(Geodesic frame). Let $M$ be a Riemannian manifold of dimension $n$ and let $p \in M$. Show that there exists a neighborhood  $U \subset M$ of $p$ and $n$ vector fields $E_1,...,E_n \in X(U)$, orthonormal at each point of $U$, such that, at $p$, $\nabla_{E_i}E_j(p) = 0$.  Such a family $E_i, i = 1,..., n,$ of vector fields is called a (local)  geodesic frame at p.   Question: I don't know how I can to prove the step: $\nabla_{E_i}E_j(p)=0$.   I am using $\exp(te_i)$ as a geodesic for this.
I'm having the same problem as it was questioned   I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n a_{il}(0)\dfrac{\partial b_jk}{\partial x_l}(0)\partial x_k(p) $$ which I can't show that equals zero.  On the linked question, there is also the indication that $\nabla_{E_i}E_j (E_k)=\Gamma^k_{ij}$. This could lead that $\nabla_{E_i}E_j (E_k)(p)=\Gamma^k_{ij}(p)=0$, but $\nabla_{E_i}E_j (E_k)(p)$ is not equal to $\nabla_{E_i}E_j (p)$ which is the one I need.  Can someone help me?
Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S This is a very easy fact we use in Group Theory,  But somehow, I wondered that whether there may be another way (other than Lagrange's Theorem) to prove that the order of an element divides the order of Group.  I attempted to go on the term "exponent" of the Group (just assume that G is a Finite Group), which we may define the least common multiple of orders of elements in G. But this exponent divides $|G|$, since the order of each element divides $|G|$. So it became a little paradox.   Shall I think that; before teaching this fact in a Course, one should teach about Lagrange's first? How can I create launchers on my desktop? In older versions it was easy to create a launcher on my desktop. All I had to do is right click on my desktop and select the "create launcher" option.  How can I create such launchers now? Using "on" vs. "in"? Consider the following sentence:     I am developing an application to be installed on Android.   And this:     This has been a major flaw in Android...   To be clear I am unsure of the usage of "in" vs. "on". When should I use "in" and when to use "on"? Local file access with JavaScript Where is it preferred to place the answer to duplicate questions? Saving Photoshop multilayer file to PDF without layers size difference Why is it that when I save a file that has many, many layers to a PDF without layers as a copy that the file size is larger than flattening the layers and then saving the file as a copy to PDF? 19Mb vs. 5Mb?? Wouldn't saving a file without layers be the same as flattening the layers and then saving the file?? Can anyone explain this? How do I make Windows 10's File Explorer open "This PC" by default? I know how to do that on Windows&nbsp;Vista and Windows&nbsp;7, and I know how to do that on Windows&nbsp;8 and Windows&nbsp;8.1, but neither of the ways work with Windows&nbsp;10; it always defaults into the "Home" view.  (I am basically reposting  for Windows&nbsp;10.) "You've earned (badge)" panel layouts are broken When I visit  I see this:     The "Congratulations! You've earned (badge)" section has dropped down completely beneath the tabs under it. The "You've earned a new privilege!" section has a gap at the bottom:     Shog9 says in comments how to reproduce this:     Gotta have earned a badge, not dismissed the notification, and be looking at your own profile. If I attempt to change my password to nothing by opening "User Accounts", the "Change" button remains greyed out:    How do I change my password to be empty? I know you can set Ubuntu to automatically log you in, but I want my password to be empty, I never want to type in a password to authenticate myself as that user.  I know that there are reasons why this might not be a good idea, but I want to know if it is even possible. I'm using Ubuntu 12.10. overset text over equal sign in in an Align environment Look at the following example  \begin{align}      &amp; x \\      \overset{\text{some long explanation}}= &amp; foo \\      &amp; = bar \end{align}   what is the best way to get the equal signs aligned? Now I have seen different solution such as adding \overset{\hphantom{\ţext{some long explanation}}}= to the beginning of every line, but I don't really think that is the best solution. Any better ideas?    UPDATE  I guess what i am actually after is   aligning the first column of the align environment right aligning the equal sign under the overset text right How do I show that all continuous periodic functions are bounded and uniform continuous? A function $f:\mathbb{R}\to \mathbb{R}$ is periodic if there exits $p&gt;0$ such that $f(x+P)=f(x)$ for all $x\in \mathbb{R}$. 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I've tried  (although I know it's for 11.10 and mentions that even 11.10 is an unsupported OS for Matlab R2011a) but without any satisfactory solution.  This is my launcher file, /usr/share/applications/matlab.desktop:  #!/usr/bin/env xdg-open [Desktop Entry] Type=Application Icon=/usr/share/icons/matlab.png Name=MATLAB R2011a Comment=Start MATLAB - The Language of Technical Computing Exec=matlab -desktop Categories=Development;   I open the dash panel and search for "matlab". This launcher is found among applications. I click it, and Matlab's splash screen shows up, but when it disappears the program doesn't start. (I've verified with htop that no matlab-processes are running in the background either.)  