In a (great) paper "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations" by Iaroslav V. Blagouchine, the following integral representation of the first Stieltjes constant $\gamma_1$ is given (on page 539):
$$\gamma...
@Chris'ssis Sorry, I'm not interested in this one. I'm more interested in knowing the closed-form of each integral in VR's problem. Anna's thought turns out to be correct, that each integral has no closed-form
Now I become more interested in knowing the answer about her claim on this one
@VladimirReshetnikov As I said, I don't know. But I'am able to prove it using a real method as a combination of them, not the separated one. — Anastasiya-Romanova 秀Dec 8 at 19:39
@Chris'ssis I referred to her comment too. How to use Hermite's integral to evaluate this integral? As far as I know, Hermite's integral can only be evaluated by using contour integration, not real method
Have you ever seen this constant? $$\log B=\lim_{n\to\infty} \left(\sum_{k=1}^{n} k^2\log(k)-\left(\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{2}\right)\log(n)+\frac{n^3}{9}-\frac{n}{12}\right)$$ This one amongs others was considered by Choi and the famous professor Srivastava. Now, if you're aware of it, you can do the series I gave you and find a closed form, otherwise you cannot do it.
Hey @DanielFischer I returned the elements like that: if (m%2==1) m=A[(m-1)/2] else m=0.5*(A[(m-1)/2]+A[m/2]) j=low for (i=0; i<high-low+1; i++){ M[i]=abs(A[j]-m) j++ } count=0 while (count<p){ min=M[0] position=0; for (i=1; i<high-low+1; i++){ if (M[i]<min and i!=position){ min=M[i] position=i } } return A[position] count++ }
@skullpatrol Well, like what you may have heard already, sometimes, I don't think I can get well and go to grad school anymore. I think the chances of me getting well are 0.1, and the chances of me getting into grad school are 0.1, so the chances of both happening are 0.01.
@Jasper I do not know what the probability of either of those are. On the other hand, I do believe that they are lower if you do not have some faith in yourself. So I hope you do, and do not worry about the probability, and try regardless.
I hope you still begin your study on Jan 1, since I think it will make you happy.
I am looking at $f:R_s \to R_s$, $f(x)=e^x$. I am not mistaken that it is continuous, open and closed, right? And yet $f(x)=-x$ is not continuous, neither open nor closed
@DanielRust Hm... there doesn't seem to be a big list question about applications of elementary linear algebra, and I'm TAing it next quarter. It might be good to have a bunch of them to construct examples with.
(I know the canonical one in linear algebra courses is modeling dynamics with Markov processes and using diagonalization to compute the $n$th state easily. But surely there's more than that.)
I don't understand your picture, @BalarkaSen, but your answer is correct. (I probably would have done away with $c$, given we know that it's just $a$; the group also has presentation $\langle a,b,c | [a,b] = 1, [a,c] = 1\rangle$).
@Huy Any applications outside of math of elementary linear algebra that don't take too long to present. (When I say elementary linear algebra, my students won't have done any more than diagonalization and the singular value decomposition by the end of the quarter.)
@evinda Not right. You return A[position], and thus the while (count < p) loop is run only once. If you replace the return with something that adds A[position] to a list, you have the problem that every iteration will find the same position. You'd need to modify your M array so that M[position] is set to something large enough that position will not be used again. And it's somewhat inefficient, $O(p^2)$, where an $O(p)$ algorithm can be used.
Because we are identifying S^1 cross {x_0} and S^1 cross {x_0}, the identification space is the square [0, 1]^2 and [0, 1]^2 with opposite ends identified as well as four of the sides from two of the squares are identified @MikeMiller
Any ideas about the series question, guys? It's do with using the cube roots of unity to find a closed form of the series below. $$\sum_{k=0}^{\infty} \dfrac{x^{3k}}{(3k)!}$$
Anybody here have any experience with basic graph theory (simple graphs only?). Is there any formula to calculate this? "Five vertices are labeled 1,2,3,4,5. In how many ways can edges be drawn between some pairs of these vertices so that the result is a connected graph?"
@Chris'ssis Mind you, I haven't tried this yet, but I'd try integrating $$\int_0^\infty\frac{\log(x+1)^3}{2x^2+3x+1}\mathrm{d}x$$ along a keyhole contour
@MikeMiller: Sure, if it's no trouble for you to wait a bit. I'll have to take the trash out in the next 30 minutes and then I'll continue if I'm not finished already. I hope I won't delay you for more than 1 hour.
@DanielFischer I do these things every day, they simply come to mind in a natural way, and all together comes from exploiting very simple things. I think Ramanujan did the same (it's just a guess).
Now I can go on and talk about a CW complex, and the next thing that will happen is Ben is going to come in and ask me whether I know what a CW complex is. I am still feeling upset over this incident from years ago.
@DanielFischer You can get a copy of her book when it is done. It is not for me though, because it would be too hard.
@DanielFischer I'm glad you appreciate my work. I'd like to write some articles first (some are about to be published) and then I try to publish a book. It's not that easy with publishing books since I have no math background. Of course, everything must be done in a rigorous way, but still, this won't be a problem. I'm only concerned with my lack of background.
@JasperLoy well the comment intended sarcasm. if @Behaviour feels i am posting silly answers/silly comments or if he has any other complaints about me he could've just wrote that much. if participation in this site brings along sarcastic offences, i will desist active participation from this point onwards.
Does anyone know if the following is true: If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can it be shown that $$\limsup\limits_{n \rightarrow \infty}(-a_{n}b_{n}) \leq -ab$$?
The only thing that I cannot stand for as regards my work is when someone suggests my work is taken from elsewhere. I prefer to be shot down than to be told (even in a nice manner) such things. I love my work very much (as happened with the proof to Au-Yeung series).
