As $5$ is a prime number, thus $\sqrt{5}$ is an irrational number.  Now I am thinking about how to prove -  If $r$ is a rational number, then how do we prove $r\sqrt{5}$ is an irrational number?  I was thinking that since $r$ is a rational number, then $r$ can be expressed as the fraction in simplified form that is $r = \frac{a}{b}$ such that $a,b \in \Bbb{Z}$ and $gcd(a,b)=1$. So $r\sqrt{5} = \frac{a\sqrt{5}}{b}$, but How can this guarantee us the irrationality of $r\sqrt{5}$?   ALso let $c =r\sqrt{5}$, then $c^2 = 5r^2$, if we could prove it is a prime number, then its square-root $c$ must be irrational and ths proved but unfortunately we donot have $c^2$ prime as it has more than one factors like $r$ , $5$
Is an irrational number times a rational number always irrational?  If the rational number is zero, then the result will be rational. So can we conclude that in general, we can't decide, and it depends on the rational number?
Should my first trip be to the country which issued my Schengen Visa? In case, if I get a multiple-entry Schengen visa in one of the embassies in Ukraine, Kyiv - should my first trip be to the country which issued the visa?  Any there any requirements of this sort? Does this condition vary depending on embassy which issued the visa?  What are the possible consequences on not visiting that country first? Does it somehow depend on country where the embassy is (i.e. non-Schengen and not EU members)? I have a long sequence of hex digits in a string, such as      000000000000484240FA063DE5D0B744ADBED63A81FAEA390000C8428640A43D5005BD44   only much longer, several kilobytes.  Is there a builtin way to convert this to a bytes object in python 2.6/3? Mean, Average and Expectation What is the precise difference between the Mean, Average and Expectation ? Is there any mathematical formaulation to show the difference ? I see that often they are used interchangeably. Reference template variable within Jinja expression Rule of thumb for number of bootstrap samples I wonder if someone knows any general rules of thumb regarding the number of bootstrap samples one should use, based on characteristics of the data (number of observations, etc.) and/or the variables included? Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials? Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?   I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases? apt-get update, unlinking Can someone explain me this  please I am a bit confused?  It's from a texbook I am reading (block cipher companion)  it says that:      each weak key has  $2^{32}$ fixed points $m$ where $\operatorname{DES}_w(m) = m$ ( $w$ is a DES weak key). In other words encrypting a fixed point gives exactly the same point. To see this consider the $2^{32}$ ciphertexts after 8 rounds of encryption for which the two 32 bit halves are equal. For a weak  DES key, the round keys of the eighth and ninth rounds are equal. Thus the intermediate texts after seven rounds and after nine rounds of encryption will be equal. Since the round keys of the 7th and the 10th rounds are equal, the intermediate texts after 6 rounds and after 10 rounds of encryption will be equal. Continuing like this it follows that the plaintext will be equal to the ciphertext.    Now did not quite get :   Why are there $2^{32}$ fixed points?  And why after the eighth round of encryption the two halves must be equal? What does the author mean by the intermediate texts exactly? And how are they being equal?   P.S.: This question is not asking for what the weak keys are. I am asking for "fixed points" not exactly weak keys. For this reason I think it is not a duplicate of any other question. There is one similar question to what I've asked for which I have checked and saw that no satisfactory answer has been given there either.  Here is the similar question with no proper answers: Prove that there exists such a set of $n$ positive integers Prove that for any positive integers $m$ and $n$ , there exists a set of $n$ consecutive positive integers each of which is divisible by a number of the form $d^m$, where $d$ is some positive integer not equal to $1$.  I don't know how to approach this question. I have a Visual Studio 2008 solution with two projects (a Word-Template project and a VB.Net console application for testing).  Both projects reference a database project which opens a connection to an MS-Access 2007 database file and have references to System.Data.OleDb.  In the database project I have a function which retrieves a data table as follows   private class AdminDatabase    ' stores the connection string which is set in the New() method    dim strAdminConnection as string     public sub New()    ...    adminName = dlgopen.FileName    conAdminDB = New OleDbConnection    conAdminDB.ConnectionString = "Data Source='" + adminName + "';" + _        "Provider=Microsoft.ACE.OLEDB.12.0"     ' store the connection string in strAdminConnection    strAdminConnection = conAdminDB.ConnectionString.ToString()    My.Settings.SetUserOverride("AdminConnectionString", strAdminConnection)    ...    End Sub     ' retrieves data from the database    Public Function getDataTable(ByVal sqlStatement As String) As DataTable         Dim ds As New DataSet         Dim dt As New DataTable         Dim da As New OleDbDataAdapter         Dim localCon As New OleDbConnection           localCon.ConnectionString = strAdminConnection          Using localCon             Dim command As OleDbCommand = localCon.CreateCommand()             command.CommandText = sqlStatement             localCon.Open()             da.SelectCommand = command             da.Fill(dt)             getDataTable = dt         End Using      End Function End Class   When I call this function from my Word 2007 Template project everything works fine; no errors. But when I run it from the console application it throws the following exception     ex = {"The 'Microsoft.ACE.OLEDB.12.0'   provider is not registered on the   local machine."}   Both projects have the same reference and the console application did work when I first wrote it (a while ago) but now it has stopped work.  I must be missing something but I don't know what.  Any ideas? Windows 10 lets you 'schedule' a reboot for later. I want to disable it.  Evidently Windows scheduled itself for a reboot last night when I wasn't looking and just closed everything I had been working on the night before.  I reboot on the regular; I don't need Windows to do that for me.  Can I disable it completely? I don't mind if it downloads everything, and then says "hey, you should reboot," but it should never reboot itself, ever.  I'm using the "Pro" edition of Windows 10. How can I restart a Windows 7 computer without screen or mouse? I've recently had problems with my work computer not recognizing either of its attached screens.  I've also had problems with my home computer "locking" the mouse to one app, and no longer acknowledging clicks on, say, the start menu.  In both cases the computer is still running, and I can close apps and so forth with the keyboard.  Restarting usually addresses the problem.  What's the simplest way to restart a Windows 7 computer by the keyboard alone?  Something consistent across all versions would be super.  In Win XP, I believe it was Ctrl-Alt-Delete; pause; Alt-R for Restart; Enter to select.  But on 7 Ctrl-Alt-Delete no longer brings up the same menu.  I'd use Ctrl-Alt-Esc for the Task Manager, but it no longer has a Restart menu (at least on Win 7 Home). How to recover data from a failing hard drive? An external 3½" HDD seems to be in danger of failing — it's making .  I've acquired a replacement drive, and want to know the best strategy to get the data off of the dubious drive with the best chance of saving as much as possible.  There are some directories that are more important than others.  However, I'm guessing that picking and choosing directories is going to reduce my chances of saving the whole thing.  I would also have to mount it, dump a file listing, and then unmount it in order to be able to effectively prioritize directories.  Adding in the fact that it's time-consuming to do this, I'm leaning away from this approach.  I've considered just using dd, but I'm not sure how it would handle read errors or other problems that might prevent only certain parts of the data from being rescued, or which could be overcome with some retries, but not so many that they endanger other parts of the drive from being saved.  I guess ideally it would do a single pass to get as much as possible and then go back to retry anything that was missed due to errors.  Is it possible that copying more slowly — e.g. pausing every x MB/GB — would be better than just running the operation full tilt, for example to avoid any overheating issues?  For the "where is your backup" crowd: this actually is my backup drive, but it also contains some non-critical and bulky stuff, like music, that aren't backups, i.e. aren't backed up.  The drive has not exhibited any clear signs of failure other than this somewhat ominous sound.  I did have to fsck a few errors recently — orphaned inodes, incorrect free blocks/inodes counts, inode bitmap differences, zero dtime on deleted inodes; about 20 errors in all.  The filesystem of the partition is ext3. I use the following command to clear a directory, of files and directories over 30 days old, and move them to an archive directory which I can delete after a few weeks if nobody asks for their files back. The target directory has subdirectories by user name, so will the archive directory.  This is the command I use:  find /path/to/directory/username/ -mtime +30 -exec mv "{}" /path/to/archive/username/ \;   I suggested a modified version of this to answer a question on ask ubuntu, another user edited the code to change the end of line \; for + as it's faster(and more correct?).   However, using + in this way works if the -exec command is ls -lh but not in the actual command that I use. If I try it with + I get an error message:  find: missing argument to '-exec'   I don't understand why it's behaving this way, or what the correct command would be. Please don't just post a command correction, I'd like to understand rather than just follow a suggestion blindly. What are the basics needed to learn quantum computing? I have a parent theme that uses correctly load_theme_textdomain() to load all the translated strings in many languages.  Then I created a child theme that uses load_child_theme_textdomain() to achieve the same thing for its strings.  There are certain translated strings for a particular language on the parent theme that I'd like to replace/override in the child theme.  I know if they were on a template file I could replace the file and simply change the textdomain for those strings, but unfortunately the ones I'm talking about are used in many places and also in the dashboard (so inside some filter/action functions).  So my question is: is there a way to replace those translated strings inside the child theme without having to replace parent template files or functions?  I don't know, maybe adding a parent-theme.mo file inside the languages folder of the child theme with only those strings translated, or something like that. Is it possible in Latex to make a triangular hat? The one that is used in math over three letters representing points to indicate a triangle.  Something like: Which transformations are canonical? Which transformations are ? So I have a 2010 15" macbook pro (MBP6,1) and I replaced the optical drive with a hard disk. I wanted to install windows 7 on the hard disk with boot camp with a USB and ISO image.  After searching I found that I need to modify my plist.info to allow my boot camp to create a bootable USB (since normally it doesn't have the option to create the USB for my model since it came with a disk drive).   After doing so and seemingly having a fine (but long) install process onto the USB. I told boot camp how much space to partition windows on and hit install.  Then the computer restarted. When it came back up it was stuck on a grey apple logo screen forever (I waited 15 minutes to ensure it wasn't just "thinking"). I force restarted and the next time it turned on it got stuck on the grey logo for a bit but then proceeded to go to a black screen that said " No bootable device - please insert a bootable device and hit any key". I tried hitting keys with he USB in but nothing happened.  I had to eventually remove the USB and reset PRAM to get it to boot into osx again  Is there any way to install windows 7 from a usb on my macbook pro using boot camp? How can I enable "clicklock"?