/sci/ - Science & Math

Forever, as you need a heat bath to work with metal.

Well, well, well, show that the series

<span class="math">sum_{n=0}^\Infty x^n / n!<span class="math">

takes the same values as e^x for all x.
(directly, not over Picard Lindelöf or some shit.)
reals to the power of reals are bitches if you can't use the series expansion imho[/spoiler][/spoiler]

Au contraire!
I have a cute little physicist friend, sporty spice, ballet dancer, notorious atheist, obsessed with rationalism. And she looks up to me like I'm her only hope.
I'm 5 years older than her and live in another country. But when we meet, she likes to dress up like sailor moon. That's a little eccentric, but hot.

Regarding the feel...
people have come up with a theory explaining the problems. It introduced the gf-energy density <span class="math">\mathcal{L}_\mathrm{gf}[/spoiler] and tells you what to do:

http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory#Mathematical_overview

>>4950470
they are classified

http://simple.wikipedia.org/wiki/Wikipedia:Basic_English_alphabetical_wordlist

etc.

see
http://simple.wikipedia.org/wiki/Wikipedia:How_to_write_Simple_English_pages

There are different interesting things I have though up at one point in this regard.
Like imagining beings that are able perceive the world from two different places at once and their resulting different understanding of the world.
Or beings as clever as humans, but which live on a different length scale. If you are big as an ant or a planet, then things seem totally different.

On which level are you asking this?

I don't know if it's done (I.e. if it's possible w.r.t. computer efficiency), but I'd assume one does numberical simulations of linearized gravity. Plugging in initial conditions and computing time evolution given by the Einstein equations.

http://en.wikipedia.org/wiki/Linearized_gravity
http://en.wikipedia.org/wiki/Gravitational_waves#Mathematics

>>4368443
Yes, I know much group theory.
I read Zee and Peskin and Weinberg.

The integral of <span class="math">x^n[/spoiler] should be <span class="math">\frac{x^{n+1}}{n}+c[/spoiler]. For <span class="math">n=-1[/spoiler], that clearly doesn't work, since then the "antiderivative" is not x-dependend anymore, which is certainly wrong.

Now what is the integral of <span class="math">\frac{1}{x}[/spoiler]. It's problematic to argue using an integral from <span class="math">0[/spoiler] to <span class="math">a[/spoiler], because for <span class="math">x=0[/spoiler], the integral will diverge.

To compute the integral, shift the whole function to the left by a constant, i.e. consider <span class="math">\frac{1}{x-1}[/spoiler].
This is minus the function which is expanded as a the geometric series
http://en.wikipedia.org/wiki/Geometric_series#Formula

check out this
http://www.wolframalpha.com/input/?i=Series[-1%2F%28x-1%29%2C{x%2C0%2C6}]

the integral is that function
http://www.wolframalpha.com/input/?i=Series[Log[x-1]%2C{x%2C0%2C6}]

Ettore Majorana, like Antoine de Saint-Exupéry, just vanished.

>>4182352
Mhm, thats actually an interesting line of though.

If I understand you correctly, you kind of describe the proof p as a series of statements s(i), following form axioms axioms a, so

p=(a,s(1),s(2),s(3),...,s(n)),

where s(n) is the theorem you wanted to proof.

What I like about your statment is how you imply that each two neighboring statements (s(i),s(i+1)) are associated with a predicate "trivial", nevertheless, the whole chain doesn't have this property. In any case, your description of a proof doesn't really fit this one

http://en.wikipedia.org/wiki/Proofs_as_programs

at least in my eyes. So I think there is some problem with that.

http://en.wikipedia.org/wiki/Statistical_mechanics#Fundamental_postulate

http://d24w6bsrhbeh9d.cloudfront.net/photo/672730_700b.jpg

People took hundreds of years to come up with the number 0. Do you believe that you'd have discovered it if you would have lived back then? protip: if thousands of other people didn't, then you wouldn't have too.

Even if it's just because of religious oppression of the fact that poeple had to work 70h a week back then and there were'nt many mathematcans around - that some were able to come up with these things is still an achievement, I'd say.

You will not be able to "sort" it in a strinct sense but here are some ingrediences

Axiom Theory
Model Theory
Proof Theory
First Order Logic
(Higher Order Logics)
Algebra
Set Theory/Cathegory Theory
Topology and fields "like" Arithmetic
Algebraic Topology and discete mathematics
Differential Geometry
Algebraic Geometry
fields "like" Geometry

>>3997946
which year/semester?

The basic idea is that you quantize not a particle (worldline four vector x(\tau) in spacetime, \tau a time parameter) but a string (worldsheet x(\tau,\sigma) in spacetime)

http://universe-review.ca/I15-51-string1.jpg

Depending on what you know about QFT, good point to start would be to talke a look at the Lagrangian

http://en.wikipedia.org/wiki/Polyakov_action

where a,b is \tau or \sigma, one parameter.

notice that when you don't consider a string, i.e. only tau, then this lagrangian basically is (m/2)v^2.

The Polyakov action is the classical (bosonic) Lagrangian function, you take it and implement its degrees of freedom in quantum mechanics. The quantization process.
A string has much more degrees of freedom than a point, because of its many embeddings in spacetime and so the quantum theory is richer too. A structure relevant to the string vibrations is the
http://en.wikipedia.org/wiki/Virasoro_algebra

It then turns out that this quantiation doesn't work in 4 dimensions. One way to see it is that the algebra doesn't close. For the bosonic string you'll need 26, for the ferminonic string (matter) you'll need 10. This is why we are talking about 10 dimensions here. But we observe only 3+1 so the idea is to model the world by a 3+1+6 dimensional manifold where the 6 dimensions are compactifyable in a way which doesn't kill the dynamics of the string, symmetries etc. Calabi-Yau manifolds (in this case one with 3 complex dimensions) are the way to go here.

>>3997963
my bad,
>where do you disagree
was directed at
>>3997877

>>3997969
it's 10 dimensions, one extra dimension for tricky M-theory stuff.
Since the manifolds are real, only the tangent space will be R^10. So "locally like R^10".

"I've had the chance, in the world of mathematics, to meet quite a number of 
people, both among my elders and amoung young people in my general age 
group, who were much more brilliant, much more "gifted" than I was. I 
admired the facility with which they picked up, as if at play, new ideas, 
juggling them as if familiar with them from the cradle -- while for myself 
I felt clumsy, even oafish, wandering painfully up an arduous track, like a 
dumb ox faced with an amorphous mountain of things I had to learn (so I was 
assured), things I felt incapable of understanding the essentials or 
following through to the end. Indeed, there was little about me that 
identified the kind of bright student who wins at prestigious competitions 
or assimilates, almost by sleight of hand, the most forbidding subjects. 
In fact, most of these comrades who I gauged to be more brilliant than I 
have gone on to become distinguished mathematicians. Still, from the 
perspective of thirty or thirty-five years, I can state that their imprint 
upon the mathematics of our time has not been very profound. They've all 
done things, often beautiful things, in a context that was already set out 
before them, which they had no inclination to disturb. Without being aware 
of it, they've remained prisoners of those invisible and despotic circles 
which delimit the universe of a certain milieu in a given era. To have 
broken these bounds they would have had to rediscover in themselves that 
capability which was their birthright, as it was mine: the capacity to be 
alone." - Grothendieck