How to import package properly
Import a module from a relative path
Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough. How to get a file's Media Type (MIME type)? How do you get a Media Type (MIME type) from a file using Java?  So far I've tried JMimeMagic &amp; Mime-Util. The first gave me memory exceptions, the second doesn't close its streams properly. How would you probe the file to determine its actual type (not merely based on the extension)? Custom action to set emblem in thunar There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue Question:  Suppose $V$ and $W$ are finite dimensional vector spaces over $\mathbb{C}$. $A$ is a linear transformation on $V$,  $B$ is a linear transformation on $W$. Then there exists a non-zero linear map $C:V\to W$ s.t. $CA=BC$ iff $A$ and $B$ have a same eigenvalue.  ===========This is incorrect==========  Clearly, if $CA=BC$, suppose $Ax=\lambda x$, then $B(Cx)=CAx=C(\lambda x)=\lambda (Cx)$, so $A$ and $B$ have same eigenvalue $\lambda$. On the other hand, if $A$ and $B$ have a same eigenvalue $\lambda$, suppose $Ax=\lambda x, By=\lambda y$. Define $C:V\to W$ s.t. $Cx=y$, then $BCx=By=\lambda y=C\lambda x=CAx$. But I don't kow how to make $BC=CA$ for all $x\in V$.   ====================================== Have Blender on about 4 machines rendering animations. Just spotted on one of the machines its rendering something different from the actual scene that I have setup.    Top window shows a different set of layers objects to whats actually being rendered in the rendering window below.  In my scene contents, layers objects 1-5 are not visible, but these are the ones that look like they are being rendered.  The only thing I can assume is that perhaps it might be because I did not save the document before clicking render, but I would not have expected that behaviour (you dont need to save your word document before printing it).  Can anyone confirm why this is happening? How to perform a full system dump/backup with a broken screen? So, yesterday my phone took a serious tumble.  It is now essentially useless, as the screen either fails to turn on, or when it does, there is nothing useful to see (just lines of primary colors, mostly).  I believe (from the fact that it still chirps at me when emails come in, etc), that the phone is still functional, otherwise.  I would like to perform a full system dump, so that I can get off any files from the phone (a TMobile SGS2 (hercules) running slimrom 4.3 build 1).  The sdcard probably isn't important, as I can just pull it, but the phone's own internal storage I should like to get a copy of.  When I plug the phone into a usb port, it 'mounts' the usb via MTP, and I can see all the files.  Unfortunately, when I do a select all | copy | paste operation, the paste op dies before transferring all files.  I've tried a couple other ADB shell operations, in hope of flipping regular USB mounting on, but nothing has worked.  Any ideas how I can get a full dump of my poor phone's contents?  Edit  Currently running a adb pull /sdcard/ ... Its copying files presently, but its too early to tell if its going to get evertying I'm expecting.  Edit #2  looks like its going to work...its taking a rather long time, but it appears to be grabbing everything I was expecting it to. Windows Batch recursive delete old files I need help with this exercise. What I've done so far is prove the exercise when $n=1$. So:  $$n=1$$ $$1!\ge2^{(1-1)}$$ $$1\ge2^0$$ $$1\ge1$$ Which is true  Therefore, now that I assume that the assumption is correct, I want to prove that with $n+1$, it will also be true. So, what I've done now is:  $$P(n) \implies P(n+1) $$ $$n!+n+1\ge 2^{(n-1)}...$$ And my  problem is that I do not know how to add the $n+1$ in the right side of the equation, therefore I'm not been able to finish the exercise.  Thank you so much for your help. If something's not very clear, please let me know. I'll try to be clearer next time :) Is it possible to paint only inside a stroke on illustrator? what i actually want to do is paint the same stroke with different colors in diferent segments of it but i can't do it...   