Is there a proof that the set of real numbers can exactly represent distances?
Why is physical space equivalent to $\mathbb{R}^3$?
Crossfading Video AND Audio with ffmpeg This Super User answer  has got me so far, but now that I have tried many solutions and seen every SO link out there, it's time to ask you all for help.  I am trying to crossfade 2 videos that are 10 seconds each in ts format with audio and video at the same time.   The steps are:   Add complex filter black screen Fade out video 1 after 9 seconds, with 1 second fade duration Fade in video 2 at 9 seconds, with 1 second fade duration Trim black filter output to 19 seconds   This works fine for just video alone, but when I attempt to add audio, I can't get past this error message:     Output pad "default" with type audio of the filter instance "Parsed_asetpts_4" of asetpts not connected to any destination   It appears I'm not connecting the audio output properly. I think I need to alter the [over] lines to include audio. But how do I do this?  `ffmpeg -i vid1.ts -i vid2.ts -f lavfi -i color=black -filter_complex \ "[0:v]format=pix_fmts=yuva420p,\ fade=t=out:st=9:d=1:alpha=1,setpts=PTS-STARTPTS[v0];\ [0:a]\ afade=t=out:st=9:d=1,asetpts=PTS-STARTPTS[a0];\ [1:v]format=pix_fmts=yuva420p,\ fade=t=in:st=0:d=1:alpha=1,setpts=PTS-STARTPTS+9/TB[v1];\ [1:a]\ afade=t=in:st=0:d=1,asetpts=PTS-STARTPTS+9/TB[a1];\ [a0][a1]amix=inputs=2;\  [2:v]scale=720x406,trim=duration=19[over];\ [over][v0]overlay[over1];\ [over1][v1]overlay=format=yuv420[outv]" -vcodec libx264 -y -map [outv] final.ts` I gave Google my phone number to verify it, not knowing that it would change it, and now my phone number is completely different? What have I done, and how do I unverify it and change my phone number back? I'd like to use epigrafica specifically for chapters and sections, but like to use times font for main texts.  \usepackage{epigrafica}          % for chapter/section \usepackage{newtxtext,newtxmath} % for main text   Is there any way to do this? I can only use pdftex. How can I remove these apps from my device? How long does it take for a black hole to form for an external observer? The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon? What is the time span of The Last Jedi? Why didn't all the Jedi at the Jedi Temple sense Anakin and his legion of Clones’ thoughts? If you go back to Attack of the Clones, Anakin and Obi-Wan sensed that poisonous alien worms were about to kill Padme. Back in Revenge of the Sith, the Jedi could take on the legion or if there were too many, they could at least escape. I want to assign a class attribute via a string object - but how?  Example:  class test(object):   pass  a = test() test.value = 5 a.value # -&gt; 5 test.__dict__['value'] # -&gt; 5  # BUT: attr_name = 'next_value'  test.__dict__[attr_name] = 10 # -&gt; 'dictproxy' object does not support item assignment Merging two arrays in .NET Is there a built in function in .NET 2.0 that will take two arrays and merge them into one array?  The arrays are both of the same type. I'm getting these arrays from a widely used function within my code base and can't modify the function to return the data in a different format.  I'm looking to avoid writing my own function to accomplish this if possible. Defining coordinate reference system with rotation in GeoServer? I am using GeoServer and have a layer in EPSG:900913 ("Google Mercator").  I need to "rotate" the map around certain point (say, 1500000, 7000000) by certain degree (say, 30 degrees clockwise). How could I define such a coordinate system based on EPSG:900913?  GeoServer's  does not work for my purposes as I need to tile the map later on.  As far as I understand this, my only option is to define an own coordinate system. For GeoServer I'd need to . The configuration seems to be straightforward, but I have a difficulty defining my rotated CRS in .  