faster than the wind katamaran This is the catamaran in question, and in a separate article(on the same thing) that it can travel 3x as fast as the wind...  Under everything I understand about physics, this does not work, you can't get more energy out than you put in, then can anyone explain this?
How to sail downwind faster than the wind? Recently a  set a record for sailing a wind-powered land vehicle directly down wind, and a speed faster than wind speed. Wikipedia has a  talking about it, but it doesn't explain exactly how it works.  I can imagine some configurations, but I don't get how a system could be set up to push forward with a tailwind (when the cart is moving slower than the wind) and push forward with a head wind (when the cart is moving faster than the wind).  Could someone help explain how this is possible?
Can you take the same feat multiple times? Word for "not consciously aware of" What's the word to convey the meaning "not consciously aware of"? python ImportError: No module named primes I have an exercise found on a list but I didn't know how to proceed. Please, any tips?  Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of $M\setminus X$ that $M\setminus C$ is connected. Is there a way to get reverse numbering on the enumerate environment? To build a list of publications, I wish to show the most recent on top of the list, while keeping the numbering similar. This can be achieved manually by doing something like:  \begin{enumerate} \item[42] pub num 42 \item[41] pub num 41 \item[...] \item[1] pub num 1 \end{enumerate}   but is there a more principled approach? Name tikz pic as if it was a node I'm confused about tikz pics. I can freely define them without issue, and even add coordinates inside that will be available by name inheritance like (-A) as for example in \pic (S) at (0,0) {myPic}; \draw (S-A) -- ++(0,0);. This is all documented around p. 255 of the pgf manual.  However I don' understand why it is not possible to give a name to a pic like the one above if no coordinate is defined inside it. By default it would be great if it could behave like a normal node.  MWE:  \documentclass{minimal} \usepackage{tikz}  \tikzset{     myBlock/.pic = {         \node[draw,minimum width=1cm,minimum height=0.5cm] at (0,0) {};     },     myBlockNamed/.pic = {         \node[draw,minimum height=1cm,minimum width=0.5cm] (-m) at (0,0) {};     }, }  \begin{document} \begin{tikzpicture}     \pic () at (0,0) {myBock}; % can't give it a name     \pic (S) at (3,0) {myBlockNamed}; % can give it a name     \draw (S-A) -- ++(2,0); \end{tikzpicture} \end{document}   The underlying point is I'm working on a control blocs library (so that we can stop the chat below with @Alenanno). For example here is the definition of a math operator  \tikzset{     mySum background/.style = {         circle, minimum size=1.5em, fill=white,     },     mySum edge/.style = {         draw=black, circle, minimum size=1.5em, line width=.5pt,     },     pics/bOp/.style = {     code = {       \node [mySum background] {};       \foreach \t [count=\i] in {#1}{         \pgfmathsetmacro{\angle}{\i*90}         \node[anchor=center, font=\tiny] at (\angle:0.45em) {$\t$};       }       \node [mySum edge] {};     },   }, } Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$   $\bf{My\; Try::}$ I have solved Using Direct formula::  $$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}$$  Now Put $n=7\;,$ We get   $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}\cdot \sin \frac{4\pi}{7}\cdot \sin \frac{5\pi}{7}\cdot \sin \frac{6\pi}{7}=\frac{7}{2^{7-1}}$$  So $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} =\frac{\sqrt{7}}{8}$$  Now my question is how can we solve it without using Direct Formula, Help me  Thanks It's my understanding that "easily" is the regular adverb for "easy", e.g. "He makes friends easily".  But in some collocations one still uses "easy", e.g. "Take it easy", "Rest easy" or "Slow down easy". Is there any rule when "easy" would be used over "easily", or does one just have to know these collocations? Making separate Python installation that can call ArcPy? Is an expanding map on a compact metric space continuous? This is a batch file in Windows.  Here is my .bat file          @echo off copy "C:\Remoting.config-Training" "C:\Remoting.config"  "C:\ThirdParty.exe"   This works fine except the .bat file leaves the command window open the whole time the "ThirdParty" application is running. I need the command window to close.   I would use the short-cut for the application but I must be able to run this copy command first (it actually changes which data base and server to use for the application).    The ThirdParty application does not allow the user to change the source of the db or the application server.    We're doing this to allow users to change from a test environment to the production environment. Why is Facebook missing from the list of ways to share a page in Safari?  That option used to exist before the latest update.  When I'm directed to the extensions box Facebook is not there. If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$.   Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in \mathrm{Spec}(A): P \supset (f)\}$  My attempt:   On one hand, if $P \in \mathrm{Spec}(A)$ and $P \cap S$ is empty then we identify $\mathrm{Spec}(A_f)=\mathrm{Spec}(S^{-1}A)=\{P \in \mathrm{Spec}(A): P\cap S= \emptyset\}$.  On the other hand, $\mathfrak{V}((f))^c=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\}$.   So the exercise is equivalent to prove: $$\{P \in \mathrm{Spec}(A): P\cap S =\emptyset\}=\{P \in \mathrm{Spec}(A): P \nsupseteq (f)\},$$ but I can't continue so I'd appreciate if somebody could help me.   Thanks in advance. How do you mock a Sealed class? can be quite a pain.  I currently favor an  to handle this, but something about just keeps feels weird.    So, What is the best way you mock sealed classes?  Java answers are more than welcome.  In fact, I would anticipate that the Java community has been dealing with this longer and has a great deal to offer.  But here are some of the .NET opinions: Sometimes when I am debugging code in Eclipse it happens that although I can see and inspect class member variables without any difficulty I am unable to inspect the values of variables declared locally within functions. As an aside, any parameters to the current function lose their 'real' names and instead one sees their values listed in the Variables window as arg0, arg1, arg2, etc but at least the values are visible.  This is occurring at present in relation to classes defined within the core JDK. I have verified that the installed and current JRE is a JDK.  Is anybody able to shed some light on this behaviour? Is it possible to save `bc` command line history? bash has a handy file .bash_history in which it saves the history of commands and on the next execution of bash the history is populated with saved commands. Is it possible to save bc command history to a file in the same way and then load it on startup so that bc history is preserved? I tried reading  and it mentions readline and libedit. From ldd /usr/bin/bc I see mine uses readline and readline has  and read_history functions. Is this functionality implemented in bc or to do it I'll need to patch bc? Is there a proof of existence of time? It seems to me that there is no such thing as time. There  is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.  Does this view of time work within the current framework of phsics?  Do physicists have an explanation/proof about time's existence? How to select a Master's thesis topic if your advisor won't suggest one? If a master's degree candidate needs to select a thesis topic, but their advisor is unwilling to offer suggestions on the matter, how should one go about selecting a thesis?   How might one select an area to study? How can one evaluate if if is a suitable topic? Does the thesis need to focus on an area which is relatively unexplored by prior research? What is the bug about: Keyword gets duplicated even when the query is not modified.  Impact: As far as i can tell - cosmetic.  How to replicate: Go to:  and keep pressing Search button. With each search there will be a duplicate python keyword appended to search box.  Best guess for what is causing it: I think you have somewhere a piece of code that appends tag as a keyword without checking if it is already present. My reasoning comes from this url:  not having the same issue - the tag is not present in it. Limit of $n$th root of $n$! I am asked to determine if a series converges or not:  $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{(2^n)n!}{(n^n)}$$  So I'm using the $n$th root test and came up with $\lim_{n \to {\infty}}\frac{2}{n}\times(\sqrt[n]{n!})$ I know that the limit of $\frac{2}{n}$ goes to $0$ when $n$ goes to infinity but what about the $(\sqrt[n]{n!})$?