$\forall n\in \mathbb N$, Let $$f_n (x)=\underbrace{\cos(\cos(\cos(\cdots (\cos}_n(x\underbrace{))\cdots)}_n$$ It is known that $$\forall x\in\mathbb R,\space \lim_{n\rightarrow\infty}f_n(x)=0.7390851332\dots$$Does the convergence uniform?
In Radian mode, continually pressing the $\cos$ function of a number causes the result to converge to $x=0.739085133$, a fixed point of $\cos(x)$. Repeating this behavior with the $\sin$ function causes the result to converge to $x=0$, a fixed point of $\sin(x)$.   What happens if this is done using the tan function? It seems that $x=0$ is a repelling fixed point with no convergence in this case.  What would be the best way to explain these results mathematically?
We add negative numbers and zero to natural sequence to make it closed under subtraction, the same thing happens with division (rational numbers) and root of -1 (complex numbers).  Why this trick isn't performed with division by zero? Can you contract the main verb '[I] have' in a sentence? Is their a way to move multiple vertices gradually at a time? I have moved multiple vertices gradually but a set of vertices at a time not all at a time, as shown in the animation below (First I moved the four vertices from top, then three, then two and at last the last vertex).  I wanted to do the same thing as shown in the animation below but all at a time. I have a computer with Windows XP installed.  My user does not have a password.  When Windows starts, it goes to the login screen. Because my user has no password, is there any way to bypass this screen, and log in automatically? Custom arrangement for fields with multiple values in Drupal 7 Should I always disconnect event handlers in the Dispose method? I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} How to implement a single instance Java application? Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. With the confirmation that elements 113, 115, 117, and 118 are indeed fundamental elements that are now to be named on the periodic table, the next question is: what is the highest atomic number possible for an element? Feynman had a go at this years ago, and he derived (according to my limited understanding) that above element-137 (informally denoted 'Feynmanium') the electrons in the nearest orbit about the nucleus would be traveling at a velocity greater than the speed of light. (Note I am considering the Bohr model of the atom in this question.) I wished to demonstrate to my friends why this is and I came up with the following explanation.  According to Bohr's quantum condition, the angular momentum of an electron about the nucleus is given by $$ L=m_evr_n=n\hslash \implies r_n=\frac{n\hslash}{m_ev},$$ where $m_e$ is the mass of the electron, $v$ its velocity, $n$ an integer, and $r_n$ the radius of the $n$th possible orbit. Because we are concerned with the nearest orbit to the nucleus, $n=1$; and thus $$r=\frac{\hslash}{m_ev}.$$   Now, according to Coulomb's law, if the electron orbits about the nucleus the centripetal motion can be described by $$\frac{Ze^2}{4\pi\epsilon_or}=m_ev^2, $$ where $Z$ denotes the number of protons in the nucleus (the atomic number), and $e$ the elementary charge. Solving for $Z$ and substituting in $r$ from above yields $$Z=\frac{4\pi\epsilon_o \hslash v}{e^2}. $$ But what is $v$? Well, the maximum velocity an electron could ever have is the speed of light, and we wish to find the atomic number associated with an orbiting electron traveling at this speed, so we set $v=c$ and obtain our final result of $$Z=\frac{4\pi\epsilon_o\hslash c}{e^2}\approx 137.521,$$ which implies that for $Z&gt;137$, the electrons at a position of $n=1$ in the Bohr model would have a velocity $&gt;c$; and thus the highest atomic number achievable on the periodic table is 137.  Again, I just want to make sure this is a correct method for deriving element-137 before presenting. Perhaps one could explain how relativity plays a role here. I know Feynman used the Dirac equation to get this result...so could anyone (subsequently of course) expatiate on this in a simplistic manner? Thanks! I'm getting the following error message:     System.QueryException: Didn't understand relationship 'AccountContactRelation' in FROM part of query call. If you are attempting to use a custom relationship, be sure to append the '__r' after the custom relationship name. Please reference your WSDL or the describe call for the appropriate names.   Select Email, Id, (Select ContactId, AccountId From AccountContactRelation) From Contact  In AccountContactRelation object there are two lookup fields for Account and for Contact.  Please advise how to avoid such error message? Does 'A countable infinite Cartesian product of countable sets is not necessarily countable' violate induction principle? We have two theorems about Cartesian product:   A finite Cartesian product of countable sets is countable; A countable infinite Cartesian product of countable sets is not necessarily countable.   