Let $\textsf{FinVec}$ denote the category of finite-dimensional real vector spaces. For each vector space $V \in \textsf{FinVec}$, choose a vector space isomorphism $\varphi \colon V \to \mathbb{R}^{\dim V}$ and endow $V$ with the standard smooth structure from $\mathbb{R}^{\dim V}$ pulled back along $\varphi$. I think (correct me if I'm wrong!) that this smooth structure on $V$ is independent of the choice of $\varphi$ up to diffeomorphism.    Definition Let $n \in \mathbb{N} \cup \{\infty\}$ (where we use the convention that $\mathbb{N}$ includes $0$). A functor $F \colon \textsf{FinVec} \to \textsf{FinVec}$ is called $C^n$ iff $$f \mapsto F(f) \colon \operatorname{Hom}(V,W) \to \operatorname{Hom}(F(V), F(W))$$ is a $C^n$ function for all $V,W \in \textsf{FinVec}$ (where the smooth structures on the hom sets are the ones described above, as these hom sets are finite dimensional real vector spaces under pointwise addition &amp; scalar multiplication).    Of course, any $C^n$ functor is also $C^m$ whenever $m \leq n$, and in particular $C^\infty$ functors are $C^n$ for all $n$. Examples of $C^\infty$ functors include:   $\operatorname{Hom}(V,{-})$ for any $V \in \textsf{FinVec}$ $V \otimes {-}$ for any $V \in \textsf{FinVec}$ The exterior algebra functor   However, I've been unable to think of any examples of an endofunctor of $\textsf{FinVec}$ which is not $C^\infty$! So:  Does there exist an endofunctor of $\textsf{FinVec}$ which is not $C^\infty$?  If the answer is yes, then my follow-up questions are:  Does there exist an endofunctor of $\textsf{FinVec}$ which is not $C^0$? If $n,m \in \mathbb{N} \cup \{\infty\}$ with $m &lt; n$, does there exist an endofunctor of $\textsf{FinVec}$ which is $C^m$ but not $C^n$?
Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$. (Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.) In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:  What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent. Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere.
Why is the mean function in Gaussian Process uninteresting? Help, I cannot get my test code coverage higher than 75%! Code below Since I upgraded to the latest version of Ubuntu, I cannot install the Java OpenJDK package needed to compile Android custom ROMs. When I run sudo apt-get install openjdk-7-jdk, the following error appears:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  openjdk-7-jdk :  Depends: openjdk-7-jre (= 7u51-2.4.6-1ubuntu4) E: Unable to correct  problems, you have held broken packages.   How do I fix this problem? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Good 1st PDE book for self study What is a good PDE book suitable for self study?   I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis.  My goal is to understand basic solutions techniques as well as some basic theory. I have made some changes to a file which has been committed a few times as part of a group of files, but now want to reset/revert the changes on it back to a previous version.   I have done a git log along with a git diff to find the revision I need, but just have no idea how to get the file back to its former state in the past. How to access the Keyboard Layout Options in 13.10+ How do I read / convert an InputStream into a String in Java? If you have a java.io.InputStream object, how should you process that object and produce a String?    Suppose I have an InputStream that contains text data, and I want to convert it to a String, so for example I can write that to a log file.  What is the easiest way to take the InputStream and convert it to a String?  public String convertStreamToString(InputStream is) {     // ??? } Why does `last` show '{' and '|' in the TTY field? How to prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$ Do I need a US visa to transit (or layover) through an American airport? I checked my  to see whether I can find something similar to the main-site flag-history for chat-flags, but didn't find anything. I recently had the (dis)pleasure of regularly having to deal with a user, that has been deleted multiple times. Whenever they show up (with the same nick-name) again I have to flag them again. I'd like to get an overview of how often I had to do that, and whether my flags were helpful for the moderators. As such I propose the implementation of a flag-history as it is already in place for the main-site for chat-flags, that request moderator intervention. To be completely