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Previous thread is >>13216886

Talk about mathematics stuff in this thread. Do not post irrelevant garbage.
Stupid questions go in the stupid questions thread, which is located at >>>/sci/sqt
>>
Would be interesting to perform a study on the habits (including drug habits) of high performance mathematicians.
>>
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0.999...wnb1
>>
>>13245710
More meme rules.
>>
>>13245710
Suggestion: remove the "stuff" after "mathematics".
>>
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How is this possible?
>>
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>>13246083
Anything's possible, if it happens.
>>
>>13246092
Just started with chapter 16 (inductions) of Basic mathematics by Sergey Lang, everything feels so unreal.
>>
>>13246083
>putting the binomial in the middle
absolutely disgusting
>>
>>13235468
>still
If you want to learn calculus just grab a precalc book then one on calculus. Maybe an easier one than apostol/spivak depending on what you're looking for
There's no need at all for all that other shit. Learn messily, get your hands dirty
>>13235514
The few times that channel has popped up here people agreed that the guy had a very weird math education. Probably not your first stop for pure math stuff
>>
>>13246083
How is what possible? That's almost the definition of the Binomial coefficient.
>>
>>13246253
How is it possible that an infinite thing is equal to a finite thing?
>>
>>13246266
I see no infinite thing in that equation.
>>
>>13246253
Mmm i know is possible and true(proof by induction) . It just amazes me how someone can come with that equation
>>
Everywhere online I see that the Yoneda embedding [math]\text{よ}:\mathscr{C}\to\mathbf{Set}^{\mathscr{C}^{\text{op}}}[/math] does not preserve all colimits, and I am trying to come up with a counterexample myself. I know that [math]C\in\mathscr{C}[/math] is mapped to [math]\mathscr{C}(-,C)[/math], but does it suffice to show e.g. [math]\mathscr{C}(E,C\sqcup D)\neq\mathscr{C}(E,C)\sqcup\mathscr{C}(E,D)[/math]?
>>
>>13246312
>how someone can come with that equation
Probably by induction, in the sense that you would expand the RHS for n = 2, n = 3, and so on until you notice a pattern.
>>
>>13246312
>>13246643
Are you guys trolling? Consider a particular term x^ky^{n-k} and count how many times it appears in the product. Each time it appears corresponds to a choice of k factors (x+y) which contribute to the x exponent (with the remaining n-k contributing to the y exponent). By definition this is n choose k.
>>
>>13246643
I think it's save to say the symbol n over k was introduced in conjunction with the expansion theorem. I.e. it's true by definition, and the formula for n over k in terms of factorials is the real theorem.
>>
Tooker posted a math for philosophers reading list in /lit/, what do you think of it,
>>>/lit/18402687

His list:
Lang - Basic Mathematics
Euclid - The Elements
The next three in any order as long as you finish all three before moving on:
Smith - The Laws of Logic
Spivak - Calculus
Hoffman, Kunze - Linear Algebra
Then:
Birkhoff - Basic Geometry (100% optional, only if you enjoyed Euclid)
Velleman - How to Prove It
Tao - Analysis I
Ross - A first course in Probability
Tao - Analysis II
Artin - Algebra
West - Introduction to Graph Theory
Hinman - Fundamentals of Mathematical Logic
Hardy - An Introduction to the theory of Numbers (optional)
>>
>>13247171
All lists are memes.
>>
>>13247171
I'm not the greatest fan of Tao and I don't know whether The Laws of Logic is as easy of a read as the others on the list
>>
How are hausdorff sets not contradictory? Infinite unions of point closed sets....
>>
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>>13245710
Tips on how to solve math problems:

SHUT UP, stop complaining, and SOLVE IT faggot.
>>
Dynamical systems chads rise up. I posted this as a thread but will repeat it here: alternative ways to Jacobian/eigenvalue method of determining stability of equilibrium? Where do you learn more?
>inb4 Poincares method
>>
For topology basis it says all intersections so is it a recursion algo or do unions suffice and is it infinite if recurs
>>
about to take my topology exam, wish me luck
>>
Wow so much fucking garbage here. Avatar/namefags still posting straight trash, OP image is stupid af.

>>13246635
You have the right idea. Technically you should show that the coproduct of the images isn't the image of the coproduct under the Yoneda embedding by showing the universal property fails, but what you said is basically it you just do this with finite sets you're essentially trying to show that x^(y+z) is not x^y+x^z. Sorry I'm too lazy to tex, congrats on getting the hiragana to show up correctly.
>>
>>13245710
I'm a first year math undrgruate and i want to learn mathematics in a fully axiomatiic fashion (Never use anything relying on facts that i haven't proved in a specific axiomatic system).
I've read terrence tao analys (proved analysyis from peano axioms) and now i'm going to learn axiomatc set theory at least up to constriction of the naturals, what would naturally follow? (like say i want to learn abstract algebra, how do i make sure that everyhting assumed in an undergraduate proposion-proof can be proven in zfc (or the objects constructed and defined from pure set theory i guess)?
>>
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This can be proved with T1 axiom? Because intersect neighborhoods
>>
>>13247692
You will 100% fail to do this. No one works this way and it's pointless to try if you want to actually do mathematics, or even just do okay in higher level classes. You haven't even taken algebra yet so trust me, eventually things go so fast that you will definitely not memorize every proof, much less do anything from the axioms up.
>>
>>13247692
That's pure autism. It's like trying to learn object-oriented or functional programming by first learning microcode. Utterly redundant.
>>
>>13247692
>caring about foundations
lol, this isn't early 20th century. Anyways the construction of the naturals (and integers and rationals etc) can lead you into group theory, or ordinal theory if you wish to delve deeper into set theory
>>
What is the definition of an axiom?
>>
>>13247907
something that you assume as being true while doing mathematics
but whats the definition of true?
and more importantly, whats the definition of mathematics?
>>
>>13247776
I don't necessary want to start from the axioms each times, just want to be sure that i have aldready proved the assumptions i'm working with with something that eventuially leads up to axioms (for exemple now that i built the reals from the peano axioms, i'm not going to sperge on real numbers properties my whole life, it's done now i 'm just working with them.Like for exempls i would have no problem read rudin principle of mathematics after that, i just want to make sure that the basic objects we're working with exist in zfc before working normally, but i don't where i can find that kind of stuff.)
>>13247813
Funnily enough, it's reading people like russel and wittgensteint hat really got made me want to do mathmatics.
>>
Have problems seeing why this works

