How to prove $f$ can be uniformly approximated on $[1,\infty )$ by a function $g$, using Weierstrass theorem.
Stone-Weierstrass theorem with $p(1/x)$
Why aren't there any fighter carriers in the Star Trek universe? All of the ships of every species seem to be medium to large vessels with multiple crew members. Why aren't there single fighter craft and carrier ships to transport them in mass?  I've just noticed the difference between Battlestar Galactica and Star Trek, where the former has primarily aircraft carriers, and the latter has battleships.   Wouldn't a huge fleet of small fighters be extremely maneuverable and possibly stealthy?  Update:  For clarity, my question is not about the Federation and their choice of ships, but the universe at large. I would think there would be some diversity in space warfare strategy, such as huge armadas of fighters with base stations / carriers. Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$ Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) &gt; 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?  My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the . It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas? I recently obtained a Seagate Expansion 3TB external harddrive. I'm trying to copy over all of my stuff to prepare for reformatting and installing a new OS. A handful of my mp3s did not transfer when I was doing my music folder. I checked them after the fact and the ORIGINAL files became corrupted somehow. I am certain of at least a few that were not corrupted in anyway before doing this. Every now and then during my transfer I got  There was an error copying the file into /media/***/Seagate Expansion Drive. Error splicing file: Input/output error   Each of these files that I check after are corrupted at some point in the duration and will not play. Why did this happen and what can I do to fix it? $f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,  $$\int_0^a f^{-1}(x) dx.$$  Reasoning  The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$  $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.  So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?  Question  Is my reasoning correct? I encountered the integral $$\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx = -0.9393323982...$$ while researching the evaluation of harmonic sums. Mathematica 11 is not able to evaluate this integral, and is not able to evaluate the underlying indefinite integral.  I considered using the Maclaurin series for the expression $\sin ^{-1}(x)^2$ to evaluate this integral. Using this Maclaurin series, it is easily seen that the problem of evaluating the above integral is equivalent to the problem of evaluating the series $$\sum _{n=1}^{\infty } \frac{2^{2 n-1} H_{n-\frac{1}{2}}}{(1-2 n) n^2 \binom{2 n}{n}}=-0.9393323982...$$ which Mathematica 11 is not able to evaluate directly.  I also considered using the Maclaurin series for the expression $\log \left(1-x^2\right)$ to evaluate the above integral. Using this Macluarin series, it is easily seen that this integral is equal to $$\frac{1}{4} \left(16 \pi  C-\pi ^3-\pi ^2 \log (4)\right)+\sum _{n=1}^{\infty } \frac{\, _3F_2\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},n+1;1\right)}{n^2-2 n^3}=-0.939332...,$$ but Mathematica is unable to evaluate the above infinite series.  What is a closed-form evaluation of $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$? What techniques can be applied to evaluate this definite integral? I accidentally closed it without voting, how can I find it? When I suggest declaring war on the target it says there's already a proposition to do so. Ideally, I'd like to write a unit test to help me "dummy check" for unexpected effects of the upcoming guest user sharing setting changes.  We'd had some controller/trigger code "gracefully handling" data it couldn't see thanks to the new sharing restrictions ... which resulted in simply "not doing" some work that it was supposed to do, which from a business process perspective isn't "graceful" at all (it was an important automation for downstream processes).  I'd like to write up a unit test with some system.assert statements validating that "data still goes where it's supposed to" when the "submit" button a public-facing Visualforce page of a community is clicked.  Unfortunately, I can't seem to make the "submit" button on a Controller for a public-facing Visualforce page work in the context of a unit test because the record whose creation it results in doesn't get its corresponding "sharing setting" record created in time for the execution context to pick it up.  I did a little proof of concept here ...  @isTest public class TestClass {      private static Id guestProfileId = Util.getProfileIdByName('Public Facing Portal Profile');     private static User guestUser;     private static Id portalGroupId = [SELECT Id FROM Group WHERE Name = 'Public_Portal' LIMIT 1][0].Id;      private static Custom_Object__c co;       private static testMethod void runTest() {         createCO();         Test.startTest();         // Note that putting createCO() inside the test and doesGuestSharingRuleExistForCO() doesn't help.         // That's what I was afraid of.         try {             doesGuestSharingRuleExistForCO(co.Id);         }         catch (Util.DeveloperException e) {             System.assert(FALSE, e.getMessage());         }         Test.stopTest();     }       private static void cacheUsers() {         guestUser = [SELECT Id FROM User WHERE profileid = :guestProfileId AND IsActive = TRUE LIMIT 1][0];     }      private static void createCO() {         co = new Custom_Object__c();         INSERT co;     }      private static void doesGuestSharingRuleExistForCO(Id coId) {         List&lt;SObject&gt; existingSharingRulesOnCO = [             SELECT Id FROM Custom_Object__share             WHERE RowCause = 'GuestRule' AND AccessLevel = 'Read' AND ParentId = :coId AND UserOrGroupId = :portalGroupId LIMIT 1         ];         if ( existingSharingRulesOnCO.isEmpty() ) { throw new Util.DeveloperException('Guest User sharing is not on for this custom object ID'); }     }      private static Boolean isCOSharingFlippedOnAtAll(){         String lookingFor = 'Custom_Object__Share';         for (Schema.SObjectType sObjectType : Schema.getGlobalDescribe().values() ) {             String sobjName = String.valueOf(sObjectType);             if (sobjName.contains(lookingFor) ) { return TRUE; }         }         return FALSE;     }  }   Am I just out of luck?  Is there no way to properly unit-test the effects (to automations relying upon "guest user" sharing settings when running as "guest user") of simulating the insertion of records into my database from a Visualforce page, due to not being able to separate out "execution context" enough and "wait" for the sharing rule record for the new record to come into existence? How to correctly assess the correlation between ordinal and a continuous variable? I'd like to estimate the correlation between:  An ordinal variable: subjects are asked to rate their preference for 6 types of fruit on a 1-5 scale (ranging from very disgusting to very tasty) On average subjects use only 3 points of the scale.  A continuous variable: the same subjects are asked to quickly identify these fruits, which results in an mean accuracy for the 6 fruits.   Is Spearman rho the best method to analyze these data and/or are there other good methods I could consider? find command with regex quantifier e.g. {1,2} I have been trying to create a find command string that will find all files that end with a number 1-99 but exclude all others.  e.g. I want to find myfile1 myfile99 but not myfile456 and not myfilebackup  The regex I'm thinking of is myfile[1-9]{1,2} but I can't get this to work with find.  find . -regex '.*myfile[0-9]{1,2}' OR  find . -iname 'myfile[0-9]{1,2}'  From what I can see it's the {1,2} part that is not working.  (by the way can you use -name -regex interchangably?)  Any help appreciated. Foolproof way to ensure Google news pulls the correct image for its thumbnails? Google news results have an acompanying thumbnail next to articles that show up in the results.    If Google's crawler can't find a thumbnail to pull from our site, it uses its next best guess from another site, therefore linking the image to another site but still uses our headline.  Example:  Headline from Reuters, Image from Livemint:   Our pages absolutely have images, they are not massive in file-size or dimensions, yet we are not having them pulled / crawled correctly.  We have read up on the suggestions from Google, and from others around the web and nothing is panning out.  Has anyone had any experience where they can ensure Google news will pull a thumbnail of our choosing? How do I get on a Minecraft PE server when it says 'invalid name'? I am having trouble getting onto a server in Minecraft PE on iOS. When I tap on the person's server it just says invalid name:    I have tried turning off and restarting the iPad, going to home screen and logging back into the server but I have had no luck and no change. Is there a reason why this is so? How to determine the cause of MacBook battery not charging? I see that the question &quot;why is my MacBook not charging?&quot; gets asked a lot. The problem is, there are multiple reasons. Possible points of failure are: charger, cable, magsafe connector, DC input board, logic board, battery. (Have I missed any?). What is the best sequence for a user to follow, to diagnose the problem? By way of example, here are the first steps that I usually try:  Make sure the charger is working. Either use your charger on a different Mac, or test a spare charger with your Mac.  Check the connection. There might be gunk in your port that's stopping the connector from lying flat.  Try resetting the SMC and PRAM.  If the computer boots, run Coconut Battery to determine battery status.   