/sci/ - Science & Math

Threadly reminder to work with physicists.

>>9757798
Why are you recruiting people from these threads to work on something so distant from mathematics? A more appropriate place would be >>>/toy/physics/.

>>9758240
>mathematics
This is not well-defined.

>>9757798
Why would I want to work with a physicist if they bought 4chan? Into the garbage it goes

https://arxiv.org/pdf/1805.07741.pdf
>100% of the zeros of the Riemann zeta-function are on the critical line
>Tatyana Preobrazhenskaya, Sergei Preobrazhenskii
>(Submitted on 20 May 2018)
>We consider a specific family of analytic functions [math]g_{\alpha,T}(s)[/math], satisfying certain functional equations and approximating to linear combinations of the Riemann zeta-function and its derivatives of the form
>[math] c_0\zeta(s)+c_1\frac{\zeta'(s)}{\log T}+c_2\frac{\zeta''(s)}{(\log
T)^2}+\dots+c_{K}\frac{\zeta^{(K)}(s)}{(\log T)^{K}} [/math].
>We also consider specific mollifiers of the form M(s)D(s) for these linear combinations, where M(s) is the classical mollifier, that is, a short Dirichlet polynomial for 1/ζ(s), and the Dirichlet polynomial D(s) is arbitrarily long and arises from substitution for w, in Runge's complex approximation polynomial for [math]f(w)=\frac1{c_0+w}[/math], of a Dirichlet polynomial being the Selberg approximation for
>[math] \frac{c_1}{\log T}\frac{\zeta'}{\zeta}(s)+\frac{c_2}{(\log
T)^2}\frac{\zeta''}{\zeta}(s)+\dots+\frac{c_{K}}{(\log
T)^{K}}\frac{\zeta^{(K)}}{\zeta}(s)[/math]
>(analogous to Selberg's classical approximation for [math] \frac{\zeta'}{\zeta}(s)[/math].
>Exploiting the functional equations previously mentioned (which are about translation of the variable s), together with the mean-square asymptotics of the Levinson-Conrey method and the Selberg-Tsang approximation theory (with some additional results) we show that almost all of the zeros of the Riemann zeta-function are on the critical line.

https://arxiv.org/pdf/1805.07741.pdf
>100% of the zeros of the Riemann zeta-function are on the critical line
>Tatyana Preobrazhenskaya, Sergei Preobrazhenskii
>(Submitted on 20 May 2018)
>We consider a specific family of analytic functions [math] g_{\alpha,T}(s)[/math], satisfying certain functional equations and approximating to linear combinations of the Riemann zeta-function and its derivatives of the form
>[math] c_0\zeta(s)+c_1\frac{\zeta'(s)}{\log T}+c_2\frac{\zeta''(s)}{(\log T)^2}+\dots+c_{K}\frac{\zeta^{(K)}(s)}{(\log T)^{K}} [/math].
>We also consider specific mollifiers of the form M(s)D(s) for these linear combinations, where M(s) is the classical mollifier, that is, a short Dirichlet polynomial for 1/ζ(s), and the Dirichlet polynomial D(s) is arbitrarily long and arises from substitution for w, in Runge's complex approximation polynomial for [math] f(w)=\frac1{c_0+w}[/math], of a Dirichlet polynomial being the Selberg approximation for
>[math] \frac{c_1}{\log T}\frac{\zeta'}{\zeta}(s)+\frac{c_2}{(\log T)^2}\frac{\zeta''}{\zeta}(s)+\dots+\frac{c_{K}}{(\log T)^{K}}\frac{\zeta^{(K)}}{\zeta}(s)[/math]
>(analogous to Selberg's classical approximation for [math] \frac{\zeta'}{\zeta}(s)[math].
>Exploiting the functional equations previously mentioned (which are about translation of the variable s), together with the mean-square asymptotics of the Levinson-Conrey method and the Selberg-Tsang approximation theory (with some additional results) we show that almost all of the zeros of the Riemann zeta-function are on the critical line.

>>9757795
Why hasn't anyone come up with a model that takes exponentiation as a primitive?

>>9758344
Wasn't there another proof submitted like 2 days ago?

>>9758362
>Wasn't there another proof submitted like 2 days ago?
This is "almost all of the zeros", not "all zeros", so it doesn't imply the Riemann hypothesis..

>>9758365
>This is "almost all of the zeros", not "all zeros", so it doesn't imply the Riemann hypothesis..
What the hell are you referring to?
The paper you linked says "100%" which isn't "almost" all, it's exactly all. And the one I am was referring to was this. https://arxiv.org/abs/1805.06746 which also states *all* and not almost.

>>9758448
>What the hell are you referring to?
see >>9758344
>we show that almost all of the zeros of the Riemann zeta-function are on the critical line.

>The paper you linked says "100%" which isn't "almost" all, it's exactly all.
This is false. See https://en.wikipedia.org/wiki/Almost_all

>>9758362
>>9758360
>>9758365
>>9758448
>>9758454
>>9758344
>100% of the zeros of the Riemann zeta-function are on the critical line, pic related
I proved that the critical line stops hazing Re(z)=0.5 somewhere between countable and uncountable infinity
On The Riemann Zeta Function
http://www.vixra.org/abs/1703.0073

For the anon that was asking whether schools ranked below 20 on US News were worthwhile, I wrote a fucking novel before realizing the thread was archived:

>Are schools ranked #21-30 on US News worth it for math?
Yes lol, US News is infamous for fucking over public schools in its undergrad rankings because of things like endowment per student and student teacher ratio, which disadvantages large public schools. Its formula also uses selectivity, alumni giving, and graduation rates, which screws public schools as they enroll more disadvantaged students.

Berkeley is ranked #21 on US news, behind fucking Rice, Emory, WashU, Notre Dame, Cornell etc, and you'd be retarded to go to any of those over Berkeley for math/engineering. Basically, any of the public schools listed #21-30 are some of the top math schools in the country, particularly Cal, UCLA (#21), UMich (#28), UNC Chapel Hill (#30). NYU, Carneggie Mellon, and perhaps USC are all very good private schools that are worthwhile for math, engineering and other sciences. However, below this the private schools kinda go to shit for math.

Basically crosslist your choices with this list to give you an idea of the strength of the math department: https://www.usnews.com/best-graduate-schools/top-science-schools/mathematics-rankings . Remember that US News fucks public schools for undergrad rankings, but also that a good math grad program doesn't necessarily make for a good undergrad. It does, however, indicate more research opportunities, a higher calibre of faculty, the possibility of taking graduate level classes etc...

>>9757795
Is he the comfiest mathematician?

>>9758805
he took this picture at RIMS
>mochizuki will never personally teach you IUTT
>you will never reach maximum comfy

Looking for an explicit expression for this series
1/2, -1/2, -1, -1/2, 1/2, 1, 1/2, -1/2...

>>9758834
I mean sequence

>>9758834
>>9758838
To elaborate this came from calculation of trace of power of adjacency matrix

The numbers are real parts of complex eigenvalues that are distributed as pictured

But I need an explicit representation of the real part only

>>9758834
Use a generating function dummy, or wolfram alpha if you're too lazy.

>>9758834
>>9758838
>>9758843
[eqn]\frac{\left ( \frac{1}{2} +i\frac{\sqrt{3}}{2}\right )^{n}+\left ( \frac{1}{2} -i\frac{\sqrt{3}}{2}\right )^{n}}{2}[/eqn]

>>9758834
is that not just cos(pi * x / 3)

>>9758908
noice

>>9758847
>>9758853
>>9758908

Anyone knows where can I download this book? Please, I really want this book.

>>9758948
just buy it you poorfag
it's a dover trade paperback, it's fucking $10

>>9758448
>The paper you linked says "100%" which isn't "almost" all, it's exactly all.
LMAO imagine being this fucking stupid.

>>9758823
>>9758805
do we know how comfy his house is?

>>9758448
Brainlets get out REEEEEEEE

>>9759133
What's so stupid about that?

>>9759521
>What's so stupid about that?
If you don't know about the basics of measure theory don't talk about probability theory.
100% MEANS "almost all", which is NOT a vague term.

>>9759536
I know what "almost all" means. Does the author use "100%" to mean it in that sense or in the sense of "all"? I didn't bother to open up the paper.

>>9759558
In the brief abstract posted above it is called "almost all".
And I doubt that any mathematician would use "100%" in a serious paper to mean "everything" and not just "almost all".

>>9759591
Well, after this I know I wouldn't ever use the whole "100%" in my papers if I were to
(1) publish anything ever
(2) publish anything suitable for the use of "100%".

>>9759600
That might be a good idea.

>>9757795
fuck you math kiddies why do I have to do integral calculus it takes so much time and writing its literally bullshit

>>9759637
you say this because it's difficult and not because it's actually retarded
you'll hate it more after realizing that 70% of the course was literally useless and you never actually solve an integral like that once you complete the course

>>9759558
>I know what "almost all" means.

>>9759769
Yes? The Lebesgue measure of the complement is 0.

I cant decide between algebra and analysis :/.

what area of maths combines algebra, geometry, and topology?

>>9760561
Almost all modern areas of math require knowledge from those branches. Stop listening to retarded undergrads.

>>9760561
>what area of maths combines algebra, geometry, and topology?
https://en.wikipedia.org/wiki/A%C2%B9_homotopy_theory

>>9760561
I'm a filthy undergrad who has studied some math on his own but from my understanding you should lean towards analysis, geometry, and topology rather than algebra since all the low-hanging fruit has been taken.

>>9760603
>I'm a filthy undergrad
Oh, it shows.

what are the most useful branches of math, in your opinion

>>9760719
>math
This is not well-defined.

>>9760719
Long division.

>>9759521
Suppose set A is uncountably infinite. Consider set B which is a subset of set A.

Suppose card(B)=countably infinite

then measure(A-B) = measure(A)

Then there's some shit about almost all, but basically almost all(100%) doesn't actually mean for all, even if the measures are the same.

Look, just stop being stupid and take a measure theory class or something.

>>9760719
Probably arithmetic since it's the only fucking thing normies can bring themselves to do.

>>9760719
Everyone here is too insecure to admit it, but the answer is clearly Computer Science.

>>9760880
Was compsci used to develop thermodynamics of an engine? Was it used to develop the theory of statics behind trusses of bridges? Or perhaps to develop the partial differential equations used to model fluid mechanics? No? Well then perhaps it's not as useful as you think.

Notice that its main use so far has been collecting data for the sake of advertising. Compsci, though very innovative in recent times, is not the end-all-be-all that you think it is.

>>9760906
compsci forms the basis that allows you to rant about muh data collection here on the internet

I'd say that's useful enough

>>9760824
I know measure theory.

>>9760880
>Everyone here is too insecure to admit it, but the answer is clearly Computer Science.
It's not a matter of insecurity, it's a matter of accuracy. Computer science is not a branch of mathematics since mathematics is a branch of computer science.

>>9761022
>math is a branch of computer science
Kill yourself nigger

>>9761036
>nigger
Why the racism?

>>9761022
>>9761036
>>9761046

>>9757795
How would you explain the Riemann hypothesis to a layman?

>>9761079
>How would you explain the Riemann hypothesis to a layman?
We wouldn't.

>>9761082
>We

>>9761114
>>9761114
>>We
Mathematicians use "we", not "I".

>>9761125
I'm not a "we"

>>9761139
>I'm not a "we"
We already knew you aren't a mathematician.

My final day as a student. It wasn't fun, but it was a reason not to do anything important. Quite sad this end of an era.

>>9760561
algebraic geometry
algebraic topology

>>9761079
Find all the zeroes of the sum [math]\sum_{n=1}^\infty n^{-s}[/math].

>>9759133
>100% is mostly all and not all.

Can someone illustrate this with an example a retard pleb like me can understand?

>>9761468
[0,1) is 100% of [0,1]

https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news
>Two prominent and very well-regarded mathematicians have isolated a specific and serious error in Mochizuki’s proof of the abc conjecture. They are preparing a detailed writeup explaining the issue, which should be available publicly in the next month(s).

what are the latest betting odds on whether anything actually comes out of this?

https://mathoverflow.net/questions/300889/what-makes-a-structure-a-mathematical-structure
>We know that a group or a topology, for example, are purely mathematical objects. But what, for example, about a turing machine, or a normal-form game? They are also structures, but one could say that they belong to computer science and ecomonics, respectively, and that they are not purely mathematical structures. But they all are rigorously defined using logic and sets, so, what makes them different to say that one is a mathematical object and another one not?

>tfw only read russian text books
Am I a brainlet

>>9761723
it's crazy how high quality russian math texts are
they blow any other nationality on the planet clean out of the water

>>9760561
Algebraic geometry and algebraic topology are the obvious candidates. Something I'm surprised hasn't been mentioned yet would be Lie groups and especially their representations because you can look at them from different angles, meaning you're free to do what you like. Maybe dunk some differential equations on them, everything goes.

I'll also plug my own question >>9761751 in case someone here knows more about quantum computing and post-quantum cryptography.

>>9761731
But it makes you feel like a brainlet desu since I can't read all non-russian books that are always recommended.

>>9761781
>differential equations
Refer to >>>/toy/.

>>9759914
>complement

>>9758797
What about schools outside of the US? How should one go about the rankings?

>>9761841
Yes?

>>9761861
do not respond to the troll that does not believe in LEM

you're either stupid or not stupid, and both of you fall in the former category

>>9761876
>you're either stupid or not stupid
What does this have to do with LEM? Do you have brain damage?

>>9761876
>LEM
not science or math

Can all math be derived from ZF(C)?

>>9762023
uhh obviously not, or else we wouldn't have different mathematical systems in which these things are not used.

You're either a troll or an orangutan who learned to use a computer.

>>9762029
r-rude :(

>>9757795
>pure mathematics

What is it with indian instructors and uploading their math lectures on youtube?

>>9761514
If it’s Scholze and Kedlaya, then Mochizuki is fucked

>>9762331
It's me and David Hansen. We will be publishing this week.

>>9762113
le ebin reddit face XDXDXD

>>9762412
le reddit meme xd

How the fuck could P possibly be equal to NP?

>>9762482
Try asking >>>/g/.

>>9762523
Isn't this the equivalent of asking the dude that installed my fiber optic internet?

>>9762531
I wouldn't know. Try asking >>>/g/.

literally all of my lost marks are due to some shitty arithmetic error and it makes me want to blow my """"""""""""brain"""""""""""" up

>>9762550
>arithmetic
Refer to some other thread, like >>>/sci/engi/.

>>9762373
If serious, is there any essentially new idea inside his mountain of theory? Nobody I've talked to has really been able to point to any particular idea/key step that cracks the problem.

>>9762573
He obviously meant he was doing arithmetic geometry.

>>9762531
Yes, but his answer will be just as good as everything you will get out of the bitter mathfags here.

>>9762482
How the fuck can you prove it isn't?

>>9762482
>>9762681
https://www.youtube.com/watch?v=hpigjnKl7nI

>not math(s)

>not well-defined

>>>(insert general or board)

Why are mathematicians so bad at banter

>>9763254
pretty sure its just one autismo

>>9763254
>ribbit frog
>>>/b/

Maybe one of you can help me with this question
>>9763850

>>9757798
This is a good set of books (quantum fields and strings)

>>9762029
You sound like babbling pseudointellectual, can you provide some examples?

>>9763853
inkscape?

>>9763853
Some tikz library? Inkscape as suggested is good enough for 2d images (export as pdf+latex), asymptote, otherwise some adobe^{pirated} software, or use a 3d graphics library with any programming language. Just pick your favourite.

>>9757795
how does one stop panicking and think clearly when faced with a difficult question on a test?

>>9764400
You have to ascend.

>>9764400
Skip it immediately if you don't know how to do it. Do all the problems you know, and then you can tell yourself
>I've already got 80% on this test and I have 30 minutes to play with these 2 problems and see what I can do

>>9761731
Have you ever followed French ENS courses ?

Anyone have Ahlfors Complex Analysis Third Edition?

Starting on page 284 he introduces analytic continuations in terms of germs and sheaves. This is my first time being introduced to the material and there is a part on page 287 that doesn't make sense and I suspect it is a typo.

Relevant section in the image.
There's no way that [math]\pi(s - s') = \pi(s) - \pi(s')[/math] because each of those terms equals zeta, right? So did he mean to have that minus sign be an equals sign?

BTW, Googling errata gives nothing.

>>9764894
https://math.stackexchange.com/q/592580

>>9764894
Someone has fucked up with the manuscript where he has written = but one of the bars has worn out so badly it looks like -. That is my conjecture.

>>9764899
Thanks.
>>9764912
Makes sense.

What is it that pure mathematicians see in PDEs?

The extent of my experience with them is a computational course (ie, no proofs) that taught (among other things) solutions to the heat equation using Fourier Series. As it stands currently I have no interesting in learning anything further about them.

What should I read if I want to see what PDEs are really all about?

>>9765153
>mathematicians
>PDEs
What sort of twisted world are you living in?

>>9765164
Isn't the majority of mathematical research done in PDEs? Hasn't Tao done a lot of work on the topic?

Don't go telling me he isn't a real mathematician now you autist.

>>9765173
Don't reply to the spammer.

>>9765173
>Isn't the majority of mathematical research done in PDEs?
No mathematical research is done in PDEs.

>>9765173
>Isn't the majority of mathematical research done in PDEs?
Certainly not the majority, but it's an active field.

>>9765173
>Hasn't Tao done a lot of work on the topic?
Mathematicians can do engineering research as well.
>>9765179
Yeah, it's a pretty active field of engineering. I'm working in it myself.

thread theme

https://www.youtube.com/watch?v=bWze_JdO57s

I want to prove that a second countable metric space X has a countable dense subset (a countable subset whose closure is X). Is this proof outline correct?

Let B be a countable basis for the topology of X. Pick one point from each member of B and form a set Q of those points (using the axiom of choice). Q is countable because B is as well. To prove that Q is dense is equivalent to proving that X - Q has an empty interior, so that every non-empty open set O that contains a member of Q. Assume the existence of such a set O. But O is a union of members of B, so Q must contain at least one point of those members, hence the intersection of Q and O cannot be empty. This is a contradiction, so no such O can exist.

I probably wouldn't doubt this proof were it not for the fact that it doesn't use the condition that X is also a metric space.

>>9765221
Second countability implies separability which is the property of having a countable dense subset. Neither of these are related to metric spaces in any way, as you can find examples of non-metrizable 2nd countable spaces and non-2nd countable metric spaces.

>>9765255
>you can find examples of non-metrizable 2nd countable spaces and non-2nd countable metric spaces
Do these examples have any actual substance or do they just follow from some unintuitive and unnecessary formalism? Cause that's what it seems like to me.

>>9765255
Ah, thank you very much. The actual problem was to prove that for a metric space, second countability is equivalent to separability. I guess the metrizability is used only in the other direction of the equivalence.

>>9765263
The Sierpinski space is not Hausdorff, so it is not metrizable, but its topology is finite, and thus it is 2nd countable. On the other hand, take the real line and equip it with the discrete metric with d(x, y)=1 for distinct x and y to get a metric but not separable, and thus not a 2nd countable space. They have substance in the sense that you don't need properties like metrizability to get results related to the countability axioms, nor does metrizability imply them.

>>9765272
Your proof for 2nd countability -> separability is correct. The other direction is similar to sequential compactness -> compactness for metric/metrizable spaces.

>>9765277
>The other direction is similar to sequential compactness -> compactness for metric/metrizable spaces.
And by this I mean that this uses metrizability, as we are dealing with a strictly weaker notion.

https://arxiv.org/pdf/1805.09587.pdf
>Associative algebras and broken lines
>Jacob Lurie, Hiro Lee Tanaka
>(Submitted on 24 May 2018)
>Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves on Broken with values in any compactly generated infinity-category C. Moreover, we show that factorizable C-valued sheaves (with respect to a natural semigroup structure on the stack Broken) can be identified with nonunital A-infinity-algebras in C. This is a first step in a program whose goal is to present an `equation-free' construction of the Morse complex associated to a compact Riemannian manifold.

>>9765277
Thanks again. I used the set of open balls with rational radii, centered on points of the dense subset as the countable basis for the other direction.

>>9765288
Good job!

>>9765292
Whore

>>9765292
>anime

>>9760824
nigger he knows measure theory and so do i
that doesn't change the fact that "100%" is ambiguous language and, naturally, would imply ALL and not almost all
if the nigger authors had any sense or decency they would not use such a provocative title for their shitty paper, opting instead to just say "Almost all zeros etc."

>>9765899
>that doesn't change the fact that "100%" is ambiguous language and, naturally, would imply ALL and not almost all
It's not ambiguous, nor does it imply "all".

>>9765902
it is surely ambiguous when the title of a paper is written in natural english and has nothing to do with probability theory, where one would reasonably associate probabilities with measures
stop pontificating, your undergraduate education is embarrassing

>>9765905
>it is surely ambiguous
There are infinitely many zeros of the zeta function, so 100% does not imply "all".

>>9765876
Don't respond to her. She's obnoxious and retarded

I'm self-studying differential geometry from Jeffrey Lee's book (not to be confused with John Lee's book, which is also a common recommendation for the topic).

This proof for the existence of cut-off functions seems to have a small error (easily fixed), but I just want a sanity check. Clearly the support of [math]\phi[/math] is equal to the closure of O, right? And hence not contained in O.

This could be fixed by choosing r < R' < R and constructing the function using R' instead of R, but it just seems odd that he would let such an obvious error slide, especially when he doubles down on it later in the proof.

>>9766490
The rest of the proof, if anyone cares

>>9764256
>muh examples
Do you know what an axiom/postulate is?

Consider this: reject Euclid's 5th postulate and replace it with a postulate stating that two parallel lines have exactly one intersection. What do you get? Spherical geometry.

Consider this: reject ZF and construct other axioms in their place. Derive what you can from your new axioms. Literally that simple, brainlet.

>>9765283
Based Lurie does it again. Absolute lad
>mfw centrally extending the Lie structure by C* and obtain a quantized description

>>9765911
>i don't know what infinity is
just stop posting, retard

Redpill me on Algebraic Geometry. I don't really have the prerequisite knowledge to jump into it, mainly because I just can't get myself interested enough in algebra to study it beyond an undergraduate level.

I don't necessarily dislike algebra, but I just don't like studying it for its own sake. I like learning about it in the context of say Algebraic Topology, but on its own I find it dry.

I have this text and I've been reading it a bit, but I just can't see how anyone can be so goddamn interested in the zero loci of sets polynomials. Like literally who gives a fuck?

>>9766973
>I like learning about it in the context of say Algebraic Topology
You don't if you claim "I just can't get myself interested enough in algebra to study it beyond an undergraduate level".
>I have this text
This text doesn't really contain much algebraic geometry. It doesn't even define varieties properly.

>>9766973
I really liked the first Algebra courses about Groups, Rings, Fields and Galois theory but when I heard most of what comes after that really does revolve around fucking zeroes of polynomials I lost all interest.

>>9767054
>most of what comes after that really does revolve around fucking zeroes of polynomials
I keep hearing this constantly. Where do undergrads get these retarded ideas from?

>>9767026
The definition of affine varieties in the text seems perfectly fine.

>>9767092
The definition is not given in that book.

>>9758797
No schools outside Harvard, MIT, Princeton, Berkeley, Stanford, Chicago are worthwhile in general, but a few other places might be okay depending what your subfield is.

>>9767074
I could imagine someone would think that if their knowledge of algebraic geometry is memes on /sci/ and the first two paragraphs of the wikipedia article, which covers at least half of this thread

>>9767092
dont respond to the spammer

>notahe will claim that the definition given in the book is not a variety, but instead some ever-so-slightly less general object that is probably irrelevant to any work you will do in the next couple of years, and probably goes by some technical jargon like quasi-affine semi compact variety or some bullshit

>>9767402
fug didnt mean to gt that

>>9767404
Who is this spammer, and how do I identify him?
He sounds like a real fucking pedantic autist.

>>9767409
whenever he says some bullshit about definitions, you know it's him because no one that has ever actually internalized and truly understood the material would actually care about some quasidefinitions

>>9767416
d-delet

>>9767402
>but instead some ever-so-slightly less general object
Not him, but I guess you're rigrht. I haven't done much algebraic geometry too, but it's nice to internalize the fact that every field or even ring can be viewed as [math]\mathbb{C}^n[/math] and not some technical "quasi-affine" dry formalism. Do you know of some good texts which use this approach?

>>9767169
Really depends on your advisor too. If you go to UCSD and have a great collaboration with Kiran Kedlaya, you shouldn’t have trouble finding a job. Same for other mid ranked schools like UIUC and Kevin Ford.

>>9767439
Right, that was what I was getting at, but in general if you can't get into one of the top 6 or so US schools, you really don't have what it takes to do pure mathematics, ability-wise. There's nothing wrong with going to a shit school (as in not top-6 or with a top advisor) if you plan to just go work in tech or something later and are just doing your PhD for fun or personal fulfillment, but your academic prospects outside the super-elite are basically nil - you just won't have the connections and pedigree to get anywhere.

>>9766490
I think this might be one of those times where the wording is slightly off but it's not technically incorrect. Maybe what we means is that the function [math]\phi[/math] is non-zero only in [math]O[/math] therefore when he says "compactly supported in [math]O[/math]" what he means is that support of the function is compact and the function is nonzero in [math]O[/math]. I feel like that's a stretch but it would fix the error, just a change of language really.

>>9767402
>probably
You have much to learn. Understandable for an undergrad, though.

>>9758344
This post makes me go REEEEEEmann.

>>9757795
We had a problem come up in a D&D session last night that we couldn't solve. The GM guessed an answer and let it slide but several of us are too autistic to let it go but not enough to actually solve it. No actual mathematicians present, just some math-adjacent guys.

The situation was a giant water wheel with 12 buckets, the direct distance between two buckets was given as 100'. We were supposed to walk over the top of the buckets (lids) when it was at the 9 o'clock position but it was stopped halfway between buckets and we wanted to know the horizontal distance of the nearest two buckets.

So, what is the chord height given the chord length and an arc angle of 30°?
I think there's enough information to calculate that isn't there?

I can sort of fumble my way to arcsin(30)=1/x ?

>>9757795 (OP)
We had a problem come up in a D&D session last night that we couldn't solve. The GM guessed an answer and let it slide but several of us are too autistic to let it go but not enough to actually solve it. No actual mathematicians present, just some math-adjacent guys.

The situation was a giant water wheel with 12 buckets, the direct distance between two buckets was given as 100'. We were supposed to walk over the top of the buckets (lids) when it was at the 9 o'clock position but it was stopped halfway between buckets and we wanted to know the horizontal distance of the nearest two buckets.

So, what is the chord height given the chord length and an arc angle of 30°?
I think there's enough information to calculate that isn't there?

I can sort of fumble my way to arcsin(30)=100/x ?

>>9767675
ok, treating it as an isosceles triangle, I get sin(15)=50/r which gives me r and working out the rest from there is easy.

i was trying to learn ML through Introduction to Statistical Learning and i felt like a fucking dumbass brainlet with all this shit about statistics. its not unbearable but im just so slow parsing the goddamn thing
and then i looked online and people are using algebraic topology (wtf??) in neural network and im like "wtf im not up to this shit"

>>9767809
Same. I've taken Calc III, linear algebra, and probability theory, but even the brainlet intro to statistical learning is a bitch to get through. There's not enough exercises either.

I honestly don't know how to proceed other than to just say "fuck it" and keep going through the chapters after I've done the exercises. P.S. the solutions to many of the exercises are online

>>9765283
>https://arxiv.org/pdf/1805.09587.pdf
cool paper

>>9767829
how far deep are you? im only at chapter 3 and this deep analysis of linear regression is pretty rough. its been some time i have done some rigorous math too

>>9766973
Algebraic geometry is about zero loci of polynomials in the same way differential (complex) geometry is about zero loci of differentiable (holomorphic) maps, i.e. it is not just that.
There are deep connections between algebraic geometry over complex numbers and complex geometry. If you study them you will see that you can replicate the same theory is those different settings (say divisors and riemann-roch).
Usually before getting into schemes you will study varieties embedded in affine/projective space and you will answer a lot of interesting questions. Then you define morphisms, and you call a variety affine if it is isomorphic to a closed subset of an affine space, same thing with projective. And then you discover deep properties of affine and projective varieties. You can study dimension, singularity, and a lot of interesting stuff and constructions (Segre, Veronese, Plucker, ..).
Maybe you picked up a book which is too devoted to polynomials, I don't know, but algebraic geometry is geometry. I also don't think you need a lot of algebra, you can safely skip the constructive theory of Groebner basis and friends. The only prerequisite is commutative algebra, which if you are an undergraduate is not hard to pick up, you only have to read the proof of few theorems (or you can read the statements and come back later).

>>9767809
Now just imagine yourself putting all this effort into learning the theory of ML and most of the guys using ML for a living don't really understand the math and just import tensorflow

>>9767867
is that really? i mean dont you need to at least have deep understanding of statistics to analyze your dataset. i thought ml engineers are basically business consultant with knowledge in CS so they can automate their thinking and scale them up

>>9767870
Honestly, I don't really know. But the impression I get is that the ML work that isn't about furthering the theory and developing new techniques is a lot closer to script kiddie stuff than it is mathematics/statistics.

>>9767860
Thanks dude.
I might just start reading Hartshorne and use a commutative algebra text as reference, treating some of the tougher theorems as black boxes.
I just want to know what this shit is all about.

>>9767870
>>9767877
The 'art' in ML is designing the models, not in the number crunching. As an MList (MLer?), I know fuck all about the maths[1], well that's not true but I read the papers, see what people say is the best technique and use that. Half the time, community consensus on things like TensorFlow just say 'this technique works best most of the time but not for X' and you just go with that, maybe do some comparisons for if the techniques for X work better on your data.

Where we earn our pay is in finding a way to model the problem as a dataset that can be learned over. This is called Representation and is hand's down, the hardest part of ML. The rest you can learn from youtube, you'll be a script kiddy but you'd be able to solve some sorts of problems. The Representation problem though, does not have shortcuts. When you do your tutorials (say MNIST), they'll have good representations that have stood the test of time and you can use them but then what happens when you want to apply those skills to making an autoencoder for identifying something completely different, like predicting an individual's next five years taxation contribution using subject's twitter hash tags and resume as source data?

Figuring out how good models work and designing ways to identify relevant features without second-guessing the learning...that's an art and it takes lots of case studies, reading of literature and experimentation to get a feel for it.
The difference between good and bad models is huge, a bad model will fail to solve even easy problems, unless the ML can adapt a bad model into a good one internally or something but even then, it's having to learn two things which is hard.

[1] I did once have to dive into maths to develop a fitness covariance method of bloat control in GP trees. That was intense but fun.

>>9767912
Algebraic geometry at a basic level is the study of geometric objects locally modeled on spectra of commutative rings, so a lot of things are proved by reducing them to commutative algebra. Atiyah Macdonald is pretty short and the first three chapters should be enough for starting out with something like Hartshorne, also check out Ulrich Görtz.

>>9761885
>Literally an example of LEM
>>9762023
>Hasn't heard of the continuum hypothesis
>>9762482
Easy, if there's a polynomial time algorithm to 3-SAT

>>9767809
>>9767829
>>9767939
Refer to the >>>/g/hetto/.

>>9767967
>Literally an example of LEM
How can something be an "example" of LEM? Are you daft?

>>9767972
>Doesn't know what LEM is
fixed

Are all of the schools on the first page of the us news math grad school rankings worthwhile for a PhD? https://www.usnews.com/best-graduate-schools/top-science-schools/mathematics-rankings

how do i go from calc3 to algebraic topology?

>>9767809
A lecturer for a ML course I did was writing this book that he shilled a bit.
https://mml-book.github.io/
No idea if it's any good but he was a cool dude

>>9761514
>>9762331
>>9762373
Big if true.

>>9765283
This fellow has a very rich imagination.

>>9768024
PhDs are a waste of time.

>>9768167
see >>9767968

>>9768150
from calculus to cohomology by Ib Madsen and Jxrgen Tornehave

>>9768188

>>9768183
Hot damn that is one T H I C C girl.

>>9767912
The first chapter of Hartshorne is too much concise. Try mixing it with Hulek (chapters 0-3) and Shafarevich (chapters 1-2). *After* you learn the theory try to work out the examples from Harris's book. There are also video lectures on youtube, search 'algebraic geometry nptelhrd'.
I'm currently waiting the publishing (5 July 2018) of Introduction to Algebraic Geometry by Cutkosky, from the toc it seems nice.
If you want to learn commutative algebra intended for algebraic geometry (w/o schemes) try 'Undergraduate commutative algebra' by Reid.
Also stacks project's chapters 'fields' and 'topology' are helpful.

