/sci/ - Science & Math

[Anonymous No.5685944]

Ask me about a math concept, and I will do my best to explain it.

Picture related.

[Anonymous No.5685947]

Whats the deal with Uniform Continuity?

[Anonymous No.5685951]

Explain double integrals to someone who hasn't taken Multivariable yet.

[Anonymous No.5685949]

Why are women bad at it?

[Anonymous No.5685957]

What should I learn as an introduction to Category Theory?

[Anonymous No.5685967]

>>5685947
Hey! That's a little vague, but here it goes. Uniform continuity is basically the idea that the continuity of a function depends uniformly on its position.

In particular, the definition of continuity of f (say, mapping \mathbb{R} to \mathbb{R} for simplicity) AT x_0, is that given any e>0 there exists a d>0 such that |x_0-x|<d implies that |f(x)-f(x_0)|<e. Or, in words, that you can the image of points under $f$ really close to f(x_0) by making the points themselves close enough to x_0.

Now, assume that f is continuous at every point of \mathbb{R} and fix some e>0. Then, think about d as a function which, given a point x_0 of \mathbb{R}, spits out a number d(x_0)>0 such that the above holds: |x_0-x|<d(x_0) implies that |f(x)-f(x_0)|<e. Now, this function really is a function! Given two points x_0 and x_1, d(x_0) may vary wildly from d(x_1)--you might need to make the points MUCH closer to x_0 to make their images closer to f(x_0), then how close you need to make points to x_1 to make their images close to f(x_1).

Uniform continuity says that this anomaly doesn't happen. That, in fact, you can take d to be a constant function! That there exists a SINGLE d>0 NOT DEPENDING on x_0 such that |x_0-x|<d implies that |f(x)-f(x_0)|<e.

This has many important properties, depending on what your tastes are. Uniformly continuous functions can be extended to the completion of the space you're working on, but continuous function can't for example (this is because uniformly continuous functions preserve Cauchy sequences and continuous functions don't).

I hope that helps! Ask further if I didn't answer with what you were hoping I would.

[Anonymous No.5685980]

>>5685944

Do you ever feel like learning/explaining math is like learning arbitrary rules to a complicated game that isn't fun to play and has no interesting philosophical implications at all?

At least its practical in the sense that digging ditches and building bridges is practical, I'll give it that.

[Anonymous No.5685981]

How do I get better at proofs?

[Anonymous No.5685982]

>>5685944
E=MC^2

[Anonymous No.5685987]

>>5685957
Good question.

I think the key is to understand that category theory, for the most part, is not something you should "jump" into. Unlike some subjects, for a pure mathematician category theory is purely a tool. So, studying pure category theory for the sake of pure category theory, in my opinion, would be like a carpenter studying a lathe in isolation, instead of studying the lathe by how it interacts with wood.

This said, a healthy understanding of the basics of category theory illuminates many disparate parts of mathematics. There is definitely a diminishing returns effect though. If you took the time to understand categories, functors, products of categories, functor categories, limits, colimits (in particular products and coproducts), natural transformations (before functor categories!), continuou and cocontinuous functors, and adjoints then you'd know more than enough for a first go.

The key to all of this is to understand WHY you care about each individual concept. Every time you learn something new, be sure to see how the concept applies to every category you can: Top, Top_*, hTop, k-Alg, Rings, Groups, etc. In particular, see how these concepts are used, to see not only why they are useful (why the fuck are you learning this if not to apply it!) but also to illuminate the object itself. For example, look back at famous theorems like Seifert Van Kampen, and realize that you are really just making a statement about a certain functor preserving pushouts.

Also, for a large part of math, especially the algebraic geometry side of the stream, it is more helpful for you to learn about abelian categories. These are, intuitively, the most general categories where you can do homological algebra. They comprise two of the most important types of categories--categories of modules, and categories of sheaves.

I hope that helps! Ask any followup questions if I missed something!

[Anonymous No.5685989]

If I have two circles that are DEFINITELY of different diameters, AND they are touching, in how many places are they touching if one is inside the other, or vice versa? How do I know for sure?

[Anonymous No.5685990]

>>5685981
You don't. You just suffer, cold and alone, like the rest of us.

[Anonymous No.5685996]

>>5685951
Hello. Imagine, as somewhat strange of a scenario this might be, you wanted to find the volume of an object, let us say a building. Moreover, let's assume that you know how to describe the building as follows: you know the shape of its base, its footprint on the ground, and at each point of its base, what is the height of the tallest point on the building over that point. Each point of its footprint then can be represented as a point (x,y) in the plane, and the height can be represented by the number h(x,y)--the height at (x,y).

The double integral then is the function that takes in h(x,y), and the footprint set R, and spits out the volume of the building V.

In other words

V=double integral of h(x,y) over R

You can attack finding this double integral in a couple of ways. You can try to stack blocks inside of your building, to get an approximation, and then keep using tinier and tinier blocks until you get the exact area. This is the exact translation of Riemann sums that you encountered in calculus one. But, somewhat more interestingly, instead of trying blocks, you could instead by to find the area of each cross section of the building, and then add up these infinitely many cross sections. This is writing your double integral as an iterated integral--an integral of an integral. Indeed, the first integral is just the area of the cross section, and the second integral is measuring the amount of area picked up as you vary over all the cross sections.

I hope this helps!

[Anonymous No.5686006]

>>5685980
Hello! Interesting question.

I feel no such way. To me math is fascinating and beautiful. The lack of philosophical or practical importance is of no concern to me--I'm having fun.

Just like I don't understand how people derive pleasure from spending 12 years practicing to speedplay Ocarina of Time in 24 minutes, or playing video games at all for that matter, you don't understand why I enjoy math.

That's fine! If we all liked math, the world would be abhorrently boring.

I hope that answers your question!

[Anonymous No.5686010]

>>5685981
This is a classic, albeit tough question. As absolutely unhelpful, as annoyingly trite as it may sound, the answer is practice. Take every chance you get to write and rewrite proofs. And REALLY write them, don't just come up with a quick outline in your head.

As someone that, in my humble opinion of course, is good at proofs, it just happens. You remember when you didn't know how to drive, and then you did it a lot, and one day you said "shit, I can start on a hill without stalling! When did that happen?" That will happen with proofs. It's a passively learned topic, something better learned gradually and naturally then "studied".

Sorry for not being able to be more helpful! Feel free to ask any followup questions!

[Anonymous No.5686009]

>>5685989
There can only ever either be one or two real points of contact (depending on the nature of their intersection) if they are of different diameters.

[Anonymous No.5686014]

>>5686009
This was actually a bad example however, because all are of equal diameters. The only way there could be more than two real points of contact is if two congruent circles were on top of each other.

[Anonymous No.5686015]

Can you explain to me why the derivative of e^x = e^x ?

[Anonymous No.5686020]

>>5685989
Hello! This is a great question! The answer is two, as other posters have mentioned, at least if you're talking about real solutions. You're basically asking for solutions to a system of equations of the form (x-a)^2+(y-b)^2=r AND (x-c)^2+(y-d)^2=R. Intuitively their intersection is a polynomial of degree two, and thus can have at most two solutions. That is just purely intuitively though. The actual proof is either A) very messy or B) fairly technical.

Hope that helps!

[Anonymous No.5686023]

Are you answering physics related questions as well?

[Anonymous No.5686032]

>>5686015
I can try!

It all depends on your definition of the function f(x)=e^x though, doesn't it? Many people would define f(x) to be the unique solution to the IVT y'=y, y(0)=1 on R, which then says that f'(x)=f(x) tautologically.

If you instead define f(x) to be \sum_{n=0}^{\infty}x^n/n!, then f'(x) would be \sum_{n=0}^{\infty}n x^{n-1}/n!=\sum_{n=1}^{\infty}x^{n-1}/(n-1)!=\sum_{n=0}^{\infty}x^n/n!=f(x).

Now, while this really does not circumvent any of the issues with the definition of f(x), here is something that usually sates people. Let f(x)=e^x, then log(f(x)) is x, and so (log(f(x))'=1. But, (log(f(x))'=f'(x)/f(x) and so f'(x)/f(x)=1 so that f'(x)=f(x).

Hope that helps! Feel free to ask for clarification.

[Anonymous No.5686035]

>>5686023
Sadly, no. :( I would be all of the money in the UAE that you probably should be answering MY questions about physics.

Sorry!

[Anonymous No.5686037]

>>5686020
I'm not OP, but it's based on the limit definition of e.
e = lim(x -> inf) (1+1/x)^x
which we can also express as
e = lim(h -> 0) (1+h)^(1/h)
Now differentiate by first principles, and when you reach lim(h -> 0) ((e^x(e^h - 1))/h), substitute the second limit for e^h.
P.S. fuck latex

[Anonymous No.5686040]

>>5685982
See here!

http://terrytao.wordpress.com/2012/10/02/einsteins-derivation-of-emc2-revisited/

[Anonymous No.5686043]

I don't get limits (category theory), even familiar examples are completely unenlightening, but I understand products and pullbacks. I hear that products and equalizers are enough to construct a limit, and equalizers are just a special kind of pullbacks, but I've never seen a construction that's not overly complicated and abstract.
So my question is: assuming the appropriate pullbacks and products exist, how do you construct a limit out of them?

[Anonymous No.5686045]

What's the mathematical formula used to determine the inner volume of a 4 dimensional tetrahedral non-gaussian prism ?

[Anonymous No.5686053]

>>5686043
Hello! That is somewhat difficult to answer in this format. See these notes though, they look like they may answer your question. Feel free to ask something more specific though:

http://www.andrew.cmu.edu/course/80-413-713/notes/chap05.pdf

Also, see the Unapologetic Mathematician's post about this here:

http://unapologetic.wordpress.com/2007/06/20/the-existence-theorem-for-limits/

Sorry!

[Anonymous No.5686054]

>>5686045
Hello! I'm not even sure what a non-Gaussian prism is!

Sorry!

[Anonymous No.5686063]

>echelon form matrices
Explain this shit.

[Anonymous No.5686066]

What intrinsic property of Euclidian geometry causes pi to be the ratio between a circle's circumference and its diameter? Basically, why is pi, pi?

[Anonymous No.5686068]

>>5686063

Haha, hello! That would be like saying "turtles, what's with those fuckers". What about it particularly? Putting a matrix in echelon form is a procedure to reduce a complicated matrix to a simpler one, while at the same time maintaining most of the vital information about the matrix. It just makes matrices easier to deal with.

Hope that helps! If not, try and be more specific.

[Anonymous No.5686069]

>>5686054
http://www.youtube.com/watch?v=MDvlO9q6qWk
cmon son

[Anonymous No.5686079]

>>5686068
>Putting a matrix in echelon form is a procedure to reduce a complicated matrix to a simpler one, while at the same time maintaining most of the vital information about the matrix. It just makes matrices easier to deal with.
It seems to do the complete opposite for me.

[Anonymous No.5686083]

Taylor Series

[Anonymous No.5686085]

>>5686079
Why? You can read off a lot of data about a matrix in Echelon form very easily?

[Anonymous No.5686081]

>>5686066
Hello! Not to get even more meta, but what is pi? I suspect you are asking about why the ratio of a circle's circumference and its diameter has representation 3.14... in the base 10 number system that we're used to?

As you can probably guess, this question is too vaguely worded to have an answer, since there are so many arbitrary decisions we've decided to make when we discuss numbers.

Perhaps this may be of some interest to you:

http://math.stackexchange.com/questions/254620/pi-in-arbitrary-metric-spaces/264312#264312

Sorry I couldn't be more helpful!

[Anonymous No.5686087]

>>5686069
>http://www.youtube.com/watch?v=MDvlO9q6qWk
Haha, I still can't help you. You might like this though!

http://www.youtube.com/watch?v=0z1fIsUNhO4

[Anonymous No.5686090]

Is there a way to formalize Category Theory? Most of mathematics branches can be formalized using a first order language and a list of axioms (for example, ZFC). However, I don't know if you can do the same thing to Category Theory in general.

[Anonymous No.5686097]

Please explain the Langlands program and what it tries to achieve.