[math]\frac{16200x-3x^{3}}{\sqrt{81000-x^{2}}}[/math]

[math]0=16200x-3x^{3}[/math]

how can you just remove the

[math]{\sqrt{81000-x^{2}}}[/math]

by setting it equal to 0?
>>
>>13248009
A fraction equals zero if and only if its numerator is zero.
>>
>>13247924
>something that you assume as being true while doing mathematics
Awful definition. Axioms are that which cannot be proved. Therefore the Riemann Hypothesis is an axiom.
>>
>>13248050
Prove that the Riemann hypothesis cannot be proved.
>>
>>13248042
if the numerator is set to zero, then you can assume the denominator is also zero?
Is that why the answers are telling me to do this?
and if the denominator is zero you can solve for x by, multiplying or dividing both sides?

i.e.
[math]0=\frac{16200x-3x^{3}}{\sqrt{81000-x^{2}}}[/math]

[math]\frac{0}{\sqrt{81000-x^{2}}}=16200x-3x^{3}[/math]

is that how that works?
Okay the reason I didn't realise this is the software didn't continue the 0= all the way down the page until the next part.
>>
update: I'm content
>>
>>13248089
oops I meant to multiply it.
>>
>>13248089
[math]0 = \frac{x}{y}[/math]
is equivalent to
[math]0 = x[/math]
because a fraction can only be zero if its numerator is zero. You can just multiply by y in the first equation to see why this is true. (0*y = 0)
>>
>>13248116
yeah, got it now.
I'm tired and frustrated...
>>
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>>13247692
>Never use anything relying on facts that i haven't proved in a specific axiomatic system
As others have pointed out, this is not really possible using set theory in under a decade, at least if you really want to be fully formal about it.
Otherwise just read a set theory book?

>now i'm going to learn axiomatic set theory at least up to constriction of the naturals, what would naturally follow? (like say i want to learn abstract algebra..
If you got, some theory, say Zermelo set theory, which has a model of the integers, you can already study algebra with a good conscious. Write down the axioms of a ring and see if you have two, three models for them in your theory.

>>13247993
>for exemple now that i built the reals from the peano axioms
well then

>>13247924
>something that you assume as being true while doing mathematics. but whats the definition of true?
Just think of all propositions as belonging to a class and those which are proven are in a special basket. The axioms are in the basket by convention and the inference rules let you put more statements into the proven basket. You don't need a more special explanation for "true" to get on proving things from axioms.
>>
>>13247993
>just want to be sure that i have aldready proved the assumptions i'm working with with something that eventuially leads up to axioms
That's how every good student does it. If you understand the subject, you'll have no problem translating everything into ZFC, There's no need to be thinking in terms of ZFC all the time.
>>
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The shape in the figure consists of a rectangle and a square. The side of the square forms 2/3rd side of the rectangle side x. Calculate the largest possible area of this shape if the perimeter of the shape is 88 cm.
>>
>>13248188
>proven
what do I use to prove the basket propositions in the first place?
I only have so many fingers I can count with anon!
>>
>>13247585
good luck anon
>>
>>13248386
If A is in the basket, and A=>B is in the basket, then put B also in the basket.
Here "A" is a variable that can be substituted with any sentence in your first order language.
>>
>>13248369
s=square side = 2x/3
2(s+y+x)=88
y=44-s-x
s^2+x(44-s-x) -> (2x/3)^2 + x(44-5x/3)
4x^2/9 + 44x - 25x^2/9 -> 44x-25x^2/9
d/dx -> 44-50x/9 = 0 at x=396/50 = 7.92
so probably that, cba checking other values or if x>y
>>
The Axiom Structure is fascinating. No one knows how to properly define them or how to even know for sure whether some statement is an axiom or not. Is there anyone studying this? Any area in logic about this?
>>
>>13248440
What are you talking about, there's nothing really ambiguous about it. Formal languages are defined - even formally
>>
>>13248369
2y + 10x/3 = 88
2y = 88 - 10x/3
y = 44 - 5x/3

area = x*y + (2x/3)^2 = x*(44 - 5x/3) + (2x/3)^2 = 44x - (11x^2)/9
set derivative to zero: 44 - 22x/9 = 0 -> x = 18
44*18 - (11*18^2)/9 = 396 cm^2
>>
>>13248440
Any mathematical statement can be an axiom.
>>
i jumped from hs algebra to calculus and am in calculus 2 but i pretty much blank or wing it out when i see trig stuff. how deep in trig should i go to cover my base in the rest of calc and later linear algebra?
>>
>>13245710
what makes something algebraic?
>>
>>13249109
Algebra is the structure of mathematics, therefore it's also the structure of nature. It's like the pillars who support a building or the skeleton of an animal.
>>
>>13249109
Roughly you don't have real valued measures. But then again, algebra creeps into everything (or the other way around).
>>
>>13248786
anon's axiom i: axiom i is not an axiom
>>
>>13249390
That's perfectly fine. You now have an inconsistent system. Have fun.
>>
>>13249109
When it has something to do with an algebra.
>>
>>13249109
An associative binary operation
>>
>>13249109
Algebra is about symbol manipulation. Wherever you're manipulating symbols, algebra is relevant.
>>
>>13247171
needs some CT; i recommend peter smith's intro book
>>
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Mathfags, please answer this question
>>13250591
>>
>>13250679
Philosophy questions go to >>>/his/
>>
Say I want to do research in topology. I am supposed to go to my masters next year -- which courses should I take? I already plan on taking algebraic topology, analysis on manifolds, differential geometry, maybe some finite geometries (I know these are more group/graph theory), stuff like that. But if I want to see what contemporary topology is about, what should I do besides the usual?