>>9768024
Yes. At schools outside the first page, you basically need a world famous advisor to have a shot at tenure track academia. If you’re just going to industry, don’t worry too much about school rank other than to make your dick look bigger

>>9768806
>w/o schemes
Is there an actually legitimate reason to do this?

>>9766973
Many interesting geometric/arithmetic objects can be described as zero sets of polynomials (algebraic curves, linear groups, solutions of diophantine equations, etc.)
Obviously, algebraic geometry *is* geometry, but as indicated by its name, it strongly relies on its objects of study having an algebraic flavor (as you will probably soon find out, the local theory of algebraic varieties is that of algebras of finite type, hence you should not be surprised to see that many theorems tend to reduce to statements about polynomial rings).
Now of course, geometry is not limited to its local theory and just as local differential geometry is basically linear algebra but has very interesting global phenomena, the same thing happens in algebraic geometry (ie. at some point, you can start thinking in a global way without thinking as much about the sometimes gory commutative algebra underneath).
But you should be aware of the fact that A. the basic theory is a lot more involved than in diff geo and B. the algebra is not going away, and it only gets more involved the deeper you get into it.

>>9768923
pedagogy ?

>>9768937
Pedagogy would be one of the main reasons against the w/o schemes approach to "algebraic geometry".

>>9768806
For the same reason you don't give to read EGA to a 5 years old kid. I'm not saying you have to limit youself to that. But it is easier to start with this approach (less prerequisites, less abstraction), and you can still do a bunch of concrete things.

This >>9768965
was meant to >>9768923 *

>>9768965
>For the same reason you don't give to read EGA to a 5 years old kid.
So inability to read and an undeveloped mind? I guess that might be a legitimate reason, it's not like those people can help it.
>less abstraction
I fail to see how the level of supposed "abstraction" is a legitimate reason.

>>9768985
>So inability to read and an undeveloped mind? I guess that might be a legitimate reason, it's not like those people can help it.
>I fail to see how the level of supposed "abstraction" is a legitimate reason.
t. person who did not studied cohomology of sheaves before knowing an example of manifold

>>9769051
Studying sheaf cohomology is a pretty good way to learn about examples of manifolds.

To the differential geometers here. Why can physicists assume there is a "momentary comoving reference frame"? To put it in more mathematical terms, why can they assume that given a timelike curve in a lorentzian manifold, there is t0 such that the the tangent vector to the curve at that point has the form (1,0,0,0)?

>>9769086
>Why can physicists assume
Maybe you should try asking the physishits themselves? Try doing that in >>>/sci/pg/ or >>>/toy/.

>>9768923
How do you get students to understand why schemes need to exist if they don't know what a variety is?

>>9769133
This is not something worth wasting much time on as there is already a lot of material about these things. Explaining the intuitive idea is enough and doesn't require developing AG in a worse manner.

>>9769164
It requires developing enough machinery so they can actually see where the classical approach breaks down or is too restrictive and what the modern definitions are accomplishing.
It's for the same reason nobody (except Dieudonne's books, who was unsurprisingly a Bourbaki memelord) does Lebesgue before Riemann even with very gifted students despite it being an unarguably better theory.

>>9768935
Well said.

>>9769086
To be more clear to the lesser autists, I want a formalization of these concepts. I have manged to bring back many of the shaky concepts physicists use, but this one strikes me ass odd. I underrstans that you can always define a movile frenet frame, but the actual velocity of the curve expresses in this frame doesn't seem to get that form readily at some point. I suppose it has to do with existence and uniqueness?

>>9768619
Do you have another copy of the photo :(

>>9768935
Thanks for the info.
What I said about not wanting to learn algebra definitely was retarded.
Really what I meant was that I don't want to study algebra purely as an end, but rather as a means to understanding its application to other areas of math. In fact, that is what excites me the most about algebra, its appearances in topology, geometry, smooth manifolds (lie groups), and physics stuff.

redpill me on the distinctions between machine learning, deep learning, and neutral networks

>>9770340
there are none

>>9770340
>>>/g/

What text did you dudes learn Galois Theory from?

>>9770456
>What text did you dudes learn Galois Theory from?
I'm not a "dude", but pic related.

>>9770458
Dude is a unisex term. No need to make a fuss to draw attention to yourself.

>>9770340
Machine learning is the general field.
Neural networks are a specific technique in machine learning (basically a composition of linear functions and non-linear activation functions).
Deep learning usually refers to neural networks with a lot of layers (many functions composed). The current consensus in ML seems to be that deep learning gives better results than wide (this just means the linear functions have higher dimensions) networks which take the same amount of computation.

>>9762523
P vs NP is a question in mathematics dumbass, Android enthusiasts are not the people to ask

>>9770461
He didn't mean he was a girl, he meant "dude" is too cool for him, he's just a loser

https://github.com/leanprover/mathlib/commit/d62bf5605ec8971d446a01af40abf9183447cb42#diff-6650f7dae83be3a52c8eb036a23d7b26R175
Proof of Rice's theorem and as a corollary the halting problem in Lean (not mine)

>>9770502
finding asymptotic running time of polynomial algorithms is not maths
this is one instance where >>>/g/ actually needs to apply here

>>9769734
Search the warosu archive.

>>9765153
Well first of all, you can meditate what you have been taught about Fourier series and try to understand wtf was going on, how it could be proved, if this could be extended to a larger class of equations, work out examples etc. Basically be curious and think mathematically.
Second of all, no one book can teach what PDEs are "all about", it is not a unified field. But you can read Arnold's "Lectures on Partial Differential Equations". PDE has deep connections to differential geometry and complex geometry
Many problems of optimization and curvature can be stated as PDE problems (not very surprising since these objects are defined in terms of derivatives of sorts, hence putting constrains on them is basically defining a PDE system).
For further reading, you can investigate minimal surfaces, isoperimetric problems, hodge theory, optimal transport, geodesic flow

>>9768959
>>9768985
>I fail to see how the level of supposed "abstraction" is a legitimate reason.
Because taking a purely top-down approach (ie. starting from the most abstract to the most familiar) is a terrible way to teach.
It is probably a good way to write a book (or at least a treaty), but people do not learn linearly.
Any reasonable person can probably remember definitions but unless they have a good idea of what the definitions are supposed to model (ie. a good set of motivating and nontrivial examples), they will have no intuition about them and no idea why anyone would be interested in it (like teaching monoids to grade school students before teaching them to multiply integers).
That's the problem with starting AG with schemes. Plus, in specific case of schemes, there is a whole lot of algebraic machinery (sheaves, cohomology, differentials, divisors, etc.) you need to introduce before you can say anything interesting and geometric.
You could literally spend a semester learning sheaves and cohomology and still be unable to say anything interesting about algebraic curves.

How many of you guys are just hobbyists? That is, not in school or working as a "mathematician" but study pure math as a hobby?

I figured I wouldn't be able to cut it in academia because I suck at networking and can't handle the stress, so I decided to go into software development instead and just study math for fun.

It's such a great hobby. Often I'd rather read a textbook than play videogames/read fiction/watch TV etc.

>>9772371
Id consider it a hobby for myself, even though I am taking classes for interest and hope to take a more quantitative role at my career. Its nice to be able to take it at your own pace, and really spend time on something if its puzzling you instead of rushing to the next thing because its part of a class schedule

I never realized how enjoyable doing proofs just for fun could be. I never loved math in school since it was always computation and memorization heavy, but loved logic based puzzles, and light programming.

why isnt logic and set theory taught in elementary and high schools? wouldnt it make sense to teach the building blocks of math before moving on to the more complex ideas of calculus?

>>9766973
Sounds like you had a very average algebra professor!

>>9772447
>logic and set theory
>building blocks of math

>>9772454
I had a very tall substitute teacher in my 8th grade algebra class, one Mr. Comey. He told us he was 6'8''.

>>9772482
What are the building blocks?

Surely there has to be a more efficient way to study?

>>9772689
it's what they already teach in elementary and high schools

arithmetic for counting and geometry for engineering/physics

>>9772702
>arithmetic
>geometry
But these are isomorphic

>>9772706
nobody asked if they are isomorphic though

>>9772706
equivalent*

>>9772886
To high school students they aren't

why are we even having this autistic discussion

>>9772886
isomorphic in the category axioms, bucko

Guys, do you love this homotopy theory course trailer?
https://www.youtube.com/watch?v=qlh6knaaVYY

>>9773125
Yes, although it's group cohomology.

>>9773128
Do you think we need more things like this to promote math and educate the society?

>>9773125
What the fuck?

>>9773154
What? The trailer. For lection course. About groups and homotopy theory. By Roman Mikhaylov.

>>9773158
It was a reaction to the style of the video. Does it even work? With attracting the right students I mean.

>>9773160
Well, that's what I would be glad to know from you. What do you think? The lections itself is at
https://www.youtube.com/playlist?list=PLIxkLc10wqUZvPfL22tq-Rs6CBmc9pY-t but in Russian only, sorry.

>>9773144
Not really. Academia shouldn't re-brand itself like this, imo. Besides, the reason we are discussing this is that the video is extremely unique, and as such worth more than a collection of multiple similar videos. I'd rather see something tying the historical and modern aspects of modern fields together by using examples the viewers are familiar with or stuff that can be visualized nicely. But this is just my opinion.

>>9773117
>axioms
No such thing.

>>9758853
or you could just call it [math]\cos(\frac{n\pi}{3})[/math]

>>9758948

http://gen.lib.rus.ec/

>>>/adv/19614354
please give me your take lads, I really need an informed opinion.

>>9774300
Why would you expect people in a mathematics thread to know about statistics or other related topics? You will have better luck asking in more appropriate places like >>>/sci/engi/ or >>>/sci/sqt/.

>>9774300
Beware, there's an autist here who likes to deem all topics he dislikes as being "not mathematics"

Is there a discord or anything for this general? Kinda wish I had some fellow autistic mathbuddies to talk with

>>9774371
Yes there are a few. Most of them suck.

I wish I didn't major in this stupid bullshit.

>>9774371
discord is just the backup for the homosexual anime posters to rp in when they get banned from here

>>9774371
>discord
>>>/v/

>>9774371
>discord

you must be over 18 to post here

>>9762482
n = 1

>>9774759
or P = 0

So I'm taking an abstract linear algebra course and suddenly so much shit in loads of subjects make sense.

Are there any other mathematical topics that that can be easily applied to a wide array of subjects?

>>9774955
>abstract linear algebra course
>abstract
As opposed to?

>>9774955
>Are there any other mathematical topics that that can be easily applied to a wide array of subjects?
Set theory

>>9774985
He clearly said "mathematical topics".

>>9774986
>He clearly said "mathematical topics".
I'm not a "he".

>>9760824
B doesn't have to be countable, it can be uncountable as well, consider for example Cantor's set C, let I be an unit interval, then m(I)=m(I-C)=1, m(C)=0 and card(C)=card(R)

>>9761796
Why? Americans read only American textbooks because they don't know any foreign language, and Russian texts are often not translated or translated to European languages like German or Polish, and even if they have English translations then they're less popular than American books

>>9774970
I dunno, ask my uni/prof. I'm guessing that it's because the course is rigorously proof-based as opposed to a STEM-tier linear algebra course.

>>9774985
Yeah, that actually sounds like a good idea. Gracias.

>>9775062
don't listen to xim, set theory is useless beyond the axiom of choice and its equivalents, apart from some exotic results that you probably won't be seeing very soon

>>9770505
I stumbled upon LEAN about 3 years ago. Now I see there is a math library.
How big is it and how many people are working on it? What's the formal logic that Lean uses anyway? A second order logic?

If the any intersection of k-sized subset made up of sets from a size n set of sets is equal to some number i, does that mean any intersection of k+1 or more sized subsets is also equal to i?

>>9775203

>>9775206
let me rephrase in simpler terms

Consider the n elements of a set represented by a simple graph with n vertices, and subsets represented by cliques, and any k cliques all only share a common clique on i vertices.

My question: Does any k + 1 or more cliques inherit this property? Furthermore does the graph have to be a pseudo-windmill graph where the universal vertex is replaced by a complete graph?

>>9772689
homotopy types

Is Mochizuki a Chad? Looks decently handsome as far as mathematicians go. But he isn't married!

>>9775881
>Looks decently handsome
Asians are not handsome.
>Is Mochizuki a Chad?
Nope. He's a yellow subhuman with a minuscule dick.

Fuck Mathematica I am so fucking tired of these fucking problems

>>9775887
Back to /pol/ dude

what do u math guys think about chemistry?

>>9775907
I don't go to /pol/.

>>9775848
>homotopy types
But those aren't even mathematical.

>>9760880
that's not a branch of math. That's applied math at most.

>>9775985
>that's not a branch of math.
Correct, because math is a branch of computer science.

>>9775911
We don't

How many years away are you guys from understanding IUT?
I doubt that I will never understand it in my lifetime.

>>9776028
Even if I were an arithmetic geometer I think there would be better uses of my time than reading several thousands of pages of terribly formatted papers that may or may not be totally useless

>>9775188
There's about 2-3 people working on the language and probably about 10 big contributors to mathlib.
There's kind of a schism in that Leo, the creator of Lean, doesn't want people to be working on maths stuff yet and isn't really willing to work with the mathlib people at all.

Lean is based on a type theory system using the types-as-propositions correspondence.
There is a pretty big emphasis on constructive maths (ie avoiding LEM and AoC).

I feel like I'm ready for my Calc II final tomorrow, but how can I feel over-prepared for it? I need a B at least.

>>9776972
If you can't get an A on a fucking first year brainlet course then you have no business being in this general

Abstract Algebra 2 exam tomorrow
Red pill me on Galois theory

>>9775203
>>9775250
awaiting response

>>9777374
>Red pill me on Galois theory
see >>9770458

>>9777402
Learn to write, fucking frog poster.

>>9777613
learn to read you caveman

>>9757795
I have a question, I'd appreciate it if anyone could answer it or point me in a useful direction:
I'm looking at the space of all positive sequences of real numbers who's sum at infinity converges to a finite number.
On this space I define the inner product - the product of two sequences, is the sum of the sequence created by multiplying the two sequences together, term by term.
I also define the linear functional of a sequence as the sum of its corresponding series.
I'm trying to prove that this is an example of a linear functional that can't be represented as an inner product with some specific vector from this space (I.E, there is no series 'a' in our space such that for any series 'b' in our space, the functional can be represented as <b, a>.

Any help would be appreciated.

>>9777830
Sequences of positive reals do not form a vector space

>>9777374
M/L/K extensions with M/K Galois. Then some element of M is in L iff it is fixed by Gal(M/L). Ther

>>9777830
You are pretty confused. First, the sequence of postive real numbers whose sum converges isnt a vector space. I thitnk you are talking about the space of sequences whos absolute value converges. This space is called l1. You cannot define an inner product here, thats an important theorem. The space of square sumable series l2 is the only hilbert space, i.e., the only space that can be endowed with an inner product.

>>9777951
>>9777981
sorry guys, I'm completely new to this world of content. so could you give me an example of an infinite dimensional inner product space and a functional with the property I described? I'm trying to wrap my head around how something like this can even be true and can't come up with an example.
thanks anyway.

>>9777987
Who told you it was true in the first place?
Other than the trivial examples where the vector space doesn't even HAVE an inner product, every functional is an inner product with respect to some vector. It's called the Riesz representation theorem.

>>9777987
What are you looking for exactly ? An inner product space and a continuous functional that cannot be represented as x -> <a,x> for some vector a, is that it ?
If your space is complete, then it cannot be done, that is Riesz' theorem.
For non complete spaces, it's not so complicated. Take for example the space C([0,1]) of real-valued continuous functions on [0,1] with as inner product [math](f|g) = \int_0^1 f(t)g(t)dt[/math].
Now, set the functional: [math]\phi(f) = \int_0^{1/2} f(t)dt[/math]. Then, it's not very hard to see that this is continuous but cannot be represented as a dot product by a continuous function (otherwise, that function would have to be 1 on [0,1/2) and 0 on (1/2,1], contradicting the continuity)

>>9777999
Thank you for the example! It helped a lot in seeing where the 'problem' might arise in these sorts of examples. Do you know where I could read up about some more examples like this? I couldn't find anything on this specific topic

>>9777999
No, that is a complete space and your integral is bounded.

>>9778142
Are you really telling me that C([0,1]) is complete wrt the inner product norm ?
Absolutely not, take the sequence of functions f_n(x) = 0 for x in [0, 1/2-1/n], (n/2)* [x - (1/2 - 1/n)] for x in [1/2-1/n, 1/2+1/n], 1 for x in [1/2+1/n, 1].
This is a sequence of continuous functions which is Cauchy with respect to the inner product norm (check it) but does not converge in *C([0,1])*.
It does converge in the completion, ie. L^2([0,1]), to the function which is 0 on [0,1/2] and 1 on. (1/2,1] but this is *not* almost everywhere equal to a continuous function

>>9778043
Well I'm not sure that there is much more to say about this particular question. In fact, the example I gave above was essentially the only obstruction that can arise.
Basically, given an inner product space V, you can always complete V into a Hilbert space H. Then, given a continuous functional f on V, there is a unique continuous functional g on H that extends f (there is this theorem that states that a uniformly continuous map on a dense subset of a metric space extends uniquely to a uniformly continuous map on its closure, and continuous linear maps are lipschitz, hence uniformly continuous) and you can apply the Riesz representation theorem to g.
This tells you that all functionals on an inner product space can be written as x -> <a,x>, but that a may only belong to the closure.
If you think about it for a sec, it also tells you that functionals of the form x -> <a,x> with a in V are dense in the dual, so it's still pretty neat.

>>9778142
Are you really telling me that C([0,1]) is complete wrt the inner product norm ?
Absolutely not, consider the following sequence of functions:
[eqn]f_n: x \mapsto \left\{\begin{array}{c c c} 0 & \text{ if } & 0 \le x < \frac{1}{2} - \frac{1}{n}\\ \frac{n}{2} \left(x - \left(\frac{1}{2} - \frac{1}{n}\right) \right) & \text{ if } & \frac{1}{2} - \frac{1}{n} \le x < \frac{1}{2} + \frac{1}{n}\\ 1 & \text{ if } & \frac{1}{2} + \frac{1}{n} \le x \le 1\end{array}\right.[/eqn]
This is a sequence of continuous functions which is Cauchy with respect to the inner product norm (check it) but does not converge in *C([0,1])*.
It does converge in the completion, ie. L^2([0,1]), to the function which is 0 on [0,1/2] and 1 on. (1/2,1] but this is *not* almost everywhere equal to a continuous function

Let

dead general desu

redpill me on the use of schemes in astrophysics

>>9778604
They're for name dropping Grothendieck and related buzzwords to get more funding and textbook sales.

>>9778604
>astrophysics
Refer to >>>/toy/.

I work at a pharmacy so shit gets boring pretty fast. so I try to come up with expressions of directions. example:

take 2 pills 3 times a day for one day
take 2 pills 2 times a day for one day
take 2 pills 1 time for one day
take 1 pill 3 times for one day
take 1 pill 2 times for one day
take 1 pill 1 time for one day

wouldnt this be a really simple summation from 1-2 where the expression was 3! or am I overthinking it

official /mg/ draw site
https://www.groupboard.com/gb/744618

>>9778999
Stop bullying Grothendieck. He was one of the greats. You're not even worthy to lick his piss.

>>9778335
Fugg I thought you meant an integrak operator on that space but with the supremum norm. Yea you are right.

>>9779291
>>>/b/
>>>/lgbt/

>>9779291
Is this the Finn?

>>9775203
If an extraterrestrial intelligence tried to parse what you just said they would send 65536 probes to survey the planet in preparation for wiping humanity out.