How is the formula for the Standard error of the slope in linear regression derived?
Derive Variance of regression coefficient in simple linear regression
Is there any way to generalise this problem? How do you keep user.config settings across different assembly versions in .net? Basically the problem is that each time the assembly version changes (i.e. the user installs a new version of the application) all their settings are reset the the defaults (or more accurately a new user.config file is created in a folder with a different version number as the name)  How can I keep the same settings when upgrading versions, since using ini files or the registry seem to be discouraged?  When we used Clickonce it seemed to be able to handle this, so it seems like it should be able to be done, but I'm not sure how. Cut points and connected space The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? Topological/Geometrical justification for $\text{CFT}_2$ being special It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n&gt;2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.   To show what I mean, consider this example: in $\mathbb{R}^n$ for $n&gt;2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n&gt;2$) but this is not true for $n=2$.   I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$. Fact  The intersection of a closed set $A$ and a compact set $B$ is closed.  Background  I wanted to prove that the intersection of closed and compact sets is compact. There might be other ways, but I am not interested in them for now and am looking to prove the above statement.  Relevant Facts  I don't have much idea what to use. $A^c$ is open. Every cover $\cup_{i\ge1}U_i$ of $B$ contains a subcover $\cup_{i=1}^kU_i$. I might want to prove $A\cap B^c$ is open. Limit points may be another points to consider but I don't get any idea in that direction.  I also don't have that $B$ is closed otherwise it would be trivial.  Some Clarification  I am aware that compact sets are not closed in arbitrary topological spaces. The reason why I was motivated to believe tha intersection of closed and compact set is compact is true is: See Matt E.'s Answer      So if $K$ is compact in an arbitrary topological space (which is just to say that   it is a compact topological space when given its induced topology) and     $L$ is closed then $K \cap L$ is a closed subset of $K$ in its induced topology   Is it true if $A$ and $B$ are not whole spaces?  I would appreciate any hints. Drawing Mechanical Systems in LaTeX I really like CircuiTikZ for drawing diagrams in my LaTeX documents. However I mostly draw mechanical and not electronic systems, which is why I'm looking for some similar way to draw mechanical systems. What I want is something to draw masses, springs, dampers and ground (boundary condition).  are typical examples:     Anybody got some suggestions or am I out of luck? What is the exact difference between an autoregressive (AR) and vector autoregressive model (VAR)?  I always thought that VAR would just be for more than two variables, until I learned that AR can also have more than two variables. I.e. I had a misunderstanding(?) that AR models are not always bivariate.  Before I write too much potentially false assumptions about the difference between AR and VAR, it would rather be interested to see if anyone can make provide a quick and good summary. In recent versions of PostgreSQL (as of Dec 2013), can we share a query between two or more cores to get a performance boost? Or should we get faster cores? What is the difference between an abstract method and a virtual method? What is the difference between an abstract method and a virtual method? In which cases is it recommended to use abstract or virtual methods? Which one is the best approach? I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible? To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$ How can I show   $\lim_{n \to \infty} a_n = 0$  $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$  I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below then it automatically would converge and hence my question will be solve. But I'm unable to show its boundedness... Or there maybe another method to prove this. Thanks Summarizing flatness of county with statistic Elevation variance per square mile I need to classify every county in the US as either hilly, moderately hilly, or flat. Accuracy is not critical, just a rough classification. I used R and followed the advice in  answer to import elevation and counties and calculated the zonal variance. This was inherently biased towards larger counties since they have more surface for hills, so I need to divide by area. If I divide variance by square miles does that reasonably represent hilliness? The preliminary results seem decent, with the highest values being places like Colorado, Hawaii, Montana, etc. In the inbox of the app, there is no way to differentiate between comments and answers as is in the global inbox of the website.  Could this be added? How to find and replace a string by increasing its numerical part? My input file is,  ami21 ami65 ami67 ami66 ami88 ami76 ami29 ami55  ami54 ami32   Using a single command line I need the output as,  ami22 ami66 ami68 ami67 ami89 ami77 ami30 ami56 ami55 ami33   I used the command  awk -vRS=ami '{$0=$0+1;ORS=RT}++n' inputfile &gt; outputfile   but I got the outputfile as ami21ami65ami67ami66ami88ami76ami29ami55ami54ami32  i.e. all strings are written in same line and without space. Could anyone suggext me some better command line. Why does the /boot partition get updated when I install software Based on some recommendations I read online, on install of a new (Ubuntu) system, I made my /boot partition separate from / and small in size (only around 50 MB). It makes sense to me that you would only need limited files in that partition, and the boot sequence only needs so much data initial data on boot so no need (or so goes my thinking) to make it much larger.   I was surprised as I started using my system and installing software was failing due to the following (or similar) issue:  gzip: stdout: No space left on device cpio: write error: Broken pipe E: mkinitramfs failure cpio 1 gzip 1 update-initramfs: failed for /boot/initrd.img-3.2.0-18-generic with 1.   I have already seen  and am planning on doing a reinstall with a larger /boot partition. My questions are:  Why does the /boot partition data get updated when I am only installing software?  obviously 50 MB is too small for my needs, so what would be a reasonable /boot partition size?  Does it even make sense to have separate boot partition? after all it could simply use all the disk space it wants in the / partition. Is there a VB.NET equivalent for C#'s '??' operator? Is there a VB.NET equivalent for C#'s ?? operator? The Bash man page describes use of ${!a} to return the contents of the variable whose name is the contents of a (a level of indirection).  I'd like to know how to return all elements in an array using this, i.e.,  a=(one two three) echo ${a[*]}   returns  one two three   I would like for:  b=a echo ${!b[*]}   to return the same.  Unfortunately, it doesn't, but returns 0 instead.  Update  Given the replies, I now realise that my example was too simple, since of course, something like:  b=("${a[@]}")   Will achieve exactly what I said I needed.  So, here's what I was trying to do:  LIST_lys=(lys1 lys2) LIST_diaspar=(diaspar1 diaspar2)  whichone=$1   # 'lys' or 'diaspar'  _LIST=LIST_$whichone LIST=${!_LIST[*]}   Of course, carefully reading the Bash man page shows that this won't work as expected because the last line simply returns the indices of the "array" $_LIST (not an array at all).  In any case, the following should do the job (as pointed out):  LIST=($(eval echo \${$_LIST[*]}))   or ... (the route that I went, eventually):  LIST_lys="lys1 lys2" ... LIST=(${!_LIST})   Assuming, of course, that elements don't contain whitespace. How can I work with file uploads during a Webdriver test? This is a canonical question, intended to provide a comprehensive answer to many related questions  During the course of a Webdriver test, I want to upload files. However, this opens a native file picker, which Webdriver doesn't seem to support interacting with. How do I interact with it? Exporting QGIS project to Mapnik XML? Is it possible to export a QGIS project to Mapnik xml?   I'm using QGIS 2.4 and Quantumnik does not seem to work with that version.