I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}f(k)}{\sum_{\mathbf{k}}f(k)}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}f(k)}{\sum_{\mathbf{k}}f(k)}\right)^2}\;.$$    Where $f(k)$ is the Fermi-Dirac distribution given by $f(k) = \frac{1}{1+e^{\frac{k^n}{k_B T}}}$ and $k_B$ is the Boltzmann constant and $T$ is the temperature.  Now, I want to do this for $d$-dimensions, so I convert $L$ sum to integral and it results in the expression  $$ L = \frac{\dfrac{\int_{0}^{\infty}k^{n}\left(\frac{1}{1+e^{\frac{k^n}{T}}}\right) \mathrm d^d k}{\int_{0}^{\infty}\left(\frac{1}{1+e^{\frac{k^n}{T}}}\right) \mathrm d^d k}}{\left(\dfrac{\int_{0}^{\infty}k^{n-1}\left(\frac{1}{1+e^{\frac{k^n}{T}}}\right)\mathrm d^d k}{\int_{0}^{\infty}\left(\frac{1}{1+e^{\frac{k^n}{T}}}\right) \mathrm d^d k}\right)^2} $$ where $ \mathrm d^dk  = \dfrac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1} \mathrm dk $ is the volume element in $ d $-dimensional space.  I want to get a simple expression for $ L $ and how it depends on $ T, n $ and $ d $. Any physical/geometrical insight into this integral could also be helpful. What can I do in Mathematica to understand these terms? I specially want to understand what happens at very low and very high $T$.
I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful.
How do I sudo a command in a script without being asked for a password? I want to turn my system on automatically every day. So I use the below code in my Python script, but sudo asks me for a password every time:  os.system('sudo sh -c "echo date \'+%s\' -d \'+ \        24 hours\' &gt; /sys/class/rtc/rtc0/wakealarm"')   How can I run this script without sudo asking for the password every time? Does it make a difference if your URL path ends in a trailing slash or not? Does it make any difference to SEO ratings what-so-ever if your url has a trailing slash:  http://www.example.com/some/slug/paths/   as opposed to one that does not:  http://www.example.com/some/slug/paths Question regarding the validity of the big bounce I have several questions regarding the "big bounce" theory. It appears to be popular among LQG researchers. My questions are as as follows.  1) How one reconciles it with the fact that it is now experimentally established that the universe is expanding in an accelerating manner?  2) The entropy of the universe should have been very very low (maybe zero) at the time of the big bang/ big bounce. What was the entropy "before" that event? Was it even lower? If it was lower then doesn't that mean entropy decreases indefinitely? On the other hand if we accept the entropy was higher then doesn't that violate the second law of thermodynamics?  3) What observational evidence can confirm pre-bounce physics? According to GR there are some points near the bang/bounce where one simply can't easily define time in a meaningful way. Even for a quantum gravity theory, is it possible for any theory to extrapolate itself "before" the "bounce"? If there are no observational consequences after the bounce shouldn't one apply Occam's razor? What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$?  The level of powers is $2020$.  First there is no parenthesis so it means 3 power of 3 which is also power 3 and so on 2020 times  Second I think that we can use Fermat's Little Theorem but I don't know how, and maybe there is a better way. Where can I ask about Software Licenses/Licensing? Is it appropriate to ask questions about software licenses/licensing on ? Or is the appropriate place  ? Or is there no appropriate site currently and we should suggest such a site on ? Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Many anime do not follow the manga history and are made shorter. However there are a few that do follow it or try to finish the history with the manga like the big three (Naruto, Bleach, One Piece), Death Note, and FMA Brotherhood. But others finish the anime without the real ending like FMA, Ouran High School, Pandora Hearts, Tsubasa Reservoir Chronicle. Why is this so? Is it because of the high cost? Or the lack of popularity? While drawing a simple 3D (fake 3D in fact) cube, I realized that the corners end up with unwanted extensions of the line. I understand this is related to the width of the lines themselves, but how can I remove them? An alternative solution of drawing the desired figure is also welcomed.  Example:  \documentclass{article} \usepackage{tikz}   \begin{document} \begin{tikzpicture}  \draw[draw=black,fill=gray!20] (0,0) --(4.8,0)-- (4.8,5)--(0, 5)--cycle; \draw[draw=black,fill=gray!20] (4.8,5) -- (4.9,4.8)--(4.9,-0.1)--(4.8,0)--cycle; \draw[draw=black,fill=gray!20] (0,0) -- (0.1,-0.1)--(4.9,-0.1)--(4.8,0)--cycle; \draw[dashed,draw=black] (4.8,5) -- (4.9,4.8)--(0.1,4.8)--(0.,5)--cycle; \draw[dashed,draw=black] (0.1,-0.1)--(0.1,4.8)--(0.,5)--(0,0)--cycle; \end{tikzpicture} \end{document} Noncoalescing and bouncing droplets if you set a dish filled with water onto a loudspeaker or other vibrator and make vibration, When it oscillates, it is possible to hold small droplets on its surface for a long time. I mean, when you put a droplet of water on vibrating water surface, it doesn't coalesce with the water and bounce on the surface of water.( see this video  )  As we know, drop reaches its minimal potential when coalescing with water ( that's why drops coalesce)  what exactly cause this? why doesn't the drop coalesce with the water? Will a PCI Express x16 3.0 card work in a PCI Express x16 slot? I have a motherboard with a PCIexpress x16 slot.  