I understand light consists of photons. Likewise it is a wave with varying intensity in the electric and perpendicular magnetic fields. Say there is a candle lit on the moon. Why wouldn't we be able to see it from Earth? Since light is quantized, would an equal distance in space correspond to a proportional loss in intensity? If so, what are the proportionality constants in the vacuum of space and in 'regular' air?
It stands to reason: if it didn't, the entire sky would be covered with stars shining blindingly day and night.  But what causes a light wave (or an electromagnetic wave) to fade if there are no celestial bodies in its way? Or, alternatively, why doesn't it fade sooner? Some stars are visible to the naked eye that are hundreds of light years away. Others, closer, can only be detected with the aid of a telescope. Why?
How to Disavow Bad Backlinks Automatically? What is the difference between sourcing ('.' or 'source') and executing a file in bash? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. How to fix java.lang.UnsupportedClassVersionError: Unsupported major.minor version Distribution of the maximum of two correlated normal variables How to initialize all members of an array to the same value? I have a large array in C (not C++ if that makes a difference). I want to initialize all members of the same value.  I could swear I once knew a simple way to do this. I could use memset() in my case, but isn't there a way to do this that is built right into the C syntax? Why can't I import a file in the init.m file? How do I change the email address associated with my Stack Exchange OpenID? I registered my Stack Overflow account using the Stack Exchange OpenID provider. I later changed my e-mail, but I cannot change the e-mail address associated with my OpenID for some reason. Why is that, and how do I change it? I logged in to Stack Exchange and it lets me change my password, but never my e-mail. Proof for two sequences producing the maximum value when sorted I am trying to improve my discrete Math skills by doing some proof exercises. But I am struggling to understand how to start proving the following hypothesis:  $$ S = {x}_1{y}_1 + {x}_2{y}_2 + \ldots + {x}_n{y}_n $$ Where ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are ordering of two different sequences of positive real numbers, each containing n elements. S takes its maximum value when ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are sorted in non decreasing order.  I was thinking that I should maybe use a proof by contradiction. Where the negation of the hypothesis is: S takes it minimum value over all ordering of the two sequences when both sequences are sorted into non increasing order.  However I am not sure my negation of the hypothesis is correct or how proceed from there. I do not want the complete proof because I am trying to learn how to do it myself but if some one can tell me how to make a start on this I would be grateful. Cannot install Ubuntu in VirtualBox due to "this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot" error I've had an issue for a while now which makes my Alt + Tab not work and function correctly.  I can't:   Click on it It displays items which are supposed to be hidden (?) It doesn't use the Aero theme   I've tried the following:   Rebooting Closing dwm.exe Updating Windows.   If there is a need for OS: I'm running Win 8.1 x64.  Example: Using Python, how do I to read/write data in memory like I would with a file? Group Membership Test for Permutations What could cause touch to fail with this error message?  touch: cannot touch `foo': No such file or directory   Note that an error due to incorrect permissions looks different:  touch: cannot touch `foo': Permission denied What is the difference between can and could in 'Can/could you please explain this to me?' There is no set of all sets.. So there are a few posts about this already, but they skip over the problem I have.  The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{x\in A : x\not\in f(x) \}$ by its definition cannot lie in the image of $f$, precluding it from being a bijection - hence power sets have strictly greater cardinality.  But now, I thought one (obvious) way to characterise the "set of all sets" would be $S=\mathcal{P}(S)$, as indeed, it contains all possible sets (perhaps here is the problem, but I don't immediately see how using this as a definition for the set of all sets is bad). Now consider simply taking the identity function $i:S\rightarrow S$. Then $B=\{x\in S : x\not\in x \}$ - the set of all sets which do not contain themselves. But this doesn't exist. It's exactly the set of Russell's paradox. If this isn't a set, and so is not in $S$, then the proof no longer offers an obstruction to $i$ being surjective.    