Prove the identity $\sum\limits_{k=1}^{n+1} \sum\limits_{m=0}^{k-1} \binom{n+1}{k} \binom{n}{m} = 2^{2n}$
proof of binomial identity involving double sum
How do I Sort a Multidimensional Array in PHP I have CSV data loaded into a multidimensional array. In this way each "row" is a record and each "column" contains the same type of data.  I am using the function below to load my CSV file.  function f_parse_csv($file, $longest, $delimiter) {   $mdarray = array();   $file    = fopen($file, "r");   while ($line = fgetcsv($file, $longest, $delimiter))   {     array_push($mdarray, $line);   }   fclose($file);   return $mdarray; }   I need to be able to specify a column to sort so that it rearranges the rows.  One of the columns contains date information in the format of Y-m-d H:i:s and I would like to be able to sort with the most recent date being the first row. Does Exchange support disposable e-mail addresses? Sometimes it is useful to have e-mail address like foo-randomnumber@example.org, where randomnumber identifies the conversation between a robot such as a mailing list server and a user. Is this possible with Microsoft Exchange? I have a question.  I have been given this proof: "For any $n$ in the integers where $n&gt;2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$."  I have gone through and actually proved this, (that the numbers are $1$ and $n-1$) but i didn't' know how to prove that $n-1$ is in fact in the set $U(n)$. Is it because two consecutive numbers are always relatively prime? Floating point arithmetic not producing exact results The black book knowledge How to hide users from the GDM login screen? I have recently added several new users, that I need for qmail. Now they appear in the box in the login screen and clutter it, and I have to scroll to find my user. How can I hide those users from the login box? If the edges of the complete graph $K_{17}$ (on 17 vertices with no three collinear) are each colored one of three colours can it be proven to have two or more monochromatic triangles? How do I get a substring of a string in Python? Is there a way to substring a string in Python, to get a new string from the third character to the end of the string?  Maybe like myString[2:end]?  If leaving the second part means 'till the end', and if you leave the first part, does it start from the start? i want to create new event in SharePoint calendar list in which I have some custom columns,i tried below code but it is not working      function bookingRoomForMeeting() {         var ctx = new SP.ClientContext(appWebUrl);//Get the SharePoint Context object based upon the URL           var list = ctx.get_web().get_lists().getByTitle("MyCalendarList"); //Get the List based upon the Title         var listCreationInformation = new SP.ListItemCreationInformation(); //Object for creating Item in the List         listItem = list.addItem(listCreationInformation);         listItem.set_item('Title', $("#txt_Title").val());         var date = $("#txt_startTime").val();         var dateTime = moment(date, 'DD/MM/YYYY HH:mm a').format("MM/DD/YYYY HH:mm a");         listItem.set_item('EventDate', dateTime);         var enddate = $("#txt_endTime").val();         var enddateTime = moment(enddate, 'DD/MM/YYYY HH:mm a').format("MM/DD/YYYY HH:mm a");         listItem.set_item('EndDate', enddateTime);         listItem.set_item('EmailId1', $("#txt_EmailId1").val());         listItem.set_item('EmailId2', $("#txt_EmailId2").val());         listItem.update();         ctx.load(listItem);         ctx.executeQueryAsync(             Function.createDelegate(this, success),             Function.createDelegate(this, fail)            );     } function success() {     alert("Completed"); } function fail() {     alert("operation failed"); }   list am using is SharePoint hosted app list,can any one tell me where am going wrong? How do i know which wheels will fit my bike? my bike is:  and everything I have came with the bike, only now the axle of the rear wheel bent and the skewer has lost its grooves.  and I would like to replace the rear wheel with the 12 x 142mm version of:   I want to reuse the old casettes and disc break. I was told to buy new rotor bolts, will these do?   Will I be able to migrate all my previous parts onto this new wheel and will the wheel fit onto my bike?  update: A replacement axle would be prefered. What axle would work for my bike? What is the most convenient way to change options for Graph[] objects? Use /testfor to make a custom command Basically, I'm trying to recreate the Faction plugins /f map in my single-player without mods as my PC doesn't run them very well. (I know it sounds strange.) I am trying to use the /testfor to test if I have written '/f map' in chat, and if I have, it would show me a custom generated map. I don't know how to word this better xD Essentially what I want is for it to test if I have written a command in chat, and if I have, display something. Once a post has been edited, you can go into the revision list, and click "View Source".  If a post has not been edited (and I don't have the edit privilege) is there any way to see the raw Markdown?  The reason I ask I because of  - it's impossible to see the source markdown, and how it's different from the rendered Markdown.  Not exactly a huge issue, just curious! Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice? Why is this power adapter transformerless? Remotely changing hard drive boot order, windows 7 I have one workstation chassis with two hard drives.  Depending on which one I boot from, I get one of two different operating systems: both are windows 7, but they have different static IPs, serve totally different purposes, etc.    The workstation has a modern asus motherboard, core i7 intel CPU, 32 GB ram.   Is there a way to remotely change which hard drive is booted from on a reboot?  Both systems have remote users, and i'd like a solution that is more elegant than being physically present and selecting boot order manually. What is NDSolve`FEM`*? Windows 8.1 missing font files after restart Some days ago I rebooted my windows 8.1 system. After the reboot all my system texts were very cryptic - showing only some strange symbols. After some investigation I discovered that my c:\windows\fonts folder was nearly empty. So I copied the font files from another windows 8.1 installation to the nearly empty folder. As soon as I had copied the files the system texts were readable again.  But after another reboot the problem occured again - the systems font folder was empty again. Does anyone know why windows deletes my font files during a restart? Representative point of a cluster with L1 distance The representative point of a cluster or cluster center for the k-means algorithm is the component-wise mean of the points in its cluster. The mean is chosen because it helps to minimize the within cluster variances (which is to say that it is minimizing within cluster squared Euclidean distance, since its the same). Now, when I want to use a different distance metric, like L1 (Manhattan distance), I would be minimizing the within cluster absolute deviations (if I choose to). I was looking at what the centers calculation would be like, because the mean won't reduce the new objective function, and I found this on the Matlab documentation of their kmeans function for the distance parameter     cityblock - Sum of absolute differences, i.e., the L1 distance. Each centroid is the component-wise median of the points in that cluster.   Now, how did they come up with the component-wise median? How does a median minimize absolute deviations? Remove rogue app from menu bar