Show that $e^x\ge x^e$ for all $x\ge0.$ I was playing around with splitting a number into parts (that add to the original number) and then multiplying the parts to get the result, and I found that the product is highest when the parts tended towards $e$. This inequality is a consequence of that. Is there a proof or intuitive explanation for this?
I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$   I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the reason behind the above inequality? I know this is math community, but I'd appreciate a more intuitive explanation than a technical one.
I'd like to "add to" an existing command by writing something like  \renewcommand{\somecommand}{\begin{something}\somecommand\end{something}}   but (not surprisingly) I get errors about being "too deeply nested".  Is there a way to take an existing command, add to it (by appending or prepending additional commands or text), and then have that "augmented" command replace the existing command? round in PHP shows scientific notation instead of full number I'm trying to do an echo of a variable containing 1400000.  so there is written: echo round(1400000); this gives 1,4E+6 instead of the full number. Anybody an idea on how to display it fully? Maintenance plan for creating transaction log backups I have a database that is set to the 'Full' recovery model, and I'm creating a maintenance plan to backup the Transaction Log every 15 minutes. A full backup is then taken once every 24 hours.  I was wondering: will the maintenance plan for the transaction logs create a lot of different files every 15 minutes, or will it by default append to the same file for 24 hours?  Also: what is preferred? I couldn't find anything on this subject (or maybe I was Googling the wrong words).  Thanks for any pointers anyone can give me.  EDIT: I also see that the checkmark "Verify backup integrity" is OFF by default. Is there a risk with turning it on? It seems to be that it should be on by default, so is there a reason for not using this? In a script I'm trying to find the Terminal.app window containing a specific tty device and raise that window to the top. Here's what I have in Applescript so far:  tell application "System Events"   set foundWin to false   if (name of processes) contains "Terminal" then     tell application "Terminal"       set theTabs to first tab of every window where tty is "$(tty)"       repeat with theTab in theTabs         if class of theTab is tab then           set theWin to (first window whose tabs contains theTab)           set selected tab of theWin to theTab           set foundWin to true         end if       end repeat     end tell   end if   if foundWin then     --RAISE THE WINDOW?!   end if end tell   Where I'm getting stuck is the "raising the window" part.  Here are some things that are not quite what I want:  set frontmost of theWin to true -- this brings the window to the front of the group of Terminal.app windows but doesn't raise it above any other windows.  tell application "Terminal" to activate -- this brings every Terminal window to the front in a big stack. I just want the one window.  tell application "System Events"     set theSysWin to first window of process "Terminal" whose name is (name of theWin)     perform action "AXRaise" of theSysWin end tell   This almost does it, but what it does is to raise the terminal window to the #2 position, still underneath the active window (if the active app is something other than Terminal.app.)  click theSysWin at {10,50} -- doesn't seem to do anything.  click first static text of theSysWin -- doesn't seem to do anything.  Is there a way to do this? It doesn't have to be in Applescript.  EDIT I did find this web page (  ) quoting an Obj-C/Cocoa call:  SetFrontProcessWithOptions(&amp;psn, kSetFrontProcessFrontWindowOnly);  I'm not familiar with either ObjC or Cocoa, and this is (ultimately) getting invoked from a shell script, so not sure where to go from there. A No-Nonsense Introduction to Quantum Field Theory SPSite.Exists returning true for a just deleted site collection Short story - I have a process that creates some sitecollection, so to test it I keep deleting and recreating the same site over and over. The process includes a check with the istruction SPSite.Exits, done to check if the sitecollection is present or should be created.  Sometime, soon after I delete the site collection, SPSite.Exists will report it as still available. Is this a cache related problem? Massless charged particles In the simple linear regression model, $\hat{\beta}$ is the sum of independent normally distributed random variables.  Is it false because in linear regression there is $\beta$ and not $\hat{\beta}$? How do you decline nouns borrowed from languages with several categories for declining nouns (or none at all)? Are the review queues planned for the iOS app? I enjoy reviewing on SO, is it currently possible to review on the iOS app? If not, is it planned to be implemented? Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant. How do you show that if $f$ is entire and if $\text{Im} f\ge0$, then $\text{Im} f$ is constant? Is this even true in the first place? If you're not used to seeing the 'low on internal storage' notification, what is it that you're doing to keep your phone's internal memory from filling up so quickly?   The 'SD Card and Phone Storage' menu in the settings doesn't really tell me much other than how much is free. And, I don't really have that many applications installed. The top 5 storage hogging apps on my phone are Gmail (18.6 MB), Twitter (17.5 MB), Facebook (12.55 MB), Maps (9.63 MB), and Swype (6.06 MB).  So, are there apps out there (besides your normal file manager ones, like Astro) that will help monitor internal phone storage? Is there something I should be doing to keep my phone storage use to a minimum? Is this metric space incomplete? Period character used for local computer name I'm wondering about the period character . used as a shortcut for the local computer name.  You can use it when logging into Windows 7, for example, to specify that the user account you're entering belongs to the local computer rather than to a domain:  .\MyLocalUserAccount   And you can use it within a Windows Management Instrumentation (WMI) query to indicate that the target is the local computer rather than a remote computer.  You can also use it when working with named pipes to for similar purposes.  However, you cannot use it as the UNC network name for the computer, like this:  \\.\MyShare   Does this character have an official name (e.g. "LocalHostToken") and when and where can it be used for this purpose? Proving that expression is equivalent to the definition of derivative Let $f$ be differentiable at $x=a$.   Prove that if $x_n \to a^+$ and $y_n \to a^-$ then:  $$\lim_{n\to \infty} \frac{f(x_n)-f(y_n)}{x_n-y_n}=f'(a).$$  Every option that I think about seems to my very trivial, so I believe that I am doing something wrong.  Both numerator and denominator approach zero as $n\to\infty$ as the case of the formal definition of derivative, but it isn't guaranteed that the limits are equal (“$\frac{0}{0}$”).   Any direction? I created a custom counter 'rtaskno' and a command '\rtask' as following:  \newcounter{rtaskno} \newcommand{\rtask}{%    \stepcounter{rtaskno}%    \thertaskno}   I like to use them as following:  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}   Later in the text, if I want to cross-reference 'Task 3', how can I do it? Where to put label? ¿Can you help me with axiom of regularity? I understand that if we define a set such that $A=\{A\}$ we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I define a set $A=\{A,b\}$ ($b$ disjoint from $A$)?  Thanks. How is a pitch supporting fast bowlers or spin bowlers? There are several algorithms to calculate distance between two latitude/longitude points.     Hubeny     But there are not so many algorithms (actually I can't finally find it) which can calculate latitude/longitude of point B from another latitude/longitude point A with bearing and distance.      I'm glad to know if there are any algorithms which is lesser accurate, but faster than Vincenty's formulae. Textbook on the Geometry of Special Relativity I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?  I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.  Thanks in advance.