It is known that electromagnetic waves with a high frequency possess a greater amount of energy than waves with lower frequencies. Why is this the case? Does it have anything to do with Planck's law?
This is probably a dumb question. I guess what I'm trying to ask is if radio waves travel the same speed as gamma rays, how do gamma photons carry more energy than radio photons? Do they spin faster? What other energy sources could they carry if they are moving the same speed through space?
POST JSON data denormalization error Prove that $a^{2^n}=1 \mod 2^{n+2}$ I would like to prove that  $$a^{2^n}\equiv 1 \pmod {2^{n+2}}$$  I tried induction but could not get it.  Thank you very much! What are the shell's control and redirection operators? I often see tutorials online that connect various commands with different symbols. For example:  command1 |  command2 command1 &amp;  command2 command1 || command2     command1 &amp;&amp; command2   Others seem to be connecting commands to files:  command1  &gt; file1 command1  &gt;&gt; file1   What are these things? What are they called? What do they do? Are there more of them?    . nodejs vs node on ubuntu 12.04 That's basically the question, is there a "right" way to implement operator&lt;&lt; ? Reading  I can see that something like:  friend bool operator&lt;&lt;(obj const&amp; lhs, obj const&amp; rhs);   is preferred to something like   ostream&amp; operator&lt;&lt;(obj const&amp; rhs);   But I can't quite see why should I use one or the other.  My personal case is:  friend ostream &amp; operator&lt;&lt;(ostream &amp;os, const Paragraph&amp; p) {     return os &lt;&lt; p.to_str(); }   But I could probably do:  ostream &amp; operator&lt;&lt;(ostream &amp;os) {     return os &lt;&lt; paragraph; }   What rationale should I base this decision on?  Note:   Paragraph::to_str = (return paragraph)    where paragraph's a string. In a  of Amstat News, the authors (Mark van der Laan and Sherri Rose) stated that "We know that for large enough sample sizes, every study—including ones in which the null hypothesis of no effect is true — will declare a statistically significant effect.".  Well, I for one didn't know that. Is this true? Does it mean that hypothesis testing is worthless for large data sets? Need clarification on the Draw 4 rule. - Player 1 lays down a Draw 4 Card and says the color is Red. - Player 2 does not have a Red card but has a Draw 4 Card. Can Player 2 lay down his Draw 4 Card and have Player 3 draw 8 cards? I've been modeling an F-15, but when I rotate the flaps on Y-Axis the Flaps offset from the wing and they look like they are about to fall out, so I end up correcting it by rotating in the X and Z-axis to align them correctly.  However when I rotate them back from the Y-Axis they look distorted again because I did rotations on the X and Z axis. I tried riding it, no help. I saw something about custom orientation to fix it but that was not clearly explained and it was done on an ideal plane that was (0,0,0); mine is pre-rotated on all axis.   I'll take scripts if needed and I'm using Blender 2.8.  When I rotate it on the Y-Axis:    Fixing it using X and Z rotation:    Rotating it back on the Y distorted it again:    Here's the wing with ailerons and flaps: As the title says. I use, among other things, the following in my lstset:  framerule=1pt, frame=tblr   Some places a page break happens in the middle of my \lstlisting. Fine. But sometimes the whole code of the listing appears on one page, but the bottom frame appears on the top of the next page!    I know that I can float a particular listing, but I have a long document with many listings, and I don't want to float all of them; Most of them also need to be exactly where they are declared. That they are broken over a page doesn't bother me that much.  Is there something that can stop this from happening, without me needing to adjust each individual listing?   had the same problem, but fixed it by removing spaces in front of keywords in a custom language definition. I too have a custom language definition, but his solution doesn't help me. I managed to recreate the situation without a custom language.  Edit and MWE  I managed to reproduce it in a minimal working example. 's a pastebin, if that's helpfull.  \documentclass[UKenglish,a4paper,11pt]{report} \usepackage{babel}  %% Adjustments \addtolength{\topmargin}{-1\baselineskip} \addtolength{\textheight}{4\baselineskip} \pretolerance = 2000 \tolerance = 5000   \hbadness = \tolerance  %% Smaller font \usepackage{mathptmx}  %% LISTINGS \usepackage[final]{listings} \lstset{     xleftmargin=1.5em,     framexleftmargin=1em,     framextopmargin=0.5ex, %%  THE FRAME:     framerule=1pt,     frame=tblr }  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \usepackage{lipsum}     % for demo \begin{document} \subsubsection{Lorem}   % some fiddling to get the "wrong spacing just right": \lipsum[1-3] \subsubsection{Ipsum}  \lipsum[1] and then some text  \begin{lstlisting} Here is a listing with three lines of code; The bottom border is placed on the next page! \end{lstlisting}  Can you believe it..?! \end{document}   This MWE is rather long. My document inputs most of the listings stuff from files written by others, so I'm not sure what all of it does. If anyone sees some lines that they are sure is irrelevant, feel free to take it out of the MWE.  "Fun fact"  An equally silly situation occurs if you remove \usepackage{mathptmx}: The whole code listing will then appear on the new page, except for the top frame which is left on the bottom of the first page! Q. Find all pairs $(p,n)$ of positive integers where $p$ is prime and $p^3-p=n^7-n^3$.   Rewriting the given equation as $p(p+1)(p-1)=n^3(n^2+1)(n+1)(n-1)$, we see that $p$ must divide one of the factors $n,n+1,n-1,n^2+1$ on the $\text{r.h.s}$.   Now, the $\text{l.h.s}$ is an increasing function of $p$ for $p\ge1$. This implies that for any given $n\ge1$, there is exactly one real $p$ for which $\text{l.h.s}=\text{r.h.s}$. For $p=n^2$, we get $\text{l.h.s}=n^6-n^2&lt;n^7-n^3=\text{r.h.s}.$ This means that either $p&gt;n^2$ or $p&lt;n^2$ must hold.  Assuming $p&gt;n^2$, it follows that the prime $p$ cannot divide any of $n,n+1,n-1$. So $p$ must divide $n^2+1$ and hence $p=n^2+1\quad (\because p&gt;n^2)$.  Substituting the value of $p$ in the given equation we get, $n^2+2=n^3-n\implies n^3-n^2-n=2$. As the factor $n$ on the $\text{l.h.s}$ must divide $2$, the above equation has a unique integer solution $n=2$.  Finally, we get $(5,2)$ as the solution to the given equation.      But how do I conclude this is the only solution possible? Also, why does'nt $p&lt;n^2$ (the case which I ignored) hold? As a bonus question, I would like to ask for any alternative/elegant solution (possibly using congruence relations) to the problem. What do you call the grouping of numbers based on the cardinality of digits in the square they produce? php--glob for searching directories and .jpg only Are getters and setters poor design? Contradictory advice seen how many distinct values does it have? When creating accounts on Windows 10, you get a choice between "Your family" and "Other users".  If you choose "Your family" you are forced to use a Microsoft account to sign in. There are various reasons why one wouldn't want to do that, such as too much sharing of private information with Microsoft, etc.  If you choose "Other users" you can create regular accounts just fine. However, if you need to setup time limits for some accounts, it is impossible to do so. It is a feature exclusive to "Your family". Not all time-restricted users I want to create are my family, and not everyone is okay with sharing their info with Microsoft.  So, is there a way to setup login time limits for the "Other users" type of accounts? Roots of polynomials in field of prime characteristic p Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in my lectures. Can someone explain why please?  Thanks. I am trying to prove  $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$  This problem is a classic, but I seem to be missing one step or the understanding of two steps which I will outline below.   $$I_n := \int\cos^n x \ dx = \int\cos^{n-1} x \cos x \ dx \tag{1}$$  First question: why rewrite the original instead of immediately integrating by parts of $\int \cos^n x \ dx$?  Integrate by parts with $$u = \cos^{n-1} x, dv = \cos x \ dx \implies du = (n-1)\cos^{n-2} x \cdot -\sin x, v = \sin x$$  which leads to  $$I_n = \sin x \ \cos^{n-1} x +\int\sin^2 x (n-1) \ \cos^{n-2} x \ dx \tag{2}$$  Since $(n-1)$ is a constant, we can throw it out front of the integral:  $$I_n = \sin x \ \cos^{n-1} x +(n-1)\int\sin^2 x \ \cos^{n-2} x \ dx\tag{3}$$  I can transform the integral a bit because $\sin^2 x + cos^2 x = 1 \implies \sin^2 x = 1-\cos^2 x$  $$I_n = \sin x \ \cos^{n-1} x + (n-1)\int(1-\cos^2 x) \ \cos^{n-2} x \ dx \tag{4}$$  According to Wikipedia as noted , this simplifies to:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) \int \cos^{n-2} x \ dx - (n-1)\int(\cos^n x) \ dx \tag{5}$$  Question 2: How did they simplify the integral of $\int(1-\cos^2 x) \ dx$ to $\int(\cos^n x) \ dx$?  Assuming knowledge of equation 5, I see how to rewrite it as  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x  - (n-1) I_{n} \tag{6}$$  and solve for $I_n$. I had tried exploiting the fact that  $$\cos^2 x = \frac{1}{2} \cos(2x) + \frac{1}{2} $$  and trying to deal with $\int 1 \ dx - \int \frac{1}{2} \cos (2x) + \frac{1}{2} \ dx$  which left me with $\frac{x}{2} - \frac{1}{4} \sin(2x)$ after integrating those pieces. Putting it all together I have:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x \left(-(n-1) (\frac{x}{2} - \frac{1}{4} \sin 2x) \right) \tag{7}$$  but I'm unsure how to write the last few terms as an expression of $I_{something}$ to get it to match the usual reduction formula of   $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ Which Windows 7 log file contains device connection/disconnection information? Windows 7 frequently plays the usb/new device connection/disconnection sound. It happens both when the computer is idle and under load. All usb device connections have been checked (though I can't rule out a bad cable yet). I thought a good way to track down the issue would be to wait for the sound and then check the log file for the latest connected/disconnected device. I just need to know which log file has this information.  Edit: I'm going to leave the answer accepted, however the issue persists. I get that sound seemingly randomly throughout the day. Anyone else have any ideas? How do I search for questions with specific tags?       For more information about searching, see &quot;&quot; in the . Should I flag/down-vote "Draft Answers"?