What is the relationship between the Gamma distribution and the chi-squared distribution?
If $$Y=\sum_{i=1}^{N}X_i^2$$ where $X_i \sim \mathcal{N}(0,\sigma^2)$, i.e. all $X_i$ are i.i.d. normal random variables of zero mean with same variances, then $$Y \sim \Gamma\left(\frac{N}{2},2\sigma^2\right).$$  I know the chi-squared distribution is a special case of the gamma distribution, but could not derive the chi-squared distribution for random variable $Y$. Any help, please?
Has a free neutron ever been shown to absorb/emit/interact with a photon? Protons only 'interact with' very high-energy photons, whether inside a nucleus or free, right?  I'm assuming the same about neutrons....  Neutrons have a small magnetic moment and a slight electric 'moment' (dipole?), correct?  They are sometimes considered to consist of a negative charge surrounding a positive charge?  I would assume that neutrons, in a nucleus or free, interact with photons even less often than protons, and that the gamma photon has to have an even higher energy than one interacting with a proton....  Lastly, I have read about 'photodisintegration', in which a nucleon is kicked out of a nucleus by a gamma ray.  It seems to happen to neutrons sometimes...  But, I am mostly curious about experimental evidence of free neutrons interacting with photons, how often it happens compared to free protons, and whether the interacting photon's energy has to be extremely high, higher even than a photon interacting with a proton... With the listings package, I would like to color the integers out of strings and comments and out of keywords too. The question for coloring numbers     . Some answer handle the numbers occurring within comments, strings or barewords, but no solution given work when a keyword contain a number. Which is annoying because C and C++ have a lot of types with a number in them, like uint32_t.  Here is a small example.  \documentclass{standalone} \usepackage{xcolor} \usepackage{listings}  \definecolor{darkgreen}{rgb}{0.18,0.54,0.34} \definecolor{maroon}{rgb}{0.64,0.16,0.16} \definecolor{darkpink}{rgb}{0.75,0.25,0.5}  \lstdefinestyle{myc++}{     language=[ISO]C++,     keywordstyle=\color{darkgreen}\bfseries,     commentstyle=\color{blue}\textit,     stringstyle=\color{darkpink}\ttfamily, %     literate=*     {0}{{{\textcolor{red}0}}}1     {1}{{{\textcolor{red}1}}}1     {2}{{{\textcolor{red}2}}}1     {3}{{{\textcolor{red}3}}}1     {4}{{{\textcolor{red}4}}}1     {5}{{{\textcolor{red}5}}}1     {6}{{{\textcolor{red}6}}}1     {7}{{{\textcolor{red}7}}}1     {8}{{{\textcolor{red}8}}}1     {9}{{{\textcolor{red}9}}}1,     escapeinside={|}{|}, %     morekeywords={int32_t} }  \begin{document}  \begin{lstlisting}[style=myc++] // In comment: 42 int32_t in_keyword = 80085; char *in_identifier|2| = "And in string 1337\n"; \end{lstlisting}  \end{document}   And the result:    The star for literate avoid to coloring the digits inside comments or strings. The escapeinside avoid coloring the digits inside a bareword when specifically marked. But I have yet to find how to color int32_t as a keyword. How do you correctly say large numbers I saw a post on  which links to a video where a person . Is he saying a large portion of the numbers wrong? Back in high school my algebra teacher was extremely picky.  If we were going to say the number 135, we would say one hundred thirty-five.  If a person said one hundred and thirty-five my teacher would interpret that to mean 100.35.  Was my algebra teacher wrong, or did Jon just waste over 80 hours counting to 100,000 incorrectly. I tried searching the web, but searching for ‘numbers and’  doesn’t work very well.  Can you point me to a reference or style guide that describes how to properly say large numbers? Is it possible for a system to have negative potential energy? With respect to both gravity and electromagnetism, to the best of my understanding potential energy is added or subtracted from a system based on the distance between two objects such as charged or massive objects.  Would this then imply that when oppositely charged objects or two massive objects are brought together within a small enough radius that the drop in potential energy would be greater than the combined mass of the two objects making the net mass/energy of the system negative? Domain Controller etiquette after a crash between Server 2003 & Server 2008 I have 2 domain controllers on a non-Internet connected network running Windows Server 2008 R2. They running RAID 5 on a 8 Hard Drive server. Sometime last week the Boot Sector became corrupt on DC1 and I was unable to recover the server. I am not going to have to rebuild the server. I have been reading about the changes from Server 2003 to Server 2008. DC1 was the main domain controller and now that it won't but it is all falling to DC2. I know that in previous versions of Windows Server you had to make on the primary and one the secondary. Since DC2 does not see DC1 anymore is there anything I need to do to DC2 as it is now the one in control? I am thinking some kind of meta data clean up but I can't find any hard information on what procedure to do. I understand as of Server 2008 domain controllers are on an equal footing and provide replication to each other so they should be the same correct? Algorithm to return all combinations of k elements from n I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. A subspace $X$ is closed iff $X =( X^\perp)^\perp$ Let $X$ be a subspace of a Hilbert space $H$.  prove:  $X$ is closed iff $X =( X^\perp)^\perp$   I do not know how to proceed. any hints would be appreciated.  thanks.  $X^\perp = ${ $y \in H$:$\langle x,y\rangle = 0 \forall x \in  X\}$. How to include SVG diagrams in LaTeX? I'm making all my diagrams in SVG format using Inkscape and then I export them to some other format (e.g. PDF, EPS or PNG). I wrote a short script that does this for me automatically, but I'd like to avoid this step, if possible --- sometimes the conversion from SVG converts the text labels into vectors, and that's annoying. Hence the question:  Is there a way to include SVG diagrams directly without conversion to an intermediate format?  Package suggestions, or instructions for specific LaTeX distribution are welcome. Tips and personal experience on which tools you have found to be working reliably would be also appreciated. I'm trying to print types like off_t and size_t.  