Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$?   It is obvious that $\frac{\zeta(2n)}{\pi^{2n}}\in\mathbb{Q}$, but since there is no closed form for the odd values, are we left to be unable to determine if the solution could have such a form?
Related to the question :  It is well known that   $$\zeta(2n) = f(2n) \pi^{2n}$$  where $f(n)$ is an function in rationals: (the denominator = OEIS : 6, 90, 945, 9450).  Apery proved that $\zeta(3)$ is irrational, and presumably $f(2n+1)$ is so for all (but the latter is open).  But what else is known about $f(2n+1)$? What sort of pattern does it share (if any) with $f(2n)$? Numerically does it fit well? Is there any clue/progress since Apery as to its form? Of course, $f(2n+1)$ may in effect divide out the $\pi$, meaning that somehow $\pi$ is not involved in $\zeta(2n+1)$ in any essential way.
Align multiple numbered aligned blocks There are many discussions over how to align multiple align environments with interspersed texts. However, I want to label multiple aligned blocks such that each number will center vertically for each aligned blocks and at the same time align across blocks.  Here is a MWE that achieves either at one time. The first part simply uses one giant align environment with some \nonumber directive. The second one use equation to wrap aligned but lose alignment info. How should I achieve the best of both worlds?  \documentclass{article} \usepackage{amsmath} \begin{document}   % Part I   \begin{align}           x^2 + y^2 &amp;= 1 \\                 z^2 &amp;= 0 \nonumber \\     x^2 + y^2 + z^2 &amp;= 1   \end{align}    % Part II   \begin{equation}     \begin{aligned}       x^2 + y^2 &amp;= 1 \\             z^2 &amp;= 0 \\     \end{aligned}   \end{equation}   \begin{equation}     \begin{aligned}       x^2 + y^2 + z^2 &amp;= 1     \end{aligned}   \end{equation} \end{document} I can't download apps from the Play Store on my Android device (maybe due to a network issue), so I'm wondering how could I download the APK file of an app on my PC from ? Privacy/Security concers for setting up business email using Gmail app on Nexus devices I need to set up my business email*(non gmail email)* on the mobile devices (Nexus 5 and Nexus7). The business has its own private mail server. The way I can do it is by using Gmail app on these devices. The question I have is does Gmail app on the mobile devices store the IMAP emails(non gmail emails) on the gmail servers? Or Gmail app is just a client and does not bother storing the IMAP emails(non gmail emails) on its servers that are not gmail emails? Thanks for the insight... Finding solutions to congruence equations Review Count - Fix It or Lose It Another meta question identified a bug that was marked as designed. To summarize the bug, the little number that appears next the Review link in the header shows an incorrect value most of the time.         I would like to propose that the feature be dropped. Honestly, I would prefer it be fixed, but the response to the question above gives me little faith that it will be. Half of the time, I click that little brown box and find out that I was lied to. There's nothing to review.  The little number no longer does the job it was created to do.        Oh sweet! Three things I can review!        Just kidding... Can I have multiple Schengen visas for future travels in my passport? I'm a citizen of Ukraine planning on visiting Italy early in October for tourism. Later that month I'm accepted as a speaker at the conference in Prague, Czech Republic. Both countries are members of Schengen agreement and a visa is required for me to visit them. Therefore, I need to visit Schengen countries 2 times and I need to get my visas ASAP (to confirm my talk to the conference organizers, as well as to avoid overpaying for hotels/flight tickets).   I've heard a number of times that a single person can't apply for a Schengen visa if already in possession of one, but I'd like to clarify this. Say, I've got a visa from Italy for the period of October 1st - October 14th. Can I get another one from Czech Republic for October 25th - October 30th before I visit Italy? Would the answer be different if I were to visit the second country (CZ) on October 13th - October 20th, thus, having a single day overlap between these visits. It would be of a tremendous help if you could link to the official documents that prove your point, but I'd certainly appreciate answers even without such links.  Both times I'm going to apply for a short-term tourist visa covering the dates of my travel. If the answer to both questions above is no - would several visits to multiple Schengen countries over a short period of time be a valid reason to apply for a single multiple-entry visa that covers the whole period from October 1st to October 30th from Italy? Prepend 0 to sublists Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for  every pair of elements in $E$, one is a subset of the other. Clearly, $E$ could be countable by considering $E=\big\{\{1,\cdots, n\}, n\in\mathbb N\big\}$. My question is the following: could we find an example that $E$ is uncountable? Many thanks for the answer! SSH to open GUI on target machine? In the top left corner of the post-race menu, there is a prompt to press + to add (or remove) one star out of six. What do these stars do? How can a function with a hole (removable discontinuity) equal a function with no hole? I've done some research, and I'm hoping someone can check me. My question was this:  Assume I have the function $f(x) = \frac{(x-3)(x+2)}{(x-3)}$, so it has removable discontinuity at $x = 3$. We remove that discontinuity with algebra: $f(x) = \frac{(x-3)(x+2)}{(x-3)} = (x+2)$. BUT, the graph of the first function has a hole at $x = 3$, and the graph of the second function is continuous everywhere. How can they be "equal" if one has a hole and the other does not?  I think that this is the answer:   Because the original function is undefined at the point $x = 3$, we have to restrict the domain to $\mathbb{R} \setminus 3$. And when we manipulate that function with algebra, the final result, $f(x) = (x + 2)$ is still using this restricted domain. So even though the function $f(x) = (x+2)$ would not have a hole if the domain were all of $\mathbb{R}$, we are sort of "imposing" a hole at $x = 3$ by continuing to throw that point out of the domain.   And then just to close the loop: Removing the removable discontinuity is useful because it allows us to "pretend" that we're working with a function that is everywhere continuous, which helps us easily find the limit. But the reality is that the function $f(x) = (x +2)$ is actually NOT continuous everywhere when we restrict the domain by throwing out the point 3. Or am I now taking things too far?   Thanks in advance!  EDIT: For anyone coming across this in the future, in addition to the excellent answers below, I also found this other question about the  helpful. Find the default application for a certain extension Given xdg-open and an extension, is there a way to get the application which xdg-open is set to for that particular extension?  For example given xdg-open and .jpg the result is eog. Update Manager is constantly offering me updates (e.g. security fixes, updates from PPAs).  How can I tell my Ubuntu installation to automatically download and install updates whenever they become available? Deploy Community Builder audience page assignments with ant migration tool? In Community Builder, if you create audiences and assign them to page variations, you cannot deploy these using change sets, according to .  In fact, Audience doesn't even appear in the list of change set components.  On the other hand, Audience metadata can be deployed using the ant migration tool, so I was hopeful that the audience assignments would also be deployable.  Unfortunately, I haven't had success in deploying any audience assignments, either for entire pages or for components on a page.  Has anyone found a workaround for deploying audience assignments for Community Builder pages?  I should say that I'm using the Build Your Own template.  If it can't be deployed, is there at least a way to get a list of audience assignments so I don't have to open every single page in the Community Builder to check?  Here's what I'm using in my package.xml manifest.  All the community builder page variations and all the audiences are created correctly, but the assignments are not.  &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;Audience&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;CustomSite&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;SiteDotCom&lt;/name&gt; &lt;/types&gt; I am writing a small KOMA-Script scrartcl document with multiple sections. In the table of contents, LaTeX fills the current line with dots between the end of the section name and the page number. Is there any way to customize this behavior? I would like to reduce the space between the single dots (i.e. use more dots in the same space). Are the datum and the ellipsoid the same? What is a Datum?     The Earth is shaped like a flattened sphere. This shape is called an   ellipsoid. A datum is a model of the earth that is used in mapping.   The datum consists of a series of numbers that define the shape and   size of the ellipsoid and it's orientation in space. A datum is chosen   to give the best possible fit to the true shape of the Earth.  in this   article.... Is it adviseable to stack internal hard disk over each other (after removing them from the casing) Would it damage or cause any interference to the non-operational (whereby electricity or power is removed or disconnected from the hard disk and it will be store inside a cupboard or container either made of plastic or metal) hard disks (IDE, SATA)? What does $\sum_{k=0}^\infty \frac{k}{2^k}$ converge to? This problem comes from another equation on another question (. I tried to split it in half but I found out that $$\sum_{k=0}^\infty \frac{k}{2^k}$$ can't be divided. Knowing that $$\sum_{k=0}^\infty x^k=\frac{1}{1-x}$$ I wrote that $$\sum_{k=0}^\infty \frac{k}{2^k}=\sum_{k=0}^\infty \left(\frac{\sqrt[k] k}{2}\right)^k=\frac{1}{1-\frac{\sqrt k}{2}}=\frac{2}{2-\sqrt[k] k}$$ But that's not what I wanted. Could anyone help me? Can a wizard under the effects of Feign Death issue telepathic commands to their familiar? A wizard has a summoned familiar within 100 feet that is currently idle and has  (PHB, p. 240) cast on them by another PC. Would the wizard still be able to issue telepathic commands to their familiar?  Specifically, while Feign Death states that they appear dead and are incapacitated and blind, they aren't listed as being unconscious.   doesn't state that issuing telepathic commands requires an action (which you can't do while incapacitated), whereas it does state that seeing through your familiar's eyes requires an action as does dismissing it.  I was DMing a game where this occurred and I hastily ruled that the wizard was unable to issue commands, but I am second guessing myself after the fact. And knowing my PCs this situation is likely to happen again. Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?  I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.  Does time have a starting point of note?