What is azimuthal symmetry?
Cylindrical vs azimuthal symmetry
Reentering serial number in Windows 7 Does anyone know how to install a new serial on a working copy of Windows 7 - I get notified that my software is not genuine but cannot reenter the serial Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring? Massive spam users and spam entries despite triple layer protection I have a long running BASH script that I am running under CYGWIN on Windows.  I would like to limit the script to run for 30 seconds, and automatically terminate if it exceeds this limit. Ideally, I'd like to be able to do this to any command.  For example:  sh-3.2$ limittime -t 30 'myscript.sh'   or   sh-3.2$ limittime -t 30 'grep func *.c'   Under cygwin the ulimit command doesn't seem to work.   I am open to any ideas. Center of a free product Let $A$ and $B$ be two groups and let $G=A*B$ be their free product. Suppose both $A$ and $B$ are nontrivial. I want now to prove that $G$ is infinite and that $Z(G)$ is trivial.  I think I can prove that $G$ is infinite using this argument: two reduced words of $A*B$ are equal if and only if they have the same length and the ith letter of the first word is the same as the ith letter of the second one. Hence one can find a reduced word of $G$ of length arbitrarily big (since we are sure it won't be equal to a shorter word). Do you think it works? Has anyone another suggestion?  On the other hand, I don't really have a clue for the second part except trying to analyze a long list of cases of reduced words and prove for each one that the only element with which they commute is the identity. Does anyone have a smarter idea?  Thanks in advance! What is the need of complex functions in wave analysis? It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$y=y_{_0}e^{i(kx-\omega{t})}$$ where $y_{_0}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular velocity and $x$ &amp; $t$ are position an time respectively. Ofcourse, I know that the function $e^{ix}$ can be written in the form $\cos{x}+i\sin{x}$ and so it is a periodic function with period $2\pi$ but my question is for what purpose we define it in terms of complex numbers? It seem to be more convenient to use real functions for real variables such as amplitude, electric and magnetic field of an electro magnetic wave, and also in quantum mechanics. What actually this interpretation means or what is the advantage of such functions? Find $\sum_{n=1}^\infty \frac{n^3}{n!}$. I have to find the sum of the following series:  $$\sum\limits_{n=1}^\infty \frac{n^3}{n!}$$  I know how to prove the convergence of this series, but how do i find the sum. I can't use the properties of derivation or integration of the series, i have to do it using the definition that a series converges iff it's sequence of partial sum converges. If $x_n \to 0$, show that $\sqrt{x_n} \to 0$ Let $x_n \geq 0$ for all $n \in N$. If $(x_n) \to 0$, show that $(\sqrt{x_n}) \to 0$  Unless I am missing something, this seems simple to me. Since $x_n$ converges for an arbitrary $\epsilon^2&gt;0$, we have $|x_n| &lt;\epsilon^2 $ for $n\geq N$. Taking square roots here we have $\sqrt{x_n}&lt;\epsilon$ for all $n\geq N$ which is what we needed to show. Is that all? If so, then why $x_n \geq 0$ is important? What is the proper way to change the DNS IP? I suspect that the name server provided by my ADSL modem/router is buggy. Whenever I browse to a website for the first time in ubuntu, resolving the domain name takes at least 15 seconds.  To work around that problem, I changed the nameserver configuration in /etc/resolv.conf from 192.168.1.1 (my ADSL modem) to 8.8.8.8 (google's primary DNS). This seems to fix the problem, but unfortunately my changes to /etc/resolv.conf are overwritten by "NetworkManager" at each startup.  What is the proper way to configure the name server IP in ubuntu 10.4? Missing whitespace in beta site statistics box "He was neither seer nor prophet" How would you explain the absence of an article? Here's a fragment from Jack London's Star Rover:     Wordsworth knew. He was neither seer nor prophet, but just ordinary man like you or any man. What he knew you know, any man knows. But he most aptly stated it in his passage that begins "Not in utter nakedness, not in entire forgetfulness.   To the best of my — admittedly sketchy — knowledge of grammar, the second sentence should read neither a seer nor (a) prophet, but just an ordinary man. Even if we accept the seer and the prophet as "poetic entities" ("father and son," "robber and robbed," "man to man," or whatever), logic would still dictate that ordinary man should be preceded by an an.  It is a well-known fact that Jack London's usage wasn't always up to par; his grammar, on the other hand, was always top-notch.  Logically, the sentence is grammatically incorrect (correct me if I'm wrong: no pun intended).  Illogically, it reads smoothly: intuitively, a native speaker understands that there's nothing wrong here.  However, those to whom English is not the first language cannot be expected to be intuitive in this case and would want an explanation. Any suggestions? How can I play mp3 files or videos? This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? In my opinion the accepted answer should permanently stick as only the original question submitter has the ability to accept. What is the intuition behind beta distribution? Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial for other people here. So I would be very thankful if answers included less specific terms and more explanation. Imagine that you are talking to your grandma :)  I'm trying to grasp the nature of beta distribution – what it should be used for and how to interpret it in each case. If we were talking about, say, normal distribution, one could describe it as arrival time of a train: most frequently it arrives just in time, a bit less frequently it is 1 minute earlier or 1 minute late and very rarely it arrives with difference of 20 minutes from the mean. Uniform distribution describes, in particular, chance of each ticket in lottery. Binomial distribution may be described with coin flips and so on. But is there such intuitive explanation of beta distribution?    Let's say, $\alpha=.99$ and $\beta=.5$. Beta distribution $B(\alpha, \beta)$ in this case looks like this (generated in R):     But what does it actually mean? Y-axis is obviously a probability density, but what is on the X-axis?   I would highly appreciate any explanation, either with this example or any other. How do I disable overlay scrollbars? It looks like Ubuntu 11.04 will come with  by default. I do like them, but I don't like the current mix of scrollbar types as many applications don't yet use those overlay scrollbars.  Is there a way to disable overlay scrollbars (without removing the overlay-scrollbar package)? How to check that a cubic polynomial is irreducible? Prove that $X^\ast$ separable implies $X$ separable My current idea is to use the dual graph, and prove that that is a bipartite graph, and so proving the statement. But I can't seem to figure out how even degree for all verticies means that it is bipartite.  Any help is appreciated. Thanks! Run Script on Wakeup?