/sci/ - Science & Math

>Zariski topology

>>11897303
So what's an array again?

/mg/ meetup when?

>>11897356
Any Washington anons? I'm available most days.

why there are still mathematical physics classes in a fucking mathematical degree? what a joke.

>>11897704
They're electives anon. You don't have to take them.
>>11897308
Stop it. I'm sick of elliptic curves.

>coHOMOlogy

>>11897704
Partial Differential Equations isn't a mathematical physics class, undergrad-kun, even if the entire class is dedicated to the heat equation, the wave equation and Laplace's equation.

>>11897710
My field.

>mathematicians claim intellectual superiority over physicists
>still haven't found a unification of relativity with QFT

>>11897729
gay fag lol

>>11897737
Say that one more time and my bf will beat you up. He's a big engineering guy.

>>11897715
it depends. 4 months on "how phisically works and why PDE arises" it's physics imho. If it were on "how do you solve the actual problem and all the mathematical stuff related" then I agree it's math.

second my classes on basic physics (lagrangian mechanincs, hamiltonian and such) are mandatory.

test

What is it about category theory that /mg/ hates, I thought it was necessary for certain graduate level maths?

>>11897759
It's a few actual haters and mostly just fun posting about it. I mean, I always make fun of abstract non-sense even though it is literally the only thing I do.

>>11897759
/mg/ doesn't hate category theory, /mg/ hates category theorists

>>11897753
>it depends. 4 months on "how phisically works and why PDE arises" it's physics imho.
Can you seriously spend four months on a PDE course without seeing properties of solutions or solution methods?
>>11897759
>necessary for certain graduate level maths?
Category theory isn't necessary for anything, it's just convenient for a lot. Saying category theory is necessary for algebraic topology is like saying set theory is necessary for real analysis, it's not wrong, but it's also completely besides the point.
It's extremely convenient for a lot tho.

How long until the USA is 50% infected?

>>11897759
category theory itself is ok. what's not ok are math beginners who think they can skip the basics by learning category theory far too early in their development and who feel all patrician about it.

>>11897792
>Category theory isn't necessary for anything,
>Saying category theory is necessary for algebraic topology is like saying set theory is necessary for real analysis, it's not wrong

>>11897308


Matlab Coding Riemannian Geometry.
Variables are symbols, functions are symbolic objects. Covered rudimentary curve theory and up to Christofel symbols for surface theory. Even got the first fundamental displayed with the gay ds^2

Carantine is a bitch.
Did I tell you guys corona is 100% fatal?

>>11897800
The phrasing is awkward, yes.
Am I wrong tho?

>>11897803
100%?
That's a big percentage. Are you leukaemia or something?

>>11897759
Trannies move to CT as easy math

>>11897820
>The phrasing is awkward, yes.
Explicitly self-contradictory within two sentences is a little worse than "awkward".
>Am I wrong tho?
Yes.

>>11897823
nope

>>11897839
>some claims are still unsourced, but it checks out

>>11897800
>>11897792
well ""technically"" you don't need anything for anything, you can always define and prove stuff from scratch. if we ditch this autistic point of view, I would say category theory is pretty much necessary for algebraic topology.

>>11897839
I feel like I could have lived without reading that.

https://youtu.be/5eBT6OSr1TI

>>11897852
Does algebraic topology really need category theory in the same way that algebraic geometry needs commutative algebra or differential geometry needs real analysis, tho?
It does need homological algebra, but considering homological algebra to be part of CT is kinda forced.

>>11897849

he meant it agrees with parts of the clinical outlook of patients in Wuhan during January, which was skimmed over in the famous Lancet article. That pic is from mid February.

brainlet question:
Is vector calculus easier than PDEs?
I fucking hate partial differential equations. I'm taking a 5 week course right now that uses Haberman's book but the professor has somehow managed to skip all of the physics and use the book exclusively for problem sets.
This is just an elective credit, and I could easily just drop this class and take number theory in the Spring for a nice pure maths class I actually enjoy, but I have to take vector calc and I am worried that if I can't handle PDEs in a 5 week course then I am too retarded for vector calc. What do?

>>11897879
hom alg is not a part of CT but you need CT for hom alg (considering that the fundamental notion is derived *functor*)
also you would probably make it farther in AT without CT than in AG without commutative algebra, yes, but it will still be nothing compared to actually knowing AT

>>11897895
What kind of dumbfuck school lets people into the PDE course if they haven't taken vector calculus? That ought to be a prerequisite.

>>11897905
mediocre state school in Texas with a math program that basically lets you skip around to whatever you like after real analysis 1. Also I've had multivariable calculus that touched like a small amount of vector calc, but Vector Calc is a different class. I think it is more like Vector Analysis because real 1 is a prereq.

>>11897759
I view category theory the same way I view my little pony to me. Its fine when used by its intended audience, mathematicians who might find it useful and 8 year old girls, respectively. But both attract austistic trannies for some weird reason.

>>11897895
vector calculus is necessary for PDE

>>11897356
Never.
>>11897706
Every thread until you like it.

>>11897356
Anyone from Scotland?

>>11897895
Anon, you can't do PDEs without vector calculus.

>>11897920
Which Texas state school is this?
Their math program sounds better than UNT, whose PDE class requires Cal III and DiffEq, but doesn't require Real Analysis 1 or Vector Cal.
I wish Texas had less abysmal math education

>>11898585
Rice, UT, UTD, Baylor and SMU all have pretty good math programs, though.

About to start my first logic and statistics classes at University level. What am I in for?

>>11898602
>Statistics isn't mat-

>>11898597
Not surprised by Rice. Baylor and SMU surprise me, though.
>UT
Would probably be top-tier, if not for the mega-classes. One of the few decent places in Texas for math people, and has some good research going on.
>UTD
It seems like it caters to engineering more than pure math, but it's definitely a good public STEM school.