clear I am not talking about the 10k user flags, which are acted on in a matter of sometimes seconds with instantly visible results, but about moderator attention flags from this menu: I'm currently trying to configure our exchange 2010 to be used as a external smtp relay. What I'm trying to do is allow certain websites that we manage to send email using our exchange server smtp relay. I was told that I needed to created a receiver connector under Server Configuration>Hub Transport. After I created the receiver connector I added the following confi in which I added the website IP address in the "Receiver mail from remote servers ect" then unders Authentication I selected Externally Secured,TLS and under permissions I selected exchange servers.   I went to one of our websites and configure it with this configuration and I'm still getting "SMTP connect failed."  Can someone help me? or what is the  proper way to get this to work. Multiple websites are going to be using this exchanger server SMTP relay in a secure way. I want to calculate the derivative of $|x|^\alpha$ with respect to $x$, where $1&lt;\alpha&lt;2$.  $\frac{d|x|^\alpha}{dx}=\alpha|x|^{\alpha-1}\mathrm{sign}(x)$  Is it correct?  And is it true that  $\lim_{x \to 0} \alpha|x|^\alpha\mathrm{sign}(x)=0$?  What with the following derivative $|x|^\alpha\rm{sign}(x)$. I would be grateful for any hints. How to test apex classes with Pagereference? I'm struggling to test a class with pagereference. Following is the class...  public with sharing class redirectOrderPartner{  public String currentRecordId {get;set;} public orders__c orderId {get;set;}  public redirectOrderPartner(ApexPages.StandardController controller) {      }     Public Pagereference go(){      currentRecordId  = ApexPages.CurrentPage().getparameters().get('orderid');      system.debug('---currentRecordId  ---'+currentRecordId  );     orderId = [select id from orders__c where name =: currentRecordId limit 1];      system.debug('---orderid---'+orderId);      PageReference pageRef = new PageReference('/apex/OrderTrackingClass?id='+orderId.id);     pageRef.setRedirect(true);     return pageRef;    }   }   Following is what I tried....    @isTest  public class redirectOrderPartner_TEST  {             static testMethod void Test(){            Account acc = new Account(Name='ABC Corp.');           insert acc;            orders__c tempOrder = new orders__c(name = '0001234567', account__c = acc.id );           insert tempOrder;            Test.startTest();            PageReference pageRef = Page.redirectOrderPartner;           Test.setCurrentPage(pageRef);            ApexPages.Standardcontroller sc = new ApexPages.Standardcontroller(tempOrder);           ApexPages.currentPage().getParameters().put('Id',tempOrder.name);           PW_redirectOrderPartner ec = new redirectOrderPartner(sc);           ec.go();                                   Test.stopTest();       }   } I get the following error message on testing..     System.QueryException: List has no rows for assignment to SObject   What am I missing? Thanks WPF CommandParameter is NULL first time CanExecute is called Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log Homology Groups of $S^n$ How to create this curved 90-degree-angle quickly There's a story my grandpa use to tell me when I was younger, about the sole survivor of of an attack on a mining vessel in space. The man lives in space for months/years, ignored by the company he worked for and never rescued. Then when he thinks he is being rescued, it's actually another set of raiders who attack him and kidnap him.  He then wakes up in captivity in some sort of prison inside of an asteroid, and has a word tattooed on his forehead. My grandpa called him Ronin (that's what the tattoo on his forehead said) but I don't know what the character's real name is. Anyways, he learns that he has the ability to teleport after having a brush with death and decides to make his way back to Earth so that he can exact revenge on the mining company who abandoned him and left him floating to die.  Does anyone know what this story or book may be? My grandpa told it to me when I was little, he said that he got it from a book he use to love reading when he was younger. I'm assuming the book is very old (possibly pre 1940's as he was born in the 20s). I can't ask him as he is no longer alive. I'm also unsure as to whether or not all the details in the story he told me, are the same as in the original.