Are there any other interesting blogs except Dan Ma's topology blog btw
>>
how do i get a cute girl to sleep with me if im a math major
>>
>>13246873
Astounding.
>>
>>13247335
Perhaps Tooker (as well as you) meant "Logic: The Laws of Truth" as opposed to "Laws of Logic"? In that case the book is quite easy; it's used for first year logic classes (in the philosophy department).
>>13247171
I would take out Euclid and Hardy because they are memes, as well as the graph theory and mathematical logic books because I have never read them. Otherwise I think it is quite fine.
>>
>>13251697
Lie about being a French literature major
>>
>>13251091
The ones that matter do.
>>
>>13251690
No one's doing research on topology anymore, people drifted into its branches like DT or AT. Pure topology is as dead as pure analysis.
>>
>>13252171
There's also "topological algebra" which is essentially tackling the algebra *inside* topology (lots of abstract nonsense) instead of bringing algebraic methods to prove stuff in topology. I don't know how to describe better, but I discovered from a guy who's actually a topological algebraist. He says he's an algebraist first, topologist second.
>>
>>13251690
meanwhile somebody is analyzing manifold of your younger sister every thursday
>>
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>>13249653
>tfw operator commutation isn't algebra

>>13251697
Show her your coq
>>
Hey, I just turned 22 and have brain damage from oxygen deprivation and long-term PCP addiction and alcohol abuse. Is it too late for me to relearn and get into math? I still find it interesting and see some threads on here that are fucking stupid, so I haven't lost all of my memory when it comes to math. I just don't know where to start.
>>
>>13252687
depends on which parts of the brain is damaged
>>
>>13252412
>yfw before surgery that manifold was my younger brother
>>
>>13252843
definitely not an isomorphism
>>
if I use an answer to a /mg/ question in research, how would I cite it?
>>
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>>13252980
>Source: my brain
>>
>>13253061
legend
>>
>>13247171
Is he retarded? If you can do all the exercises in Hoffman and Kunze, Velleman is not something you're gonna learn anything from. Also, spivak is an analysis textbook. 80% of Tao will be review. Better List IN ORDER

1. Lang
2. Velleman
3. Calc Textbook of your Choice. (Maybe postpone multivariate calc until you've learn more linear algebra. Lang should cover what you need tho).
4. Friedberg
5. Abbott
6. Jaynes
7. Algebra Book of your choice. Doesnt matter which as long as its not gallian or fraleigh, but is somewhat popular.
>>
hi
>>
>>13253594
Hey :)
>>
Are there any recommendations to get started in TQFT and topological strings in particular ?
Something akin to the book by Katz' enumerative geometry & strings, or Vafa's mirror symmetry.
>not maths
>>
How do I learn LaTeX?
>>
>>13253733
Just pick something to type up. E.g. I am converting all my Analysis I note to LaTeX from paper.
>>
>>13253733
use it
>>
>>13253791
but how do I do that? I downloaded one of those text editor things and I typed some latex commands but it didn't work. instead of returning the formulas it just returned the commands on the pdf.
>>
>>13253800
Yeah that's no good. Either look for an online guide to intro latex or get a book on intro latex. Fyi sci uses mathjax not latex, it's mostly similar but not 100%
>>
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>>13247524
Pic rel is a great source for this kinda stuff. Cant actually remember the answer tho as I took the course over a semester ago
>>
>>13252940
akchyshually males and females are isomorphic
>>
Any physicists here? As a personal project I want to restart my education in mathematics but with a focus towards theoretical physics. Do you guys have any recommendations for textbooks on this?

So far my plan is to do
Calculus,
Real analysis,
Topology,
Multivariable calculus,
Linear algebra,
And Differential geometry.

