@robjohn back to the math :) So it seems that conditioning on the first inner product being zero, you expect there to be more zeros in w
@robjohn umm. I am not honestly sure if I do log out each time. But I know that even if you do, and then someone else wants to log in as someone else, you often have the situation where different parts of the SE network have different IDs for you
@robjohn on the math.. my intuition is that you are more like to get 0 inner product if you have more 0s in the vector. So if you know you got 0 inner product before then this skews the number of 0s you are likely to have
Hrm, I need some guidance with some sub-part of a big exercise question. I want to show the Sylow-2 subgroup of $S_7$ is isomorphic to $D_4 \times C_2$, and then count the number of elements of order $4$ in such a sylow-2 subgroup and conclude it can't have $Q_8$ as a subgroup. Anyhow, I am stuck in the first phase.
I understand that all the elements of the sylow-2 subgroup of $S_7$ must be permutations whose order is either 2 or 4, and necesserily not all of order 2.
I have no clue how to proceed, I just need a little kick in the right direction though.
Uhm... what could I say besides that it is something I encountered in a new study of mine? (I'd like to avoid telling others what I'm up to; is it bad?)
You mean one part for each inequality? I think they really belong to the same problem, so I wouldn't think of asking another question
Hello @robjohn !! Can I ask you further about the exercise:
Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a Lebesgue measurable $E\subset R^d$ with $0<m(E)<+\infty$ we have that $L^{p_2} \subsetneq L^{p_1}$.
@Khallil I dunno, I didn't write down the steps. I got to $\displaystyle \int_0^1\frac{\operatorname{Li}_1(w) \sqrt{\operatorname{Li}_0(w)}}{w}\mathbb{d}w$, but I did the rest in my head and wrote down the answer I got.
What's the significance of $f(t, \epsilon)$, @Eric?
Ah, I see. So you'll be a semester behind those that started before you then, @teadawg1337?
I mean, if $t \geq f(t, \epsilon)$, then you can arrive at that inequality. Otherwise, it doesn't seem possible at first sight. I might be wrong, @Eric.
@RobertCardona This is a classical problem, and I was thinking of a different approach now: Is it useful to consider $$\int_0^{\infty} \frac{\sin^2(x)}{x} \ dx$$?
If you're interested. The problem came up as follows: I was trying to come up with $f \in L^1$ such that $\widehat f \notin L^1$. I chose $f = \chi_{[-1, 1]}$, and $\widehat f$ looks something like what I started with, but to show that it isn't in $L^1$, I need to show the integral of it's absolute value isn't in there.
@RobertCardona You can also simply observe that $$\int_{k\pi}^{(k+1)\pi} \left\lvert \frac{\sin x}{x}\right\rvert\,dx \geqslant \frac{1}{(k+1)\pi} \int_{k\pi}^{(k+1)\pi} \lvert \sin x\rvert\,dx = \frac{2}{(k+1)\pi}$$ for $k\geqslant 0$, and since the harmonic series diverges, $\frac{\sin x}{x} \notin L^1$.
I looked up the description of the Mass Market Paperback (the one that's cheapest in the link you've provided) on Amazon: "Paperback edition published in China. This book is completely in English with totally the same contents."
"... completely in English with totally the same contents."
@Chris'ssis yes. $-6$. I thought Mathematica might have some trouble with that one. I am sure it won't be able to handle $$\lim_{a\to\infty}a^6\left(6+a^4\int_0^\infty \frac{\cos(ax)}{1+x^3}\mathrm{d}x\right)$$
During the study of some integrals I came across a very interesting integral, that is
$$\int_1^{\infty} \int_1^{\infty} \frac{\Gamma(x+1)\Gamma(y+1)}{x y \Gamma(x+y+2)} \ dx \ dy$$
Apparently nothing seems to work, but I bet you can do much more than me. So, what
tools would you recommend me t...
@DanielFischer Speaking of those users, your account and two of your comments were imported to Physics Overflow. I don't know how you feel about this, but I thought you should be aware.
Their import practices are OK by the CC license we have. I'm more displeased by the account import than the content import, but both are properly attributed. You might see the recent discussion on Meta.MO that made me aware of this practice.
@daOnlyBG I don't quite get what you mean. The definition I know (and the usual one) is that for an ideal $I$, $f\in \sqrt{I}$ if $f^n \in I$ for some $n$.
hmm. it looks like Cech nerve of any open cover of the solenoid is the uncountable graph of $|\mathbf{Z}_p|$-many vertices obtained from taking the inverse limit of circles with $p^n$-th roots of unities marked (the nodes) and the arcs joining them as edges.
OK, so here's what (I think) works: Let $f^n \in IJ. Thus, f\in \sqrt{IJ}.$ This implies that $f^n$ is of the form $h_1\cdot i_1\cdot j_1, h_2\cdot i_2\cdot j_2,..., h_k\cdot i_k\cdot j_k$. If this is correct, I'll let you know what else I have-if not, a correction would be appreciated
@teadawg1337 I don't know what's in Apostol or Zorich, so I can't say
@MikeMiller I had an excellent teacher for complex analysis. We could have used almost any book, I think. I think we used Ahlfors, in that class. Perhaps it was Rudin for that class and then Ahlfors was for the graduate Complex Analysiis course that I took soon after
@robjohn Apostol covers Riemann-Stieltjes integrals, infinite series/products, sequences of functions, Lebesgue integration Fourier series, and residue calculus (among other things). Not sure about Zorich, I just started skimming through the PDF an hour ago
Consider the even, $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{1}{\pi}\int_0^{\pi} \sqrt{1-k^2 \sin^2 x}\,dx$$ as considered in this earlier question. In comments for that problem, the foll...
also, is there a better name for "a periodic function whose Fourier coefficients are polynomials in a parameter"? i suspect not, but it'd be cute if there was
Two comments. First - yes, for some $n$. Second - the $h_n$ are redundant, because $I$ is an ideal; so by definition $h_ki_k\in I$, so you could just not write them at all.
