Why does solid turn to liquid deep in the Earth?
Why has Earth's core not become solid?
Creating intuition about Laplace & Fourier transforms Subgroup of $\mathbb{Q}$ with finite index Density of $\mathbb{Q}^n$ in $\mathbb{R}^n$ I am wanting to find the correlation between two variables: a categorical vegetation variable (with categories 1-4)  and a continuous variable of erosion (with values anywhere between 0.2 and 9). I think I would be required to use a non parametric test to find the correlation, but bit of a newby to stats so could anyone suggest the best test to use? I also wanted to know how sample size can affect these tests, ie with a sample size of 100 or 1000? When a ball moves to the right, friction acts to oppose the motion, in other words, to the left. However, when a car travels around a bend, the friction acts in the perpendicular direction to the car's velocity and provides the centripetal force. I just cannot understand why friction would act in that direction. Get PHP to stop replacing '.' characters in $_GET or $_POST arrays? If I pass PHP variables with . in their names via $_GET PHP auto-replaces them with _ characters. For example:  &lt;?php echo "url is ".$_SERVER['REQUEST_URI']."&lt;p&gt;"; echo "x.y is ".$_GET['x.y'].".&lt;p&gt;"; echo "x_y is ".$_GET['x_y'].".&lt;p&gt;";   ... outputs the following:  url is /SpShipTool/php/testGetUrl.php?x.y=a.b x.y is . x_y is a.b.   ... my question is this: is there any way I can get this to stop? Cannot for the life of me figure out what I've done to deserve this  PHP version I'm running with is 5.2.4-2ubuntu5.3. Difference between $\sum_{i=1}^{k}{\frac{s_i}{kn_i}}$ and $\frac{\sum_{i=1}^{k}{s_i}}{\sum_{i=1}^{k}{n_i}}$ I am getting confused at how to calculate the average probability.  Suppose we repeat a kind of binary survey $k$ times each of which was done on a completely separate sample group.  For each $i^{th}$ group, $i=1,2,3,...,k$, Let $n_i$ be the number of samples and $s_i$ be the number of positive results. With this we know that the probability $p_i$ of the positive result for the $i^{th}$ group is $s_i/n_i$  With my ignorance, I happened to use two ways to calculate the average probability in my work wrongly assuming that they are the same:  $$\frac{p_1+p_2+p_3+...+p_k}{k}$$ and $$\frac{s_1+s_2+s_3+...+s_k}{n_1+n_2+n_3+...+n_k}$$  I don't know which is the correct way to calculate the average probability. So could you please explain the difference of these two and when to use which? My character has a bite attack from his race.  What happens if he gains another bite attack from class abilities or spells?  Does the damage increase?  Can he make more than one attack?  Or is it simply that only the better version applies?  I would prefer answers from a Pathfinder context but 3.5e is also acceptable. Disable the activation of the menu bar when Alt is pressed in Windows 7 In Windows 7 how can I disable the function that activates the menu bar when Alt is pressed?  Are there some registry values to modify this behaviour? If $a^\phi = 1$ then the order of $a$ divides $\phi$ I am trying to filter items with a stored procedure using like. The column is a varchar(15). The items I am trying to filter have square brackets in the name.  For example:  WC[R]S123456.  If I do a LIKE 'WC[R]S123456' it will not return anything.  I found some information on using the ESCAPE keyword with LIKE but I do not understand how to use it to treat the square brackets as a regular string. I'm using the built in Screen Sharing tool in El Capitan to control an iMac from a Macbook Pro in full screen mode. Everything works beautifully except that many special keys and gestures are intercepted by the local Mac and not sent to the shared Mac. Specifically:   Ctrl-Arrow (Desktops and Mission Control) F3 (Mission Control) F4 (LaunchPad) Three finger swipe (Switch between Desktops, Mission Control, Exposé) Two finger swipe from left side (Notification Center) Pinch with four fingers (Launchpad)   I can't find any way to switch between desktops on the remote machine other than manually opening the Mission Control app from the Finder or Dock. Disabling the gestures on the local machine doesn't help; they still don't get sent to the remote machine.  