[Anonymous No.5686099]

>>5686083
Hello! Taylor series should just be thought of, in a sense, as a polynomial that, in some sense, best approximates your function at the point you expand the Taylor series at. Think about it this way. You say to me "Anon, fuck, I hate this function, it's so damn nasty." to which I say "Oh you! If you don't want to deal with that function, perhaps dealing with a polynomial might be more to your taste!" to which you say "Fuck you talking about? I want to know the exact value of my function at this point, or at the very least, the value approximated REALLY well" to which I say "Haha, of course! That's why you don't pick any polynomial, you pick the polynomial which best approximates your function near that point! This is the Taylor polynomial! If you take larger and larger degree Taylor polynomials your function's values are better and better approximated by the Taylor polynomials. If you want maximal accuracy, equality, you need to take the limit of these Taylor polynomials--you then get the Taylor series" to which you say "Kay".

[Anonymous No.5686100]

>>5686087
> mobius video
blew my mind the first time I saw it. You never realize how simple things are once you know the proper methods

[Anonymous No.5686107]

>>5685944
I want a result that says how well you can approximate a function on R^n, *and its derivatives* with polynomials.

Like this paper

http://link.springer.com/article/10.1007%2FBF02432855#page-1

but on R^n not R.

Do you know of any such results?

And it can't be for C-infinity functions because I need explicit bounds in terms of the sobolev norm on f.

[Anonymous No.5686109]

>>5686090
Not OP, but you can formalize CT in first order logic:
http://ncatlab.org/nlab/show/fully+formal+ETCS#the_theory_of_categories_23
From there you add axioms of the categories you're interested into (in that page, the axioms needed to formalize ETCS)
That said, that formalization is quite ugly, it's prettier to formalize CT as a multisorted first order theory (i.e., with more than one universe of discourse, you usually want one for objects and one for morphisms). This kind of formulation is usually done in type theory, see also this paper for more:
http://arxiv.org/abs/1303.0584
(past the title, it's pretty easily readable)

Then, there's the obvious formalizations in terms of higher category theory, of course.

[Anonymous No.5686112]

>>5686097
You know what? Every time I do one of these /sci/ question things, I always wonder if this question will be asked--it just seems like a natural one!

I don't have anywhere near the level of expertise to answer this question well. That said, I can at least say a few words, and throw some links at you.

Because I don't know what level you are at, I am going to take a somewhat lowbrow attempt at a very quick answer. Basically, in the kingdom of the representation theory of a certain class of groups, there are two classes of peoples: the algebraic and analytic representations. This means that there are two types of representations which pop up, those that come from analytic considerations, and those that come from algebraic considerations. The Langlands program, in a sense, tries to unite these two. Showing that there is a correspondence between these two classes of representations, and seeing how exactly the correspondence works. Of course, the Langlands program is far from being completed (I've had experts in the field tell me its unlikely to be done in the next 60 years).

If you'd like to read more, and you have a relatively high level of background, I would highly suggest reading this fantastic article of Gelbart

<<http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183551573>>

which segues nicely into this book:

<<http://www.amazon.com/Introduction-Langlands-Program-Joseph-Bernstein/dp/0817632115>>

Hope that helps!

[Anonymous No.5686119]

>>5686109
>http://arxiv.org/abs/1303.0584
>>5686090

I'm glad other anon was here to help. I know too little about logic to try and pretend to answer this question.

Sorry!

[Anonymous No.5686123]

>>5686107
>http://link.springer.com/article/10.1007%2FBF02432855#page-1
Hello! I am far, far from an expert on numerical analysis. I would suggest asking this at one of the many fantastic online math forums for a real answer!

Sorry!

[Anonymous No.5686120]

>>5686112
Thanks, much appreciated. One of my profs that I've known for a while has spent most of his career working on this so I've always wondered about it.

[Anonymous No.5686125]

>>5686123
ok thanks anyway

[Anonymous No.5686127]

>>5686120
Hello! This is one of those times in which it's a shame 4chan is anonymous. I'd be curious as to whom this professor is--but I don't want you to give any identifiers away.

[Anonymous No.5686133]

>>5686109
Thank you, kind anon.

[Anonymous No.5686131]

Hello,

Fixpoints seem like they are a pretty awesome tool. You work in various completely different kinds of settings (e.g. a Banach space or a complete partial order), and you have fixpoint theorems that basically have the same structures. One theorem gives you the existence and uniqueness of a fixpoint for some classes of functions (e.g. contractions or Scott-continuous functions), one theorem tells you that you reach this fixpoint by iterating the function from anywhere you want, etc.

I guess I don't really have any precise question but do you have any insight on why spaces that do not seem to have anything in common topologically still have "almost the same" theorems about fixpoints? Is it possible to intuitively justify that studying sufficient conditions for a function to have a unique fixpoint (or least fixpoint, or similar things) will provide interesting results in most frameworks?

[Anonymous No.5686141]

>>5686131
Hello! Fascinating question! I don't know precisely what you're asking. What spaces do you reference? Banach spaces? The Banach fixed point theorem is pretty intuitive.

Do you mind clarifying your question a little more? Thanks!

PS, there is the notion of a fixed-point space--a space where every continuous self-map has a fixed point. See here:

http://en.wikipedia.org/wiki/Fixed-point_space

[Anonymous No.5686163]

>>5686141
> PS, there is the notion of a fixed-point space--a space where every continuous self-map has a fixed point.

Wow, that's cool! Thank you.

I don't really know how to be more precise with my question because it is not precise in my mind either. It's just something I've been wondering without really being to formalize a proper question in my head. It's actually been hinted at me when I was introduced to http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem and the professor mentioned the similarities with the Banach one, which I was clearly not thinking about at all, being in a discrete math course and having left all my complex analysis far behind me at that point.

And that's really what struck me. I wouldn't have expected what I guess you could call topological similarities between a Banach space and something that looked like the complete opposite in terms of the "continuous vs discrete" scale.

Anyway, thank you for the link. I'll be thinking about it when my mind wanders.

[Anonymous No.5686173]

>>5686163
Haha, yeah.

PS, you didn't mention probably the most famous, and most profound of all the fixed point theorems:

http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem#Statement

Glad I could help!

[Anonymous No.5686283]

What is a natural number? How would you define N other than by {1,2,3...}?

[Anonymous No.5686309]

Explain to me functional integration.

[Anonymous No.5686354]

>>5686309
>not using 'please'
stay pleb, faggot

[Anonymous No.5686371]

>>5686283
Not OP, but Google "Peano axioms".

[Anonymous No.5686389]

OP, please give me a good intuition of what a tensor is

[Anonymous No.5686409]

What's a real number?

[Anonymous No.5686420]

>>5686409
The limit of a convergent sequence of rationals.
For example, pi=lim(3, 3.1, 3.14, 3.141, 3.1415, ...).
This probably isn't a very satisfying explanation so I'd recommend finding a real analysis textbook and reading its construction of the real numbers.

[Anonymous No.5686423]

>>5686389

[Anonymous No.5686426]

>>5686389
>>5686423
I doubt you'll find one anywhere, honestly.

[Anonymous No.5686447]

>>5686420
I should have said Cauchy instead of convergent, sorry.

[Anonymous No.5686475]

Adjunctions in category theory.I've been learning category theory (mainly for categorical logic) for a year,and still don't grasp adjunctions

[Anonymous No.5686498]

>>5685944
What are degrees of freedom, really?

[Anonymous No.5686544]

Spinors. Pleaseeeee. Specifically the geometric meaning.

[Anonymous No.5686549]

>>5686010
not OP, but can I add something to that?

I too feel very comfortable about proofs, and it comes naturally once you ask yourself: why? is that always true?

Try to prove very intuitive things, and you'll get the idea of how some proofs work:
in analysis, Rolle theorem or the mean value theorem are very intuitive.
In algebra, try proving that a projection p is orthogonal if and only if |p(x)|<|x| for all x (in finite dimension at least)
And so on and so forh.
Don't stick with only one branch of math... Try to expand your horizons

[Anonymous No.5686561]

>>5686475
Kan defined an adjunction F -| G for functions F : C -> D and G : D -> C as a bijection <span class="math">\frac{FA \to B}{A \to GB}[/spoiler] that is natural in A and B.

[Anonymous No.5686563]

>>5686561
functors

[Anonymous No.5686564]

>>5686475
Given <span class="math">F : \mathcal D \to \mathcal C[/spoiler], <span class="math">G : \mathcal C \to \mathcal D[/spoiler] Kan defines an adjunction <span class="math">F \dashv G[/spoiler] as a bijection <span class="math">\frac{FA \to B}{A \to GB}[/spoiler] that is natural in <span class="math">A[/spoiler] and <span class="math">B[/spoiler].

[Anonymous No.5686567]

>>5686564
Naturality means the following squares commute:

<span class="math">
\begin{array}{ccc}
\mathcal C(FA,B) & \overset{\sim}{\to} & \mathcal D(A,GB) \\
\downarrow & & \downarrow
\mathcal C(FA',B) & \overset{\sim}{\to} & \mathcal D(A',GB) \\
\end{array}
[/spoiler]

where the down arrows are the hom functors <span class="math">\mathcal C(Ff,B)[/spoiler] and <span class="math">\mathcal D(f,GB)[/spoiler]

[Anonymous No.5686568]

>>5686564
Naturality in A means given <span class="math">f : A \to A'[/spoiler] the square pictured commutes

[Anonymous No.5686570]

>>5686568
and naturality in B given <span class="math">g : B \to B'[/spoiler]

[Anonymous No.5686572]

>>5686568
mistake this should be <span class="math">f : A' \to A[/spoiler]

[Anonymous No.5686573]

Gauge theory and local gauge invariance

[Anonymous No.5686584]

>>5686568
>>5686570
from these naturality squares you can put A=GB in the first and run 1 from the top right through it to deduce that <span class="math">\bar f = F f \circ \bath 1[/spoiler], a similar thing with the other square gives <span class="math">\bar g = \bar 1 \circ Gg[/spoiler] where bar denotes transposing through the adjunction.

You can inspect these two bar 1's and find they are natural transforms <span class="math">1 \to FG[/spoiler] and <span class="math">GF \to 1[/spoiler], call them the units <span class="math">\eta,\varepsilon[/spoiler]. This is another equivalent way to define an adjunction.

[who fucking cares No.5686583]

>>5686498

good question +1

[Anonymous No.5686586]

>>5686584
<span class="math">\bar f = F f \circ \bar 1[/spoiler]

[Anonymous No.5686587]

>>5686586
Now you could study some simple examples of adjunctions (like free and forgetful functors) before learning more theory about them.

[Anonymous No.5686590]

>>5686354
Why are you so mean? ;_;

[Anonymous No.5686594]

>>5686389
I had a teacher once who explained tensors as 'objects that transform as tensors'. Fun guy.

[Anonymous No.5686617]

Do you think Grothendieck really enjoys the isolation?

[Anonymous No.5686624]

>>5686617
There is some value to isolation and it is his character but I think he's ill and probably struggling quite seriously.

[Anonymous No.5686629]

>>5685980
you don't learn arbitrary rules in math, you try to figure them out.

[Anonymous No.5686630]

>>5685944

Is who wants to be a millionaire simply a game of luck?

[Anonymous No.5686632]

>>5685980
I feel like that about some areas of math more than others, there's one area of math I feel really platonic about and that 'just arbitrary rules' would never taint it

[Anonymous No.5686682]

Slightly off topic here but I'll ask anyway.

I'm going to study math at University next year and was wondering if you have any tips or things you wish you had known when you started learning.

What makes a great mathematician etc?

Also what's your job/what are you studying for?

[Anonymous No.5686689]

mandelbrot set

[Anonymous No.5686690]

>>5686689
Wwhat about it?

[Anonymous No.5686695]

>>5686583
>lel i posted dat name again

[Anonymous No.5686706]

>>5686704
example?

[Anonymous No.5686704]

Why do some people who study mathematics focus on axioms and stem their arguments from that making arbitary "definitions" for the things they're talking about while other groups have a list of standardized definitions that they memorize and treat as flawless and important as axioms?

[Anonymous No.5686713]

>>5686389

Hello! This is somewhat tricky, because a tensor is entirely intuitive to me, but I fear that you mean something different than I have in mind. Namely, for me a tensor is just an element of a tensor product. So, to understand tensors we need only understand tensor products. The tensor product of two vector spaces (or modules) is just the vector space (resp. module) which turns bilinear maps into linear ones. In other words, suppose that V and W are two vector spaces. Their tensor product T=V\otimes W is the vector space such that bilinear maps VxW--->X are in one-to-one correspondence with linear maps T--->X.

