Light is a non-mechanical wave. It means that it does not require a medium to travel. But the characteristics of the medium to affect light. That s why refraction occurs, right?  I also read somewhere that the mass density of a medium does not affect it's speed of light, but instead it's optical density does so.  What would a quantitative definition of optical density be? Why is it not related to mass density and how does it affect the speed of light?  Why does the presence of more mass in less volume (more mass density) not affect the speed of light?  Thanks!
The optical density of a medium is quantified by its refractive index $n$. Does it have any correlation with the mass/number density of the medium? In other words is an optically denser medium must have a greater mass density compared to an optically rarer medium?
If $A$ and $B$ are nonempty sets, prove that $A \times B = B \times A$ if and only if $A = B$.   Proving the first direction of this is easy. That is, if $A = B$ then it is obvious that $A^2 = A^2$. I am wondering how to prove the other direction now. Maybe proving it by contradiction will be easier than directly. Necessary and sufficient condition for the matrix $A = I - 2 x x^t$ to be orthogonal Let $x$ be a non zeo (column) vector in $\mathbb{R}^n$. What is the necessary and sufficient condition for the matrix $A = I-2xx^t$ to be orthogonal? Setting-up SSH-tunnel for VPN? Is it alright to use lowercase "i" or should you always use "I" (uppercase)? Say you have a stick which breaks randomly into three pieces (we can choose the points randomly). What is the probability that the area is greater than or equal to $0.4$?   I can see it has something to do with Heron's formula but I just can't put t together. Calculate bounding box coordinates of a selected polygon with QGIS I just manually (and somewhat inaccurately I'm sure) retrieved the bounding box coordinates for a county polygon, which is a selection from a larger layer of county shapes. I'd like to be able to use a QGIS plugin or other function to calculate this quickly and not-by-hand. This is almost certainly possible but I couldn't find anything with a Google search Word for "things which exist" I've noticed that in comments, when you create `code region` with backticks, the backslash will still work as an escape character within that region (but it should not, IMHO).  To clarify:   In comments, `\\` will be rendered as \, but it should be rendered as \\ In the post body itself, `\\` will be rendered as \\ (i.e., correctly)   This is especially confusing when you enter regular expressions into a comment, since backslash-escaping is very common there, and confusion increases if you are dealing with a regex that is double-escaped for representation in a string.   At some point you lose track how many backslashes you must make so the comment renders correctly, or you don't notice fast enough (or at all) and the comment contains wrong code where the same markup in a question or answer would come out as perfectly correct.   It's just that comments can't be corrected anymore after the grace period, leaving you with the choice of either deleting the comment, thus crippling the conversation, or engaging in a trial-and-error game of writing another comment with even more backslashes to explain the issue.  My question is two-fold:   Will this error be corrected? If yes, existing comments should probably be searched and "fixed" by a routine that tries to do the right thing (this is probably somewhere between hard and impossible) Or, can I rest assured that never ever the behavior of the backslash in comments is going to be changed, so I can start working around this issue by hand, knowing that they won't start displaying incorrectly at some point in the future? I am new to QGIS. I am looking for any plugin or any manual method in QGIS for finding minimum distance (perpendicular distance) from a point which is part of one shapefile to nearest line among many lines which is part of another shapefile.   MMQGIS is not useful for this purpose. Is there a way to calculate the improper integral $\int_0^\infty \big(\frac{\ln x}{x - 1}\big)^2 dx$?  What have I tried:  $$\int_0^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_0^1 \Big(\frac{\ln x}{x - 1}\Big)^2 dx + \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2 dx = \int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx + \int_1^\infty\Big(\frac{\ln t}{\big(\frac{1}{t} - 1\big)t}\Big)^2dt = 2\int_1^\infty \Big(\frac{\ln x}{x - 1}\Big)^2dx$$  From the above statement we can conclude, that the integral does indeed converge, as $0 \leq (\frac{\ln x}{x - 1})^2 \leq \frac{C}{(x - 1)^{\frac{3}{2}}}$, for some constant $C$. However, this new form of the integral is not much helpful in finding its exact value.  I also tried substituting $z = \ln x$ , then $dz = \frac{dx}{x}$. Then the integral becomes reduced to $2\int_{0}^{\infty} \frac{z^2}{e^z(e^z - 1)^2}dz$, which, unfortunately, also does not seem to be any easier. Max Planck, arguably the leading theoretical Physicist of his day, apparently used of experimental data to formulate his equation for , which was a major step in the birth of quantum mechanics. For this seminal event of the 20th century, I would expect a detailed explanation of how Planck arrived at his conclusion that energy MUST be quantized. I've even read his Nobel lecture but was non the wiser.  Can some kind person please explain it, or give me even just a hint? Adding a reference between Eclipse Java projects As a non-native speaker, I cannot grasp any difference between the expressions "here we go" and "there we go": both expressions seem to underline an event that is going to happen immediately.  