Geometric interpretation of covariance
Analogy of Pearson correlation for 3 variables
Why does the odd part of the function integrate to $0$? When evaluating the integral $$\int^1_{-1}\frac{\cos(x)}{1+e^{1/x}}dx,$$ the author says that the odd part of the function integrates to $0$ over $-1$ to $1$. I have calculated the even part of the function ($\frac{1}{2}\cos(x)$), so the integral is equal to $\sin(1)$. But why exactly does the odd part integrate to $0$? I have noticed after deleting an original page in WordPress with lets say a permalink of your_domain.com/contact/ and then trying to recreate the same page with the same name/slug, my WP is generating a "-2" at the end of the link, like: your_domain.com/contact-2/.  I don't understand why, since the original page was deleted. Why is WordPress treating new pages as if the original still exists? How to I clean WP caches, if such a thing even exists?Thank you ahead of time for clearing this up for me. How to determine the size of a package while using apt prior to downloading? When using apt-get install &lt;package_name&gt;, and there are dependencies that need to be downloaded, the terminal outputs names of additional packages and total size, and asks for confirmation before downloading.   But, when dependencies are satisfied and nothing but the named package needs to be downloaded there is no size output and no confirmation.  When using Synaptic, I can see the total size that new packages that will use after installation but no way to see the size that needs to be downloaded, except to go from package to package and use properties to see the compressed size.  I would like to know if there is a way to see the size of a package(s) in terminal and Synaptic prior to downloading and installing it/them? How does friction stop a car? I'm unable to understand how friction causes motion in cars and also is responsible for stopping them. When it starts the impending motion of tyres is in the backward direction that's why friction is in the forward direction and the car starts moving. But when the car engine turns off, the car stops after some time, but the condition of tyres are the same as in the previous case and friction must be in the forward direction. How to fix task manager memory column if it shows lies I have a problem in showing memory in task manager with memory and paging columns. Why it shows so small size usage? And maybe anyone knows how to fix it?  Here is a screenshot(working set, memory, page pool) sorted by most consuming: Using Perl to change the url homepage for Internet Explorer Counterexample to Leibniz criterion for alternating series I have this statement and I need to say if it is true or false:  Let $\{a_n\}$ be a real sequence.  $$\lim_{n\to +\infty} a_n = 0 \quad \implies \quad \sum_{n=1}^{\infty}(-1)^na_n \quad \text{converges}$$  I know, from the Leibniz criterion that:     If $a_n \to 0$, $a_n$ is decreasing and positive then $\sum_{n=1}^{\infty}(-1)^na_n$ converges   From this fact I believe that the statement is false but I couldn't come up with an infinitesimal sequence that isn't decreasing and for that reason is divergent. I tried some function with $sin(\frac{1}{n})$ without any luck. Any help would be very appreciated, thank you! How can I get sql string, generated by Linq2SQL ORM? If $S$ is a finitely generated graded algebra over $S_0$, $S_{(f)}$ is finitely generated algebra over $S_0$? Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$. Why don't my female dwellers give birth to children? I am playing Fallout Shelter on my IPad. There were some women who have already gave birth, but two female dwellers are carrying their babies for about 18(!) hours. I know that this process takes three hours, so this is not normal. What do I do? I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? Access is denied because the NTFS volume is already exclusively opened Young Adult book about kids being bred for genetic purity in a world where races no longer exist I'm looking for a book I read in Junior High (probably) about a world where everyone was basically the same race.  One genetically pure male and female teenager of each race still exists.  They are transporting a genetically pure boy to meet his genetically pure female counterpart and accidentally introduce him to a genetically pure female of a different race.  And, of course, chaos ensues.  Any thoughts? How to fix the weird surface in fluid object What is the argument selector tracking a defined score (not [score_min=])? What if I want to make a command like this:  execute @e[score_A=10] ~ ~ ~ summon Creeper   So that the entity must have a score of 10 (not less, not more)?  Does it even exist that or a [score_A_max=] argument?  I know about scoreboard players test, but i guess it is not the case to use it. I'm trying to use Linq to return a list of ids given a list of objects where the id is a property. I'd like to be able to do this without looping through each object and pulling out the unique ids that I find.  I have a list of objects of type MyClass and one of the properties of this class is an ID.  public class MyClass {   public int ID { get; set; } }   What I want to do is write a Linq query to return me a list of those Ids  How do I do that given an IList&lt;MyClass&gt; such that it returns an IEnumerable&lt;int&gt; of the ids?  I'm sure it must be possible to do it in one or two lines using Linq rather than looping through each item in the MyClass list and adding the unique values into a list.  Any help in creating an elegant solution would be much appreciated! 74190 counter problem So, I got my 74190 counter shows number like this  9-8-7-6-5-4-3-2-1-0-5-4-3-2-1-0-5-4-3-2-1-0-5-4-3-2-1-0 and so on.  My questions:   What can I do to change the output count down into 5-4-3-2-1-0-5-4-3-2-1-0 and so on? When it changes 0 into 5 it moves so fast. How could I fix this one? Where does Mac OS X come from? Automatically make or update a copy real-time on another hard drive volume whenever files are saved to a particular folder We have a view for a type of node that includes a Private File.  We cannot see a way to add an exposed filter, such as "Contains any word", and have it filter based on the file's name.  Are we missing something obvious?