What are the units for rate of movement through time?
What is the speed of time?
Calculating determinant with different numbers on diagonal and x everywhere else I'm having troubles solving the following determinant:  $$\left|   \begin{array}{cccc}     a_1 &amp; x &amp; \ldots &amp; x \\    x &amp; a_2 &amp; \ldots &amp; \vdots \\     \vdots &amp; \ldots &amp; \ddots &amp; x \\     x &amp; \ldots &amp; x &amp; a_n \\   \end{array} \right|$$  I have tried coming up with recurrent formula, but there's always one lingering row and column. I also tried finding values of x for which determinant is 0, hoping I could get a nice polynomial solution, with no luck.  By subtracting first row from every other I get somewhat nicer result, but there's still that one row/column which I can't solve. If all other columns are added to the first one and then I subtract the first row, then the column doesn't depend on x anymore, but I haven't found 1..n much more useful. As part of an early quest, my PCs are likely to face off against a wizard.  The problem here is that the stat block for Mage NPCs listed in the Basic Rules has a Challenge Rating of 6, while the PCs are going to be five Level 1s. So, instead of messing with the stat block and risk weakening it too much/too little, I'm choosing to create a Level 1 Wizard PC and use it as an NPC for this quest.  The problem with that is that, as far as I can tell, there aren't any designated rules for assigning either Challenge Ratings or XP values to a PC. So, should I just give the wizard a Challenge Rating equal to his level (and thus have him give 200 xp when killed), or is there a better way to do this?  (This is different from , because he's asking for a way to assign a level to an NPC, while I'm asking for a way to assign an XP value/Challenge Rating to a PC (for use as an NPC). Similar questions, but different.) Xubuntu Dropbox icon fail Do superheroes/villains like Quicksilver or Flash with super speed get tired due to running? Usage of "shall we?" What does it mean and where would I use it? Validity and Equivalence of two definitions of the real exponential function The Problem :  We state the following two definitions of the real exponential function from the . We're interested in showing that the two definitions are valid $($i.e. the defining sequence/series does converge to a unique real number$)$ and that the two definitions are equivalent.   I'm stuck at a couple of points $($which are described in highlighted lines$)$. Any help would be much appreciated. Thank you!     Definition $1$. The exponential function can be defined as the following limit of a sequence $$\exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$$      Definition $2$. The exponential function can be defined as a  $$\exp x := \sum_{n = 0}^\infty \frac {x^n} {n!}$$   My Progress and two places where I'm stuck : Essentially the solution consists of three parts,  namely validity of definition $1$, validity of definition $2$ and equivalence of the two definitions.     Validity of Definition $1$. I'm stuck here! Can we show that the sequence $(a_n)$ given by $a_n = \left(1+\frac{x}{n}\right)^n$ converges for every $x \in \mathbb{R} ??$ Is it eventually monotone and bounded $??$   Validity of Definition $2$. The  of the power series is $$r=\lim_{n \to \infty}\left| \frac{\frac{1}{n!}}{\frac{1}{(n+1)!}} \right|=\lim_{n \to \infty}\left| \frac{(n+1)!}{n!} \right|=\lim_{n \to \infty}(n+1)=+\infty$$ Thus the infinite series in the right-hand side of Definition $2$ converges to a unique real number for all $x \in \mathbb{R}$. Hence the definition is well-defined.  Equivalence of Definition $1$ and Definition $2$.  There's this  of Definition $1$ $\implies$ Definition $2$ in the Pr$\infty$fWiki page, but it's kind of under construction and I'm not really convinced by it. So I decided to try to take my own shot at it.  For all $n \in \mathbb{N} \cup \{0\},$ let  $$T_n=\left(1+\frac{x}{n} \right)^n, ~S_n=\sum_{k=0}^n \frac{x^k}{k!}$$ We have to show that $\lim_{n \to \infty} T_n = \lim_{n \to \infty} S_n$  Now, \begin{align} T_n &amp;= \left(1+\frac{x}{n}\right)^n\\ &amp;= 1+n\cdot\frac{x}{n}+\frac{n(n-1)}{2!}\cdot\frac{x^2}{n^2}+\cdots +\frac{n(n-1)\cdots 1}{n!}\cdot\frac{x^n}{n^n}\\ &amp;= 1+x+\left(1-\frac{1}{n}\right)\cdot \frac{x^2}{2!}+\cdots +\left(1-\frac{1}{n}\right)\cdots \left(1-\frac{n-1}{n}\right)\cdot \frac{x^n}{n!} \end{align}  Clearly, $$S_n-T_n=\left\{1-\left(1-\frac{1}{n}\right)\right\}\frac{x^2}{2!}+\cdots +\left\{1-\left(1-\frac{1}{n}\right)\cdots \left(1-\frac{n-1}{n}\right)\right\}\cdot \frac{x^n}{n!}\geq 0$$     I'm stuck at this point. Can we show that $S_n-T_n \leq B_n$ such that $B_n \to 0$ as $n \to \infty ??$ How do you performance test JavaScript code? CPU Cycles, Memory Usage, Execution Time, etc.?  Added: Is there a quantitative way of testing performance in JavaScript besides just perception of how fast the code runs? Noticing that the "Human Verification" image did not show, I found that the src, , shows that sstatic.net is giving an SSL vertificate for *.stackexchange.com.. Save entire process for continuation after reboot I developed an algorithm for a fairly hard problem in mathematics which is likely to need several months to finish. As I have limited resources only, I started this on my Ubuntu 12.04 (x86) laptop. Now I want to install some updates and actually restart the laptop (the "please reboot" message is just annoying).  Is there a way to save an entire process including its allocated memory for continuation beyond a reboot?  Here is some information about the process you might need. Please feel free to ask for further information if needed.   I called the process in a terminal with the command "./binary &gt; ./somefile &amp;" or "time ./binary > ./somefile &amp;", I cannot really remember. It's printing some debug information to std::cerr (not very often). It's currently using roughly 600.0 kiB and even though this will increase, it's unlikely to increase rapidly. the process runs with normal priority the kernel is 3.2.0-26-generic-pae, the cpu is an AMD, the operating system is Ubuntu 12.04 x86. it runs since 9 days and 14 hours (so too long to cancel it ;-) ) Casting a Somatic, non-Material spell while holding a focus In  a side issue came up of whether you can cast a spell with Somatic components but no Material components while holding a focus in the hand used for Somatic components.  My reading of the rules sees "A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus--but it can be the same hand that he or she uses to perform somatic components." - it doesn't say you have to be holding the focus for the purpose of avoiding materials.  However, @Adeptus and @enkryptor argue that - since that rule is in the section on Material components - you don't consult it in for spells without Material, and fall back to the Somatic rule "the caster must have free use of at least one hand".  (I also appeal to the logic of fluff - why should it be harder to cast a spell with fewer components?)  Is there an official RAW or Word Of God on this matter? Can you cast a material-less, somatic spell while holding a focus in your only free hand? "Has/Have/Had" as a main verb and auxiliary verb I need help regarding the use of has/have/had both as a main verb and as an auxiliary verb.  In the sentence below, the word has acts as the main verb. I would normally follow it with the phrase "so do I."        Kenny has a car, and so do I.      On the other hand, the word has in the following sentence acts as an auxiliary verb. I would normally follow it with the phrase "so have I."        Kenny has bought a car, and so have I.      a) Are sentences 1 &amp; 2 above correct? I'm under the impression that so do I is used when has/have/had is the main verb of a sentence, and so have/had I if it's an auxiliary verb in a sentence.  b) Does the verb has/have always need a supporting "auxiliary verb" (either implied or mentioned)? (e.g., Kenny has a car = Kenny does have a car.) How long does a ban or suspension on an SE site last? I've tried to make some searches about ban and suspension but the data I found doesn't seem to correspond to my experience. I also cannot find a FAQ on the help section.  Can you explain me how these mechanics work?   I've just found  for a review ban duration to be extended automatically at every new suspension . But it's speaking of 2, 7, 30 days.  I've heard from a moderator that the suspension duration is 7, 30, 365 days. Is that correct? Can a moderator change the duration of a suspension? I need to get a history of which processes have run on my Windows machine, and when they were run.  However, I cannot use any third-party software, as I cannot guarantee that it will always be running.  Is there any way to get this information only using Windows built-in functionality? How to print a string at a fixed width? Most of what I have read about retry operations is that they are vaguely complicated and indepth. I have written the code below which seems to accomplish this without too much confusion - I'm wondering if there are any pitfalls I am not seeing here.   