/sci/ - Science & Math

Explain to me Poincaré's conjecture please.
Also, can you tell me why the cardinality of R is bigger than that of N? Is it because the former can contain the latter and not vice versa?
Moreover, how do we know there are infinitely many transfinite numbers and that they are discrete (i.e. No infinite gap between any two of them)? Is it by construction? By contradiction? By instinct? How?
Final question is set theory really important aside from the logical foundation of Math? (e.g. Russel's Principia Mathematica.)

>>5701127
Hello! That is a lot of questions, so allow me to answer each really glibly.

Poincare Conjecture: There are many notions of equivalence in topology. Two of the most common are homotopy equivalence and homeomorphism. The latter catches ALL of the topological aspects of a space, whereas the former focuses more on aspects of connectivity. For example, the unit circle and the punctured plane R^2-{0} are not homeomorphic (one is "compact" and one isn't) but they are homotopy equivalent--you can continuously deform the punctured plane into the circle. As this illustrates, in general, homotopy equivalence is a much coarsers, less restrictive form of equivalence then homeomorphism. The Poincare conjecture states the somewhat surprising result that for 3-manifolds related to the 3-sphere (the unit sphere in 4-dimensional space) this is actually not so. Namely, a 3-manifold is homeomorphic to the 3-sphere IF AND ONLY IF it is homotopy equivalent to it. In other words, for 3-manifolds, only their connectivity aspects differentiate them from the 3-sphere.

#R>#N: I mean, intuitively, yes, but how does one prove this? There are many, many proofs of the fact that N has cardinality strictly greater than R's. Probably the most straightforward is to prove that <span class="math">\mathbb{R}[/spoiler] is equipotent to <span class="math">2^\mathbb{N}[/spoiler]--the power set of <span class="math">\mathbb{N}[/spoiler]. See below.

Infinitely many infinite cardinals: The short answer is Cantor's theorem which says that the cardinality of any set <span class="math">X[/spoiler] is strictly greater than the cardinality of its power set <span class="math">2^X[/spoiler]. Because of this, we can just keep taking more and more power sets of <span class="math">\mathbb{N}[/spoiler] to get infinitely many infinite cardinals. The discreteness you mention is NOT true. For the sake of space, I advise you to read the wiki article on the continuum hypothesis to see that not only is this description not true, it's not false--it's undecidable in our normal system of logic.

>>5701146
(continued)

Importance of set theory: The importance is simply this: set theory is the medium upon which math is conveyed. There are attempts to change this, but as it stands, all of math (practically) is immersed in, and composed of, the language of sets. It's easy to forget this sometimes because the actual set theory doesn't really get in our faces too often, but we take for granted the fact that nearly every time we attempt to make a mathematical statement, we are doing it in the context of sets.

I hope this helps!

>>5701147
It helps a lot.
I read your answer twice and you really know your stuff. Too bad it's 12:30 and I gotta wake up in 6:00, I really want to discuss more since people like you are very rare (Obviously very smart, knowledgeable AND willing to help others.)
Thank you for your answer and I understood everything except for the topology a little bit. Not a problem though, I guess, I will read an introductory book on it soon along with naive set theory.
Bump.

>>5701146
>Namely, a 3-manifold is homeomorphic to the 3-sphere IF AND ONLY IF it is homotopy equivalent to it.
The poincare conjecture is a lot weaker, actually: a simply connected 3-manifold is homeomorphic to a 3-sphere.
What you said is true, but generally, homotopy equivalence is a stronger condition.

>>5701156
Hello! Thanks for fact-checking, but the two are equivalent I think. This is from wiki:

"An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it."

It could be wrong, but this is the statement that I think of.

Also, I assume you meant to say "*homeomorphism* is a stronger condition"!

Thanks

Is there a quick easy way to check if a function or homorphism is bijective?

ｷﾀ━━━━(ﾟ∀ﾟ)━━━━ｯ!!

Just passing by to say I love your threads.

>>5701181
Stated that generally? Not really. In more specific contexts, possibly: for example, if you're working with homomorphisms of abelian groups (or anything else where you have a reasonable notion of "kernel"), then to show a homomorphism is injective, it's enough to show that the kernel is zero, i.e., only zero gets mapped to zero.

Oh, by the way, I'll be checking this thread every now and then; I can answer some questions, too, though I probably don't know as much as OP about some topics.

>>5701181
Hello! A group homomorphism? Are you wondering if there is a surjective analogue of "a group homomorphism is injective if and only if it has trivial kernel"? Well, the answer is yes, at least in abelian groups, but you're not going to like it. You can always formally dualize this statement (in an abelian category [ignore this if those words aren't familiar]) to get the statement "a group homomorphism is surjective if and only if it has trivial cokernel". That said, the cokernel of <span class="math">f:G\to H[/spoiler] is merely <span class="math">H/f(G)[/spoiler] :) So, you really only have to show that <span class="math">H/f(G)[/spoiler] is trivial to show <span class="math">f[/spoiler] is surjective! Of course, as you are most likely way to smart to see through my encouraging smilies and exclamation points, this is really bullshit. It is no easier (in general) to show that <span class="math">H/f(G)[/spoiler] is trivial, then to show that <span class="math">H=f(G)[/spoiler]. That said, this really is the correct analogue.

Besides this, the answer is unfortunately no, I do not know a method of showing surjectivity that allows you to feel like you're "cheating the system" of some work, the same way the injectivity/kernel trick does.

Sorry!

Can you explain how the infinite sum over 1/n^4 is equal to pi^4/90?

I understand how to prove it using a Fourier series, but there is another method using a contour integral, with an integrand of 1/z^4*(tan(pi*z))^-1 that' I'm having trouble with. Are you familiar with this?

>>5701188
Hello! What you said is one-hundred percent correct. Just one little nit-pick, the kernel of a homomorphism exists for homomorphism between any groups--not just abelian, and the same trick you mention works. If you see my response, the fly in the ointment for working with non-abelian groups comes when you try and dualize the notion of kernel, where you do, in fact, need to assume abelian.

Thank you very much by the way for helping answer some questions--I'm sure you'll have a lot to add :)

>>5701185
Thank you very much! I am not entirely sure what that ascii art is (a man with abnormally long arms hugging?), but I appreciate the sentiment all the same!

>>5701112
What is a surface integral and how is it related to surface area?
Why do I need to paramaterize my surface? Can't I just represent it as a function?

>>5701197
Hello! Yes, there is a complex analysis way of evaluating your mentioned some. In fact, there is a contour integral way of evaluating, in one fell swoop (by letting k remained a fixed variable), to evaluate <span class="math">\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2k}}[/spoiler] for all <span class="math">k[/spoiler]!

See here: http://www.mat.uab.cat/matmat/PDFv2009/v2009n06.pdf (page 15)

I hope that helps1

>>5701193
You should perhaps mention that when the group is finite, it sufficient to show injectivity and that the groups are the same order.

Alternatively, you could try the obvious choice of finding an inverse.

>>5701205
Hello! A surface integral is the double integral version of a line integral. Namely, you want to do a double integral over something that "looks" like a flat piece of the plane (at least locally), and thus to integrate it you first need to flatten it.

The surface area of a surface can be found using a surface integral. The formula is found on the wiki page.

The reason you need to parameterize your surface is captured in the above first paragraph. We can't integrate our surface without "flattening" it out onto the plane first. Think about your parameterization of the surface as the process of flattening your surface.

I hope this helps!

>>5701201
Er... right, of course kernels make sense for arbitrary groups. Thanks for catching that. I've been working through Hartshorne this semester, so my mind's in the commutative realm, and I was thinking of the natural generalization to modules and other abelian categories.

Actually, for (arbitrary) groups, it seems like enough of a fragment of the generalization would work to still make sense for checking surjectivity. The problem is just that f(G) might not be a normal subgroup, but even so, we can look at the cardinality of the set of (left or right) cosets H/f(G), which is 1 if and only if the map is surjective.

In the fall, I will be going to a well-regarded engineering school as a freshman to most likely study Materials Science and Engineering.

My question: Will I ever understand anything in this thread?

>>5701210
That sounds perfect, but the url isn't working for me. If you don't have another link that's fine, I'll go ahead and look for the method. Thanks

>>5701214
Aha! Very true. This is actually a very useful trick in ALL parts of mathematics. And, of course, this really has nothing to do with group theory, this is merely the Pigeon Hole Principle. An injective map between finite sets of the same cardinality is surjective.

Of course, the statement with all instaces of "sujrective" and "injective" switched is also true

Good call!

How do you prove xlnx tends to zero as x goes to zero without L'hopital's rule?

>>5701227
yay, a question in here where I at least recognize some of the words.

>>5701217
Hello! Working on Hartshorne can make anyone's brain a little droopy--so no issue.

Yeah, you are absolutely right. I was trying to stay in the realm of groups (all our objects being groups), but you're absolutely right. There is really no need to require <span class="math">H/f(G)[/spoiler] is a group, only that it is a singleton!

Thanks!

>>5701164
>Also, I assume you meant to say "*homeomorphism* is a stronger condition"!
No, (strong) homotopy equivalence to an n-sphere is a stronger condition than being simply connected.
There are spaces of same dimension that are simply connected but aren't homeomorphic (wedge of two 2-spheres vs a 2-sphere). Simply connected just means the fundamental group is trivial, while homotopy equivalence implies, in particular, that all homotopy groups are isomorphic.
It does not change the poincare conjecture, but it is a significant difference.

>>5701112
If you don't mind me asking, who are you and why do you know math so good?

>>5701214
Right, good point. That ties in nicely to a few nice classes of objects where maps are injective if and only if they're surjective; finite sets of the same size (and therefore also finite groups, rings, fields, etc. of the same size) and finite-dimensional vector spaces of the same dimension are the two best-known examples.

There's also the closely related fact that distance-preserving maps from a compact metric space to itself are bijective. These three cases are discussed in depth (albeit in an extremely abstract way) here:
http://golem.ph.utexas.edu/category/2011/12/the_eventual_image.html

>>5701227

Hello! Let's make the substitution <span class="math">x\mapsto \frac{1}{t}[/spoiler], so that we're really trying to evaluate the limit <span class="math">\displaystyle -\lim_{t\to\infty}\frac{\log(t)}{t}[/spoiler]. Note then that for <span class="math">t\gg0[/spoiler] we have that <span class="math">e^{\sqrt{t}}=1+\sqrt{t}+\cdots>\sqrt{t}[/spoiler] and so takin logs <span class="math">\sqrt{t}>\log(\sqrt{t})=\frac{1}{2}\log(t)[/spoiler]. Thus, <span class="math">0<\frac{\log(t)}{t}<\frac{2\sqrt{t}}{t}[/spoiler] and so the Squeeze Theorem gives it to us.

I hope that helps!

>>5701230

How do you prove xlnx tends to zero as x goes to zero without L'hopital's rule?

>>5701238
see
>>5701236

>>5701232
Hello! This is certainly true, homeomorphism is not saying that their fundamental groups are isomorphic. I'm confused with your objection. Homeomorphism clearly is stronger than homotopy equivalence. Perhaps we are using different phrases?

I hope we can settle this

>>5701219
Hello! You will likely understand some of this stuff, but not a lot of it. This should not concern you in the slightest though, since 99% of the math I do/people may ask about, is not in the slightest applicable to anything you will do.

Which school, btw, if you don't mind me asking.

>>5701235
Hello! It is interesting to note that the only reason your example of finite dimensional vector spaces works is because of the finite set case. Namely, by fixing bases, we can reduce all problems about f.d. vector spaces, to maps between finite sets.

And yes, the case of isometries on compact metric spaces is certainly true, and is a classic problem in many first books on analysis!

Thanks fo pointing this out~

>>5701233
Who are you is obviously a bit of a sticky-wicket question. Why am I so good at math? I am a math major, going to graduate school next year, and spend a large amount of my time doing mathematics. I want to be a mathematician, so for me it's a hobby, passion, and job--it's hard to avoid having some amount of proficiency with it.

I hope that helps!

>>5701233
The answer OP gave here >>5701253 is also completely true of me.

Except I wouldn't use the phrase "sticky-wicket". And I don't really care if someone figures out who I am.

>>5701267
Hello! Plot twist: I'm American and you're British.

I hope that clarifies things!

>>5701154
> (Obviously very smart, knowledgeable AND willing to help others.)

They are not that rare. I mean, very smart people are rare, knowledgeable people are rare, but the combination of being very smart and knowledgeable doesn't reduce the chance that one will also be willing to help others.

However, if you add "Posts on /sci/", then you may be onto something. People on /sci/ who attempt to look smart and knowledgeable usually are in quest for e-attention, not trying to help anyone.

>>5701275
>People on /sci/ who attempt to look smart and knowledgeable usually are in quest for e-attention, not trying to help anyone.
Those are usually trippeople.

>>5701267
Hello! Actually, as a followup, if you don't care if people find out who you are, what school are you heading to next year?

What are the top 5 mathematical notations you would change if you could?

Statistics never really predict anything, do they?

>>5701273
...I think you got that backwards. I live in the States. Guessing you're British, then?

>>5701280
I'm going to UW-Madison. (By the way, if you don't mind telling me where you're going next year, I put a throwaway address in the email field.)

>>5701112
Hi, OP, love your threads; I'm just getting my feet wet with sequences and I understand simple sequences like a_n=2n+1 therefore a_1=3 and so forth where you are able to find a_1 once you've supplied n; I'm having trouble grasping sequences where they are totally dependent on n, such as instances where you are given information such as a_0 and a_1 and a sequence such as a_n=a_(n-1)-a_(n-2).

Could you help me grasp these kinds of sequences?

(Please pardon my writing as I don't really know how to use LaTeX but _ is subscript)

>>5701219
>In the fall, I will be going to a well-regarded engineering school as a freshman to most likely study Materials Science and Engineering.
lol Wow. You sound like me, except I'll be a sophomore next year.

>>5701275
Oh great math guy, I have a question, though not really related to a concept. If I want to be an effective high school teacher and possibly teach math without it being my major, what class(es) do you think it absolutely essential I take to be able to give good insights into higher fields of mathematics or connect lower concepts to something higher? (I hope that makes sense) As of right now, I have firm knowledge of single and multivariable calc, ordinary and partial differential equations, some statistics, and I'll be taking linear algebra next semester. Do you think number theory would be necessary? Any thoughts (I do plan to take an introductory course in topology and differential geometry later on)? In other words, what math do you wish your math teachers knew in high school so they could tell you how the concepts related or what good they were for?

THanks

>>5701287
Hello! Fascinating question! That's tough, here are some that come to mind:

1. Despite this being AGAINST popular opinion (I know, I'm too edgy for you, or the block text version of it) I prefer the fraktur notation for the ideals of a ring. So, instead of <span class="math">I[/spoiler] being an ideal of a ring, I would much prefer <span class="math">\mathfrak{a}[/spoiler]. I like this, because outside of Lie algebras, Fraktur letters aren't common, and unstylizied letters are a dime a dozen. It's nice to cut down on overloading of notation.

2. I don't know why, but the implies symbol for spectral sequences has always annoyed me.

3. I do not like it when people don't unitalicize operators like GL, SO, or Gal, etc.

4. I don't like pi being used for homotopy groups. Pi is just overloaded as it is, and unlike homology where, if you put coefficients, there is no ambiguity <span class="math">\pi_n(X)[/spoiler] could mean so many things.

5. I hate, for some odd reason, <span class="math">\ln[/spoiler]. It should be <span class="math">\log[/spoiler]. Saying natural log just annoys me. And if anybody says "lynn" for natural log, I secretly put them on my shit list.

Ihope that answers your question!

>>5701291

Haha, I am not. I am also American. Sticky wicket is just a British phrase--that's why it was a plot twist.

UW-Madison! Nice, going to hobknob with Ellenberg I guess.

What's the difference between pure and applied math as a field of study?

>>5701304
Yeah, Ellenberg seems like a great guy. Seems he has quite a good reputation, since like half the people I talk to about UW-Madison mention him.

>>5701293
Hello! Thanks for the kind words.

Think about recurrence relations, sequences given in terms of previous terms, as a rule. For example, the recurrence relation <span class="math">a_n=a_{n-1}-a_{n-2}[/spoiler] is the rule "To find the nth term, you need to take the n-1st term and subtract the n-2nd term from it". So, now imagine if I said to you "Dear anon, what is the 73rd term of the sequence?" You'd be like "Oh shit, that's easy! Just take the 72nd term and subract from it the 71st term" and then it hits you--you don't know what either of the 72nd or 71st term is. Ok, well no problem! To find the 72nd term, you merely take the 71st term and subtract from it the 70th term. Ok, well you see the issue here. You don't know any of these terms either. Thus, you keep repeating your mantra "to find the nth term you take the n-1st term and subract from it the n-2nd term", until you've reduced the term of the sequence you know from 72, to 71, to 70, to 69, all the way down to needing to know the 3rd term. You then go "well, fuck, I don't know what the third term is!" But then, it dawns on you--you've reached the promised land. Namely, the third term is just the second term minus the first term (the golden rule) and you DO know what those terms are! Thus, you can finally figure out the 3rd term! But then, since the fourth term is just the third term minus the second, and now you know both of these values, you can get the fourth value. In fact, using the same idea, you can work your way up--find the fifth term using the ones you've already found, use this newly found fifth term and the previously found 4th term to find the 6th term, etc. until you get all the way back up to the 73rd term. From where you will be triumphantly able to announce "Anon, the 73rd term is..." well, whatever you got.

I hope that makes sense! Please ask for clarification if it does not!

>>5701320
Yeah, he's a super cool guy. I had the pleasure of talking to him at the last JMM conference. UW is a great school, and I'm sure you'll be very sucessful there. PS, I sent that email.

>>5701313
Hello! I apologize, but I am not well-equipped to answer that question. I know so little of applied math, that even attempting to act like I know the difference would be an affront to the applied field.

I apologize again! Perhaps some other, more well-informed, anon can answer this question.

can you explain the taniyama-shimura conjecture?

only heard about it through Andrew Wile's proving of Fermat's theorem, but i never learned what it was.

would rather a "personal" lesson rather than a google search

>>5701236
Your substitution for x=1/t is wrong it should be ln(1/t)/t not ln(t)/t

>>5701324
Hey, thanks for the response! This is excellent in explaining these types of sequences; I do understand that you can methodically work your way all the way back up to the 73rd term in this case; however, what if you could produce an equation such that you could find a_73 with ease?-- given a_0 and a_1; surely this would be possible; possibly through sigma notation?

>>5701334

Hello! That is a tall order for an anonymous image board. I will assume that you have some amount of knowledge about number theory/algebra/etc.

To every elliptic curve <span class="math">E[/spoiler] defined over <span class="math">\mathbb{C}[/spoiler] there is a numerical invariant assigned to it--the <span class="math">j[/spoiler]-invariant. A priori we only know that <span class="math">j(E)\in\mathbb{C}[/spoiler], and, in fact, <span class="math">j(E)\in k[/spoiler] for some subfield <span class="math">k[/spoiler] of <span class="math">\mathbb{C}[/spoiler] if and only if <span class="math">E[/spoiler] is "defined" over <span class="math">k[/spoiler] (meaning that <span class="math">E[/spoiler] is isomorphic to a curve with coefficients in <span class="math">k[/spoiler]).

The Taniyama-Shimura conjecture states roughly that those elliptic curves with rational <span class="math">j[/spoiler]-invariant (i.e. those which are isomorphic to equations with rational coefficients, those showing up in Diophantine geometry) are covered by <span class="math">X_0(N)[/spoiler]--which is a modular curve (it's a classifying space for elliptic curves with some torsion data).

In particular, it says that you can always find a rational map <span class="math">X_0(N)\to E[/spoiler], which is necessarily a covering.

So, intuitively, the Tayinyama-Shimura conjecture says that those elliptic curves which come from number theory situations (Diophantine equations) can be though about as "collapsings" of the well-understood modular curve <span class="math">X_0(N)[/spoiler] for some <span class="math">N[/spoiler].

I hope that was somewhat satisfactory! If not, there is a whole book dedicated to teaching you what Wiles proved, building all the necessary background material, and even giving something to the tune of 8 different versions of the Modularity Theorem.

The book is found here:

http://www.amazon.com/First-Course-Modular-Graduate-Mathematics/dp/038723229X

>>5701343
Hello! There is a minus sign there :) Tricky, eh?

Do I get my question answered? ;_;
>>5701294

>>5701341
Hello! I can do you one better, at least in this case.

