I have been thinking about what would happen if by chance the photons collided with each other. I heard once that electromagnetic waves pass through each other without any interference.  I wanted to know if I am correct?  Does the collision produce sound?
Could someone explain to me what the collision of two photons would look like? Will they behave like,   Electromagnetic waves: they will interfere with each other and keep their wave nature Particles: they will bounce like classical balls   I assume that energy of that system is too small to make creation of pairs possible.
Theoretical Proof of Pauli's Exclusion Principle I'd like to do it in Python. What I'd like to do in this example in C: #include &lt;stdio.h&gt;  int main() {     int i;     for (i=0; i&lt;10; i++) printf(&quot;.&quot;);     return 0; }  Output: ..........  In Python: &gt;&gt;&gt; for i in range(10): print('.') . . . . . . . . . . &gt;&gt;&gt; print('.', '.', '.', '.', '.', '.', '.', '.', '.', '.') . . . . . . . . . .  In Python print will add a \n or space, how can I avoid that? Now, it's just an example, don't tell me I can first build a string then print it. I'd like to know how to &quot;append&quot; strings to stdout. Who is bound to the Prime Directive? In the galaxy there are a fair number of pre-warp societies. A Federation ship is not to contact them. But what about other, non-Federation species?   If for example a Ferengi ship passes by a pre-warp planet, would they be allowed to contact it? What would a Federation ship observing this contact have to do? Are they obliged to intervene? Are they allowed to? Or were they merely allowed to "strongly discourage" the attempt of the Ferengi?  If it is legal to use force to stop the Ferengi, than in essence the Ferengi were bound to the Prime Directive too. And if so: does this go for all Federation territory? Is this confined to the Federation territory? If Janeway observes such an incident, what are her rights to intervene? Or does she have the duty to do so? Zero to the zero power – is $0^0=1$? How to get into Walker in the Wastes as a level 3 Wizard I am building a Wizard character that focuses on using Greater Drought from the Walker in the Wastes prestige class.  The campaign is starting at level 12, so I need to figure out a way to build a Wizard 3/Walker in the Wastes 9.  I am planning on using Southern Magician and Extra Spell to get around the spell requirement, but I'm having trouble qualifying for Heat Endurance by level 3.  We are using a variant rule where, at every level divisible by 3, we get 3 feats instead of 1. At level 3, I'll have 1 feat from level 1, and 3 feats from level 3.  How can I build a character with 11 total levels of Wizard spellcasting and the Greater Drought Walker in the Wastes ability by level 12?  My current character sheet can be found . missing $ inserted error for equation I get the missing $ inserted error for the following code:  \begin{equation}  C = $ \frac{2}{\omega^2N^2\mu\pia} $  \end{equation}   I have tried out different changes based on Googling for this error, but nothing seems to help. Any suggestions would be appreciated. difference between "function foo() {}" and "foo() {}" I can define bash functions using or omitting the function keyword. Is there any difference?  #!/bin/bash  function foo() {   echo "foo" }  bar() {   echo "bar" }  foo  bar   Both calls to functions foo and bar succeed and I can't see any difference. So I am wondering if it is just to improve readability, or there is something that I am missing...  BTW in other shells like dash (/bin/sh is symlinked to dash in debian/ubuntu) it fails when using the function keyword. 1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why? If the topology of a space  has a countable base, then each base contains a countable subfamily which is also a base. Although Ubuntu 16.04 release notes mention that:  The default VIM package has been built against python3 instead of python2.  Yet after I installed vim with the following command:  sudo apt-get install vim   And checked:  vim --version   It showed:  -python -python3   Is it a bug? How do I get vim with Python support? Is it completely against the Java way to create struct like objects?  