Prove that, for all positive integers $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$
Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
Capitalize fields of study? I am having trouble loading csv files into layers using QgsVectorLayer().  I know this looks like a repeat, but I have carefully read     and    but still cannot get it work.  An example of my data file, test.csv:  Index,y,x 0,37.7646618,-122.4960858 1,37.777494,-122.416311 2,37.717707,-122.3997056 3,37.732284,-122.498118 4,37.749998,-122.392333   My code:  import os import urllib  import pathlib from qgis.core import QgsVectorLayer  local_path = 'C:\dev\FCMS_conflation\data\TMC_nodes.csv' node_layer = 'test'  abspath = os.path.abspath(local_path) params = {'delimiter': ',',           'decimalPoint': '.',           'xField': longitude,           'yField': latitude}  uri = "%s?%s" %(pathlib.Path(abspath).as_uri(), urllib.unquote(urllib.urlencode(params)))  layer = QgsVectorLayer(uri, node_layer, "delimitedtext")   The formatted URI looks like:  [] uri &gt;&gt;&gt; 'file:///C:/dev/FCMS_conflation/data/TMC_nodes.csv?decimalPoint=.&amp;delimiter=,&amp;xField=longitude&amp;yField=latitude'   I have tried many variations of the parameters, including the crs and type. But no matter what I try, I cannot create a valid layer.  [] layer.isValid() &gt;&gt;&gt; False   I have verified that I can add the file using the QGIS gui. I even tried using the uri provided by layer.metadata from within the Python plugin.   I am using QGIS 2.10.1 on Windows 7 64-bit machine. Which LaTeX to install on Linux? Which LaTeX distribution would you recommend for using on Linux from command line? I am looking for one with most complete coverage of packages. Plotting several 2d functions in a 3d graph I am trying to reproduce a similar figure to the following one. My first guess was to use pgfplots but I didn't find any example that I could use as a starting point. At the moment I am unsure if pgfplots is actually the right package or I should rely on something else e.g. TikZ.   Since I don't really know where to start, I would appreciate any piece of code that points me in the right direction.    EDIT  Here is some dummy data for the plots.  plot1   plot2   plot3   plot4 0       0       0       0 3.466   2.058   0       0 4.262   2.976   0.001   0 3.822   3.168   0.006   0.008 2.953   2.936   0.019   0.063 2.065   2.492   0.046   0.265 1.332   1.977   0.092   0.734 0.797   1.478   0.164   1.508 0.443   1.045   0.268   2.44 0.228   0.698   0.412   3.219 0.107   0.438   0.598   3.524 0.046   0.256   0.831   3.219 0.017   0.138   1.109   2.44 0.006   0.067   1.429   1.508 0.002   0.029   1.78    0.734 0       0.01    2.141   0.265 0       0.003   2.479   0.063 0       0.001   2.736   0.008 0       0       2.808   0 0       0       2.465   0 0       0       0       0 I use the highlight mode in vim to copy a few characters.  I then want to paste more than once.  My current technique does not work well.  Sample text: Linux Solaris Irix HP-UX  Suppose I want to copy the word Linux, then paste over Solaris and Irix.   Place cursor at L in Linux Command v (for visual hilite), then e (for end-of-word), then y (for yank/copy) Now Linux is on my "vim clipboard" Move cursor to S in Solaris (first instance) Command v (for visual hilite), then e (for end-of-word), then p (for paste) Text is now: Linux Linux Irix HP-UX, but now Solaris is on my "vim clipboard" Move cursor to I in Irix (second instance) Command v (for visual hilite), then e (for end-of-word), then p (for paste) Text is now: Linux Linux Solaris HP-UX which is not what I expected.   I resort to using highlite/paste with the mouse (via X Terminal).  Surely, I can do this better.  How? When are "because", "since","for" and "as" interchangeable? (Google) Title of page not updated, everything else did My website changed significantly this month.  Amongst the changes was the website's title.  Two weeks ago, I finally used Webmaster Tools and submitted the page.  So far, Yahoo and Bing have updated the title in the search results, but not Google.  I've resubmitted via Webmaster Tools the website, and the bots have crawled the page, and have added "content keywords".  But the title still doesn't match.  The "content keywords" lists many of the words in the new title (but contains keywords of the existing title also).  Must say that 4 of the top 5 key words in the "significance" list are the new title's.  Still, no change.  Any remedies for this problem?  Is adding the "alt" tag for the logo significant enough to trigger the change?  There isn't an appropriate place to put the company name between  tags on the front page.  Any suggestions?  Thanks. Possible Duplicate:        When I click 'Unanswered' in the menubar, I get a list of questions, all of which have been answered. Does it just mean there's no accepted answer? If so, can we make that more obvious? Can we get some way of finding really unanswered questions? Fundamental Theorem of Calculus. Fundamental Theorem of Calculus says  Let $f$ be an integrable function on $[a,b].$ For $x$ in $[a,b],$  let $F(x)=\int_{a}^{x}f(t)dt.$ Then $F$ is continuous on $[a,b].