Simulating a Brownian Excursion Process using Software
Simulating a Brownian Excursion using a Brownian Bridge?
Arc length parameterization lying on a sphere Add desktop notifications for Chrome or Firefox For example, the radiation dominated cosmology, the energy density of radiation is proportional to $a^{-4}$ and the volume is proportional to $a^3$, where $a$ is the scale factor. So the total energy of radiation is propotional to $a^{-1}$. So where is the loss of energy of radiation? Is it because the gravitational field has the energy? Does $\nabla_aT^{ab}_{\rm matter}=0$ represent the conservation of energy and momentum of matter field in GR? Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction I recently proved that  $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$  using mathematical induction.  I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs. Does the speed of light vary in non-inertial frames? The speed of light is the same in all inertial frames.  Does it change from a non-inertial frame to another? Can it be zero?  If it is not constant in non-inertial frames, is it still bounded from above? You are here, but where's here? I find the .youarehere a styling for the main navigation links causes the text to be a little difficult to read on the also-blue header background. Could we give it a little bit of background colour or something else that would provide more contrast?  For example:    which is the product of the following CSS*:  .nav ul li {     margin: -2px 10px 0 0;     padding: 5px; }  .nav ul li.youarehere {     background-color: rgba(0, 80, 112, 0.25);     border-radius: 5px; }  .nav ul li.youarehere a {     color: #A1E9FF; }   *For bonus points, also reduce the line height for the codeblocks! Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd I read somewhere that     $(a^n - b^n)$         It is always divisible by $a-b$.   When $n$ is even it is also divisible by $a+b$.   When $n$ is odd it is not divisible by $a+b$.      and     $(a^n + b^n)$         It is never divisible by $a-b$.   When $n$ is odd it is divisible by $a+b$.   When $n$ is even it is not divisible by $a+b$.      I wonder what's the proof for this.  First postulate is clear. $(a-b)$ would or would not be a factor. Any light on others? Equation mode, labelling equations I have the following   \[K^{-1}\left\|\sum\limits_{n=1}^N a_ny_n\right\|    \leq \left\|\sum\limits_{n=1}^N a_nx_n\right\|    \leq K\left\|\sum\limits_{n=1}^N a_ny_n\right\| (\dagger)\]   But the dagger doesn't appear at the right hand end of the page. How can I Get it so that it does appear on the end of the page? the \hfill command doesn't seem to work. Thank you. Can wizards create their own spells? In the Harry Potter series, are wizards capable of creating their own spells, or are they limited to the spells that they're taught/learn?  I seem to remember something about Snape creating sectumsempra, but I'm not sure if that really is the case. How will multiple GPUs improve Cycles' rendering performance? I have an NVIDIA Titan GPU and I'm able to add another card, but I wonder whether  Blender will benefit from more than one GPUs?  Should I do it as SLI or should I just do it as an added card? I just need some help in understanding how Blender will use either SLI or a second GPU.  Note that I only use Cycles. Rest mass of the photon Why is the rest mass of the photon is taken to be zero?  If rest mass of the photon is non zero, then it leads to infinite energy of the photon (by $E=mc^2$ equation) and there is a violation of conservation of energy. Is this a reason?  Also, is there any thing called the relativistic mass of the photon? I was using , implemented in scikit-learn, for my research work. But I have no good understanding of this.   Can anyone please give a simple, good explanation of one class SVM? This may be a simple question, but how do I use files starting with "-" with command line programs?   For example,  I am trying to use pdfgrep on a file named -2013-01-01.pdf but it complains that there are no options defined for 2, 0, 1, 3 and so on.  Changing the filename to something that does not start with "-" solves it, but this is not an option since the files aren't created by me. I simply want to check for a specific change I know is coming. For a simple circuit with a battery supplying a voltage $V$ to a capacitor, let us assume that the charge on the capacitor is $Q$. Now, the work done by the battery or the energy supplied is given by the relation:  $$W=QV$$  But the energy stored in the capacitor is given by:  $$U = \tfrac12 QV$$  The value of $Q$ as well as that of $V$ should be the same in both the equations.  Now my question is, where is the other half of the energy that the battery supplied? How to use "fit" to frame the nodes and labels Typically Apt packages are served over bare HTTP and the downloaded contents are verified with GPG signatures. This makes it easy for  to proxy the HTTP connections and add the downloaded packages to its cache. apt-cacher-ng is unable to directly MITM an HTTPS connection created by Apt. It is possible to  apt-cacher-ng, but then the contents are not cached--they are re-downloaded from the remote server every time. This quickly gets annoying when rebuilding Docker containers that install many packages from Apt repositories served over HTTPS. Is there a way to get apt-cacher-ng to cache packages downloaded from HTTPS repositories? I'm reading Nano: The Essentials by T. Pradeep and I came upon this statement in the section explaining the basics of scanning electron microscopy.     However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. At such velocities, one needs to do relativistic correction to the mass so that it becomes[...]   We all know about the famous theory of relativity, but I couldn't quite grasp the "why" of its concepts yet. This might shed new light on what I already know about time slowing down for me if I move faster.  Why does the (relativistic) mass of an object increase when its speed approaches that of light? I've noticed that the majority of Ubuntu help websites present apt commands in the following format:  sudo add-apt-repository ppa:[ppa-address] sudo apt-get update sudo apt-get install [application-name]   But I know that apt commands like this one work too:  sudo add-apt-repository ppa:[ppa-address] &amp;&amp; sudo apt-get update &amp;&amp; sudo apt-get install [application-name]   Is there a particular reason that most sites present in the first format? Are there known issues with chaining commands with &amp;&amp;, or to put it another way are there any benefits/drawbacks to using either method?  Further, other then loss of internet, are there any circumstances in which an apt command might fail? I am aware    but I ask this regarding SE-2.0 sites, as the example I will take gaming:  Both answers on the question  make perfect sense to me, i.e. on the one hand this might overuse the tagging system but on the other hand it would help a lot for filtering.  Therefore I suggest a system which, in this case, would internally store the platforms to which a tag is relevant and   if a question is tagged with one of these platforms, nothing special will happen and the question is to be considered platform specific (e.g. a bug which only appears on the PS3 version) if the question is not tagged with a platform, the only result will be that the tag filter mechanism works a bit different compared to what would happen if the question was (invisibly) tagged with all platform tags:   If the invisible tagging is definite on interesting or ignore (i.e. there is not at least one ignored and one interesting tag) the usual mechanism works. if the tagging is ambiguous (i.e. contains both interesting and ignored tags), no filtering happens at all    The actually attributed tags must of course override this behaviour. That way questions about a game one likes but which is available for a platform one doesn't care about will still be highlighted unless specifically tagged with that platform, but no additional tags have to be used to filter a game which is only available on ignored platforms.  This can of course easily be extended to other sites, e.g. the  may be interesting both general and for c++ but totally uninteresting if the question is only about a JavaScript implementation.  I am only unsure whether search should include those hidden tags, too. What is the process for a moderator to request reinstatement or appeal a removal?