Given two points A, B on the unit sphere in R^3, does d(A, B) = cos^-1(dot(A, B)) give a metric? The only non trivial property should be triangle inequality. But I am not so sure if it holds or not.
@user19536 It holds. If you want an intuitive demonstration, consider that the shortest curve on the sphere connecting A to B is an arc of great circle, which has length given by your formula.
The concatenation of arcs AB and BC is a curve connecting A to C; as such, it cannot be shorter than the arc AC. (But, if you want a watertight proof without extra concepts like geodesics, you'll probably have to do some computations with cos(a+b) etc.)
How can I calculate the eigenvectors of this matrix:
[(1, 3) , (3, 2)]
I calculated the eigenvalues , I got :
Lambda 1 = 4.541381265149109
Lambda 2 = -1.5413812651491097
But, now I don't know how to get the eigenvectors.... When I create new matrix after I subtracted Lambda value from all t...
it seems you've learned a fair amount of algebraic number theory, algebraic topology, algebraic geometry, differential geometry, lie and representation theory in a relatively short span of time
well, I wasn't looking into computationally actually doing it, just the fact that unique factorizations exist (and that fractional ideals form a group in the first place)
eventually I can hopefully get some more precise intuition of what the class group "is"
do you got some nice examples for the snake lemma ?
user40730
hey, short question: Given a linear map $L\in \mathcal L(\mathbb R^n,\mathbb R^m)$, $L$ is totally differentiable, but is $DL(x)=L$ or $DL(x)=L(x)$? (I am a little confused about it at the moment; the definition of the total derivative gives me $L$... just want to be sure ;) ) Thx!
@AlexanderGruber Shiver me timbers, awesome ahead mate.
How does the fact that $$ \sum_{n=1} \frac{1}{n(2n+1)} = 2 - 2 \log 2 $$ follow *easily* from the sum value $$ -\log(2) = \sum_{n=1} \frac{(-1)^n}{n} $$ ?
both $\sum_{n=1}^\infty \left(\frac{1}{2n}-\frac{1}{2n+1}\right)$ and $\sum_{n=2}^\infty\frac{(-1)^n}{n}$ represent the value of $1/2-1/3+1/4-1/5+\cdots$. the first just "compresses" the series a tad, as $(1/2-1/3)+(1/4-1/5)+\cdots$.
By doing two digits per iteration <- poor english. I was working on getting the details to showing that $\psi(1/2) = - \gamma - 2 + 2 \log 2 $, and were working out the details. Thanks.
you would say the grouping of terms is strange. but really, if you think that grouping is strange, you have a pretty low bar (i.e. standards) for "strange," as that grouping is actually pretty straightforward and easily noticeable, if one writes things down explicitly on paper or mulls over what things stand for directly, say.
By mistake i wrote out the first 5 terms of $2/(2n+1) - 1/2n $, and then rearranging the terms to notice the wanted pattern. But ofcourse I could not find it..
@PeterTamaroff Maple gives me an answer using $\text{Bessel}$ and $\text{Struvel}$ functions. Do you want an answer using purely elementary expressions?
There's a problem (#10.23) in Apostol's Mathematical Analysis of which I am having a rough time solving:
Let $F(y)= \int_{0}^{\infty}\frac{\sin xy}{x(x^{2}+1)}dx$ if $y > 0$. Show that $F$ satisfies the differential equation $F''(y)-F(y)+\frac{\pi }{2} = 0$ and deduce that $F(y)= \frac{1}{2}\pi(...
How can I calculate the eigenvectors of this matrix:
[(1, 3) , (3, 2)]
I calculated the eigenvalues , I got :
Lambda 1 = 4.541381265149109
Lambda 2 = -1.5413812651491097
But, now I don't know how to get the eigenvectors.... When I create new matrix after I subtracted Lambda value from all t...
