how to create a vector that has a fixed number of points per decade?
How can I get exactly 5 logarithmic divisions of an interval?
Proving the so-called "Well Ordering Principle" Is there anything wrong with the following proof?  Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member.  Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, for if $B$ contained even one $b\in\mathbb{N}$, then that $b$ would satisfy $b=min(B). $ Contradiction, end proof.  I get the sense there is something wrong here, but I can't seem to define exactly what. Also, is there a context where one can simply take this principle as an axiom, and not have to prove it? After all, it is extremely intuitive. Thanks. When should I make my players actually roll for stuff? I've tried to focus my group heavily on Role Playing, perhaps too well given some of my recent problems, which are Orthagonal, and linked mostly to provide historical context.        A lot of the answers to these questions revolve around "have off the rail's multiple angles" and "don't make people roll for information they must have". That seems well understood to me at this juncture, in fact I have been doing just that, and I think it's causing some other problems.  I have only rarely made people roll for anything (combat aside) anymore, unless it's absolutely non critical. I try to get the players to Role Play their investigations and give them information when they look in the right places, and their are multiple right places.  This results in skills, especially social/mental, being completely inessential, and players not gaining important information because they aren't poking around the right area at all, or asking people who aren't interested in talking to them because they're currently a bunch of random Joe's.  The players aren't stuck at this time, I'm just trying to get better and not have functionally dead scenes, that don't rely on the players to play "what is the GM thinking" nor do I want to just give them every answer.  The reality is, I'm not sure when to make them roll to find things out. what criteria? or other advice, can I use to decide that the players are better off making a Roll to move forward than to purely use Role Playing?  update This question focuses on the Social and to a lesser degree Mental aspects, as it's usually pretty easy to tell when a Physical Roll is needed, even out of combat. It's harder to tell when to make people roll for things they can talk their way through.  I'm playing World of Darkness 2nd Edition, but answers not specific to that are welcome. Q. Find all pairs $(p,n)$ of positive integers where $p$ is prime and $p^3-p=n^7-n^3$.   Rewriting the given equation as $p(p+1)(p-1)=n^3(n^2+1)(n+1)(n-1)$, we see that $p$ must divide one of the factors $n,n+1,n-1,n^2+1$ on the $\text{r.h.s}$.   Now, the $\text{l.h.s}$ is an increasing function of $p$ for $p\ge1$. This implies that for any given $n\ge1$, there is exactly one real $p$ for which $\text{l.h.s}=\text{r.h.s}$. For $p=n^2$, we get $\text{l.h.s}=n^6-n^2&lt;n^7-n^3=\text{r.h.s}.$ This means that either $p&gt;n^2$ or $p&lt;n^2$ must hold.  Assuming $p&gt;n^2$, it follows that the prime $p$ cannot divide any of $n,n+1,n-1$. So $p$ must divide $n^2+1$ and hence $p=n^2+1\quad (\because p&gt;n^2)$.  Substituting the value of $p$ in the given equation we get, $n^2+2=n^3-n\implies n^3-n^2-n=2$. As the factor $n$ on the $\text{l.h.s}$ must divide $2$, the above equation has a unique integer solution $n=2$.  Finally, we get $(5,2)$ as the solution to the given equation.      But how do I conclude this is the only solution possible? Also, why does'nt $p&lt;n^2$ (the case which I ignored) hold? As a bonus question, I would like to ask for any alternative/elegant solution (possibly using congruence relations) to the problem. I'm looking into a restful design and would like to use the HTTP methods (POST, GET, ...) and HTTP headers as much as possible. I already found out that the HTTP methods PUT and DELETE are not supported from the browser.  Now I'm looking to get different representations of the same resource and would like to do this by changing the Accept header of the request. Depending on this Accept header, the server can serve a different view on the same resource.  Problem is that I didn't find a way to tell my browser to change this header.   The &lt;a..&gt; tag has a , that can have a mime type, looked like a good candidate but the header was still the browser default (in Firefox it can be changed in about:config with the network.http.accept.default key). Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$ If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$.  