Let $f: S^1 \to S^1$ be a continuous map, where $S^1$ is the 1-sphere. Prove that if there is an homotopy between $f$ and a constant map, then $f$ has a fixed point.   I don't know how to prove it. I try with the Brouwer fixed point theorem, but I think that it doesn't work in $S^1$
How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$  (a) has a fixed point and that (b) has a point $x$, such that $f$ maps $x$ to its antipodal point $-x$?
Strictly increasing sequence $\{ a_k \}$ of positive integers such that $\sum 1/{a_k}$ is finite Is there a "smallest" divergent sum? Whenever I open my files, my hidden files show up. I can hide them but when I open up my files again, they're showing again. How do I make the hidden files invisible permanently? Where has the trailing newline char gone from my command substitution? The following code best describes the situation.  Why is the last line not outputting the trailing newline char?  Each line's output is shown in the comment.  I'm using GNU bash, version 4.1.5          echo -n $'a\nb\n'                  | xxd -p  # 610a620a              x=$'a\nb\n'   ; echo -n "$x" | xxd -p  # 610a620a      echo -ne "a\nb\n"                  | xxd -p  # 610a620a x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # 610a62 Insert result of a parallellized select query into a table, in Postgresql According to ,      "Even when it is in general possible for parallel query plans to be   generated, the planner will not generate them for a given query if any   of the following are true:       The query writes any data or locks any database rows. If a query contains a data-modifying operation either at the top level or within   a CTE, no parallel plans for that query will be generated. This is a   limitation of the current implementation which could be lifted in a   future release."   Indeed, when I try to insert result of a parallel SELECT query into a table (either by SELECT.. INTO either by WITH..SELECT..INTO, the query is not executed as parallel query.  My question is: Is there any way to trick the Postgresql so that a SELECT query is executed as parallel query and then its result inserted into a table? I have a website with login page that works normal. The problem is that once logged in, the user still has access to the login page form to enter, which should not happen.  I would like to know a wordpres plugin that redirects this user to another page when they access it after login. Are DSLRs a dying breed, making now the time to switch to a mirrorless camera system? I came across this  on Stuck in Customs today that makes a strong case to hold off too much investment in DSLR gear, and instead switch to mirror-less cameras like the Sony NEX. The author prefers to call such cameras 3rd generation cameras to prevent any bias.  The crux of the argument lies in the smaller size and faster shooting speeds of the new cameras, while the sensor size is an area where they seem to be lacking (Sony &amp; Samsung cameras are APS-C sized though). Currently the big makers - Nikon &amp; Canon - have practically no presence in this market (Nikon seems to be doing something with the V1). However, they seem to have slowed down the introduction of new entry level DSLR models over the last year as well.  So, as DSLR users looking to build and enhance their kit (there are many like me who've jumped the bandwagon over the last couple of years, and are just starting to build their kit), what are the advantages of sticking with the DSLR system rather than switching to the 3rd gen cameras?  IMO, one point that could cause a massive switch in the DSLR base would be if Nikon &amp; Canon announced mirror-less models compatible with their current lens lineups, but that is a hypothetical scenario.  P.S. There is a similar question on mirror-less cameras, but that seems to be from the perspective of a person starting out rather than a person already invested in the DSLR system. How to delay command blocks until player respawns? I have a command block line set up so that it detects for a players death, then gives them weakness for 600 seconds. It applies weakness to soon before they get out of the respawn time. If there is not way to delay a command is there a way to detect a player respawn? Why does Math.Floor(Double) return a value of type Double? How can I get query string values in JavaScript? example:   Does Chocolatey run the native program installer? If I prefer simplicity, should I avoid packages with (Install) in their name? Strange Pop-up appldnld.apple.com:80 I received a strange Pop Up message. The unsettling part is password will be sent in clear text.  Is my phone affected with any malware or app virus ? How do I change my login provider(s)? Add increase/decrease tab indent buttons to the editor toolbar These should work by adding spaces or removing spaces from the beginning of the current line or every line between the start and end of a text selection (inclusive).  (edited this question to supplement ) I've used following codes and no luck  1  //Hide admin footer from admin function change_footer_admin () {     return ' '; } add_filter('admin_footer_text', 'change_footer_admin', 9999);  function change_footer_version() {     return ' '; } add_filter( 'update_footer', 'change_footer_version', 9999);   2  function wpbeginner_remove_version() { return ''; } add_filter('the_generator', 'wpbeginner_remove_version');   3  function my_footer_shh() {     remove_filter( 'update_footer', 'core_update_footer' );  } add_action( 'admin_menu', 'my_footer_shh' );   Still I see those texts and version on my footer.   So what is the accurate way to remove these? I've added these code snippets into my child theme function.php A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help! I was caught cheating on an exam, how can I minimize the damage? I got caught cheating in a two-hour engineering exam consisting of 100 multiple choice questions which was done online through the  e-learning software. I feel like my life is almost over.  Having not studied well and having no time to study, I decided to cheat. I know there are no excuses for cheating. Before the exam I dug up research about moodle exams, and it turned out you can take the exam anywhere you want to as long as you have the quiz password. Taking advantage of this, I went up to my engineering library and asked a close friend who was taking the exam to send me the quiz password before he starts.  What I had prepared:   Calculator (was not allowed during the quiz). Printed material (consisted of more than 200 pages). Google search engine (on a laptop ready to help). Whatsapp (to ask questions for a friend who took the quiz last year).   As I sat down, I did the first 50 questions, suddenly the quiz froze and moodle told me:      you are not allowed to take the exam from this location.   While I was leaving, one instructor responsible for the course (there are 5) came and found me. She accused me of cheating, took my mobile and made come with her to a huge office where two instructors searched all my mobile (took the name of my friend who gave me the password), and started talking to me, asking for all details of this crime. They took and confiscated my phone and looked at everything: Whatsapp conversations, all my emails, and images. The instructor took the mobile from my hand without asking me and kept it with her. It seems that the instructors are going to report me as well as my friend who sent me the password for the exam.  The university's Student Code Conduct said that cheating will result in one of the following: a Dean's Warning, Suspension, or Expulsion. If any of that happens, my future is over. A dean's warning will cancel my financial aid. A suspension would be for at least two years, and coming back would require a lot of work. An expulsion will be definite.  I've learned the lesson about cheating, now how can I fix this? How can I prevent the five instructors from reporting me? It's obvious that I should speak to them, but what should I say? What can I do? My future is almost over, but many of you are teachers and instructors here, what can I do to fix this?    EDIT: Wow, it has been almost 3 years. I would like to update on how I dealt with the situation.   Lesson: It was obvious to never ever cheat under any circumstances, and it was not whether you'd get caught or not.    Consequences: I received a Dean's warning but the financial aid wasn't revoked. However, I couldn't stay in the department as engineering was not for me and I was even ashamed to stay in the university.  What happened next: I transferred to an ABET-accredited Computer Science institution and I graduated with distinction in 2.5 years. It was extremely stressful as I had taken 6 major courses (18 credits) in one semester, but I had to do it.  Future plans: I'm going for an MSc in Computer Science in St Andrews next year. Is there a way to prevent those pesky .DS_Store files from being ever created? Is there a way (eg. a binary hack into Finder) to stop it from creating .DS-Stores on LOCAL Volumes?   I am using Leopard, therefore these don't work:    (requires Lion)  (requires Snow Leopard)   Also,  slows down my computer. Modified pattern does not see the pattern color option Why do sunbeams diverge even though the sun is much more than a few kilometers away? Consider this picture of sun beams streaming onto the valley through the clouds.    Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this gives the result that the position of the light source is being a few tens of km away at most.   What is wrong with the analysis?