$$A(x^2+y^2)+Bx+Cy+D=0$$  is the equation of a circle in the plane provided $A\ne0$. The equation above can be written as the determinant $$         \begin{vmatrix}         x^2+y^2 &amp; x &amp; y &amp; 1 \\         x_1^2+y_1^2 &amp; x_1 &amp; y_1 &amp; 1 \\         x_2^2+y_2^2 &amp; x_2 &amp; y_2 &amp; 1 \\         x_3^2+y_3^2 &amp; x_3 &amp; y_3 &amp; 1 \\         \end{vmatrix}         = 0, $$ where $$         A =         \begin{vmatrix}         x_1 &amp; y_1 &amp; 1 \\         x_2 &amp; y_2 &amp; 1 \\         x_3 &amp; y_3 &amp; 1 \\         \end{vmatrix},         B =         \begin{vmatrix}         x_1^2+y_1^2 &amp; y_1 &amp; 1 \\         x_2^2+y_2^2 &amp; y_2 &amp; 1 \\         x_3^2+y_3^2 &amp; y_3 &amp; 1 \\         \end{vmatrix}, $$ $$         C =         \begin{vmatrix}         x_1^2+y_1^2 &amp; x_1 &amp; 1 \\         x_2^2+y_2^2 &amp; x_2 &amp; 1 \\         x_3^2+y_3^2 &amp; x_3 &amp; 1 \\         \end{vmatrix},         D =         \begin{vmatrix}         x_1^2+y_1^2 &amp; x_1 &amp; y_1 \\         x_2^2+y_2^2 &amp; x_2 &amp; y_2 \\         x_3^2+y_3^2 &amp; x_3 &amp; y_3 \\         \end{vmatrix}. $$ $P_1(x_1, y_1, z_1)$, $P_2(x_2, y_2, z_2)$, $P_3(x_3, y_3, z_3)$ are 3 points defining the circle and $P(x, y, z)$ is an arbitrary point.
Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged like so:  $\left|\begin{array}{ccccc} x^2+y^2+z^2 &amp; x &amp; y&amp;z&amp;1\\ x_1^2 + y_1^2 + z_1^2 &amp; x_1 &amp; y_1 &amp; z_1 &amp; 1\\ x_2^2 + y_2^2 + z_2^2 &amp; x_2 &amp; y_2 &amp; z_2 &amp; 1\\ x_3^2 + y_3^2 + z_3^2 &amp; x_3 &amp; y_3 &amp; z_3 &amp; 1\\ x_4^2 + y_4^2 + z_4^2 &amp; x_4 &amp; y_4 &amp; z_4 &amp; 1\\ \end{array}\right| = 0$   yields the equation for the sphere.  Then one need only re-arrange terms into the more familiar form to find the center and radius.  This works fine.  My question is why.    This same approach also works for one or two dimensions.  I'm guessing it also works for finding hyperspheres in higher-dimensional spaces as long as you have a corresponding number of points.  But where did that determinant form come from?  Is there an intuitive meaning for what that relationship is saying?
How can I install Windows software or games? Can .exe and .msi files (Windows software) be installed in Ubuntu? The final state of 1000 light bulbs switched on/off by 1000 people passing by There are  1000  light bulbs and 1000 tutors. All light bulbs are off. Tutor 1 goes flipping light bulb 1,2,3,4... tutor 2 then flips 2,4,6,8... tutor 3 then 3,6,9...etc until all 1000 tutors have done this.  What is the status of light bulb 25? 93? 576?  Is there a general solution to find out the status of any light bulb?  How many light bulbs are on after all 1000 tutors have gone by?  Not homework but a UK university interview question. Please help. Equivalent definitions of convexity for $f\in\mathcal C^1(\mathbb R^n)$ I noticed many identical questions on this site relating to equivalent definitions of convexity, see for example , , , , , ,  and there might be more. So I wanted to create a &quot;big list&quot; of equivalent definitions of convex functions, I will start with some equivalent definitions and post my proofs in my answer below. Feel free to add any other equivalent definitions. This question can then serve as an abstract duplicate. Claim. Let $n\in\mathbb N$ and $f\in\mathcal C^1(\mathbb R^n)$. Then the following are equivalent:  $f$ is convex, i.e. $f(\lambda x_1 + (1-\lambda x_2))\le\lambda f(x_1)+(1-\lambda)f(x_2)$ for all $\lambda\in[0,1]$ and $x_1,x_2\in\mathbb R^n$. For all $x,y\in\mathbb R^n$, $\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq0$. For every $x,y\in\mathbb R^n$, $f(y)\geq f(x)+\langle y-x,\nabla f(x)\rangle$. How can I upload files asynchronously with jQuery? Is there a limit of $\cos(n!)$? I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks Can't plot with gnuplot on my Mac I am trying to plot with gnuplot on my Mac (OS X 10.8.5). I have installed X11 and XQuartz 2.7.4 and after that I installed gnuplot, but unfortunately gnuplot couldn't plot.   Simple plots like the following fail to render and have no error message to help understand what is amiss:   [1/10/13 $gnuplot &gt;plot sin(x) JAVA Bad variable reference? Receiving strange output - possibly memory address? Infinite algebraic extensions of $\mathbb{Q}$ I need help to solve the following exercise:   If $K$ is an algebraic extension of $\mathbb{Q}$ (finite or infinite), then we let $O_{K}$ denote the ring of algebraic integers in $K.$ If $\frak{P}$ is a prime ideal of $O_{K}$ then $\frak{P}∩\mathbb{Z}$ is a prime ideal of $\mathbb{Z}.$ Moreover, if $\frak{P}$ is non-zero, then so is $\frak{P}∩\mathbb{Z}$ . Thus $\frak{P}∩\mathbb{Z}=$$ p\mathbb{Z}$ for some prime $p.$ Show that $O_{K}/\frak{P}$  is a field, and is an algebraic extension of $\mathbb{F}_{p}$; then show that it is Galois over $\mathbb{F}_{p}$ with abelian Galois group.  Now suppose that $K/F$ is a (finite or infinite) Galois extension. Let $\frak{P},$  $O_{K}$ be as before and put $\frak{p}=\frak{P} \cap $$O_{F}$, a prime ideal of $O_{F}.$  Show that $\mathrm{Gal}(K/F)$ acts transitively on the set of prime ideals of $O_{K}$ above $\frak{p}.$   Thanks in advance I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable or in some way? For example, does there exist a function continuous on $\mathbb Q$ and discontinuous on the irrationals? Equivalence Relation with dividing x and y integers Define $x\sim y$ means 5 divides $(x - y)$ for $x$ and $y$ integers. Show that is an equivalence relation.  Equivalence relation means it satisfies reflexity, symmetry, and transitivity.  reflexive: $x\sim x$ means 5 divides $x$  symmetry: $x \sim y \rightarrow y\sim x$ means 5 divides $x - y$ and 5 divides $y - x$: $$5/(x - y) = 5/(y - x)$$ so symmetry is satisfied.  I am not sure if I am right here and I am lost on how to prove it is transitive any suggestions would be greatly appreciated What books would you recommend to start learning about software testing I'm trying to learn more about software testing, so I want to know which books you would recommend on this topic How does electron move around nucleus? I need to get a nice picture about how electron moves around nucleus? I find concept of probability and orbitals quite difficult to understand? Suspending with root on USB I have a laptop running Ubuntu 14.04 from a root file system on USB storage. This is not working well, because after wake up from suspend, ext4 will frequently try to write to the root file system before USB is ready.  Here is what I see in the kernel log when this happens, notice how I get a bunch of I/O errors on sda1, and then one second later the USB storage drive is finally detected by the kernel.  [ 2826.517419] wlan0: associated [ 2826.517452] IPv6: ADDRCONF(NETDEV_CHANGE): wlan0: link becomes ready [ 2827.575371] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 1733735 (offset 0 size 0 starting block 12629950) [ 2827.575380] Buffer I/O error on device sda1, logical block 12629694 [ 2827.575400] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3148603 (offset 0 size 8192 starting block 12844470) [ 2827.575404] Buffer I/O error on device sda1, logical block 12844212 [ 2827.575411] Buffer I/O error on device sda1, logical block 12844213 [ 2827.575448] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3015015 (offset 0 size 90112 starting block 6588832) [ 2827.575453] Buffer I/O error on device sda1, logical block 6588576 [ 2827.575461] Buffer I/O error on device sda1, logical block 6588577 [ 2827.575465] Buffer I/O error on device sda1, logical block 6588578 [ 2827.575469] Buffer I/O error on device sda1, logical block 6588579 [ 2827.575473] Buffer I/O error on device sda1, logical block 6588580 [ 2827.575477] Buffer I/O error on device sda1, logical block 6588581 [ 2827.575481] Buffer I/O error on device sda1, logical block 6588582 [ 2828.857284] sd 0:0:0:0: [sda] No Caching mode page found [ 2828.857293] sd 0:0:0:0: [sda] Assuming drive cache: write through   At first there is no visible indication outside of the kernel log, that the problem has triggered, but if I let Ubuntu stay running beyond this point, then the file system will get errors and eventually switch to read only mode. At that point I have to reboot into recovery mode and run fsck.ext4 manually from a root shell in order to repair the file system.  Is there some setting I can change such that access to the root device after waking from suspend can be delayed until the USB drive is ready? Do gravitational lenses work on gravitational waves? When to choose checked and unchecked exceptions In Java (or any other language with checked exceptions), when creating your own exception class, how do you decide whether it should be checked or unchecked?  My instinct is to say that a checked exception would be called for in cases where the caller might be able to recover in some productive way, where as an unchecked exception would be more for unrecoverable cases, but I'd be interested in other's thoughts. Virtualbox Server/Client topology I used python add in to make a tool bar in Arcmap that edits features on a map. It works on a file geodatabase, but not on an ArcSDE version one. I've been getting an FDO error: 0. The solutions that I've found all involve opening a workspace, but I'm getting an error doing that as well. Here is the relevant portion of my code:          outFolderPath = r"U:\tempStuff"         outName = "Stuff.sde"         databasePlatform = "SQL_Server"         instance = "SQL Server"         arcpy.CreateDatabaseConnection_management(outFolderPath, outName,                                                   databasePlatform,                                                   instance)         print "Centroid of buidling:(%r,%r) Side length: %r" % (xCoordinate,                                                                 yCoordinate,                                                                 bob)         print "workspace is %r" % outName #outName is workspace          #Start edit session with an undo/redo stack then start         #edit operation         print "Starting editing"         edit = arcpy.da.Editor(outName) #outName is workspace Line 86         edit.startEditing(True, True)         edit.startOperation()          #stuff happens          cursor = arcpy.da.InsertCursor( fc, ["SHAPE@"])         cursor.insertRow([stuff])             #stop the edit operation and then save changes         edit.stopOperation()         edit.stopEditing(True)          del cursor         print "Building should be drawn"   This is the exact error I've been getting:  Traceback (most recent call last):   File "C:\Users\153289\AppData\Local\ESRI\Desktop10.3\AssemblyCache\{0BEF4E61-6332-4BCA-A908-959CA3C0B4E7}\leetScripts_addin.py", line 86, in onClick     edit = arcpy.da.Editor(outName)#outName is workspace RuntimeError: cannot open workspace   What's am I doing wrong in my code? I would like to see a full how-to guide on how to install Ubuntu. Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things. Determining if trees within forest gaps are clustered using R? The attached dataset shows approximately 6000 saplings in approximately 50 variable sized forest gaps.  I am interested in learning how these saplings are growing within their respective gaps (i.e. clustered, random, dispersed).  As you know, a traditional approach would be to run Global Moran's I.  However, aggregations of trees within aggregations of gaps seems to be an inappropriate use of Moran's I.  I ran some test statistics with Moran's I using a threshold distance of 50 meters, which produced nonsensical results (i.e. p-value = 0.0000000...).  The interaction among the gap aggregations are likely producing these results.  I have considered creating a script to loop through individual canopy gaps and determine the clustering within each gap, although displaying these results to the public would be problematic.    What is the best approach to quantify clustering within clusters?