What is the correct placeholder for printf() that is portable?  Or is there a completely different way to print those variables? How to remove/apply all modifiers of one object in python Lets assume you have 1 object with 4 modifiers added to it. I am curious how one would approach the task to apply all or to delete all modifiers of a just selected object that has many modifiers already added.  The console reads out the modifier delete as  bpy.ops.object.modifier_remove(modifier="Box Right ")   So "Box Right " I might have to replace with something like active.object.modifier.name and then work each modifier off?  This would be very helpful when you want to generate a static mesh for animation or 3D printing. Creating Workflow -> Error When I try to create a new WF in my Sharepoint Designer I get following error:     The list of workflow actions on the server references an assembly that   does not exist. Some actions will not be avaliable. The assembly   strong name is "Microsoft.SharePoint.WorkFlowActions,   Version=14.0.0.0,Cultureneutral,PublickeyToke=   Does any on know how to fix this problem? How to get distance when acceleration is not constant? I have a background in calculus but don't really know anything about physics.  Forgive me if this is a really basic question.  The equation for distance of an accelerating object with constant acceleration is:  $$d=ut        +\frac{1}{2}at^2$$  which can also be expressed  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}$$  (where x(t) is the position of the object at time t)  That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed?  The acceleration is obviously not constant, but what about the change in acceleration?  Is it constant?  I suspect not.  And then what about the change in the change of acceleration, etc. etc.?  In other words, how does one know how many additional terms to add in the series?  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}+\frac{\mathrm{d^3}x}{\mathrm{d}t^3}\frac{t^3}{3}+\frac{\mathrm{d^4}x}{\mathrm{d}t^4}\frac{t^4}{4}\cdot etc. \cdot ?$$ Is it "5–6 weeks are a lot of time" or "5–6 weeks is a lot of time"? I was just copyediting  and my native English speaker Sprachgefühl told me I had to correct the grammar of one sentence:     ... 5–6 weeks are a lot of time ...   by changing the are to is. But as I was doing so I started wondering why is it that in this case it seems that I have to make the verb disagree with the plural subject?  So is my feeling for English going bad or if I did the right thing, how could I explain this to somebody who's learning English for instance? What are the differences between the original Kaito Kid specials vs. the new Kaito Kid 1412 series? I am currently watching . Should I watch the other  anime films?  What's the difference between this version and the 1412 version?  And another question, what's the meaning of 1412? Endnotes do not be superscript and add a space Can we prove that every entire one-to-one function is linear? How to prefix post permalinks with /blog/ - without affecting custom post types? I have several custom post types, which are properly structured as /%post_type%/%postname%/, however I wish to have the same effect for regular blog entries: /blog/%postname%/  If I prepend the permalink structure with /blog/ it affects the custom post types as well, so currently my permalink structure is simply /%postname%/  Which is the recommended way of doing this? I had a solution involving some rewrites, but the permalink shown in the post editor area does not reflect it. Say I have a cologne bottle of 100 ml that I want to carry in my hand luggage on a plane trip.  100 ml is the limit so it should be fine for the transport regulations (if not, please let me know!).  The problem I always encounter is regarding the storage of the bottle: how to prevent it from leaking?  What I normally do is to put it in a plastic bag, so that the possible loss remains there. It is fine, only that the bottle ends up quite wet.  What would be a better approach? Suppose I have some theorem in the paper  Section 1 Theorem 1.1. Let ...  And then later in the paper I want to recall the theorem by reprinting it  Section 4 We recall Theorem 1.1: Theorem 1.1. Let ...  What's the proper way to do this? Starting yesterday, I have a very odd problem on my Windows 10 desktop-- my CPU is constantly running at 100% (according to Task Manager), and it isn't throttling down. Because of this, it's constantly running at 3.3 GHz and the temperatures are rising to the point where the fans kick in at full force. If I go into the Windows power settings screen and manually change the processor power maximum to 50%, then it's capped at 50% frequency (~1.5 GHz), but it never goes below the maximum possible frequency.  The weird thing is, when I look at the process list, there isn't really anything obvious that's triggering it. None of the processes are abnormal, and though the process CPU percentages do add up to 100%, it doesn't make sense to me that each individual process is taking up as much CPU as Task Manager says it is. Everything seems to be "using" a significantly higher amount of CPU resources than it normally does.  As an attempt to diagnose the problem, I've also tried using Process Explorer. The really weird thing is that Process Explorer shows only 1-2% usage when idle, with the System Idle Process taking up the remaining ~98% as I would expect. Again, nothing looks out of the ordinary in the process list, but the processes are showing the levels of activity I would expect in Process Explorer.  I've already run both Windows Defender and Malware Bytes (a virus was my first guess), but nothing has turned up yet.  This same issue also occurs in Safe Mode, so I'm guessing that it's more along the lines of something to do with Windows itself...  So, to recap:   Windows 10 Desktop Intel 970, 3.3 GHz, 6 cores Task manager and resource monitor graphs shows 100% usage total No individual process is taking up significant resources in the resource monitor list CPU is not throttling down automatically based on the dynamic load, causing high temperatures and the fan to kick in Process Explorer is only showing 1-2% CPU usage   Before I go uninstalling everything, how would I figure out what's causing the issue if nothing weird is showing up in Task Manager or Process Explorer?