did my teacher mean by this? I literally don't know how to proceed

>>11077718
Looks like a desperate plea for a new whiteboard

>>11077732
Europoor

>>11075809
I studied exclusively on khan academy to go back to school and none of the problems they give you are anything close to the algebra in the first few chapters of the calculus textbook. Same with the placement test, I did very well, but simplifying d/dx (arccos[b+acosx/a+bcosx]) ? There is nothing even close to that on KA.

Are there any guides on heavy duty algebra simplification? My professor and the alternate professor that sometimes does the teaching constantly make simple algebra mistakes and NEVER simplify anything to the degree that this textbook does. She even said 'don't simplify'--I guess since you can end up with many different answers they don't want to have to work through all of your steps to see if the answer is equivalent.

>>11077718
x is multiplication and : is division.
Just do the operations in the usual order.
Also, don't try to get anyone here to explain basic arithmetic, it won't work.

>>11077826
I appreciate the reply but did you see how she wrote the second one? What the fuck is [2^2 x 2^3 x x 3^5 ??? Maybe she wrote the X again for continuity because she's tarded?

>>11077851
>she wrote the x again for continuity
Yes.

>>11076822
NOBODY IN THIS THREAD CARES YOU MENTALLY UNSTABLE TRANNY. JOIN THE REST OF YOUR POPULATION AND KILL YOURSELF!

>>11076836
*In the integers.*

>>11078327
Calm down, incel.

>outgrew khan academy and patrickjmt
I'm all alone now bros. I'm scared

>>11077851
you're supposed to write your operation on both lines like that idiot

>>11078525
you have no right to speak stupid tranny faggot. I already regret responding to your original post because if there is one thing preventing you degenerates from killing yourselves, its attention, good or bad. now make like the rest of your people and kill yourself. this world is better off without mentally ill pseudo-intellectuals like you.

>>11078592
you're not a mathematician and you'll never be one. you'll always be mediocre and struggle to accomplish even a modicum of original thought, regardless of your performance in classes.

>>11076822
I have no clue but I got an honest question: Why not study graph theory instead of category "theory", a.k.a. language? You have dots and arrows and it's actually focused on these, not on fancy words.

>>11077029
>rudin isn’t difficult at all
if you’re good at formal maths yes its just about the right difficulty for a freshman math major. most people in public school would never be able to complete the more difficult exercises in Rudin’s intro text. You are misinterpreting the intention and topic of the post your vituperatively responded to (and used as an opportunity to talk about yourself, fag).

>>11078713
How do you prove the Yoneda lemma with graph theory?

>>11078720
you're a fucking moron. if someone is not good at formal math, why are they taking analysis in their first semester? lmao.
the "rudin is so le hard :A(" thing is just a meme. stop coping.

>>11078720
>fag
Why the homophobia?

>>11078713
>I have no clue but I got an honest question: Why not study graph theory instead of category "theory", a.k.a. language? You have dots and arrows and it's actually focused on these, not on fancy words.
Category theory is isomorphic to graph theory.

>>11078741
The question that was asked was pure cat-theory. Im not saying category theory isn't a nice language for abstract algebra, all I'm saying is that if you enjoy diagram chasing and the combinatorical aspects of cats in and for itself, you might be happier with graph theory.

>>11078871

The diagram chasing often comes down to more than combinatorics though. Like, it's about the nature of the arrows. It's about algebra.

>>11079027
Not that anon, and I agree that graph theory is inadequate for most categorical constructs, but on the other hand I see no reason not to just use quivers, while then defining categories as quivers equipped with a partial binary operation (application) satisfying associativity and unital axioms.
This approach is justified by >>11076284 (not me either), the occasonal dropping of these axioms (e.g. the free diagram construction), as well as the ability to turn the associators and unitors into explicit (BHK) constructs in higher-order category/quiver theory.
And of course, this doesn't stop you from concretizing or enriching the quiver with algebraic semantics whenever you need to.

Any good modern intro linear alg book?

>>11070554
>that hair
hmmmm

>>11079424
Linear algebra is the same shit everywhere. There's 500 intro books on the subject, a couple new ones coming out every month, and they're all pretty much identical. Just use whatever's conveniently available.

>>11070543
Hey frens,
I'm an absolute retard when it comes to math. I'm actually a fucking mouthbreather because I ignored it in high school. Best way to self-learn everything that came in high school and eventually up?

In what ways are the solution curves to second (or higher) order differential equations different to odes? How can you tell if the solution is for a first or second order equation just by looking at solution curves with initial values?

What are the prerequisites to studying type theory? Any book recommendations?

>>11079452
*I meant different to first order DE
as an example, would it be possible for this to be a family of solutions for a first order differential equation?

>>11079457
I think type theory is best learned by using it rather than reading. At least, I haven't seen books introducing type theory to mathematicians other than the HoTT book, which would give you a very biased and distorted view of what type theory has been for the last few decades.
Try proving some basic logical facts in Haskell, Idris or Agda's type systems. The prerequisite for reading about type theory is understanding what you can prove and what you can't, and how it's justified by that kind of computation.

>>11078713
>I have no clue
No need to mention this. We can tell.

>>11079481
for a first order ODE (without singular solutions) there is exactly one solution passing through each point. so the solution curves cannot intersect and the (t,x) plane is foliated into disjoint curves. this is obviously not true for higher order ODEs.

>>11079424
Peter peterson linear algebra.

>>11075849
>Can you imagine spending months and months on something only for no one to give a fuck?
That’s how I feel anytime I attend a conference. These are supposed to be attended by experts of the subject but even then, most people don’t understand a thing at most talks and start napping or reading their own stuff.
That is the most soul-crushing part I think. Working so hard at something that no one else will understand.

>>11070728
lmao all the math grad students in my school look like that

>>11070803
go to after school office hours bruh

>>11079685
>That’s how I feel anytime I attend a conference. These are supposed to be attended by experts of the subject but even then, most people don’t understand a thing at most talks and start napping or reading their own stuff.
>That is the most soul-crushing part I think. Working so hard at something that no one else will understand.
I know your pain anon. Being completely honest, a HUGE part of being a mathematician is that you're doing some autistic shit, probably without any real world application whatsoever, and which NOBODY cares about except maybe a few other people in the whole world. Math is fun and all, but one needs to seriously consider this when making the choice of doing mathematics for a living.
>t. second year math phd student just slowly beginning to realize this

>>11079424
Artin's Algebra. Definitely the best book. It's a general algebra (meaning abstract algebra) book which covers all the linear algebra you need as an undergrad.

>>11077663
You should really be reading up the derivations of these (or derive them yourself by thinking about real situations with collections of objects). It's no use just rote learning them.

>>11079457
https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/

>>11077259

that thumbnail tho

>>11070543
why is he squatting?

>>11079503
>agda
>not coq
you just made a french student somewhere shed a tear

>>11078648
stop projecting tranny fag. I can almost guarantee you are a cslet that took up category theory because he wants to sound intelligent without putting any effort in.

>>11078713
because he is not a real mathematician. He is a cs brainlet who is envious of the purity of mathematics so after a quick google search he decided category theory was what he wanted to study while taking his HRT injections.

>>11078648
dilate

>>11080013
cope, i'm an operator algebras and dynamical systems bro (and i am very confident in my male gender)
>>11080019
why the transphobia?

Guys help

Suppose A = {1,2,3} and define a relationship R = {(a,b) in AxA : 2a+b is prime}
Prove that R is a function with domain A

Wouldn't it not be a function? Since (1,1) and (1,3) are both in R, there isn't a unique image for a=1.

Or am I just retarded?

>>11080078
you're right

How can I expand e^(-ax) in Laguerre polynomial?

can someone explain prime numbers to me

>>11079928
Heh, I don't like the syntax and I think it's too much of a theorem prover and too little of a programming language, so I'd rather get people to learn type theory through a more Haskell-like language, but sure, Coq and other theorem provers are fine, too.

>>11070543
men with long hair are really hot, why can't more men have long hair?
>>11080126
they're spooky because there's no way to predict them.

>>11080126
If you have any natural number, sometimes you can write it as the product of two natural numbers and sometimes you can't.

>>11080131
>men with long hair are really hot
Stop being a faggot who likes faggots with long hair.

Inb4 "why the homophobia".

