I learned many years ago that according to Ohm's law, current is equal to voltage divided by resistance. Now if superconductors have zero resistance then the current should be infinite.  Moreover the power should be zero too as $P=I^2R$.  In superconductors, current can pass without applied voltage. Since current is flowing so there is definitely some energy flowing in the wires but VIt=energy and V is zero so there is no energy ?  How to solve this discrepancy?
states that the relationship between current ( I ) voltage ( V ) and resistance ( R ) is  $$I = \frac{V}{R}$$  However  cause the resistance of a material to go to zero, and as I understand it, as $R \to 0$, $I \to \infty$. Does this present a problem for Ohm's law?
It just struck me that Utility Companies world-wide supply domestic users at either 50/60Hz. It could have been selected to be 10 Hz, or 100 Hz ... whatever  Was 50/60 chosen at random? Is there some rationale involved (efficiency, easier to construct/control etc) in restricting domestic supply to either 50/60 Hz?  In other words, if I were designing a power system, why would I pick 50/60 Hz instead of some other frequency? There are a lot of poorly drawn schematics here. A few times people have actually asked for critiques of their schematics. This question is intended as a single repository on schematic drawing rules and guidelines can point people to. The question is  What are the rules and guidelines for drawing good schematics?     Note: This is about schematics themselves, not about the circuits they represent. Prove that $\phi(n) \geq \sqrt{n}/2$ So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$. But how can we get the lower bound? I tried using the same expression for $\phi(n)$ but it seems to not really give me the inequality. Can you help? What is the probability of a biased coin coming up heads given that a liar is claiming that the coin came up heads? A biased coin is tossed.   Probability of Head - $\frac{1}{8}$  Probability of Tail - $\frac{7}{8}$  A liar watches the coin toss. Probability of his lying is $\frac{3}{4}$ and telling the truth is $\frac{1}{4}$. He says that that the outcome is Head. What is the probability that the coin has truly turned Head?  My Attempt:  I used the formula: $$P(A \mid B) = \frac{P(A\cap B)}{P(B)}$$  $\ \ \ \ \ \ $P(it is head GIVEN liar said it's head) = P(it's head AND liar said it's head) / P(liar said it's head)  or, P(it is head GIVEN liar said it's head) = $\frac{ \frac{1}{8} \frac{1}{4} }{ \frac{7}{8}\frac{3}{4} + \frac{1}{8}\frac{1}{4} }$ [using a probability tree will be helpful here]  or, P(it is head GIVEN liar said it's head) = $\frac{1}{22}$  The Question: Is the method I used wrong in any way? Some others I have talked to are saying the answer will be $\frac{1}{4}$. Their reasoning is this: since the liar lies 3 times out of 4 and he said it is head, then the probability of it being head is 1/4. So who is right? What will be the answer? How to make equally sized windows in corners of the screen? Subadditive sequences and the limit $\sup_{n\to\infty} a_n / n$ Let $\{ a_n, n\geq 1\}$ be a sub-additive sequence, i.e. for any $m,p\geq 1$ $$ a_{m+p} \leq a_m + a_p. $$ .Prove that $$\lim_{n\to\infty} \frac{a_n}{n} = \inf_n \frac{a_n}{n}.$$ I guess the first step is to state $$a_{n} = a_{mp+q} \leq a_{mp} + a_q$$ but I'm not sure where to go from here.... As the topic is there exist a homeomorphism between either pair of $(0, 1),(0,1],[0,1]$ Determine the number of functions $f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd. Problems:Determine the number of functions $$f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$$ satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd.   My Attempt: Each integer in domain has $4$ choices and therefore the total number of functions is $f$ is $4^{1999}.$ Since there are an equal number of functions that yield odd or even result, we can directly write that the number of functions satisfying the above condition to be $2*4^{1998}.$ I am unsure about this last claim and would like to prove it (if it is true.)Also is the answer to this problem correct?  Edit $1$: I tried to proceed further in the following manner:   For a given mapping $f$ let the number of integers assigned to $2000,2001,2002$ amd $2003$ be $p,q,r$ and $s$ respectively, where $p,q,r,s\in \mathbb{Z}^+$. Therefore the sum in question can be written as $f(1)+f(2)+...+f(1999)=2000p+2001q+2002r+2003s.$ Clearly the sum will be odd iff $q+s$  is odd. Also note that $p+q+r+s=1999\Rightarrow p+r$ is even. On the other hand if we let $q+s$ as even then $p+r$ must be even. Now the number of values of $p,q,r$ and $s$ for which the above two lemmas hold is the same and therefore if we let $f$ to be the function which maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $q+s$ is even and $g$ to be the function  hich maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $p+q$ is even we will get a bijection.(Maybe!) windows 7 desktop icons: new icon placement When moving/saving/etc a new file on the desktop, the icon appears second (just after "Computer"). I find it annoying, since I want my "special" icons to be first (such as the recycle bin, "Network", etc.).  Can this behavior be changed? Why do people go to the Wall in Game of Thrones? Is the wall a bad or a good place to be? Some people are sent there as punishment, others go because they want to (Jon Snow), others to get away from harm (Arya).  If I am not completely correct, it is because I have only started watching the series recently, and it is a bit confusing!  Please also correct me if I am wrong with any facts! I'm trying out  with gnuplot to generate the data.  This works by having pgfplots write some stuff to an external file, run gnuplot on the file, and then use the generated data to plot the graph.  The problem I'm having is that I get different behaviour under xelatex as pdflatex or lualatex.  In xelatex, the data file is littered with ^^I whereas in the other two formats these are tabs.  It appears that the code ^^I is getting stored as a macro which then gets written out to the file, so possibly there's something funny going on with the ^^I syntax: xelatex is reading it literally.  Indeed, when I do \show\pgfplots@TAB (the macro in question) then for pdflatex and lualatex then I get a literal tab whereas for xelatex I get ^^I.  So how do I get a tab in xelatex?  And why isn't \gdef\pgfplots@TAB{^^I} working?  (The previous line in the code, by the way, is \catcode`\^^I=12 - I don't know if that has anything to do with it.) I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height) How can I get back to Blender's default theme? Even though I am relatively new to Blender I am recording a tutorial for a Blender exporter. I have been advised that I should use the Default colour scheme (for the UI). I have no idea which one that is. Can anyone tell me? Can you reduce yourself, crawl into the Tarrasque's airway, and try to collapse its lung by enlarging inside of it? A player tried to use the spell  on his PC while in the mouth of the Tarrasque to shrink himself, then go deeper into its airway and to try to collapse its lung by enlarging himself inside of it. I imagine there's no real rule for this but I felt silly just denying it, because it seemed like a cool idea that you could only probably do with a creature as large as the Tarrasque.  Can you crawl into a gargantuan creature’s mouth or into its airway via reduce/enlarge and attack its brain or internal organs? Why isn't the center of the galaxy "younger" than the outer parts? I understand that time is relative for all but as I understand it, time flows at a slower rate for objects that are either moving faster or objects that are near larger masses than for those that are slower or further from mass.  So, the illustrative example I always see is that if I were to leave earth and fly around at near light speed for a while or go into orbit around a black hole the time I experience would be substantially shorter than for those I left behind at home on earth and I'd come back to find that I've only aged however long I felt I was gone by my own clock but that on Earth substantially more time would have elapsed.  Following through on this model, the stars in orbit around the black hole at the center of the Milky Way are aging much much slower (relative to us), right?  So does it not follow that the center of the galaxy is some appreciable (I have no idea how to go about putting this in an equation so will avoid guessing at the difference) amount "younger" than the stuff further away from the center?   If this is not true, could someone please explain why not, and if it is true, can someone please point me to where I can calculate the age of the center of the galaxy :-)  And to be clear... what I'm asking is... if there was an atomic clock that appeared at the center of the galaxy when the center was first formed, and we brought it through a worm hole to earth today - how much time would have elapsed on that clock vs. the age we recon the galaxy currently is? (13.2 billion years) I have been around on Twitter for awhile and I regularly find people using "Please to retweet this" or "Please to help me", etc. Is this proper English? I do not think prefixing the infinitive form with a 'to' is necessary here at all. "C" sizeof with a type or variable I was just chatting on the Ask Different chat - and the reputation numbers for me and the others chatting don't seem to match any other reputation count I can see on the sites here.  Is there some calculation being made? These numbers are the highest of any other combined reputation I have seen. For me - I'm listed over 10k - but no other aggregate reputation counter I can see is above 8k.  Perhaps I'm unaware how reputation is calculated, but the people there suggested a bug report in case this isn't intended.  (after three lines of text are drawn - a number shows beneath our icons in the left hand side) I'm being forced to use my Macbook more and more with my metered 4G connection. I really need to keep data usage limited. Windows has an  mode which really helps with this.   What's the best way to handle this issue in the OSX world? Perhaps some use of the firewall? Is there any sort of logging happening as to what is using the network?  Thanks for your help :-) The problem is, when I open applications I need to always right click, select run as admin.     When I don't do it getting unexpected application errors. BTW, I turned off UAC.   It's annoying. Is there any way to change behavior of double left-click to act as "Run as administrator" action? I mean to make "Run as administrator" action by default. Thx in advance