class SomeData1 {     public int x;     public int y; }   I can see a class with accessors and mutators being more Java like.  class SomeData2 {     int getX();     void setX(int x);      int getY();     void setY(int y);      private int x;     private int y; }   The class from the first example is notationally convenient.  // a function in a class public int f(SomeData1 d) {     return (3 * d.x) / d.y; }   This is not as convenient.  // a function in a class public int f(SomeData2 d) {     return (3 * d.getX()) / d.getY(); } Why doesn't the last digit method work for divisibility by 4? I wish to explain to a grade 5 kid, the answer to the question - 'When can we know that the observing the last digit of a number will not be sufficient to determine its divisibility by a given N?'  For example - It doesn't work for 3, because we get all sorts of digits, i.e. from 0 to 9, at the units place. The same reason works for 7 and 9.   But what would be a convincing argument for 4 and 8? Even though the last digits form a pattern - 0, 2, 4, 6 and 8 - but we're still not sure about its divisibility. Don't Have A .minecraft Folder on Windows 10 PC Clearing Flag Bits By Writing 1 Proof of a proposition about recursion definition (Terence Tao's Analysis I) Exercise 3.5.12. Let $ f: \mathbb N \times \mathbb N \to \mathbb N$ be a function, and let c be a natural number. Show that there exists a function $a:\mathbb N\to \mathbb N$ such that $$a(0)=c$$ and $$a(n++)=f(n,a(n))\;for\;all\;n\in \mathbb N,$$ and furthermore that this function is unique. Tao gives the hint: first show inductively, by a modification of the proof of Lemma 3.5.12, that for every natural number $N\in \mathbb N$, there exists a unique function $a_N:\{ n\in \mathbb N:n\lt N\}\to \mathbb N $ such that $a_N(0)=c$ and $ a_N(n++)=f(n,a(n))$ for all $n\in \mathbb N$ such that $n\lt N$.  In my opinion, we could solve it by proving that $\forall n\in \mathbb N,a(n)\in \mathbb N$ according to mathematcial induction.  The property P(n) is:$$\forall n\in \mathbb N, a(n)\;is\;unique$$ P(0) is true. And if P(n) holds, P(n++) is true because different natural numbers must have different successors. $$a(n++)=f(n,a(n))$$ $$a(m++)=f(m,a(m))$$ $$n++=m++\Rightarrow n=m\Rightarrow a(n++)=a(m++)$$ It's similar to the proof of  Proposition 2.1.16 (Recursive definitions): Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to the natural numbers. Let $c$ be a natural number. Then we can assign a unique natural number $a_{n}$ to each natural number $n$, such that $a_{0} = c$ and $a_{n{++}} = f_{n} (a_{n})$ for each natural number $n$.  This proof indicates both the existence and uniqueness of the function $a$. I don't know why Tao says it's informal.  But regarding the hint, how do we get the function f even if we've got the conclusion from the hint. Also it seems like the function a(n) can be diffentent considering different N. I don't know if the hint is correct. Plurals of acronyms, letters, numbers — use an apostrophe or not? I make use of the view: "taxonomy term". Is there an easy way to also show the children of a term, next to the nodes beloging to this term?  Example:  Cars - Volvo --- Family_cars --- Sports_cars - Honda --- Parts --- ... Trucks - ...   When you see the "cars" page I want to show all articles about cars (but not about Volvo, Honda, ...) AND I want to show the children (Volvo, Honda, ...) When you click one of these I want to show all the articles belonging with them, again with their children. consider the following subsets of complex plane $$\Omega_1=\left\{c\in\Bbb C:\begin{bmatrix}1&amp;c\\\bar c&amp;1\\ \end{bmatrix}\text{ is non-negative definite } \right\} $$  $$\Omega_2= \left\{c\in\Bbb C:         \begin{bmatrix}         1 &amp; c &amp; c \\         \bar c &amp; 1 &amp; c \\         \bar c &amp; \bar c &amp; 1 \\         \end{bmatrix} \text{ is non-negative definite } \right\}$$  Let $$\bar D=\{z\in\Bbb C:|z|\le1\}$$ Then 1) $\Omega_1=\bar D,\Omega_2=\bar D$  2). $\Omega_1\neq\bar D, \Omega_2=\bar D$  3). $\Omega_1=\bar D, \Omega_2\neq\bar D$  4). $\Omega_1\neq\bar D, \Omega_2\neq\bar D$  How to proceed? Thank you. Export nodes from Drupal 4? I am looking into updating a Drupal 4 site. The Node Export module does not seem to have a release suitable for Drupal 4. Can anyone recommend an alternative module or procedure?  The site is running Drupal 4.6.3. Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus