When two photons annihilate each other through destructive interference, where does the energy go?  Perhaps this is only answered by ‘wave type’ questions and answers? Does the conservation of energy get violated?
What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong?
Opposite of "straight talk" What is the opposite for the straight talk idiom? How do I best call the activity when someone makes a very long preamble before he says what he wants? I'm trying to install pandas. When I do, it's automatically installing to the python installation for v 2.7.   How do I force pandas to install to the python 3.5 installation? Spam iCloud Calendar Invitations Fixed Field of $\sigma, \tau$ Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. Let $\DeclareMathOperator{\aut}{Aut}\sigma, \tau \in \aut(K)$ s.t. $$\sigma\left(\frac {f(x)}{ g(x)}\right)=\frac {f(1/x)}{g(1/x)}\qquad \&amp;\qquad \tau\left(\frac {f(x)}{ g(x)}\right)=\frac {f(1-x)}{g(1-x)}.$$ Find the fixed field $F$ of $&lt;\sigma, \tau&gt;$. Determine $\DeclareMathOperator{\gal}{Gal}\gal(k/F)$. Find an $h \in F$ so that $F=k(h)$.   Now for the first automorphism $\sigma$, I think if I concentrate on $k[x]$ it will give me some idea. Now, if $f(x)=a_0+a_1x+a_2x^2+ \cdots + a_nx^n \in k[x]$ then if $x^n \sigma(f(x))=f(x)$ the case will be $a_i=a_{n-i}$ $\forall i=0(1)[n/2]$. So all the funtions in $K$ which will be fixed by $\sigma$ are of the form $\frac {f(x)}{ g(x)}$ where $f(x)=a_0+a_1x+a_2x^2+ \cdots + a_nx^n$ and $g(x)=b_0+b_1x+b_2x^2+ \cdots + b_nx^n$ such that $a_i=a_{n-i}$ $\forall i=0(1)[n/2]$ and $b_i=b_{n-i}$ $\forall i=0(1)[n/2]$.   Please check whether am I right or not?  And any ideas for the rest of the part? What does: Error Broken count >0 mean? I have the message:   Error Broken count &gt;0 Check if you are using third party repositories. If so disable them, since they are a common source of problems. Furthermore run the following command in a Terminal: apt-get install -f Details: The following packages have unmet dependencies: firefox-globalmenu: Depends: firefox (= 19.0+build1-0ubuntu0.12.04.2) but 19.0+build1-0ubuntu0.12.04.1 is installed   Any suggestions? How can I model flips until N successes? You and I decide to play a game where we take turns flipping a coin. The first player to flip 10 heads in total wins the game. Naturally, there is an argument about who should go first.  Simulations of this game show that the player to flips first wins 6% more than the player who flips second (the first player wins approx 53% of the time).  I'm interested in modelling this analytically.  This isn't a binomial random variable, as there are no fixed number of trials (flip until someone gets 10 heads). How can I model this?  Is it the negative binomial distribution?    So as to be able to recreate my results, here is my python code:  import numpy as np from numba import jit   @jit def sim(N):      P1_wins = 0     P2_wins = 0      for i in range(N):          P1_heads = 0         P2_heads = 0         while True:              P1_heads += np.random.randint(0,2)              if P1_heads == 10:                 P1_wins+=1                 break              P2_heads+= np.random.randint(0,2)             if P2_heads==10:                 P2_wins+=1                 break     return P1_wins/N, P2_wins/N   a,b = sim(1000000) The following is a rigorous proof of De Moivre's theorem by means of mathematical induction. The theorem put simply is that:  Any complex number, $z = a+bi$, on a cartesian plane can be expressed in polar form, where $a=r\cos\theta$ and $b=r\sin\theta$ and $r$ is the absolute distance from the origin to the point $z$. In light of this, the expression for the complex number $z$ in polar form is $z=r(\cos\theta+i\sin\theta)$ In order to find the $n^{th}$ power of $z$, the following rule applies: $$\displaystyle z^n=(r(\cos\theta+i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$$ where $n\in\mathbb N$ (I have confined $n$ to the natural numbers given that I intend on using mathematical induction to prove the theorem.  Base step: We must prove that what is stated is true for $n=1$, hence $$z^1=r^1(\cos 1\cdot\theta+i\sin1\cdot \theta)$$ $$=r(\cos\theta+i\sin\theta)$$ which is true. Next we assume it is true for $n=k$, and therefore proceed to the inductive step. Inductive step: When calculating $z^{n+1}$, it's the same as calculating $z^nz$, hence $$z^{n+1}=(r^n(\cos n\theta + i\sin n\theta))(r(\cos \theta+i\sin \theta))$$ $$=r^nr(\cos n\theta+i\sin n\theta)(\cos\theta +i\sin \theta)$$ $$=r^{n+1}(\cos n\theta\cdot\cos\theta+\cos n\theta \cdot i\sin\theta+i\sin n\theta\cdot\cos\theta+i^2\sin n\theta\cdot\sin\theta)$$ from here, $i^2=-1$, and therefore $+i^2\sin n\theta\cdot\sin\theta$ becomes $-\sin n\theta\cdot\sin\theta$. In addition to this, the following trigonometric identities will be used: $$\sin(n\theta + \theta) = \sin n\theta\cdot\cos\theta+\cos n\theta\cdot\sin\theta$$ $$or$$ $$k\sin(n \theta+\theta)=k\sin n \theta\cdot\cos \theta + \cos n\theta\cdot k\sin\theta$$ $$and$$ $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta-\sin n\theta\cdot\sin\theta$$ by using these, the original equation has now become $$=r^{n+1}(\cos(n \theta + \theta) + i\sin(n \theta + \theta))$$ then, after factoring out $\theta$, the theorem becomes $$=r^{n+1}(\cos((n+1)\theta)+i\sin((n+1)\theta))$$ which completes the proof. Question: is this proof correct all the way through? Or have I missed any conditions? If all is well in the case of this proof, could somebody aid me in proving it for $\{n|n\in\mathbb R,n\geq1\}$ limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$ Do we have negative prime numbers? Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$ alias multi commands as one alias I made an alias for youtube-dl  with some options  like that  yd="youtube-dl -cit -f 18"   so I can download it like   yd URL   I want to modify it to show only the two lines from the output using   grep -e "Downloading video #" -e ETA  I can make it in two aliases using the |   but is there a way to make it one alias Loopback to forwarded Public IP address from local network - Hairpin NAT This is a  about Hairpin NAT (Loopback NAT).   The generic form of this question is:  We have a network with clients, a server, and a NAT Router. There is port forwarding on the router to the server so some of it's services are available externally. We have DNS pointing to the external IP. Local network clients fail to connect, but external work.   Why does this fail? How can I create a unified naming scheme (DNS names which work both locally and externally)?   This question has answeres merged from multiple other questions. They originally referenced FreeBSD, D-Link, Microtik, and other equipment. They're all trying to solve the same problem however. In the sentence below, are the usages of apostrophes correct?     In research about parent stereotypes, the correlation between parents’ beliefs about their early elementary age children’s’ mathematics abilities and the children’s actual abilities increased as children aged. Retrieving Current Location and Zip using Geolocation Button Why does the direction of a dipole moment go from negative to positive charge? An  is defined as $p = q\times d$ (for two point charges $\pm q$ separated by a distance $d$).    What is the physical meaning of this quantity?  Why does the direction of the electric dipole moment go from the negative charge to the positive charge? Another way of saying this, why electric dipole moment is a vector quantity? When is $A- B = B- A$? Hello I am taking a Discrete Mathematics course and having some trouble with this question about sets:  Under what conditions is $A- B = B- A$  Diagram:   Diagram:   Maybe I'm understanding incorrectly, but how can A-B and B-A be equal if they contain elements that are not in each other? Reverse (parse the output) of Arrays.toString(int[]) Is there in the JDK or Jakarta Commons (or anywhere else) a method that can parse the output of Arrays.toString, at least for integer arrays?  int[] i = fromString(Arrays.toString(new int[] { 1, 2, 3} ); Can I turn off macOS Catalina's internal display without close the lid while using an external monitor? I often find  syntax very helpful, e.g. process substitution like in diff &lt;(sort file1) &lt;(sort file2).  Is it possible to use such Bash commands in a Makefile? I'm thinking of something like this:  file-differences:     diff &lt;(sort file1) &lt;(sort file2) &gt; $@   In my GNU Make 3.80 this will give an error since it uses the shell instead of bash to execute the commands. I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:      "The rank of a matrix A is the number   of non-zero rows in the reduced   row-echelon form of A".   The lecturer then explained that if the matrix $A$ has size $m  \times n$, then $rank(A) \leq m$ and $rank(A) \leq n$.   The way I had been taught about rank was that it was the smallest of    the number of rows bringing new information the number of columns bringing new information.    I don't see how that would change if we transposed the matrix, so I said in the lecture:  "then the rank of a matrix is the same of its transpose, right?"   And the lecturer said:   "oh, not so fast! Hang on, I have to think about it".   As the class has about 100 students and the lecturer was just substituting for the "normal" lecturer, he was probably a bit nervous, so he just went on with the lecture.  I have tested "my theory" with one matrix and it works, but even if I tried with 100 matrices and it worked, I wouldn't have proven that it always works because there might be a case where it doesn't.  So my question is first whether I am right, that is, whether the rank of a matrix is the same as the rank of its transpose, and second, if that is true, how can I prove it?  Thanks  :) For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ converge?