/sci/ - Science & Math

>>8515730

What are Morse functions again? I remember hearing about them years ago in a lecture on the proof of the Gauss-Bonnet theorem.

>>8515730
>mathematics general on a math board
fuck off with your thinly veiled blog thread

>>8515825
Morse functions are functions which have only non-degenerate critical points. We use them to study the topology of smooth manifolds using Morse theory.

>>8515831
the other 'math threads' on here are like borderline grade school material

>>8515844
the complex analog is nice too
https://en.wikipedia.org/wiki/Picard%E2%80%93Lefschetz_theory

>>8515847
Indeed it is! I was going to work through a book on Morse homology and then read one on Picard-Lefschetz theory! Know any good books for the latter?

>>8515858
I learned from this one

help

>>8515947
Thanks! I will check it out then. Is Volume 1 also a decent read?

>>8515979
Yep, just about anything with Arnold's name on it is worth a read

>>8515992
Perhaps I will read them both then, thanks for the suggestion!

>>8515963
this is gibberish

>>8516007
nope
http://www.msri.org/people/members/defthy07/exercises/exercisesday4.pdf

anyone know math problems with cash rewards (or something else) other than the millennium prize problems?

>>8516320
i've also found the Erdos problems (https://en.wikipedia.org/wiki/Paul_Erd%C5%91s#Erd.C5.91s.27_problems)) and the Hutter prize (https://en.wikipedia.org/wiki/Hutter_Prize))

Working on translating motivic integration into something school children can understand.

>>8515730

1. Working on a few rote problems in communication complexity.

2. Working to be able to use standard cryptographic primitives as a form of subliminal channel.

3. Developing a practical compression scheme that doesn't rely on stochastic assumptions.

It's not strictly mathematics (I'm a graduate student in computer science), but I'm genuinely enjoying my work right now.

>>8515730
Well my root-finding numerical code for a system of gap equations isn't converging, and there's literally nothing that can be extrapolated from the analytic expressions since they're in a form as open as my asshole after a night out.

On the other hand the paper relating the holonomy formalism and the principal bundle formalism of gauge theory turns out to be not what I had anticipated. The author defines the generalized holonomy map [math]H: L_x \rightarrow G[/math] as a homomorphism on path-equivalent curves [math]L_x \ni \gamma: [0,1]\rightarrow M[/math] based at [math]x[/math] to a Lie group [math]G[/math], which makes this rather distinct from the braid group approach. I don't think the connection between the braid group and Chern-Simons theory approach of particle statistics can be established just by this paper. There will still be work needed to be done in order to relate what the generalized holonomy map describes to what the braid group describes.
I never had high hopes for this project desu. If a world-renowned physicist tells you that your idea is a dead-end, he might be right.

>>8515730
I heard Naive Lie Theory is good... very visual using only matrices to build intuition for the big theorems. What are you using?

I'm searching for nice applications of nonstandard analysis.

>>8517065
Pissing of Halmos et al is a great application.

>>8517105
The Bernstein-Robinson theorem is of course on the top of my list.

Why are L^p spaces interesting to consider beyond L^1, L^2 and L^inf ? I mean I guess it's natural to ask about their properties since L^1, L^2 and L^inf are nicely behaved and I know that they're interesting from a functional analytic standpoint (Banach spaces, reflexive, blabla) but I have never concretely used them for anything.
Are there any problems that naturally call for L^p regularity for p other than 1,2, inf ?

>>8515730
Reading a book about Analysis and another about Proofs in parallel. Too bad there are no problems, just exercises.

I have a somehow dumb question about the theorem of Bolzano-Weierstrass.

It says that if, for any n, we have [math]a<u_n<b[/math], then there is at least one sub-sequence that converges towards a limit [math]l[/math].

My question is, if there's no obligation for this number of sub-sequences, is it possible to have an infinite number of these ?

>>8517819
Of course. If your sequence (u_n) is convergent, then each of its subsequences converges to l

>>8517822
Now, if you were asking if the number of different limits can be infinite, the answer is still affirmative, as you can see by taking [math](u_n)[/math] to be an enumeration of the rationals between a and b.
This sequence will have, associated to each number x in [math][a,b][/math], a subsequence converging to x.

>>8517822
The point is that this sequence might not be convergent

[math]u_n = (-1)^n[/math] follows the theorem : we have [math]-1.5 < u_n < 1.5[/math], and there are two l, [math]l_1, l_2[/math], that verify :

For any [math]\epsilon > 0[/math], I can find a [math]p[/math] so that there is at least one [math]n \geq p[/math] that verify [math]|u_n - l| < \epsilon[/math].

For both [math]l_1[/math] and [math]l_2[/math]

>>8517837
Ok, thanks. That was what I was asking.

>>8517788
In general [math]L^p[/math] spaces for [math]p\neq 2[/math] are uninteresting as they don't have a natural inner product structure on them.
However the Sobolev space [math]H^1 = S_{1}^{2}[/math] are useful for analyzing the stability of the solutions of equations of motion in quantum and classical field theories. The solutions [math](\psi,\psi_0)[/math] are required to lie in [math]H^1 \oplus L^2[/math] or the Cauchy problem given by the equations of motion wouldn't be well-posed and the existence wouldn't be guaranteed for arbitrary times (Strocchi 2008). This gives rise to the Swieca regularity conditions that the quantum fields need to satisfy so that your QFT is defined, which leads to the famous Haag's theorem.

>>8517788
I think Sobolev's embedding theorems are an example.
PDEs are not my thing, but some well-posedness theorems might require different regularity for the initial data to be applied.

>>8515730
trying to pass intro to analysis 1
thought i was good at calc
turns out im not lol

>>8517861
>>8517863
Thanks guys, I had forgotten about Sobolev spaces but actually it's a very good point. L^p spaces aren't really all that useful by themselves but Sobolev spaces have very strong properties

>>8517861
I don't think that's a good example, since H^1 is THE sobolev space and doesn't really have to do with any L^p other than p=2.
Does quantum mechanics ever deal with any other vectorspaces than hilbert spaces?
A more general space [math] W^{1,p} [/math] (once weakly differentiable and the weak derivative is in L^p)
is needed for example for PDEs, where some of the terms don't have l^2 regularity

>>8517861
Is it necessary to be very rigorous on proofs before reading Algebra on your own or is a vague training as EE enough ?

>>8518153
>Does quantum mechanics ever deal with any other vectorspaces than hilbert spaces?
Nope. The principle of quantization a la Dirac will naturally give you a Hermitian line bundle [math]B \rightarrow M[/math] over the classical symplectic phase space, the sections of which are square integrable complex-valued functions.

>>8518153
>Does quantum mechanics ever deal with any other vectorspaces than hilbert spaces?

Not as the "space of states".

I'm studying the properties of the Haar integral at the moment. This is pretty cool, to be honest.

If you take a bunch of lines and you connect their midpoints together, you get a new set of lines with more than you started with. Turns out the whole process is just really weird vector addition and the curve you end up with after a shitload of cycles is just a really weird derivative of the set of lines you started with, if you treat them like vectors.

Pic related

>>8518518
Very cool but fuck you for distracting me from studying for finals. Now I have to try and program this.

>>8516320
I'll give you $10 to prove I'm gay

>>8518523
Easy.

Try to post a new thread on 4chan. If it goes through, you are now OP and therefore a faggot.

QED

>>8518529

What's your PayPal?

>>8518534
j701758@mvrht.com

>>8518522
heh, had the same idea and did it in 3d
only problem is creating a big enough data set of starting points without it being too "regular"

>>8518553
Damn I don't think Mathematica's Manipulate function does 3D locators.

>>8518553

Fuck man you stole my idea, haha. I was going to use a 3d rendering tool I wrote to do some 3d stuff. Cool pic anyway though, if anything it's basically a spoiler.

>>8518630

I hope you realize that 3d plotting is a basic out-of-the-box functionality provided by both Mathematica and Matlab.

Not to discourage you or anything--small projects like these are good practice--but you don't need to reinvent the wheel unless you really need to.

>>8518648

Nah i was just going to plug in my program to the tool. Reinventing the wheel is a hobby of mine and it's pretty fun if you appreciate really digging into how the 'basic functionality' of many programs and tools actually works and is implemented.

Also it seems that if you calculate this continuous midpoint transformation on an open polygon with height 2A and width A, you get a cycloid.

Randomly came across this video. Thought this guy was really clear and concise.

https://www.youtube.com/watch?v=sMqI1esWpEs

>>8515730

I'm an undergraduate self studying out of this bad boy. None of my professors or peers do logic, so I'm completely alone.

Skipping around a bit, but I just did the proof that if a is a measurable cardinal, then a is the a'th inaccessible cardinal.

Why don't more people love this stuff? The rest of math is utter trash in comparison to logic.

>>8519040
>Self-studying

I'm amazed that some people manage to do this. I never ever managed to get a subject from a book alone, without the least bit of explanation from a teacher.

Fuck being dumb.

>>8519048

MUH DICK

>>8519048

I share your pain. I think it takes a certain level of stubbornness that I just don't have.

>>8519040
hook a nigga up?

>>8515730

Is it normal to not have the slightest fucking idea what you guys are talking about most of the time? I don't bother to look these things up anymore because I at most only ever get a superficial understanding that I forget not long after.

Am I just especially impaired or what?

>>8519147
1. is litteraly jus limits arithematics

>>8519040
>the rest of math is utter trash in comparison to logic.
With opinions like that, now we know why you are completely alone.

>>8517129
I am glad. :)

>>8519181
It's normal to not understand what majority of mathematics we talk about here. Thoguh don't let that stifle your learning in math. You should definitely be eager to learn more, despite the buzz-word-dropping that occurs in threads on /sci/.

>>8519040
>The rest of math is utter trash in comparison to logic.
Model theory is the subject I want to study the most but don't have the time. But having bad attitudes might be related to why mathematicians don't get along with logicians.

>>8519181
>Is it normal to not have the slightest fucking idea what you guys are talking about most of the time?
Yes, especially when they're using categorial language.

>>8519040
Reading about formal logic right now, loving it. Though it might not be as technical, it has some interesting historical backgrounds and quotes.

>>8519181
Most people here don't know shit about shit, don't worry about it.

>>8515730
Studying my ass off in algebra. I'm taking graduate algebra as an undergraduate and while it's hard, the thing that's killing me is just the volume of stuff I have to know.

>>8519181
You can't even begin to comprehend why you don't understand shit about mathematics until you've studied abstract algebra and analysis.

>>8518559
Aren't those just Bezier-Curves with factor 1/2?

HELP ME DO STOCHASTIC PROCESSES ITT:

>>8519407

>>8519897
I thought Bezier curves were supposed to match the derivatives and the end points.

>>8519147
These are all pretty easy. Problem 3 follows from problem 2, problem 4 is just testing definitions and 5 is just conformal transofmations.
>due on April

First time on this board. You guys are a bunch of fucking nerds, I need help. What are good books to study Algebra 2 and Trigonometry.

>>8520006
Can you give rough outline of those courses?

>>8519927

Only the original guy's curve was technically correct. The others didn't continue the transformation at the end points.

>>8520584
But his doesn't have derivatives matching the endpoints either.

how do i make myself like other areas of mathematics
i really enjoy algebra, favorite is probably ring theory/commutative algebra
but i absolutely fucking hate analysis and analysis-related things
it's not that i'm bad at it, in fact, i'm quite good with most analysis and topology stuff
but i still hate it

can i train or trick myself to like analysis?

>>8521828
Not that good of a mathematician, but I really like Analysis for some reason.
I find it really intuitive, it usually combines a lot of maths domains and helpful "tricks" (like how the n-sqrt(x) is actually x^1/n, trigonometry, Taylor series,...)

Find a textbook with funny functions/series to study. In one of my tests in Analysis, we had to study the nature of series of general term 1/n, n being any integer not containing 9. I genuinely enjoyed that test.

>>8521861

>analysis
>intuitive

analysis is like 90% counterexamples to things that you previously thought were intuitive

>>8521940
This

The continuum was a mistake

>>8521940
I mean that even if you don't have the rigorous proofs, you can still deduce things with accuracy.

Like if I know a sequence/series converges for n -> +infinity, it will converges for 2n -> +infinity.

>>8521828
You just think you are good at analysis. You are not actually.

>>8521828
>i'm quite good with most analysis and topology stuff
>but i still hate it
You make topology cry. Why don't you like it? Maybe you would like algebraic topology.

>About to get my ass kicked by Functional Analysis because I can't remember basic definitions

How do you guys do it? I'm finding it harder and harder to focus these days

>>8516434
I haven't studied motives much. Is it just a port of fibre integration into some category of motives?

currently writing a paper on orthogonal art galleries
>the whole art gallery problem thing

>>8522222
Checked. I rate your repeating digits 3/10 because your post was boring. Better luck next time!

Writing an introductory section of my thesis on quantum walks. Currently cherrypicking terminology from Markov chains and quantum probability to give background for my arguments.

This is the kind of stuff I'm building up to:
https://www.youtube.com/watch?v=zzASv4G9bNA

>>8521940
That's when they teach it to you, to teach you caution. But after that you can go back to working with your intuition, keeping in mind the fact that that intuition can deceive you.

>>8521828
Be curious and try to think of real-life phenomena that you can model with math or try to find proofs of interesting results that use analytic tools (if you like number theory or geometry, it shouldn't be hard to find interesting results that require analysis).
Also, as a general rule, don't say you "don't like" areas of mathematics until you're at least in grad school. You are going to need everything you are currently learning, and then some, so you can't afford to be picky about what you learn at the moment.

>>8521828
>>8521828
Be curious and try to think of real-life phenomena that you can model with math or try to find proofs of interesting results that use analytic tools (if you like number theory or geometry, it shouldn't be hard to find interesting results that require analysis).
Also, as a general rule, don't say you "don't like" areas of mathematics until you're at least in grad school. You are barely scraping the surface of each topic and you are going to need everything you are currently learning, and then some, so you can't afford to be picky about what you learn at the moment. It's like a grade schooler saying they like to add but not multiply.

Which of the basic calculus texts cover Big O and Little O notation? Last week we coverd this in a lecture about infinitesimally small limits, but standart texts like Spivak don't cover it.

What is the hardest part of Calc II?
How can I make it easier?

>>8520584
Oh I see what was going on. You have to include the endpoints in the new list.

>>8522297
I thought it was already solved for orthogonal galleries. Did I mistake it for a more concrete case or what?

>>8522598
Also I tried doing something other than the midpoint (replacing "Mean" with "((1 - t) #1 + t #2) & @@ # &") and this is what it looks like as t goes from 0.01 to 0.95.

Hosted on pomf because slightly too big for 4chan:
https://a.pomf.cat/jaxtzr.gif

There's a weird optical illusion that makes it look like the points jump at the end when they loop around but I've compared the two frames and they're in the same locations.

>>8522548
Really not that much to cover about it, once you know the definitions.

>>8522658
The points along the curve seem to follow the same trajectory when adjusting the interpolation distance as they would when adjusting how many iterations are done.

Compare:
https://a.pomf.cat/xmtveg.gif
https://a.pomf.cat/qyzslx.gif

Okay I'm done.

>>8521828
>favorite is probably ring theory/commutative algebra
>but i absolutely fucking hate analysis and analysis-related things


Try combining them.

https://anagrams-seminar.github.io/hdr/kahler.pdf

http://www.ihes.fr/~celliott/D_modules.pdf

http://www.math.harvard.edu/~gaitsgde/grad_2009/

etc.

I've been studying for my diff eq final. Series solutions to ordinary points and regular points are fucking annoying. It's not a hard concept, but I can't find a source that explains it clearly.

>>8521828
This >>8522153. Most algebraists think they're hot shit for proving equalities but they have no idea how complicated the objects analysts deal with are.

>>8522655
From my research, a maximum boundary is proven but the minimum question still remains.
I'm a 2nd year undergrad so it's really just a research and recite kind of thing

>>8522692

It looks like deviating the 'interpolation' from .5 just reduces the effectiveness of the continuous transformation.

>>8522692
>>8522912

Also I did the transformation on a Hilbert curve.

>>8523305
Wow freaky

>>8523305
Nothing particularly interesting happens to the 7th iteration of the Sierpinski arrowhead after 300 iterations.

Same for 200th 11-dragon.

>>8523840

What tool are you using to do this? Your the man with these great images.

>>8523919
Mathematica.

Here's a PDF and notebook file with all the crap I've done:
https://a.pomf.cat/cvsupn.pdf
https://a.pomf.cat/wdcxnp.nb

Probably not too surprising but you can approximate this whole process pretty nicely with a low-pass filter.

Pic related is the 300th 3-peano overlaid onto the first 40 coefficients of the DST of its (adjusted to make it periodic) data on the left, and on the right is the 300th 7-arrow overlaid onto the first 80 DST coefficients.

Here's a PDF with the all of the computations
https://a.pomf.cat/aafklx.pdf

What went wrong /sci/?

>>8526174
He failed to reach enlightenment even after going wizard mode.

B^)

https://arxiv.org/abs/1612.02664
> A Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers

>>8526273
>Jordan
>physics
>gen.math
>mfw

>>8526174
dat resemblance tho

>>8526273
>>8526293

What is this? Is this tip-top Arabian bait or what? How did he upload this to arXiv?

What am I missing?

I'm trying to solve the problem z^2+z(12+10)i-(37-40i)=0. z is complex so I want to write z^2 as (a+bi)^2 and put everything else on the right side. I complete the square and expand some but I never get closer than (z+6+5i)^2=81+200i which I don't know what to do with. Any hints to what I'm supposed to do?

>>8527015
is there a reason you're not using the quadratic formula?

>>8526983
Just a regular crackpot

>>8527018
Eh, it's shit and I don't like it? I guess.

>>8526273
>https://arxiv.org/abs/1612.02664
>> A Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers
The correct place for that is viXra, not arxiv.

>>8527015
Complete the sqaure brainlet

>>8527015
>(z+6+5i)^2=81+200i
You have it there, compute the two solutions of the square root and you have it.

>>8526983
Anyone can upload to arxiv, it's not peer-reviewed. However if you're clearly a crackpot then your shit will be moved to gen.math, which no one takes seriously.

>>8515963
Just pop it into the calculator m8

>tfw too intelligent to do maths

How is everyone studying for exams?

>>8529428
>He studies for his exams

Brainlets I swear.

>>8529428
Could be better, could be worse. I have trouble finding motivation, but I understand my stuff.

>>8529450
I would agree with this.

>>8529450

The problem I have with motivation is that I'm usually happy with an overall understanding of things, and since I always take notes and pay attention, I always have what I want by default, so then studying becomes a bit of a chore, but I can't rely just on memory because I know how that usually does for my marks, and since I don't know what to do with my life, I want to maximize my chances for a nice Masters program.

I guess I just don't have any ambition in life.

>>8515963
Your first mistake was studying category theory.

>>8529488
How do people even get ambition?

>>8529503

I think believing you can make it in the first place is a big (and probably the first) requirement. Then there's perseverence, patience, a good work ethic, having somebody to fall back on when needed, knowing how to plan, adapt and learn, a sense of urgency (it has to be motivating, not debilitating), being honest with oneself, etc...

And then you just have to not be a brainlet, which is a requirement people like Tao like to forget when they give advice on their blogs, because then they'd feel like undeserving assholes.

"No guys! You can totally be as smart as I am even though I'm a gifted kid who lived in the perfect environment to turn into an immense genius! You just have to bee urself and like maths a lot!"

>>8529562
>believing you can make it in the first place
This is probably the hardest part.
>perseverence, patience, a good work ethic
I guess I have these.
>having somebody to fall back on when needed
What do you mean by this?
>knowing how to plan, adapt and learn, a sense of urgency (it has to be motivating, not debilitating)
Check, I guess.
>being honest with oneself
And what do you mean by this?

>>8529585
>>having somebody to fall back on when needed
>What do you mean by this?

Life's tough, you sometimes need friends, family and/or some kind of tutor/mentor to talk/vent/ask for advice on career/life. You know, the usual thing.

>>being honest with oneself
>And what do you mean by this?

You can't hope to make anything of your life if you keep on living a lie, can you?

Curious to know what you think of the rest of my post.

>>8515963
You did not post the whole question.

>>8529503
The most common ambition I see in the STEM community is to prove themselves to their peers.

Probably has something to do with childhood bullying and/or inadequacy.

>>8516050
>http://www.msri.org/people/members/defthy07/exercises/exercisesday4.pdf

The issue is that H is being defined implicitly.

The exercises are not solvable without that information.

>>8516326
>>8516320
Those prizes never seem to pay out.
The organizations appear criminal in their lack of transparency. There is too little oversight to prevent someone on the review committee from plagiarism.

>>8516434
I would try to explain it in terms of how many toppings can fit on a doughnut.

>>8529596
Yeah, that's what I was pretty certain you meant.

About the rest of your post, I'm not quite sure what Tao writes in his blog, but I agree it is not a good thing to tell anyone they can solve some difficult things with only hard work. It just doesn't work like that for most people. I'd call it a waste of potential, too, as there are smaller things to look at, and these people wasting their years trying to prove the Riemann hypothesis or some similar statement could look at those instead.

>>8529619
So, you are saying STEM ambition is peer pressure of some kind? Or to make dad proud, that could be one reason, too. That's an interesting point of view.

>>8529645
I think it's potentially a result of receiving a lack of attention during developmental years.

Look at Newton for example.

>>8529684
Yes, that is quite possible.

Have a real analysis and PDE exam coming up on tuesday and wednesday and I just can't concentrate on studying. I really have to force myself.

>math general
>99% of the posts are undergrads complaining about finals and blogging

>>8522222

Cue cards for EVERY example/problem type and all definitions and little tricks/nuances necessary for understanding.

How do I git gud at Optimization? I absolutely fucking loathe it and the lecturer is boring as all hell, but I'd really like to pass this class with a good grade. The topics we cover are Linear Programming, Integer Linear Programming and a bit of graph optimization. Any tips on resources to learn this?

>>8529723
That's fine. I don't mind.

>>8529715

Leave now, unplug the internet, take breaks, don't stress, try to have a clear and focused mind, and godspeed.

>>8529723
>WOW this thread is shit!
>I better just whine instead of making it better

Does anyone here have undergrad research experience? I want to know how's the experience.

>>8529859
Great experience, even if you don't do anything particularly special for the good of all mathkind.

Definitely try it!

>>8529890

And how does one *make* it? Do I just go to some prof and be like "Yo whaddup, how's it goin'?". I don't have much to offer and the initial knowledge requirement seems huge. Give me specifics anon!

>>8529902
Most schools have a research program that you can do it through. If not, just go to a prof you either enjoy (his teaching or research) or one you think would be a good supervisor, and just ask if they know of any possible way you could do a small research project under them.

>>8529636

>>8529757
same

I was an undergrad once. Except the thai cartoons kinda bothers me sometimes.

>tfw got a 82 on my DE test because I didn't go over the linearly independent definition
>used Wronskian
>0/10
REEEEEEEEEEE

How are you folks today?

How do mathematicians work today? let's say they have to prove something (in order to prove another result). do they just look it up and use a lemma they found in another paper without understanding its proof or context?

>>8532026
Sometimes results will be used without really having a deep understanding of the proof (if the proof is long and difficult you won't always want or be able to waste the time learning it by heart) but it's very dangerous to start applying things when you don't understand the context they came from.

Using things with a shaky understanding of which situations they're actually meant to work in is just begging for a huge fuckup

>>8532168
cf. all of calculus done shortly after Newton & Leibniz.

>>8515730
All done with finals. Took applied stats, linear algebra, a course covering the 2nd actuarial exam, and a number theory course.

>>8515730
Procrastinating from studying for Calc 2 final as we speak.

Yet preemptively girding my loins for Calc 3 next term. Is it as hard as they say, /sci/?

Thinking about going for an engineering masters degree. Doing some study myself at the moment with calculus and while I never did take calculus I remember a lot of the stuff I'm seeing in the beginners stuff from high school, just using different terms.

And the terms themselves are really the things that are giving me issues. I guess I'll get used to them eventually though.

>>8532382
who told you calc 3 was hard?
If you know what a vector in R^n is and you know calc I-II there's virtually nothing really new in calc 3

>>8532427

>who told you calc 3 was hard?

I think his name was WahIHateSeries McDuff

>>8532434
Calc 3 is pretty easy compared to the shit in calc 2.

>>8515730
What's going on in Theorem 2.5? Is the purpose of [math]\theta_I[/math] essentially just to check if the ordering of the input vectors matches a certain ordering prescribed by [math]I[/math]? Does it basically check whether or not [math]I = J[/math]? He calls the [math]\theta_I[/math]s elementary k-tensors. What's so significant about them?

>>8532455
Basically, it tells you that given a basis for the vector space V, you can construct a convenient basis of the space of k-tensors on V, which he does in the proof.
It's the same idea as the proof of the fact [math]\mathcal L(V,W) \simeq M_{n,p}(K)[/math] if [math]\dim V = p, \dim W = n[/math].

>>8532455
>I looked at tensor calculus without taking a proof-based lin alg course
Please end yourself.

>>8518523

You're posting on 4chan /sci/. Hence, you are a faggot.

>>8519129

It's funny, when people tell me "oh u must be smart if you major in math" my replying is always "I'm not that smart. JUst very stubborn".

>>8520006

Algebra 2 for Dummies

Trigonometry for Dummies

Not even memeing, that's how I learned both of those classes

>>8533263
agreed

>>8529745
Someone helps me

What are the best books to learn about Optimization?

>>8533263
This. This is the way to go

Does anyone have a table with all known (co)homology and/or homotopy groups for all the standard examples of manifolds?

For example, if I wanted to know the 4th singular cohomology group of SO(n) with coefficients in the rationals, I don't want to compute that shit tediously by hand.

>>8534198
thats kind of sad

>>8535200
agreed, imagine how much of a brainlet you'd have to be to not be able to compute any of those homology or homotopy groups on your own

>>8534198
probably not

there's no definitive list of "standard manifolds"

also if i remember correctly, the cohomology of SO(n) is computed in one of the appendices to hatcher's book

How do I prove that a^x is inversable/bijective and therefore that log exists?

>>8536213
Well, first of all notice that [math] a^x [/math] is strictly increasing on its domain, so therefore an injection. It's also a surjection on [math] (0,\infty) [/math].

>>8536228
From a human perspective these things are quite obvious. I understand it intuitively but have a hard time proving it mathematically.

>>8536233
Suppose a>1 and y>x. Then y=x+r for some r>0, and a^y=a^(x+r)=(a^x)(a^r). Since a>1 and r>0, a^r>1, so (a^x)(a^r)>a^x, that is, a^y>a^x. If 0<a<1, then a=b^(-1) for some b>1, and now, for y>x, a^y=b^(-y)<b^(-x)=a^x, by what was shown for a>1. Thus, a^x is strictly monotonous for 0<a with a=/=1, and therefore it is injective. Do the surjectivity by showing there is an x such that a^x>M for any M>0.

>>8536213
a^x is defined as [math] e^{x*ln(a)}[/math]
the exponential function is invertible, and adding ln(a) just stretches the function

>>8536280
>>8536306
Thanks guys!

Hello guys

I have a question about proofs. Which seems the most rigorous/natural to you to prove that, say, A < B

1) Establish that A < B <=> C < D, C < D being a trivial or easy to prove inequality, and then concluding that A < B

or

2) Say that C < D, establish that C < D => A < B and conclude here.

Also, how can I prove that :
[eqn] \int_{n}^{n+1} \frac{1}{t} \, \mathrm{d}t = \int_{0}^{n+1} 1 + ... +\frac{1}{t-1} + \frac{1}{t} \, \mathrm{d}t \; - \; \int_{0}^{n} 1 + ... +\frac{1}{t-1} \, \mathrm{d}t \;[/eqn]

I tried changing variable t' = t-1 in the second integral, but it makes it "start" at -1. I handwaved it saying that log(x) is not defined with x inferior or equal to 0, but that's not rigorous at all.

I have a quick question.
Can you guys tell me what happens here, and why it gives the result it does?:

2^(-1) = 0.5

>>8536532
[math]2^-1[/math] is the same as [math]\frac{1}{2}[/math]

[math]2^-2[/math] would be the same as [math]\frac{1}{2^2}[/math]

>>8536539
Fug, I meant [math]2^{-1}[/math] and [math]2^{-2}[/math]

>>8536539
>>8536543
Alright. Thanks a bunch, man.

>>8536505
2 is, strictly speaking, easier, since to prove that A < B <=> C < D you have to prove that C < D => A < B (that's the <= part).
Generally speaking the fact that A < B => (true inequality) is extraneous because that doesn't say anything about whether or not A<B is true. Although sometimes you might like the stronger result.

I don't understand your integral in the bottom. t is a continuous variable and yet you've set up the sequence 1,2,...t-1, t in the denominators. What are you thinking is in between the dots?

>>8536566
I'm doing the integral of a sum.

Can somebody help me, step by step, to get from:

A = B * C

to

B = C^(-1) * A

Really hope you can help me, haha. Thanks in advance.

>>8536601
You didn't answer my question. I can see the plus signs, I can see it's "a sum".

What is 1...1/(pi-1)+1/pi supposed to be in your integral?

anyone here does self study?
how do I learn multiple subjects simultaneously? the only way I can get myself to focus on a subject is by studying it and nothing else, otherwise I just read a chapter from book A, a chapter from book B (et cetera) and end up absorbing very little

>>8536642
A = B * C. Therefore, B = C^(-1)* A. The proof is obvious and is left as an exercise.