I am wondering how to apply a rotation around certain point onto a CRS like Google Mercator:  PROJCS["WGS84 / Google Mercator",    GEOGCS["WGS 84",      DATUM["World Geodetic System 1984",        SPHEROID["WGS 84", 6378137.0, 298.257223563, AUTHORITY["EPSG","7030"]],        AUTHORITY["EPSG","6326"]],      PRIMEM["Greenwich", 0.0, AUTHORITY["EPSG","8901"]],      UNIT["degree", 0.017453292519943295],      AXIS["Longitude", EAST],      AXIS["Latitude", NORTH],      AUTHORITY["EPSG","4326"]],    PROJECTION["Mercator_1SP"],    PARAMETER["semi_minor", 6378137.0],    PARAMETER["latitude_of_origin", 0.0],    PARAMETER["central_meridian", 0.0],    PARAMETER["scale_factor", 1.0],    PARAMETER["false_easting", 0.0],    PARAMETER["false_northing", 0.0],    UNIT["m", 1.0],    AXIS["x", EAST],    AXIS["y", NORTH],    AUTHORITY["EPSG","900913"]]   My questions, specifically:   How to write a WKT which transform an existing CRS? My guess would be that I need a new PROJCS wrapping an existing one and adding a PROJECTION clause. How would I found out the projection id (like Mercator_1SP above) and the required parameters (the PARAMETER clauses)? Can I "reference" EPSG:900913 in CRS WKT instead of copy-pasting the whole PROJCS clause? I can type:  dirname ~/home/blah/file.zip   for instance, and this works fine alone, but when I use this syntax as the parameter for some command it always reads 'dirname' as the desired directory. i.e.:  cd dirname ~/home/blah/file.zip bash: cd: dirname: No such file or directory   Basically, how do I get 'dirname file' to be read as one entity? Meaning of 'const' last in a function declaration of a class? I go to change my birthdate on Roblox, but nothing shows up, the dropdowns are just blank with no Month, Day or Year:  It also shows me a pop up asking to put in the Missing Details to improve the experience, but no matter how many times I put in my age it doesn't save and asks it again. Does python have the ability to create dynamic keywords?  For example:  qset.filter(min_price__usd__range=(min_price, max_price))   I want to be able to change the usd part based on a selected currency. Microsoft.Office.Interop.Excel doesn't work on 64 bit Prove $\sqrt{2} + \sqrt{5}$ is irrational I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. How can I add left aligned text to an equation? I have a document I am trying to copy to learn TeX. Here is what I have encountered: How can I have the "or" in the equation. This is what I have right now:  \[ f = ma; \]  But $a$ is the change in velocity,  \[ f = m \frac{dv}{dt};\] \[ f = m  \frac{d^2y}{dt^2};\]   Edit: Trying out Stefan's answer It isn't any of the words on this  and I believe it started with the letter 'e' (though please offer your word even if it doesn't).  Example sentence: Because he researched what he had heard from his friends, he was able to refute many of their _________________.  Factoid seems to be the closest one, but the word I am looking for only refers to a belief that is false. Pgfplots: How to fill area under a curve in a 3d plot similar to \closedcycle in 2d How can I fill the area under a curve in a 3d plot, similar to the \closedcycle command in ordinary 2d plots? Applying this command does a filling which is not rotated.  Minimal working example:  \documentclass{article}  \usepackage{tikz} \usepackage{pgfplots} \begin{document}   \begin{tikzpicture}      \begin{axis}[     xmin=0,xmax=3,     zmin=0,zmax=2     ]         \addplot3[red,domain=0:1,fill=blue,opacity=0.5,samples y=0] (2,x,x^2) \closedcycle ;      \end{axis}   \end{tikzpicture} \end{document}   In 2D it looks like this  \documentclass{article} \usepackage{tikz} \usepackage{pgfplots} \begin{document}   \begin{tikzpicture}   \begin{axis}[domain=0:1     ]         \addplot[red,domain=0:1,fill=blue,opacity=0.5,samples y=0] {x^2} \closedcycle ;   \end{axis}   \end{tikzpicture} \end{document}   How can I easily apply this to a 3D plot?