Is it possible to slow down the speed of light  in a vaccum?
First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity?
Why can array constants only be used in initializers? Validity of work-energy theorem in presence of non-conservative forces? I had this "almost bonus" question on the final in Group Theory recently: prove that there is no such group $G$ which would satisfy Aut$(G)\cong \mathbb{Z}_n$, where $n$ is an odd integer. I don't have much certainty if this proof is OK, and one's opinion would be appreciated.  Here's my attempt:     Suppose that there exists such a group $G$ that satisfies the above condition. Since $\mathbb{Z}_n$ is cyclic, Aut$(G)$ is also cyclic and $G$ is abelian, which implies that Inn$(G)\cong \{e\}$. Thus Aut$(G)=$Out$(G)$. (I don't think this fact is important here though). Now, since an automorphism sends a generator to a generator, and since each automorphism is completely determined by such mapping, |Aut($G$)| must be of factorial order. But an integer factorial is always even. However, $|\mathbb{Z}_n|$ is odd, which is a contradiction. Therefore, no such $G$ exists.   I'm afraid that my proof is not very nice, but at least a genuine attempt was made. This is supposed to be related to the  (my justification for the independence tag). In Williams' Probability with Martingales, 2BCL is proven and then the following is given as an exercise:  Prove $S \doteq \sum_{n=1}^\infty p_n &lt; \infty \implies \prod_{n=1}^\infty (1-p_n) &gt; 0$ assuming $0 \leq p_n &lt; 1$. Hint: Show that $S &lt; 1 \implies \prod_{n=1}^\infty (1-p_n) \geq 1 - S$.   Proving the hint: I tried to find $\{a_n\}$ s.t. $$\prod_{n=1}^\infty (1-p_n) \geq \prod_{n=1}^\infty e^{a_n} = e^{\sum_{n=1}^\infty a_n} \geq 1 - \sum_{n=1}^\infty a_n \geq 1 - S &gt; 0$$ That is, find $\{a_n\}$ s.t.: 1a. $1 - p_n \geq e^{a_n}$ 1b. $a_n \leq p_n$ I thought of $a_n = \ln(1-p_n)$ (a reason why $p_n &lt; 1$, I guess) Is that right? Edit: Actually, assuming my proof is right, is $S &lt; 1$ used in the proof? If so, where?  From Williams book: One of the new search hints is wrong How to control a motor with only relays and push button? Hello I´ve been working in this project for some days, I need to make a circuit with only relays that can make a dc motor start, stop and a reverse. The thing is that when you push the start or the reverse button the motor need to keep running, I can´t keep it pushed.  This is as far as I can get. SW1 turns the engine on, SW5 turns it off and SW7 reverse the motor but only if it is already running. What i need is that the reverse button also turns the engine on. Please help.  notes: -i can only use three push buttons and as many relays as needed. Yes is a school assignment.  THANKS! Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$? Near Earth vs Newtonian gravitational potential Newton's Law of Universal Gravitation tells us that the potential energy of object in a gravitational field is $$U ~=~ -\frac{GMm}{r}.\tag{1}$$  The experimentally verified near-Earth gravitational potential is  $$U ~=~ mgh.\tag{2}$$  The near-Earth potential should be an approximation for the general potential energy when $r\approx r_{\text{Earth}}$, but the problem I'm having is that they scale differently with distance.  $(1)$ scales as $\frac 1r$.  So the greater the distance from the Earth, the less potential energy an object should have.  But $(2)$ scales proportionally to distance.  So the greater the distance from the Earth, the more potential energy an object should have.  How is this reconcilable? How to reinstall default packages In an attempt to uninstall wine I ran the command sudo apt-get purge wine* which as you may have guessed removed many of the default packages. I am wondering if there is a way to use the installation disk to reinstall the default packages without overwriting the file system. Not sure if it makes a difference or not, but I am duel booting Windows 8 and Ubuntu 14.04 which was updated from Ubuntu 12.04.  Thanks in advance. Remove numbering from Section and Subsection I would like to scale an image file for each dimension differently for example 50% of the x dimension or 25% of the y dimension. Thank you very much. Regular cast vs. static_cast vs. dynamic_cast Currently I have a script.sh file with the following content:  #!/bin/bash wget -q http://exemple.com/page1.php; wget -q http://exemple.com/page2.php; wget -q http://exemple.com/page3.php;   I want to execute the commands one by one, when the previous finishes. Am I doing it in the right way? I've never worked with Linux before and tried to search for it but found no solutions. What is the difference between a "link function" and a "canonical link function" for GLM What's the difference between terms 'link function' and 'canonical link function'? Also, are there any (theoretical) advantages of using one over the other?  For example, a binary response variable can be modeled using many link functions such as logit, probit, etc. But, logit here is considered the "canonical" link function. A couple years ago I ran upon a  video demonstrating how researchers used x-rays given off by tearing tape off its spindle in hopes to miniaturized and cheapen future x-ray devices. As of late I have been wondering what interactions are causing this. I understood from the video that the electrons are being ejected due to the added energy to the system (peeling). This instinctively pointed me towards QED where interactions between electrons produce a photon for short periods. 1)Are those same photons from electron interaction the resulting x-rays? 2)Are the frequencies of the photons being limited by the $\Delta$energy$\Delta$time uncertainty, and/or $\Delta$position$\Delta$momentum uncertainty (at the vertex of the ripping tape, where high densities of electrons may accumulate) ? Note: I've also considered that the van der Waals force may come into consideration, not certain though. Looking for a QED based descriptions/answers. Any elementary references to this work is appreciated. I'm trying to remember the name of a series of books about a boy who works with dragons.  Dragon's blood (or maybe it was slobber?) is acidic, and the children who wrangle these dragons often end up with deep pock marks.  On this world, days are like ours but nights get very cold, requiring anyone stuck outside to find a shelter until dawn (not the false dawn)  The other thing I remember is that in one of the books by slicing open a dragon and spending the night, the character emerged with special dragon powers. Cannot see OpenLayers under Plugin in QGIS? Although I have installed the OpenLayers plugin in QGIS 2.4, I cannot see it in the list when I click on Plugins.    Does anyone else have this issue? How to render a 30K image with cycles? I have to render an image of 27000 x 7000 for a 150dpi print. The scene is quite big, but cycles preview and a render in 2700x7000 works as expected.  If I hit render my memory raises up to 20gb, but my cpu usage is at 5%. Why cycles does not using the full cpu power during the render process?  Here are my rendersettings:  Pathtracing Samples:3000 Tilesize: 64x64 Bounces: Max 12 | Min 3 Diffuse: 4 Glossy: 4 Transmission: 12 Volume: 1   Is there a way to optimize things? Comparing two integers without any comparison Is it possible to find the greatest of two integers without any comparison? I found some  solutions:   if(!(a/b)) // if a is less than b then division result will be zero. {     cout &lt;&lt; " b is greater than a"; } else if (!(a-b)) // we know a is greater than or equal to b now.  check whether they are equal. {     cout &lt;&lt; "a and b are equal"; } else     cout &lt;&lt; "a is greater than b";   But if(c) or if(!c) is a comparison to zero. In addition it doesn't work for negative numbers. In fact I need a solution that avoids any if statement. Instead I should use switch statements and arithmetic operators. ThanX. Labelling figures in the tikz package Consider the following piece of code, which uses the tikz-feynman package:  \usepackage{tikz-feynman, contour} \tikzfeynmanset{compat=1.1.0}  \begin{document} \begin{center}     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (c) -- [fermion] (b)                 -- [fermion] (d)                 -- [fermion] (a)                 -- (c),         };     \end{feynman}     \end{tikzpicture}     %     \feynmandiagram[baseline={(current bounding box.center)}, vertical'=a to b,] {         {   [edges={fermion}]             a -- b [label=0:\(\mu_{3}\)]             -- c [label=-180:\(\mu_{4}\)]             -- d [label=-180:\(\mu_{1}\)]             -- a [label=0:\(\mu_{2}\)],         },         {   [edges={photon}]             a -- ap [particle=\(p_{2}\)],             b -- bp [particle=\(p_{3}\)],             d -- dp [particle=\(p_{1}\)],             c -- cp [particle=\(p_{4}\)],         },     };     %     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (a) -- [anti fermion] (b)                 -- [anti fermion] (d)                 -- [anti fermion] (c)                 -- (a),         };     \end{feynman}     \end{tikzpicture}     \end{center} \end{document}   This piece of code draws the following figure:    I would like to label each figure. I do not want a caption, but just a label beneath each diagram. The label should say Diagram I, Diagram II, Diagram III.  How do I implement this into my code?