>>11080131
>they're spooky because there's no way to predict them.
Yes, there is. The distribution of prime numbers is relatively well-understood. We can predict prime numbers very well.
>>11080135
>If you have any natural number, sometimes you can write it as the product of two natural numbers and sometimes you can't.
Wrong. You can always write a natural number as a product of natural numbers.
>>11080126
Prime numbers are natural numbers which don't have any divisors except 1 and itself (assumed to be distinct from 1). 2,3,5. They matter because of FTA, which states that every number can be uniquely factored into prime numbers, up to reordering of the factors. This helps reduce the complexity of many many problems about numbers. For example, FTA is a great way to prove that sqrt(n) is irrational for a squarefree n.

>>11080138
"NOOOOOOOO YOU MUST LIKE WOMEN! YOU MUST LIKE WHAT I LIKE!"
kys
>>11080143
Sorry. But there is no efficient formula for primes right?

>>11080150
There are plenty of very efficient formulas for the primes. Literally hundreds of them. What do you want your formula to do?

>>11070554
>sudoku
nooooo ;_;

>>11080150
I cannot wait for the day where mentally ill faggots like you are rounded up and butchered. I hope you know that nobody in your life likes you. Your parents are disappointed and if I could get away with it, I would personally strangle you.

>>11080154
They are algorithms though.
>>11080168
Maybe you should become gay as well because you obviously aren't getting any pussy.

>>11080176
based gaybro
>>11080168
str*ights are too stupid to achieve anything in math, especially disgusting inc*ls and cucks*rvatives like you. go back to posting in your engineering general str*ightoid. math is for gays, based acebros, and women only.

>>11080150
>"NOOOOOOOO YOU MUST LIKE WOMEN! YOU MUST LIKE WHAT I LIKE!"
Big Yikes.
If you don't like women you are a genetic failure and should end your existence.

>Sorry. But there is no efficient formula for primes right?
Shut up fag.
There are about a billion methods for factoring numbers, what do you even mean by efficient?
You can factor numbers dozens of digits long in seconds.

>>11080114
>>11080114
>>11080114
>>11080114
>>11080114
>>11080114
Help.

>>11080143
>Wrong. You can always write a natural number as a product of natural numbers.
My natural numbers start at 2.

>>11080176
>Maybe you should become gay as well because you obviously aren't getting any pussy.
That's like cutting of your feet because you got a scratch on them.

>They are algorithms though
There is no difference between "algorithm" and "formula". Neither of these terms is well defined.

>>11080205
I do not understand the question.
Laguerre polynomials are a sequence of polynomials.

>>11080214
I mean Associated Laguerre polynomials.

>>11080176
neither are you. as a matter of fact, you probably smoke meth and are bound to get aids at some point in your life.
>>11080187
there is nothing based about being gay. dont fool yourself.
>>11080204
this.

>>11080217
No clue what that might be.
http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
What does "expanding in a polynomial" even mean?
They are a fixed sequence of polynomials.

>>11080226
>What does "expanding in a polynomial" even mean?
He probably means a Fourier expansion.

>>11080225
I think Floer was hot and there's nothing you can do about it, cry harder /pol/

>>11080232
What does that have to do with laguerre polynomials?

>>11080208
Still wrong.

>>11080241
Write 3 as a product of two natural numbers.

>>11080238
im not crying faggot. stop projecting, it is very cringe. we all know you fan yourself with your hands when you cry you filthy faggot. I hope that you die a slow painful death due to sepsis you contracted while buttfucking your poz partner.

>>11080114
>>11080214
>>11080226
Goddamn, this general is ignorant. Laguerre polynomials form a Hilbert basis of [math]L^2(\mathbb R^+, e^{-x}dx)[/math], hence for any square-integrable function [math]f[/math] on R^+ with respect to this inner product, we have:
[eqn]f = \sum_{n \ge 0} \langle f, L_n\rangle L_n = \sum_{n \ge 0} \left(\int_0^{\infty} f(x)L_n(x)e^{-x} dx\right) L_n[/eqn]
In the case in question, you need to compute each [math]\int_0^{\infty} L_n(x) e^{-(\alpha+1)x} dx[/math], which looks suspiciously like a Laplace transform. This is already done on Wikipedia.

>>11080268
>Goddamn, this general is ignorant.
Yes. I am very ignorant that by "expand e^(-ax) in Laguerre polynomial" someone meant "express e^(-ax) in some basis".

Also write your fucking weighted Lebesgue measures like they are fucking measures.

>>11080241
>>11080250
Still waiting...

>>11078713
Category theory is to graph theory as algebra is to analysis. With one you find and relate nice properties and with the other you deal with pathological bullshit like the Weierstrass function or the Petersen graph.

>>11080250
3 = 3*1.

>>11080293
Not true, analysis is undoubtedly a part of mathematics although not very pleasant for some people while graph theory is a pointless thing existing solely for the purpose of soothing undergrads that are too stupid to understand actual mathematics yet.

>>11080319
Are you stupid?
>>11080208

>>11080326
>graph theory is a pointless thing existing solely for the purpose of soothing undergrads
But it is actually important in CS applications.

>>11080326
2edgy4me
(pun actually not intended)

>>11080346
I don't know? Am I the one claiming 1 is not a natural number?

>>11080349
Almost no importance for real life engineering problems either because besides couple of elementary "theorems" there are not so much of a general theory, any graph fag just doing research on very special kind of graphs by very mundane, unintresting methods. In real problems where graphs appear, you almost never can impose these neet restrictions that make your graphs fall in one of these well studied classes, which makes all this stuff worthless.

>>11080367
No, you didn't read the conversation.

>>11080436
>caring about applications to engineering
fuck off subhuman bugman

>>11080479
Calm down npc, im not one of these people.

>>11070543
Ive got a calc 3 test tomorrow, wish me luck bros. Shoutout to the FTC of Line Integrals

>>11080524

Good luck! Get plenty of sleep, and remember to bring gum to the test. :)

>>11076417
how to get a job that does use my degree?

Is there anyway to recover results about finite matrices through the lens of functional analysis?

>>11076991
I do but I'm not famous.

>>11080159
yeah man i donno, it's crazy. he was brilliant and then he offed himself at 34. i donno why, and this guy is the only person to ever offer a theory:
>>11079436
and i have a hard time believing Floer was a tranny. i mean, aside from the sick hair, i don't see any hints of that. back in the 80s and 90s plenty of cool totally chad dudes had long hair, rockstar aesthetic

>>11080764
Depends on which result.
But a lot of stuff for finite dimensional operators still works for compact operators.
If you want a specific result, iirc Fredholm proved the multiplicativity of the trace for compact operators.

>>11080138
But long hair is simply better looking. Nothing wrong with saying that. Unless you want to look the same as half the male population.

>>11080925
>half
thats being awfully generous to normies

>>11081131
not the guy you were replying to, but floer was based and i think he had long hair because in his time plenty of chad rockstars had long hair. remember “hair bands”? times change and now long male hair is out of style but for floer it was chad, pic related

>>11080925
It's impractical and annoying. Choosing it makes you a fag.

>>11080764
Yes. Half of functional analysis is proving statements trivial or basic in finite dimensions, in infinite dimensions.

>>11081474
>It's impractical and annoying.
>being so stupid you don't know that hats exist
>being so fragile that your own hair annoys you

>>11070543
New Thread
>>11081694
>>11081694
>>11081694

Euler's identity doesn't make sense when I try to algebraically solve for [math]e^{i}[/math] so there's clearly something I'm doing wrong. Does [math]e^{i} = -e^{-\pi}[/math] or am I misunderstanding some of the liberties taken by the [math]e^{i\theta}[/math] representation of complex numbers?

>>11081702
e^(i) = e^(i1) = cos(1) + i*sin(1)
that is nothing like -e^(-pi)

>>11081763
Yeah I understand that it's wrong, I just don't understand where. My derivation was
[math]e^{i\pi}=-1[/math]
[math]e^{i}*e^{\pi}=-1[/math]
[math]e^{i}=-e^{-\pi}[/math]
which doesn't make sense. I was wondering more specifically which of these operations is illegal.

>>11081778
The second, retard.

>>11081778
you know how to use latex but you dont know that exp(ab) = exp(a)*exp(b) is a non sense

>>11081778
>tfw not big brain
don't you add exponents when multiplying the same base?

I just found out my professor is nuts, holding tests every other week and having them be a prerequisite for taking the final exam.
Suggest me an introductory logic book that I can go through in 2-3 weeks.

>>11082960
>I just found out my professor is nuts

don't you hate that? when you suddenly realize the person you're dealing with is not actually a rational person who listens to reason?