Prove that $(\Bbb{R},+)$ is isomorphic to $(\Bbb{C},+)$.   I don't know how to proceed in this question. On doing a customary internet search, I found that this can be proven using Axiom of Choice. Does there not exist an explicit isomorphism?
Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?  I know that there is a bijection between $\mathbb{R}$  and $\mathbb{C}$, and  asks whether they are isomorphic as abelian groups, are they referring to the additive abelian group? If so is there any simple isomorphism I can find? I know nothing about Hamel basis. Thanks.
Script at Startup Rendering an animation the way 3D view looks Is there some way to render exactly whats in the camera view in the 3D view? By that I mean just like a screenshot really, all paths, lamps, and cameras, just like things look in the camera view. The reason I can't just take a screenshot is that I need an animation and it would be too cumbersome to take 1000+ screenshots! Multiple I2C devices with the same address I have multiple devices with the same address, and only one I2C bus. If I don't want to/can't use an address multiplexer, will the following arrangement work? (sorry for the messy schematic)          &ndash; Schematic created using   SW1 and SW2 are microcontroller I/O, and are pulled low when we don't want to talk to the corresponding IC, and set to high impedance when we do want to talk to that IC. Can anyone see any problems with this, apart from with clock stretching? Assuming you choose diodes with a small voltage drop, like a schottky.  Edit:   Also assuming that the VIH and VIL all worked out with the diode drop and the VOH VOL.   The diodes could also go on the data lines instead, not sure if there would be any benefit/difference (maybe some I2C devices do something special when the clock goes low even if it's not their address? It's probably safer to not clock at all i.e. have the diodes on the clock lines. What do the different colors mean in ls? What do the different colours in Ubuntu's ls command mean? For example, when I type the ls command in one of my folders, I get one of the files in light green, the other (which is a folder) in blue with green highlighting.   What do those colours mean, and there is any manual about all the colours? How to use grep on column? I want to apply grep on particular column in column command.  myfile.txt  3,ST,ST01,3,3,856 3,ST,ST02,4,9,0234 6,N1,N101,2,3,ST 6,N1,N102,1,60,Comcast 6,N1,N103,1,2,92   My Command:  column -s, -t &lt; myfile.txt | grep -w "ST"   Here I want to only grep the pattern ST in 2nd column. How to do this ?  Expected Result:  3       ST            ST01              3            3            856 3       ST            ST02              4            9            0234 I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$  I know that $$2^n = \sum\limits_{k=0}^n \binom{n}{k}$$ but that didn't help me I created a simple service layer polygon in network analyst (ArcGIS Desktop 10.2) with no hierarchies set in the street network. Now I don't understand why is this polygon created along some particular streets?   Why isn't is properly spread out? For example, why doesn't the polygon cover the areas/streets I have marked in red? Obviously those marked areas are also within 1 km.  The impedance is 1 km and 2 km respectively.   The streets are also well connected individual line elements as you can see the junctions. How to use find command with wildcard when current directory contains a match? Possible Duplicate:          I'm having some confusion with the find command.  If I type:  find . -name *.xml   or   find ./ -name *.xml   I receive a list of all the xml files in subdirectories of the current directory, UNLESS the current directory contains an xml file, in which case I get:  find: paths must precede expression: foo.xml Usage: find [-H] [-L] [-P] [-Olevel] [-D help|tree|search|stat|rates|opt|exec] [path...] [expression]   What is the correct way to search for files with wildcard when the current directory may contain a match? Guaranteed lifetime of temporary in C++? Sequential Update of Bayesian I am currently reading Murphy's ML: A Probabilistic Perspective. In CH 3 he explains that a batch update of the posterior is equivalent to a sequential update of the posterior, and I am trying to understand this in the context of his example.  Suppose $D_a$ and $D_b$ are two data sets and $\theta$ is the parameter to our model. We are trying to update the posterior $P(\theta \mid D_a, D_b)$. In a sequential update, he states that, $$ (1) \ \ \ \ \ \ \ \  P(\theta \mid D_{a}, D_{b}) \propto P(D_b \mid \theta) P(\theta \mid D_a) $$ However, I am slightly confused as to how he got this mathematically. Conceptually, I understand that he is saying the posterior $P(\theta \mid D_a)$ is now a prior used to update the new posterior, which includes the new data $D_b$, and is multiplying this prior with the likelihood $P(D_b \mid \theta)$. Expanding the last statement out, I have, $$  P(D_b \mid \theta) P(\theta \mid D_a) = P(D_b \mid \theta) P(D_a \mid \theta) P(\theta) $$  but are we allowed to say $P(D_a \mid \theta) P(D_b \mid \theta) = P(D_a, D_b \mid \theta)$ in order to make the connection in (1)? I was coding a physics simulation, and noticed that I was using discrete time. That is, there was an update mechanism advancing the simulation for a fixed amount of time repeatedly, emulating a changing system.  