I am readind about Heisenberg's Uncertainty Principle and although the main formula most texts use is  $$\Delta x\Delta p_{x}\geq \frac{\hbar}{2}$$  or equivalently  $$\Delta x\Delta p_{x}\geq \frac{h}{4\pi}$$  I have seen it written as  $$\Delta x\Delta p_{x}\geq \hbar$$  or  $$\Delta x\Delta p_{x}\geq h$$  First of all, are all those correct formulas? If so, then how are they equivalent with each other?
Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why?
In addition to whatever files I may choose to exclude, what files does the system exclude by rule or classification?    It might make sense not to backup Trash or Caches, but I'm looking for a way to tell what files are skipped "behind the scenes." Knowing if the system maintains an exclusion list or exclusion pattern would be very helpful in managing larger groups of macs. Windows update giving error 80070002 and manual update not working how to update? I have been trying to update my pc from 20h1 to 20h2 but the windows update shows error message 80007002 and when I manually install the update using windows 10 update assistant it downloads completely and then even starts install and gets stuck at 87 percent on the screen where it says do not turn off. What to do? This is the log file. Matching Profile found: FindRollbackFailure - 3A43C9B5-05B3-4F7C-A955-88F991BB5A48 SetupDiag version: 1.6.0.0 System Information:     Machine Name =      Manufacturer =      Model =      HostOSArchitecture = 1033     FirmwareType = UEFI     BiosReleaseDate = 20160513000000.000000+000     BiosVendor = A12     BiosVersion = A12     HostOSVersion = 10.0.19041     HostOSBuildString = 19041.1.amd64fre.vb_release.191206-1406     TargetOSBuildString = 10.0.19041.621 (vb_release_svc_prod1.201014-1758)     HostOSLanguageId =      HostOSEdition = CoreSingleLanguage     RegisteredAV = Quick Heal Internet Security     FilterDrivers = FileInfo     UpgradeStartTime = 12/20/2020 2:54:38 PM     UpgradeEndTime = 12/20/2020 5:30:40 PM     UpgradeElapsedTime = 02:36:02     CV = Pi3IdU3SlEqpFx93     ReportId =    Error: SetupDiag reports rollback failure found. Last Phase = Pre OOBE Boot Last Operation = OOBE boot apply Error = 0xC1900101-0x4000D LogEntry:   Refer to &quot;https://docs.microsoft.com/en-us/windows/desktop/Debug/system-error-codes&quot; for error information.  Last Setup Phase: Phase Name: Pre OOBE Boot     Phase Started: 12/20/2020 4:37:50 PM     Phase Ended: 1/1/0001 12:00:00 AM     Phase Time Delta: 00:00:00     Completed Successfully? False   Last Setup Operation: Operation Name: OOBE boot apply     Operation Started: 12/20/2020 4:37:50 PM     Operation Ended: 1/1/0001 12:00:00 AM     Operation Time Delta: 0:00:00:00.0000000     Completed Successfully? False Check which IPs or web-sites an app is communicating with? Usually they do it behind the scenes. So I'd love to know where apps are actually sending data and from whom they are actually getting responses. I added a user account to the admin group and discovered I added wrong user. So I now have to remove the account from the admin group.   How can I remove an account from a group without deleting the user? My installer build "signs" a DLL using a Code Signing certificate during the build process.   I've noticed that if I try to build twice in succession, the second build fails because the DLL is already signed so signcode chokes. Obviously I can fix this by signing a copy of the DLL in the build, but the problem intrigued me:   Is it possible to "unsign" a DLL, and if not, why not...? Problem with ListVectorPlot3D Real Time LaTeX? Possible Duplicate:          I'm looking for an application in which I can write pure LaTeX in one half of the screen and on the other half to see the compiled document in real time. This seems really like an obvious way to do LaTeX authoring but it doesn't look like any software is able to do it. LyX is not what I'm looking for. I want to edit the actual LaTeX and see it rendered in real time. Why isn't this already being done somewhere? How to mount an internal HDD on start-up? when is this series convergent. Let $(x,y)\in \mathbb{R^2}$ and consider the series   $$\lim_{n\rightarrow \infty}\sum_{\ell,\,k=0}^n \frac{k^2 x^k y^\ell}{\ell!}$$  Then for what values $(x,y)$ the series converges. I have absolutely no idea how to deal with it. Why are the characters on the keyboard all capitalized? Why are all the letters (A, B etc) and all the writings (Home, Page Up etc) on the keyboard, capitalized?    Especially, the letters, because a new user could rightly expect uppercase letters to be entered upon pressing those keys to find the contrary. Can a shell script set environment variables of the calling shell? I'm trying to write a shell script that, when run, will set some environment variables that will stay set in the caller's shell.  