Add a URL in faces message JSF 2.0
Embedding a link (or other html) in a JSF message
This is Exercise 3.1.A. in      Suppose that $\pi: X\rightarrow Y$ is a continuous map of differentiable manifolds. Show that $\pi$ is differentiable if differentialble functions pull back to differentiable functions, i.e., if pullback by $\pi$ gives mpa $\mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$.   Let $f: V\subseteq Y\rightarrow \mathbb{R}$ be a differentiable function on an open subset of $Y$. Let $(U,\phi)$ be a chart of $X$, $(V,\psi)$ be a chart of $Y$. Then $f\circ\pi(\phi^{-1})$ is differentiable and $f\circ\psi^{-1}$ is differentiable. We need to show that $\psi\circ\pi\circ\phi^{-1}$ is differentiable. I don't know how to prove this.   Can we say that since $(U,\psi\circ\pi)$ constructs a chart of $X$, by the compatibility, $\psi\circ\pi\circ\phi^{-1}$ is differentiable? This seems true but this does not use the fact that $f$ is differentiable.  Sorry I have no background of Differentiable Manifolds. Any help would be appreciated.  I also saw this post: , but I don't understand how $\gamma$ is smooth in the answer. $\gamma=\beta\circ\rho$, where $\rho$ is defined to be smooth, but $\beta$ is not. Variable number of arguments in C++? Help me to solve this problem please..  Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 &lt; y &lt; \infty$, zero elsewhere. Show that $Z_1 = Y_{(2)}$ and $Z_2 = Y_{(4)} − Y_{(2)}$  are independent. Hint: First find the joint p.d.f. of $Y_{(2)}$  and  $Y_{(4)}$. a, b are integers. p is prime. I want to prove: $(a+b)^{p} \equiv a^p + b^p \pmod p$  I know about Fermat's little theorem, but I still can't get it I know this is valid: $(a+b)^{p} \equiv a+b \pmod p$ but from there I don't know what to do.  Also I thought about $(a+b)^{p} = \sum_{k=0}^{p}\binom{p}{k}a^{k}b^{p-k}=\binom{p}{0}b^{p}+\sum_{k=1}^{p-1} \binom{p}{k} a^{k}b^{p-k}+\binom{p}{p}a^{p}=b^{p}+\sum_{k=1}^{p-1}\binom{p}{k}a^{k}b^{p-k}+a^{p}$ Any ideas?  Thanks! Integral of Wiener Squared process I don't have a background of stochastic calculus.   It is known  that definite integral of standard  from $0$ to $t$ results in another Gaussian process with slice distribution that is normal distributed with mean equal to $0$ and variance $\frac{T^3}{3}$ i-e  $$ \int_0^{t} W_s ds \sim \mathcal{N}(0,\frac{t^3}{3})    $$  Question:  What if we square the standard Wiener process and then integrate i-e $$ \int_0^{t} W_s^2 ds \sim ?    $$  Would that be scaled Chi-square distributed ? Does $\sum_{n=1}^\infty \frac{\cos{(\sqrt{n})}}{n}$ converge? How do I check if an array includes a value in JavaScript? What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? Is there a word for people who always act too friendly in order to curry favor with their acquaintances? Is there a single noun or adjective for a person who is always flattering friends and acquaintances, not only to be liked and accepted but, at other times, to curry favor with them as well ?  I'm not looking for regionalisms but a word that will be understood in any English speaking country. Does the 2020 Apple M1 MacBook Pro support 2x daisy chained Thunderbolt Displays? The Apple Silicon based  supports Thunderbolt version 3 and I'm wondering if anyone has tried connection one to an ? Of course you'd need to use one of Apple's Thunderbolt 3 to Thunderbolt 2 converters. I also understand that this version is limited to supporting only a single 4K or 6K display, but since the original Apple Thunderbolt Display is only 2K I’d want to try and daisy chain these. Can I run more than one external display from the Apple Silicon M1 CPU/GPU based MacBook Pro (or Air)? Drawbacks of making public which journal a manuscript is submitted to, while it is under review? Many researchers list out their submitted manuscripts on their website. Some specify the journal name, like     Smith J, Oliver J (under review) My awesome paper. Nature.   while others don't, like     Smith J, Oliver J (under review) My awesome paper.   One reason I can think of as to why some don't mention the journal name is that they may get embarrassed in the case where the current journal rejects the paper. Then, the "circle" will know this paper gets rejected from Nature first and then ends up in an inferior journal.  Besides this, are there any other considerations preventing people from mentioning to which journal they submit a paper to?  P.S.: I am asking because my advisor is a researcher who only mentions accepted  papers on his webpage. As a student hunting jobs, I feel it is advantageous for me to mention my submitted papers on my webpage. I hesitate to do so, as I worry I may make my advisor unhappy by announcing in which journal our papers are currently under review. I just came across this code and a few Google searches turn up no explanation of this mysterious (to me) syntax.  Hashtable^ tempHash = gcnew Hashtable(iterators_);  IDictionaryEnumerator^ enumerator = tempHash-&gt;GetEnumerator();   What the heck does the caret mean? (The gcnew is also new to me, and I asked about that .) How to adjust the gap between figure and caption within the same subfloat My question is:  How to adjust the gap between figure and caption within the same subfloat  Not the gap between two subfloat.  For the following example:  \begin{figure}     \centering     \subfloat[A]{\includegraphics[ scale=0.5]{images/a.jpg}}     \vspace{-10pt}     \subfloat[B]{\includegraphics[scale=0.5]{images/b.jpg}}     \vspace{-10pt}     \caption{}\label{fig:online_test} \end{figure}   I got the following figure:    The sub-caption of (a) is too close to the sub-figure B. I hope the caption (a) to stay closer to sub-figure A. How can I adjust the space between the caption and figure within a subfloat?  Thank you all for helping me and I would prefer a solution for subfloat, instead of using other packages. But I would also appreciate a solution of using other packages.   I know my question looks similar to . But please make it clear. I'm asking for how to adjust the gap between figure and caption within a subfloat, not the gap between two subfloat. That is, the gap between the caption "a" and the figure "A"  Thank you. Identical creatures entering the battlefield Proof a $(2n-1)$-compact manifold I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold.  How give the charts. For $n=1$ and $d=2$ not distinguish which 1- manifdold is? What is the principle of underdetermination? While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Using same modulus for RSA Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back?? I want to know how to get the X and Y position of HTML elements such as img and div in JavaScript relative to the browser window. A locally constant sheaf on a locally connected space is a covering space; Proof? Proof that exponential function's limit definition increases monotonically? It is well known that we can define $e^x$ by the following limit  $$e^{x}=\lim_{n\to\infty}\left(1+{x\over n}\right)^n$$  I would like to show that the RHS sequence is always less than or equal to $e^x$ for all $-n\le x\le0$ and $n&gt;1$, and what I had currently done is to study the property of this sequence (which is defined as $f(x)$)  $$f(x)=\left(1+{x\over n}\right)^n$$  I also find its first and second derivative $f'(x)=\left(1+{x\over n}\right)^{n-1}$ and $f''(x)={n-1\over n}\left(1+{x\over n}\right)^n$ and show that they are strictly positive for all $x\in[-n,0]$. As a result $f(x)$ is monotonically increasing and concave up. By plugging in the end points I found that $f(0)=1\le e^0$ and $f(-n)=0&lt;e^{-n}$, I wonder if these conditions allow me conclude that $f(x)\le e^x$ for all $x\in[-n,0]$.