$ If $f$ is continuous at $x_{0}$ in $(a,b),$ then $F$ is differentiable at $x_{0}$ and $F^{'}(x_{0})=f(x_{0}).$  Now my question is why we are saying that this is "Fundamental Theorem of whole Calculus".  It is clear that it connects intergal and differential calculus. But i am not getting the exact concept, why this is so important as its name is fundamental theorem of calculus. Why we says that it is fundamental theorem of calculus?  Please suggest me. Thanks in advance. I would like to prove the following inequality, for $n=0,1,2,...$,   $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof. How do filament LED bulbs work, looking very similar to incandescent bulbs? Filament LED bulbs use very narrow strips containing the LED, which look similar to filaments of an old incandescent bulb:  They look like a normal bulb with large "old-style" coal filaments, and are of about the same size and shape as the common standart E27 socket bulb.    There are two basic variants of these bulbs:     One "plain" as described, looking impressively similar to a classic bulb The other has a white coating on the glass, about 15mm wide next to the metal socket.   How are these filaments build? I could imagine the it's "pretty normal" LEDs, but lots of them, directly bonded to something? (There are references to COB)  How does the heat management work? Ii have seen speculations about filling the bulb with either helium or "some patented organic gas" - could that be enough for cooling?  How does the driver - which seems to be very small - work?  In the variant with white coating - hiding some components, I assume - is already not that much space, and in the plain variant, there seems to be almost no space for something nontrivial.  Below you see some bulbs of the white coating variant, switched on:   (Images with the permission of this )    Note how the LED filament bulbs are not so much similar to the tungsten filament bulbs recently being phased out - they are more resembling the long outdated carbon filament bulbs: Are objects in PHP passed by value or reference? Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. Git SSH error: "Connect to host: Bad file number" Do iOS app visits count to your consecutive visits I watched this movie when I was a kid and never discovered it again.  It’s about a boy whose father left him after his divorce from the mother, the kid enters a world somehow where he has a big sword he needs to keep searching for precious stones to fill it with. it has certain holes that look like each stone shape! I also remember the kid going to eat with his mythical (dragon-like friend), and he gets stuck in the restaurant because the crazy restaurant lady won’t let him leave without finishing huge amounts of food!  The kid also gets bullied so I’m guessing the the swords journey is to teach the kid how to deal with all of this... How to burn or mount an ISO file? Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)  SO this is for abstract algebra and I am really struggling with this. Here are some of the definitions and theorems I think would apply.  -an integer a divides and integer b if there is an integer q such that b=aq -Let a and b be integers, with a>0. Then there exist unique integers q and r such that b= aq+r and 0&lt;=r  The question says hint: use 2 cases but I'm not really sure what that means. I'm guessing it wants me to look at it when a is congruent to 0 and when it's not? so i tried to do that but just ended up stuck. below is as far as i got with that attempt  Proof: Case 1 There should be some information about letting variables exist in certain number sets here  assume a is not congruent to 0 mod 3, then 3 does not divide a-0, therefore a-0 does not equal 3q for some q in the integers  I was trying to get this to wind down to thus 3 divides a^2-1 but even tho it seems true (I couldn't find a counter example) I couldn't figure out how to prove it with what I have.  Next i tried reversing it thinking I could contradict it or something.  I think this would be case 2? assume a is congruent to 0 mod 3, then 3 divides a-0 and a-0=3q for some q in the integers, then 3q=a since 0 is the additive identity, therefore 3 divides a,  and again stuck and my brain hurts a little..  Can anyone help me figure out how to work this. What is the "correct" way of embedding text into math mode? Lets say I have something like:   \[ Let x = number of cats \]   (This is a contrived example, but pretend we are actually inside a long align environment block...) "Let" and "number of cats" need to be written like normal text, while x= needs to be math type text.  I've seen this:  \[ \mbox{Let } x = \mbox{ number of cats} \]   and this:  \[ \textrm{Let } x = \textrm{ number of cats} \]   They seem to produce equivalent output, but I'm not sure which is the "correct" one. Also, they are not ideal in that the extra space needs to be placed inside the text mode block, otherwise it is ignored as math mode and butts the text right up against the math.  Ideally there would be some command which allowed embedding text into an equation, which correctly inserted leading and trailing space around the block as required without adding space inside the text block.  How does one do that "correctly"?