I want to calculate this integral:$\int_{-\infty}^{\infty} \frac{c(\theta)}{(1 + x^2)^\theta} dx $  it's easy to do, when $\theta$ is given: For example when $\theta$ is 1, then integral is equal to $\pi$  But how to calulate integral, when \theta is not given?  And $c(\theta)$ is a constant, that is dependent on $\theta $
Does any one know the exact expression of the integral,  $$E_n(s)=\int_0^\infty \frac{r^{n-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s&gt;n, n\in \mathbb{N}$$ or more generally, $$E_a(s)=\int_0^\infty \frac{r^{a-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s&gt;a, a\in \mathbb{R}$$ For the special case $s=n, n+2$ I find out by induction that $$ E_{n-1}(n)=\frac{\omega_{n-1}}{2\omega_{n-2}}~~\text{and}~~E_{n-1}(n+2)=\frac{\omega_{n-1}}{2n\omega_{n-2}}. $$ where $\omega_{n-1} = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$ is the  surface measure of the n-dimensional sphere of $\mathbb{R}^n$. Further result is welcome
Why is ANOVA taught / used as if it is a different research methodology compared to linear regression? ANOVA is equivalent to linear regression with the use of suitable dummy variables. The conclusions remain the same irrespective of whether you use ANOVA or linear regression.   In light of their equivalence, is there any reason why ANOVA is used instead of linear regression?   Note: I am particularly interested in hearing about technical reasons for the use of ANOVA instead of linear regression.   Edit  Here is one example using one-way ANOVA. Suppose, you want to know if the average height of male and females is the same. To test for your hypothesis you would collect data from a random   sample of male and females (say 30 each) and perform the ANOVA analysis (i.e., sum of squares for gender and error) to decide whether an effect exists.  You could also use linear regression to test for this as follows:  Define:  $\text{Gender} = 1$ if respondent is a male and $0$ otherwise. $$ \text{Height} = \text{Intercept} + \beta * \text{Gender} + \text{error} $$ where: $\text{error}\sim\mathcal N(0,\sigma^2)$  Then a test of whether $\beta = 0$ is a an equivalent test for your hypothesis. How does $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ make sense? In Multivariable Calculus, the professor said that in order to compute the angle $x$ between two vectors $v$ and $w$, we use the formula: $\cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$  He also mentioned "Projection" but I didn't understand much of it. The only thing I got was that we can create a right triangle by drawing a perpendicular line from one vector to the head of the other and compute the angle like that using $\cos$, but I don't understand why we're multiplying vectors, or dividing by their magnitudes…how do those operations make sense?  Also, in the specific example, the result of $\cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ ended up being $1/2 = \pi/2$. Where does $\pi/2$ come from here? I can't understand how we go from $1/2$ to $\pi/2$. I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ones. Map a network drive to be used by a service Suppose some Windows service uses code that wants mapped network drives and no UNC paths. How can I make the drive mapping available to the service's session when the service is started? Logging in as the service user and creating a persistent mapping will not establish the mapping in the context of the actual service. What is the difference between sourcing ('.' or 'source') and executing a file in bash? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. What are the fault tolerances on the FDA food handling guidelines? Sometimes I have suggested that food that sat out a little longer than optimal may be safe to eat.    Granted, this is based on my experience, and I always qualify that it shouldn't be served to guests (and by extension, customers) and that one should determine their own tolerance to risk, but I want to know if there is information as to the methodology of the FDA guidelines.  Specifically, what is the contamination rate at n hours, and what are the initial conditions of the food and environment?  How much risk is there REALLY (in incidences per n-population and such) in a standard kitchen for food that has sat out 50%, 100%, 200%, etc. longer than recommended? Does Class need to implement IEnumerable to use Foreach This is in C#, I have a class that I am using from some else's DLL. It does not implement IEnumerable but has 2 methods that pass back a IEnumerator. Is there a way I can use a foreach loop on these. The class I am using is sealed. What is the word that is used to describe the oversimplification or non-academic discussion of a complex issue?   Typically, I'm looking for a word like pedantic, or puerile, except that the word covers the fact that the discussion is particularly non-academic. I know that such a word exists since I have used it before, but it has skipped my mind. Ping is a a program owned by root with the user id bit set.  $ ls -l `which ping` -rwsr-xr-x 1 root root 35752 Nov  4  2011 /bin/ping   As I understand it, if a user runs the ping process, then the effective user id will change from the real user id (i.e. the user id of the person who launched the process) to the user id root. However when I try this and look at the output of ps to see if the ping process is running as the root user, I still get the real user id showing.  ps -e -o user,ruser,euser,cmd,args | grep ping sashan   sashan   sashan   ping -i 10 -c 1000 www.goog ping -i 10 -c 1000 www.google.com How can be solved the equation $x^3+x-1=y^2$ in positive integers?    I know this equation defines an elliptic curve, but this seems to be a non-elementary way to solve this question. Is there a more elementary solution?   By the way I found three solutions: $(1,1), (2,3)$ and $(13,47)$. Is this related to the law group property of elliptic curves? Thanks for further answers. Tokenizing strings in C I have been trying to tokenize a string using SPACE as delimiter but it doesn't work. Does any one have suggestion on why it doesn't work?  Edit: tokenizing using:  strtok(string, " ");   The code is like the following  pch = strtok (str," "); while (pch != NULL) {   printf ("%s\n",pch);   pch = strtok (NULL, " "); } When is it appropriate to use SQL Server Developer Edition? Do you use SQL Server Developer Edition on server-class machines in DEV and STAGING environments?  I am working on a large project where (if it passes the proof of concept stage) we will have several large geographically distributed enterprise class database servers running SQL Server 2012 Enterprise Edition.  We will have a production environment will initially have 3 servers, our Staging environment will have a minimum of 3 servers, and our Development environment will have 1 server (hosting three instances).  I was under the impression that we would only need to acquire Enterprise licences for the actual production servers, and we could get by with developer edition in our developer and staging environments because they are not "production".  Other sources have told me that we need to have an enterprise licence on all of the above mentioned machines, and that the developer edition is only meant for a single developer on their workstation.  Since developer edition has all the sexy features of Enterprise, I can't really see the value of it on a workstation class machine, especially for developing and testing the type of high availability system we are building.  If we have to fork out Enterprise licences for a dev server, that will just about kill our proof of concept stage, thus killing the project.  Forcing an enterprise licence on a staging environment will make management just want to skip staging altogether. IBM Slack links broken Does anyone know where I can find working Slack invite links to any of IBM's Slack pages? The Slack links on  are all broken. Thank you. what is the difference between IBE and ABE schemes Can somebody explain in simple words. real life example light help and limitations of these schemes etc. Schengen Visa Application: what are some helpful tips for preparing? Plugin or brush for transforming a photo to a fractal in gimp? Folks,  I've been looking all over for a plugin or brush that will help me take a photo and reduce it to that 'electric look' (or fractal?) that i'm seeing in various places. I'd vastly prefer one for Gimp.  Does anyone know how these are being done? I'm seening references to fractalius for photoshop, but what about gimp?  Examples: First, my apologies if this has already been asked/answered.  I wasn't able to find this question via search.  My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1, Example 1.1 on p. 2.  In the second version of the proof, showing that sets A and B do not have greatest or lowest elements respectively, he presents a seemingly arbitrary assignment of a number $q$ that satisfies equations (3) and (4), plus other conditions needed to show that $q$ is the right number for the proof.  As an exercise, I tried to derive his choice of $q$ so that I may learn more about the problem.  If we write equations (3) as $q = p - (p^2 - 2)x$, we can write (4) as   $$ q^2 - 2 = (p^2 - 2)[1 - 2px + (p^2 - 2)x^2]. $$  Here, we need a rational $x &gt; 0$, chosen such that the expression in $[...]$ is positive.  Using the quadratic formula and the sign of $(p^2 - 2)$, it can be shown that we need   $$ x \in \left(0, \frac{1}{p + \sqrt{2}}\right) \mbox{ for } p \in A, $$  or, for $p \in B$, $x &lt; 1/\left(p + \sqrt{2}\right)$ or $x &gt; 1/\left(p - \sqrt{2}\right)$.  Notice that there are MANY solutions to these equations! The easiest to see, perhaps, is letting $x = 1/(p + n)$ for $n \geq 2$.  Notice that Rudin chooses $n = 2$ for his answer, but it checks out easily for other $n$.  The Question:  Why does Rudin choose $x = 1/(p + 2)$ specifically?  Is it just to make the expressions work out clearly algebraically?  Why doesn't he comment on his particular choice or the nature of the set of solutions that will work for the proof?  Is there a simpler derivation for the number $q$ that I am missing? In a bash shell script, writing a for loop that iterates over string values Can we have a list of all the Stack Exchange domains somewhere, for firewall purposes? Manually change filetype icons in Windows 7 I'm trying to find the dialog tab in Windows 7 that allows a user to change the icon used for a particular file type.  In Windows XP it would be under the Folder Options properties dialog.    Where is it on Windows 7?  I only want to change the icon of a specific filetype, to anything I want but this tab is no longer present in Folder Options of Windows 7.