Which two points are we talking about in Fermat's principle? Are those points decided by light or decided by us? Can we take any two points?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:  In analytic (Lagrangian) mechanics, the derivation of the Euler-Lagrange equations from the principle of least action assumes that the start and end coordinates at the initial and final times are known. As a consequence, any variation on the physical path must vanish at its boundaries. This conveniently cancels out the contributions of the boundary terms after integration by parts, and setting the requirement for minimal action, we obtain the E.L. equations.  This is all nice and dandy, but our intention is finding the location of a particle at a time in the future, which we do not know a priori; after we derive any equations of motion for a system, we solve them by applying initial values instead of boundary conditions.   How are these two approaches consistent?
Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges.   I see  that,  $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} \frac{1}{n+\frac{k}{n}}.$ When $n$ gets sufficiently large the contribution by the $ \frac{k}{n} $ term is diminishing and $ a_{n} &lt; \sum_{k=1}^{n} \frac{1}{n} = 1 $.   Thank you. Only generation-I Pokemon are available, however it seems that the types which did not exist in Gen-I do exist in Pokemon Go (e.g. Dark type moves).   So I'm wondering which type chart the game is using for move effectiveness.    According to Bulbapedia, in Gen I Bug attacks were super-effective against Poison, but in Gen II-V Bug attacks are not very effective against Poison. What is it in Pokemon Go? Similarly, in Gen II-V Ghost attacks are not very effective against Steel, but in Gen VI Ghost attacks are normal effective against Steel. Does anyone have a Magnemite, is it Steel type? Does it resist Ghost? Does STAB exist in this game? How do I check if a string contains a specific word? I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files? Can Minecraft Pocket Edition play with a desktop Minecraft? Are there any areas that should be avoided and are there any steps I should take to protect myself from radiation? Why does "start winword" work but just "winword" doesn't? Assume Microsoft Office is installed. Consider the following batch file:  REM the following line successfully launches Microsoft Word start winword  REM none of the following commands work to launch Microsoft Word winword winword.exe winword.com winword.bat winword.cmd winword.vbs winword.vbe winword.js winword.jse winword.wsf winword.wsh winword.msc   What is it about the "start" instruction that allows the launching of Microsoft Word using the name "winword"?  My curiosity stems from my interest in the mechanics of the "start" verb - I don't need suggestions on other ways to start Microsoft Office.  Please don't suggest that I add the Microsoft Office directory to my PATH environment variable - I know what that would do.    How is start finding "winword" when cmd cannot find it directly? Get referenced node fields in twig In Drupal 8, I have a content type, that has an "entity_reference" field and I wish to access the referenced node's fields in twig template. the kint() output for the field is:   In this list I can't find the referenced node's field values. How can I (if at all possible) load the node or node fields/values in twig?  Best,  Alari Is there a limit to the number of Power Armor sets you can keep? I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep? I've installed Windows 10 in BIOS mode and use MBR. How can I convert it to use UEFI and GPT without reinstalling Windows 10? Is this possible? QR code generation library in javascript I often have to write up reports based on the analysis of some data. I use R to analyse the data and export tables, figures, and text. This is then included into a LaTeX document either using input or Sweave ().  However, when I collaborate with others, I sometimes need to provide a document in Open Office / MS Word format.  Question:  Thus, assume the simplest scenario   I have a LaTeX document with text, tables, and figures I need to export this reliably into Open Office or MS Word format: this includes mathematical formulas, table formatting, and quality figures) I don't need to go back from MS Word to LaTeX   What is a reliable, efficient, and preferably free process?  Initial Thoughts:   Export to HTML,  LaTeX - LyX - ODF Use odfWeave  that deals with some specific issues associated with LaTeX to Open Office conversion    I was hoping that there is an expert out there who has worked out a good system already. `testl` eax against eax? I am trying to understand some assembly.  The assembly as follows, I am interested in the testl line:  000319df  8b4508        movl   0x08(%ebp), %eax   000319e2  8b4004        movl   0x04(%eax), %eax   000319e5  85c0          testl  %eax, %eax   000319e7  7407          je     0x000319f0     I am trying to understand that point of testl between %eax and %eax? I think the specifics of what this code isn't important, I am just trying to understand the test with itself - wouldn't the value always be true? Finding functions such that $f(g(x)) = f(x)g(x)$ Let's assume the functions in question are continuously differentiable. If $f(g(x)) = f(x)g(x)$, then: $$f'(g(x))g'(x) = f'(x)g(x)+f(x)g'(x)$$ I attempted to find a particular solution by letting $g(x) = x$. Then, $$f'(x) = xf'(x) + f(x)$$ $$df = xdf + fdx$$ $$\frac{df}{f} = \frac{dx}{1-x}$$ $$\ln(f(x)) = -\ln(1-x)+C$$ $$f(x) = \frac {A}{1-x}$$ Where $A$ is a constant. This solution was fairly unsatisfying to me, and upon checking it, I see it doesn't actually work, as letting $g(x) = x$ leads to $f(x) = xf(x)$. I was wondering if there were better answers, or at least a less trivial way to go about the problem. How can I reset my Ability Points? I'm also curious about being able to reset my points, should I wish to change my ability configuration later.  So far (10h-ish) I've only seen one way, which is a potion sold by Kiera Metz for 1000g. However she's no longer available as a store, due to a story choice, so I'm not sure where else I could buy this potion.  Where might I buy this potion, and is it the only way to reset your ability points? What is the benefit of review a scientific paper as a referee? What is the shortest way to compute the last 3 digits of $17^{256}$? What is the shortest way to compute the last 3 digits of $17^{256}$ ?  My solution: \begin{align} 17^{256} &amp;=289^{128} \\ &amp;=(290 - 1)^{128}\\ &amp;=\binom{128}{0}290^{128} -  ... +\binom{128}{126}290^2 - \binom{128}{127}290 + \binom{128}{128} \end{align}  Computed the last 3 terms whose last 3 digits gave the last 3 digits of $17^{256}$.  Is there any shorter method to do this(which requires much less computation) ? Is there any way to tell Google to display the updated title instead of displaying the old blog title in search results? As a matter of optimization, a few weeks ago I modified the way the titles of my blog posts are displayed (I made a change also in the title of the blog). Then I used the Google Webmaster Tools to re-index the old pages and update titles. Immediately I could update the titles and posts appear correctly on Google.  The problem is that now the titles of posts began to appear incorrectly in the search, using the old blog title. Some posts continue to display the updated title, but many recent posts show the old title.  I've tried re-index the pages with Google Webmaster Tools, but the titles remain unchanged in the search results.  Is there any way to "force" Google to display the correct titles? How do American English and British English use the definite article differently? I decided to make sure that I know this important difference between American and British English, so I wrote what I have found out so far and I would be grateful to anyone who reads this and tells me whether I am wrong, or not.  In British English when people say  to hospital or in hospital when  talking  about somebody being there as a patient they don't use the definite article : "I had to go to hospital", "She spent two weeks in hospital". And the meaning is that somebody was there as a patient.   If then for some other reasons British English speakers  will use the definite article which will change the meaning itself, I noticed that, in American English, native speakers often use the  the and if they need to show that somebody is in church to pray,  in school as a student, in hospital as a patient, in prison as a prisoner, they use 'in' and not 'at'. Do American English speakers use 'at' like British English speakers use 'the' to give the sentences a different meaning?    Are my sentences correct? Do they show American English usage?   He is in the school. (enrolled as a student) He is at the school. (for some different reasons) He is in the hospital. (as a patient) He is at the hospital. (visiting somebody) He is in the church. (to pray) He is at the church. (for some different reasons) He is in the university. (as a student) He is at the university. (not as a student) He is in the college. (as a student) He is at the college. (Not as a student) He is in the prison. (as a prisoner) He is at the prison. (not as a prisoner) Are there any studies on the use of justified text in web apps? How does justification affect screen legibility?  Example of justified text: