If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite?
If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.
Google Apps Script Javascript compatibility error I have successfully created . Handy to avoid multiple calls to GAS services whit ordinary getRange calls.  While the code perfectly works in javascript, it throws an error in Google Apps Script upon saving.  The offending code is:  return matrix.slice(startRow, endRow + 1).map(i =&gt; i.slice(startCol, endCol + 1))   GAS seems rejecting code in the .map function. I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas? When I press the power button on my Epson WorkForce WF-3520 printer, it starts up normally, but shuts down within 30 seconds. Ten seconds later, it starts itself and the stop/start cycle is repeated again and again. This has only started happening today (6th December 2016). What did the general public know about Darth Vader? When Darth Vader first becomes Palpatine's apprentice, very few know of his existence and connection to the Emperor: the members of the Trade Federation (whom he dispatches) and Yoda, Kenobi, Organa, and their associates are all that seem to know at the end of Episode III.  Ultimately, the Resistance would know about him as they would be fighting him, and Organa could have been able to inform them about his identity and the fact that he is a Force user. The military of the Empire would know of his existence, but even high-ranking members likely wouldn't know his past or identity. However, they would know about the fact that he is a Force user due to his tendency to use his "sorcerer's ways" to choke people.  So, how much would the general public have known about Vader? Would they have known him as just another military leader, or would they have known he was Palpatine's right-hand man? Would they have known about his ability to use the Force, and therefore the possibility that he was a Sith? on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$ I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct.   First, when I actually multiplied out $\vec{B} (\vec{A} \cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B}) $ I noticed that I got the same thing. That is, the two quantities are equal. This makes sense since they are dot products. But that further implies that the expression goes to zero.  Well, that means that $\vec{A} \times (\vec{B} \times \vec{C})$ would have to be zero too. And when I thought about it it occurred to me that $\vec{B} \times \vec{C}$ describes a vector perpendicular to both B and C; it's orthogonal. But $\vec{A} \times$ (any vector) will describe a vector perpendicular to (any vector) and in this case, that's $\vec{B} \times \vec{C}$. That is going to be parallel with the vector described by $\vec{B} \times \vec{C}$ itself and that further implies that the cross product of the two is equal to zero. Which is equal to the BAC-CAB quantity above.   So, what is the big honking flaw in my reasoning here? :-) How to tell Google to ignore URL parameters and forget about any already crawled? How to create a filter in Gmail to identify messages forwarded from Zimbra? Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device. How do I install OpenStack? How do I install ? Is there an easier way other than official installation instructions? From axiom of regularity, how can I prove $x \notin y \lor y \notin x$ cgi and tomcat im trying to run a cgi script (.cgi) with tomcat. I am getting the below error and cant find out whats wrong. I know i should really use apache and mod proxy but this really isnt my area of expertise so im taking the easy way out!  Thanks for any help.   java.io.IOException: Cannot run program "perl" (in directory "C:\Java\tomcat\webapps\my_app_name\WEB-INF\cgi"): CreateProcess error=2, The system cannot find the file specified     java.lang.ProcessBuilder.start(ProcessBuilder.java:459)     java.lang.Runtime.exec(Runtime.java:593)     java.lang.Runtime.exec(Runtime.java:431)     org.apache.catalina.servlets.CGIServlet$CGIRunner.run(CGIServlet.java:1705)     org.apache.catalina.servlets.CGIServlet.doGet(CGIServlet.java:597)     javax.servlet.http.HttpServlet.service(HttpServlet.java:627)     javax.servlet.http.HttpServlet.service(HttpServlet.java:729)     org.tuckey.web.filters.urlrewrite.UrlRewriteFilter.doFilter(UrlRewriteFilter.java:738)     org.apache.struts2.dispatcher.FilterDispatcher.doFilter(FilterDispatcher.java:416) "Which" vs. "what" — what's the difference and when should you use one or the other? Most efficient way of booking academic supervision appointments I'm wondering if there's a more efficient method of booking an academic supervision appointment. The method I experienced was a schedule on the professor's door, first-come-first-serve.   How do we employ technology that is the most efficient (reduces scheduling time, permitting all students to view sign ups)? Should we use 'Title Case' or 'Sentence case' for headlines and buttons? I am wondering if we should use  or  for buttons and headings in websites and web/mobile apps? Title Case: Manage my Account Delete my Account  Sentence case: Manage my account Delete my account  In  I found the recommendation 'choose your style and be consistent'. This is it? Aren't there any other (additional) recommendations or sources. Like for specific use cases or apps where the one style would be preferred over the other? Edit: The question refers primarily to English, but differences in other languages are also good to know. Discouraged by simple Bayesian Question My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has.  I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info.   My understanding from stats way back when is that $p(x|\omega_1)$ is stating the probability of $x$ given $\omega_1$ for $0\le x \le 2$ and zero other wise. Same obviously if given $\omega_2$. If I understand, the priors are the probability of $\omega_1$ and $\omega_2$ which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated. Quantum mechanics textbooks that use path integrals I'm looking for a textbook in quantum mechanics that relies heavily on Green functions and the path integral formalism to supplement my QM books. I want to do some calculations using alternative methods (path integrals, green functions, Lagrangians etc). Hide or disable form options? Possible Duplicate:          On a create/edit form, where the options available to the end user will depend on their previous input, in order to give the user some guides on how to fill the data, we shall:   Disable the controls and enable them only when the pre-requisites are meet. Hide them instead Do nothing Other options   Example: when asking who is the legal representative of someone, the form will contain a combobox with the following options: the own person, their parents or someone else. Only in the case the user chooses someone else, he was to fill a textbox telling who.  Which option will provide the best user experience? How to exclude some files from filename expansion mechanism in bash? $\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums? Is there a word for colloquial forms of address?