>>11898617
Higher education, for the most part, in Texas is pretty good; it's the K-12 that's abysmal. I remember reading that a lot of Texas colleges like applicants from New England because the school system up there is just overall much better.

>>11894661
I understand that. But a line isn't a single point, so why is it that a geometric vector is described as a line segment?

>>11898630
I don't know how Texas K-12 can be so abysmal. Some of their policies are counterproductive to a ludicrous extent.
The ranking system is probably the greatest offense. It encourages students to take easy classes with good weights rather than challenging/fun classes (or debate, orchestra, band, etc.) It also hurts standards for colleges, since they're forced to admit high-ranking students by default.

Does anything in particular make the New England school system better?

>>11898692
Here's a good brief write up (https://www.govtech.com/education/k-12/3-Ways-New-England-States-Have-Advanced-Competency-Based-Education.html).).

>>11898703
Sounds like a focus on students developing skills, combined with collaboration between teachers/school administration and policymakers. Might not be feasible in Texas, depending on politics, but it definitely sounds like a more effective model of public schooling than Texas', with the focus on poor standardized tests like the STAAR.

>>11898739
Teaching to the test is a terrible mentality, in my opinion, since it only stifles the way a teacher can engage with their students because they need them to know certain knowledge in a certain way. This sort of approach is a big reason why I feel that disliking math has become a badge of honor in American society because rather than actually learn math, students learn to vomit back algorithms with little to no understanding of where they came from. From my understanding, this is something that CCSI tried to alleviate, but that's just a series of guidelines and it's still subject to standardized testing.

>>11898747
I don't know anybody who thinks teaching to the test is a good mentality. At the greatest extreme, I could see it justified as a byproduct of funding measures, but never as an end in itself.
Money is probably the source of most problems facing the American education system, considering that some schools are critically under-supplied. I haven't heard any arguments for funding strategies outside of "teach to the test" or "throw money at the problem until it disappears" with little differentiation. The politicization of teachers' unions is a problem in this regard, particularly in red states like Texas.

>>11898585
UNT, you got it. I guess I just assumed this class required real 1, but I guess we are worse than I thought. To any young /sci/ lads, I know they give out mad scholarshipbux but pls avoid this school and go to A&M or something.

>>11898756
The thing I don't get is that the US spends, per capita, more than any other country (I think) on education and yet we still have such mediocre results. Then on the other hand, the US has most of the prestigious and best academic institutions in the world.

>>11897954
>>11898530
>>11897905
Thank you for the replies lads, I'm dropping, taking vector calc, and hopefully will return to Haberman's hellscape in the future to learn this stuff.

>>11898783
I always thought of US education as having both extremes compared to Europe. While Europe might have better quality education overall, the US still has the best, but also the worst.

>>11898800
>the US still has the best, but also the worst.
That also describes our healthcare.

>>11898782
I'm sorry for your loss. The fun part of UNT math is the zoomers from TAMS who plunder it for an Ivy-tier resume.
>>11898783
K-12 in the US has a lot of problems, and they're not going to go away until major reforms happen

>>11897356
london anyone?

>>11897356
Let's have a Neetup instead

>>11897308
Speaking of elliptic curves, here's a good problem.

>>11898804
The occasional TAMS chad in my 4000 level classes so far have made me hopeful for the next generation.

>>11898823
TAMS kids in math are bro-tier, but they're a massive minority compared to the Bio and CS ones.
A lot of them can get really entitled, especially when their grades are concerned. They hated top-tier profs because Cal 2 was hard for them (i.e. the class average wasn't an A or B and a couple people failed)

t. former TAMS student

Most accelerated programs are traps. Take your time for fucks sake.

>>11898602
>What am I in for?
Wild Tablemania, truth tables and distribution tables.
Also some other stuff without tables, if you're lucky.

>>11898858
This.

>>11898319
near Aberdeen/Dundee

>>11898858
ngmi

>>11898875
Made for being kept barefoot and pregnant.

>>11897458
Yeah. If you come anywhere near me I’m going to send pieces of you to the other posters in this general, pussy.

Why is math so fucking hard? Fuck you all.

>>11898979
You will not be missed.

>>11898983
Have sex.

>>11898990
die, simp

>>11899001
epic

>>11898818
Here’s a similar image for fermat’s Last theorem

>>11898783
That's because Americans meme their own institutions to be the best

Just doing a quick survey, but when did you guys take functional analysis?

>>11899155
In the year 2012

>>11899181
I meant in which semester did you take functional analysis

>>11899024

Let pear be any number in its domain, and likewise let apple and banana be any of the numbers in their domain. Then pineapple is simply the sum of the choices. The existence of pineapple is assured by the fact that addition on fruit is a ring.

Working through Stein and Shakarchi and came upon this.

Why is there normally the obsession with bounding quantities with a clean epsilon at the end of the proof, at the expense of cluttering up the rest of the proof with wonky epsilon expressions?

It’s so much nicer to use plain epsilons throughout the proof—especially when the final bound, even if not pure epsilon, is so obviously dependent entirely upon epsilon.

>>11899220
4th semester. But my university only offered it in summer semesters

>>11899245
Assuming that it's not just one of those cases of "I do it that way because Rudin did it that way STFU" (which I'm sure is the real reason for most authors) the best real reason I can think of is that it provides more quantitative information to use the wonky epsilon expressions.

You certainly can set all your starting data to epsilon and then end up with some big ugly fuckheap that happens to go to zero, but if I ask the question "here's [math]\varepsilon_0[/math] , what delta do I need" now you don't know enough to answer.
Even if the stated goal is only to know IF something converges, it's nice to actually have explicit control over how it's converging, especially when the expense is cluttering up the proof a little with a few fractions.