Also... I've seen some tutorials that include editing a gradient but that would be very complicated and so would be to make a new stroke over this one, I just hope theres an easyer way XD  Edit per comment:  thats cool but not really what im looking for, what i want is lets say my stroke has the shape of a burger i want part of i to be bread color and part of it to be meat color... not one stroke over the other... i want something like this, imagine the jelly worm is my stroke: BibTeX handling of the Dutch "van" name prefix with natbib How is prototyping done in a professional EE environment? Do you do your prototypes on breadboards, on copper clad boards, Manhattan style, or do you just draw schematics, order PCBs made and assembled and work with them, hacking here and there if necessary? Identify $P(2014)$ if $P(i)=1/i$ for every positive integer $1\le i\le2013$ Query to find GUID's "not in" null containing subquery returns no results When I open the file in Vim, I see strange ^M characters.  Unfortunately, the world's favorite search engine does not do well with special characters in queries, so I'm asking here:   What is this ^M character? How could it have got there? How do I get rid of it? Ackermann function - how to calculate the number of times it calls itself Purely for my own amusement I've been playing around with the .  The Ackermann function is a non primitive recursive function defined on non-negative integers by:  $A(m,n) = n+1$, if $m=0$  $A(m,n) = A(m-1,1)$, if $m&gt;0$ and $n=0$  $A(m,n) = A(m-1,A(m,n-1))$, if $m&gt;0$ and $n&gt;0$  I'm not interested in values of the function itself, rather I'm interested in the number of recursions the function takes to arrive at a value. For example:  $A(1,2) = A(0,A(1,1)) = A(0,A(0,A(1,0))) = A(0,A(0,A(0,1))) = A(0,A(0,2)) = A(0,3) = 4$  That's 6 steps, so if we define $RA(m,n)$ as the number of recursions to calculate $A(m,n)$, then $RA(1,2) = 6$.  I made a short Excel macro to calculate $RA(m,n)$ for any given $m$ and $n$. The problem I have is that $RA(m,n)$ grows even faster than $A(m,n)$ and I quickly run out of stackspace. The table below shows $RA(m,n)$ for very small values of $m$ and $n$. The numbers shown for $RA(3,10)$ and $RA(4,1)$ are the number of steps before crashing!    A formula for $RA(1,n)$ is trivial, and it is relatively easy to see that $RA(2,n) = 5(n+1) + 4n(n+1)/2$.  So my questions: can anyone see an explicit formula for $RA(3,n)$? More generally, does a formula exist for $RA(m,n)$, in the same way that $A(m,n)$ can be expressed using Knuth arrows?   For the low values shown of $m$ and $n$, $RA(m,n)$ looks about order of magnitude larger than $A(m,n)$. Can we prove that $RA(m,n) &gt; A(m,n)$ for all $m,n$. Intuitively, it should be.  For bonus points, can $RA(m,n)$ be expressed in terms of $A(m,n)$? Getting a stack overflow exception when declaring a large array My internet connection is very slow. I suspect someone in the network is using a BitTorrent client which takes huge amount of bandwidth.  Is there a way to detect who is using my bandwidth (network sniffing tools etc)?  I should have pointed out that I have no access to the router, what I am looking for is a way of detecting someone's MAC/IP if she uses Bittorrent. When voting to migrate an off-topic question, why is the list of sites limited? How to call drupal_bootstrap from custom file in subfolder? I'm trying to call drupal_bootstrap from 'temp.php' in a subfolder called 'custom' my code as follow:  chdir('../'); define('DRUPAL_ROOT', getcwd()); require_once DRUPAL_ROOT . '/includes/bootstrap.inc'; drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);   Once I call the temp.php I get Drupal homepage.   I also tried an explicit physical path but with no use, code as follow:  define('DRUPAL_ROOT', '/var/htdocs/drupal7'); require_once '../includes/bootstrap.inc'; drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);   temp.php located under the following folder structure:  - var -- htdocs --- drupal7 ---- custom  ----- temp.php What's the projective limit of these polynomial rings ? Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m&lt; n)$.   Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ?