Show that $\lim_{m \rightarrow \infty}(\frac{4^m}{2m \choose m} \mathbb{P}(X = m + k))^m = e^{-k^2}$
Finding $\lim_{n\to\infty} a_n^n$ where $a_n=\frac{(n!)^2}{(n+k)!(n-k)!}$
Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)&gt;9$   This is how I tried doing it:  Let $p= a + b + c,$ and $q=\frac1{a}+ \frac1{b} + \frac1{c}$.  Using AM>GM for $p, q$, I get:  $$\frac{p+q}{2} &gt; {(pq)}^{1/2}$$ $$\sqrt{(a + b +c)\bigg(\frac1{a}+ \frac1{b} + \frac1{c}\bigg)} &lt; \frac{\big(a+\frac1{a} + b+\frac1{b} + c+\frac1{c}\big)}2$$ And for any $x\in \mathbb{R}, \space \space x+\frac1{x}≥2.$ Thus,   $$(a + b +c)\bigg(\frac1{a}+ \frac1{b} + \frac1{c}\bigg)&lt;9, $$ which is the opposite of what had to be proven. What did I do wrong? What causes atoms to have their specific colors? As linguistics student, I have to put more than one thing on a character every now and then. For example, I need to type ā̆. I thought I could simply do that by typing \={\u{a}}, but then the macron is placed above an empty space in front of the character.  How can I solve this? Thanks  P.S. I'm using MiKTeX 2.9, running on Windows 8; in case it might be of use. Is there a word with phobia suffix for fear of waking others up? I usually come home after basketball at like 12:30 a.m. and at this time my mom and sister usually sleep (my Dad though is usually up). When I come home, I don't go to sleep right away and will sometimes eat or do something else. When I do so, I do it in stealth, so that I don't wake up anyone. I do out of two things, to be polite as I do not want to wake anybody up and also the fact that if I make a loud noise and wake my mom up, my mom will get mad and lecture me. Does this count as a phobia? In addition, if it does what would the name of it be. I have seen the options such as a4paper, a5paper, letterpaper, etc... to go in the square brackets after \documentclass, but what I want is 8.25 inches width by 10.75 inches height for one document and 6 inches width by 9 inches height for another one. How is this done? How does RewriteBase work in .htaccess Proving that kernels of evaluation maps are generated by the $x_i - a_i$ I want to prove that the kernel of the evaluation map $s_a : \mathbb{C}[x_1,\dots,x_n] \rightarrow \mathbb{C}, x_i \mapsto a_i$ where $a = (a_1,\dots, a_n) \in \mathbb{C}^n$ is the ideal generated by $\{x_1 - a_1, \dots, x_n - a_n\}$.    The proof in the book first shows it easily for the case $a = 0$, and says that by substitutions of variables $x_i' = x_i - a_i$, you can show it for the rest of the cases.  I know that $f \in \ker s_a$ iff $f(a) = 0$.  And $\phi : \mathbb{C}[x_1,\dots, x_n] \rightarrow \mathbb{C}[x_1 - a_1, \dots, x_n-a_n], x_i \mapsto x_i - a_i$ is a surjective ring homomorphism.  Please give a hint and not the full answer.  Thanks. register_globals error in php I was stuck up with the error   directive 'register_globals' is no longer available in PHP in unknown on line 0   when tried to check the php version using "php -v" after enabling register_globals in php.ini file. I am not getting any php version info by doing so. Instead it throws the above mentioned error.After turning off this option, php info works quite well. It is very essential for me to have register_globals to be turned on.How can I have this corrected.  my php.ini is as follows:   ; Default Value: None ; Development Value: "GP" ; Production Value: "GP" ; http://php.net/request-order request_order = "GP"  ; Whether or not to register the EGPCS variables as global variables.  You may ; want to turn this off if you don't want to clutter your scripts' global scope ; with user data. ; You should do your best to write your scripts so that they do not require ; register_globals to be on;  Using form variables as globals can easily lead ; to possible security problems, if the code is not very well thought of. ;  register_globals = On  ; Determines whether the deprecated long $HTTP_*_VARS type predefined variables ; are registered by PHP or not. As they are deprecated, we obviously don't ; recommend you use them. They are on by default for compatibility reasons but ; they are not recommended on production servers. ; Default Value: On ; Development Value: Off ; Production Value: Off ;  register_long_arrays = Off  ; This directive determines whether PHP registers $argv &amp; $argc each time it ; runs. $argv contains an array of all the arguments passed to PHP when a script ; is invoked. $argc contains an integer representing the number of arguments ; that were passed when the script was invoked. These arrays are extremely ; useful when running scripts from the command line. When this directive is ; enabled, registering these variables consumes CPU cycles and memory each time ; a script is executed. For performance reasons, this feature should be disabled ; on production servers. ; Note: This directive is hardcoded to On for the CLI SAPI ; Default Value: On ; Development Value: Off ; Production Value: Off ; register_argc_argv = Off  ; When enabled, the SERVER and ENV variables are created when they're first ; used (Just In Time) instead of when the script starts. If these variables ; are not used within a script, having this directive on will result in a ; performance gain. The PHP directives register_globals, register_long_arrays, ; and register_argc_argv must be disabled for this directive to have any affect. ; auto_globals_jit = On Can I successfully bake previously-frozen bread dough? I prefer my bread freshly baked (who doesn't?), but my mixer prefers kneading two loaves at a time to just kneading one. I have tried to freeze the second loaf before baking it, but when I thaw it, it does not rise well before baking. I've tried thawing it entirely in the refrigerator and then letting it rise on the counter, but that did not seem to work. Does anyone have a good technique or suggestion for this? I'm trying to use the Html.DropDownList extension method but can't figure out how to use it with an enumeration.  