The class uses a top-level schedulable class, which calls a queueable class to make the callout, and if the callout is unsuccessful, it just schedules the top level class again five minutes later. After testing it appears this works indefinitely.   A simple string token, assigned at the first invocation, is also passed between the classes, and I have tested and found that the token does persist, even when runs are scheduled for a later time.   Thanks for helpful thoughts!  public class SelfServe implements Schedulable {      public String Token = null;      public void execute(SchedulableContext ctx) {                  RetrieveMachineDetails Job = new RetrieveMachineDetails();             Job.Token = Token;             System.enqueueJob(Job);      }           public class RetrieveMachineDetails implements Queueable, Database.AllowsCallouts {          public String Token = null;          public void execute(QueueableContext context){                                            Http http = new Http();             HttpRequest request = new HttpRequest();                             request.setEndpoint('https://app.endpoint.com/api');                     request.setMethod('POST');                     request.setHeader('Content-Type', 'application/json;charset=UTF-8');                                      request.setBody('{"query":  "$message_=\'METRIC\' &amp;&amp; $tier=\'customer\' "}');                                                                                                                                                                                    HttpResponse response = http.send(request);                            if (response.getStatusCode() == 200) {                 //do stuff                  }              else { //retry                  DateTime Dt = system.now().addMinutes(5);                                String day = string.valueOf(Dt.day());                                        String month = string.valueOf(Dt.month());                                          String hour = string.valueOf(Dt.hour());                                       String minute = string.valueOf(Dt.minute());                                                       String year = string.valueOf(Dt.year());                                   String ScheduledTime = '0 ' + minute + ' ' + hour + ' ' + day + ' ' + month + ' ?' + ' ' + year;                  SelfServe Job = new SelfServe();                     Job.Token = Token;                 System.schedule('SelfServe Job'+Dt, ScheduledTime, Job);              }         }        }  } A positive integer $n$ is called a balancing number if  $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$  for some positive integer $r$.     Problem: Find the general solution (closed-form expression) of equation $(1)$, in other words, find $f_1(m)=n, f_2(m)=r$ where $f_1, f_2$ are solution of $(1)$ for any integer $m \in \mathbb Z$.   I have asked a  with a constrain involving Pell's equation. What is the use of percent signs (%) at the end of lines? (Why is my macro creating extra space?) Awesome WM menu missing Firefox After sudo apt-get install awesome awesome-extra and logging into it (with no previous configuration that I know of), there's no menu entry for Firefox. All my other stuff is there - Mutt, Meld, jEdit, BOINC, etc., only Firefox is missing. I have no ~/.config/awesome/ directory, so there should be no override in place. Differences between MATCH FULL, MATCH SIMPLE, and MATCH PARTIAL? I've noticed a MATCH SIMPLE and MATCH FULL, but I don't understand what they do. I see default is MATCH SIMPLE; but, how do the other MATCH clauses to the FOREIGN KEY constraint function? Why does latex stretch small sections across the whole page vertically? I'm writing my bachelor-thesis and have a couple of small sub-sub-sections explaining conversions, like in the following  \subsubsection*{Assignment}  This document only explains non-compound assignments. The other kind, i.e \texttt{a += b}, is supported by the language too, but is not yet implemented.   \begin{itemize} \item If the first is an \texttt{auto} variable, no conversion takes place when assigning       the right hand side to the variable designated by the left hand side.  \item Otherwise, an assignment conversion takes place converting the given value to the type       of the designated variable.  \item Assignment requires an lvalue as the left hand side, and yields an lvalue result.  \end{itemize}  \subsubsection*{Multiplication} % ...   But latex renders it quite ugly with much space in between the items. I think it does so to fit the whole page. How can I tell it not to do so? Is what I'm doing right at all?