If I add  Terminal=true   to the launcher file, the program starts OK, and opens a terminal as well as Matlab. However, both the terminal and Matlab itself show up in the Launcher area, with the Matlab icon, so it looks like I have two Matlab instances running when really it's only one. (Actually, they show up as two different programs, and not just two instances of the same - the icons are independent, not grouped together.)  This is definitely not optimal. I had hoped to create a launcher I can lock to the launcher area, and then that same icon will be the icon for the active Matlab instance when the program is running.  How do I create a launcher for Matlab that works as expected?  Update: I was apparently a bit unclear on my symptoms, I'll try to clarify a little. I've also tried some suggestions from the answers, and further investigated what happens. My current setup (a launcher file with Terminal=true and Exec=matlab -desktop -nosplash &amp;) renders the following behavior:   I open Dash by pressing the Windows key on my laptop, and search for "matlab". It finds the launcher named "MATLAB R2011a". I click it. A terminal window opens, using the icon I referred to in the launcher file. Almost immediately, MATLAB's splash screen also opens, using the same icon (and thus being grouped with the terminal window in the launcher). The splash screen disappears and, so does one of the icons in the launcher. The MATLAB desktop environment opens, using a different version of the icon which is displayed next to the icon for the terminal window (not grouped with it).   I can lock the terminal window's icon to the launcher and successfully start MATLAB by clicking it, but it doesn't feel optimal that I start the program with one icon, and switch to it with another. I've also tried the following:   Exec without the ampersand &amp; in the launcher command, but it didn't make a difference. Executing matlab -nosplash manually from a terminal still shows the splash screen. (What, then, does the nosplash option really do?) Since a few months, on one of the desktop Ubuntu 14.04 systems I use, CPU has regularly been going up to 80-100% for no apparent reason. htop would usually attribute responsability to file system processes, such as nemo or ntfs. To get CPU back down I would kill these process, which in some cases would lead to loss of access to the file system and a consequent restart.  Days ago I noticed that whenever this situation happened another process was always present also using high CPU: mediascanner-service-2.0. Some research lead me to  that simply advised its removal. So I did.  The end result has been unexpectedly positive: not only are the high CPU load episodes gone, the system is altogether much faster. The speed up of the Lens is dramatic: finding an application like Calc now takes 2 seconds at most when previously it took some 20 seconds. Finding a file takes now some 5 seconds, whereas before it took in the order of 30 seconds. Heavy programmes like Eclipse are starting up faster and workspaces are more fluid. There seems to be also an overall reduction in the frequency of disk access.  What is exactly this mediascanner2.0 package? Is it part of Unity 7 or a dependency that gets installed a posteriori? Is it possible to prevent its installation?  Update: following  it is possible to track this package down to indicator-session:  $ sudo aptitude why mediascanner2.0 i   indicator-session         Recommends indicator-applet (&gt;= 0.2) | indicator-renderer i A unity8                    Provides   indicator-renderer                             i A unity8                    Recommends unity-scope-mediascanner2                      pi  unity-scope-mediascanner2 Depends    mediascanner2.0                                  Apparently, mediascanner2.0 (and the Unity 8 shell) are part of the base 14.04 system. Left to answer is the exact function of this rogue package. $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}$ using residues I'm trying to find $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx$ where $\alpha&gt;0$ is real.  My approach was to take an integral along the real line from $1/R$ to $R$, around the circle counterclockwise to $-R$, along the real line to $-1/R$, and then around the circle clockwise to $1/R$.  I have encountered 2 problems with this:   This path encloses one pole, at $z=\alpha i$.  I found the residue at $z=\alpha i$ to be $\frac{\ln(\alpha)+i\pi/2}{2\alpha i}$.  However, this gives me that $\int_0^\infty \frac{\log(x)}{x^2+\alpha^2}dx=\frac{\pi(\ln(\alpha)+i\pi/2)}{2\alpha}$.  Since I have a real function integrated over the real line, there cannot be an imaginary part.  Where did I go wrong?  (Also, doing a few examples, the correct answer seems to be $\frac{\pi\ln(\alpha)}{2\alpha}$, the same as I have but without the imaginary part.) At first chose my path so instead of going all the way around the upper semicircle, it only went 3/4 of the way around, as I wanted to avoid anything that might go wrong with the discontinuity of $\log(x)$ at the negative real axis.  When I do this, though, I get a different answer than before (the denominator of the fraction is $\alpha(1-e^{3\pi i/4})$ instead of $2\alpha$.  What am I doing wrong that gives me different answers? Why are all known distributions unimodal? How does the CS:GO ranking system work? In Ruby on Rails, how do I format a date with the "th" suffix, as in, "Sun Oct 5th"? I want to display dates in the format: short day of week, short month, day of month without leading zero but including "th", "st", "nd", or "rd" suffix.  For example, the day this question was asked would display "Thu Oct 2nd".  I'm using Ruby 1.8.7, and  just doesn't seem to do this.  I'd prefer a standard library if one exists. 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