@AshleyDavies you're trying to do a substitution, so let u=2x, and so the dx must be replaced with du/2 in the integral, which is where the 2 factor comes from
@SohamChowdhury being a subset and a group with the same operation is the same as being a subgroup. and also your subset is a normal subset, it's just not a subgroup :-)
At the cylinder of the picture there are static pressures from environment fluid of density $\rho$. If we neglect the atmospheric pressure, calculate how much force and how much torque is needed so that the cylinder balance.
In my notes there is the solution, but I haven't really understood ...
@AntonioVargas: just now i was looking at the arxiv preprint which your talk seemed to be based on, and noticed that part of what you're interested in there were certain bessel functions, and their integral representations
...and yesterday, in my research, the following integral (and the question of its analytic continuation) came up: $$\int_{y}^\infty \frac{y^2-x^2}{e^{x}\pm 1}\,dx $$
which integrates to (up to a multiplicative constant) $\sum_{n=1}^\infty (\pm 1)^n \frac{y}{n}K_1(n y)$
that integral representation is for $y>0$, i should say. so the summation of bessel functions should be understood as analytically continued to the rest of the complex plane
so if you've built up some applicable expertise in that realm, i'd love to know :)
Is there a standard method for finding expansions in $N$ of sums like
$$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$
beyond the first term?
It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} \mathrm
d x + \mathcal O(N) = \pi N^2/4 + \mathcal O(N)$$
but finding $S(N) \approx \pi N^2 /4 +...
@SohamChowdhury For some reason, when I think of "god", I think it would be more of a jock type, and thinks you are crazy for saying the oxymoronic phrase "mathematical beauty".
btw, the reason i'm defining it for $y>0$ is that, for $y<0$, the way that integral shows up is a bit different: the lower limit become $-y=|y|$
but rather than have a lower limit which is nonanalytic in $y$, i figure it's better to start with $y>0$ and analytically continue the bessel function sum
For large $y$ you can probably get the leading order in terms of a cosine or sine sum. I was lucky enough there to only need to evaluate it for $y$ like $2\pi i n$
@AntonioVargas: for reading after your flight---the two things i really care about re: that sum of bessel functions are 1) how does the prefactor of the logarithm behave? It should be some infinite sum over bessel functions $I_1$. 2) what's the behavior along the imaginary axis? for any particular $K_1$ the function gives decaying oscillations along the imaginary axis, but what about after resumming? pretty sure something nontrivial happens
i find that kind've dumb too. i'm actually somewhat sympathetic to religion in general, but for reasons which make me entirely unsympathetic to that kind of thing
Morphisms is just the generic term for, well morphisms. homomorphisms are typically specific to the situation, like linear transformation, or homeomorphism. But morphisms of blah is just as clear if not more so
I saw some interesting debates among two algebraists when I was in my undergrad. One likes homomorphism, the other one says that morphism is what people actually say. Another one was whether the dihedral group of order $n$ is supposed to be $D_{n}$ or $D_{2n}$. Dang, I hope both of those people aren't leaving soon :/
Random question, I forgot what $S_n$ is. Isn't it the set of all permutations $(1, \dots, n) \to (1, \dots, n)$ - and a permutation is just a bijection... right?
@KarimMansour I'm too OCD about proving everything. Could barely get through chapter 1, since I found myself using axiom of choice to prove the left-inverse iff injective, right-inverse iff surjective proofs
Let $A,B$ be $n\times n$ singular real matrices such $ker A\cap ker B=\{0\}$, how could I show that there exists $x\in \mathbb R$ such that $ker (A+xB)=\{0\}$?
@PaulPlummer I am trying to prove that R/I is a field iff I is maximal using maps, so I look at $A\rightarrow A/I$. Assuming that $A/I$ is a field, then if there exist an Ideal $J$ such that $I \subseteq J \subseteq R$. First we have $\phi(J)$ is an ideal of $A/I$, so has the form $<0+I>$ or $A/I$, I am stuck at the case $ \phi(J) = R/I$
@Gato the way to think about this is with lattice-correspondence. something is a field if there are no ideals between it and 0, and I is maximal in A if there are no ideals between I and A. Now, lattice correspondence puts the ideals in A/I with ideals between A and I.
@Clarinetist I think piano and guitar do arpeggios better than most other instruments, because real legato isn't possible (although it is to some extent on a guitar).
@SohamChowdhury Renaissance - traditional "species" counterpoint - is extremely easy. But you start getting me into Baroque counterpoint, I'm completely lost
@Ted: Here's an interesting question. Suppose I have a fixed-point-free involution on $S^n$. Is there some embedded $S^{n-1}$ that's invariant under the involution? I think it's true.
@MikeMiller I need to prove that $\Bbb{Z}/10\Bbb{Z}$ is isomorphic to $\Bbb{Z}[i]/(1+3i)$. I consider a homomorphism $\Bbb{Z}$ to $\Bbb{Z}[i]/(1+3i)$. The kernel is the set $\{x\in $\Bbb{Z}$ : \phi(x)=\overline{0}\}$, and $\overline{0}=0+(1+3i)$, and $(1+3i)=\{(1+3i)(a+ib) : (a,b)\in \Bbb{Z}\}$. Am-I correct ?
Hi guys, I was here earlier asking for help with integration - and managed to solve the question after someone pointed out a flaw in my reasoning
I just looked back at my answer and I noticed something I believe to be a mistake, yet I still arrived at the correct answer.
I integrated 1/rt(3-4x^2) using definite integration and the, er, are they called boundaries? of integration were 3/4 and 0. I substituted in u as 2x, so that it was 1/rt(3-u^2) and then used a formula I had given for arcsin(x/a), ending up with 1/2 arcsin(2x/rt3)
I simply then just evaluated it by substituting in 3/4 as x and running it through my calculator; correctly giving a sixth of pi. I now realised that when I did u = 2x, I never changed the boundary from 3/4 to 6/4, but despite this my answer came out correct - Why is this? :/
We were given a velocity field and we have to calculate the vector field of the local acceleration, the acceleration because of the transport and the total acceleration at the time $t=0$.
At the cylinder of the picture there are static pressures from environment fluid of density $\rho$. If we neglect the atmospheric pressure, calculate how much force and how much torque is needed so that the cylinder balance.
In my notes there is the solution, but I haven't really understood ...
Is there a standard method for finding expansions in $N$ of sums like
$$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$
beyond the first term?
It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} \mathrm
d x + \mathcal O(N) = \pi N^2/4 + \mathcal O(N)$$
but finding $S(N) \approx \pi N^2 /4 +...
Let $A,B$ be $n\times n$ singular real matrices such $ker A\cap ker B=\{0\}$, how could I show that there exists $x\in \mathbb R$ such that $ker (A+xB)=\{0\}$?