Is there any way to get special keys and gestures passed to the shared machine? How to add citation for a whole paragraph? If I write a report and a whole paragraph is based on one source but not cited verbally it seems unnecessary to include the reference after every sentence:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua (source 1). Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat (source 1).  Now I found a Phd thesis that puts the reference after the last dot of the paragraph to indicate that it is based on this very source:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. (source 1)  Is this a common practice? Is it generally understood? What are other ways to add the reference for a whole paragraph? Why does water pouring from a glass sometimes travel down the side of the glass? How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$? Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise? Where in the file system are applications installed? I have installed a few Android applications both with the adb install MyApplication.apk command and via a webserver (like an alternative to Android Market).  But where in the Android file system on my phone is the MyApplication.apk file placed? Are applications installed with adb install not placed in the same directory as applications downloaded from the Market?  I found some applications on /system/app but I can't recognize them as some downloaded from Android Market or the one I have installed with adb install.  Where on the system is applications that I have installed located?  My phone is not rooted, and I am running Nexus One with Android 2.2.1. What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15 Question: Are my proofs below valid?   In both cases we are using: $f:A\to B, g: B\to C$  Notation of your type converted: $(g\circ f)(x)=g(f(x))=xfg$    If $fg$ is injective what can be said about $f$ and $g$?  If $fg$ is injective then $f$ is injective:  Proof:  $(x_1fg = x_2fg \implies x_1=x_2)$  $x_1fg=x_2fg$ is clearly true when $x_1f=x_2f$  Which requires $x_1=x_2$, hence $f$ is injective.  If $f$ were not injective, then $x_1fg\ne x_2fg$  $\blacksquare$    If $fg$ is surjective what can be said about $f$ and $g$?  If $fg$ is surjective then $g$ is surjective:  Proof:  $(\forall c\in C)(\exists a\in A) |( a(fg))=c$  This means for $g$ every element must have had been mapped to. How can this be proved Mathematically?  well $(af)g=c$ is surjective, hence $(af)=(b\in B)$ and $bg=c$  $(\forall c\in g)(\exists b\in B)|(bg=c)$  Hence $g$ is surjective.  $\blacksquare$ How to simplify the summation kk! without using induction? Survival probability up to time $n$ in a branching process. Let $\{Z_n : n=0,1,2,\ldots\}$ be a Galton-Watson branching process with time-homogeneous offspring distribution $$\mathbb P(Z_{n,j} = 0) = 1-p = 1 - \mathbb P(Z_{n,j}=2), $$ where $0&lt;p&lt;1$. That is, $Z_0 = 1$ and $$Z_n = \sum_{j=1}^{Z_{n-1}}Z_{n,j}$$ for $n\geqslant 1$. Let  $$T=\inf\{n : Z_n=0\} $$ be the extinction time of the process. I want to find $$t_n := \mathbb P(T&gt;n), $$ i.e. the probability that the process survives up to time $n$. I found the following recurrence:  $$t_{n+1} = p(2t_n - t_n^2)$$ (with $t_0=1$) from this mathoverflow question:   I checked the recurrence for small values of $n$, and confirmed that $t:=\lim_{n\to\infty} t_n$ satisfies $$ t = p(2t - t^2),$$ both in the case where $t=0$ $\left(p\leqslant\frac12\right)$ and where $t=2-\frac1p$ $\left(p&gt;\frac12\right)$. So I'm fairly confident this recurrence is valid. However, I have no idea how to solve it.  For context, this problem comes from Adventures in Stochastic Processes by Sidney Resnick:   From @Did's comment, it appears to be intractable to find a closed form for $\mathbb P(T&gt;n)$. I find it curious that the question would be asked were it not, though. I met an interesting issue about C#. I have code like below.  List&lt;Func&lt;int&gt;&gt; actions = new List&lt;Func&lt;int&gt;&gt;();  int variable = 0; while (variable &lt; 5) {     actions.Add(() =&gt; variable * 2);     ++ variable; }  foreach (var act in actions) {     Console.WriteLine(act.Invoke()); }   I expect it to output 0, 2, 4, 6, 8. However, it actually outputs five 10s.  It seems that it is due to all actions referring to one captured variable. As a result, when they get invoked, they all have same output.  Is there a way to work round this limit to have each action instance have its own captured variable?