Data set size, how many to have a significant result?
What is a reasonable sample size for correlation analysis for both overall and sub-group analyses?
How do we compare the size of numbers that are around the size of Graham's number or larger? When numbers get as large as Graham's number, or somewhere around the point where we can't write them as numerical values, how do we compare them?  For example:  $$G&gt;S^{S^{S^{\dots}}}$$  Where $G$ is Graham's number and $S^{S^{S^{\dots}}}$ is $S$ raised to itself $S$ times and $S$ is Skewes number.  It appears obvious (I think) that Graham's number is indeed larger, but how does one go about proving that if both numbers are "so large" that they become hard to compare?  More generally, how do we compare numbers of this general size?  As a much harder problem than the above, imagine a function $G(x,y)$ where $G(64,3)=$ Graham's number.  The function $G(x,y)$ is as follows:  $$G(x,y)=y\uparrow^{(G(x-1,y))}y$$  Where $G(0,y)$ is given.  I ask to compare $G(60,S)$ and $G(64,3)$ Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? Change keyboard layout (English UK) on command line to English US Dataset collapsing/reducing C++ virtual function from constructor Why is the electric field inside a conductor zero in equilibrium? My textbook says the field inside a conductor must be zero in order for the system to be equilibrium and therefore there must be no excess charge inside.  Their proof:  1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero.  2) Therefore the net flux is zero, implying the charge inside is zero.  3) If there is no charge inside, all excess charge must lie on the surface.  My problem:  "Since the system is at equilibrium, all points on the surface must have an electric field of zero."  Why?  Consider an conductor with no excess charge. To simplify things, say the conductor consists of a 100 free electrons and a 100 ions of charge 1$e^+$. The conductor is neutral and at equilibrium.  Now consider the field on one specific electron. There a total of 99 other electrons and 100  ions. So there would be a net field acting on that specific electron. So even though the system is at equilibrium, the net field on the individual particles is non-zero.  So a system can be at equilibrium, even if the net field inside is non-zero. What am I misunderstanding?  Another example: Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate. Mark or don't mark required fields if all are required? If all fields in a form are required should they be marked somehow (eg. with an asterisk)?  I see this done a lot and find it redundant?  There could be a difference in big forms and small forms, but in big ones I would add a note at the top that all fields are required, but I wouldn't necessarily do that on small ones.  What do you think is the right approach? What's the best way to use SOAP with Ruby? A client of mine has asked me to integrate a 3rd party API into their Rails app.  The only problem is that the API uses SOAP.  Ruby has basically dropped SOAP in favor of REST.  They provide a Java adapter that apparently works with the Java-Ruby bridge, but we'd like to keep it all in Ruby, if possible.  I looked into soap4r, but it seems to have a slightly bad reputation.  So what's the best way to integrate SOAP calls into a Rails app? How do moving charges produce magnetic fields? How to extract code of .apk file which is not working? What is spin as it relates to subatomic particles? Is there a filesystem that is "friendly" to both windows and Linux? Prove that even + odd is odd. Prove that a even number + odd number = odd number  Let $x$ be the even number, let $y$ be the odd number.   From the definition of odd numbers, $y + 1$ is even.   Let:  $$x + y = z$$  Suppose $z$ is even.   $$x + y = z$$  It follows that:  $$x = x + y - y = z - y$$  But I cant think of any other contradiction here. The issue is with the assumption I suppose. Any help is appreciated! Can toddlers be taught to safely navigate stairs? I'm studying the Darboux definition of integrability, which I completely explained . There's an exercise that asks me to prove that the Darboux Integrability is equivalent to Riemann Integrability, but this Riemann integral is defined as the following:  It first defines a 'pointed' partition (I don't know how to say it in ensligh) by the following: a 'pointed' partition $[a,b]$ is a pair $P^*=(E,ξ)$, where $P=\{t_0, t_1, \cdots, t_n\}$ is a partition of $[a,b]$ and $ξ = (ξ_1, \cdots, ξ_n)$ is a list of $n$ chosen numbers such that $t_{i-1}\le ξ_i\le t_i$ for each $i=1,\cdots ,n$.  Now, the Riemann Integral is defined as:  $$\sum(f,P^*) = \sum_{i=1}^n f(ξ_i)(t_i-t_{i-1})$$  (I didn't understand the notation for the left hand side of the equation, by the way)  Finally, I'm asked to prove the following: given $f,g:[a,b]\to \mathbb{R}$ integrable functions, for the entire partition $P=\{t_0, \cdots, t_n\}$ of $[a,b]$ let $P^* = (P,ξ)$ and $P^{\#} = (P, η)$ be pointed partitions of $P$, then:  $$\lim_{|P|\to 0}\sum f(ξ_i)g(η_i)(t_i-t_{i-1}) = \int_a^b f(x)g(x) \ dx$$  I guess here I need to prove that the Riemann Integral of the product of two functions if the darboux integral of $f(x)g(x)$, but it seems too obvious, I just need to verify that $f,g$ are integrable, then their product is too, isn't it? I'm pretty sure this should be a hard question. Is there another interpretation that I'm missing? Print total free memory from `free -m` I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? Sketching base characteristics of first order pde $$xu_{x} - uu_{t} = t$$   with intial data: $u(1,t) = t, −\infty &lt; t &lt; \infty$  So I want to sketch the base characteristics and I'm not too sure what I'm doing.  characteristic equations: $\frac{\partial{x}}{\partial{\tau}}=x$, $\frac{\partial{t}}{\partial{\tau}}=-u$, $\frac{\partial{u}}{\partial{\tau}}=t$  initial conditions:  $x(0)=1$, $t(0)=\xi$, $u(0)=\xi$  $x=e^\tau$ so $\tau=lnx$  Notice that:  $\frac{\partial^2{u}}{\partial{\tau^2}}=\frac{\partial{t}}{\partial{\tau}}=-u$,  This gives me the second order ODE  $\frac{\partial^2{u}}{\partial{\tau^2}}+u=0$  Solving I get:  \begin{align}   u&amp;=c_{1}\cos(\tau) + c_{2}\sin(\tau)\\ t&amp;=c_{1}\sin(\tau) - c_{2}\cos(\tau) \end{align}  From here I don't know what I'm doing? Transferring Fallout 4 Saves (PS4) ReadList can't read from file containing high-precision Reals