What you have written down, in your example, is something called a homogenous linear recurrence relation with constant coefficients. These have, in fact, a closed form (see here: http://en.wikipedia.org/wiki/Recurrence_relation#Linear_homogeneous_recurrence_relations_with_constant_coefficients)) This should remind you of, if you have taken it, the solution to certain linear differential equations. That is because recurrence relations can be thought, somewhat rigorously in fact, as discrete versions of differential equations.

If this doesn't answer your question, let me know!

>>5701294
Hello! Wow, great question!

I think the most crucial thing to take, for a high-school math teacher, would be analysis. It is most likely that calculus students would be the ones asking about the relation of their subject to higher mathematics. Analysis, or at least a first course in analysis, is nothing but the rigorization of what you learned in calculus, and so would fit the bill perfectly. Taking an abstract algebra course would also not hurt, but it may be somewhat of an overkill.

I have to admit that this is not something that I have thought about before, but it is an amazing question. I always found my high school teachers to be lacking, in general, in terms of math. They didn't know how to point me in the correct directions to fulfill my somewhat precocious interests. In this way, having the broadest possible knowledge of mathematics would be optimal.

I hope this helps!

>>5701348
Oh i didnt see it, srry.

>>5701342

Sure helped me a lot more.

Thanks i'll be sure to check out that link also

Can you prove the continuity of the indefinite integral?

>>5701378
Hello! Sure I could, but why do you ask? I'll give you a hint. The key property is that <span class="math">\displaystyle \int_a^b f(x)\, dx\leqslant (b-a)M[/spoiler] for any <span class="math">M[/spoiler] such that <span class="math">M\geqslant f(x)[/spoiler] for al <span class="math">x\in[a,b][/spoiler].

Good luck!

>>5701354
Hey again, I really appreciate your answers; however, I'm distraught that I can't seem to decipher a proper application of an equation to use from that page in order to find very large values of n (like <span class="math">a_500[/spoiler]) given <span class="math">a_n[/spoiler]=<span class="math">a_{n-1}[/spoiler]-<span class="math">a_{n-2}[/spoiler] where <span class="math">a_0[/spoiler]=4 and <span class="math">a_1[/spoiler]=7.

I'm really sorry about my incompetence in this matter; this might be a tall order for someone like myself with only a background in pre-calculus.

>>5701373
thx for the cool picture :)

>>5701385
Sorry; that should read <span class="math">a_{500}[/spoiler]

>>5701385
Hello! Using the link I gave you, you can prove that the following is true for your recurrence relation with initial values:

<span class="math">\displaystyle a_n=10 \frac{\sin(\frac{\pi n}{3}){\sqrt(3)}+4 \cos(\frac{\pi n}{3})[/spoiler]

I hope that helps!

>>5701390
Fuck. That should read

<span class="math">a_n=\frac{10}{\sqrt{3}}\sin(\frac{n\pi}{3})+4\cos(\frac{n\pi}{3})[/spoiler]

>>5701362
Analysis, noted. Tyvm

My school seems to break math up into two main types: analysis and algebra. I see that number theory, linear algebra, and the like are in the algebra category. So I guess I'll see to it that I take a few from both categories. Thanks a bunch!

>>5701299
Hey, that looked like a cool exercise. Makes me want to contribute, with elementary notations.

As much as people may think math notations are now universal, they aren't, and really depend on where you learned and who you read. In France, a lot of notations differ from the common English ones:
- We use 42,13 to denote the number 42.13. This is honestly a bad notation. The comma is much more likely to be ambiguous than a dot. The only time when the dot can really create an ambiguity is that <span class="math">a\cdot b[/spoiler] is a*b, but people don't really use <span class="math">42\cdot 13[/spoiler] for the product of 42 and 13.
- This forces us to use ";" as our sequence separator, or to write intervals. These two are really annoying.
- On the other hand, I think our notation for open vs closed intervals is better. We write [a;b] for the closed interval (instead of [a,b], not much difference), but we write ]a;b[ for the open one (and ]a;b] / [a;b[ for the open on the left / open on the right ones), which makes more sense to me than (a,b), especially since (a,b) is overloaded as it is also the pair (a,b). Sure, having brackets facing outward may be confusing grammatically speaking, but not much more than having mismatching brackets like (a,b].

I also don't like the fact that <span class="math">2\sum a+b+c[/spoiler] is ambiguous. In particular:
<span class="math">2(\sum a+b)+c[/spoiler] could be <span class="math">2((\sum a)+b)+c[/spoiler] or <span class="math">2(\sum (a+b))+c[/spoiler],
<span class="math">2\sum (a+b)+c[/spoiler] could be <span class="math">2\sum ((a+b)+c)[/spoiler] or <span class="math">2(\sum (a+b))+c[/spoiler].

An optional "end" symbol for sums which are ambiguous would be cool, otherwise the only way to be very clear is to use <span class="math">(\sum (a+b))+c[/spoiler] with 4 brackets.

Also I couldn't agree more with
> I do not like it when people don't unitalicize operators like GL, SO, or Gal, etc.
and have to add "max" and "min" to the list. People to to use \max and \min, but LaTeX has no building \argmax and \argmin and many don't know how to define a new math operator.

How can I imagine the tenth dimension?

Why is euler's number significant?

>>5701390
Wow, that is actually 100% correct; I'm really thankful and impressed with your responses! If you don't mind me asking, (Since I don't think I've learned it yet) What kind of crazy sorcery did you use to change <span class="math">a_n[/spoiler]=<span class="math">a_{n-1}[/spoiler]-<span class="math">a_{n-2}[/spoiler] into <span class="math">a_n[/spoiler]=<span class="math">\frac{10}{\sqrt{3}}[/spoiler]+4cos(<span class="math">\frac{npi}{3}[/spoiler]) ?

Derivation?

>>5701414
Hello! I don't have quite enough space/time to explain this, but here is a helpful link. I hope it will be enough!

http://www.fpp.edu/~milanb/tpmeh/mehanika/lect18.pdf

>>5701406
>I also don't like the fact that 2a+b+c is ambiguous.
you write whats in the sum on the right and whats out of it on the left
<span class="math">c+2\sum a+b[/spoiler]

>>5701411

Hello! That is an extremely tough, and somewhat unanswerable question. It comes up in the limiting process of interest, etc.

Perhaps one of the more novel answers would be the following. Out of all the differential equations with initial values, probably one of the most obvious is f'(x)=f(x), and f(0)=1. It turn out that this is precisely e^x.

Sorry I couldn't be more helpful!

Prove e is an irrational number

>>5701373
what the fuck am i looking at

>>5701419
>http://www.fpp.edu/~milanb/tpmeh/mehanika/lect18.pdf

Excellent, I'll be reading this; I appreciate it.

>>5701424
This is a trivial application of the Lindemann-Weierstrass theorem... :)

More seriously, I'm sure you know that this is somewhat involved. One of the quickest proofs can be found here:

http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem

>>5701409
Not OP. Here are my two cents.

Visualizing many dimensions in an Euclidean space is not trivial but you can get a pretty good idea by introducing weird dimensions like color, time, brightness etc to the classic spatial dimensions. You can also think with binary hypercubes which are non-intuitive in many regards, but still tractable.

The real issue with "imagining the 10th dimension" is that it's not what you actually want. What you want is understanding how having our 3D or 4D universe be a part of a 10D universe means we can have cool stuff like wormholes etc. And that implies understanding that our 4D universe isn't some kind of 4D cube in a 10D universe. It's a 4D "shape" which "folds".

Imagine that your universe is 2D, placed inside a 3D universe. If your 2D universe is a plane, you won't have any kind of wormholes or anything. Going from one point to another in the 3D universe will be the same as in the 2D one.

If however, your 2D universe can be obtained by taking a plane, and folding it to the left, then to the right, then to the left etc, stacking layers over each other, then the shortest way from the bottom layer to the top layer in the 2D universe is to go across the bottom layer, get on the 2nd one, go across it, get on the 3rd one, go across it, etc. If you are allowed to leave your 2D universe and use the whole 3D universe to move, then you can jump from one layer to another without having to go across it first.

So yeah, to visualize the topological effects of being part of a higher-dimension universe, think with a small amount of dimensions: you'll reach roughly the same conclusions in an easier manner.

>>5701422
I know, but this is sometimes no desirable. For instance:

- c could also be a sum or a product,
- I could be trying to match the order in which the corresponding terms were in the previous line of my derivation.

>>5701423
Not OP.

Let's say you're playing good old "Guess my number between 0 and 1010" game. You ask your questions instantly and you analyze the answers instantly, but the guy answering them is a bit slow. A question is of the form "Which of these intervals contains my number? [a0;a1), [a1;a2), [a2;a3), ..., or [a_{k-1},a_{k})?", where you choose k.

In the most basic case, k=2 and you are just doing basic dichotomy, splitting the set in half every time. Of course, if you use k=10, you need far less steps to win. However, the guy answering your questions takes a longer time to think when k grows. In fact, he needs k seconds to answer when you have k intervals.

How should you choose k to maximize the speed at which you find the answer?

If you check what happens, taking k>4 is pretty bad: the fact that you need less steps to converge if really offset by the fact that the steps are more expensive. For instance, let's say you need M steps when k=2 (M*2 = 2M seconds), then for k=4 you will need M/2 steps which are twice as expensive (M/2 * 4 = 2M seconds as well). For k=8, you will need M/3 steps (M/3 * 8 = 2.7M seconds). And as k grows, you keep needing more and more time.

So 2, 3 and 4 are decent values (3 being the best of the three). But what if non-integer values of k made sense? What would be the best then?

The best would be e (the reason being that e is the minimum of k ↦ C*k/ln(k), which describes the ratio of the cost per turn divided by the number of turn needed, for a fixed original interval length).

Basically, e is the growth rate that gives the best tradeoff between precision and complexity per step.

>>5701434
Hello! Fantastic response!

>>5701437
Thank you :)

I wrote that in a <span class="math">red dit[/spoiler] thread two weeks ago but the thread was a bit old and it went unnoticed; I'm glad someone liked it here.

The thread had roughly the same question about the relevance on e. Find it here: /r/math/comments/1brnvy/

Can you explain the Continuous Wavelet Transform?

And if you have time, the Hilbert Transform?

Despite going over them thrice over three different semesters, I /still/ have a hard time understanding epsilon-delta proofs of limits. convergence, etc. Any advice? Recommended reading? Anything?

>>5701608
read baby rudin

>>5701112
Not a question about a math concept, but I hope you'll answer nonetheless.

What have been/are your favorite math textbooks? It doesn't matter if they're introductory or advanced or anywhere in between. Just some textbooks that you've studied from that you would recommend to someone else learning the same stuff.

(not trying to restrict the question to OP either - I just figured the textbooks someone as well-versed as OP used must be at least ok)

I want to thank you OP, a couple days ago you helped me in another thread in a post regarding ring theory. It took a bit for me to digest it but it just clicked for me earlier while I was taking a shower.

Keep up the good work, it's much appreciated.

Second year engineering here and having some serious trouble with Phasors. More specifically addition (because that's where I'm stuck in my notes).

Could you explain to me the procedure for phasor addition? (Both trig and complex if possible)

Which are the fields you're most interested in.
Which are the field which don't interest you at all.
Which are the subjects outside of math you like.
And do you have any specific ideas for topics or approaches in mathematical disciplines you see yourself working on in the future?

>>5701112


Is Who wants to be a millionaire just a game of luck?

I mean, there's 33.34% chance of being right unless the person happen to know the answer, and lifelines can only get you so far into the game.

>>5701666

Let's say there are 15 questions with 4 choices each. Let's also say there are 3 lifelines: call-a-friend, 50-50, and ask-the-audience.

For each of the 15 questions, there is only 1 correct answer, and 3 incorrect answers, so immediately your chances are 1/3 = 33.34%. You can use each lifeline once. Using phone-a-friend, let's say gives you a 95% chance the person you call is correct. So 1 question is 95%. Similarly, we can say ask-the-audience gives you a 95% chance. Finally, 50-50 will remove 2 wrong answers, so you have 1 right answer and 1 wrong answer, so you basically know the answer. To recap:

Question 1: 33.34%

Question 2: 33.34%

Question 3: 33.34%

Question 4: 95% <--- ask-the-audience

Question 5: 33.34%

Question 6: 33.34%

Question 7: 33.34%

Question 8: 33.34%

Question 9: 95% <--- phone-a-friend

Question 10: 33.34%

Question 11: 33.34%

Question 12: 33.34%

Question 13: 33.34%

Question 14: 33.34%

Question 15: 100% <--- 50-50

Total: 690.08%

Divided by 15 questions = 46.00%

So basically, it's more a less a game of chance, but more than half the people on the show will lose, so it's not very fair. That show is making tons of money off of people.

Could you give me some insight about why the laplace transform works?

In ZFC, what exactly is the difference between the axiom schema of restricted comprehension and the axiom schema of replacement? They seem like they would do the same thing to me.

>>5702767
Separation justifies the notation { x ∈ X | φ[x] }, the subset of X such that φ holds for all its elements. φ can be any predicate.
Replacement justifies, for an entire functional relation φ, the notation { φ[x,Y] | x ∈ X }, the set of all sets Y such that φ[x,Y] holds for all x ∈ X.
Separation can only form subsets of a set, but replacement can form bigger sets.

So my understanding of the proof of countability of the rationals is that it's the specific case of a general result--if you have a union of countable sets of n-tuples then you can construct a sequence of these sets and then make a bijection from the naturals to each element (i.e. each set of n-tuples) of this sequence.

So when we count the rationals we really count a countably infinite union of sets, and each natural number corresponds to one set in this sequence which contains an ordered pair representing a rational number?

I know this isn't really a question--I just wanted to put the explanation in 'my own' words to see if I get what's going on.

>>5701603
Hello! Sadly, no. I know absolutely nothing about either. That said, I emailed a friend of mine that is focusing in that direction--hopefully he will have something enlightening to say!

Sorry again!

>>5701112
topos theory...what is it all about?

>>5701608
Hello! Let me take a crack at explaining it. It's not likely I'll succeed if three other sources have failed, but it's worth a shot.

Think about delta epsilon proofs as a game. To make thing's concrete, let's play the game of proving that <span class="math">f(x)=2x[/spoiler] is continuous at <span class="math">x_0=0[/spoiler]. Ok, so here's how the game works. For whatever reason, you do not believe that <span class="math">f[/spoiler] is continuous at <span class="math">0[/spoiler]. Thus, you are convinced that you can find some number <span class="math">\epsilon>0[/spoiler] such that, no matter how hard I try, I CANNOT find a number <span class="math">\delta>0[/math\ such that whenever a number <span class="math">x[/spoiler] satisfies <span class="math">|x-x_0|=|x|<\delta[/spoiler] (recalling that in our case <span class="math">x_0=0[/spoiler]) that <span class="math">|f(x)-f(x_0)|=|f(x)|<\epsilon[/spoiler] (noting that <span class="math">f(x_0)=f(0)=0[/spoiler]). In words, you are claiming that you can find some amount of "closeness" (the epsilon) so that no matter how hard I try, I cannot find a corresponding measure of closeness (this is the delta) such that two points being close in my sense, guarantees that their images are close in your sense.

Ok, so now that the rules of the game are set, we begin our, no doubt to be epic, battle. You start off with the saucy value of <span class="math">\epsilon=\frac{1}{2}[/spoiler]. In other words, you're claiming that I CANNOT find a number <span class="math">\delta>0[/spoiler] such that <span class="math">|x|<\delta[/spoiler], taking values of <span class="math">x[/spoiler] really close (<span class="math">\delta[/spoiler] close in fact) to our point of interest <span class="math">x_0=0[/spoiler], guarantees that <span class="math">|f(x)|<\frac{1}{2}[/spoiler] (the image of these points close in the domain goes to points <span class="math">\frac{1}{2}[/spoiler] close in the image).[/spoiler]

>>5703528
(continued)

To your choice of <span class="math">\epsilon=\frac{1}{2}[/spoiler] I exclaim "what a pitiful choice! Clearly <span class="math">\delta=\frac{1}{4}[/spoiler] works!" And indeed, we see that if choose <span class="math">x[/spoiler] within <span class="math">\frac{1}{4}[/spoiler] of <span class="math">x_0=0[/spoiler], we find points whose images are within <span class="math">\frac{1}{2}[/spoiler] of <span class="math">f(x_0)=f(0)=0[/spoiler]. Indeed, to prove this, if <span class="math">|x|<\frac{1}{4}[/spoiler] then <span class="math">|f(x)|=|2x|=2|x|<\frac{2}{4}=\frac{1}{2}[/spoiler].

Rocked, but still resolute, you decide for another bold move--<span class="math">\epsilon=\frac{1}{28\pi}[/spoiler]. Not expecting such a strange number, but feeling able to rise to the challenge, I pause for thought. After about five seconds, I proclaim "<span class="math">\delta=\frac{1}{56\pi}[/spoiler] works!". And, indeed, if <span class="math">|x|<\frac{1}{58\pi}[/spoiler] then <span class="math">|f(x)-f(x_0)|=|f(x)-f(0)|[/spoiler]<span class="math">=|2x-2\cdot 0|=2|x|<\frac{2}{58\pi}=\frac{1}{28\pi}[/spoiler].

Getting desperate, you throw your ace up the sleeve at me "Ok, anon, if you're so smart, I'm not even going to TELL you which <span class="math">\epsilon[/spoiler] I picked. But you need to find your corresponding <span class="math">\delta[/spoiler] nonetheless!" While you think you have beaten, you have not. For, there has been a pattern I have been following. Every time you have chosen some <span class="math">\epsilon[/spoiler], whatever that is, whether it be <span class="math">\frac{1}{2}[/spoiler] or <span class="math">\frac{1}{28\pi}[/spoiler], I always knew how to construct my <span class="math">\delta[/spoiler]--it was always just <span class="math">\delta=\frac{\epsilon}{2}[/spoiler]. Indeed, this finishes the battle since if <span class="math">|x-x_0|=|x|<\frac{\epsilon}{2}[/spoiler] then <span class="math">|f(x)-f(x_0)|=2x-2\cdot 0|[/spoiler]<span class="math">=2|x|<\frac{2\epsilon}{2}=\epsilon[/spoiler]. Thus, the pattern works REGARDLESS of the actual value of <span class="math">\epsilon[/spoiler].

>>5703531
(continued continued)

This is how delta-epsilon proofs work. You imagine that you have an adversary that claims he can choose a teensy-weensy enough number, so that no matter how hard you try, you cannot find another number, such that if points are that close in the domain, their images will be that teensy-weensy amount of close in the image.

But, the key part of the game, is that to prove he CANNOT find such a number, you can't just do it for specific values of teensy-weensy, say <span class="math">\frac{1}{2}[/spoiler] or <span class="math">\frac{1}{28\pi}[/spoiler], but that you can do it for ALL values of teensy-weensy. Meaning that the value of <span class="math">\delta[/spoiler] you pick is a function of/is dependent on what teensy-weensy he picks, but that it always exists.

I hope this helps!

>>5703496
could you explain what the derivative is? from what i know it can be described as the tangent line at a point but are there any other ways you can describe it

Complex impedance in RLC circuits please!
Or if not, an explanation of the complex plane in general

whats the mathematical formula for a girlfriend and not wanting to kill myself every night

>>5703528
>the saucy value of 1/2

love that you chose to word it this way. i really enjoy a sense of humor in math writing--i find spivak is the best for this, e.g.,
>...again no finite collection of sets in O will cover (0,1). Although this phenomenon may not appear particularly scandalous...

>>5703520
Hello! As I'm sure you're well aware, that is a very difficult question, and one that I am far from the most informed or well-equipped to answer.

I can give you the basic idea though. Basically, people wanted to move the normal definition of sheaves away from the classic definition of "a functor on the open sets of some topological space, which satisfies some gluing properties" to a more general definition. In other words, they didn't want to have to restrict their sheaves to having domain equal the open sets of some topological space, and instead, try to replace it with a general category.

The reason for this is clear--we want to extend local-global principles to situations other than those arising from topological spaces. Of course, there is an obvious issue with this. Namely, while a presheaf on a general category makes utter and total sense, it is the gluing and local determinacy properties that caused an issue. Namely, there was no notion of locality, or, more egregiously, of an open cover so we can formalize "gluing". Grothendieck, in his infinite wisdom (Amen), came up with the appropriate notion of open covers for a general category. This is called a Grothendieck site on the category. I don't have time to go through the axioms, but they follow what you'd expect, see the wiki article for more.

Ok, now that we have a notion of local determinacy and gluing, we can actually consider the subcategory of all <span class="math">C[/spoiler]-valued presheafs on a our category <span class="math">X[/spoiler] (i.e. just the functor category <span class="math">C^{X^\mathrm{op}}[/spoiler]) consisting of sheaves. THIS is the topos over that site.

>>5703567
(continued)
Summarizing, intuitively, we want to do sheaf cohomology/formalize local to global principles on a general category, and so sites are the replacement for locality and open coverings, in the context of our category, normally given by the underlying topological space. This site then allows us to intelligently discuss sheaves on our category, and the category of these sheaves is a topos.

I hope that helps!

>>5701632
Hello! Not a problem--always glad to help!

Happy mathing!

>>5703549
Not OP. Just a tool that exploits the fact that in sinusoidal steady state, voltages and currents are related to their derivatives, through each device, by linear relations, and they also satisfy linear constraints across the circuit (sum of voltages is null, currents are equal in the series case). You can represent those linear operations (which are in dimension 2) with 2x2 matrices, which can also be seen as operations on complex numbers. Using complex impedances just trivializes the computations that would be annoying if you were working with functions with like <span class="math">\sin(\omega\cdot t)[/spoiler] everywhere.

>>5703563
I like these kind of jokes in article titles, like
"Division by three" by Conway (from which I read only the "division by two" part but it was pretty cool and easy to understand for someone with little knowledge in that topic), or "On <span class="math">O_n[/spoiler]", etc.

>>5703573
ah alright, you learn something new everyday.
Thanks!

>>5703535

Hello! Sure! But, there is much literature on this out there, and there are certainly more informative sources.

The key to understanding the derivative, is to eschew the number definition, and to replace it with the linear function--the function whose graph gives your tangent line! Namely, instead of thinking about the derivative of a function <span class="math">f(x)[/spoiler] at <span class="math">x_0[/spoiler] as just being the number <span class="math">f'(x_0)[/spoiler] instead think about it as the linear function <span class="math">Df_{x_0}(x)=f'(x_0)x[/spoiler]. Then, the linear function <span class="math">Df_{x_0}[/spoiler] is the linear function that BEST approximates (in a technical sense) the function <span class="math">f[/spoiler] at <span class="math">x_0[/spoiler].

Think about it this way. General functions suck, in a lot of ways. That said, linear functions, those that are just multiplication by a fixed number (e.g. <span class="math">L(x)=17x[/spoiler]) are extremely simple. It would be fantastic if we could find some way to forget these general nasty functions, and replace them with these clean linear ones. Of course, globally, at every single point of our domain, this is impossible--no linear function will work, functions are just too crazy, and vary to widely on a large scale for this to be true. That said, if we instead of trying to do it EVERYWHERE, try to do it only really, really close to a specific point <span class="math">x_0[/spoiler], then this is possible...almost. Namely, while we cannot say that our linear function EXACTLY models our ugly non-linear function, even near our fixed point <span class="math">x_0[/spoiler], we can get a damn good approximation. Ok, so what linear function does this damn good approximation of <span class="math">f(x)[/spoiler] at <span class="math">x_0[/spoiler]. As I said above, it is <span class="math">Df_{x_0}(x)=f'(x_0)x[/spoiler].

I hope that helps!

>>5701659
I apologize, but I have no idea what a phasor is. I know almost nothing of applied mathematics, let alone engineering.

Sorry again!

>>5703288

Hello! This is morally right, but technically wrong. Unfortunately, you do not have a bijection since a) what finite tuple hits infinitely repeating number such as 1/3, and b) representations of numbers in terms of decimals aren't unique.

But, let's see if we can adapt your proof to something that is intuitively the same. So, you want to prove that <span class="math">\mathbb{Q}[/spoiler] is countable. Since it's infinite, you already know that it's cardinality is AT LEAST of countable size, and thus if you can show it is AT MOST of countable size, then you'll be golden. To prove that <span class="math">\mathbb{Q}[/spoiler] has at most countable size, it suffices to find some countable set <span class="math">X[/spoiler] and a surjection <span class="math">X\to\mathbb{Q}[/spoiler]. Well, there is an obvious surjection <span class="math">\mathbb{Z}\times\mathbb{N}\to \mathbb{Q}[/spoiler] given by <span class="math">(a,b)\mapsto\frac{a}{b}[/spoiler]. But, why is <span class="math">\mathbb{Z}\times\mathbb{N}[/spoiler] countable? Well, because it is a countable union of countable sets! Indeed, <span class="math">\displaystyle \mathbb{Z}\times\mathbb{N}=\bigcup_{n\in\mathbb{N}}\mathbb{Z}\times\{n\}[/spoiler]. Since each <span class="math">\mathbb{Z}\times\{n\}[/spoiler] is countable, since it's clearly equipotent to <span class="math">\mathbb{Z}[/spoiler], and we're taking a countable union of such sets, we're home free.

I hope that helps!

>>5701662
Hello!

Fields I'm most interested in:

Complex differential geometry, algebraic geometry, number theory.

Fields that don't interest me:

Applied math, statistics. That's about it--almost all other parts of math fascinate me!

Subjects outside of math:

Ha, not a lot. If I had to choose, I would probably say history and philosophy.

As for your last question, I am not sure what you mean. Could you narrow the scope of the question a little perhaps?

Thanks!

>>5701112

Tell me what Truth means in math, what does it correspond to and how do you access it?

>>5703561
How about an inequation: significant other ≠ happiness.

Really though, go kickass. Do something amazing. The rest will follow naturally. My mother told me something once that has always stuck with me, and has rung more and more true the longer I live "No one can ever truly love you, until you learn to love yourself".

Go out there and be something friend! I know you can!

>>5701112
can you explain tensors to me? thanks

Yes, I have a question. I understand how there are different "levels" of infinity but it doesn't make sense that one set of literal infinity can be bigger or smaller than literal infinity. How can endlessness be bigger than endlessness?

>>5703624
Hello! While I am not a big fan of the guy himself, this vide should be of great help!

http://www.youtube.com/watch?v=elvOZm0d4H0

>>5703616

Hello! In what context, I have explained this before in another thread.

>>5703603
Thanks for the explanation, I will have to go back and study the proof more carefully.

I think the thing that is tripping me up is the idea of a tuple 'hitting' an infinitely repeating number as you put it. I thought it is just a matter of having a set containing the ordered pair (1,3) and then whether you are thinking about just the pair (1,3) or the fraction 1/3 is incidental

>>5703614
ok since it's obvious you have no formula for this could you at least tell me the most sure-fire way to kill myself in terms of statistical probability?

>>5703585
interesting explanation, thanks.

1 more question. how do you explain a "corner" on a graph. graphs are continuous but not differentiable at these right?

>>5703585
>>5703585
interesting explanation, thanks.

1 more question. how do you explain a "corner" on a graph. graphs are continuous but not differentiable at these right?

>>5703638
I believe helium exit bags are the goto thing here.

Hi there! I was wondering if you could conceptually explain Galois theory, or at least why it's useful. I'm currently taking an abstract algebra course, and I have no intuition about this subject.

>>5703668

Hello! Sure. It relies on a simple principle: take difficult problems in field theory, and turn them into easy problems in group theory. Namely, a field extension <span class="math">L/K[/spoiler] can be thought about as an algebraic object (the WHOLE extension, both parts simultaneously). A good way to study the algebraic object then may be to study its "automorphism" group <span class="math">\mathrm{Aut}(L/K)[/spoiler]. But, what precisely does this mean? What precisely do we mean when we want to think about the extension as a single algebraic object.

The key concept is the notion of an algebra over a field. Namely, note that if <span class="math">L\supseteq K[/spoiler], then <span class="math">L[/spoiler] is both a field AND a vector space over <span class="math">K[/spoiler] (where the scalar multplication is just the restriction of the ring multiplication on L, so that one variable just takes in K values). Thus, an "automorphism" of <span class="math">L/K[/spoiler] should be a map which both respects the field(ring) structure and vector space structure. Namely, an automorphism should be a ring map <span class="math">\sigma:L\to L[/spoiler] which is bijective, and which is also <span class="math">K[/spoiler]-linear. With this definition, the automorphism group <span class="math">\mathrm{Aut}(L/K)[/spoiler] is called the *Galois group* of the extension, and is denoted <span class="math">\mathrm{Gal}(L/K)[/spoiler].

Now, it somewhat of a miracle, well at least at first glance, that the Galois group of an extension could really tell you *that* much about the extension. I mean, automorphism groups are often informative, but not earth-shatteringly so. In the case of nice (read *Galois*) extensions (these are just extensions where the Galois group governs a particular large amount of structure) the Galois group tells us SO much about the extension.

The piece de la resistance of a beginning course in Galois theory is the Fundamental Theorem of Galois Theory, which says that the Galois group of a (Galois) extension completely dictates the lattice of subextensions of the extension <span class="math">L/K[/spoiler].

>>5703697
(continued)
Namely, suppose that I have you a particular example--say the extension <span class="math">\mathbb{Q}(i)/\mathbb{Q}[/spoiler], and said "ok, anon, find ALL fields <span class="math">L[/spoiler] such that <span class="math">\mathbb{Q}(i)\supseteq L\supseteq \mathbb{Q}[/spoiler]" (such a field is called a "subextension"). You probably would have no idea how to proceed. The amazing fact is that the subextensions of <span class="math">L/K[/spoiler] (where the extension is Galois) are in one-to-one correspondence with the subgroups of <span class="math">\mathrm{Gal}(L/K)[/spoiler].

Moreover, not only is their a bijection, but it is a) constructive (so you can actually PRODUCE all of the subextensions of your extension using the Galois group), it is b) something called a Galois connection, meaning that it respects the order theoretic aspects of the lattice of subgroups and lattice of subextensions (it is an adjoint functor between the two categories).

So for example, <span class="math">\mathbb{Q}(i)/\mathbb{Q}[/spoiler] is a Galois extension, whose Galois group has a simple isomorphism type <span class="math">\mathrm{Gal}(\mathbb{Q}(i)/\mathbb{Q})\cong\mathbb{Z}/2\mathbb{Z}[/spoiler]. Well, how many subgroups of <span class="math">\mathbb}{Z}/2\mathbb{Z}[/spoiler] are there? Well, there are just as many subextensions of <span class="math">\mathbb{Q}(i)/\mathbb{Q}[/spoiler] by the Fundamental Theorem. Since there are always two subextensions of a non-trivial extension, namely <span class="math">L=\mathbb{Q}(i)[/spoiler] and <span class="math">\mathbb{Q}[/spoiler], and there are exactly two such subextensions by what we have just said, these are, in fact, *all* of them.

To summarize, Galois theory is the exploitation of the unreasonable power that the Galois group of a nice (read Galois) extension of fields has over the structure of the extension itself. We turn hard theorems about field theory (find all subextensions) to easy group theory problems (find all subgroups of its [necessarily finite, for a finite extension] Galois group).

I hope that helps!

what is the quotient space in linear algebra?

Eigenvectors and values please! You would be my hero!

>>5703657
>>5703654
>>5703654
>>5703654
Maybe I can field this one...

A more strict definition of differentiability is concerned with how well a graph can be approximated at a certain point by a linear function. This could go by different names--Newton's Method, a 1st degree Taylor polynomial, linear approximation, etc. But you have probably heard of one of these if you've taken any calculus.

Think of a function that you intuitively consider to be smooth. y=x^2 is pretty damn smooth looking isn't it? But what do we mean by that? Well if you get an equation for the linear approximation of y=x^2 at any point on the graph, your approximation will be reaaallly accurate when you're right on the point, a little less accurate up or down the curve, and pretty useless when you get really far away.

So how far away do we have to be for it to get useless?

Well think of y=|x|. An unsmooth function if I ever saw one. Say you wanted to approximate y=|x| right around its kink at the origin. Well the line y=0 would do the job if you were right at the origin, but the second you move even the tiniest bit away the value of y=|x| is nowhere near the value of y=0.

To sum it up, a function's differentiability is a stricter way of quantifying how bad the linear approximation becomes in relation to how far away you are from the point you're trying to approximate. On a corner (intuitively not smooth) you can't get a good approximation unless you're right on top of it. And indeed, a rigorous definition will show that corners are not differentiable. Conversely, on a smooth looking curve like y=x^2, the approximation isn't too bad until you get really far away and the strict definition will agree with your intuition--y=x^2 is differentiable everywhere.

>>5703703
Hello! It is the same quotient construction in other parts of algebra. It is intuitively, the act of "crushing" part of your vector space down to the zero space. So, in the vector space <span class="math">\mathbb{R}^3[/spoiler] with the subspace <span class="math">V=\{(0,a,0):a\in\mathbb{R}\}[/spoiler], the quotient space <span class="math">\mathbb{R}^3/V[/spoiler] is the new vector space which is just <span class="math">\mathbb{R}^3[/spoiler], but we have now declared that all elements of <span class="math">V[/spoiler] are to be zero, and that all equations which follow from this are true (e.g. if <span class="math">v\in V[/spoiler] and <span class="math">w\in\mathbb{R}^3[/spoiler] then <span class="math">w+v=w[/spoiler] since <span class="math">v[/spoiler] "equals zero" in this new space).

I hope this helps! If there is something more specific you were curious about, please feel free to ask!

>>5703704
Hello! Which aspect were you looking for? Theoretically why are they important/what do they mean or practically, how do I find them, etc?

I have answered this in another thread. It's a shame this "no memory" thing sometimes.

>>5703717
How they are defined, in english! And how to find them

Hey OP, if you don't mind me asking, what job prospects are there for a math major? I've always adored math and chemistry and wanted to go into it in college but never thought I'd be able to find a job. I havn't entered college yet, so I was just wondering before I do enter college and have my major set in stone.

I took a course in Differential Equations and Linear Algebra a couple semesters ago (I was 16, so I have plenty of time to catch up), and we had to accept some methods at face value because the proof was outside the scope of the class -- I hated that. Could you explain how Laplace transforms work? I understand how to use them, but not exactly why or how they accomplish what they do.

>>5703740
I am: >>5703750
And I have considered becoming a professor of mathematics. It would be a lot of fun to teach upper-division math because the students aren't complete shitheads. I wouldn't like grading papers though, so for the time being I'm studying computer engineering.

>>5703697

Awesome! Thanks so much.

>>5703752
I would love being a teacher, and I'm pretty good at it (basically have to explain how everything works and how to solve problems to most people because the teacher is bad at it) but the problem for me is I have far too much sympathy. I'm the kind of person that feels bad for squashing a little bug, and whenever possible I always spare the bugs life and let it go on chilling enjoying life. I'd never be able to fail someone in college and have them have to retake the class or something like that, I just couldn't do it.

>>5703752
Can't guarantee you'll get to teach upper-division courses.

Fresh meat often gets stuck teaching classes like calc for science or (heaven forbid) calc for arts and social science.

Then one prof I have (master's degree) and a friend of mine (recent phd) usually only teach 200 level courses in the math or applied math department.

Then there are people with masters and phd's from harvard or something like that who have been at the school for 5-10 years. They get to teach the advanced level version of calculus, algebra/linear algebra, as well as the upper year pure or applied math courses in their area.

But believe me there is NO SHORTAGE of shitheads/unappreciative students in these math for sciences/social sciences courses

>>5703777
That's basically what I wanted to avoid, which is why it's a back-up. What does it take, beyond a PhD to teach 400+ level courses?

Polynomial time

>>5703791
From what I reckon it must be a combination of skill and seniority. When I look at the profs who are teaching the advanced level calculus and also get to teach real analysis, functional analysis, etc. (as well as grad-level courses on these topics) they're the ones who have been at the university for a long goddamn time and have a pretty long list of publications in these subject areas too.

For someone in academia with pure passion for their field I can only imagine what a joy it must be to teach upper division or advanced level courses to highly motivated students who want to engage with the material and ask incisive questions. What I'm saying is don't think the entire department doesn't realize what a privilege that is--if somebody's getting to teach these courses they're probably gonna be the one who really knows more than anybody else about these topics.

And in general, you have to have an intense command of a topic to be able to teach it to advanced undergrads or grad students. Somebody who just took a course a couple years ago is probably not ready to teach it, that kind of thing. That would be where seniority comes in.

>>5701615
Time to read off a sizable chunk of my bookshelf, then.

Linear algebra:
- Axler, "Linear Algebra Done Right"

General algebra:
- Artin, "Algebra"
- Isaacs, "Algebra"
- Lang, "Algebra"

Analysis:
- Spivak, "Calculus"
- Rudin, "Principles of Mathematical Analysis"
- Rudin, "Real and Complex Analysis"
- Gamelin, "Complex Analysis"

Commutative algebra:
- Atiyah & MacDonald, "Introduction to Commutative Algebra"
- Eisenbud, "Commutative Algebra with a View Toward Algebraic Geometry"

Algebraic geometry:
- Smith et al., "An Invitation to Algebraic Geometry"
- Shafarevich, "Basic Algebraic Geometry"
- Harris, "Algebraic Geometry"
- Hartshorne, "Algebraic Geometry"
- Eisenbud & Harris, "The Geometry of Schemes"
- Vakil's "Foundations of Algebraic Geometry" notes: http://math.stanford.edu/~vakil/216blog/

Number theory:
- Serre, "A Course in Arithmetic"
- Ireland & Rosen, "A Classical Introduction to Modern Number Theory"

Also, lots of good recommendations here:
http://www.ocf.berkeley.edu/~abhishek/chicmath.htm

>>5703796
The average life span of a starved parrot.

>>5703796
The land speed of an unladen swallow.

>>5703830
Hello! There are other additions I would add, but let me add on to the subject you have least of (weighted by my interests :) )

Number Theory:

Algebraic:

Lorenzini's "An Introduction to Arithmetic Geometry" (HIGHLY suggested for a first time through of algebraic geometry and algebraic number theory. Absolutely fantastic)

Neukirch's "Algebraic Number Theory" (What you should tackle after Lorenzini. This is the truly modern way to attack number theory, from a geometric point of view. An invaluable book)

CFT:

I have discussed this in another thread. My suggestions would be:

Algebra II by Falko (This will mostly give you the necessary background on local fields and its ilk needed to tackle local class field theory, and then takes you right into LCFT. A great book)

Milne's Notes on CFT (A modern approach to the subject. Decides to pursue things via the sledgehammer that is Tate cohomology. Definitely dense, but definitely comprehensive).

Neukirch's CFT book (Slightly novel approach, but one of my favorites. Highly recommended)

Elliptic Curves:

Washington's Elliptic Curve Book (Slightly less advanced than the ones following this, but perhaps the perfect level to start. Gives you a much broader view of the subject than Silverman and Tate's "Rational Points", but doesn't overwhelm you with a need to be comfortable with algebraic geomtry)

Baby (Arithmetic of) and Papa (Advanced Topics in) Elliptic Curves by Silverman (The truly modern approach to elliptic curves. These books will kick your ass, especially Papa, but if you are able to finish them, you will have a true understanding of one of the most fundamental objects in number theory/mathematics in general)

>>5703914
(continued)
Modular Forms:

Diamond and Shurman's Modular Forms and Elliptic Curves (While there are many books on modular forms, this is my favorite. Not only does it give you brisk stroll through all the major topics in the study of modular forms, it has purpose. The goal of the book is to describe the Modularity Theorem (the thing Wiles actually proved) in a variety of different ways, from different perspectives. An absolutely indepsenable book for an aspiring number theorist).

I'm trying to remember the concept of a two-volume book on something called...

centrinions?

I don't remember what they were called, and it's bugging the shit out of me.

The opening chapter is about vector calculus, and the concept is commonly used in graphics processing. I think it ends with -inions.

>>5703944
quaternions?

>>5703951
Jesus christ, thank you. I'm embarrassed to say I googled triconions. I was close.

Why are C*-algebras and compact Hausdorff spaces the same?

>>5704177
Not OP, but surely you mean commutative C*-algebras?

http://en.wikipedia.org/wiki/Gelfand_representation#Statement_of_the_commutative_Gelfand-Naimark_theorem

>>5704177
>http://en.wikipedia.org/wiki/Gelfand_representation#Statement_of_the_commutative_Gelfand-Naimark_theorem
Hello! They are not *literally* the same. But, the category of commutative C*-algebras is equivalent to the opposite category of compact Hausdorff spaces. This is, as the other poster mentioned, a consequence of the Gelfand representation.

This is the starting place for a lot of K-theory btw!

What is a Hilbert space and why is it so improtant?

Hi OP, I'm a freshman majoring in computer science and planning to double major in applied mathematics. I do like math, although the sheer tediousness of some of the stuff that I am doing in calculus is less than appealing. I would also say I have no talent in math. Do you think I should stop trying to pursue a degree in math?

>>5704215
As a theoretical CS major who did a lot of math as well, I strongly recommend that you keep studying math as long as possible. The people in my classes that didn't study math were usually as good at the non-theoretical parts of CS like programming, but they really lacked the formal capabilities when it came to theoretical subjects like semantics or abstract interpretation, and even subjects in between theory and application but with statistics requirements like machine learning, classification, clustering etc (which really don't require that much knowledge of stats, but if you haven't done math for years, chances are you'll be lacking the ability to handle that).

>>5704193
Hello! Once again, it is unfortunate that I don't have old threads I've done saved.

Let me start with a mathematical mantra: linear algebra good, not linear algebra bad. Linear algebra is one of the few realms of mathematics that we have a, pretty much, complete theory of--at least in finite dimensions. In finite dimensions, linear algebra often ends up reducing to set theory--of *finite* sets.

In infinite dimensions, things start to get sticky. In particular, we can't say nearly as much in the infinite-dimensional case as we could in the finite dimensional case. Wanting to still have a well-behave theory, we introduce the infinity-wrangler AKA analysis. Since we can't, algebraically, deal with infinite dimensions to well, we introduce an analytic restriction on them. Namely, not only do we have a vector space, but that vector space comes with an inner product which gives induces a complete metric--an extremely strong condition. It turns out that this is precisely the condition we need to carry over many of the proofs in finite dimensions (where ALL inner products induce (topologically all the same) complete metrics).

Besides the above, they are important for the simple reason that many abstract notions are important--they're everywhere. Hilbert spaces abound, but the most important are probably measure spaces <span class="math">L^2(X,\mu)[/spoiler], and subspaces of. This single class of Hilbert spaces (which, in a sense, actually constitutes ALL Hilbert spaces) appears in analysis (in obvious ways), in complex differential geometry (so, so many <span class="math">L^2[/spoiler]-estimate/Hormander's method proofs), and elsewhere.

I hope that helps!

>>5704215
Hello! It is hard to give advice, not knowing your full background. I would follow what makes you happy, and consult those who are in a better position to give you substantive advice.

Sorry!

>>5704773
>theoretical CS major
>who did a lot of math
HAHAHAHAHA, no.

>>5705967
Hello! Please refrain from being condescending. I see no reason to doubt that this person was a theoretical CS person, who also did a lot of math. Some of my best friends who are in graduate school for math, started out as CS, even graduating with, a CS major.

Thanks!

>>5705973
>theoretical CS
That in itself is an oxymoron.

>>5705991
Hello! That is entirely untrue. There are many theoretical parts of computer science. Now, whether or not you want to make an argument of the form "yeah, but if its theoretical, its not CS its really just _____", then don't. There is no hardset definition of CS, and so we should go by the common parlance of what is "CS", which really amounts to "what is taught in a CS major". With this definiton, there are some highly theoretical "CS" topics.

Thanks again!

>>5705995
>There are many theoretical parts of computer science
Not in undergrad programs

>"what is taught in a CS major". With this definiton, there are some highly theoretical "CS" topics
... watered down and hand-waved to hell or just mention in passing

>>5705995
Thanks for coming to my defense, OP, but really, this is 4chan, just ignore the trolls and they will realize they are wasting their time... Or maybe not, I'm not sure, maybe they actually feel like CS majors are so insecure that they actually get mad at this kind of trolling?

I feel very confident that while my math knowledge isn't that of a graduate math major, it beats that of anyone criticizing the math abilities of CS majors, so until someone in the same post both shows a deep non-googled math understanding and that anti-CS attitude, I am simply going to assume that they know neither CS nor math.

It also really amuses me that it started by me stating how important it is to avoid relying purely on CS courses and how studying math in parallel is important. Seems like a post that would actually go in their direction, does it not?

>>5706032
> >There are many theoretical parts of computer science
> Not in undergrad programs

I'm sorry, who said anything about undergrad programs? Do you not realize that undergrad math and graduate math also differ by quite a margin in terms of the required abstraction level? Thankfully neither math nor CS are restricted to what you learn in undergrad courses.

>>5706041
>I'm sorry, who said anything about undergrad programs
people don't call getting graduate degrees in a program "majoring in it"

>>5706371
My apologies for not being a native English speaker. I'm a post-doc in Information Theory with a PhD in Information Theory and CompSci and I did my undergrad in Math / Physics and my grad school in CompSci / Math. Better?

This is a general question, not necessarily for the OP.

How does one without a formal university education of math learn to think like a mathematician, I am studying chem E, but I want to try to learn some real math one day. There is no way I could afford/ spend the time to get a second degree in math, so it would be completely self taught.

>>5706388

There are a lot online learning resources (complete curriculum, with problem sets and video lectures for free) MIT, Stanford, and many others. Check out their Youtube pages. Also check out: https://www.coursera.org/

>>5706388
Hello! My biggest suggestion would be to frequent math forums. As another poster has suggested, there are vast amounts of resources for actual learning, but what you are trying to replace is peer-to-peer/instructor interaction. In this way, forums will not only allow you to ask questions and get answers, much the same way an instructor would act, but see the questions of other people learning the same material, and the answers to their questions (in a sense, replacing your peers).

I hope that helps!

>>5706414
>math forums
Which ones do you visit?

I'm going through a paper, and there's a statement along the lines of

>Let ( , ) be an inner product on both a vector space W and its dual space W*

I'm not quite sure how the same inner product can work on both a space and it's dual. ( , ) provides a map from W x W to the real numbers, but how can we use that to get a map from W* x W* to the reals?

>>5706422
Hello! Most of them :)

>>5706423
Hello! I'm not familiar with this either, could you give more context? I mean, one sensible thing to do is to fix a basis <span class="math">\{v_1,\ldots,v_n\}[/spoiler] for <span class="math">V[/spoiler], and then consider the dual basis <span class="math">\{v_1^\ast,\ldots,v_n^\ast\}[/spoiler]. You could then define <span class="math">\langle v_i^\ast,v_j^\ast\rangle:=\langle v_i,v_j\rangle[/spoiler] and extend by bilinearity. But, of course, this is non-canonical. I'm having difficult seeing how to canonically induce an inner product. If I think of something, I'll write back.

>>5706429

The paper's talking about representation spaces, and polynomials functions on the representation space which are invariant under the group action.

I think your suggestion would work, it would give something which would perform as required. Thanks!

>>5706429
Hello! Me again. I don't think that there is any naturally induced inner product on the dual space. Holistically, I expect this to be true since one would assume (or hope!) that this association would come from some natural isomorphism between the bilinear forms on <span class="math">W[/spoiler] and the bilinear forms on <span class="math">W^\ast[/spoiler]. Of course though, this former object is <span class="math">(W\otimes W)^\ast[/spoiler] and the latter object <span class="math">(W^\ast\otimes W^\ast)^\ast[/spoiler]. Since these two functors don't have the same variance, they can't be isomorphic.

Check back with any more context--maybe there is something lurking there that will give us a canonical identification.

>>5706435
Hello! If you're happy with this, then there is really no need to consider my last post explaining why there isn't a canonical choice (probably).

So, it's a paper on invariant theory (of polynomial rings)? Very interesting stuff! Why, if I may ask, are you reading this paper?

Happy mathing!

>>5706446

Good observation about how there can't be a natural isomorphism, it makes sense. I'm now pretty sure that the paper was simply talking about inducing an dual inner product relative to a chosen basis.

Yes, the paper is about the invariant theory of polynomial rings, combined with some semi-algebraic geometry. It's "Defining Orbit Spaces by Inequalities" by Procesi and Schwartz. I'm starting a masters degree on that topic later this year, and this is one of the papers my supervisor gave me to look through. Most of it makes sense, except that one early bit about the dual inner product. Thanks for your help!

>>5706388
>How does one without a formal university education of math learn to think like a mathematician

Get a copy of Rudin and keep reading it until everything becomes crystal clear

>>5706464

Fantastic stuff. I wish you the best of luck! Also, remember to not overheat in the Tropics of semi-algebraic geometry ;)

>>5706466
Hello! Op here. I am not sure if this is some /sci/ injoke that I am unaware of, but what is with everyone's fascination with Rudin? Why do people that this is somewhow the measuring stick of one's mathematical ability? It seems quite nonsensical to me. If you could explain, that would be great!

Thanks!

Hello OP,

have been reading your thread, and I must say I'm thoroughly impressed with your ability to answer most (if not all) of the community's questions. Was wondering if you knew of the careers with the best paying salaries with a degree in applied math and statistics?

>>5706429

Giving a non degenerate bilinear form on a (finite dimensional) vector space V is the same as giving an isomorphism V->V*. Its inverse is an isomorphism V*->V=V** and hence gives a non degenerate bilinear form on V*.

>>5706488
Hello! Sure. Right, which is precisely what I said in essence. This is still non-canonical, because without the presence of an inner product, you are choosing a basis form this isomorphism

>>5706486
Hello! Thank you for your kind words. I honestly have no idea. I have known for a long time that I was going to go into academia, and so haven't ever checked any such statistics.

Sorry! This seems like the ultimate question for your advisor, etc.

>>5706496
Sorry that the question was off topic from the thread. Was wondering if you'd be able to explain what the concept of "forcing" is? Realized this was used to prove that the continuum hypothesis couldn't be proved with modern maths (please correct me if I'm wrong) but I have no idea how this concept works. Thanks in advance! Also, do you have a website/blog I'd be able to follow? Completely blown away by your experience and overall kindness (something very rare on this website)

>>5706488

It is completely canonical. Your variance argument is wrong: on the category of vector spaces with isomorphisms as morphisms, the dual space is also a covariant functor: send f to f transpose inverse. It is for this functor that the bijection is natural.

>>5706498

Hello! That doesn't make any sense--why would you consider the category with isomorphisms as morphisms? This is not only the non-standard category of vector spaces, but is slightly strange, since the skeleton of this category is discrete--kind of weird. The standard category has just regular linear maps as morphisms, and the dual map is transpose, which is contravariant.

I hope I am not misunderstanding you!

>>5706497

Hello! Thanks, once again, for the kind words! Isn't that the million dollar question. Forcing is a somewhat magical term of model theory that, because of things like Cohen's proof, has made its way over the logic/math bridge to people like me and you. I could tackle it, but it seems silly considering I know the perfect reference.

If you are not aware, the AMS has a fantastic set of "What is..." articles. The point of these articles is to explain, to mathematicians/mathematical enthusiast, what some of the buzz words in contemporary mathematics are. Lucky for you, and me :), there is one on forcing.

Take a look, and if you have any follow-up questions and/or this article isn't at an appropriate level, let me know!

http://www.ams.org/notices/200806/tx080600692p.pdf

>>5706498

> why would you consider the category with isomorphisms as morphisms

Because non degenerate bilinear forms are only functorial in isomorphisms.

> This is not only the non-standard category of vector spaces, but is slightly strange, since the skeleton of this category is discrete--kind of weird

Not really. There are lots of automorphisms.

This construction (identifying non degenerate inner products on a space with those on the dual) is standard.

>>5706509

Hello! I'm not really sure what you mean. You have a bifunctor <span class="math">\mathbf{Vect}\times\mathbf{Vect}\to\mathbf{Vect}[/spoiler] taking two spaces to their space of bilinear forms. If you restrict to the diagonal of the product category you get a functor. So, I don't know what you mean by the lack of functorality.

Regardless, I find it shocking to argue that the obvious category structure on vector spaces isn't just linear maps. It may be a lack of knowledge on my part, but I have never seen the category you mention in nature. Moreover, the category, and the dual space functor sending an iso to its inverses transpose just seems like a artifical way to force covariance of the dual space functor.

As to your second point, of course. It's not discrete, I meant to say totally disconnected. I, personally, want to secretly think about the skeleton of vector spaces to be <span class="math">\mathbb{N}[/spoiler]--so that when we pass to isomorphism classes we get what we should--the pre Grothendieck completion of <span class="math">K_0[/spoiler].

I'm sorry that we are disagreeing!

>>5706509

I'll repeat myself:

> Because \emph{non degenerate} bilinear forms are only functorial in isomorphisms

>>5706527
Hello! Ok, well, I am not entirely sure what you're getting at with that. It's ok though--the important thing is that guy got the answer he needed!

>>5705962

Hello again! How important would you say it is for my grasp of calculus to be for my upper division classes, should I choose to go there?

>>5706551
Hello! I am not entirely sure what you mean. Could you rephrase that please?

Thanks!

How is Fermat's last theorem proved?

>>5701112
ziarinsky topology

>>5706589
The Zariski topology on...what? A variety? Spec?

Thanks for clarifying!

>>5706586
Hello! There is a great explanation of that here:

http://math.stanford.edu/~lekheng/flt/wiles.pdf

>>5701409
>>5701430

In the past I found the idea of more than 3 dimensions very difficult to understand.

However I realized that the stock market can be easily used to imagine a multidimensional space.

R^1 can be thought of as the price of a stock.
R^2 are the prices of two different stocks.
This is handy because R^n is just n stocks.

I find this much easier than trying to imagine colors etc.

Maybe others can think of other analogies.

>>5701112
>a question about a mathematical concept
let X be the set of all people who do khan-academy-like instructional videos for abstract algebra
let u be you and
why isn't u in the set X?

>>5706616
Hello! Haha, thanks for the compliment. A write a lot of note-like things, but have never ventured into doing actual lecture videos. It is mostly because I don't have the proper setup, or at least not a setup that would meet my requirement, to partake in such a doing.

That said, if you have a question about abstract algebra, feel more than free to ask it now. I would love to explain something that may be bothering/confusing you.

Thanks!

PS you may want to see my answer above about the importance of Galois theory.

>>5701112
is p=np?

>>5706668
Hello! Not only do I not know, I don't think I fully understand or appreciate the problem. So, unfortunately, that's all I got for ya.

Sorry!

>>5706628
it's really not just Abstract Algebra. From my experience, mathematics is not a hard subject if you have someone to translate the concepts to you.

>>5706671
Ha! First time I've actually seen someone on /sci/ admit that they don't fully appreciate the problem P=NP. Considering that most people who claim that they know it can't even define P or NP properly, I find it very ironic that it comes from someone with a knowledge of math they will most likely never reach.

Props to you, OP.

Does 0.999... equal 1 or not?

in linear programming simplex method

min c'x
Ax = b
x>=0

why is it required b >= 0 as well?

>>5706672
Hello! I guess that is somewhat true. Although, it takes some work to actually "get it" sometimes. There are many times somebody explains something to me, and I can parrot the words back at them, but it takes me a little longer to "get it".`

>>5706675
Hello! Thanks for the props!

>>5706676
Hello! I believe this is an injoke of /sci/, and so I am not sure if you are serious. Assuming you are, the answer is yes. The possible point of contention really comes from what we "mean" by the repeating decimal .9999...

If we take the real numbers as the metric space completion of the rationals, then .999... is the equivalence class of the Cauchy sequence <span class="math">\{a_n\}[/spoiler] where <span class="math">a_n[/spoiler] is <span class="math">.9999[/spoiler] with <span class="math">n[/spoiler] nines. In this version of <span class="math">\mathbb{R}[/spoiler], two such equivalence classes of Cauchy sequences are equal, if their difference is a nullsequence (converges to zero). We think about the rationals as living inside of this set of Cauchy sequences by letting a rational number be represented by the constant sequence associated to it.

So, when we claim that .99999... is equal to 1, what we are really saying is that the sequence <span class="math">1-a_n[/spoiler], as defined above, tends to zero, which it does.

I hope that helps!

>>5706743
Hello! Yes.

I hope that helps!

Explain Fourier transforms to me.

I'm coming to the end of my Physics masters and I'm still just bluffing my way through anywhere they're required by remembering the solutions.

>>5706747
Have you ever taken LSD?

>>5706556

I don't feel super comfortable with calculus, how do you think that will affect me later on?

>>5706697
Try an example using b < 0. I think you'll end up with a negative optimal solution, which doesn't really make sense.

problems with zero

GO!

Is it possible to be very good at mathematics (good enough to make a career of moderate success) without ever having mastered computation?

Is it unheard of for someone working in abstract, advanced areas of mathematics who can't do multiplication in their head?

where can i learn intermediate/top level math?

>>5706809
In the library

>>5706799
x=y(y)
x=y^y
There you go.

>>5706804

Not OP, but this is perfectly normal. Abstract math is more about ideas than performing calculations of that basic sort.

Here's a trick for multiplying numbers from 10 to 19:

find: 15 x 16
Step 1) 15 + 6 = 21 -> 210
Step 2) 5 x 6 = 30
Step 3) 210 + 30 = 240
Result) 15 x 16 = 240

>>5706778
Hello! This has been asked before. This is not somewhere where I have expertise, and so it's prudent of me to refer you to someone who does:

http://www.math.ucla.edu/~tao/preprints/fourier.pdf

>>5706782
Hello! No, never. Drugs don't appeal to me.

>>5706804
Hello! Well, if it is any indication, I am absolutely terrible at mental calculation

How do I find the constants for an ODE who's solutions are Bessel's Equations for cases where nu is an integer and y(0) and y'(0) are known?

>>5708550
Hello! I am not sure. I have never taken a class in differential equations, at least not a non-theoretical class. I would highly suggest you take this to one of the many fantastic online math forums.

Sorry!

What does the zeta function have to do with prime numbers?

Also, have you considered using a tripcode so that people in this thread can easily identify you?

>>5708579
Hello! I assumed that my jovial greetings, pretty face, and superlative answers would identify me enough.

Anyways, fantastic question, albeit a tough one. I don't quite have the time right now to do it justice.

The importance of the zeta function in number theory is the realization that <span class="math">\zeta[/spoiler] is <span class="math">\zeta_\mathbb{Q}[/spoiler] where <span class="math">\zeta_K[/spoiler] is the zeta function of a number field. In particular, <span class="math">\zeta[/spoiler] encodes in its mere definition, and some of the identities it satisfies, deep facts about number theory.

Instead of trying to say big words at you about this, perhaps two examples of concrete things the zeta function tells about primes will sate you.

There is a classic identity involving the zeta function that you may well be aware of. Namely <span class="math">\displaystyle \zeta(s)=\prod_p (1-p^{-s})^{-1}[/spoiler]. While this may just seems like a cool fact at first, it is actually somewhat deep. In fact, this identity of functions (really of formal power series) is EQUIVALENT to the Fundamental Theorem of Arithmetic (every natural number, save 1, is the unique product of primes). Indeed, when one expands this infinite series, one arrives at the sum of all terms of the form <span class="math">\frac{1}{(p_1\cdots p_r)^s}[/spoiler] where <span class="math">r[/spoiler] varies from <span class="math">0[/spoiler] (when we get 1) to infinity, and the <span class="math">p_i[/spoiler] run over all possible primes. The fact that this sum is the same as <span class="math">\displaystyle \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}[/spoiler] tells us that every n appears as one of the <span class="math">p_1\cdots p_r[/spoiler] and, in fact, appears ONLY ONCE. This is, indeed, the fundamental theorem of arithmetic.

>>5708594
(continued)

Secondly, you can consider $\zeta(z)$ as a function on a certain subset of the complex plane. This function has a non-removable singularity at $1$ though. What this means concretely is that $\displaystyle \lim_{s\to 1}\zeta(s)=\infty$. Ok, so what? This seems obvious since, everyone knows the Harmonic Series converges. Consider the product <span class="math">\displaystyle \prod_{p}(1-p^{-s})^{-1}$ which, as we have claimed, is equal to <span class="math">\zeta(s)[/spoiler]. Now, when we take the limit of this infinite product as <span class="math">s\to 1[/spoiler] we better get infinity, since this product is equal to the zeta function, and we claimed the zeta function had a singularity at <span class="math">s=1[/spoiler]. Think about it though, if there were only finitely many primes, say <span class="math">p_1,\ldots,p_n[/spoiler], then <span class="math">\displaystyle \lim_{s\to 1}\prod_p (1-p^{-s})^{-1}=\lim_{s\to 1}(1-p_1^{-s})\cdots (1-p_n^{-s})^{-1}[/spoiler], or, in other words, <span class="math">(1-p_1^{-1})^{-1}\cdots(1-p_n^{-1})^{-1}[/spoiler]. That surely doesn't look infinite to me. Thus, the fact that the zeta function has a singularity at <span class="math">s=1[/spoiler] implies that there are infinitely many primes!


I hope that helps![/spoiler]

>>5708595
Fuck.

The content of that post, is that since this product over the primes is equal to the zeta function, it better equal infinity when we plug in s=1. But, if there were finitely many primes, we certainly see that this infinite product DOES NOT converge. Thus, the fact that the zeta function has a singularity at s=1 implies that there are infinitely many primes.

I hope that helps!

>>5701425

the three lines are made by the centers of the three circles each of which is "rolling" along the inside of another circle (counter)clock-wise depending on how the circle it is in is rotating

>>5708598
>>5708595
>>5708594
Holy shit, thats pretty cool