setenv FOO foo   in csh/tcsh, or  export FOO=foo   in sh/bash only set it during the script's execution.  I already know that   source myscript   will run the commands of the script rather than launching a new shell, and that can result in setting the "caller's" environment.  But here's the rub:  I want this script to be callable from either bash or csh.  In other words, I want users of either shell to be able to run my script and have their shell's environment changed.  So 'source' won't work for me, since a user running csh can't source a bash script, and a user running bash can't source a csh script.  Is there any reasonable solution that doesn't involve having to write and maintain TWO versions on the script? C Scanf suddenly stopped reading in values General solution of expected value of E(f(X))? This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable with some distribution $p$ and we need to evaluate $\mathbb{E}[f(X)]$. Here, $f(X)$ may or may not have the same distribution as $X$. In my case, if $f(X) = e^{sX}$, where $s&gt;0$. I would work out this problem by letting $Z = e^{sX}$ and   $\mathbb{E}[f(X)] = \mathbb{E}[Z] = \int_{\infty}^{\infty}z \hspace{0.2em}p(z)dz$.   Now I find the distribution of $Z$ as: $P\{ Z \leq z \} \implies P\{ e^{sX} \leq z \} \implies P\{ X \leq \frac{\log(z)}{s}\} = \int_{-\infty}^{\frac{\log(z)}{s}}x.p(x)dx $  thereby substituting this into the expectation one gets $\mathbb{E}[Z] = \int_{\infty}^{\infty}z\hspace{0.2em} \left(\int_{-\infty}^{\frac{\log(z)}{s}}x\hspace{0.2em}p(x)dx\right)^{\prime} dz$  According to the Law of Unconscious Statistician , this approximates distribution of $Z$ as $p$. I understand the backward proof through calculus, I don't follow the measure theory approach. For example I don't follow how this works if say $X \sim Gamma$. Is there a source I can read through for someone with no measure theory background? How does electricity propagate in a conductor? Is it only red, green and blue that can make up any color through additive mixture? I'm reading about color vision and have some trouble understanding the motivation for why the  was suggested in the first place. The book I'm reading ("Psychgology: The science of mind and behavior") states:     At the beginning of the nineteenth century it was discovered that any colour in the visible spectrum could be produced by some combination of the wavelenghts that corresponds to the colours blue, green and red in what is known as additive color mixture.   From the explanation in the book, it seems like this somehow should be a reason for also supposing that the human retina was composed of cones sensitive to the colors green, red and blue respectively. I guess this would be a valid argument if it was only red, green and blue that, through additive mixture, could make up any color.  Is it only red, green and blue that, through additive mixture, can make up any color? I am trying to install Ubuntu 12.10 for quite some time, and passing hurdles one by one. Now I am in a situation as follows.  I have got a PC and 10 GB HDD which will be totally dedicated to Ubuntu so no option of Wubi and dual boot.  I was trying to install from DVD, but it is getting stuck at "Out of frequency" error. So I had to adapt for USB boot option. But my PC is USB non bootable, so workaround is "Plop Boot Manager". So I am doing the installation procedure as follows:   starting  from a CD drive which is having plop installed. opting for for USB boot in plop options. booting begins from USB. monitor eventually gives "out of frequency" error press Shift+Alt+F1 to get the terminal. open the grub with sudo nano /etc/default/grub. do necessary changes. sudo update-grub.   Now here I am getting error as follows:   /usr/sbin/grub-probe:error:failed to get canonical path of /cow.     My system is   P4 3.06 GHz, 1 GB ram , 10 GB HDD without an OS, monitor CRT lg StudioWorks (7 years old). Mobo Mercury P4 266a NDMx (865 equivalent). The whole system is perfectly in working condition under XP, but it is USB non bootable, and all other devices working perfectly.  What should I do next? windows command prompt change directory is now changing directory Were there any improvements to the second Death Star (other than the change in exhaust port) that we know of? How to add user to a group from Mac OS X command line? trying to add a user to a group from the command line but can't figure out how?   Specifically this is on Mac OS X Server 10.5.8  like to add user to 'admin' and 'wheel' groups Where do the Python unit tests go? If you're writing a library, or an app, where do the unit test files go?    It's nice to separate the test files from the main app code, but it's awkward to put them into a "tests" subdirectory inside of the app root directory, because it makes it harder to import the modules that you'll be testing.    Is there a best practice here?