>>8536649
Sorry, I completely written this the wrong way.

It's the sum of 1/k, k going from 1 to t.

And I'm actually very wrong in how I written this. Should be :
[eqn] \log(n) = \int_{0}^{n} \frac{1}{t} \, \mathrm{d}t = \int_{0}^{n} 1\, -1\, + ... +\frac{1}{t-1} - \frac{1}{t-1} +\frac{1}{t} \, \mathrm{d}t \;[/eqn]
[eqn]= \int_{0}^{n} 1\, + ... +\frac{1}{t-1} +\frac{1}{t} \, \mathrm{d}t \; - \int_{0}^{n} 1\, + ... +\frac{1}{t-1} \, \mathrm{d}t \;[/eqn]

Then there's a step that's missing
[eqn]= \int_{0}^{n} 1\, + ... +\frac{1}{t-1} +\frac{1}{t} \, \mathrm{d}t - \int_{0}^{n-1} 1\, + ... +\frac{1}{t-1} +\frac{1}{t} \, \mathrm{d}t [/eqn]

[eqn]= \int_{n-1}^{n} 1\, + ... +\frac{1}{t-1} +\frac{1}{t} \, \mathrm{d}t [/eqn]

>>8536669
The more I read this the more I'm thinking I did a terrible mistake. At this point it's more about telling me where my reasoning is wrong rather than correcting me.

>>8536675
should be starting from 1 instead of 0 m8
[eqn]\int_1^n \frac{1}{t} \, \mathrm{d}t = \left[\log (t)\right]_1^n = \log(n) - \log(1) = 0[/eqn]

>>8536666
>The proof is obvious
Not to me. It's okay if you don't want to tell me the rule that allows this behaviour, but could you point me in the right direction?
I know that it's true, but I don't know the rule that allows it.

>>8536695
> log(n) - log(1) = 0 *
should be [math]\log(n) - 0 = \log(n)[/math] obviously

>>8536699
Thanks, and of course, 1/0 is not defined.

And after that, how can go from line 2 to line 3 ? I will have an integral going from 1 to n, and another from 0 to n-1. I wouldn't be able to subtract them.

>>8536675
The problem in what you're doing is that 1/1,1/2...1/(t-1),1/t doesn't actually exist as as progression unless t is an integer

Since you're integrating t that's usually not true, so the "sum" doesn't really mean anything

I'm not sure exactly what you're trying to accomplish by rewriting log in this way so I can't point you towards what you should be doing.

>>8536711
Then don't substitute t' = t-1 since you know you'll just be producing nonsense. Why not try t+1 instead?

Are you trying to bound the harmonic numbers or something?

>>8536661
I do. My method is simply reading a lot of something for a day or a few, then moving on to some other topic. My mind is then working on what I have read, so that a few days later revising it I will get a lot more out of it than the first time. Then I just repeat this until I have a firm enough grip on the subject. This is assuming I have to study something else at the same time, otherwise it'd just be concentrating on the stuff full time.

>>8536714
The objective is to show that 1 + 1/2 + ... + 1/n - log(n) converges, knowing that :

If f(n) is a positive and decreasing function, then for any n > 1, we have

[math] f(n) - \int_{n-1}^{n} f(t) \, \mathrm{d}t < f(n+1) - f(n) [/math] (proven in a previous exercise).

I tried to show that log(n) can be written as the integral between n-1 and n of the sum.

But now I see how wrong I was actually...

>>8536740
I'm retarded. My sum is not a function.

>>8536675
When you do a change of variable t'=t-1 you still have n-1 terms, so the first integral is 1+...+1/t which has n-terms, while the second integral is 1+...+1/t with only n-1 terms.

>>8515963
why should I learn about category theory?

Who here /haskell/?

>>8526273
>The mathematicians hope that the poof of this hypothesis to contribute in defining the locations of prime numbers on the number line.
This is the last sentence I'd read before throwing the paper in the trash.

>>8535200
Reading through the rest of the page (https://www.quora.com/What-is-the-hardest-thing-you-have-ever-done)) without feeling sad is pretty hard as well. (if you want to find his comment, you have to hold down the end key for a solid minute)

i can finish my applied math degree in 3 years taking a light course load.
OR
i can do a math/cs degree in 4 years but it'll be a heavy course load every semester + summers

is it worth the extra effort to learn cs formally? or should i just learn a couple programming languages on my own?