To prove the first theorem, we can show   Firstly, we can prove $A\times B$ is countable,  Secondly, suppose it is true for $n-1$, i.e. $X=A_1\times A_2\times \ldots \times A_{n-1}$ is countable, we prove the case for $n$ by the fact $A_1\times A_2\times \ldots \times A_{n-1}\times A_{n}=X\times A_{n}$ is in fact the product of two countable sets, in above we know it is still countable.   The above method to prove the first theorem is exactly the mathematical induction, which should show the theorem is true for all $n\in\mathbb{Z}_{+}$, so it should works for countable infinite unions.  However, we may use Cantor diagonal method to prove at least $\{0,1\}^w$ is not countable, thus the second theorem is also right. Does this means that the mathematical induction is wrong?    EDIT: After the following discussions, I have the answer:  The key point is that any element $n\in\mathbb{Z}_+$ is always a finite number, thus induction principle only guarantee the theorem is true for any $n\in\mathbb{Z}_+$, i.e. for any finite number.  When discuss a countable infinite union, it is infinite number of unions, this infinite number no more belong to $\mathbb{Z}_+$, thus can not be guaranteed by induction principle. Question about Riemann integral and total variation Why do you need to change your position rather than just focal length to affect perspective? Why must you change your distance from an object to change your perspective of it? Why is it not sufficient to change the focal length of your lens (the zoom level) to change  your perspective? Can you create images like this:    when you have something like  0 = green (#54ff00) 1 = white (#ffffff) 2 = red   (#ff0000) 3 = blue  (#0048ff)  Image (Python list of integers defined above): [[2,0,0,0,0,0,0],  [0,3,0,0,0,0,0],  [0,3,2,1,1,0,0],  [0,3,2,2,2,1,1],  [0,3,2,0,0,1,0],  [0,0,0,0,0,1,0],  [0,0,0,0,0,1,0]]   with TikZ?  How far I got  The following MWE defines all colors, produces a grid of the correct size (though the size is determined by hand):  \documentclass[varwidth=true, border=2pt]{standalone} \usepackage{tikz} \usepackage{xcolor}  \begin{document}     \newcommand\n{7}     \definecolor{green}{HTML}{54FF00}     \definecolor{wite}{HTML}{FFFFFF}     \definecolor{red}{HTML}{FF0000}     \definecolor{blue}{HTML}{0048FF}     \begin{tikzpicture}         \foreach \x in {1,...,\n}{             \foreach \y in {1,...,\n}{                 \begin{scope}[shift={(\x,\y)}]                   \draw [fill=green] (0,0) rectangle (1,1);                 \end{scope}             }         }     \end{tikzpicture} \end{document}   Of course, I could simpy make a lot of adjusted                   \begin{scope}[shift={(\x,\y)}]                   \draw [fill=green] (0,0) rectangle (1,1);                 \end{scope}   but I've wondered if this can (with reasonable effort) be done in TikZ / LaTeX.  My problem is giving TikZ the color array. I don't have any idea how to do this. I only know two types of looks in TikZ:  \foreach \x in {0,1,2,3,4,5}{...} \foreach \number in {1,...,\n}{...}   I have never seen a nested over a 2D-array. All I have seen so far were 1D-Arrays of tuples (with fixed tuple size). The only 1-manifolds are $\mathbb R$ and $S^1$ I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof? No package php-imap available on CentOS 7 I'm running a dedicated server with CentOS 7 64bit, here is the output when trying to install IMAP:    yum install php-imap   Loaded plugins: fastestmirror   Loading mirror speeds from cached hostfile    * base: ftp.ciril.fr    * extras: ftp.ciril.fr    * updates: ftp.ciril.fr   No package php-imap available.   Error: Nothing to do       Also the same when disabling exclusions:    sudo yum install --disableexcludes=all php-imap Loaded plugins: fastestmirror Loading mirror speeds from cached hostfile  * base: ftp.ciril.fr  * extras: ftp.ciril.fr  * updates: ftp.ciril.fr No package php-imap available. Error: Nothing to do Text spirals with TikZ I just ran across this, and I'm curious about whether it can be replicated in TeX...    Ideally one would have the spiral getting tighter and tighter as one got to the centre, as in    I know TikZ has the ability to fit text along a path, but I'm not sure how one might make text get smaller and smaller.  The closest I can find is , but that uses manually inserted font commands.  Edit: I got what I wanted using mwibrow's answer: How can I stand on the ground? EM or/and Pauli? What is Ruby's double-colon `::`? What is the correct formula for gravitational time dilation for a satellite in a circular orbit? In the Wikipedia article on  I see two formulae for the gravitational time dilation. One contains $\sqrt{1 - r_0/r}$ but the other $\sqrt{1 - 1.5 r_0/r}$. What formula should I use to calculate the time dilation due to gravity inside a satellite? The speed induced time dilation is of course just special relativity.  With the first of these two formulas I can perfectly reproduce the graphs on time dilation as a function of $r$,  I find that the two contributions to time dilation cancel at 1.5 Earth radii, all numbers are OK... But why not the second formula?