I though that was interesting, and now believe the real world must behave just like my program does. It is actually advancing forward in tiny but discrete time intervals? Explaining to an 8-year-old why nothing can travel faster than light How to set HttpResponse timeout for Android in Java I have created the following function for checking the connection status:  private void checkConnectionStatus() {     HttpClient httpClient = new DefaultHttpClient();      try {       String url = "http://xxx.xxx.xxx.xxx:8000/GaitLink/"                    + strSessionString + "/ConnectionStatus";       Log.d("phobos", "performing get " + url);       HttpGet method = new HttpGet(new URI(url));       HttpResponse response = httpClient.execute(method);        if (response != null) {         String result = getResponse(response.getEntity());         ...   When I shut down the server for testing the execution waits a long time at line  HttpResponse response = httpClient.execute(method);   Does anyone know how to set the timeout in order to avoid waiting too long?  Thanks! An improved \rowstyle that takes the cell contents as an argument It is possible to format the column entries of a tabular in an argument-style fashion using the lrbox environment to   store the column cell in a box (say) \mybox; and supply \mybox as an argument to some other macro.   Here is a short example illustrating this fact*:  \documentclass{article} \usepackage{array}% http://ctan.org/pkg/array \begin{document} \newsavebox{\mybox} \begin{tabular}{c&gt;{\begin{lrbox}{\mybox}}c&lt;{\end{lrbox}\fbox{\usebox{\mybox}}}c}   One &amp; Two &amp; Three \\ \hline   1 &amp; 2 &amp; 3 \\   4 &amp; 5 &amp; 6 \\   7 &amp; 8 &amp; 9 \\ \hline \end{tabular} \end{document}     The above example uses the  to achieve this column-specific formatting. More specifically, each cell in the column is supplied as an argument to \fbox. Does there exist an equivalent automation that would yield    \documentclass{article} \begin{document} \begin{tabular}{ccc}   One &amp; Two &amp; Three \\ \hline   1 &amp; 2 &amp; 3 \\   \fbox{4} &amp; \fbox{5} &amp; \fbox{6} \\   7 &amp; 8 &amp; 9 \\ \hline \end{tabular} \end{document}   without having to specify \fbox for each entry in the row?  As a start, I know that the  provides \rowfont[&lt;alignment&gt;]{&lt;font specification&gt;}. However, this only applies macros &lt;font specification&gt; to each cell in a row and these macros can't take the cell contents as an argument. Moreover, you need to use the tabu environment in order to use this. So that's a little restricting. Also, trying to decipher the  macros for colouring a tabular or array row is just confusing.  The UK TeX FAQ entry on  modifies the row entries via column specifications and therefore requires the addition of "subtle" identifiers like  \begin{tabular}{|$l|^l|^l|}   to initiate $ and propagate ^ whatever row style is defined. Is there a way around this or at least a cleaner alternative that doesn't use columns to format entries in a row? In particular, a solution should provide an lrbox-style flexibility, since the contents of the cell in the to-be-styled row could be anything, even a paragraph (say).  * There are other alternatives, as indicated by the post . Why isn't an infinite, flat, nonexpanding universe filled with a uniform matter distribution a solution to Einstein's equation? How can I create launcher icon for PhpStorm? I have PhpStorm on my PC, to launch it I have to go to phpstorm/bin/ and launch phpstorm.sh, how can I pin this to launcher so that I dont have to go into folders every time? I want to know if I can pass an argument with an alias command.  for example:  alias d="dmesg|grep -iw usb|tail -5"    Now d will print the last 5 lines. If I want to use d to print a different number of lines, I have to make change in the alias command declaration of d again.  Is there any way I can modify the declaration of an alias so that I don't have to retype the declaration to change the number of lines. Like incorporating passing the number of lines as an argument while declaring alias for d? Or is there some other method to solve this? How do I translate strings in Twig templates? I am working on a (my first) Drupal 8 project witch is a multilingual site. Drupal 8 really delivers in terms of native support for content translation.  However, I have a few strings set directly in some if the theme .Twig template files printed out like ex.   {{ 'Read more'|trans }}   I am really stuck on where to handle the translation for strings like these within the Drupal 8 installation.  What steps do I need to take to be able to translate these strings into the language currently selected for on the site? Do I need any additional modules other than the ones included in Drupal 8 core?  Steps I have taken:   Activated the Content Translation &amp; Interface Translation modules Added Twig strings like {{ 'Read more'|trans }} Flushed cache and refreshed pages containing the strings Checked admin/config/regional/translate for the strings with no result $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$ How to reduce fireflies in Cycles? I often use Cycles' built in Glass BSDF shader to achieve realistic refraction effects, but in many cases, it seems that Cycles' glass caustics creates an unfortunately large number of fireflies (pixels that are abnormally bright relative to their surroundings). In some models of mine, even when I render for 3000 or more samples/pixel, my final image is still covered in fireflies. While I can improve this in many cases by increasing the total amount of light available in the scene, this loses a lot of the potency of the refractive effects I'm looking for, especially in scenes with low light levels.  So my question is, how can I reduce the amount of fireflies while using Cycles' caustics without sacrificing the quality of the refractive effects?  Note: I am aware that Cycles' caustics will perform much better once Cycles has full support for bidirectional path-tracing, but it looks unlikely that that will become , unfortunately.