@Semiclassical I saw the bounty you put on my question; appreciate it.
no problem. figured that since i'm the one who initially got the ball rolling on that topic, me putting a bounty on it was appropriate
though i'm not sure what kind of response it'll get anyways. it feels like a question which is almost-but-not-quite more suited for MathOverflow than MSE
figuring out the line between questions for MSE v. MO is something i have a hard time judging, though, so take that with a grain of salt
@MikeMiller gotcha. By the same reasoning, then for some polynomial $g$, $g^n \in \sqrt{I \cap J}$ implies $g^n$ has the form of whichever polynomials are in both I and J.
Let $a_1 \ge 0$ and $a_2 \ge 0$ be real numbers and let $n_1 \ge 0$ and $n_2 \ge 0$ be integers. Finally let $m\ge 1$ be another integer. By using the method of generating functions I have shown that the following identity holds:
\begin{equation}
\sum\limits_{i=0}^{m-1} \prod\limits_{\xi=1}^2 \bi...
During the study of some integrals I came across a very interesting integral, that is
$$\int_1^{\infty} \int_1^{\infty} \frac{\Gamma(x+1)\Gamma(y+1)}{x y \Gamma(x+y+2)} \ dx \ dy$$
that is obtained by manipulating
$$\int_0^1 \operatorname{li}(x) \operatorname{li}(1-x) \ dx$$
Apparently nothi...
i'm not sure what's terribly confusing: he shows that $li(x)$ can be written as an infinite series in (shifted) Legendre polynomials (whose coefficients are evidently relatively 'nice'), uses this to do the same for $li(1-x)$, then multiplies them together and uses orthogonality to compute the integral
@PedroTamaroff Which one? The Lebesgue number one I already did.
And the other I don't recall.
Lebesgue number lemma is not hard. It says that for any given open cover of a compact space, any point has an open ball that fits into some elt of the open cover.
So assume not and collect a bunch of points. the whole space is compact, so the sequence converges to something. and as that something is contained in some elt of the open cover, there is a disk around it that fits inside that set.
however we can choose (as the sequence converges to that particular point) a disk around each point of the sequence that fits in the disk that in turn fits inside some open set of the cover.
Hey guys! Do you know any good lecture notes on Binary Integer Programming? I'm not really looking for a complete reference, just a quick introduction to the main methods.
Uh. "Hey guys and girls" would have been better. Sorry for that!
@MikeMiller I'd be more interested in a fuzzier version of that question: What implications of RH and $\not$RH do people consider as (heuristic) evidence for or against RH?
@Semiclassical But I claim this is either too vague or already answered. I would be astonished if there weren't questions on MSE and MO about "Why should I believe RH?"
@r9m Yeah, I got your point perfectly. I also asked myself these days for what reasons I'd like to be a moderator for free? I don't find any reason to ever want such a position for free. I'm also very curious about the real face of the things with this site, but time will reveal it definitely.
@r9m I don't bother too much with these things, I'm here only for having some math conversations with a few people.
@Chris'ssis I might like to be a moderator some day considering the fact that this site has aided my growth since my participation tremendously !! currently I lack knowledge or the level of maturity required for such a responsibility ! but some day sure I'd like to run for a diamond star ;)
@r9m That back to the aid you're talking about I personally understand it by the questions and answers I posted. As far as I know, to be a moderator is a totally different story, it requires a lot of resources, time, but well, you know better how to spend your resources. To be honest, presently I don't know who will be able to beat you for such a position. Really! :-)
@Chris'ssis 'A leader is best when people barely know he exists, when his work is done, his aim fulfilled, they will say: we did it ourselves.'-Lao Tzu ;)
yo man @G.T.R ... got any mind boggling integral inequalities for me ? :-)
@r9m First I wanna be a great leader of my thoughts, feelings, and all things related to my life, and then to be the leader of the others. To be a moderator is not really to be a leader in my opinion, you're restricted to a set of actions, and possibly only apply what you're told to apply. Never that kind of leader Lao Tzu talked about. World has changed much in the meantime ...
@r9m Let $E$ be the set of $C^{1}$ real functions over $[a,b]$. Prove that $\forall \epsilon>0, \exists D>0,\forall f\in E$, $\sup_{x\in [a,b]}f^2(x)\leq D \int_a^bf^2+\epsilon \int_a^bf'^2$
@r9m when you're done, prove that if $f:[0,1]\to \mathbb R$ is a continuous concave function with $f(0)=1$ then $\int_0^1xf(x) dx\leq \frac{2}{3} \left(\int_0^1 f(x)dx\right)^2$
@G.T.R for these inequalities (related to quadrature and stuff) I think it suffices to verify it for piece wise linear functions (concave arrangement in this case) and claim its true for the limiting case .. but I'll try and see if there is any smarter approach :-)
@teadawg1337 that $\Gamma(5)$ on the top is a complete give-away ! ;) ;P
During the study of some integrals I came across a very interesting integral, that is
$$\int_1^{\infty} \int_1^{\infty} \frac{\Gamma(x+1)\Gamma(y+1)}{x y \Gamma(x+y+2)} \ dx \ dy$$
Apparently nothing seems to work, but I bet you can do much more than me. So, what
tools would you recommend me t...