I want to purchase a card called AMD Radeon HD 7770, which uses PCIe version 3.0.  Is that backwards compatible? What does the [Flags] Enum Attribute mean in C#? From time to time I see an enum like the following:  [Flags] public enum Options  {     None    = 0,     Option1 = 1,     Option2 = 2,     Option3 = 4,     Option4 = 8 }   I don't understand what exactly the [Flags] attribute does.  Anyone have a good explanation or example they could post? Use character string as function argument Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure Show that the image of the curve $\Gamma\in\mathscr{C}^{1}\left([a,b]\to\mathbb{R}^{d}\right)$ has d-dimensional Lebesgue measure zero (of course, $d\geq2$).  This can be proved using the absolute continuity of $\Gamma'$ (since $[a,b]$ is compact and $\Gamma'$ is assumed continuous, hence in $L^{1}([a,b])$) together with the fundamental theorem of calculus to obtain an $\epsilon$-small cover of $\Gamma$ by balls.  But I am trying to prove this using more elementary means (i.e. without integration).  Intuitively, since $\Gamma$ is smooth, we ought to (for fine enough partitions) be able to cover $\Gamma$ by boxes which arise from its tangent line.  And by taking the partition of $[a,b]$ to be finer and finer, the "tangent box" cover ought to also get smaller and smaller.  More rigorously, the vector-version of the mean value theorem can be applied: $$|\Gamma(t_{i-1})-\Gamma(t_{i})|\leq(t_{i}-t_{i-1})|\Gamma'(t_{i}^{\star})|\leq M_{i}\Delta t$$ where $t_{i}^{\star}\in(t_{i-1},t_{i})$ and $M_{i}=\sup_{t\in[t_{i-1},t_{i}]}|\Gamma'(t)|$ which exists and is finite since $\Gamma'$ is continuous.  But to me, it's not quite clear how to rigorously construct a cover by boxes from here.  NOTE:  In the proof I mentioned (using integration), essentially you sum the left hand side over all partition intervals of uniform length $\delta$ (which depends on the $\Gamma'$), and "integrate" the right hand side.  Actually, to be more specific, for each $\epsilon&gt;0$ there exists a $\delta&gt;0$ such that $||P||&lt;\delta$ implies $\int_{t_{i-1}}^{t_{i}}|\Gamma'(t)|dt&lt;\epsilon$ (e.g. absolute continuity).  This allows you to define numbers $\epsilon_{i}=\sup_{t,\bar{t}\in[t_{i-1},t_{i}]}|\Gamma(t)-\Gamma(\bar{t})|\leq\epsilon$ so that $\sum_{i=1}^{\#P}\epsilon_{i}\leq||\Gamma'||_{L^{1}([a,b])}$.  Then you can use these $\epsilon_{i}$ to put balls at each point $\Gamma(t_{i})$ of radius (say) $2\epsilon_{i}$, thus giving you an $\epsilon$-small cover.  Again though, this is harder to establish without integration theory. What would you call someone who imposes on other people's generosity? What would you call someone who isn't afraid to ask for money or any kind of favor or who misinterprets someone's generosity for a consistent resource for what they need? Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions. just wondering if anyone has ever tried embedding and actually integrating any js engine into the .net environment. I could find and actually use (after a LOT of pain and effort, since it's pretty outdated and not quite finished) spidermonkey-dotnet project. Anyone with experience in this area? Engines like SquirrelFish, V8..   Not that I'm not satisfied with Mozilla's Spidermonkey (using it for Rails-like miniframework for custom components inside the core ASP.NET application), but I'd still love to explore a bit further with the options. The command-line solutions are not what I'd need, I cannot rely on anything else than CLR, I need to call methods from/to JavaScript/C# objects.  // c# class public class A {     public string Hello(string msg)     {         return msg + " whatewer";     } }  // js snippet var a = new A(); console.log(a.Hello('Call me')); // i have a console.log implemented, don't worry, it's not a client-side code :)   Just to clarify - I'm not trying to actually program the application itself in server-side javascript. It's used solely for writing custom user subapplications (can be seen as some sort of DSL). It's much easier (and safer) to allow normal people programming in js than C#. Where, precisely, does feedback on Community ♦ generated flags go to die? For the most part, I treat "Community" like any other user when handling system raised flags. If they are useful, I mark them as such. If they are noise, I dismiss them as such. The Community user has a static flag weight, so our reaction to them is more or less feedback for developers.  Is this feedback from moderators network wide used in any way to help improve the accuracy of system raised flags? Or, would clicking either button suffice?  I'm asking because our reasoning behind declining a flag is now more precise. Does dismissing some as noise help improve the criteria that ultimately generates the flag?  Edit  Yes, thank you Skype, I realize that  asked a similar question . Error Code: 1215. Cannot add foreign key constraint (foreign keys) Suppose the proper length of a train is longer than that of a bridge, and the bridge can't bear the total weight of the train but can bear it partially. As the train goes very fast, it becomes shorter than the bridge and it can be entirely on the bridge. As a result, the bridge will collapse. However, in the train's frame, the train is even longer than the bridge, and it is only partially on the bridge; so the bridge will not collapse. What is the solution to this paradox?