hypotheses testing
What follows if we fail to reject the null hypothesis?
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$. Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set.   (Can we use definition of compact set for this problem? ) The exchange of photons gives rise to the electromagnetic force Pardon me for my stubborn classical/semiclassical brain. But I bet I am not the only one finding such description confusing.  If EM force is caused by the exchange of photons, does that mean only when there are photons exchanged shall there be a force? To my knowledge, once charged particles are placed, the electromagnetic force is always there, uninterruptedly. According to such logic, there has to be a stream of infinite photons to build EM force, and there has to be no interval between one "exchange event" to another. A free light source from an EM field? The scenario is really hard to imagine.  For nuclei the scenario becomes even odder. The strong interaction between protons is caused by the exchange of massive pions. It sounds like the protons toss a stream of balls to one another to build an attractive force - and the balls should come from nothing.  Please correct me if I am wrong: the excitations of photons and pions all come from nothing. So there should be EM force and strong force everywhere, no matter what type of particles out there. Say, even electrical neutral, dipole-free particles can build EM force in-between. And I find no reason such exchanges of particles cannot happen in vacuum.   Hope there will be some decent firmware to refresh my classical brain with newer field language codes. I feel that I might have a complete misunderstanding here. If DNA has two strands, how does the machinery of RNA transcription determine which one to transcribe? Test if a package (or package option) is loaded [french]{babel} (aka frenchb) is incompatible with some other packages, such as tikz, since it tries to insert \thinspace before semi-colons.  In order to make these two play nice, you have to call \shorthandoff{;} in the document.  Now I have a package that uses tikz and I would like to locally disable the semi-colon for babel, instead of disabling it for the whole document. I could hack the .sty of the package to do that, knowing that I'm using frenchb, but I would rather make something clean and portable.  Is there a way in my package that I can test if frenchb is loaded and disable semi-colon management with babel locally, so the package doesn't have to be hacked?  Edit:  Martin's solution gives me something like this in general:  \RequirePackage{ifthen} \newcommand{\@babel@loaded}{0} \AtBeginDocument{\@ifpackageloaded{babel}                                                    {\renewcommand{\@babel@loaded}{1}}                                                         {\renewcommand{\@babel@loaded}{0}}                                                       } \newcommand{\somecommandusingsemicolon}{   \ifthenelse{\@babel@loaded=1}{\shorthandoff{;}}{}     % code with semi-colon here }   and in the specific case of tikz:  \AtBeginDocument{\@ifpackageloaded{babel}   {%     \g@addto@macro\tikz@installcommands{%       \shorthandoff{;}%     }     \g@addto@macro\tikz@uninstallcommands{%       \shorthandon{;}%     }   }   {\relax} } Decode escaped Url without using HttpUtility.UrlDecode Why does the natural ring homomorphism induce a surjective group homomorphism of units? I'm trying an old problem here:   Suppose $n\mid m$, and I have a natural ring homomorphism $\varphi\colon \mathbb{Z}/m\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ defined by $\varphi(j)=j\mod{n}$. I can verify that this is a ring homomorphism, but why does it induce a surjective group homomorphism on the unit groups $(\mathbb{Z}/m\mathbb{Z})^\times\to(\mathbb{Z}/n\mathbb{Z})^\times$?  My question in particular is why is it surjective? I take $k\in(\mathbb{Z}/n\mathbb{Z})^\times$. Then there exist integers $s,t$ such that $sn+tk=1$. Since $n\mid m$, I can also write $m=na$. So multiplying through I get $sm+tka=a$. I'm lost here. How can I find a unit in $(\mathbb{Z}/m\mathbb{Z})^\times$ which maps to $k$ to show surjectivity? How many maximum CPUs does Ubuntu support by default? I think this is kernel dependent and probably will change over time depending on the kernel a release uses, correct me if wrong  I'd like to know two things for all the currently supported Ubuntu releases:  How many maximum CPUs can Ubuntu handle (by default) at the same time with a standard desktop kernel? Is that count different from a server kernel and 32bit/64bit systems?  Where can I find that information on my system? Top Down RPG Movement w/ Correction? I would hope that we have all played Zelda: A Link to the Past, please correct me if I'm wrong, but I want to emulate that kind of 2D, top-down character movement with a touch of correction. It has been done in other games, but I feel this reference would be the easiest to relate to.  More specifically the kind of movement and correction I'm talking about is:   Floating movement not restricted to tile based movement like Pokemon and other games where one tap of the movement pad moves you one square in that cardinal direction. This floating movement should be able to achieve diagonal motion. If you're walking West and you come to a wall that is diagonal in a North East/South West fashion, you are corrected into a South West movement even if you continue holding left (West) on the controller. This should work for both diagonals correcting in both directions. If you're a few pixels off from walking squarely into a door or hallway, you are corrected into walking through the hall or down the hallway, i.e. bumping into the corner causes you to be pushed into the hall/door.   I've hunted for efficient ways to achieve this and have had no luck. To be clear I'm talking about the human character's movement, not an NPC's movement. Are their resources available on this kind of movement? Equations or algorithms explained on a wiki or something?  I'm using the XNA Framework, is there anything in it to help with this? Cartesian product of two $\sigma$-algebras is generally not a $\sigma$-algebra? When reading the book Measures, Integrals and Martingales written by R.L. Schilling, I saw a statement as below: $$\mathcal{A} \times \mathcal{B}:= \{A \times B: A\in \mathcal{A}, B\in \mathcal{B}\}$$ where $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras, is not $\sigma$-algebra in general.  However, I cannot construct a counterexample. Could anyone offer help here?  Kind regards What do you feel when crossing the event horizon? Pressure variation in a sound wave Use physical harddisk in Virtual Box Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? Metapost label based on variable value I want to label a scale based on a variable value and I wrote a metapost code like this.  $ := 1; for i = -.3cm step .6cm until 3.6cm:     label.bot(str$,(i,-3.7cm));     $ := incr($); endfor;   It just labels "$" instead of its value. How to output the variables value? How can I get crash data (stack traces at least) from my Android application? At least when working on my own device being retrieved by cable, but ideally from any instance of my application running on the wild so that I can improve it and make it more solid. Why isn't it $E \approx 27.642 \times mc^2$? Why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides?   Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?  For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?  Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.  ... and so forth.    I'm not well educated on matters of physics and so I feel a bit sheepish asking this.   Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.    EDIT: I just wished to give a little more context as to where I was coming from with this question based on some of the responses:  @Jerry Schirmer comments:      You do have an ugly factor of $2.997458 \times 10^8 m/s$ in front of everything. You just hide the ugliness by calling this number c.   These are not the types of "ugly constants" I'm talking about in that this number is the speed of light. It is not just some constant needed to balance two side of an equation.  @Carl Witthoft answers:      It's all in how you define the units ...   Of course this is true, we could in theory hide all sorts of ugly constants by using different units on the right and the left. But as in the case of $E=mc^2$, I am talking about equations where the units on the left are consistent with the units on the right, irrespective of the units used. As I mentioned on a comment there:     $E=mc^2$  could be defined using units like $m$ in imperial stones ($\textsf{S}$), $c$ in cubits/fortnight ($\textsf{CF}^{−1}$) and $E$ in ... umm ... $\textsf{SC}^2\textsf{F}^{−2}$ ... so long as the units are in the same system, we still don't need a fudge factor.   When the units are consistent in this manner, there's no room for hiding fudge constants. Only charge capacitor when button pushed then turn on LED momentarily with capacitor when button released I am trying to make circuit to turn on an LED momentarily. I would like to charge the capacitor when the button is pushed, then when released turn on the LED momentarily with the capacitor. Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number?  I tried to use the definition: $$\frac{1}{R}=\limsup\left|\frac{a_{n+1}}{a_{n}}\right|$$ but I wasn't successful. As the title say, I want to prove that  $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$  i.e.  $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n)  \quad (\star)}$. In a book I found that in order to prove it, we need the following claim:  Claim: Let $\displaystyle{\zeta \in C^{\infty} ([0, \infty)) }$ such that $\zeta (t)  =1 $ for $ t \leq 1 $ and $\displaystyle{ \zeta (t) =0 }$ for $ t \geq 2$. For $R&gt;0$ and  $x \in \mathbb R^n$ define $\displaystyle{ \eta_R (x) = \zeta \left( \frac{|x|}{R} \right)}$. If $ \displaystyle{ u \in W^{k,p} ( \mathbb R^n) }$ with $1 \leq p &lt; \infty $ then, $\displaystyle{ \eta_R u \to u }$ in $W^{k,p}(\mathbb R^n) $ as $ R \to \infty$.  If I prove the claim then the $(\star)$ follows using mollifiers. But unfortuently I have stuck, with the proof of the claim.  Any ideas? What (if any) countries are either open to or have set definitive plans to allow international tourism? Let $G$ be a group and $\{H_i\}_{i\in I}$ a family of subgroups. I would like to find a condition that will imply that $\cup_{i\in I} H_i$ is a subgroup. I know that it's not true in general, I need help to find this condition.  Any help is welcome.  Thanks a lot.