Each other post I've seen on the matter uses Cantor's theorem to show such a set doesn't exist. My query is to find that something else that must be happening, since I think the proof has issues for this particular set. I'm expecting other issues to arise with the definition $S=\mathcal{P}(S)$... Can we interchange phrases " What's the date today?" and "what day is it today?" What is the Euler Totient of Zero? Wolfram MathWorld defines the Euler Totient function as follows:  The totient function phi(n), also called Euler's totient function, is defined     as the number of positive integers &lt;=n that are relatively prime to (i.e., do  not contain any factor in common with) n, where 1 is counted as being  relatively prime to all numbers   By this definition would $\phi(0)=0$? I'm writing my thesis using MiKTeX and WinEdt 9 for Windows. I write the different chapters individually, and then I have a "masterfille" with the documentclass "report" where I use the command "input" to load the individual chapters. In other words there is only a preamble in the "masterfile".  I have a very general problem which is the following: No matter what kind of mistake I do whatsoever, I will first receive the correct error message, say I forgot to end a table or something (it really does not matter). If I correct this mistake and try to recompile without using the option "Erase Output Files" in Winedt, I will always and very quickly receive a new error message, often containing a mysterious "Undefined control sequence". Having to do this every time and to recompile plus run the Biber (L+B+L+L+D) is very time consuming, since the document is very large. Any ideas what the problem could be? Here is the preamble of my "masterfile" (I am probably including irrelevant information, sorry for that):   \documentclass[12]{report} \setcounter{tocdepth}{2} \usepackage{layout} \setlength{\voffset}{-0.75in} \setlength{\headsep}{5pt} \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{csquotes} \usepackage{authblk} \usepackage [T1]{fontenc} \usepackage[backend=biber, style=authoryear-comp]{biblatex} \usepackage{xyling} \usepackage[super]{nth} \usepackage{xspace} \usepackage{cgloss4e} \usepackage{abstract} \usepackage{array} \usepackage{multirow} \usepackage{hyperref} \usepackage{xr} \usepackage{gb4e} \noautomath  \let\eachwordone\itshape   \makeatletter \pretocmd{\@footnotetext}{     \@noftnotefalse\setcounter{fnx}{0}%     \renewcommand{\thexnumi}{\roman{xnumi}}     }{}{} \apptocmd{\@footnotetext}{     \@noftnotetrue     \renewcommand{\thexnumi}{\arabic{xnumi}}     }{}{} \@ifpackageloaded{bidi}{% \pretocmd{\@LTRfootnotetext}{     \@noftnotefalse\setcounter{fnx}{0}%     \renewcommand{\thexnumi}{\roman{xnumi}}     }{}{} \apptocmd{\@LTRfootnotetext}{     \@noftnotetrue     \renewcommand{\thexnumi}{\arabic{xnumi}}     }{}{} } \makeatother     \addbibresource{ref.bib}    \title{The origins of Old Romance V2: A comparative study of Late Latin, Old French and Middle High German} \author{XXX} \date{\today}   \begin{document}  \tableofcontents   \chapter{Old French} \input{chapters/OldFrench2}  \printbibliography  \end{document} I have scripts that expect specific drive letters for hard drives. Windows sometimes assigns the different drive letters (e.g. I expected my Maxtor drive to be E:, but it's now F:, therefore my scripts will fail), depending on the order of how the drives are plugged into my laptop.  In Windows 7, how do I assign a permanent drive letter to a drive? Finishing a proof: $f$ is injective if and only if it has a left inverse I've already done a lot of searching (in particular: ) to try to prove this statement:      $f: A \to B$ is injective if and only if it has a left inverse. [I'm going to also assume $A$ and $B$ are nonempty.]   My attempt so far:     $\Leftarrow$ Suppose $f$ has a left inverse and $f(a_1) = f(a_2)$ for    $a_1, a_2 \in A$. Then there is a function $g: B \to A$ such that   $g\left(f(a_1)\right) = a_1$ and $g\left(f(a_2)\right) = a_2$. Since   $g$ is well-defined, it follows that $g\left(f(a_1)\right) =  g\left(f(a_2)\right)$, and hence $a_1 = a_2$.         $\Rightarrow$Now   suppose $f$ is injective. Then $f(a_1) = f(a_2) \implies a_1 = a_2$.   Furthermore, since $A$ is nonempty, there is an element $x \in A$.   Define $g: B \to A$ by  \begin{equation*} g(b) = \begin{cases}  f^{-1}(b), &amp; b \in \text{Im}(f) \\ x, &amp; b \in A \setminus  \text{Im}(f)\text{.} \end{cases} \end{equation*} This mapping is   well-defined, since if $f$ is injective, $f^{-1}(b)$ is unique.   My main question: does this imply that if $f: A \to B$ is injective that any mapping $g: \text{Im}(f) \to A$ is bijective? Otherwise, I don't understand why we can even use $f^{-1}$ on an element. [Please read the link above for more details - in proof 1.]