In essence, the tensor product T of V and W, allows us to turn mutlinear algebra involving VxW into linear algebra involving T. This is extremely useful because linear algebra=easy and mutlilinear algebra=hard.

Anyways, that's what makes sense to me. But, as I said, when people ask questions like "what is a tensor" they usually have the physics definition in mind. Not only do I have pitifully poor knowledge of physics, but I have never understood their definition of a tensor, although I have never really tried.

I hope this was, in some way helpful!

[Anonymous No.5686717]

>>5686498
Hello! This is a good question, one I may not have the best answer to.

To me, the degrees of freedom of a set X, is roughly the minimal amount of variables need to describe X fully. If you are privy to the notion of a manifold, it's analgous to the dimension of X as a manifold--how many parameters are needed to describe the object (at least locally).

As degrees of freedom is not a well-defined (in the technical sense) idea, I don't know how much more I can say about it.

Sorry! Hope that helps!

[Anonymous No.5686718]

Do you have a reference describing the complexityof different types of integrals?

I'm currently wondering to what extend you can build programming languages to reduce the complexity of certain theorems.

And do you see any relation between geometry and statistics?

[Anonymous No.5686719]

>>5686706

Uhh, I'm having trouble thinking of a good one off the top of my head, but for example, consider the normal subgroup.

It can be defined like this.
H is a subset of the group G, g is an arbitary element in G.
H=gHg^-1
or
Hg=gH

Both are true if and only if the other is true. So, it can be said that one implies the other. However, in some cases I've had courses where they require you to memorize specifically one of those and even ask in exams to give the same definition they used and marked incorrectly if you said an alternative. I'm sure there are better examples really. Maybe talking about odd numbers as either a=2k+1 or a=2k+3 or so on.. They're equivalent and more importantly everyone already knows what you mean when you say something is an odd number, why then are specific arbitrary definitions given so much importance?

[Anonymous No.5686722]

>>5686719
that's just stupid exams being marked stupidly, nothing to do with math.

[Anonymous No.5686726]

>>5686722
So once I get into higher math I won't be required to memorize a specific list of definitions? I'll be free to use my own on the fly (assuming they're easy to understand and correct)?

[Anonymous No.5686729]

What is the meaning of identity?

[Anonymous No.5686734]

>>5686682
Hello! Me again. I am training to be a mathematician. I am going to graduate school next year (I am a senior in college).

I will try and get to your question about what makes a good mathematician later!

[Anonymous No.5686733]

>>5686682
Hey! This isn't too off-topic, this is a great question.

There is one huge tip that I have, but it is one that you should only half listen to. It is extremely easy to get caught up in the "I need to understand ALL of the prerequisites before I study subject _____". For example, you decide not to take differential geometry some term because you haven't had Lie groups, and know that knowing Lie groups would be helpful. Or, you pass up the chance to take class field theory because you really don't know local fields as well as you should. It's extremely easy to want to learn everything from the ground up. The simple truth is that you can't. There just isn't enough time in the world for you to go and learn absolutely everything needed to TOTALLY be prepared for any subject. Sometimes you really just need to say "fuck it", jump into a subject a little outside of your prereq comfort zone, and be baptized by that fire.

That said, it's also important to know when you need to learn something well. What I mean, is that in the above example, you say to yourself "screw it, I don't know local as well as I'd like to study class field theory, but I'm going for it anyways!" During your study of class field theory, you pick up the local field theory you need to know, but you don't really get a full broad understanding of the subject. This is ok sometimes! Not only did your decision to learn class field theory allow you to learn class field theory, but it also gave you a hands-on approach to learning local fields--you see where they are used in mathematics, and why they are important. This type of learn-by-seeing is great, it allows you to understand how things relate in mathematics, which is often more valuable than the actual subject itself. But, as the first sentence of this paragraph alluded to, sometimes this type of on-the-fly learning of a subject is not enough, and you need a more systematic approach to learn everything.

[Anonymous No.5686735]

>>5685944
Can you please explain to me the complexification of a vector space?

[Anonymous No.5686802]

>>5686717
Could you make it simpler? ;_;
I feel guilty for passing my stat tests and this question still haunts me.

[Anonymous No.5686809]

Can you show that lim(1+x/n)^n =e^x using e=lim(1+1/n)^n? Or at least show me to a place where I can see how thats done. Without natural logarithms if possible, I feel like that's kindof circular.

[Anonymous No.5686859]

>>5686802

Not OP, but Wikipedia says: "The number of independent ways by which a dynamical system can move without violating any constraint imposed on it, is called degree of freedom. In other words, the degree of freedom can be defined as the minimum number of independent coordinates which can specify the position of the system completely".

So, consider a ball on a 2D surface, not moving. Then its state is specified by two coordinates, x and y, or polar coordinates what have you. Now allow the ball to move. Its state no longer consists of its position, but also its velocity, which adds another two degrees of freedom; x-velocity and y-velocity. If the ball had some sort of restraint, say it had no y-velocity, then the degrees of freedom is 3; x, y and x-velocity.

[Anonymous No.5686863]

What's an eigenvalue? I know like WHAT it is, but I feel like there's tons of uses for them and I don't understand the importance of them.

[Anonymous No.5686871]

can you explain the countability and separation axioms in a way that makes them easy to remember?

[Anonymous No.5686879]

>>5686863
I'm not sure, but I think it's a value assigned to something used to describe "itself" or its "proper" value so you can do stuff to it.

[Anonymous No.5686882]

>>5686498
Degrees of freedom are the number of combinations available to an object or system. Rotation, sliding. It's the dimensional variables necessary to describe the system.

I saw it exampled on wikipedia as the configurations of a ladder. You can't just stand it any angle you want, it must be perpendicular to the wall, but can slide along the wall. That'd be one degree of freedom, one continuum of configurations.

I'm afraid I can't offer anything more technical.

[Anonymous No.5686883]

What is a dot product?

[Anonymous No.5686887]

Can you solve equations in your head by visualising them?

[Anonymous No.5686901]

>>5686882
>>5686859
Okay, thanks guy I kinda get this part, but what does it have to do with statistics? I mean, if I have some sample size, why is it when I compute for something, the degrees of freedom is (most of the time) n-1?

[Anonymous No.5686963]

Why is cosine used when calculating work and multiplying vectors? Why not sine? Why is sine used for the things it is used for? Why not any other trig function?

Shouldn't this be obvious? Why can't I get an intuitive understanding of the implications if triangle side ratios?

[Anonymous No.5687000]

>>5686963
because you are underage and you haven't been taught math properly yet.

When scalar product, cosinus appears precisely because it is defined as such... (the cosinus of an angle formed by two unitary vectors is their scalar product)
It just so happens that in the very particular case of a right triangle, you have the side ratios...

Sine is used for the things it's used for because it is defined using the triple product, and the vector product is defined using it too... the link is kind of obvious here

[Anonymous No.5687002]

Can you name a few non-babby math topics?

[Anonymous No.5687021]

>>5685944
Is that my cocaine?

[Anonymous No.5687022]

you ignored this:
>>5686718
:(

also, how to grok the differences in the different types of languages in the Chomsky Hierarchy?

Does the following claim come close:
-Finite State Machine (accepting regular languages) can't remember things.
-Pushdown automata (accepting context-free grammars) can remember a natural number.
-Linear-bounded automata (accepting context-sensitive grammars) ...can... keep somehow track of more stuff? (I imagine it can it do simple computations about it's expressions and their relations to neighbors?)
-Turing machine can all computable functions for language recognition.

[Anonymous No.5687029]

>>5687022
Not OP, actually I think OP is gone.

LBA are just Turing machines with finite memory, like real computers.
The PDA analogy is incomplete: it's a FSM that can memorize things and then recall them in reverse order, i.e. it has a stack. When you push the same thing it's like keeping a natural and the act of pushing/popping becomes adding/subtracting 1 from 1.

[Anonymous No.5687041]

>>5687029
do you only have one stack, and (I ask because {a^n b^n} is recognisable) is it the case that you can only check with the stack that you pushed and poped the same amount of times --- or can you actually save e.g. the number 9001 and then access that value later again.

And still, in which sense is the machine for the context sensitive language stronger. What does that linear space imply compared to the simple pushdown automata? What are the informations you can save and how can you access them?

[Anonymous No.5687047]

What's the highest number you can count?

[Anonymous No.5687059]

>>5687022
>-Pushdown automata (accepting context-free grammars) can remember a natural number.
no retard, {a^n b^n c^n} is unrecognizable

[Anonymous No.5687062]

>>5687041
>do you only have one stack
yes, take 2 stack U,U and move them like so ⊂| |⊃ and you have an infinite tape (a Turing Machine)

[Anonymous No.5687065]

>>5687041
You only have one stack, yes. You can recognize {a^n b^n} with one stack easily, and you only need a natural number for that: your first state is "push, if next is a repeat else go to second state", your second state is "pop, if stack is empty and no more input accept, else if next is b repeat, else reject".
But yes, you can push arbitrary values and can check the top of the stack when changing state.

The machine for context sensitive languages is the LBA, right? It can read and write data on the (finite) input tape arbitrarily, PDA can only read and write the top of the stack.

[Anonymous No.5687071]

>>5687047
I literally lold like an idiot for more than a minute without apparent reason.. Awesome, thank you.

[Anonymous No.5687074]

>>5687059
thx. but you're unfriendly.
>>5687062
thx. but I don't understand the "like so" part
>>5687065
thank you.

does anyone of you know (a reference for) computational complexity for "normal" expressions like integrals w.r.t. to the integrand?
I'm specifically interested in a hierarchy of which integrals, infinite sums and also differential equations in terms of computational requirements.

[Anonymous No.5687435]

>>5686809
[/math]
e^x=(\lim_{n\rightarrow \infty} (1+1/n)^n)^x=\lim_{n\rightarrow \infty} (1+1/n)^{nx}=\lim_{nx \rightarrow \infty}(1+x/(nx))^{nx}=\lim_{t \rightarrow \infty}(1+x/t)^{t}
<span class="math">
this works for <span class="math"> x > 0 [/spoiler].
you can prove the other cases yourself if you want[/spoiler]

[Anonymous No.5687438]

>>5686809
<span class="math">
e^x=(\lim_{n\rightarrow \infty} (1+1/n)^n)^x=\lim_{n\rightarrow \infty} (1+1/n)^{nx}=\lim_{nx \rightarrow \infty}(1+x/(nx))^{nx}=\lim_{t \rightarrow \infty}(1+x/t)^{t}
[/spoiler]
this works for <span class="math"> x > 0 [/spoiler] . you can prove the other cases yourself if you want

[Anonymous No.5687458]

whats the equivalent method of solving systems of linear equations(matrices) for non linear systems of equations

or is that a stupid question.

[Anonymous No.5687467]

>>5687059
They can remember a natural number, but in a destructive manner: calling back that memory destroys it.

[Anonymous No.5687468]

>>5687458
Usually, finding a way to transform them into a system of linear equations.

[Anonymous No.5687475]

explain irrational numbers rationally

[Anonymous No.5687481]

>>5687468

can ya give an example

you don't have to if you dont wanna.

[Anonymous No.5687485]

>>5686863
Not OP. When I think "eigenvalues", I think two things:

- How does my linear map behave when I take its successive powers? If for instance it is an invertible map and all the eigenvalues are >1, then I know the norm of the n-th power will go to infinity exponentially with n. If they are all <1, then it will go to 0 exponentially with n. Things like that.

- They are a way to represent an operation which should have dimensions n*n by only n values, in a different base. If you can do many operations in the same modified base and they all end up being on diagonal maps, they become much easier. Doing the product of two diagonal maps takes 2n operations, while in general when the maps are not diagonal, it takes roughly n^3 operations.

[Anonymous No.5687496]

>>5687475
since this was mentioned, can someone tell me if its true that there are infinitely many irrational numbers between any two rational numbers?

and are there infinitely many rational numbers between any two irrational numbers?
IF not, then does that mean that two irrational numbers can be "next" to each other, such that there are no other numbers between them?

Could somoene define " the positive number that comes after 0 " by just saying there must be an irrational number which fits teh description?

[Anonymous No.5687515]

>>5687496
> since this was mentioned, can someone tell me if its true that there are infinitely many irrational numbers between any two rational numbers?

Yes. I can even give them to you.

If you give me x and y two rationals, consider the set

<span class="math">\{x + (y-x)\alpha | \alpha \in [0,1]\backslash \mathbb{Q}\}[/spoiler]

If you already know there are infinitely many irrationals between 0 and 1, we're done, I gave you one irrational between x and y per irrational between 0 and 1.

If you're not convinced about the fact there are infinitely many irrationals between 0 and 1, but you know there are infinitely many integers and you know that there exists an irrational number z (for instance you know that z=pi is irrational), then let n be the smallest integer greater than z. Let's also assume that z is positive (if not, take -z instead, still irrational). Consider the sequence

z/n, z/(n+1), z/(n+2), z/(n+3),...

All of them are between 0 and 1. All of them are irrational, because if z/n = p/q for p and q two integers, then z=(pn)/q and pn and q are both integers, so z would be rational.

Also there are infinitely many of them.

[Anonymous No.5687519]

>>5687475
pic related

>>5687496
>Could somoene define " the positive number that comes after 0 " by just saying there must be an irrational number which fits teh description?
No, real numbers possess the archimedean property that there's no infinitesimally small or infinitely great element.

[Anonymous No.5687528]

>>5685944
how do i write a power series for <span class="math">e^x - x - 1[/spoiler]
not homework it was on the exam i just wrote (and likely failed)

[Anonymous No.5687536]

>>5687528
sorry that should be <span class="math">\frac{e^x - x - 1}{x}[/spoiler]

[Anonymous No.5687561]

>>5687536
<span class="math">
e^x/x=\frac{1}{x}+\sum_{n=0}^{\infty}\frac{x^n}{(n+1)!}
[/spoiler]
<span class="math">
\Rightarrow e^x/x-1-\frac{1}{x}=\sum_{n=0}^{\infty}\frac{x^n}{(n+1)!}-1=\sum_{n=1}^{\infty}\frac{x^n}{(n+1)!}=\sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+2)!}
[/spoiler]

[Anonymous No.5687570]

>>5687528
>>5687536
<span class="math">
e^x/x=\frac{1}{x}+\sum_{n=0}^{\infty}\frac{x^n}{(n+1)!}
[/spoiler]
<span class="math">
\Rightarrow e^x/x-1-\frac{1}{x}=\sum_{n=0}^{\infty}\frac{x^n}{(n+1)!}-1=\sum_{n=1}^{\infty}\frac{x^n}{(n+1)!}=\sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+2)!}
[/spoiler]

[Anonymous No.5687574]

>>5687528
>>5687536
>>5687561
>>5687570
fuck that shit

[Anonymous No.5687573]

>>5685944

> >>5687277
>>5687310

Write it.

[Anonymous No.5687576]

Im not retarded. its called visual thinking!

[Anonymous No.5687597]

It's a Michael Caine look-alike!

[Anonymous No.5687599]

What are schemes and why do we need them?

[Anonymous No.5687605]

>>5687599
this?

[Anonymous No.5687616]

>>5687574
is the series for <span class="math">\frac{e^x}{x}[/spoiler] one that im supposed to know?
and what happens when the starting index changes from 0 to 1? how does the -1 get absorbed into the sum to change the index?

[Anonymous No.5687619]

>>5685944
Explain to me Geodel's incompleteness theorem/s.

[Anonymous No.5687620]

Can the complex numbers be expressed on a number line?

[Anonymous No.5687628]

>tfw 18 and doing below babby-tier math
>tfw I do fine, but not excellent as in I don't have a crystal clear understanding of everything I do
>tfw I don't understand shit in this thread
>tfw my head hurts
Will I ever make it, guys?

[Anonymous No.5687656]

>>5687628
Math is a language. Dont try to understand. Just learn to read.

[Anonymous No.5687652]

>>5687616
>is the series for e^x/x one that im supposed to know?
you should know the series for e^x. you can divide that by x
>and what happens when the starting index changes from 0 to 1? how does the -1 get absorbed into the sum to change the index?
the 0th term is equal to 1. so you can just leave the 0th term out and start with the first.

I just realized, that I made it way too complicated.
<span class="math">
e^x-x-1=\left (\sum_{n=0}^{\infty}x^n/n! \right )-x-1=\left (1+x+x^2/2+x^3/6 +... \right )-x-1=x^2/2+x^3/6 +... =\sum_{n=2}^{\infty}x^n/n!
[/math
so
<span class="math">
(e^x-x-1)/x=sum_{n=1}^{\infty}x^n/(n+1)!
[/spoiler][/spoiler]

[Anonymous No.5687662]

>>5687616
>is the series for e^x/x one that im supposed to know?
you should know the series for e^x. you can divide that by x
>and what happens when the starting index changes from 0 to 1? how does the -1 get absorbed into the sum to change the index?
the 0th term is equal to 1. so you can just leave the 0th term out and start with the first.

I just realized, that I made it way too complicated.
<span class="math"> e^x-x-1=\left (\sum_{n=0}^{\infty}x^n/n! \right )-x-1=\left (1+x+x^2/2+x^3/6 +... \right )-x-1=x^2/2+x^3/6 +... =\sum_{n=2}^{\infty}x^n/n! [/spoiler]
so
<span class="math"> (e^x-x-1)/x=sum_{n=1}^{\infty}x^n/( n+1)! [/spoiler]

[Anonymous No.5687672]

>>5687620
If you take it as a set, yes, using the bijection between R^2 and R.
If you take it as an ordered field, no, that would imply complex numbers admit a totally ordered field structure.
Also, I'm redefining ordered field to mean partially ordered field here.

[Anonymous No.5687679]

>>5687656
Worst advice ever.

[Anonymous No.5687698]

>>5687662
thanks alot anon this sequences and series gobblygook bewilders me
i tried to get the answer starting from the geometric series for some stupid reason

[Anonymous No.5687729]

>>5687515
thank you very much. Do you think you'd also be able to have a go at the two follow up questions?

>are there infinitely many rational numbers between any two irrational numbers?
>IF not, then does that mean that two irrational numbers can be "next" to each other, such that there are no other numbers between them?
>>5687519
yeah that's what I thought. 0.00...001 is just zero, but still I find it hard to resolve that with the knowledge that there are infinitely many rational numbers between any two rational numbers.

I mean the fact that there are infinitely many irrationals between any two rationals must mean that the number that comes after 0 must have hte property that it is irrational, right? and if it has this property that it is irratioanal then aren't we acknowledging that it exists?

[Anonymous No.5687756]

Intuitively speaking, what is a Manifold?

[Anonymous No.5687764]

Does 0.999... = 1 or < 1?

[Anonymous No.5687761]

>>5687729
Between any two rational numbers there's infinitely many rational numbers, too. Same for irrationals. The notion of successor of a real number (or even a rational number) doesn't make any sense.

[Anonymous No.5687819]

>>5687599
I'm neither OP nor the guy currently explaining. I'm new. Let me try:

Schemes were primarily invented to generalize varieties. What's a variety? Well, it gets technical, but superficially speaking a variety is the zero set of a family of polynomials over an algebraically closed field. There happen to be different kinds of those, particularly nice ones which are called affine varieties, as well as more complicated ones called projective or more generally quasi-projective. A general variety is usually taken to be any quasi-projective variety, and those turn out to be made by patching together affine varieties. In any case, it turns out that affine varieties correspond in some strict sense (hint hint categories) to finitely generated algebras over an algebraically closed field. Alright, so the theory of varieties reduces locally to the theory of affine varieties which reduces to the theory of finitely generated k-algebras. The generalization comes in wanting to use any ring, instead of a finitely generated k-algebra. So a certain topological space with special structure (a so called locally ringed space) is associated to any commutative ring with 1, its so called spectrum. And of course it works in such a way that if we go back to the classical case we (almost) get a direct generalization (doesn't work quite, gets technical). Locally ringed spaces which are isomorphic as locally ringed spaces to spectra of rings are then aptly called affine schemes. Glueing together many such then yields what is called a scheme. So formally a scheme is a locally ringed space which has an open cover of locally ringed spaces which are isomorphic as such to spectra of rings. Now general schemes are way more general than the classical variety counterpart, but it turns out that one can characterize exactly the properties needed to obtain again the classical object. A variety is then a quasiprojective integral separated scheme of finite type over an algebraically closed field.

[Anonymous No.5687820]

>>5687819
What's nice about schemes, is that there's a certain correspondence with commutative rings, which means that much of commutative algebra can be used in studying these schemes (which as we have seen also include our classical nice varieties).

[Anonymous No.5687821]

explain the concept of hilbert space

[Anonymous No.5687828]

This is relatively simple, but oh well.

I understand why any non-zero number to the power of 0 is 1, but I still don't get how 0^0 is 1.

[tn No.5687829]

>>5687729
>are there infinitely many rational numbers between any two irrational numbers?
Yes. You can use roughly the same proof. If you give me two irrational numbers x and y with x<y:
- Call d the distance y-x between them,
- I choose n = ceil(1/d) (the smallest integer greater than or equal to 1/d),
- Now consider alpha = ceil(x*n) / n. We have that alpha is rational and that x*n <= ceil(x*n) < x*n+1. Therefore y <= alpha < (x*n+1)/n = x + 1/n <= x+1/d = x+(y-x) = y, so alpha is between x and y.

Now, consider a rational z between 0 and y-alpha. alpha+z is rational, and alpha+z is between x and y.

There are infinitely many rationals between 0 and y-alpha (every term of the 1/n sequence when n is greater than 1/(y-alpha), for instance), so there are infinitely many rationals between x and y.

>IF not, then does that mean that two irrational numbers can be "next" to each other, such that there are no other numbers between them?
>such that there are no other numbers between them
Consider two numbers x and y such that x<y. Then x < (x+y)/2 < y. There is always a real number between any two distinct real numbers.

[Anonymous No.5687890]

Lebesgue Integral
People talk about it, say that its so much better, use it in proofs, but never once have i seen this shit being actually applied
for example integrate exp(x), x=-infinity..t and 1-x*x, x=-1..1

[Anonymous No.5687901]

>>5687890
not OP but you got it: it's only useful in proofs or very very particular cases.
In most cases, the Riemann integral (the usual integral) is equal to the lebesgue integral.
And if it's not, chances are you couldn't calculate it.

Actually, you cant' even integrate sin(x)/x using lebesgue integral.

[Anonymous No.5687908]

Could you talk about infinitely small numbers/points. Every time I start to think about it my brain breaks down. It is bigger than nothing, yet smaller than everything. Is it like normal infinity, that there are "smaller" versions of infinite small numbers?

[Anonymous No.5687929]

>>5686735
Hello! Yes.

Formally, the complexification of a real vector space <span class="math">V[/spoiler] is the <span class="math">\mathbb{C}[/spoiler]-space given by <span class="math">V\otimes_\mathbb{R}\mathbb{C}[/spoiler].

More intuitively, the complexification of <span class="math">V[/spoiler] corresponds to the most natural and minimal way to make <span class="math">V[/spoiler] into a complex vector space. It is intuitively the process of "extending scalars".

Complexifications often come up when one has an object of the form "the real ____" and wants to transform it into "the complex ___". For example, the complexification of <span class="math">\mathbb{R}[x][/spoiler] is (naturally) <span class="math">\mathbb{C}[x][/spoiler]. The complexification of the real cotangent space/bundle of a complex manifold is the complex cotangent bundle, etc.

I hope that helps!

[Anonymous No.5687935]

>>5687002

Hello! I don't know at what level is "non-babby". Some classic topics learned by someone interested in number theory are: algebraic number theory, class field theory, elliptic curves, modular forms, quadratic forms. For someone interested in, say, complex geometry one learns about plurisubharmonic theory, complex analysis (go figure), PDEs, metric and differential geometry.

If you want more, I can provide you with more. I can also name really obscure advanced topics if you're just looking for those (e.g. Arakelov theory)

I hope that helps!

[Anonymous No.5687948]

>>5686863
Hello! The answers you've been given so far have been great, so let me just make a quick addition. Mathematics, by and large, is the process of attacking a difficult problem by breaking it into smaller, easier to deal with pieces, and then piecing them back together.

In particular, we often take an object X and decompose it into its simplest, irreducible pieces. For example, we often brake the natural numbers into their irreducible pieces--the primes.

Undoubtedly the simplest type of linear transformations are the scalar matrices (maps of the form <span class="math">\alpha I[/spoiler] for <span class="math">I[/spoiler] the identity matrix). So, in a perfect world of puppy dogs and rainbows, we'd be able to take any linear transformation <span class="math">T[/spoiler] and decompose it into a "sum" of these scalar operators. Unfortunately, this is not the case. Such matrices are called "diagonalizable". But, being tenacious as we are, we decide not to give up on attempting this decomposition, even though we may not always get a full result. This is the idea of eigenvalues and eigenvectors. To at least try this type of process, trying to see how the vectors that your transformation does act nicely upon (multiplies it by a scalar) have an effect on the transformation as a whole!

I hope that helps!

[Anonymous No.5687953]

>>5686883
Hello! This is a moderately vague question.

"A" dot product <span class="math">\langle\cdot,\cdot\rangle[/spoiler], on a real vector space <span class="math">V[/spoiler] is a positive definite bilinear pairing <span class="math">\langle \cdot,\cdot\rangle:V\times V\to\mathbb{R}[/spoiler].

Intuitively, you can think about a dot product as measuring a "weigthed angle" between two vectors--as measuring a combination of both how large a vector is, as well as the angle they face.

I can probably be more helpful if you're more specific.

I hope that helps!

[Anonymous No.5687962]

>>5686871
Hello! I'm not entirely sure what you mean? Do you mean like a memorization technique to remember first and second countable, and T_0-T_5?

Well, people outside of set-theoretic topology really only care about T_0,T_1, and T_2. So if this is not for a class, I'd just remember those. Anyways, let me try to explain T_0 and T_2. Think about the separation axioms as being levels of "topological distinguishability". Let me try and analogize this. Pretend that a set <span class="math">X[/spoiler] is an old man, and that the topology <span class="math">\mathcal{T}[/spoiler] on <span class="math">X[/spoiler] is <span class="math">X[/spoiler]'s glasses. In particular, poor old Mr. <span class="math">X[/spoiler] can only distinguish between points of itself if and only if he can "separate" with <span class="math">\mathcal{T}[/spoiler], or in other words, if <span class="math">\mathcal{T}[/spoiler] can tell them apart. This is necessary because a topology is built so that the only way to tell if two objects associated to <span class="math">X[/spoiler] are different is whether <span class="math">\mathcal{T}[/spoiler] can say "the first object is in this open set I can see, and the other object is not--they must be different".

With this in mind, T0 just says that the topology is a good enough set of glasses to tell points apart--the topology can tell you if two points are different.

Similarly, T2 is when the topology is a good enough set of glasses to not only tell when points aren't equal, but to tell you that they are far apart--they can't be close since I can see a set around the first point which the second point isn't in, and vice versa!

I hope that helps! I can answer more if you want--just let me know.

[Anonymous No.5687967]

>>5686887
Hello! This depends on the context. I can manipulate equations and do proofs fairly well in my head, but I am pretty terrible at arithmetic.

Does that answer your question?

[Anonymous No.5687970]

>>5687496

Hello! Someone else has already given you a good answer, so let me give another, slightly more sophisticated answer.

If <span class="math">p,q\in\mathbb{Q}[/spoiler] are distinct, say <span class="math">p<q[/spoiler], then the interval <span class="math">(p,q)[/spoiler] is uncountable. If there were no irrational numbers between <span class="math">p[/spoiler] and <span class="math">q[/spoiler] then <span class="math">(p,q)[/spoiler] would be a subset of <span class="math">\mathbb{Q}[/spoiler] and thus countable--a contradiction! In fact, this shows that between any two rational numbers (and, in fact, any two real numbers) there are *uncountably* many irrational numbers.

Hope that helps!

[Anonymous No.5687974]

>>5686729

Hello! This is fairly vague, so I don't know precisely what you're asking. Assuming you mean "identity" in the sense of algebra, then identity is intuitively something that, well, does nothing! Namely, it is an operation which does absolutely nothing when it acts on other objects.

Without further clarification, I can't be more specific. Sorry!

[Anonymous No.5687975]

>>5687047
Hello! Two cookies.

[Anonymous No.5687980]

Are spinors and tensors just higher dimensional vectors?

[Anonymous No.5687984]

>>5687756
Hello! This is a fantastic question! A manifold is a space which locally looks like Euclidean space. This is the common response, yeah? While it is true, there is one tiny piece of information missing. A manifold is a space that locally looks like Euclidean space, but such that the way in which the local Euclideaness manifests itself is tame enough so that one can piece local results together to form global results.

In other words, not only does your space locally look like Euclidean space, but there is a "nice" way to transition between these different localities. A priori, if someone said that a space looked locally Euclidean you would know that if plopped down at any point <span class="math">x_0[/spoiler] of the space, nothing would look much different than Euclidean space, at least in close proximity to you. That said, if you start walking, you might leave the locality of <span class="math">x_0[/spoiler] that looks familiar. Now, don't be too worried! You will just be entering the locality of some other, new point, say <span class="math">x_1[/spoiler], which also looks Euclidean. The only fear is that transitioning from the locality of <span class="math">x_0[/spoiler] to that of <span class="math">x_1[/spoiler] may be wholly unsettling. A manifold is a space where this isn't true--not only do things look locally like higher dimensional home, but moving between these localities is not too bumpy.

I hope that helps!

[Anonymous No.5687992]

Are Lp-spaces "flat"? What's the definition of flat space?

[Anonymous No.5687996]

>>5687619
Hello! Allow me to disregard your question, and answer the, at least to me, more intuitive Godel completeness theorem.

A theory T is nothing more than a language where you have a certain set of symbols, for which you can decide whether or not certain sentences in these symbols are "true" (follow from the theory).

A model M for T is then a particular realization of the set of axioms laid out in T. Say, if T was the theory of rings, then a model for T would be an actual ring!

Now, suppose that you had a sentence <span class="math">\varphi[/spoiler] in your theory, and you wanted to prove that <span class="math">\varphi[/spoiler] is true. There is one "obvious" way to prove that the theorem is true--to formally deduce it from the axioms in T.

That said, there is another way one may try to attack proving that <span class="math">\varphi[/spoiler] is true. Namely, for each model <span class="math">M[/spoiler] of <span class="math">T[/spoiler] let <span class="math">M(\varphi)[/spoiler] be the truth value of the statement <span class="math">\varphi[/spoiler] in the model <span class="math">M[/spoiler]. The first such proof is called syntactic, and the second such "proof
is called semantic.

For example, take, once again, the theory <span class="math">T[/spoiler] of commutative rings. The statement <span class="math">\varphi[/spoiler] may be <span class="math">(x+y)^2=x^2+2xy+y^2[/spoiler]. Now you could prove this syntactically just by using the ring axioms, it follows straight from the axioms. You could instead that <span class="math">(x+y)^2=x^2+2xy+y^2[/spoiler] for every <span class="math">x,y\in R[/spoiler], for every commutative ring <span class="math">R[/spoiler].

Intuitively, a syntactic proof is one where you prove it just using the symbols and axioms of your theory--you prove it independent of what ring you're working in. A semantic proof is one where you prove it in the case of every single specific ring, but each time you prove it, you make reference to this specific ring you are using.

[Anonymous No.5687999]

>>5687996
(continued)

It is obvious that <span class="math">\varphi[/spoiler] being syntactically provable implies that <span class="math">M(\varphi)[/spoiler] is true. The shocking content of Godel's completeness theorem is that the converse is true. If you can prove that a statement is true about every specific model of a theory, maybe each time of which you use facts specific about the ring you are working in, then you can prove it in a way that DOES NOT mention any specific ring.

Truly amazing.

I hope that helps!

[Anonymous No.5688005]

>>5687599

Hello! As you have already been given a fairly technical answer, let me just give you a token answer, simple but powerful. Schemes are the manifolds of algebraic geometry.
>>5687819


If you have any other questions, feel free to ask!

[Anonymous No.5688008]

>>5687821
I have explained this in another thread I did like this. There's no way to recover such a thread, is there?

[Anonymous No.5688010]

>>5687828
Hello! My answer is definitely a function of how you define <span class="math">a^b[/spoiler], but let me try one on you.

If <span class="math">m[/spoiler] and <span class="math">n[/spoiler] are two natural numbers, then <span class="math">m^n[/spoiler] is the number of functions <span class="math">\{1,\ldots,n\}\to\{1,\ldots,m\}[/spoiler]. Thus, in this sense, we see that <span class="math">0^0[/spoiler] is the number of functions <span class="math">\varnothing\to\varnothing[/spoiler]]--which there is exactly one!

I hope this helps!

[Anonymous No.5688011]

>>5687984
Manifolds describe transitions?

ooor can they be something as them selfs?

[Anonymous No.5688013]

>>5687992

Hello! I have no idea. Could you give some context? I have never heard the concept "flat" applied to Hilbert spaces?

Sorry! Please report back--I'd be interested to learn what is meant by this.

[Anonymous No.5688014]

>>5688011
Hello! I am not entirely sure what you mean by this? Manifolds are spaces which are cut up into overlapping pieces, each piece looking like Euclidean space, and such that the overlaps (transitions) are nice.

Does this answer your question?

[Anonymous No.5688016]

What kind of syntax u use? Like matlab? Theres some more general form?

[Anonymous No.5688017]

>>5688010
Me again. That LaTeX screw up should read "<span class="math">\emptyset\to\emptyset[/spoiler].

[Anonymous No.5688021]

>>5687597
Hello! Haha, that is very funny. That is JP Serre, arguably one of the greatest mathematicians of the twentieth century. See here:

http://en.wikipedia.org/wiki/Jean-Pierre_Serre

[Anonymous No.5688025]

>>5688021
>That is JP Serre, arguably one of the greatest mathematicians of the twentieth century.
Any open lessons out there?

[Anonymous No.5688028]

>>5688025
Hello! What do you mean by this?

[Anonymous No.5688029]

>>5688028
teaching, talking about hes work, etc.

[Anonymous No.5688033]

>>5688029
Ah. This is an absolute classic:
(1/3):

http://www.dailymotion.com/video/xf88b5_jean-pierre-serre-writing-mathemati_tech#.UWyal6vSM5s

(2/3):

http://www.dailymotion.com/video/xf88en_jean-pierre-serre-writing-mathemati_tech

(3/3):

http://www.dailymotion.com/video/xf88g3_jean-pierre-serre-writing-mathemati_tech

[Anonymous No.5688037]

How would you go about creating a model wherein a self-correcting mechanism comes into being by natural evolution and ends up being robust?

[Anonymous No.5688043]

>>5688037
By starting with nothing

[Anonymous No.5688047]

>>5688037
Hello! I have no idea what the majority of those words mean!

Sorry!

[Anonymous No.5688049]

Can you do siple dissection of SU groups ?

[Anonymous No.5688055]

>>5688049
Simple dissection? Does SU groups mean special unitary groups? Can you be more specific please?

[Anonymous No.5688061]

>>5688013
I have no idea either, but the n-dimensional Euclidean space is the L2-space over R^n and it's called a flat space (as in not elliptical/hyperbolic/curved)
I was wondering whether there's a formalization of that and if all Lp spaces are "flat". Probably my question doesn't make any sense, but how do you formalize the curvature of a (metric? normed?) space?

[Anonymous No.5688067]

>>5688055
yes. special unitary groups are groups.. of what exactly?

[Anonymous No.5688091]

Cramer's Rule

[Anonymous No.5688133]

I think I'm in love with OP...

But I have a question!
Why must every first order formal language have a dyadic relator, the "="?
Functors and constants are optional, infinite variables are mandatory, okay. But relators, at least you must have the "=". Why?

[Anonymous No.5688150]

Can you recommend a good book for learning class field theory? I plan on learning it next year, but I haven't figured out what book to use yet.

[Anonymous No.5688156]

>>5688133
>But relators, at least you must have the "=". Why?
Do you? It's a convention to include =, but it can be omitted just fine.

[Anonymous No.5688157]

>>5688150
in this vein, I need to learn differential geometry this summer.

I've already been working my way through Munkres for the topological background, but can you recommend me a topic specific book?

[Anonymous No.5688185]

>>5688008
Some third party websites archive /sci/. I used to visit installgentoo.net's archives, but it appears to be down.

[Anonymous No.5688197]

>>5688047
It doesn't matter to what the terms refer.

It's a question that really has no answer.

[Anonymous No.5688210]

>>5688197
I don't know, I think that while the vocabulary anon used is pedantic and vague, it may be a valid question. Maybe what he means is something like "You are given a task to solve with a machine, and you can use rules that mimic evolution by evaluating the performances of the machine and breeding in such a way that new generations of machine are in a sense closer to the good machines from the previous generation than to the bad ones. Some randomness must be allowed for "bio-diversity". How do you design your first machine and your set of rules so that the evolution process converges to some kind of stable optimal point?".

If I'm understanding this right, then many iterative optimization algorithms can be described under this framework.

[Anonymous No.5688242]

I realize this question is considered low-level, especially when compared to all these other questions being asked, but could you please explain to me what Fourier Transforms are, and how each part of it works? I've tried reading into it, but to no avail; your simple explanations will really clarify things.

Thanks

[Anonymous No.5688255]

WHY school just want us sitting and boring watching some guy typing that A = A .... i hate classroom

[Anonymous No.5688292]

" Anonymous 04/15/13(Mon)22:22 No.5688280
>>5688274
A hole is quite ambiguous when you want it exist in 3D space. What if you hit the hole side on? etc
>>
Anonymous 04/15/13(Mon)22:27 No.5688286
okay, look
I'm not sure if you've checked out metaphysics or not but this is what I can offer
it has recently been discovered that physical matter is only an effect after a certain frequency has been achieved, meaning that everything is pure sound if that abstract shit makes any sense.
But the thing is, if this has been discovered and theorized(if it were true) on Earth, then a black horrrr would not be a hole.
This analogy is derived by Einstein's theory of the 4 relative dimensions and some of the reactions to it that have been publicized.
A black inverse, is what I call it will be the break down of energy into a cycle that which is directed inward, by the inverse following certain rules, just as a frequency attained on Earth for solid matter and life, it creates a realm that "disintegrates" "matter" and by hypothesis is recycled through the inverse grid (resemblance of a path shown in theory) and sent back out into the "other side", a real question to ask, would be does the entire item come back in something can be valid for speculation as an item and how much comes back out."

[Anonymous No.5688291]

>>5688067
Hello! The <span class="math">n^\text{th}[/spoiler] special unitary group, often denoted <span class="math">SU(n)[/spoiler], is the subgroup of <span class="math">\text{GL}_n(\mathbb{C})[/spoiler] consisting of unitary matrices (i.e. isometries, or matrices whose inverse is equal to their conjugate transpose) which also have determinant one.

These can be realized as the matrices which stabilize the natural inner product on <span class="math">\mathbb{C}^n[/spoiler]. From this it's trivial to see that the special unitary group is a closed subgroup of <span class="math">\text{GL}_n(\mathbb{C})[/spoiler], and thus by the closed subgroup theorem, is a Lie subgroup.

What else would you like to know about it? I hope this helped!

[Anonymous No.5688293]

>>5688291
Hello, me again. I don't know what LaTeX on here is weird. The first unidentified TeX command is nth, and the rest are GL_n(C).

[Anonymous No.5688296]

>>5688061
Hello! I still am a little lost. I know what it means to have a flat metric on a smooth manifold, but I haven't the faintest idea what this would mean in this context. Do you have a pointable-to reference where I can see this word used in context?

Thanks!

[Anonymous No.5688300]

What are your favourite math books?

[Anonymous No.5688303]

Laplace transforms

[Anonymous No.5688304]

>>5688091

Hello! Cramer's rule is a fantastically simple, but fantastically useful theorem/fact. It tells you that you can express the entries of the inverse of a matrix as determinants of minors of the matrix divided by the determinant of the matrix.

Not only does it have fantastic practical applications, in finding the inverse of matrices, but it has many interesting theoretical applications. Here are two that immediately come to mind:

1. Cramer's rule, the explicit form it gives to the inverse of a matrix, is what allows one to show that if one starts with a smooth function and takes a regular point, then the inverse function theorem actual gives you a smooth inverse locally, instead of just C^1.

2. Another way of phrasing Cramer's rule is that Aadj(A)=adj(A)A0=det(A)I where adj(A) is the adjugate matrix (google it!). This allows one to generalize the classic result "a matrix with entries in a field is invertible if and only if its determinant is nonzero" to "a matrix with entries in a commutative ring is invertible if and only if its determinant is a unit".

I hope this helped!

[Anonymous No.5688311]

>>5688296
I think that's exactly what I want to know. What's a flat metric?

[Anonymous No.5688313]

>>5688133
Haha, oh you!

Are you asking why a theory in first-order logic must always include the relator =? Hmm, I am not sure that I am well-equipped to answer this. I have dabbled in model theory, but mostly in the context of number theory. To me it seems as though equality is the baseline operatrion, the one thing which remains constant throughout all of matheamtics, so it seems natural to include it every theory.

Does that, at all, answer your question?

[Anonymous No.5688324]

>>5688150

Hello! Great question. There are are many fantastic texts, depending on how you want to attack the subject.

In particular, there is the older approach of attacking global class field theory first, this involves some sophisticated analysis for example, and then deducing local results from this. The more modern approach does local class field theory first, mostly using the sledgehammer that is Tate cohomology, and then deducing global results by reducing to the local case.

My personal bias, and this may just be because that is how I learned/am learning it, is to attack the local--->global approach. To me that makes more sense. For this approach, I either suggest the fantastic notes of Milne (freely available here: http://www.jmilne.org/math/CourseNotes/CFT.pdf)) or the book called "Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics" by Falko. I would actually opt for the second, unless you have a good background in valuation theory/local fields.

If you really want to attack CFT from the more historical point of view, I would suggest Nancy Childress's book on the subject. It is a concise, novel, and fairly cleaned up account of the historical way of attacking CFT.

If you are looking for something different than above, you can try Iwasaw's famous text on local class field theory. He attacks the problem not via the machinery of group cohomology, but by the somewhat novel use of Lubin-Tate Formal Group Laws. This treatment, while technically less demanding, is definitely less enlightening (this is how Milne proves some of LCFT in his first chapter). I would definitely read it though if you get a chance.

Please feel free to ask any followup questions!

[Anonymous No.5688327]

hey nerd, calculate how many punches its going to take for you to give me your lunch money

[Anonymous No.5688328]

>>5688311

Hello! Well, in short, it's a Riemannian manifold whose curvature form is identically zero.

I don't think this applies to L^p space, since it is not even a topological manifold!

[Anonymous No.5688330]

>>5688327
Hello! 0 I'd buy you lunch, and then we could chat about math.

[Anonymous No.5688333]

>>5688313
> equality is [...] the one thing which remains constant throughout all of matheamtics,
Equality is the least constant concept in mathematics, everyone cares about different kind of equalities.
(unless you mean equivalence relations in general)

[Anonymous No.5688334]

Are numbers platonic forms?

[Anonymous No.5688339]

>>5688333

No, I mean actual equality. In most of the standard theories, even though there is a notion of "equality" which isn't actual equality, similar to the notion of isomorphism in a category, there is always actual equality. Internally we always want to discuss when two things are equal. No?

[Anonymous No.5688343]

>>5688157
Hello! Sure, I'd love to? Which Munkres book exactly--calculus on manifolds? It would be helpful to know what level of differential geometry you are hoping to learn. Riemannian geometry? Curves and surfaces? etc.

Thanks

[Anonymous No.5688345]

Who do you think is the greatest (or smartest) pure mathematician of all time? And how old are you?

[Anonymous No.5688348]

>>5688345

That is one of the oldest, and most contested questions discussed at tea time at every university. I honestly don't know. There are standard answers like Grothendieck, or Von Neumann, or Gauss. I really am not sure, to be honest with you. I, personally, was always impressed with how consistently ahead of his time Hilbert was.

I am a senior in undergrad--you can do the math :)

[Anonymous No.5688358]

Prove that <span class="math">a^2 = b^2 \implies a = b \lor a = -b[\math] using only basic properties of fields.[/spoiler]

[Anonymous No.5688362]

>>5688358
Prove that <span class="math">a^2 = b^2[/spoiler] implies <span class="math"> a = b \lor a = -b[/spoiler]

[Anonymous No.5688366]

>>5688343
Just "Topology"
I quite like his writing style though, I might look into his "Analysis on Manifolds" unless you have something better in mind.

Specifically, I want to learn it for the applications to optimal control theory (and to a lesser degree, the physics applications), so that I can contribute on a level beyond rudimentary experimentation in the lab I work in. I'll be a senior undergrad next year, so I have a little experience in analysis, algebra, and topology.

I've heard good things about Munkres' and Spivak's books. I need to find something that's ideal for self study at a fairly basic level though.

[Anonymous No.5688367]

>>5688362
Hello! Without further qualification, this is not true! For example, in the ring R[x,y,z]/(x^2,y^2,z^2), one has that x^2=y^2=0 but x does not equal \pm y.

I hope that helps!

[Anonymous No.5688371]

>>5688339
There's no absolute notion of equality, so you'd have to settle on a formal system strong enough to formalize most mathematics and use its equality.
You'd think that's what we do with ZFC, but we really don't: we treat the set of natural numbers as one thing but there's more than one construction of them, same for real numbers and other objects. Most mathematics is foundations-invariant.

[Anonymous No.5688375]

>>5688366
Hmm, this is tough. As I may have mentioned in this incarnation of this thread, I am not, by any means, knowledgeable about applied math. Thus, I am not exactly sure what you need.

If you're looking for a nice, simple introduction to differential geometry of curves and surfaces, that has a gentle caress, I would suggest the book Elementary Differential Geometry by Pressley.

Take a gander at that, and see if it looks to your linking.

Hope it help!

[Anonymous No.5688377]

>>5688367
Sorry, got lost in this misformatted post.

a and b are elements of a field, so that example shouldn't work?

[Anonymous No.5688380]

>>5688371

Hello! I hate to say this, especially when you have already professed your love for me, but you are quickly and steadily moving out of my league. The amount of logic I know extends, as I've said, to some basic model theory I've used in studying number theory. We are making a beeline towards something resembling descriptive set theory, something I know absolutely nothing about. So, unfortunately, I don't know what to tell you.

I'm sorry :( If you have something less logicy, I might be able to chat with you though!

[Anonymous No.5688382]

>>5688377
Ah! Ok. Just note that a^2=b^2 implies that a^2-b^2=0, but a^2-b^2=(a-b)(a+b) in any commutative ring. If you're in an integral domain, so for example a field, this implies that either a-b=0 or a+b=0. I think you can finish from there :)

[Anonymous No.5688388]

>>5688382
Thanks, I feel silly now that you mention that!

[Anonymous No.5688391]

>>5688380
I'm not the guy that asked the question (but I love these threads), I just wanted to be pedan-I mean, to clarify things.

[Anonymous No.5688405]

>>5687628
Hello! Don't be discouraged! I was very behind in math in high school, like VERY behind, but now I am..well..not so behind. Never give up my friend :)

[Anonymous No.5688408]

You helped me once, so fuck it, why not again. This is a bit more general though.

I'm trying to understand how solving for a mixed strategy nash equilibria works in a two player normal form game.

I understand the arithmetic procedure typically employed (set the probability of you choosing one outcome so that your opponent would receive the same expected payoff regardless of what they chose).

I have no idea why this works though. Normal form games aren't adversarial, right? So why does the row player choose his probability based on the column player's expected payoff?

[Anonymous No.5688413]

>>5688408

Hello! I apologize profusely, but I know nothing of game theory. I would suggest that you post this question at one of the many fantastic online math forums to get a real response.

I apologize!

[Anonymous No.5688420]

>>5688413
No worries mate, great thread!

[Anonymous No.5688424]

Is point-set topology poison for the mind? Why would you ever want to do point-set topology?

[Anonymous No.5688429]

>>5688424

Hello! Haha, that is a fair question. Point-set topology is, to most people, one of the very few classes they will take where it is purely a tool. Nothing you learn in a point-set topology class is of much real interest (there are SOME things, but most people would say the majority of PS topology, at least at the basic level, is really of novel interest). That said, the tool is extremely powerful. Whether you do algebra, analysis, algebraic topology, number theory, differential goemetyr, whatever, pointset topology is implicit in most things you do. Ideas like compactness, connectedness, separation axioms, etc. are so pervasive, that to not learn them, is to miss large part of the modern mathematical vernacular.

TL;DR Buck up--it'll be worth it. Besides, there are *some* interesting results in PS topology (e.g. every compact space is the image of the Cantor set, Tychonoff's theorem, Bing's metrization theorem, etc.)

I hope that helps!

[Anonymous No.5688460]

OP, I'm not sure if you missed it, but:

>>5688242

[Anonymous No.5688470]

>>5686015
>>5686032
Sneaky. You could also use the binomial expansion of the limit definition. The first term of the expansion will cancel the constant you get from factoring e^(x) in the numerator and the second term will cancel the denominator and all other terms go to 0 when taking the limit going to zero.

[Anonymous No.5688487]

>>5688334

you missed this one as well, OP

[Anonymous No.5688490]

>>5688460
Not the poster, but I'd love to read this as well. OP is great at explaining things.

[Anonymous No.5688502]

If I have all the derivatives of a function at a point, is there a way to reconstruct the entire function?

[Anonymous No.5688510]

>>5688502
its called a Taylor series. look it up.

[Anonymous No.5688525]

>>5688510
I mean without taylor / infinite sums

[Anonymous No.5688553]

>>5688324
Thanks! That book by Falko looks really useful (and cheap, too). I was already aware of Milne's notes, and I have a little background in local fields, but I might start with Falko anyway.

[Anonymous No.5688572]

What do you need to know in order to guess a function (polynomial or otherwise) based on a set of points in said function and how many points are needed?

[tn No.5688577]

>>5688293
jsmath lacks a lot of stuff. \mathrm is a better alternative than \text for inserting non-italicized letters which are not text, within an equation (and works in jsmath).

The other things missing are any convenient way to do tabulars / arrays, and the fact that if you write a very long formula without putting spaces in your code (for over 35 successive symbols), 4chan inserts a "you're allowed to linebreak here" special symbol in the middle of the formula before passing it to its server-side jsmath processor, which itself just replaces that symbol with a space, so you often end up with stuff like "\frac{1}{2}" being turned into "\fra c{1}{2}" and errors telling you that "\fra" isn't recognized.

[Anonymous No.5688582]

>>5688408
Not a game theory expert by any mean, but I think it's intuitive to understand why, if there is an equilibrium, applying this strategy would converge to it. Basically, the equilibrium must be a fixpoint for these iterations. Under some kind of continuity argument, you can most likely prove that it converges to it.

I have no idea what the arguments are to prove that there exists such a fixpoint in the first place, though.

[Anonymous No.5688602]

>>5688460
>>5688490
Hey friends! I am not entirely sure what what level you are asking this question. If you are looking for a somewhat sophisticated answer, I can't do any better than Tao, see here: http://www.math.ucla.edu/~tao/preprints/fourier.pdf

If this does not satisfy your needs, let me know!

[Anonymous No.5688606]

>>5688577
Thank you very much friend!

[Anonymous No.5688605]

>>5688334

Hello! Besides knowing that platonic forms were due to Plato (go figure ;) ) I know nothing of them--I vaguely remember something from my Freshman philosophy class. I don't think I can say anything insightful about this, I apologize!

Sorry!

[Anonymous No.5688637]

>>5688429
Yeah, but why would you do *point-set* topology? I thought topology was this arcane unfathomable thing until I discovered pointless topology.

[Anonymous No.5688638]

>>5688637
Hello! I can't tell if you're being serious, since there is a legitimate way of doing "pointless" point-set topology.

[Anonymous No.5688648]

What is étale cohomology?

[Anonymous No.5688650]

>>5688525

Sounds like you would have to integrate to get back to your original function; you would lose constants, the only way to get those back would be to have initial conditions. IE: If someone hits a baseball with velocity components V = 10m/s i + 10m/s j. You could reconstruct a position vs. time function starting by integrating your acceleration / time function.

a(t) = G
v(t) = Gt + Constant <---- reconstruction begins here

p(t) = 1/2Gt^2 + C*t + K<---- initial position

[Anonymous No.5688676]

>>5688648
Hello! This is, as I'm sure you are aware, an extremely difficult question. One, sadly, I cannot answer at this point (hopefully soon!). I understand that it is the "correct" cohomology theory for varieties.

I have found this page, and links contained therein, to be more than helpful, I hope the same can be said for you:

http://mathoverflow.net/questions/6070/etale-cohomology-why-study-it

[Anonymous No.5688731]

>>5686097

The Langland's program is a huge subject with many branches, but from the point of view of a classical algebraic number theorist the goal is to generalize Gauss' law of quadratic reciprocity.

Here is the motivating problem: given a polynomial f(x) with integer coefficients, how many roots does f mod p have, as we vary p? For example, if f(x)=x^2-m, there is 2 roots if m is a square mod p, and 0 roots if it isn't (or 1 root if p divides m.) Gauss' law of quadratic reciprocity tells us which possibility occurs, in terms of p mod 4m.

What about more general f? If f is abelian (that is its Galois group is an abelian group) then class field theory gives us an answer: there is an integer N such that the number of roots of f mod p only depends on p mod N.

Langlands has a conjecture (a special case of his reciprocity conjecture) for what happens for a general polynomial f. It involves the existence of an "automorphic form" attached to f which "knows" the answer to our original question. You might imagine something like this: you solve Laplace's equation on some hyperbolic manifold and compute a Fourier expansion of the solution. The "pth Fourier coefficient" tells you the number of roots of f mod p.

[Anonymous No.5689066]

Which lectures do you visit, which books do you read?
What are you currently working on?
What is your area of interest and what do you want to do?

Also, these questions:
>>5686718

[Anonymous No.5689339]

>>5688602
>http://www.math.ucla.edu/~tao/preprints/fourier.pdf


Thanks! But can you explain this conceptually, because I don't know too much calculus?

[Anonymous No.5689358]

Can you explain linear transformations, spans of it, in a simpler way?

[Anonymous No.5689464]

could you give me a sort of "learning roadmap" (topics to learn, books to read) in order to have a good understanding of the representation theory of lie algebras? for reference, the only relevant subjects i've studied are undergraduate topology and general algebra at the graduate level (i've read most of dummit & foote)

[Anonymous No.5689482]

Is it possible to divide by zero?

[Anonymous No.5689530]

>>5689339
the cool things about the fourier transformation is, that it turns differentiation into multiplication (in the sense, that <span class="math"> \mathit{F}(\frac{\partial }{\partial x}f)_{(y)}=iy\cdot F(f)_{(y)} [/spoiler] and that this is a reversible action, so you don't loose information.
this way you can solve a whole bunch of differential equations
If you have done some linear algebra, then F^{-1}\left (\frac{\mathrm{d} }{\mathrm{d} x} \right )F is conceptionally the same as diagonalising a matrix.

[Anonymous No.5689535]

>>5689339
the cool things about the fourier transformation is, that it turns differentiation into multiplication (in the sense, that <span class="math"> \mathit{F}(\frac{\partial }{\partial x}f)_{(y)}=iy\cdot F(f)_{(y)} [/spoiler] ) and that it's a reversible action, so you don't lose information.
this way you can solve a whole bunch of differential equations
If you have done some linear algebra, then <span class="math"> F^{-1}\left (\frac{\mathrm{d} }{\mathrm{d} x} \right )F [/spoiler] is conceptionally the same as diagonalising a matrix.

[Anonymous No.5689674]

>>5687628
the longer you work at it the easier it'll become, things tend to make more sense when you visualize them in different ways

[Anonymous No.5689731]

>>5689464

Hello! Before I attempt such an answer, would you mind telling me why? That is a very specific, and uncommon desire.

[Anonymous No.5689753]

I'm doing ring theory and I'm having trouble understanding the annihilator ideal. The textbook only has a small blurb on it and I have no idea wtf it's saying. Could you give me any intuition for this ideal, or is there any advice you could offer for ideals/ring theory in general?

I've a number of vague questions due to having a good intuition for group theory but a poor one for ideals. Is every normal subgroup of the additive group in a ring also an ideal?

By the way, my textbook uses the convention that a ring is a ring with multiplicative identity, rings without it are referred to as 'general rings'. As I understand others also refer to 'general rings' as rngs, 'psuedo-rings', or even just rings and instead specifying the prior as 'ring with identity'. Above I'm speaking only about rings with identity.

[Anonymous No.5689762]

Why the fuck does calculus get easier as you go?

I got a C in calc 1, a B in calc 2, and I'm pretty sure I'm going to verge on an A in calc 3.

What the fuck? I haven't changed anything.

[Anonymous No.5689819]

>>5689753
Not OP. Atiyah has some good examples in the opening playing with the annihilator to give you the feels for it. Think about where 0 divisors go in ring homomorphisms.

My intuition about ideals comes obviously from a normal subgroup, cf multiplication and conjugation. What do each of these have to do with self-maps? What wouldn't get preserved if you tried to mod out by an additive subgroup that didn't absorb multiplication? A subgroup that wasn't normal? Thinking about the constraints with ones I already know help me "feel" these things.

[Anonymous No.5689822]

>>5689753
Also remember as a group, all additive subgroups are normal, it's the absorption of multiplication that distinguishes it as an ideal.

[Anonymous No.5689831]

>>5685944
GR. Seriously, mathematically.
INB4 HURR NOT DOING PHD COMBINATORICS 300k
Ok, if you insist, Groups Rings and Modules

[Anonymous No.5689887]

>>5689762
In my "experience" (I read around on the internet, compare course descriptions and course notes between universities, compare multiple offerings of the same topic at different universities...) calc 3 either means
a) multiple integrals, partial derivatives, blah blah blah very computation-heavy
or
b) basic topology and rigorous generalization of calculus from 1 variable to 2 or more, basically the final stepping stone before taking real anal.

is your calc course a) or b)? did you have to write proofs about differentiability? or prove that a limit DOES exist using multivariate squeeze theorem? when you got to the integrals section were all the theorems proved or did everything just start saying 'the proof is beyond the scope of this course?'

does that answer your question?

[Anonymous No.5689912]

>>5689819

Thank you very much for your personal insight. This all seems like a good direction to head in. I do have a couple questions though. What does cf multiplication refer to? I don't think I've ever heard this term anywhere. As far as groups go I understood it didn't make sense fundamentally to try and mod out by a non-normal subgroup, as far as an additive subgroup that doesn't absorb multiplication however, I'm having trouble thinking of any examples. The textbook I'm dealing with has a few examples, but they're very general, not specific at all and no examples of flawed attempts are shown. Could you give me the name or some more info on the book by Atiyah you mentioned? I think we're only using the one I have because a professor at our University wrote it.

>>5689822

>Also remember as a group, all additive subgroups are normal, it's the absorption of multiplication that distinguishes it as an ideal.

This isn't the case though, not every subgroup of C6 (eg. Z/Z6={[0], [1], [2], [3], [4], [5]}) for example is normal. Only {[0], [2], [4]} and {[0], [3]} (C3 and C2 respectively) are.

I think an issue might be that I don't fully grasp what people mean by "absorb multiplication". I've seen this explanation used on the internet, but never in my textbook.

[Anonymous No.5689968]

>>5689912
>quick thing: those are the only subgroups of Z/(6), and if G is abelian H<G, gH=Hg, so all subgroups of an abelian group must be normal, with respect to the addition

The "multiplication" is obviously mult in the ring (def of ring with unity is group under +, monoid under "."). Much like if H normal gHg^-1 =H, multiplication on all elements in an ideal (note- multiplication on an abelian group is an endomorphism, and a ring can be identified as a subset of the rng of endomorphisms on an abelian group quite easily, cf self maps by conjugation for groups), also keeps the ideal fixed.

That is, like g*H=H for all g iff H normal (* conjugation), we also have r.I = I for all r iff I ideal ("." The ring multiplication). An ideal absorbs the multiplication of the ring by being fixed under it (this multiplication by each elemt endomorphic as well), just as a normal subgroup absorbs conjugation by the group, by staying fixed.

Atiyah is just intro to comm algebra.

Hope this is helpful i'm hobbyist so there're probably better (less redundant) ways of saying this

[Anonymous No.5689971]

>>5689968
Typing on a phone, clearly

[Anonymous No.5689985]

>>5689912
Sorry last accidental bump, cf just means confer, which is just a Latin abbreviation for bring together or over, effectively just meaning "compare". As I said typing on a phone saving characters and aggravation.

[Anonymous No.5690044]

>>5685980

You're not learning arbitrary rules, you're discovering the rules that were always there.

[Anonymous No.5690049]

>>5686006

i lol'd

[Anonymous No.5690798]

>>5689731
I'm very interested in the subject. i took a sort of investigative course that covered the representations of sl(2_C) and sl(3_C), and i really liked the methods involved, specifically the interplay of linear algebra, group theory, and some topology. i just want to learn more about the subject.

[Anonymous No.5691171]

>>5685944
What is a "moment"? E.g., wikipedia
> The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments.
wut

[Anonymous No.5691187]

[MATH]\frac{d}{dx}\left(\cos^{-1}\left(sin{x}\right)\right)[/MATH]

find x

[Anonymous No.5691190]

<span class="math">\frac{d}{dx}\left(\cos^{-1}\left(sin{x}\right)\right)[/spoiler]

find x

[Anonymous No.5691205]

>>5687967
Do you actually see them or are you just able to do it?

For example I can solve some algebraic expressions and integrals in my head, but bits of information keep sliping away and it's difficult to remember.

However my mum, who isn't a mathematician can solve a lot of complex algebraic expressions in her head because she can actually "see" the equations in front of her. So if she can solve it on paper, she can solve it in her head.

I was wondering how important this skill is to have, whether you can do it and whether you can teach yourself to do it?

Thanks.

[Anonymous No.5691207]

>>5691171
laplace space is a frequency plane defined by real and imaginary axes. the real component is the moment, in angular frequency omega, while the imaginary component is the phase shift kai of the waveform by way of Euler's formula.

[Anonymous No.5691211]

>>5691207
now it became so much clearer.

[Anonymous No.5691213]

>>5686015
y = e^x
ln(y) = ln(e^x)
ln(y) = x
1/y*dy/dx = 1
dy/dx = y
y = e^x.

[Anonymous No.5691215]

What's a good book to work through as an introduction to group theory for a physics major?

[Anonymous No.5691216]

Could you answer my question about how to be a great mathematician? Any tips you have will do just fine.

[Anonymous No.5691222]

>>5691215
What difference does being a physics major make

[Anonymous No.5691223]

>>5691222
Physics majors just play with cayley diagrams and basic principles of group theory, they don't do proofs or anything.

[Anonymous No.5691224]

>>5691223
Oh interesting...

[Anonymous No.5691225]

>>5691223

not to mention Lie groups, pretty important

[Anonymous No.5691514]

>>5691190
Cow alien circus butterfly x panda zoo entropy

Find x

[Anonymous No.5691576]

>>5691514

x = Chicken flying over butterfly dancing in moonlight.

[Anonymous No.5692491]

so if i want to prove that a set is NOT compact all i need to do is think of one particular open cover and show that it does not admit a finite subcover?

[Anonymous No.5693264]

How does the intuitive concept of homology as counting k-dimensional holes in a space relate to the actual process of using chain complexes, boundary operators, and the quotient group definition of the homology group itself using kernels and images?

Why do we even use groups like the fundamental groups and homology to count holes instead of just using numbers?

[Anonymous No.5694152]

Heels friends! I hadn't realized this thread hadn't 404ed! I've been busy of late, but if you're willing to be patient, I will do my best to answer all outstanding questions!

Sorry for the inconvenience!

[Anonymous No.5694174]

>>5694152
Hurry my time is running out!

[Anonymous No.5694175]

>>5694152
Better start a new thread. This one is about to hit the bump limit with the next post.

[Anonymous No.5694178]

Polynomial time

[Anonymous No.5694179]

>>5694175
hehe bump

[Anonymous No.5694184]

not even this thread could infinity

[Anonymous No.5694185]

Teach me something interesting from topology plox?

[Anonymous No.5694186]

this is stupid

[Anonymous No.5694188]

damn

[Anonymous No.5694190]

>>5694186
You are stupid.

That'll show you.

[Anonymous No.5694446]

>>5685980
There is always a reference frame 'R' which renders some given subject 'S' arbitrary and meaningless, regardless of the value of 'S.'

[Anonymous No.5694731]

>>5689831
Hello! What is GR?

[Anonymous No.5694738]

>>5691205
Hello! Hmm, this is a good question. I think I am somewhat of an interesting case. I talk to friends, peers, and my dad about math, and when I get into it, my mouth goes like a duck's ass. I don't really picture what I'm saying a lot (sometimes) it just flows forth like some unholy avalanche.

That said, for geometric ideas, yes, I do picture things.

So, in summary, while I can, and do "see" equations and pictures well if I try, I don't often rely on this.

I hope this answers your question!

[Anonymous No.5694741]

>>5691215
Hello! That is a tough question since I do not know precisely what group theory physicists need to know. That said, it has been somewhat of a cottage industry in the last twenty or so years to capitalize on physicists need to know group theory, and so thankfully there have been written many books about the subject.

This one, for example, looks decent: http://www.amazon.com/Introduction-Tensors-Group-Theory-Physicists/dp/0817647147/ref=pd_bxgy_b_img_y

I hope that helps!

[Anonymous No.5694749]

>>5691216
Hello! The way you are phrasing this, seems to imply that you mean "can you give me tips that would make me good at researching".

While this is a very tough question, I think there is one, somewhat easy to achieve thing that helps. You need to do mathematics long enough so that you understand the global aims and goals of the subject. In particular, you need to learn word association. Often times the greatest advancements in mathematics are when someone notices that two disparate ideas actually sit as good bedfellows in a larger, enlightening theory.

To be able to connect things though, you need to be able to recognize patterns. You need to be able to say "hey, even though I am doing subject X, which is quite a way away from subject Y, this thing I am doing in X really, really reminds me of something I do in Y. Perhaps I can find a connection between the two, or if not, maybe adapt the methods to handle the situation in Y to my situation in X".

For example, a large part of algebraic geometry was gotten by people's realization that many of the problems in number theory and geometry are analgous to problems in complex manifold theory. They then said "well, if these problems are similar, maybe I can adapt the methods of complex geometry to fit my cases!"--and thus was born algebraic geometry.

The key to all of this, and where this, somewhat meandering response is going, is that you need to learn A LOT of mathematics. Now, I know what you're saying "of course, dumbass--everyone knows that". But, all learning is not created equal. There are students who go into grad school knowing EVERYTHING about one subject--say functional analysis. While this is great, and while it may get them closer to resarch level math sooner, not having a BROAD base will hurt them when it comes to drawing analogies in their field to other fields they, well, never learned. So, I think that for an aspiring mathematician, breadth is the way to go.

I hope that helps!

[Anonymous No.5694765]

>>5693264
Ah! This is a fascinating question.I believe what you're asking is why we don't care just about the rank of a homology group, say, instead of the group structure. Why not just tensor everything with Q and take dimensions? This is done, and is important--you're talking about the Betti numbers of a space. And while these are very important, they, in a very good sense "count holes", and are much easier to compute, the lose a lot of information.

The actual algebraic information contained in $H_i(X,\mathbb{Z})$ vs. $H_i(X,\mathbb{Z})\otimes\mathbb{Q}$ is huge! In other words, assuming that we're dealing with a space with f.g. homology (say compact), the subgroup $H_1(X,\mathbb{Z})_\text{tors}$ (the torsion subgroup) counts, in a sense, "fixable" holes. The existence of loops which are non-contractible if you loop around them once, but are contractible if you do them twice (google Dirac's Belt Trick for an illustration of this).

The issue is that "counting holes", at least in the sense that one thinks of this at a firs go, is a bit of a lie--that's not what we're doing. The holes that you are most likely picturing are precisely the "unfixable" holes, and yes, if you are just interested in the unfixable holes, then there really is not any issue in eschewing $H_1(X,\mathbb{Z})$ for $H_1(X,\mathbb{Z}))\otimes\mathbb{Q}$. But, except for the convenience of computation, why would you want to do this? All you are doing is cutting off valuable information!

[Anonymous No.5694769]

>>5694765
>>5693264
(continued)

For example, $H_1(\mathbb{RP}^2)=\mathbb{Z}/2\mathbb{Z}$. So, the real projective plane has a fixable hole of "degree 2". Even though this is isn't a "real hole" in the sense you are thinking it is still valuable topological data that allows us to deduce many things. The most simple of which may be that $\mathbb{RP}^2$ isn't contractible! That said, if you passed from $H_1(\mathbb{RP}^2,\mathbb{Z})$ to $H_1(\mathbb{RP}^2,\mathbb{Z})\otimes\mathbb{Q}$ you'd merely get the trivial group--we've lost information, and now we can no longer conclude that $\mathbb{RP}^2$ is not contractible.

Depending on where you end up going in mathematics, this question will come up many times. For example, if $K$ is a number field, do we *really* care about the class group $\text{Pic}(K)$ as a group? Most of the time we really only care about the class number $h_K=|\text{Pic}(K)|$. What does the extra algebraic information give us? Well, the actual answer to this question is the subject of another response but the answer is definitively "something". In general, forgetting structure always loses something (duh)--what it loses is not always so clear.

I hope this helps!

[Anonymous No.5694771]

>>5692491

Hello! Absolutely. Although, this is sometime a pain. Depending on the context, there are often trickier, but less annoying ways to verify non-compactness. For example, it is a fact that $X$ is compact if and only if every continuous function <span class="math">X\to\mathbb{R}[/spoiler] is bounded. So, it suffices to find a non-bounded real valued function on <span class="math">X[/spoiler]. Other tricks abound.

[Anonymous No.5694790]

>>5693264

Hello! Me again. I didn't address your first question, as to how the homology of the singular chain complex of a space <span class="math">X[/spoiler] intuitively is measuring "holiness".

Allow me to ask you to mention the two spaces <span class="math">X=\mathbb{R}^2[/spoiler] and <span class="math">Y=\mathbb{R}^2-\{(0,0)\}$. Clearly <span class="math">X[/spoiler] has no "holes" whereas <span class="math">Y[/spoiler] does. So, how does $H_1$ pick up on this? Well, what does <span class="math">\ker\partial_1[/spoiler] give us? If you look at the definition, you will see that basically the kernel of the first boundary map are just closed paths in your space--i.e. continuous functions from <span class="math">[0,1][/spoiler] to your space that begin where they end. The image of <span class="math">\partial_2[/spoiler] are really the closed paths in your space that occur as the "boundary" of some filled in disc, intuitively.

Thus, the quotient <span class="math">H_1(X)=\ker \partial_1/\mathrm{im }\partial_2[/spoiler] is measuring to what extent do there exist closed paths in our space that don't arise as the boundaries of filled in discs. Thus, for example nothing goes wrong with <span class="math">X[/spoiler]. EVERY closed path is the boundary of a filled in disc--just ill in the boundary of the path. We cannot do this with <span class="math">Y[/spoiler] though! The unit circle is a closed path which isn't the boundary of a closed disc since we can't fill the unit circle in--the missing origin prevents this. Thus, we see that <span class="math">H_1(X)=0[/spoiler] but <span class="math">H_1(Y)\ne 0[/spoiler] which indicates the non-holiness and holiness of <span class="math">X[/spoiler] and <span class="math">Y[/spoiler] respectively.

Also, note that the varying dimension of homology groups measure the varying dimensions of a hole. Namely, the hole in <span class="math">Y[/spoiler] is a one-dimensional hole, intuitively because it prevents one-dimensional manifolds (curves) from being the boundaries of things, but the hole in <span class="math">S^2[/spoiler] is two-dimensional since it prevents a two dimensional manifold (the sphere iteself) from being the boundary of something.

I hope that helps![/spoiler]

[Anonymous No.5694792]

>>5694790
Well, fuck. I don't know how to fix that hahah

[Not OP No.5694970]

>>5694771
Let me just mention another useful characterization of compactness: A topological space X is compact if and only if, for all topological spaces Y, the projection <span class="math">X \times Y \to Y[/spoiler] is a closed map (i.e., it maps closed subsets to closed subsets).

Here is a proof:
http://ncatlab.org/toddtrimble/published/Characterizations+of+compactness

The reason this is a nice characterization comes from algebraic geometry: the correct generalization of compactness is a "proper map", which is defined in essentially the same way, but using the tensor product (or technically, the scheme-theoretic product, which is locally the tensor product of rings) instead of the topological product. (Literal topological compactness is fairly useless in algebraic geometry, because the spectrum of a ring is *always* compact.)

[Anonymous No.5695312]

>>5685944
up

[Anonymous No.5695339]

bump for greatness

[Anonymous No.5695345]

>>5695339
the mathematical version would in duce current so no

[Anonymous No.5695433]

>>5685944
is

[Anonymous No.5695435]

>>5695339
bump limit hit, make a new thread if you wish