Any recommendations for physics books/other math topic's to have a solid understanding of physics (can be computational heavy as well, though I'm much more confident writing proofs)?
>>
>>13254828
Theoretical physics is meaningless. It could be condensed matter all the way to string theory.
As a string theorist, my advice would be to first get a solid grasp of classical mechanics through Arnol'd book.
Then pick your favourite algebra book, Artin being enough. Pick a book on representation theory, any, but stop at the level of representations of semi-simple groups. You do not need more until later in your career.
Then you can dabble in QFT. Skip the operator formalism and go straight for path integrals with Srednicki.
After that some GR is necessary. The notes by Carroll are alright, but since you want math use Choquet-Bruhat books.
That should be your core couraes for a theoretical physics degree.
If you want to do rigouros physics, forget about it. The amount of knowledge necessary is way too important, you will end up studying your whole life and produce nothing.
>>
>>13254828
>theoretical physics
not a real science
they spend their careers making conjectures about nonexistent objects, that way they can't ever be found to be wrong.
>>
>>13254956
thank you for this, very helpful as I have literally zero experience with physics.
>>
What are some math books that are so complex that they look like schizobabble?
>>
>>13255255
Try doing some math carefully and you will soon be writing "schizobabble".
>>
>>13255255
most stuff on set theory I guess?
>>
>>13254774
reread the statement
>>
>>13255255
categories for the working mathematician
>>
If I belong to people of color, does it mean I will never do math as a white man?
>>
So regardless how much I try everything can be dismissed with a contemptuous "you are not white".
>>
>>13255868
>>13255878
skin color is a social construct
>>
>>13256220
it's an evolutionary adaptation to an hot climate.
Black people also have wider noses, higher stature and extremely curly hair for the same reason, so the differences are more than just skin color
>>
>>13256441
...and they shouldn't be relevant on a societal basis, like hair color, eye color, etc.
recognizing that we've physical differences from each other doesn't mean that we aren't still just humans.
>>
>>13256441
This is complete bullshit. It's white skin that is the evolutionary adaptation because as we moved into regions with less sun paler skin make it easier for the body to absorb the weaker sunlight to generate vitamin D it was hard to get from any other source.
>>
>>13256473
They're both evolutionary adaptations but I know what you mean: black skin came first, white skin is the result of a mutation
That's because all of our ancestors came from Africa (very hot climate)
>>
>>13245710
Any good books on probability?
>>
>>13256633
Probability-1 - Shiryaev
>>
>>13249109
That it has operations and objects on which the operations are defined.
>>
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hi, i feel like an utter retard, can someone point me in the right direction as to how i would solve this question? i am looking at the laws of logarithms and i feel like im missing something

cheers
>>
>>13258142
choose some times and look at the ratios between them
>>
>>13258142
log(V) = log(A) - alpha t
simply substitute V and t by the table, and two columns give you a system of two equations you can solve for log(A) and t
>>
>>13247764
why the fuck you ruined the book writeing you crappy draws in the page?
>>
>>13247171
>spivak calculus

Absolutely based anon, but not for brainlets.
>>
>>13253697
literally google it bro
>>
TIL: https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p
>>
hello /mg/ bros, I have a few questions
a) for first intro to abstract algebra, I notice a book of abstract algebra by pinter recommended a lot, but I also see people trash the book as too easy and recommend artin instead, which should I use for self study? (answers available online somewhere would be nice)
2) an analysis book after doing understanding analysis by abbot (idk should I go to multivariable analysis or go to measure theory or anything else)
>>
>>13261009
I liked Gallian for undergrad algebra.
>>
>>13261009
gallian is also apparently too easy (ie does sylow theorem wayyy at the end)
>>
>>13261058
>>13261043
wrong reply
>>
>>13260959
interesting thanks
>>
>>13261058
Idk I went from Gallian to Lang without much difficulty so if you think that's too easy just move on to a graduate level text.
>>
my mom is not accepting my dream of becoming a mathematician, she wants me to be an engineer, what do i do?
>>
>>13261009
a) I'd say Fraleigh, and if that's too easy for you, try Herstein's. That should do the job.
b) I'd recommend moving into real analysis (metric spaces, Lebesgue integration) my recommendations are Real Analysis by Carothers or Tao's Analysis II. If those are too hard (or you want good problems) try Rudin's Principles of Mathematical Analysis
>>
Is this a fucking book recommendation thread, how many people are there wanting texts
>>
>>13261486
People are too dumb to read Amazon reviews or to just search on Warosu or Fireden.
>>
>>13261529
me want spoonfeedy
>>
>>13261529
because people have their own individual circumstances and reading the review of some MIT jew who has top professors and tutors helping them isn't the case for everybody? Half of the time they're like "lel read rudin for your first time" which should give you an idea of why if you have your own circumstances you ask for those circumstances
>>
why is Serre's A Course on Arithmetic so short
>>
Do you take notes while working through a textbook? If so, what kind of stuff do you write down?
>>
>>13261891
What's so hard about counting? lmao
>>
>>13261901
The definitions, results and theorems.
>>
>>13261901
I write down all the key points/explanations/theorems, and obviously while going through the exercises
>>
Can someone recommend me a good textbook for learning advanced math?
>>
>>13262292
Advanced math is a very broad term. What exactly do you want?
>>
>>13262292
A Transition to Advanced Mathematics - Smith, Eggen
>>
>>13262304
triple integrals
>>
>>13261901
Sort of. I don't write anything down while reading, I just sit with the book and stare at it.
After reading I wait a bit and then try to re-explain everything to myself on a sheet of paper. I think writing things down at some point is necessary, at least for me. When I feel like I understand something well in my head that's usually maybe 60% of the way to actually understanding it properly. Making everything explicit is the only way to figure out what's missing.
>>
>>13262304
You know, the sort of stuff that professional mathematicions work on
>>
>>13261424
Fuck her.
>>
>>13262635
Ok, you've made your bait obvious now.
>>
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Example 4.
First, why is the height of 0/1 0?
And how is assigning an integer to rational numbers from their height a one to one mapping, you can have multiple rationals with the same height
>>
>>13263284
reread the definition of height
also reread the paragraph regarding the mapping
>>
>>13245710
test
>>
>>13245710
Does symmetry at two different ranges imply convergence of the elements between them?
>>
>>13246083
Multiplication symmetry.
>>
>>13263326
Your colleague is a perpetually smug homosexual man?
>>13263471
Wrong person.
>>
>>13263471
No.
>>
https://en.wikipedia.org/wiki/Bayes_classifier#Proof_of_optimality

Brainlet here, I don't get this at all. Is there a better source to explain it?
>Why (b)
>Why is R*=min
>>
>>13254319
Thanks homie. I actually read a good chunk of this book quite a while ago. I'll give it another go
>>
>>13254973
>they spend their careers making conjectures about nonexistent objects, that way they can't ever be found to be wrong.
Sounds a lot like math nigga
>>
How do computational tools assist you in your research? Any anons in the mathematical life sciences that rely on sims + other computational resources to do your research? How is that going?

>>13261009
Confirmed that Pinter is rubbish and too easy. I liked Knapp's Basic Algebra
>>
Hello, I need to relearn (first experience was really bad) calculus, and I have about 30-40 days for that, what do you guys suggest? I tried apostol and while I liked thinking about the proofs and whatnot the pacing was a bit too slow and, unfortunately, don't have the time to go through ~100 proofs related exercises before reaching the meat of it the book, i.e.: the first chapter, integrals.
>>
>>13264076
Just.....solve some problems anon. Here's an integral for you to solve:
Sqrt(tan x) dx
>>
>>13261009
DUMMIT-FOOTE
>>
I would like a hint for this question that has stumped me for two hours now.

Let [math]\mathscr{C}[/math] be a complete and cocomplete category, and let [math]F:\mathbf{Arr}(\mathscr{C})\to\mathscr{C}\times\mathscr{C}[/math] be the functor sending an object [math]f:c\to c'[/math] to the pair [math](c,c')\in\mathscr{C}\times\mathscr{D}[/math] and morphism [math]f\to g[/math] to its components [math](h,k)[/math]. Construct a left adjoint for [math]F[/math].

I know that [math]\hom((a,b),Ff)=\hom((a,b),(c,c'))[/math] will have cardinality [math]|\hom(a,c)|\times|\hom(b,c')|[/math], so I need to construct a functor [math]G[/math] that will give us as many morphisms [math]G(a,c)\to f[/math]. There isn't an obvious way to me to send a pair to a morphism, since there could be none or many. So I attempted sending [math](a,b)[/math] to the canonical morphism [math]a\coprod b\to a\prod b[/math], but I've been stuck there. I also tried considering other morphisms like [math]1_a\times 1_b[/math] but those gave me the same issues when trying to count morphisms.
>>
>>13264831
I'm doing every exercise in the books I go through so I'm not doing books that'll take me a year to go through
>>
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wait what? multiply a row by a row? how do i prove myself this
>>
S \ K = (S \ A) ∪ (A \ K), with S, K, A sets, K ⊆ A, for every S.

Isn't this blatantly incorrect?? Take one x in A but not in K and not in S. Then it is the LHS but not in the RHS.
>>
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can someone explain to me what happened to
-c1x and c1 in second and third equation?
>>
Maybe a dumb question, but is there any deeper reason as to why the algebraic closure of the R and the result of turning R2 into a field are the same? Is there something special about C or is it just a coincidence?
>>
>>13263326
wasnt yukaris identity known anyway?
pretty sure he posted his own papers sometimes.
>>
>>13265613
>Isn't this blatantly incorrect??
yes
>>
>>13266439
Turning a vector space of dimension 2 into a field requires a quadratic extension, which requires adjoining the square root of a field element. In the case of a real closed field like R, all the positive elements have square roots already, so we must adjoining the square root of a negative number x, which is equivalent to adjoining the square root of 1 (because the square root of -x is already in our field)
>>
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>>13264863
>C×D
>G(a,c)
dem typos

F is the right adjoint here? (just asking for the nonobvious convention)

I don't see how the complete and cocomplete property come into play, sadly
>>
>>13263986
No, mathematics is axiomatic.
>>
Can anyone give me a good resource for understanding fourier transforms better? I'm trying to figure out their application in neural networks. FNet replaces standard convolution layers with fft -> pointwise product, and I understand that they're taking advantage of the convolution theorem, but they discard one of the domains and I don't understand how that is supposed to effect things. Their paper was not very rigorous in terms of the math (they didn't even mention the convolution theorem by name,) so it's inspired me to do my own research. Unfortunately I failed calculus twice in college and barely passed the third time, so I am having a tough go at it.
>>
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This proof is suspicious, right? The author has just proven the sum, difference, product, and quotient rules of limits. However, in this proof he attempts to use it when he says that since G(x) -> 0, then the limit of g(x) = limit of f(x). But we require already to know that the limit of f indeed does exist to use that fact, and the statement of the theorem does not mention the existence of the limit of f, indeed, it tries to show it exists.
>>
>>13267471
You know all serious contemporary mathematicians have dropped formalism, right?
>>
>>13267530
f(x) has a limit by assumption
>>
>>13267530
reread the theorem
>>
>>13267541
>>13267544
Sorry, meant "assume the limit of g exists".
>>
>>13267559
g(x) = G(x) + f(x)
>>
>>13267566
Ah, I didn't think about that. I thought he was arguing along the lines of
[math]
\lim G(x) = 0 = \lim (g(x) - f(x)) = \lim g(x) - \lim f(x)
[/math]
which is why I had an issue with it.
>>
this thread sux
>>
How do I show the tensor product of two vector spaces is co-ordinate invarient? I understand way it is, but Im struggling to wrap my head around tensor and tensor product notation and don't know how to express it.
>>
>>13268905
Sorry, I mean basis in varient. That is, if I change the basis of Vectors spaces V and W, it changes the basis of VxW but doesn't change the abstract tensor product
>>
>>13267588
why would that matter
substitute the limits of f and G in
>>
Is math monk even possible at the graduate level or is communication with peers required?
>>
>>13268991
required if you want to progress
>>
>>13268991
Why would you be at the graduate level if you didn't enjoy math?
>>
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>>13268905
>>13267559
If you speak of the whole space, then this just says you want a bijection, right? The question is a bit tiresome since various people will use different definitions of vector space. Basically, write down what you mean by basis change in V, in W and V tensor W and then show that on V tensor W it's a bijection?
>>
How can I turn [math]\Product_{k>=1} \left( \Product_{j=1 .. k} \frac {1}{1-x^{kj}} \right) [/math] into a formula for a specific k?
>>
If [math] A: D(A) \subseteq H \longrightarrow H [/math] is an unbounded/densely defined operator on a hilbert space [math] H [/math], then what is meant by [math] D(A^{-1}) [/math]?
It could obviously be the range [math] R(A) [/math], but in the paper I'm reading the authors use it like a dual space, so it seems like [math] D(A^{-1}) = D(A)' [/math] where [math] D(A) [/math] is equipped with the graph norm [math] \|x\|_{D(A)}^2 = \|x\|_H^2 + \|Ax\|_H^2 [/math]. This makes a Banach space and similarly to the way [math] W^{-k,p} [/math] refers to the dual of [math] W^{k,p} [/math] this might be what they mean by this notation.
But then what does applying [math] A^{-1} [/math] to elements of [math] D(A^{-1}) [/math] mean?
>>
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>>13269250
What do you mean by specific k? The index k is not free in your formula

I expect the answer will go through
https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Generating_function

Btw. the TeX is

\prod_{k\ge 1} \left( \prod_{j=1,2 \dots , k} \frac {1}{ 1 - x^{kj} } \right)

[math] \prod_{k\ge 1} \left( \prod_{j=1\dots k} \frac {1}{ 1 - x^{kj} } \right) [/math]

Although I would write

\prod_{k>0} \prod_{0<j\le k} \dfrac {1}{ 1 - x^{kj} }

[math] \prod_{k>0} \left( \prod_{0<j\le k} \dfrac {1}{ 1 - x^{kj} } \right) [/math]

I find that multi-set of exponents j·k interesting btw.
Worth thinking about the multiplicities, I'll take an empirical look

>>13269268
I don't know - but maybe the points in the range that are injective images, so that the preimage is unique?
Where does D(A)' come into play when you state that norm?
>>
>>13269351
Well they write stuff like
[math]
D(A) \subset H = H' \subset D(A^{-1})
[/math]
And the way they use [math] D(A^{-1}) [/math] strongly suggests that it is a space of linear functionals and strictly bigger than [math] H [/math].
But then they also write [math] A^{-1} x [/math] for [math] x \in D(A^{-1}) [/math] and I cant figure out how this is defined.
>>
>>13269268
Maybe [math]D(A^{-1}) = \{ (x, Ax) \in D(A) \times H \}[/math]? Explains the norm and explains taking [math]A^{-1}[/math] (returns [math]x[/math]).
>>
>>13269424
>But then they also write A−1x for x∈D(A−1)
That's not too nice since the right expression is a proposition and the left one, one would think, is a vector.
I don't know. I think if you look into the references you may find a more basic elaboration that defines the terms

Which text?

>>13269250
>>13269351
So looking at
[math]\prod_{k>0} \prod_{0<j\le k} \dfrac {1}{ 1 - x^{kj} }[/math],
I found it convenient to factor out the j=1 case and write it as

[math]\phi(x) \prod \prod_{k>0} \prod_{1<j\le k} \dfrac {1}{ 1 - x^{kj} }[/math],

where the the left factor (the function [math]\phi [/math]) is
https://en.wikipedia.org/wiki/Q-Pochhammer_symbol

and the right hand factor then equals
[math] \prod_{n\in I} \dfrac {1}{ 1 - x^n }[/math],

where I is a multiset with the following multiplicities
{0: 0, 1: 0, 2: 0, 3: 0, 4: 1, 5: 0, 6: 1, 7: 0, 8: 1, 9: 1, 10: 1, 11: 0, 12: 2, 13: 0, 14: 1, 15: 1, 16: 2, 17: 0, 18: 2, 19: 0, 20: 2, 21: 1, 22: 1, 23: 0, 24: 3, 25: 1, 26: 1, 27: 1, 28: 2, 29: 0, 30: 3, 31: 0, 32: 2, 33: 1, 34: 1, 35: 1, 36: 4, 37: 0, 38: 1, 39: 1, 40: 3, 41: 0, 42: 3, 43: 0, 44: 2, 45: 2, 46: 1, 47: 0, 48: 4, 49: 1, 50: 2, 51: 1, 52: 2, 53: 0, 54: 3, 55: 1, 56: 3, 57: 1, 58: 1, 59: 0, 60: 5, 61: 0, 62: 1, 63: 2, 64: 3, 65: 1, 66: 3, 67: 0, 68: 2, 69: 1, 70: 3, 71: 0, 72: 5, ...}
or
0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 1, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 5, 0, 1, 2, 2, 1, 3, 0, 4, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 3, 0, 3, 3, 1, ...

I checked oeis.org for those values and for some mysterious reason, this is almost but not quite
>Number of ways to write n as n = x*y*z + x + y + z where 1 <= x <= y <= z <= n.
(you can google it)

Pic related for the routine and a tool to evaluate it at any X, correctness not guarantted
>>
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>>13269453
Btw. Pochhammer symbol look like pic related on [-1,1] and around x=0 this and your function will be essentially the same, with
0, 0, 0, 0, 1, 0, 1, 0, 1, 1...
differing by a small factor (1-x^4)·(1-x^6)···
>>
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>>13269453
Come to think of it, the sequence must be the variant of
http://oeis.org/A001055
when looking at the factorings into exactly 2 factors as in 8=2•4 or 12=2•6=3•4
>>
>>13269720
Which makes me ask two things:
Firstly, is there a suitable concept of Bell number for multiset?
Secondly, (given that we know that primality testing is of polynomial complexity and factorization is hard), is the in-between task of finding the number of prime factors (bonus: together with their multiplicity) as hard as the factoring itself?
>>
Is Category Theory a slippery slope?
>>
>>13269898
I vote no.
Not sure how to interpret that tho.
>>
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>>13269453
>>13269446
Here is an excerpt: Note the line
"We denote by [math] \langle \cdot, \cdot \rangle [/math] also the dual pairing between [math] D(A^{\frac{\alpha}{2}}) [/math] and [math] D(A^{-\frac{\alpha}{2}}) [/math]".
Come to think of it, since these are fractional powers of operators, maybe the [math]A^{-1}[/math] doesn't refer to inversion...
>>
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>>13270085
This is where A is defined
>>
>>13270085
Is it secret what text this is or what?
>>
>>13270085
>>13270091
Holy shit you're fucking stupid.
Don't you know your basic functional calculus?
>>
>>13270190
no :(
>>
>>13270193
Then go learn it. Go read your way through Lax's Functional Analysis. Soak in functional calculus for self-adjoint operators.
>>
>>13270178
It's Martingale and stationary solutions for stochastic
Navier-Stokes equations by Franco Flandoli and Dariusz Gatarek but I think this is the relevant part.
>>
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Just found that the cofounder of Telegram is a mathematician.
His brother is $17.2 billion

https://en.wikipedia.org/wiki/Nikolai_Durov

>>13270204
This looks related

https://math.stackexchange.com/questions/2074948/domain-of-the-fractional-power-of-a-linear-operator
>>
>>13270203
wtf either I'm blind or this doesn't address the issue at all. I know how you can define a functional calculus for unbounded selfadjoint operators, but I don't know what's the standard way of interpreting the negative power of an operator as a dual object.
>>
>>13270241
>His brother is $17.2 billion
sounds like a painful birth
>>
>>13270274
Anooooooooon, if you have an unbounded self-adjoint operator [math]A[/math], then [math]\langle x, y \rangle = \langle A^{-1}Ax, y \rangle = \langle Ax, A^{-1} y \rangle[/math]
Obviously all still the same if you take powers.
>>
>>13270640
What I understand is that there is an embedding
[math] D(A^{-1]) \longrightarrow D(A)' [/math]
via
[math] y \mapsto \langle x, y \angle [/math]
since
[math] | \langle x, y \rangle| \leq \|Ax\| \|A^{-1}y\| \leq \|x\|_{D(A)} \|y\|_{D(A^{-1})} [/math]
where the domains are equipped with the graph norm, but I don't see why this embedding would be an isomorphism.
What I ultimately need to get is how
[math] H \subseteq D(A^{-1}) [/math] can make sense.
>>
I'm having trouble with this:
Let [math]X[/math] a compact topological space and [math]f:\mathbb{R}\rightarrow X[/math] a closed and continuous function. Show that there exist a [math]x\in X[/math] such that [math]f^{-1}(x)[/math] is infinite.

I tried by contradiction, supposing that for every [math]x\in X[/math], [math]f^{-1}(x)[/math] is finite. Then [math]f^{-1}(x)[/math] is compact (since it's finite it must be both closed and bounded on [math]\mathbb{R}[/math]). Thus [math]f(f^{-1}(x))[/math] is closed and also compact in [math]X[/math].
Since this happens for every [math]x\in X[/math], the family [math]\mathcal{C}=\{U:U=X-f(f^{-1}(x))}[/math] is an open cover for [math]X[/math].
The cover [math]\mathcal{C}[/math] has a finite subcollection which covers [math]X[/math].

I'm not sure if I going the right way with this or if it the right idea. Anyone has better ideas?
>>
>>13270747
How about this:
Let [math] (U_i)_{i \in I} [/math] be an open covering of [math] \mathbb{R} [/math]. Then [math]( f(U_i)_){i \in I} [/math] is a family of sets that cover [math] im(f) [/math], and since [math] f [/math] is closed these sets are open in the subspace topology of [math] im(f) [/math]. Because [math] im(f) [/math] is a closed subset of a compact space, it is compact. Hence there exists a finite subcollection [math] f(U_1), ..., f(U_n) [/math] which is still a cover.
Now assume that [math] f^{-1}(x) [/math] is finite for all [math] x \in X [/math]. Can you use that to pull the covering back to [math] \mathbb{R} [/math] and thereby show that [math] \mathbb{R} [/math] is compact?
I don't know myself but you might wanna try.
>>
>>13270747
Since [math]f[/math] is closed, [math]f([n, \infty))[/math] is always closed. Then [math]\bigcap _{n \in \mathbb{N}} f([n, \infty)) \neq \emptyset[/math], concluding the proof.
>>
Let x represent what you choose for your life, and let life be the real number line. If you choose sin, it's like if you chose a number in the line, for example x=2. It's a single point, bounded. However, if you do not choose sin, it is the negation of the previous choice: x/=2. This is boundless, infinite and free. This is the christian freedom.
>>
>>13271352
wooow, that is so elegant...
>>
>>13268991
The problem with isolating yourself doing graduate work is that a fair amount of the body of knowledge is either only available informally through obscure lecture notes or simply isn't written down anywhere and is only communicated personally among other experts
There's a reason PhD students need to be supervised for years instead of just being handed a problem when they enter and told to come back when they're done
>>
>>13245710
I'm self studying math and I'm starting calc 2, I'm currently learning U-substitution. I want to know where I go after calc 2. Is multivariate calc the best option for the next place to go?
>>
>>13271398
Have you learned linear algebra yet? You should know some linear algebra before you start multivariable calculus. As long as you do, that's the usual place to go next.
>>
>>13271402
haven't started yet. I heard linear algebra done right is the best book for it. But I don't know any good options other than that.
>>
>>13271362
Get with the times anon, you're supposed to post something like "KINO" or "CHADCHAD-SAMA I KNEEL" or "My knees are bending on their own".
>>
>>13271407
LADR is sort of a meme. It's a good book but it's overhyped. It's not significantly different from any other theoretical linear algebra book, it's just a bizarrely successful marketing ploy where he swapped out a handful of determinant-based proofs for hackjob workarounds and somehow managed to convince everyone this is a revolution in linear algebra pedagogy.
It's still a good book though. Hoffman/Kunze is the more traditional "classic" book on the subject. Lang's Linear Algebra is also a great book; it doesn't go super deep compared to others but it covers most of the really important stuff and it's a faster read.
>>
>>13271434
Good to know, I'll go with LANG's then since I want to implement linear algebra for computations mostly.
>>
Where do i begin for statistics and probability? Currently in DiffEq and I'm going to learn some
Stat on the side since I'm in CS/Math and taking stat theory next term. Anything besides khan academy?
>>
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>>13261009
>Lang's Algebra
>no prior knowledge
>>
>>13271811
Durrett
>>
Is the vector space of initial conditions to a linear system of ordinary differential equations over the reals isomorphic to the vector space of solutions/
>>
>>13272274
>isomorphic
You mean in bijection?
I suppose if you fix the value (say t_0) at which the initial condition ought to be given, as well as the interval on which the solution should live, then yes?
>>
>>13272274
>>13271489
If you have existence and uniqueness of a solution for every initial condition in your space of initial conditions, then the Solution operator will be a linear map from the space of initial conditions into the space of solutions.
Now I dont remember the best existence and uniqueness results for linear ODEs right now but I'm pretty sure for say constant coefficients and a maximum finite time T with the initial conditions being sufficiently regular for the ODE the solution operator is an isomorphism.
>>
>>13272694
>the Solution operator will be a linear map
?
>>
>>13272774
Okay so say you have existence and uniqueness of solutions. If f is the solution with initial data f_0 and g is the solution with initual data g_0, then f + g is the solution with initial data f_0 + g_0 since the equation is linear.
>>
>>13272924
the solution operator is a linear map which for an initial data f_0 gives you a solution f with that initial data
>>
How do I get to understand quantum theory at least to a basic level? I keep reading a ton of stuff/theories/whatever bullshit that a lot of time tries to explain shit with quantum theory, and I have no idea if they're just using quantum theory as a cop-out as something magical or if they're for real.
>>
>>13272988
To elaborate, I just want to know enough quantum theory to be able to spot bullshit.
>>
>>13272992
>>13272988
This is no easy question. Try Hall's "Quantum Theory for Mathematicians", it's a nice introduction, and go from there.
>>
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>>13273090
Quantum theory for mathematics s is as bad as functional analysis for engineers.
Persons who studied X reading "Y for X's". Is always dubious, even if some X liked the book.
A shoemaker who at one point took an interest in dancing might write "Dancing for Shoemakers" where various aspects known and relevant for Shoemakers, on material, durability, wear ability are emphasized. Another shoemaker may read it and even love the book. Going away with they idea that they now know about dancing. But they learned shit about dancing, they instead learned about the shoemaking specifics in a dancer context.
I like the author but I don't think there's much physics in the book.
>>
>>13272988
Just study functional analysis at the level of grandpa Rudin and then read a QM book for physicists
>>
>>13247471
Wow... What a DICK!
>>
>>13271352
I don't get it.
>>
>>13273195

I can't even understand the introduction of the first level of Rudin.
I understand that for [math]p^2 = 2[/math], there is no [math]p[/math] which is rational. However, the proof doesn't make sense to me.
In his example (1.1), the very first point (1.1.1) states that if there were such a [math]p[/math], we could write [math]p = m/n[/math] where [math]m[/math] and [math]n[/math] are integers that are both not even. (Both not even meaning both are odd or only one of them being even?) Ok, we can pick 15 and 3 to be m and n.

(2): [math]15^2 = 2(3)^2[/math]. This is where I get lost. These are obviously not equal. He also states that "This shows that [math]m^2[/math] is even", which it's not in this case (225). Also where does the multiplication by 2 come in the right side of (2)?
>>
>>13273658
Please don't take that post too seriously, it wasn't intended to be so. As for your doubts
1. "not both even" means at most one of those can be even.
2. if p were rational, it could be written as a quotient of integers and you could cancel factors of 2 from the numerator and denominator until at most one of them was even (i.e. not both even). It's not that because p is rational, you get to choose arbitrary m and n such that p = m/n.
>>
>>13273658
>that are both not even. (Both not even meaning both are odd or only one of them being even?)
>both not even
reading comprehension is a bitch
>>
>>13273608
He uses the definition of a closed function to determine that all the [math]f([n, \infty)[/math] are closed. He then uses https://www.planetmath.org/ClosedSubsetsOfACompactSetAreCompact to conclude they're also compact, but omits this out of lazyness.
Then he uses https://en.wikipedia.org/wiki/Cantor%27s_intersection_theorem to conclude that [math]\bigcap _{n \in \mathbb{N}} f([n, \infty)) \neq \emptyset[/math].
Then, any [math]x \in \bigcap _{n \in \mathbb{N}} f([n, \infty))[/math] needs to be achieved infinitely often, for obvious reasons.
>>
>>13273658
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irrational-numbers/x2f8bb11595b61c86:proofs-concerning-irrational-numbers/v/proof-that-square-root-of-2-is-irrational



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