>>11899344
That’s fair, thanks anon.

I tend to only be bothered by this in lecture, because it often makes for unwieldy expressions that obscure the critical parts of the proof.

What are all the prerequisites for algebraic geometry?

>>11899438
Algebra and Geometry

>>11899438
Algebra and geometry

>>11899467

>>11899477
>>11899466
Likewise

Maths is for gaylords.

>>11898608
Jesus

>>11899155
I had several classes:
- Banach spaces & Hilbert spaces in 2nd year and 3rd year
- Functional analysis (topological vector spaces, distributions, spectral theory) 1st year of MS
- Operator algebras 2nd year of MS

>>11899155
4th semester too. Earlier really makes no sense, unless you do a very serious first 2 semesters where you do Lebesgue integration in Analysis.

>>11899155
4th semester ug

I'm actually surprised many of us took functional analysis in 4th semester. I mean some algebra people never even take it in undergrad. I feel like our sample is biased right now.

>>11898608
so wat?

http://biochemical-pathways.com/#/map/1

this too is a nice graphics, but it is not math.

>>11899155
After measure theory and lebesgue integrals (2nd year bachelor). And then more extensively on the third year in a functional analisis class.

>>11898818
kek, this one is an absolute cunt of a problem >>11899024
this one is too obvious though

>>11898943
Agreed.

Best Python plotting library?

>>11900082
Why are you even asking this here? Do you think we actually know?
>>>/g/sqt

>>11900082
Doesn't matplotlib exist or something? Who cares about "best?"

>>11899155
fifth semester

>>11900082
Matplotlib "just works".
For high quality you want a python -> csv -> LaTeX -> pgfplots -> PDF pipeline.

If the identity [math]X\to X[/math] of a topological space can be factored as [math]X\to Y\to X[/math], does it follow that [math]X\to Y[/math] is a homeomorphism onto its image?

>>11900276

No, the map X to Y could be a subspace of Y homeomorphic to X.

Y could be X U {a single isolated point}, and the map Y to X could send that isolated point to the empty set.

>>11900285
>the map Y to X could send that isolated point to the empty set
uhhh
>the map X to Y could be a subspace of Y homeomorphic to X
>>does it follow that XY is a homeomorphism onto its *image*

>>11900276
Yes. The other poster is retardo.

>>11900276
Wait, why is the map necessarily continuous? I.e. X is (0, 1), Y is a disjoint union of (0, 1] and (0, 1), and the map splits X into two intervals and then recombines them. Do you want the map to be continuous?

>>11900447
No, when I talk about topological spaces I make sure to speak strictly about set-theoretic maps

>>11900450
Perhaps you should speak more precisely if you take such umbridge with my question. My sincerest apologies that I failed to interpret "factor" within the correct generality you had in your head.

>>11900468
The word is 'umbrage'.

>>11897736
>>still haven't found a unification of relativity with QFT
Isn't string theory basically mathematical masturbation and a unified theory?

>>11898677
Pls respond

>>11900646
cuz

>>11900646
>(You)

Where is the nice animeposter when I need him? ;_;

>>11900488
Yes, you're right. Of course. Thank you.

>>11900683
Which one of them?

>>11900683
>animeposter

>>11900691
Hint, Yukari anon isn't one of them.

>>11900700
Yes but I am an anime poster and some people (totally not me phone posting) have called me nice. Thus, it would be of great help if the person asking would specify a bit, so I would know whether it is someone else or me.

>>11900691
>>11900699
I know there's a couple but I really now nothing other than they post anime and is nice
Also they may be the same one that writes in a neat way

>>11900683
Me?

>>11900717
Well, that has also been said about my posts. Is it merely the lack of my posts' stimuli hitting your eyes or do you need some insight on something with the somewhat high of a risk of disappointment?

>>11900717
There are quite a few of us here. Is there anything in particular that you know about them (I'm the undergrad animeposter), or maybe a catchphrase (like /gnmg/-anon)?

>>11898783
Some places have funding problems. Some places get money thrown at them but have no fucking idea how to spend it.
The best example of this is the Kansas City school desegregation: https://www.cato.org/sites/cato.org/files/pubs/pdf/pa-298.pdf (note that the Cato Institute is a libertarian-right political organization, but this is the best article I can think of that covers it).

I can attest from attending an American public school (specifically in Florida) that they get money all the fucking time and spend it on frivolous crap that nobody uses (like buying a set of apple computers when everything requires Windows).

>>11900726
I just assume that there are only ever three other people on /mg/ at one time; OP, an anime posting grad student, and a severely idiotic undergrad (likely non-math major).

>>11900721
Well I had my shameful geometric vector questions. It's boring, i know. They can be considered as points if their beginning is the origin (in some spaces only apparently), fine, and they are also (directed) line segments because (I guess) they have a beginning and an end point and as such you can put a line through them. Is what I said thus far correct? So, does a geometric vector also codify a set of points (as a line segment)? Which then somehow are equivalent to other geo vectors with the same lenght and direction?
>>11900726
Dunno. But anyways, I quite like that anime girl and I save the images when they get posted sometimes.

>>11899929
4th semester is probably an indication of yuros talking, but I think a large chunk of (the non-retarded) students in America take it either 3rd or 4th year undergrad. I took in 3rd year and my graduate focus has literally 0 analysis of any kind in it.
At my uni it was one of the "nominally graduate but the class is 60% undergrads" courses. I think it was probably the most populated graduate course by volume.
FA is one of the common grounds of math. Pretty much everybody except the category trannies enjoys it.

>>11900749
>(like buying a set of apple computers when everything requires Windows).
Public school admins fucking LOVE Apple for some reason. My high school had a broken pool for over 2 years while I was there and in that time they bought an entire roomful of like 30-40 mac desktops for the music department.

>>11900788
>Banach spaces

>>11900683
What do you need him for?
>>11900700
Yes, Yukarifag didn't post anime, why did you feel the need to mention that?

>>11900801
>My high school had a broken pool
wtf your school had a POOL
t. canadian whose school had like 350 students

>>11900786
>Well I had my shameful geometric vector questions.
The most shameful question is that one never asks but is tormented by for the rest of their life.
>They can be considered as points if their beginning is the origin (in some spaces only apparently), fine, and they are also (directed) line segments because (I guess) they have a beginning and an end point and as such you can put a line through them.
Yeah, you get the line segments with arrow tip pointing to some direction from the points and then realising that you can move either of its end points freely (as long as you do that to the other one also). This follows from the fact that the change in each coordinate is the same whether you start from the origin or some other point.
>Is what I said thus far correct?
Seems good.
>So, does a geometric vector also codify a set of points (as a line segment)?
Both (I mean they are equivalent so yeah) codify a set of points, namely a line. You can take any vector [math]v\neq 0[/math] and define [math]L_v = \{ rv \ |\ r\in\mathbb{R}\} [/math] to get a line going to the direction of the given vector through the origin. If you know the length of your vector, you can use that and this set to obtain an explicit set of a points on a line segment starting at the origin and ending as far from it as you want. To move it around, you just do the translation like you had the lines y = kx + c back in school to move it around in your Euclidean space. A thing worth noting though: a translation is not a linear map unless it is the identity!
>Which then somehow are equivalent to other geo vectors with the same lenght and direction?
Yes. The point is that your geometric vector anywhere is determined by the change in each coordinate, and this change is independent of the points themselves.

I hope I managed to ask your questions somewhat sufficiently. I am assuming we are talking about the Euclidean spaces. I hope the English isn't too broken.

Why do mathtards have all the fun? AAAAAAAAAAAAAAAAAAAA WHY CAN'T UNDERGRAD CS BE GOOD?

I want to have sex with anime trannies.

>>11900872
I am going to give you an idea. Since you are an undergrad at the moment, why don't you switch after getting your bachelor's degree? It's not like you are 24 or anything, so you can still do that. There was a guy who used to be an electrical engineering on the same AT course with me.

>>11900872
>I want to have sex with anime trannies.
We're always waiting for someone...

>>11900872
>I want to have sex with anime trannies.
I though they had them in CS too
> WHY CAN'T UNDERGRAD CS BE GOOD?
Curriculum is made for brainlets. What do you think would make it better?

>>11900872
What if I told you that CS was applied math?

How much of a brainlet am I if I fail at complex analysis?

I have no problems with any other branch of math. Functional analysis, stochastic processes, differential geometry, algebraic topology - all comes easy to me. But as soon as I see "analytic continuation" or "zeta function", my IQ suddenly drops by 100 points and I feel stupid as fuck

>>11900908
Complex analysis fundamentals are easy enough that you can quickly enter to modern topics faster, and these topics start using shit from many different fields, it's much more difficult to construct the objects you are working with and there is less agreement on the literature as to how to define things or construct them. Just defining the Riemann zeta function is a mess of it's own. Because it has so much structure, the amount of problems than you can imagine is far greater.

>>11900908

>Europeans learn algebraic geometry in the first grade
american bros...

>>11900908
relatable. i think most of the "confusion" for undergraduate courses is that complex are treated like "the bad bois" of fields and never actually used untill you have a full course on it. other ideas are introduced much slower imho (i dunno just the idea of a group is treated for a year before actually doing something in group theory, and it pops up everywhere, same for banach spaces, and so on.)

>>11900895
I would if I could.

>>11900856
>Both (I mean they are equivalent so yeah) codify a set of points, namely a line
Yeye, that's what I was trying to say and ask, I'm just not very good at communicating in general.
>A thing worth noting though: a translation is not a linear map unless it is the identity!
Makes sense, the zero vector wouldn't stay at 0 right? Someone mentioned affine spaces with reference to this, but I don't know anything about them. Also I saw somewhere that you can represent a translation in n space with a shear in n+1 space, is that true?

Thank you so, so much for this anon. It's been really helpful, I mean it.

>>11900937
>French people learn topos theory in kindergarten
euro bros...

>>11900995
>Russians are only admitted to kindergarten if they can pass the comprehensive Verbitsky exam
French bros.....

>>11900985
Why not? I am not familiar with the American system.

>>11900990
>Makes sense, the zero vector wouldn't stay at 0 right?
Precisely. That's the immediate failure.
>Someone mentioned affine spaces with reference to this, but I don't know anything about them.
If you want a quick rundown, maybe (a very maybe-ish maybe) Rotman's AT book and its chapter on affine stuff (the second chapter should I not be completely mistaken).
>Also I saw somewhere that you can represent a translation in n space with a shear in n+1 space, is that true?
Probably yes. It's a linear map and you get one extra dimension to play with.
>Thank you so, so much for this anon. It's been really helpful, I mean it.
No problem <3

Threadly reminder to bully physicists.

>>11901030
>physishits don't know what cohomolgy or a fundamental group is
OH NONONONO

>>11901015
Monetary issues, time and much more.

>>11900990
https://en.wikipedia.org/wiki/Affine_space#Definition
Not him but you should look at affine spaces to understand it I think. In the definition they distinguish the set of points and the actual vectors. Just notice that [math]A[/math] can be ANY set, you can take your set to be the same set as your vector space [math]\vec{A}[/math], and the action is just usual vector addition. But because you are not looking at the set of points as a vector space, you have the freedom to label the points as you wish and there is no need to have a distinguished [math]0[/math] vector in your set of points "only in the set of translations". That's how physicists model classical space, because setting that, idk, the center of the eart should have the label [math](0,0,0)[/math] is completely arbitrary, i.e. you can set you coordinate system (or way to label such points) however you want. However the displacement vectors relative between points is not arbitrary but a universal to all of us (in classical mechanics), and so as you can see in how they define "substraction" in terms of the universal vectors. In classical euclidean geometry, you don't work with any coordinate system, the objects are universal and putting a coordinate system is a matter of choice.

Serious question, why can't professors add CSS to their websites? Pure HTML is an eyesore.

>>11901045
But what about the cohomology of the fundamental group?

>>11901046
I see. That sucks.

>>11901062
>But what about the cohomology of the fundamental group?
uh no that's a bit cringe tbqhfamalambakadesusenpai

Can somebody run through the difference between a group being finitely generated vs. finitely presented? Does one imply the other?

>>11901061

as a grad student with eyesore vanilla html for their website:

i am lazy

>>11901075
finitely presented imply finitely generated. Read the fucking definition.

>>11900809
>Yes, Yukarifag didn't post anime, why did you feel the need to mention that?
2hou is anime.

>>11901125
Touhou is generalised anime.

>>11901125
Touhou is a group acting on a set of anime.

>>11900719
Yes!

>>11900767
>and a severely idiotic undergrad (likely non-math major).
That's me! Although I do have a degree in mathematics.

>>11901128
Anime in the sense of distributions?

>>11901075
A group is finitely generated if there is a finite set of generators. It is finitely presented if further, there are finitely many relations between the generators. For example, take the free group on two generators, [math]G=\langle x,y\rangle[/math], which essentially consists of finite products of the elements [math]x,y,x^{-1},y^{-1}[/math], such as [math]xy^3xyx^{-1}[/math] or [math]xyxyxyxyxyxy^{-1}[/math]. This is obviously finitely generated, since there are two generating elements, and it is finitely presented, since there are no relations between the elements other than [math]xx^{-1}=e=yy^{-1}[/math]. Now instead consider the relations [math]xy^n(xyx)^n=e[/math]. That is, every time you see an element of [math]G[/math] containing an expression of the form [math]xyyxyxxyx[/math], you can contract it to [math]e[/math]. Now do it for every [math]n\geq 2[/math]. There is no way to express these infinitely many relations with just finitely many relations. So if we now take the quotient of [math]G[/math] by the subgroup generated by this set of relations, we get a non-finitely presented group that is finitely generated.

If [math]f:X\to Y[/math] is a morphism of (Noetherian) schemes, [math]V\subset Y[/math] an open affine, and [math]f(x)\in V[/math], is it true that [math]f(y)\in V[/math] for all specializations [math]y\in \overline{\{x\}}[/math]?

>>11901143
not that anon, but nice explanation, I've learned something today

prove that if norm tv = norm t*v then t is normal, not using axlers shite proof

>>11901055
Thank you anon, that helps.

>>11901179
I have no idea how Axler proves it.
Anyhow, [math]0 = \langle A v, Av \rangle - \langle A^* v, A^* v \rangle = \langle (A^*A - A A^*) v, v \rangle[/math], so [math]A^* A - A A^*= 0[/math].

>>11901239
I'm talking about that last step, how you take it out of the inner product
his proof for that lemma is not intuitive at all

>>11901253
Oh, that.
For [math]A[/math] self-adjoint, [math]||A|| = \max _{||v|| = 1} | \langle Av, v \rangle |[/math]
Essentially because the norm equals the largest eigenvalue, but I recall there being a simple proof which I may or may not remember and then post.

>>11901253
The inner product is non degenerate thath means that if [math]<w,v>=0[/math] for all [math]v\in V[/math] then it must be that [math]w=0[/math].

>>11901283
I'm not familiar with the max |v|=1 <av,v> symbol, what does it mean

>>11901322
that's not true, w can be a 90 degree rotation if its not self adjoint... but it is self adjoint so whats the deal? what does non degenerate mean? why does it imply what you said?

>>11901322
What you say is true, but in that case w depends on v, so it is irrelevant.

>>11901143
original asker, thanks for the eplanation

>>11901322
>>11901344
In fact take A= [0, -1; 1, 0] and see that your reasoning leads to a false conclusion.

>>11901331
It's the matrix norm, if you don't know what that is or won't help you understand the proof.

>>11901283
Almost remembered it.
Assume [math]A[/math] is self-adjoint and positive. It's easy to see that [math]||A|| = \sup_{||v|| = 1, ||w|| = 1} \langle Av, w \rangle[/math].
[math]A[/math] effectively achieves those for some [math]v[/math] and [math]w[/math] (proof omitted)(use how it achieves the suprema in the normal definition of the norm).
Since [math]A[/math] is positive, we have that [math]\langle A(v-w), v - w \rangle = [ \langle Av, v \rangle + \langle Aw, w \rangle ] - [ 2 \langle Av, w \rangle] [/math], which needs to be greater than zero, because of the positivity, and also smaller than zero, because of the suprema.
Now trying to recall the trick for the not positive case.

>>11901363
>proof omitted
A continuous function assumes it's maxima and minima on a compact set.

>>11901372
>Hilbert space
>compact

>>11901372
sup is taken over all [math]\|v\|=1[/math] and [math]\|u\|=1[/math], so product of unit spheres, which should be compact if [math]V[/math] is finite-dimensional
probably not the case for infinite dimensions

>>11901377
>Finite dimensional Hilbert space
>Not compact

But seriously, how do you prove the infinite dimensional case?

>>11901385
fuck me, meant to reply to >>11901377

>>11901389
Assume [math]||Av|| = ||A||[/math] with [math]v[/math] a unit vector.
Then [math]w = \frac{Av}{||A||}[/math] does the job.
>>11901385
>he forgot Heine-Borel exists

remind me why people study this linear shit

>>11901408
>Assume ||Av||=||A||||Av||=||A|| with vv a unit vector.
I mean, that seems like the important part of the proof and nothing you can just "assume".

