Fourier Series Analysis
Why do harmonics occur when you pluck a string?
Why was mixing declarations and code forbidden up until C99? Run cURL commands from Windows console Is there a way to install cURL in windows in order to run cURL commands from the command prompt? How to sum 2 vectors in spherical coordinate system I have 2 vectors, and each defined by 3 coordinates: radius, azimuth and zenith How i can sum it? Help me pls guys. What i have done: I found azimuth and zenith of resultant vector, by decomposition to 2 polar vectors And it gave me 2 radiuses: radius of resultant vector for azimuth polar sum and radius of resultant vector of zenith polar sum. What can i do with them? Does a logistic regression maximizing likelihood necessarily also maximize AUC over linear models? How to set a letter to the margin of the page and position it vertically according to alphabetical order? Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros How many levels are there for Steam Trading Card badges? With the new Steam Trading Cards, I decided to craft the badge for CS:GO. I got this:    I then noticed that this is only a level 1 badge, and there was a level 2 badge to get. So I crafted it too:    Now it says there's a level 3 badge to get next. How many levels are there in total? Case insensitive regular expression without re.compile? In Python, I can compile a regular expression to be case-insensitive using re.compile:  &gt;&gt;&gt; s = 'TeSt' &gt;&gt;&gt; casesensitive = re.compile('test') &gt;&gt;&gt; ignorecase = re.compile('test', re.IGNORECASE) &gt;&gt;&gt;  &gt;&gt;&gt; print casesensitive.match(s) None &gt;&gt;&gt; print ignorecase.match(s) &lt;_sre.SRE_Match object at 0x02F0B608&gt;   Is there a way to do the same, but without using re.compile. I can't find anything like Perl's i suffix (e.g. m/test/i) in the documentation. Proof that the largest eigenvalue of a stochastic matrix is $1$ The largest eigenvalue of a  (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.  Wikipedia marks this as a special case of the , but I wonder if there is a simpler (more direct) way to demonstrate this result. I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going? Simplest or nicest proof that $1+x \le e^x$ The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is accessible to students with limited calculus experience, and does not involve too much analysis of different cases. Why can't I import "pygtk" with Python 3.2 from PyDev? Can anyone offer any help with this..? I read it at school around 1976 but cannot remember the title nor the author. Thank you in advance!  A lone astronaut called - I think- Steve is stranded on a distant planet when his ship runs low on fuel.   His only companion is - again I think - a parrot. The fuel his ship uses is a filament or wire which is kept on a spool.   On exploring the locale he has landed in he discovers a huge building half-buried in a mountainside. He enters the building and discovers it is a museum filled with thousands of glass cases containing preserved specimens of various life forms as well as machines.   There is a robot curator that does repairs and, as the astronaut finds out, prepares cases for new exhibits.   On analysing the wire in one of the machines he discovers that is suitable as fuel for his ship so he winds a length of it onto his ships spool and leaves the planet just ahead of the robot collecting him and his parrot as museum exhibits... Can I mix monk traditions? I'm wanting to mix archetypes (traditions) for my monk. I'm wanting specifically the 3rd and 17th level feats from Way of the Open Fist and the 6th and 11th level feats from the Way of the Shadows. I know we can't multiclass into the same class to gain both but in the PHB page 79 2nd column "Monastic Traditions" reads "Three traditions of monastic pursuit are in common in the monasteries scattered across the multiverse." I'm fine with that but where I question is "Most monasteries practice one tradition exclusively, but a few honor the three traditions and instruct each monk according to his or her aptitude and interest."   So my question would be can I build a monk that is split like that without homebrewing to much? What is the preferable setup for MiKTeX to keep all packages up-to-date easily? Is there a way to disable a specific JS script from a specific website? I want to disable a specific JS script from a specific website without disabling the javascript completely. Is there a way to do that ? I gave my friend  as a brainteaser; while her attempted solution didn't work, it raised an interesting question.  I flip a fair coin repeatedly and record the results.  I stop as soon as the number of heads is equal to twice the number of tails (for example, I will stop after seeing HHT or THTHHH or TTTHHHHHH).  What's the probability that I never stop?  I've tried to just compute the answer directly, but the terms got ugly pretty quickly.  I'm hoping for a hint towards a slick solution, but I will keep trying to brute force an answer in the meantime. Recall the following relevant definitions. We say that   $b$ is divisible by $a$ in $R$, or $a\mid b$ in $R$, if $b = r a$ for some $r\in R$. $a$ and $b$ are associates  in $R$ if $a\mid b$ and $b\mid a$, (or, equivalently, if $aR = bR$). $u\in R$ is a unit if it has a multiplicative inverse (a $v\in R$ such that $uv=vu=1$). $a$ and $b$ are unit multiples in $R$  if $a = ub$ for some unit $u\in R$.   Given these definitions, my question is,     If $R$ is a commutative ring with unity and $a,b\in R$ are associates in $R$, are $a$ and $b$ unit multiples in $R$?   I was told that this not always true. But I encountered some difficulties in finding a counterexample. Complex integral over a circle In an exercise, I'm supposed to assume that $|a| &lt; r &lt; |b|$ and prove that $\displaystyle\int_{\gamma} \frac{1}{(z - a)(z - b)} dz = \frac{2 \pi i}{a - b}$, where $\gamma$ is a circle of radius $r$ centered at the origin with positive orientation. So I had the idea to express this integral as $\frac{1}{a - b} \displaystyle\int_{\gamma} \frac{1}{z - a} - \frac{1}{z - b} dz$. Then I tried to evaluate each of these separately, but I don't really know what to do. I get $\displaystyle\int_{0}^{2 \pi} \frac{i r e^{it}}{r e^{it} - a} dt$ for the first term. How am I supposed to integrate this? I can't substitute $u = re^{it}$, can I? How to test randomness (case in point - Shuffling) First off, this question is ripped out from  question. I did it because I think this part is bigger than a sub-part of a longer question. If it offends, please pardon me.  Assume that you have a algorithm that generates randomness. Now how do you test it? Or to be more direct - Assume you have an algorithm that shuffles a deck of cards, how do you test that it's a perfectly random algorithm?  To add some theory to the problem - A deck of cards can be shuffled in 52! (52 factorial) different ways. Take a deck of cards, shuffle it by hand and write down the order of all cards. What is the probability that you would have gotten exactly that shuffle? Answer: 1 / 52!.  What is the chance that you, after shuffling, will get A, K, Q, J ... of each suit in a sequence? Answer 1 / 52!  So, just shuffling once and looking at the result will give you absolutely no information about your shuffling algorithms randomness. Twice and you have more information, Three even more...  How would you black box test a shuffling algorithm for randomness?