How is the illusion of light streaks achieved in this photo?
How can I create light trails without a blurry subject?
Determinant of a matrix whose $(i,j)$th element is $|i-j|$? Question states: "Let $D_n$ denote the determinant of the $n$-by-$n$ matrix whose $(i,j)$th element (the element in the $i$th row and $j$th column) is the absolute value of the difference between $i$ and $j$. Show that $D_n = (-1)^{n-1}(n-1)2^{n-2}$."  It seems like that I can get a recurrence relation (especially the question was from a book chapter dealing with recursion). In trying to get a recurrence relation, here is what I have done so far:  (Let $R_k$ denote the $k$th row and let $(i,j)$ denote the element/spot of $i$th row and $j$th spot)  1) replace $R_1$ with $R_1-(R_2+R_{n-1})$ to get $R_1^*$  2) replace $R_1^*$ with $R_1^*+R_{n-2}$  3) If I am not mistaken, I should have $-2$ on $(1,1)$, $2$ on both $(1,n-1)$ and $(1,n)$, and all other element in the first row as $0$.  4) take out $2$ from the first row, and expand the determinant about the first row  5) I should have $D_n=2(-D_{n-1}+(-1)^n B_{n-1})$, where $B_{n-1}$ is the determinant of a not-easily-reducible mayrix whose last column consists of $n-1$ ones followed by one $-1$.  How should I go about from now on? Once I get a recurrence relation, it would be pretty easy to get the explicit formula for $D_n$ using induction.  P.S. This question apparently is a question taken from 1969 Putnam Competition Is there a quicker way to turn on WiFi Hotspot? is there a way to automate (or at least reduce the number of steps) to turn on or turn off the WiFi hotspot on my Galaxy 3.  I have to    unlock  press the menu button  press settings press More Settings Tethering and portable hotspot Slide portable hotspot on. Exponent and abelian groups What exponent $e$ can guarantee the group to be abelian? Are there any known results except the case $e = 2$? Thanks for help. What is an Electron? I am a n00b to physics. What physics I know comes from high school and popular science books.  At school we were taught about atoms and what make up atoms, but we weren't actually told what the particles were!  So sure an atom has electrons orbiting around a nucleus made of protons and neutrons, but what exactly is an ?  And if we don't know, how can we fire them from guns, like they do in experiments ... etc  Any clarification on this would be great. User is suspended network-wide, except not really A user has been active on one of the sites recently, even though they're concurrent suspended network-wide on the rest of the network. How is this possible, and is it possible to evade a network suspension on only one site? Suppose my speed is normally 30 feet. I get hit with a Slow spell, reducing my speed to 15 feet, and then I get knocked prone. How many feet of movement do I need to spend in order to stand back up from prone? How many feet of movement do I have left over once I've stood up?  What if I am also hit with a Ray of Frost spell while slowed, reducing my speed to 5 feet for 1 round? Am I able to stand up at all? Symbol Table from Apex Class In an earlier question  I asked in general about the symbol table for an Apex class.  Using the Tooling API wrapper from  I am able to get part way there. But am unGodly confused.  If I go to the developer console, query editor, check the "Use Tooling API" box and run this query:  Select Id, Name, symbolTable FROM ApexClass Where Name Like '%NewCaseListController%'   I get back an ID, Name and [object Object] for the symbol table, meaning there is a symbol table object there.   But if I try the same query in Apex code in the execute anonymous window I get the error:   System.CalloutException: Web service callout failed: Unable to parse callout response. Apex type not found for element location   If I do this in Apex I can get a result back, and according to the code for the class the result should be of type ToolingAPIWSDL.QueryResult. The QueryResult should contain a member consisting of a list of sObject_x records sObject_x[], unless I am mis-reading what that means.  ToolingAPIWSDL.QueryResult result = tooley.query('select Id, Name from ApexClass WHERE Name Like \'NewCaseListController\'');   I can take the result and apparently put the sObject_x object in it like this:  List&lt;ToolingAPIWSDL.sObject_x&gt; objList = new List&lt;ToolingAPIWSDL.sObject_x&gt;(); for(Integer i = 0; i&lt;result.records.size(); i++){     objList.add(result.records[i]); }   At least that does not throw an error and I can write:     System.debug('objList: ' + objList[0]);  And get a result, but how do I access the symbol table? Or more to the point, how can I get a list of methods and variables? I am assuming the symbolTable is the only way? Optimum Bone To Water Ratio For Pork and Beef Stocks I am tinkering with my stock recipes.   For pork stock I have been using 8 cups of water over a roasted trotter, a roasted backbone section with about 4 inches of rib attached, and a 1/4 inch slab of roasted pork belly.   For the beef stock I have been using 8 cups of water over an equivalent amount of roasted beef bones (knuckles and backbone/rib).   This gives a very flavorful stock and, for cost reasons, I want to determine how much I can reduce the concentration (by adding more water) without reaching too watery a taste.   Does anybody have any guidance on this topic? What is wrong with the following "proof" that $e=1$? Let's analyze this expression   $\lim_\limits{n\rightarrow\infty} (1+\frac{1}{n})^n$  It's the definition of $e$ which, as we know is not equal to $1$. So what is wrong with the following "logic":  As $\lim_\limits{n\rightarrow\infty}(a_{n}b_{n}) = \lim_\limits{n\rightarrow\infty}(a_{n})\times\lim_\limits{n\rightarrow\infty}(b_{n}) $ and $\lim_\limits{n\rightarrow\infty}(1+\frac{1}{n}) = 1$, we can say that $\lim_\limits{n\rightarrow\infty} (1+\frac{1}{n})^n=1^n$, which is equal to $1$.  I know something's wrong there, but the question is - what? Is $0$ a natural number? Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers. If "No One" uses a face from the Hall of Faces to become someone then how did he do this In season five of Game of Thrones, when Arya becomes blind there is a very interesting detail most of us might have missed.   Let's not stretch the question:    Arya returns after assassinating Ser Meryn Trant Jaqen H'ghar "No one" comes in the scene, Waif is present as well Arya and Jaqen talks, he appears to be giving the poison to Arya but drinks it and dies. Arya cries, says he was her friend when Waif asked her something (I don't recall what) Waif removes her face, reveals she is actually Jaqen Arya removes face of Jaqen from the dead body, keeps removing it until her own face appears.   What sorcery is going on in here? How could they use a face of a person who is still alive. They remove the faces of a dead person for a reason? So that the face can be used by the Faceless-men.   Is there something I am missing or can the Faceless-men do these kinds of magic as well? I converted a raster image to vector in Illustrator CS6 using Image Trace then the Expand option, though when I save as an svg the image looks embossed. I suspect it's due to the layer background being white and the paths not lining up, but I'm unsure how to fix it or prove that theory without manually having to adjust every path.   This is how it looks in Illustrator before I export it:    The image looks like this when saved/exported with art optimized option: 0 byte file created in a simple script I have a script as below:  #!/bin/bash df -k | tr -s " " "," | awk 'BEGIN {FS=","} {print $1,$5}'|sed 1d &gt; file1.txt while read partition percentUsed do if [ $percentUsed &gt; 75 ] then echo Partition: ${partition} space is ${percentUsed} else echo Pration: $partition: OK!! fi done &lt; file1.txt   The script is executing properly, however, it is creating a zero byte file by the name 75. Any ideas why this is happening?  $ sh diskUsed.sh Partition: C:/Users/Public/Documents/CYGWIN/bin space is 75% Partition: C:/Users/Public/Documents/CYGWIN/lib space is 75% Partition: C:/Users/Public/Documents/CYGWIN space is 75% Partition: C: space is 75% Partition: H: space is 91%   $ ls -lrt total 2 -rwxrwxrwx 1 diwvibh Domain Users 284 Nov 21 04:17 diskUsed.sh -rw-r--r-- 1 diwvibh Domain Users 133 Nov 21 04:29 file1.txt -rw-r--r-- 1 diwvibh Domain Users   0 Nov 21 04:29 75 On my mac I have 2 accounts and the both are normal account. So in this moment I don't have a Administrator account, only one that was in the past a administrator, but now is not.  How I can have again a Administrator account? Why have header files and .cpp files? Can spells or spell-like abilities do non-lethal damage? I'm working on a Paladin/Warlock combination, based off of , and as part of this, I noticed Miniman's remark that Repelling Blast won't be that useful, because as a Paladin, you want people up close and personal when you're in a combat situation, not further away.  However, as a Paladin, your creed sometimes leads you to not want to engage in full-blown combat with a potential adversary -- you may want to drive them off or incapacitate/capture them instead.  With this, I wonder: can Repelling Blast, or other spells and spell-likes (such as invocations), be made to do non-lethal damage?  If so, I'm thinking that it'd be useful in exactly that sort of situation -- halting someone/pushing them away without having to wade into a melee with them or blow them to smithereens. Is "of" instead of "have" correct? Unable to get a generic ResponseEntity where T is a generic class "SomeClass" Recently the keyboard and trackpad on my 2015 13" MacBook Pro have stopped working. This is the second laptop to do this, the same issue happened about 6 months ago and Apple gave me a new one when they could not figure it out.  The issue usually starts with force touch not working then progresses to the keyboard and trackpad not being recognized at all. Currently the power button works and force touch works but all other keys and the trackpad don't work.  When this happens there are a lot of errors in the console regarding the AppleHSSPIController.  4/21/16 6:56:47.000 AM  kernel[0]   Error: AppleHSSPIDevice::start failed to get device descriptor 4/21/16 6:56:47.000 AM  kernel[0]   Error: AppleHSSPIController::_enumerateDevice Start failed 4/21/16 6:56:47.000 AM  kernel[0]   Error: AppleHSSPIController::_handleEnumerationFailureGated Giving up on device   You can see the full log here: How to accumulate values of each two rows across a line in linux?