>>11901417
Because they are interested in:
PDEs, control theory, differential geometry, or a million other areas where you need this "linear shit".

>>11901283
>>11901239
BTW.
Assume that [math]A[/math] is self-adjoint and [math]\langle Av, v \rangle = 0[/math] for every [math]v[/math].
Then [math]0 = \langle A(Av+v), Av + v \rangle = \langle A^2 v, Av \rangle +2 \langle Av, Av \rangle + \langle Av, v \rangle = 2 \langle Av, Av \rangle = 0[/math], which implies it's the zero operator.
Which removes the need for the earlier estimate.

>>11901331
Not sure what you are getting at, [math]w[/math] was fixed in my expresion. What it means is if a vector is ortogonal to ALL vectors in an inner product space this means the vector must be [math]0[/math].
>>11901344
>>11901351
Oh yea it really is just positive definitness, I assumed non degenerancy was an axiom and it is not really immediate. Suppose there is an eigenvector [math]v\neq 0[/math] such that [math](A^*A-AA^*)v\neq 0[/math], then as it is self adjoint it only acts as multiplication on this eigenvectorand you have that [math]\Lambda<v,v>=0[/math], from positive definitness (which follows from non degeneracy) it implies [math]v=0[/math] I don't know if you can prove it without using the spectral theorem (or something equally strong) though.

>>11901428
That works so forget >>11901430

>>11901420
Classical Banach space result.
Just google it tbqh, the suprema in the definition of the norm is achieved.
>>11901433
K.

>>11901436
>Classical Banach space result.
Well, at least in the finite dimensional case you can get by very easy...

>>11901430
you're right, i was conflating w with Tv. there truly is no vector that is orthogonal to all vectors except for the 0 vector, because scalar multiples of all vectors exist

>>11901436
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA FUCK I COMPLETELY HALLUCINATED THAT RESULT
Eh, you can still just mess around with suprema.

>>11899438
Linear algebra
Real analysis and complex analysis(riemman surfaces)
Topology
Algebra
Homological algebra
Commutative algebra

>>11901451
>AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA FUCK I COMPLETELY HALLUCINATED THAT RESULT
You mean the max isn't generally assumed?
Then I too was hallucinating.

>>11901488
Nope.
Consider [math]\mathcal{l}^2[/math] and the operator [math](Av)_n = (1 - 1/n)v_n[/math]. Norm smaller than one, can't achieve one.

Why is enumerative geometry so hard bros

>>11901517
don't know, could you enumerate the reasons?

Why is enumerative reasoning so hard bros

>>11901516
Indeed. Well, at least I got a bit less stupid...

>Norm smaller than one
You mean that the norm of the operator is equal to one, I assume.

Does anyone know the exact form of the IMO?
I know each team is given 6 problems, 3 problems each day. I thought they brainstorm together and give a single solution per team but looks like each participant turns in his own solution and they are graded individually?

>>11901687
its an individual contest, the team with the highest total score wins

>>11901424
How do I get into control theory?

>>11901687
>I thought they brainstorm together and give a single solution per team
very few contests work that way. I only know of ARML that does that. All olympiads (and the Putnam) are solo contests where the individual scores get grouped together into teams

>>11901692
https://math.stackexchange.com/questions/2031926/self-study-control-theory#:~:text=To%20gain%20a%20basic%20background,Linear%20operator%20theory

Is there a special name for groups like [math]\mathbb{Z}_p[/math], where [math]\forall g \in G, \langle g \rangle = G[/math]?

Brainlet idea here, but please entertain my ravings for a moment.

So you know how we denote the [math]n[/math]-fold cartesian product of a set [math]X[/math] with itself as [math]X^n[/math]? Is there any reasonable way to assign a meaning to [math]X^n[/math] for [math]n \leq 0[/math]?

>>11901718
what about cyclic groups of prime order ?

>>11901718
Literally only Z_p and Z do that.
Assuming that with "for any g" you actually mean "any g not the identity".

>>11901735
Slight nitpick, not technically true for [math]\mathbb{Z}[/math], though the subgroup generated is isomorphic to [math]\mathbb{Z}[/math].

>>11901736
>slight nitpick
No, no, it was a real mistake, my bad.

>>11901735
doesnt Z_p1 X Z_p2 X...X Z_pk do that too

>>11901742
no (1, 0) does not generate the whole group

>>11901731
Not for strictly negative values, because in that case we would have [math]X^0 = \emptyset[/math] since [math]X^0 \times X^n = X^{n+0} = X^n [/math], then [math]X^{-n} \times X^n = X^{n-n} = X^0[/math], so you would have to find a set [math]X^{-n}[/math] such that its cartesian product with [math]X[/math] is the null set, which is absurd for any [math]X \neq \emptyset[/math].

>>11901742
Consider the following:
if a finite cyclic group has that property, then choosing any non-identity element gives you a field structure, so the order is prime.

>>11901752
X^0 = {*}

>>11901754
what do you mean by this, unironically?

>>11901752
Ok that makes sense. Is it standard to take [math]X^0 = \varnothing[/math]?

>>11901745
cheers, forgot about that
otherwise lagranges theorem would break

>>11901756
>>11901758
X^0 is not empty. [math]X^0 = \{\emptyset\}[/math] .

>>11901758
yeah no, I'm retarded, replace nullset with any set with one element

>>11901758
One element set so the cartesian product of X with that is essentially just X.

>>11901761
>Lagrange's
Cauchy's.

>>11901770
yeah really there wasn't enough theorems named after cauchy, we really needed one more

>>11901762
>>11901764
>>11901769
Ah of course, since [math]\varnothing \times X = \varnothing[/math].
Thanks for the replies, thought this was a stupid idea but I still got something out of asking it.

>>11901772
Burnside's lemma was known by Cauchy decades before Burnside published it. We should rename it Cauchy's lemma.

>>11901778
yeah, let's name 90% of math after Cauchy, Euler, Gauss, Riemmann and Grothendieck. Excellent idea.

>>11901772
abel pls

>>11901772
What the fuck did he mean by this?
If the order isn't prime, it's divided by some non-trivial prime, and Cauchy's theorem then guarantees an element with that order, which can't be the entire group, violating the original hypothesis.

>>11901779
>let's name 90% of math after Cauchy, Euler, Gauss, Riemmann and Grothendieck.

>>11901779
This but unironically.

>>11901779

>>11901785
for me, its grothendieckian small prime distribution theory

>>11901785
>>11901787
>>11901788
>name paper "a proof of a theorem by euler"
>use cauchy's theorem to prove gauss' construction
>deduce, using grothendieck's formula, that riemann's theorem is true
>conclude

>>11901779
This, but just Euler.
We can then convince the future generations that all of mathematics was discovered by Euler and that they'll never even remotely measure up to his monstrous achievement, so they might as well give up on maths.

>>11901802
But I already feel like this!

>his PhD thesis hasn't been given a nickname by the wider mathematics community
never gonna make it

>>11901802
>We can then convince the future generations that all of mathematics was discovered by Euler
It more or less was, to be fair

>>11901807
>publish one (1) paper in the tohoku math journal
>when people now refer to "the tohoku", they mean his article
how did he do it? He literally cucked the biggest japanese math journal...

>>11901807
co-author of big baz ere
ama

>>11901807
One of the profs at my undergrad campus wrote one of those theses that are referred to just as "X's thesis" and everybody remotely near the field has been citing their copy for decades.
Funnily enough he never did anything even close to as good as his thesis again. I guess either he got crazy lucky in his PhD or after he got fast-tracked to tenure he decided to chill.

Fuck why is picking modules so hard, I want to do so many of them
Any advice/suggestions?
btw recommended load is 4 per term, maximum is 10 in total

>>11901834
Measure Theory, Curves and Surfaces, Topology and Manifolds on the first term, Complex Analysis, PDEs, Functional Analaysis II and Algebraic Topology on the second term.
No need to thank me.

>>11901834
>T1
Commutative Algebra
Measure Theory
Manifolds
>T2
Galois Theory
Theory of PDEs
Complex Analysis

>>11901834
>E1 F1 G6 H5
>77 D5 H6

/gnmg/

>>11901852
>>11901863
>>11901864
thanks for the suggestions
btw I kinda want to get into cryptography after uni so I was thinking more algebra stuff
but I think that number theory stuff usually requires complex analysis and algebraic geometry as well

also I'm kinda interested in differential geometry the year after, which requires manifolds (general relativity sounds cool)

Shilov and Axler's takes on determinants are so opposite that it's actually rather funny

>>11901884
Who would you sooner trust. This guy...?

>>11901892
...or this guy?

I've just gone through calculus but not covered the more 'advanced' parts (I assume Calc III in the US?), can I go straight on to real analysis and linear algebra? Or is it worth doing more calculus from something like Schaum's Advanced Mathematics for Engineers and Scientists, and Advanced Calc?

>>11901900
What do you mean by "advanced" parts? Calc III in the US usually means vector stuff, you ought to know something about vector calculus.

>>11901874
>I kinda want to get into cryptography
Take a course on elliptic curves, then.

>>11901900
You should take linear algebra before vector calculus (at my university, it's a pre-req) and after vector calculus, you should go to analysis I.

>>11901821
What did he do in that thesis that people keep coming back to it for? Was it just an annoying proof that no else needs to write down because it's convenient just to cite the same source?

>>11901906

>>11901906
>>11901928
Elliptic curves is the year after

>>11901904
I touched on it, yeah. But I haven't gone further than the basics. Surely I don't need more calc if I just work through How to Prove it?

>>11901918
AFAIK, the main reason it gets cited so much is that there are many extraordinarily useful little lemmata in it. He did have some very solid results in there too, but in getting to them he basically created an entire toolbox for working with the subject, so it just happens endlessly that people want to prove some little step and the answer is "oh it follows right away from lemma 3.2 of X's thesis"

Nicolas Bourbaki and Bertrand Russell.
Are there authors more rigorous, formal?
I want to understand Math, not just calculate.

>>11901948
you type like you collect fedoras

>>11901892
>>11901897
Well...
I like Axler's book more in any case

>>11901941
That makes sense; good exposition and footholds along the way. He could have turned it into a book.

>>11901948
you wont be able to learn math if you care this much about your appearance
maybe try again once youre older

When you guys were undergrads, did you have different notebooks for book exercises and notes?

>>11901936
Honestly Analysis I was where I learnt how to prove things.
tbf my uni doesnt do lectures for analysis I, we basically just have booklets where we have to fill in the proofs ourselves
I can link the booklets if you want

>>11901992
Yes.

>>11901950
>>11901981
Sorry for trying too hard to fit in.
From what I`ve read, the only unified Math is Bourbaki`s Elements. A Treaty of Special Mathematics is taught in French places as an introduction. Im thinking of reading some of Russells books before them.

>>11897343
A set of indexed sets of indexed elements?

>>11901997
yes pretty pls

>>11902079
http://homepages.warwick.ac.uk/~masdbl/w1.pdf

replace the w1 in the url with anything from 1-10
do all the exercies
note these did take pretty much everyone on my course the whole week to do each week

>>11901864
what is this? chess moves?

>>11902082
Thank you very bong, I will add this to my list of resources

>>11897458
>Washington
Do you go to UW anon? Are you one of the undergrad fuckos who pals around in the discord?

>>11901834
Just do commutative algebra 10 times

>>11901900
How have you done calc 3 but not lin alg?

Love field extensions, lads.

>>11901852
Analysis, Geometry and Topology is easily the best combo you can study. OP, listen to that comment

>>11901813
He cucked the entire field of algebraic geometry. Truly a man ahead of his time.

Math-phys fag here, I feel like the more I grow in my math education, the harder it is for me to read physics sources.
I did both math and physics in undergrad, and moved to math for grad. In my senior year of undergrad I was ok with both math and physics sources, but as a grad student in math now I can't read a physics text without being bewildered at how badly they fuck up the math.
I wanted to keep that physics connection when I started, and now I feel like I'm losing it.
Any other anons experience this? Is this a bad sign?

>the most beautiful math identity

>>11902561
>Phoneposter

/gmmg/

>>11902118
I wish.

>>11901992
No.

>>11901948
My diary, desu.

>>11902553
At least you don't try this hard to be funny. I am pretty sure I've heard this guy live, or some other Italian doing same stuff and desperately putting in jokes everywhere, and it was la cringia. https://arxiv.org/abs/2007.07215

>>11902884
Good morning.

>>11899155
My program is long as fuck but I studied a bit of Banach spaces on my multivariable calculus course in 2nd year. The proper functional analysis course following the classical textbooks like Lax comes after Measure theory in like 4th year (program is 6 years long for reference).

>>11898818
what's the name of this problem again?

Can someone help me with this integral?

Evaluate [math] /int_C sinh 3z \, dz [/math] Where [math] C [/math] is the contour from [math] z = 0 [/math] to [math] z = 2 [/math] along the curve [math] y = e^{-x}sin{4 \pi x} [/math]

I know I need to use the definition of [math] sinh z [/math] but I'm just lost on how to set this up. Any help?

>>11903124
Integral is: [math] \int_C sinh 3z \, dz [/math]

>>11902884
good morning

>>11903124
>>11903126
[math]\sinh 3z = \frac{e^{3z} - e^{-3z}}{2}[/math] is an entire function, so the value of the integral depends only on the endpoints of the curve

>>11901834
hello fellow /warwick/

>>11901948
Bertrand Russell didn't do math, of course any formal logic or type theory book is more formal than a math book.

I'm going to read EGA, FGA and SGA, entirely. What am I in for?

Love abelian groups lads.

>>11903197
nut cracking

>>11903197
Interested in this too, bump
What order are they meant to be read in and would something like Hartshorn be a prerequisite?

How hard would you guys say it is to get a post-doc position? I am kind of afraid of getting my PhD next year. I guess I can always move back with my parents and be an "independent researcher" (lmao).

Why is 168 such a based number?

>>11903246
I hate them

>>11897356
Could be anytime but only if you bring Coronachan with you

>>11903343
Make your case. 57 makes that bald faggot grot cry

Have to pick a final paper for my math major. Applied math (where I can pick special rel, quantum, fluid dynamics...) or Hilbert spaces

>>11903328
FGA is a sketch (seminar notes) of what became EGA, SGA is something else. EGA is the super duper general version of Hartshorne if you will. It has no prerequisites except commutative algebra and topology. SGA is more specialized and not as detailed (it was a seminar after all).
I think you should read SGA after EGA but I don’t think it explicitly refers to it (they were written at the same time I think, maybe some of SGA even predates EGA)
Either way it is probably a bad idea

just started reading munkres' topology the part on topological spaces and bros wtf is this shit where are the silly shapes and donuts

>>11902209
No

>>11903328
The prereqs are basic knowledge of French, category theory at the level of any modern homological algebra textbook, homological algebra, commutative algebra and topology. Hartshorne is a condensed Noetherian version of EGA

>>11903442
You have to wait until you take an algebraic topology course.
And by this time you will hate topology.

>>11901900
Do linear algebra before real analysis.

>>11903442
There is a reason why people say point set topology is autistic and dry. Also, what the other anon said.

>>11903184
What year are you?

>>11903479
that would make me doxxable

>>11903481
Hey I've said I'm going into 3rd year, was just wondering if you were picking modules as well

I've just noticed that Russel's paradox requires third removed.
Does this mean you can you do intuitionistic set theory with unrestricted comprehension?

>>11903486
Wait, no, never mind, it doesn't require LEM.

>>11880476
From a previous thread, I think I worked out the connection between topological spaces and "topology"
Topology is about topological properties, which are properties that are invariant under homeomorphisms, like genus

>>11903472
Why?

>>11903493
It's not clear what you're asking. The coffee mug "pop math" is a prominent feature of topology.

I mean yeah, if you don't look at homomorphisms when doing topology, then that's just some random subset of the powerset.
What did your module introduce topological spaces for, if not to speak of continous maps?

>>11903493
Yes, sure. Topology is the study of topological spaces up to homeomorphism.

>>11903519
What exactly is the topological space when talking about a torus

>>11903543
[math]S^1\times S^1[/math]

so does multiplicity of a root in complex polynomial 'twist' the surface around the hole?

>>11903511
Definitely do linear algebra before differential equations. You use a lot of linear algebra in solving DEs

>>11903696
Yes.

>>11903709
cool ty
then the surface around the other roots, are they twisted more too, or is it only locally and for large values

>>11897308
Why independent subsamples give non-correlated models (so called bagging technique in machine learning) ?
Can this be proved rigorously or it's just another "empirical" bs ?

>>11903361
>>11903467

https://github.com/ryankeleti/ega

I found this, and there's an 'EGA 0' section for prerequisites; will this be enough? Or do I need to read all of Bourbaki?

>>11903921
new thread
>>11903921

>>11903921

>>11903921

>>11903916
Dont read EGA, retard