Is there any subtle usage detail that I am missing? Is static HTML the fastest? I have been researching several different techniques to deliver web content fast, like nginx and node.js.  When it comes down to performance optimization http expires headers and gzip come to mind. After that come common techniques like sprites and CDNs etc.  But I am wondering.  What would be the very fastest technique to load a website?  Is static html the very fastest that can be? Or is there something else even faster? On Ubuntu 12.04 LTS on a Dell Vostro 2420 laptop, each time I start Ubuntu, the screen brightness is set to the maximum level, which gives a high glare. (I have recently installed Ubuntu on the laptop, but did not change any default settings related to brightness, IIRC.) I am currently lowering the brightness manually each time after starting, using the System Settings icon on the left of the screen. Looking for a permanent way to set it to a lower value. How to choose between learning algorithms I need to implement a program that will classify records into 2 categories (true/false) based on some training data, and I was wondering at which algorithm/methodology I should be looking at. There seem to be a lot of them to choose from -- Artificial Neural Network, Genetic Algorithm, Machine Learning, Bayesian Optimization etc. etc., and I wasn't sure where to start. So, my questions is:  How should I choose a learning algorithm I should use for my problem?  If this helps, here is the problem I need to solve.    The training data: The training data consists of many rows like this:    Precursor1, Precursor2, Boolean (true/false)   The run I will be given a bunch of precursors. Then,     I choose an algorithm A from different algorithms (or dynamically generate an algorithm), and apply it on every possible combinations of these precursors and collect the "record"s  that are emitted. The "record" consists of several key-value pairs*. I apply some awesome algorithm and classify these records in to 2 categories (true/false). I will generate a table that has the same format as the train data: Precursor1, Precursor2, Boolean   And the whole program is scored based on how many true/false I got right.  *:"Record"s will look like this (hope this makes sense)  Record         [1...*] Score -Precursor1             -Key -Precursor2             -Value   There are only a finite number of possible Keys. Records contain different subset of these keys (some records have key1, key2, key3... other records have key3, key4... etc.).  I actually need 2 learning. One is for step 1. I need to have a module that looks at the Precursor pairs etc. and decide what algorithm to apply in order to emit a record for the comparison. Another is for step 2. I need a module that analyzes the collection of records and categorize them into the 2 categories (true/false).    Thank you in advance! To prove that $f_n(x) = \frac{nx}{1+n^2x^2}$ does not uniformly converge to $f(x) = 0$ on $[0,1]$ My Approach. To prove the given statement it is sufficient to show that $$\exists \epsilon &gt; 0, \exists x \in [0,1], \forall N \in \mathbb{N}, \exists n &gt; N $$ $$ \Rightarrow | f_n(x) - f(x)| &gt; \epsilon$$    Let $\epsilon = \frac{1}{10}, x = \frac{1}{n}, n =N+1$. Hence $| f_n(x) - f(x)| = \frac{1}{2} &gt; \epsilon = \frac{1}{10}$. Hence, we have shown for every $N \in \mathbb{N}$, how to choose $n$ such that $| f_n(x) - f(x)| &gt; \epsilon$. Thus, $f_n(x)$ does not converge uniformly on $[0,1]$.    Is there any logical flaw or wrong steps in my proof. Any help would be appreciated. I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go.  In the truth table, where $A \implies B$ we obtain this result:  A   B   T/F ___________ T | T | T T | F | F F | T | T F | F | T   Now, my confusion here is regarding why $A\implies B$ given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.  Now... then, why is B true despite the fact that A is false? Or rather, why is the statement B given not A, True. The truth of B is implied from the truth of A, but A is a false statement. So it's very contradictory that this may be, and in fact, if you go through a real life example, you would find that the third statement seems to be false.  Say $A$ is the statement that it is dark outside, and $B$ the statement that the sun is not on our side of the earth. If it is not dark outside, than stating that the sun is on the other side of the planet must be wrong (in my example i'm going to claim that the sun is the only light source).  Side note:  I just had a thought, if A is false, does that mean you can conclude... mmm, whatever? I mean, B is implied via the truth of A, if A is false, B is NOT implied, but then B could be EITHER true or false, I suppose. Is this correct?  However, would that not mean that A does not imply anything when false. Would that not set the 3rd and 4th statement to be undefined? How coherent light is created for single slit experiment? How do I make sure that a socket bound to a port is properly released on process exit such that the port can be reused without bind() failing with EADDRINUSE? I've written a tiny program which just creates a socket, binds it to a fixed port, waits for a connection and then immediately terminates. When I rerun the program, the bind() call fails with EADDRINUSE, but if I wait a few minutes, it succeeds.  Is there a way I can explicitly "unbind" the socket, thereby freeing the port number?