This is where I get a bit stuck. For example, I have a machine here that boots fine when plugged in, but not from battery. Disconnecting the MagSafe causes the machine to switch off. Sounds like it's a dead battery, except that Coconut reports the battery as detected, in good condition, but not charging. What steps should I take next? Specifically, I'm asking for a generic algorithm to help determine the source of the fault (preferably without swapping out lots of components one by one)? Need help to install persistent Ubuntu on USB drive Possible Duplicate:          I am a new user of Ubuntu. I would like to install a persistent Ubuntu 11.04 to my USB stick, and it should be able to work as a guest OS running on Windows so that I can boot it on other computers other than the one which I used for the installation. I have used several creators such as unetbootin, however from my understanding it can only create Live Linux which I am unable to save my configurations and files. If it's possible I would like to bypass the BIOS, that is to say that I can just load from the virtual machine without having to restart the computer. Thanks in advance!    Thanks for all the replies! Hi Mike, I am able to create a virtual machine and run it from my USB drive using a portable virtualbox. How do I go about from there, do I have to proceed with the installation? Could you elaborate more on the steps for the installation? Thanks again! Is there a cleaner proof of convergence for this almost-telescoping series I was wondering how the convergence of the series $$\sum_{k\geq1}\frac{(k+1)^y-k^y}{k^x},$$ depends on $x$ and $y$, where $x$ and $y$ are real numbers. I've been able to show that it converges if and only if $y\leq x$, by distinguishing nine cases; whether $x$ and $x-y$ are positive, negative or zero. If desired I can include a sketch of my proof. My question is whether there is a proof that works for all real $x$ and $y$? Or at least a proof that seems less artificial? My trackpad is broken so I can tap-to-click but i cannot drag and drop. This is really a problem when it comes to doing certain things.   Does anyone know how to program a sticky key to turn a click into a click-select that can then be dragged about?   I can't resize my browser or open window and I can't drag applications into the folder when installing etc.  free solutions only please.  A mid 2009 macbook-pro 13". 2.53 Ghz Intel Core 2 Due.   Running os lion (grudgingly). I'm a fan of extension methods in C#, but haven't had any success adding an extension method to a static class, such as Console.  For example, if I want to add an extension to Console, called 'WriteBlueLine', so that I can go:  Console.WriteBlueLine("This text is blue");   I tried this by adding a local, public static method, with Console as a 'this' parameter... but no dice!      public static class Helpers {     public static void WriteBlueLine(this Console c, string text)     {         Console.ForegroundColor = ConsoleColor.Blue;         Console.WriteLine(text);         Console.ResetColor();     } }   This didn't add a 'WriteBlueLine' method to Console... am I doing it wrong? Or asking for the impossible? I'm trying to downgrade my iPhone from the current iOS to an older one, as per .  But every time I try, I always receive this error:     The iPhone could not be restored. This device isn't eligible for the requested build.     I have tried many times with and without DFU mode. Any ideas regarding this? I would like to declare functions in tikz for multiple uses of the same function within the code.  Sadly, it seems that the tikz "declare function" is not compatible with the french option in babel. Indeed, the following code (that I simplifed on purpose) runs fine :  \documentclass[english]{article} \usepackage{babel}                       \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \tikzset{declare function={Carre(\t)=\t*\t;}}     \draw plot [domain=-1:1] (\x,{Carre(\x)}); \end{tikzpicture} \end{document}   But when I replace   \documentclass[english]{article}   with   \documentclass[english,french]{article}   compilation generates an error. This is what the log file says :     Missing character: There is no = in font nullfont!   Missing character: There is no @ in font nullfont!   Missing character: There is no @ in font nullfont!      Runaway argument?   -1:1] (\x ,{Carre(\x )}); \end {tikzpicture} \end {document}    ! Paragraph ended before \pgfmath@local@@functions was complete.                          \par    l.18       I suspect you've forgotten a `}', causing me to apply this   control sequence to too much text. How can we recover?   My plan is to forget the whole thing and hope for the best.   Before posting this, I updated all packages without success.  Any help welcome! Why do we use 'did' with questions using the simple past tense? I can't seem to edit and save config files in the root system (eg: /etc/dhcp/dhcpd.config); it just says "permission denied" or the save option is blocked in the text editor.  I'm pretty sure I'm a root user, but how do I make sure? If I am a root user, how do I fix this?