Let's say I have an enumeration like this:  public enum ItemTypes {     Movie = 1,     Game = 2,     Book = 3 }   How do I go about creating a dropdown with these values using the Html.DropDownList extension method?  Or is my best bet to simply create a for loop and create the Html elements manually? When it comes to , I often read recommendations like "Use adb logcat to find some more details". Looking for a good way to backup all my apps including their data,  refers to adb backup and adb restore. In several places one can find ways to remotely do on an Android device with the use of , such as transferring files using adb push or adb pull, accessing the  via adb shell, and more. So ADB seems to be a good thing to have for an Android user.  But how to get this "ADB thingy" on my computer? Not being a developer, installing the entire  seems a bit overkill. Is there a more minimalistic approach available? Why does the guestbook chatbox not produce any network activity? I was trying to test if I was able to automatically send a message through curl by inspecting the network activity on chrome and getting the curl command, but I found out that in spite of messages showing up in the gestbook chat there was no any network activity. Why? Minimal cut edges number in connected Eulerian graph. I have to prove that Connected graph is Eulerian Graph if and only if the number of edges in minimal cut is even.  Implication from left to right. Well if $G$ is Eulerian Graph then degree of each vertex must be even. It's oblivious if i try to disconnect graph by removing all edges connected to the vertex with minimal degree. But minimal cut can be done also by dividing graph into 2 connected graphs (each graph with more then one vertex) and i have no idea how to deal with it.  Implication from right to left - completly no idea. Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$? If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$, then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$ hold? I know  it holds if we integrate over an interval of the form $[a,b]$ but not sure about improper integral! Horizontal spacing issue in \listoftheorems using amsthm + thmtools I've seen in every introductory material on LaTeX and in a lot of code of other people (event experts) that the \newcommand command is used by wrapping the first argument in braces, even if it seems unnecessary to me.   The braces are consistent with the usual way arguments are passed to LaTeX commands (since usually the braces are used even if the argument is made of a single token, for consistency and clarity), but in this single case they appear very ugly to me and in my opinion they lower readability.  But these are opinions, while the question is technical: is there any technical reason why one should write this:  \newcommand{\mycmd}[1]{Hello #1}   instead of this:  \newcommand\mycmd[1]{Hello #1} I have a list of arbitrary length, and I need to split it up into equal size chunks and operate on it. There are some obvious ways to do this, like keeping a counter and two lists, and when the second list fills up, add it to the first list and empty the second list for the next round of data, but this is potentially extremely expensive.  I was wondering if anyone had a good solution to this for lists of any length, e.g. using generators.  I was looking for something useful in itertools but I couldn't find anything obviously useful. Might've missed it, though.  Related question: How do I get a module path? I'm creating a module in Drupal 8 and I need to define a relative path to mymodule to pass it on a JavaScript file like this.  $scope.getPlatformIcon = function (_platform) {      var modulePath = "/modules/mymodule"      switch (_platform) {          case "youtube":          case "twitter":          case "instagram":          case "facebook":              return modulePath+"/img/"+_platform+".png";      }      return false; };   In Drupal 7 I could use the following.  var modulePath = Drupal.settings.mymodule.basePath;   For the Drupal 8 root directory, I could use the following.  drupalSettings.path.baseUrl   I didn't find how to get the module path.  How can I do it with Drupal 8? #!/bin/bash echo "$(curl -s -I https://google.com|grep Server)" abc   returns:  abcer: gws   and bash -x shows:    ++ grep Server ++ curl -s -I https://google.com ' abco 'Server: gws  abcer: gws   but    echo abc "$(curl -s -I https://google.com|grep Server)"`    returns  abc Server: gws   In other words, if I add text AFTER the curl command, it gets mangled, but it's ok if I add it before curl. What is happening? I cannot figure it out. Can I use my Windows 7 license in a dual-boot configuration? Possible Duplicate:          I've purchased a retail license of Windows 7 Ultimate for my machine at home.  I'm thinking of creating a dual-boot install:   64-bit install for work apps (need the RAM for virtual machines and RAMDisk) 32-bit install for entertainment apps (Media Centre, games, etc)   How does retail licensing work in such situations? Can I activate the same license on both installations?  Edit:  I would like to add that I want to have separate partitions for work and play.