I have to prove that every homogeneous manifold is complete.  Let $p,q$ points on a Riemannian homogeneous manifold. I want to prove that there exist a minimizing geodesic join $p,q$ or equivalently, that every geodesic can be extended forever, that is the same that the $\exp_p$ is defined to all $T_pM$ for at least one point $p$.  My idea is the following: We know that there is an open ball in $T_pM$ with some radius $\epsilon &gt; 0$ such that $\exp_p$ is a diffeo in some open set on $M$. In particular, the image of the diffeo has the property that every point can be join to $p$ by a minimizing geodesic with length less than $\epsilon.$   Once $M$ is homogeneous, then every point admits a ball with the same radius and same property. This can seen noting that given two arbitrary point there is an isometry that "connects" both. Then I can transfer the geodesics from a point to another.  Ok, then:  For every point I can produce a ball with radius $\epsilon$ with the property that for every point on the ball there exist a minimizing geodesic join two points.  How can I conclude then that I can extend every geodesic? I know that for every point I can produce geodesics with the same length, how to finish?
I came across this proof while looking for hints on my homework, and I think it's only gotten me more confused. This is from .  Lemma 5.4 If $(H,h)$ is a homogeneous Riemannian manifold, then $(H,h)$ is complete. Proof. By the Hopf-Rinow Theorem, it suffices to show that $(H,h)$ is geodesically complete. Thus suppose that $c:[a,1) \rightarrow H$ is a unit speed geodesic which is not extendible to $t=1$. Choosing any $p \in H$, we may find a constant $\alpha &gt; 0$ such that any unit speed geodesic starting at $p$ has length $\ell \geq \alpha$. Set $\delta = \min\{\alpha/2, (1-a)/2\} &gt; 0$. Since isometries preserve geodesics, it follows from the homogeneity of $(H,h)$ that any unit speed geodesic starting at $c(1-\delta)$ may be extended to a geodesic of length $\ell \geq 2 \delta$. In particular, $c$ may be extended to a geodesic $c:[a, 1+\delta) \rightarrow H$, in contradiction to the inextendibility of $c$ to $t=1$.  How exactly does it &quot;follow from homogeneity&quot; that we are able to extend unit speed geodesics? Isn't that what we're trying to show in the lemma? And how does considering geodesics based at $c(1-\delta)$ lead us to extending $c$ to $c:[a,1+\delta) \rightarrow H$?
Time complexity of nested for-loop Possible Duplicate:          Well, the question is in the title. I always had the impression that "talk to someone" refers to situations when some information must be conveyed to someone else, while "talk with someone" refers to cases that involve a more complex two-sided interaction. E.g., I "talk to my colleague" to tell him that I'm going for a lunch break (no response is required apart from maybe an acknowledgment), while I go and "talk with my colleague" if I have to discuss something with him in detail. Is this correct, or is there any difference between the two phrases, or is there no difference at all? I have two user profile fields "Last Name" and "First Name". I also have a user list views. I want to expose "Name" as filter which can search for both Last Name and First Name. How can I do single filter of combination of those two fields? Can I create it from views UI?  One possible way is to create another profile field "Name" which will be hidden in the form. On user save, I would combine the two field values into the "Name" field and then expose it as filter in the views. But this solution is hard coding and it needs to write hooks. On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ Ok, so PHP isn't the best language to be dealing with arbitrarily large integers in, considering that it only natively supports 32-bit signed integers. What I'm trying to do though is create a class that could represent an arbitrarily large binary number and be able to perform simple arithmetic operations on two of them (add/subtract/multiply/divide).  My target is dealing with 128-bit integers.  There's a couple of approaches I'm looking at, and problems I see with them. Any input or commentary on what you would choose and how you might go about it would be greatly appreciated.  Approach #1: Create a 128-bit integer class that stores its integer internally as four 32-bit integers. The only problem with this approach is that I'm not sure how to go about handling overflow/underflow issues when manipulating individual chunks of the two operands.  Approach #2: Use the bcmath extension, as this looks like something it was designed to tackle. My only worry in taking this approach is the scale setting of the bcmath extension, because there can't be any rounding errors in my 128-bit integers; they must be precise. I'm also worried about being able to eventually convert the result of the bcmath functions into a binary string (which I'll later need to shove into some mcrypt encryption functions).  Approach #3: Store the numbers as binary strings (probably LSB first). Theoretically I should be able to store integers of any arbitrary size this way. All I would have to do is write the four basic arithmetic functions to perform add/sub/mult/div on two binary strings and produce a binary string result. This is exactly the format I need to hand over to mcrypt as well, so that's an added plus. This is the approach I think has the most promise at the moment, but the one sticking point I've got is that PHP doesn't offer me any way to manipulate the individual bits (that I know of). I believe I'd have to break it up into byte-sized chunks (no pun intended), at which point my questions about handling overflow/underflow from Approach #1 apply. Pipe to multiple files in the shell I have an application which will produce a large amount of data which I do not wish to store onto the disk. The application mostly outputs data which I do not wish to use, but a set of useful information that must be split into separate files. For example, given the following output:  JUNK JUNK JUNK JUNK A 1 JUNK B 5 C 1 JUNK   I could run the application three times like so:  ./app | grep A &gt; A.out ./app | grep B &gt; B.out ./app | grep C &gt; C.out   This would get me what I want, but it would take too long. I also don't want to dump all the outputs to a single file and parse through that.  Is there any way to combine the three operations shown above in such a way that I only need to run the application once and still get three separate output files? I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. How to get list of all sobject records modified today in the system? I need a list of all sobject records that got modified today. this is not just for one object but for all objects in the system. Is there a simpler way to do this? Bitlocker won't initialize on a 4TB drive with 4K sectors When I created my account, I used Facebook Connect as my only login type and I can only log in to my account using it.  I wish to create a Stack Exchange login to be able to log in with instead of Facebook Connect, but I can't find how to do this. It seems that the option for creating a Stack Exchange login is only present when creating a new account? Why is spawning threads in Java EE container discouraged? One of the first things I've learned about Java EE development is that I shouldn't spawn my own threads inside a Java EE container. But when I come to think about it, I don't know the reason.  Can you clearly explain why it is discouraged?   I am sure most enterprise applications need some kind of asynchronous jobs like mail daemons, idle sessions, cleanup jobs etc.  So, if indeed one shouldn't spawn threads, what is the correct way to do it when needed? Connection pooling in PHP What is the difference between "information on", "information of", "information about" "Information on something", "information of something", "information about something" — these three usages all sound having the same meaning for me. Are there any differences? How to tell the difference between linear and non-linear regression models? I was reading the following link on non linear regression . My understanding from reading the first section "Nonlinear Regression vs. Linear Regression" was that the equation below is actually a linear regression, is that correct? If so why?  $$y = b_1x^3 + b_2x^2 + b_3x + c$$  Am I also to understand that in non linear regression multicollinearity isn't an issue? I know that multicollinearity can be an issue in linear regression so surely if the model above is in fact a linear regression there would be multicollinearity? When discussing a treaty or international agreement, which is correct?  "There are 100 states party to the treaty." (for example, as used )  Or:  "There are 100 state parties to the treaty." (for example, as used )   Or: "There are 100 states parties to the treaty."  (for example, as used ) Can't update drupal core to 7.27 What's the best way to use matching methods with multiple treatment groups? I'm assessing the impact of an intervention on an outcome. For my first analysis, I used the MatchIt package (see code below) to match treatment and control groups and to compare outcomes for the treated vs control. The Treat variable is 0 or 1.  nn_match &lt;- matchit(treat ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df)   In my second analysis, I'd like to assess the effect of different treatment levels (for example, I have Treatment A, Treatment B, and Treatment C) on the outcome, but I'm stuck on the best way to proceed. Does it make sense to run separate matching analysis for each treatment group while excluding the observations from the other treatments groups?   nn_treatA_match &lt;- matchit(treat_A ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df_minus_other_treatBC) nn_treatB_match &lt;- matchit(treat_B ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df_minus_other_treatAC)   Or is there a way to run the analysis with the entire dataset and match controls to each treatment at the same time? Other posts have mentioned the twang package, but I'm not sure if that's what I should go with as I'd like to compare outcomes for each treated group to the control and not just across treated groups. How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I'm trying to do an incremental backup using cpio but it seems that the busybox version doesn't work the same as the normal one? This is what I've got so far:  find data/Backup -mtime -2 | cpio -cm media/USB_FLASH_2  I origianlly used:  find . -mtime -2 | cpio -cm /media/USB_FLASH_2  However, I've seen people say busybox doesn't like definitive paths while using cpio. Unsure about that though.   It returns an error that basically says I'm using cpio wrong. I can't find anyone using the busybox cpio to do things like this, does that mean i can't do it? I've looked at using rsync as well but again, it doesn't do what i would expect.  For reference I'm using PuTTY to connect through SSH to a ReadyNAS 102.  After using find . -mtime -2 | cpio -o | { cd /media/USB_FLASH_2 &amp;&amp; cpio -imd; }  i get the output:   BusyBox v1.20.2 (Debian 1:1.20.0-7) multi-call binary.  Usage: cpio [-dmvu] [-F FILE] [-H newc] [-tio] [EXTR_FILE]...  Extract or list files from a cpio archive, or create an archive using file list on stdin  Main operation mode:         -t      List         -i      Extract EXTR_FILEs (or all)         -o      Create (requires -H newc)         -d      Make leading directories         -m      Preserve mtime         -v      Verbose         -u      Overwrite         -F FILE Input (-t,-i,-p) or output (-o) file         -H newc Archive format Rounding Percentages