Entropy of an ideal gas is defined as the logarithm of the number of possible states the gas can have multiplied by Boltzmann's constant:  $${\displaystyle S=k_{\mathrm {B} }\log \Omega .}$$  In deriving the Maxwell-Boltzmann distribution, we initially start by counting a finite number of states, so this definition of entropy makes sense. But in the end we say that the number of possible states is so high that we can acctually say the distribution is continuous. But if the distribution is continuous, the number of possible states is infinite. So why is entropy not always infinite when a continuous distribution is used?
How could it be that the entropy of gas is not infinite?  If we use the formula $S=k\ln(\Omega)$ we get infinity because there are an infinite number of possible situations (because there are infinite possibilities of position and momentum for each molecule).
Finite Index of Subgroup of Subgroup Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.  So this is basically a bijective proof? The number of cosets of $H$ in $G$ equals the number of cosets of $K$ in $G$ times the number of cosets of $H$ in $K$ by the multiplication principle?   Reference: Fraleigh p. 103  Question 10.35 in A First Course in Abstract Algebra How to add multiple relative hyperlinks to single point in ArcGIS for Desktop with Basic level license? Can Superman see through his eyelids? Google AJAX API - How do I get more than 4 Results? Evaluation of OptionValue[] Is there a more accurate way to create a Javascript timer than setTimeout? How do I run .sh scripts? I'm looking to build a guitar pedal switcher based on a Raspberry Pi or Arduino (have not yet decided, but I have an unused Raspberry around and I lean towards that). Anyway, digital logic.  Now, I need to switch audio signal obviously and I need to do it quietly (both physical silence and electrical silence). Please note that the signal is line level throughout most of the build, except for the input which is much lower. I will probably have 2 or 3 loops before the preamp so those may not encounter line level signal.  From what I read and know, so far, mechanical relays are not really an option due to the clicking noise they make. I also read something about FETs, but that's a bit out of my league and I would appreciate any info on these.  PS: I will probably use different power rails for the digital and analog paths.  PS2: The whole thing will be mounted in a 1U rack enclosure, so please tell me if I should take into consideration any thermal issues. Does $E = mc^2$ apply to photons? Photons are massless, but if $m = 0$ and , then $E = 0c^2 = 0$. This would say that photons have no energy, which is not true.  However, given the formula $E = ℎf$, a photon does have energy as $ℎ$ is a non-zero constant and photons with a zero frequency do not exist.  What am I missing here, or does the famous formula not apply to photons? Getting "Authentication is required" when trying to access the Play Store Over the past week, whenever I try  to open the Play Store/Market app on my phone, the app thinks for a while and then displays this error message:     Authentication is required. You need to sign into your Google Account.   But I am indeed signed into my Google account, as seen in other Google applications running on the phone, and as seen if I open the menu in the Market app and click the "Accounts" button.  I suspect this might be related to the fact that over the past few months I've had two different Google accounts synced to this device, and I have recently removed one of them from the phone. No other Google application has had any problems with removing one account.  Does anyone know how I can solve this issue? I can still install applications by using the market website (now "Google Play"), but this is still a big inconvenience. I also think I stopped getting updates. Using Regex to generate Strings rather than match them I want to reinstall Ubuntu from scratch.   I have read ( and ) that it is recommended to have 3 partitions for the desktop installation: root, home and swap as having a separate home and root can simplify the process if you want to upgrade or reinstall Ubuntu (if you break something by mistake).  But there is no type of formula (that I can find) on how big each partition should be. I know this is somewhat dependant on what you plan to install. I want to have flexibility down the line to install new software without having to worry about shifting the partition sizes around.  Does anyone know of a formula or percentage breakdown that they have used to successfully calculate partition sizes?  i.e. x% of space will be allocated to the swap, y% of space to the root partition and z% of space to the home partition The PS4 is now released in the EU, but due to shipment shortages is not available for sale unless pre-ordered a long time ago. Luckily, it seems as though Dualshock 4 controllers are availble for sale in a number of shops already.  I've long decided to buy the PS4 as well as a second Dualshock 4 controller, so I've been considering buying the "second" controller now in order to replace my old Xbox 360 controller, whose D-Pad is useless and whose RB button no longer responds correctly. But before I buy the Dualshock 4 early, I wanted to know how well my PC will support it.  I already know that , but what exactly does work, and what doesn't?  Can I expect the same key-mapping on the PC as on a PS4?  Basically, can I replace my Xbox 360 controller with a Dualshock 4 and still play games like Batman Arkham Asylum/City/Origins as if I were using the old Xbox 360 controller? Entity Query field condition for Webform submissions Understanding how references and labels work How can I run blender from command line or a python script without opening a GUI? I'm interested in using Blender for producing images, 3d files, or calculating geometry on a server.  Can I make blender run a python script without opening a GUI?  If not, can I incorporate Blender's python API into my own python script without running a GUI? Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ Find the maximum value of $f(x)$ when $0 \leq x \leq 1$ and $$ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$$   I'm guessing that the Fundamental Theorem of Calculus can't be used because the integrand is a function of $x$.  Any help would be greatly appreciated. Are there any reputable studies on spanking? SQL Server Dependencies Is there an easy way to chase down table/stored procedure/function dependencies in SQL Server 2005+?  I've inherited a giant application with lots of tables and even more stored procedures and functions that are long and interlinked.    At the end of the day is there a way to build a dependency tree?  Ideally what I'm looking for goes in both directions:  For a table/procedure - what depends ON it?:  Show me all the stored procedures that eventually reference it (ideally in a tree view such that sub procedures nest out to the bigger procedures that call them)  For a procedure - what does IT depend on?:  Show me all the procedures and tables that a given procedure will (or could) touch when running.  It seems this tool shouldn't be that hard to make and would be incredibly useful for DB maintenance generally.  Is anyone aware of such a thing?  If this doesn't exist, why the heck not?  The built-in functionality in Management Studio is nice but the information does not appear to be complete at all. Possible Duplicate:          The FAQ specifies that:     Use your downvotes whenever you encounter an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect.   But if I find a correct question, closed (or not yet) as duplicate, is downvote a correct behavior for Stack&nbsp;Overflow policy?