So what is the MGF of the ratio distribution $\displaystyle\frac{X}{Y}$? Execute script after specific delay using JavaScript Is there any JavaScript method similar to the jQuery delay() or wait() (to delay the execution of a script for a specific amount of time)? Why don't my female dwellers give birth to children? The Mac editor Latexian has built in methods for compiling using latex+dvips, pdflatex and xelatex.  Furthermore, it assumes bibtex and makeindex to deal with bibliographies and indexes, respectively.  How do I customize Latexian to use, e.g. LuaTeX for compiling or Biber for bibliographies? Should log file streams be opened/closed on each write or kept open during a desktop application's lifetime? While browsing on Safari and listening to iTunes I come across a video and then press the "play" button on the keyboard (to stop the music while I watch the video) it starts/stops the video instead of the music (I have no control over the music).  How can I quickly play/pause my music without affecting the video? How should I install HPLIP binary/proprietary plugin/driver for my HP printer or printer+scanner? Was Philip K. Dick able to watch a complete version of Blade Runner before his death or just unedited material?  The Blade Runner release date according to  and IMDb was 25 June, 1982. Philip K. Dick died a few months before, on March 2, 1982. Is a Schengen visa/transit visa required for a transit flight through two Schengen airports? I am travelling form Delhi (DEL) to San Francisco (SFO), I have to make a connection flight in Vienna that would take me to Frankfurt (FRA). My flight to SFO is from FRA, I have one ticket for entire journey and the flights are operated by Lufthansa or its subsidiaries.  Do I need a transit visa? My flight arrives at Vienna international airport and leaves from the same, and for second leg it arrives at Frankfurt international airport.  I asked this question as an addendum to my  but was advised to make a new post.  I hold Indian passport and valid US visa. Is there a general way to change the font of a particular symbol without switching packages?  For example, suppose I'm using Computer Modern for my entire document, but I want the "subset" operator to look as it does in mathabx.  How can I redefine it to appear in this way? Difference between 'struct' and 'typedef struct' in C++? I am trying to find the name of a direct to TV movie, possibly a 2-3 part miniseries which I believe aired on the SciFi channel in the 90s. I don't remember who starred in it, but the movie opened with an astronaut attempting reentry whose shuttle ends up exploding. He awakes in a large bubble of air in a river and after reaching the surface and swimming to shore, meets several other people, from seemingly many different times throughout history (including one alien, who died on Earth).  Over the course of the movie, we learn the world is the afterlife, where everyone goes when they die. The major plot involves launching a steamboat (built by Mark Twain, no less) to travel up a river to the mountains, where the beings responsible for bringing everyone to this world are.  Does anyone know what the title of this movie/miniseries is? Short story with competitors complaining to government about a new very popular synthetic food $\mathbb S_n$ as semidirect product In  note, I've read that $\mathbb S_n$ is a semidirect product of the alternating group $A_n$ by $\mathbb Z_2$. So I am trying to define a morphism $\rho: \mathbb Z_2 \to Aut(A_n)$ to show that $\mathbb S_n \cong A_n \rtimes Z_2$. I would appreciate suggestions on how could I define the morphism. Thanks in advance. How to check which package created a user? It is suspected that some packages have scripts run at install time which create user accounts in /etc/passwd. How can I check which package created a specific user? What can I do when getting "We are no longer accepting questions/answers from this account"? If you are reading this, you may have been affected by one of our post quality bans. If so, it is important that you  to understand what has happened and how to regain the permissions that you lost. While trying to ask a question, one could see:  We are no longer accepting questions from this account. See  to learn more.  Likewise, for answers:  We are no longer accepting answers from this account. See  to learn more.   Why am I getting this message? Are deleted posts taken into account too? Is a question/answer ban the same as a suspension? How do I avoid getting a question ban? How long do I have to wait before I can post again? What can I do to release the ban? How can I reactivate my account? Can I simply create a new account? I'll just ask somewhere else on the SE network, and they'll migrate my question to the correct site! Will a ban on one Stack Exchange site affect my standing on other sites in the network? Does this apply to meta sites too? My account is in good standing. Why am I still blocked?