if that's the case, though, then samuel's objection returns. the lower bound $n\geq \sqrt{12345678}\approx 3513.64$ implies $n^3\geq (12345678)^{3/2}\approx 43378289042.64$ which exceeds 9876543210 by about a factor of 4
so if $n$ is big enough that $n^2$ contains digits 1-8 exactly once, then $n^3$ has at least 11 digits and therefore must have at least one repeated digit
well, so it's not singular homology, it's the homology of some chain complex. this chain complex has a map called the U-map on it. i wanted to see if the U-map sends certain elements where it should, but i was having a lot of trouble doing so because i repeatedly kept writing down representatives for homology classes that weren't actually... closed
so they weren't representatives of much anything
to actually get proper representatives i needed an infinite sum, which the authors hadn't indicated, and i don't really want to write down
@TedShifrin May I ask you a question about smooth manifolds? If $f\notin C^\infty(p)$, then can we define $\partial x_i^p(f)=\frac{\partial}{\partial x^i}|_p(f)$? Here is my question on main site. Thanks!
@Simeon as Eric Stucky mentioned, to make sense of "smooth coordinate chart", you need to have your manifold embedded in some Euclidean space and then say that the charts are smooth maps. so if you don't have that and you're working with manifolds as abstract topological spaces, you'd want the transition maps to be smooth
if your manifold is embedded in some Euclidean space (i.e., is a subspace of some R^m) then both the definitions are equivalent.
We generally don't ignore whether something is defined just because we have put as "if" before something where it occurs
I have just never seen this done (now I need to think about whether this is the same as being a monomorphism in the category of sets with relations, since this would be another natural generalization)
Could I get a second opinion on this question? I feel like what they say in comments doesn't match what they've written in terms of what they're after.
what makes it worse is that this is the third day in a row that i've had something go wrong. (though the other times it went bad during the run, not after)
I mean exams starting next week is a panicking situation. Especially if you don't feel like working, where you feel more panic because of not getting work done.
Statistics, physics, mathematics. To take physics I have to take chemistry though so blergh to that but it's not so bad anyway because physical chemistry is nice.
All three of the above relate to mathematics in a very direct, straightforward way. Best option I can get. That's why I want to study them.
The trouble is that it doesn't help that the textbook I am using does everything in a vague fashion. It's a relief that I have a companion textbook with me which has the rigorous definitions, etc.
I like intuitions, but I also like to have the definitions jotted down.
right, it's slightly confusing when to apply what. a motivating example i keep in mind is the two hunter problem: both hit a target with some probability, so if a target dies after the two shoot together, whose shot had better probability of hitting the target?
although one can really just convert almost everything to urns and balls and do combinatorics there. it just gets a bit tedious sometimes.
right, right, that's close to how i think about it.
I just realized P(A \cap B) = P(A) * P(B|A) is the rigorous foundation for the fact that choice of multiple balls from urns being simultaneous or one-after-another is irrelevant in the appropriate kind of urns-and-balls problems.
group representations are pretty common in inorganic, for instance
there's a lot more linear algebra happening in experimental chemistry than they let on in high school, although that is admittedly kind of an extension of the standard ODE story.
obviously there's quantumy things and stat mech, which are either chemistry or physics depending on who you ask :P
The recovery of molecular structure from spectroscopy data isn't actually a "math problem" (at least, not the stuff I know about), but it has the same sort of feel to me as good textbook exercises.
Also @Balarka: Don't count out biology, please :) Also I wrote some things here, and this blog is entertaining and also written by an actual mathematical biologist.
Basically, what I'm saying is that I went though high school and most of college thinking that pure math was the only "real" game in town, and physics was somehow distinguished in its use of serious mathematics, and I don't want you to make the same mistakes :)
u16, do you know of Max Tegmark?
I'm probably getting that name horribly wrong. (Edit: yup, fix't)
[Hmm, apparently Eisen hasn't been writing as much about his field as he used to :/ ]
According to my notes:
The convolution is in $L^1$. Indeed
$$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) g(x-y)| dx dy= ||f||_{L^1(\mathbb{R}^n)} ||g||_{L^1(\mathbb{R}^n)}$$
First of all, the formula o...
According to my notes:
The convolution is in $L^1$. Indeed
$$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) g(x-y)| dx dy= ||f||_{L^1(\mathbb{R}^n)} ||g||_{L^1(\mathbb{R}^n)}$$
First of all, the formula o...
