Finite semigroups in Algebraic structure
Semigroups and solutions of equation
What is the purpose of the Assemblies node in Web.Config? When creating a new ASP.NET AJAX Web Application in Visual Studio 2005, the default web.config contains the following section (inside the compilation node):  &lt;assemblies&gt;     &lt;add assembly="System.Web.Extensions, Version=1.0.61025.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35"/&gt; &lt;/assemblies&gt;   Whether or not the assemblies node exists doesn't seem to affect the AJAX functionality.  Update panels, etc work fine regardless of whether or not the reference to System.Web.Extensions is included in Web.Config.  What's the purpose of this section, and can it be safely removed? Summation of Central Binomial Coefficients divided by even powers of $2$ Whilst working out  the following summation emerged: $$\sum_{m=0}^n\frac 1{2^{2m}}\binom {2m}m$$ The is equivalent to  $$\begin{align} \sum_{m=0}^n \frac {(2m-1)!!}{2m!!}&amp;=\frac 12+\frac {1\cdot3}{2\cdot 4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}+\cdots +\frac{1\cdot 3\cdot 5\cdot \cdots \cdot(2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot 2n}\\ &amp;=\frac 12\left(1+\frac 34\left(1+\frac 56\left(1+\cdots \left(1+\frac {2n-1}{2n}\right)\right)\right)\right) \end{align}$$  and terms are the same as coefficients in the expansion of $(1-x)^{-1/2}$.  Once the solution  $$ \frac {n+1}{2^{2n+1}}\binom {2n+2}{n+1}$$ is known, the telescoping sum can be easily derived, i.e.    $$\frac 1{2^{2m}}\binom {2m}m=\frac {m+1}{2^{2(m+1)-1}}\binom {2(m+1)}{m+1}-\frac m{2^{2m-1}}\binom {2m}m$$  However, without knowing this a priori, how would we have approached this problem? How does end-of-year interact with mutual fund prices (if it does)? Should I expect prices of mutual fund shares to go up in anticipation of dividends, down after dividends have paid out ... or just trust that the market has already reacted to that and the price is basically fair throughout the period?  (I've generally been paranoid and tended not to play with my funds around this time of year, but that may be pure superstition.)  Clarifications: USA, open-ended index funds are my immediate interest, but the answers would be most general if we could cover some of the other varieties (and, if folks are so inclined, locations). How do I use jpnevulator to capture and log the serial traffic between an application and hardware serial port? OK, I've been googling for hours so I obviously have not been able to understand the answers to the various questions that have already been asked about this subject.  I am hoping that, by asking the question again in a more specific way, I will be able to get an answer I can understand.  I have some application running in linux that communicates with an external device attached to a serial port.  I want be able to capture and log the data sent in both directions between the application and the device, with timestamps at the beginning of each line in the file.  As a test case, I am using minicom as the application I want to monitor, connected via a null modem cable to another computer also running minicom.  I have already confirmed that, if I type characters on either of the computers, characters appear in the other computer's terminal.  So far, so good.  I then found this question:    In the answer to this question, the program jpnevulator is suggested.  However, when reviewing the man page, I could not figure out the right way to use jpnevulator to get what I want.  Here is what I tried to do:  First, I opened a terminal window and typed the following command:  $jpnevulator --tty=/dev/ttyS0 --pty --pass --read --ascii --timing-print --file=serial.log   I saw the output:     jpnevulator: slave pts device is /dev/pts/18  I then opened another terminal window and typed the following command:  minicom -D/dev/pts/18 -b115200   Minicom opened without complaint.  However, when I typed characters in either terminal (local and remote), nothing appeared in either terminal.  jpnevulator only logged the data written to /dev/pts/18.  My expectation is that jpnevulator :   reads data from /dev/pts/18 and "passes" this data on to /dev/ttyS0 while also writing this data to the specified file. reads data from /dev/ttyS0 and "passes" this data on to /dev/pts/18 while also writing this data to the specified file.   I am aware of the remark in the faq that says "Jpnevulator was never built to sit in between the kernel and your application. I'm sorry."  However the very same faq states in the second paragraph down: "Now with a little bit of luck some good news: A little while ago Eric Shattow suggested to use pseudo-terminal devices to sit in between the kernel and your application."  That is the approach I am trying to take but I am having no success.  What am I missing?  Thanks to all in advance.  Cheers, Allan  p.s. I was successfully able to capture the back and forth traffic using the socat method mentioned in the existing question I referenced but this method did not offer any way of timestamping the traffic.  I was hoping that jpnevulator would have provided this for me. I'm working on moving my current server setup to newer hardware, and migrating from ubuntu karmic koala to lucid lynx. Currently i'm using gw6c (compiled from the gogo6 website, as opposed to the version from the repositories) to get ipv6 access for my systems. On the karmic koala system, i used simple init.d script to get the ipv6 client started   #! /bin/sh /usr/local/gw6c/bin/gw6c -f /usr/local/gw6c/bin/gw6c.conf   I figured since this runs at any runlevel, it should translate to  respawn console none  start on startup  stop on shutdown  script exec /usr/local/gw6c/bin/gw6c -f /usr/local/gw6c/bin/gw6c.conf emit free6_ipv6_started end script   this works fine started from initctrl, but it apparently fails to start when it boots. - its status being stop/waiting. It works fine (and respawns) when started otherwise.Any ideas on where i'm going wrong, and what would be the appropriate 'start on' arguement?  EDIT: the exact error is 'init: gw6c main process (xxx) ended with status 8' followed by the process respawning , with xxx being a PID i suspect. I'm also half suspecting this is cause gw6c starts before networking does, and i need my eth0 up before gw6c is How do I get a file extension in PHP? This is a question you can read everywhere on the web with various answers:  $ext = end(explode('.', $filename)); $ext = substr(strrchr($filename, '.'), 1); $ext = substr($filename, strrpos($filename, '.') + 1); $ext = preg_replace('/^.*\.([^.]+)$/D', '$1', $filename);  $exts = split("[/\\.]", $filename); $n    = count($exts)-1; $ext  = $exts[$n];   etc.  However, there is always "the best way" and it should be on Stack Overflow. Review sections to be always in the same place How to render a background image in 2.8? I'm trying to "add" an 3D object onto a photo. I have added the photo to the scene in three ways so far, but none of them render (eevee):   Camera > Background Images Empty a. Add > Empty > Image (Empty gets added to current Collection) b. Add > Image > Backround (Empty gets added to Scene Collection)   These all show up in the 3D View and I was able to align the camera and the object so its perspective matches the scene in the photograph. But when I Render the scene, I only see the object, not the background image.  For the latter, it's simple: Empties are not rendered at all. There are few questions about this, since 2.8 has changed how it handles background images. The most popular seems this one (it includes references to others):      But I seem not to be able to find a method to have the background image render.  Another method might be to use the Compositor (Nodes) to Mix or Alpha Over an Image with the Render Layers. But this does not give the same alignment as achieved with the background image (also this requires that the World background does not get rendered (Alpha channel); for Eevee, this can be achieved by checking Film>Transparent).  What other options are there?  Docs: Is there any way to keep the URLs in the same form they are saved within the .bib files when using biblatex-chicago with Biber? For instance, the entry  @misc{RussianEmpire.1855,  author = {Russian Empire},  year = {1855},  title = {Treaty of Shimoda},  url = {https://ru.wikisource.org/wiki/Симодский_договор_с_Японией_(1855)}, }   results in the following output:    While there certain are benefits to converting the link itself, this sort of output in the footnote makes it both ugly and difficult to re-enter the link when reading a printed copy. How would one proceed about to receive the output along the lines of ""?  Minimal working example:  %!TEX program = xelatex \documentclass{article}  \usepackage{fontspec} \usepackage[russian,british]{babel} \newfontfamily\russianfont[Script=Cyrillic]{CMU Serif}  \usepackage[CJK, Cyrillics]{ucharclasses} \setTransitionsForCyrillics{\russianfont}{} \usepackage{csquotes} \usepackage[labeldate,backend=biber]{biblatex-chicago} \addbibresource{bibliography.bib} \usepackage[unicode,hidelinks]{hyperref}  \begin{filecontents}{bibliography.bib} @misc{RussianEmpire.1855,  author = {Russian Empire},  year = {1855},  title = {Treaty of Shimoda},  url = {https://ru.wikisource.org/wiki/Симодский_договор_с_Японией_(1855)}, } \end{filecontents}  \begin{document}  Cyrillics support test: Тест\\  \cite{RussianEmpire.1855}  \end{document}   Output: Bending of light - photon's inertia instead of mass Using classical mechanics, the formula for gravitational attraction is   $$F = G\frac{m_1m_2}{r^2}.$$   This formula does not work for photons, and we need to use Einstein's theory of gravity to account for that, as the photon is massless. What if, instead of using the mass of the photon, we rather used the inertia of it?  The momentum of a photon is     $$p=\frac{h}{\lambda}.$$  Thus, using that, we can calculate its supposed 'mass' (inertia) as momentum is also equal to mass times velocity. So 'mass' of a photon is     $$m=\frac{p}{v}=\frac{h}{\lambda  c}.$$   If we use this mass as the mass of a photon, can we use Newton's equation for gravitational attraction? I'm trying to display opportunities associated with an OpportunityLineItem on a Visualforce page on the OpportunityLineItem object   I have a problem in the select I do not know how to retrieve the OpportunityLineItem name in this request I have hard code the OpportunityLineItem name  public List&lt;OpportunityLineItem&gt; getOpp() {     return [SELECT TotalPrice, Opportunity.Name, Opportunity.StageName,                     Opportunity.Account.Name, Opportunity.id, Product2.Name              FROM OpportunityLineItem               WHERE Product2.Name='GenWatt Diesel 10kW'     ]; } Why no ooVoo on ubuntu? Is it possible to have a monoid with a proper class as the collection of elements? My question if it possible for us to have, for example, a monoid(any algebraic structure really) with a class as the collection of the elements, instead of a set.  An example would be the class of sets and set union as addition. Clearly, it is closed under the operation, is associative and it has an identity element, so it is a monoid. Is it possible to say so? I know it is defined as $ dP \wedge dx  $ or explicitly,  $$ \frac{\partial P }{\partial y } dy \wedge dx   . $$  The question is, could it be $dx \wedge d P $?  Or  $$  \frac{\partial P }{\partial y }  dx \wedge dy ?    $$  I know the question must be very naive or even stupid. But I am indeed confused. Any way to add CSS for a single page/node? I'm cleaning up my big crazy style sheets (possibly pertinent to a future question) and I'm wondering the best way to add custom CSS to a specific node or page. In particular, my work site's home page is a Panel Page and it has a bunch of different styling. Right now the CSS is just included with the main theme style sheet. Is there a way to say, &quot;if this is node Foo, then add foo.css&quot;? Is  what I'm looking for? I might be interested in generalizing this to other nodes/sections/etc, but at the moment I just want to handle this one item. how to have a custom zsh auto_cd command Instead of the normal cd I've created my own function  function cd {     builtin cd "$@" &amp;&amp; ls -F }   This works great for when I use commands like cd someDir/subDir but not when I change directories via zsh's auto_cd someDir/subDir.  Is there a way to customize what command gets called for auto_cd? What is included in Wolfram Extras? Using "of" or apostrophe, which is more common? Choosing research ideas to include in a statement of purpose If I could stop ALL motion would I stop time? I've never really understood time. Time is not a force. It isn't energy or matter. Time doesn't MAKE anything do anything. Time doesn't make a clock tick (motion or batteries do). Time doesn't make the earth revolve around the sun (motion/gravity does). I don't think time exists. People say time is used as a form of measurement but aren't we just measuring movement? Even atomic clocks measure the movement of electrons that orbit an atom's nucleus as they "jump" back and forth between energy states. So my question is if I were to stop ALL motion (even movement of electrons that orbit an atom's nucleus) would I stop time? Is time just masquerading as the measurement of motion?  EDIT: This question is not a duplicate of "Does time freeze at Absolute Zero?" Because I am NOT talking about temperature. I am speaking about the motion of ALL things. I am asking if ALL things (photons, energy, matter, forces) stop moving would time stand still? I would have to say yes it would. If ALL motion were to stop for a million years and then start again no one would even know. Atomic clocks wouldn't even miss a step.