Prove that $C(n,0) + C(n+1,1) +\dots+ C(n+r,r) = C(n+r+1,r).$
Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$
How to run a server on port 80 as a normal user on Linux? I have Googled about a solution for quite some time, but couldn't find an answer.   I am on Ubuntu Linux and want to run a server on port 80, but due to security mechanism of Ubuntu, I get the following error:      java.net.BindException: Permission denied:80   I think it should be simple enough to either disable this security mechanism so that port 80 is available to all users or to assign required privileges to the current user to access port 80. When I'm generating a List of Figures in my project, all figures are grouped by chapter. I would like to have the whole list without any additional new lines between different chapters.  How it looks like:  Figure 1.1  Figure 1.2  Figure 2.1  Figure 3.1  Figure 3.2   How it should:  Figure 1.1  Figure 1.2 Figure 2.1 Figure 3.1  Figure 3.2   I tried to apply the following solutions but failed completely :        My main file:  \documentclass[12pt,oneside,titlepage,utf8x]{report}  \usepackage{times,a4wide,multirow,tabularx,graphicx}  \usepackage{psfrag,fancyhdr,longtable}  \usepackage[T1]{fontenc} \usepackage{float} \usepackage[utf8]{inputenc}  \usepackage{url} \usepackage{color} \usepackage{verbatim} \usepackage{fancyvrb} \usepackage{natbib} \usepackage{relsize}  \usepackage{caption} \captionsetup[longtable]{belowskip=10pt}  \usepackage[left=4cm,top=2cm,right=3cm,bottom=2.5cm]{geometry}  \renewcommand{\chaptermark}[1]{\markboth{\nouppercase{\thechapter.\ #1}}{}} \pagestyle{fancy} \headheight 30pt \rhead{} \lfoot{} \cfoot{\thepage} \rfoot{}  \setlength{\parindent}{2em} \setlength{\parskip}{1em} \linespread{1.3}   \begin{document}  \begin{titlepage} \maketitle \end{titlepage}  \tableofcontents  \chapter{First} \label{sec:First} \input{chapter1/chapter1}  \chapter{Second} \label{sec:Second} \input{chapter2/chapter2}  \chapter{Third} \label{sec:Third} \input{chapter3/chapter3}  \relsize{-1} \listoffigures \normalsize  \bibliography{praca_mgr} \bibliographystyle{plain} \nocite{*}  \end{document} How does LSTM prevent the vanishing gradient problem? LSTM was invented specifically to avoid the vanishing gradient problem. It is supposed to do that with the Constant Error Carousel (CEC), which on the diagram below (from ) correspond to the loop around cell.  (source: ) And I understand that that part can be seen as a sort of identity function, so the derivative is one and the gradient stays constant. What I don't understand is how it does not vanish due to the other activation functions ? The input, output and forget gates use a sigmoid, which derivative is at most 0.25, and g and h were traditionally tanh. How does backpropagating through those not make the gradient vanish ? ref to subsection number only is there a way to reference the subsection part only? Example:  \documentclass{article} \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section. \subsection{One} \subsection{Two} \subsection{Three}\label{subsecthree}  \end{document}   Usually gives (without the stars):  1 This is my section For further information see subsection **1.3** within this section.   But I want:  1 This is my section For further information see subsection **3** within this section.   Is that possible?  [Edit] To make it more clear: I am looking for a command that works like \ref, but prints (the arabic number of) the value of the subsection counter at the position of the \label.  ======================= == Solution in LaTeX ==  There is another solution below. This is a different solution I came up with  (I cannot post an answer to my own question, so I put it here):  \documentclass{article}  \newcommand{\labelsubseccounter}[1]{     \renewcommand\thesubsection{\arabic{subsection}}     \addtocounter{subsection}{-1}     \refstepcounter{subsection}     \label{#1}     \renewcommand\thesubsection{\thesection.\arabic{subsection}} }  \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section,  which is called \ref{onepointthree}, not \ref{onepointfour}. \subsection{One} \subsection{Two} \subsection{Three}\label{onepointthree}\labelsubseccounter{subsecthree} \subsection{Four}\label{onepointfour}  \end{document}   (I would like to insert the compiled code, but don't know how, sorry) How to escape @ characters in Subversion managed file names? What is the result of $ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $ and why? How can I translate CMYK to Pantone colors? In particular, how do I translate CMYK 90C 85M 80Y 63K to a Pantone color? I've searched the net for such information and found different command lines, like these ones:  sudo apt-get remove application sudo apt-get remove application*  sudo apt-get remove --purge application sudo apt-get remove --purge application*  sudo apt-get purge application sudo apt-get purge application*   So, what is the correct way? Is it necessary to use that "*"?  After that, I also found these commands:  sudo updatedb sudo locate application sudo rm -rf (file/folder name) Why three-phase power? Why not a higher number of phases? Is there a reason, beyond historical reasons, that three phases has become the dominant number of phases?  I am aware of the advantages against one phase and two phase, namely the reduced amount of conductor needed, and that motors can provide torque when stalled (and less pulsation).  Is this solely due to diminishing returns, with only a small increase in smoothness of torque application, at the cost of increased complexity (increased number of wires (albeit of smaller CSA)).  To be clear, the phases are all evenly distributed, that is, five phases separated by 72 degrees. $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property] What are the options for traveling to the airports in Paris? Linear Combination of Random Normal Variables In order to prove that the linear combination of two independent normal distributions(say Z=X+Y) is normal, i am using their MGFs to show that the linear combination also has a similar mgf. This works okay when Z = X+Y. But when Z=X-Y, i run into trouble :  E[$e^{tZ}$] = E[$e^{t(X-Y)}$] = E[$e^{tX}$]E[$e^{-tY}$]  Now i cannot put this as $M_{x}$(t) * $M_{y}$(t) because of the negative sign. Is there evidence to suggest that Supreme Leader Snoke is Darth Plagueis? In Episode III,  explains to Anakin the power of the Dark Side to save people from death. He mentions  - the Sith who discovered this. Sidious then claims he died, which seems odd, given that kind of power.   Then in Episode VII (The Force Awakens) we find a Sith Lord who has appeared from no where. There seems to be two loose ends that fit nicely together.   My question is: Is there evidence to suggest that Supreme Leader Snoke is Darth Plagueis? I try to delete Action.005 permanently.    But I don't know how. If I click on the X    then it is still in the list.    I already researched without success, I even managed to loose all my animations by attempting to apply a fix, so I want to be cautious now... How can I do this? How to select lenses for outdoor sports photography? I would like to get into sports photography, and would probably want a better telephoto lens so I can get better close up shots.  What other factors should I keep in mind as I make my choice? Does IS (or VR) matter?  What is a good focal length?  I'll probably be shooting mostly little league baseball/softball, church softball leagues, and kids soccer. Printing labels along with equation numbers Working with documents with huge amount of equations it becomes cumbersome to go back and remind yourself what you labeled a certain equation to (I try to name them in a smart way, but there are too many).  Is there a convenient way to have equations print the label along with the tag/number, so that I can easily check the label in the compiled document?  Something like this would be perfect:  x = y               (14.1) eq:my_label Problem:  Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow  rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the Hilbert Schmidt Norm.  Please read my solution and tell me whether it is correct. If not, let me know where the mistake is.  Proof of the implication $\Leftarrow $:   If $rank(A)=0$, then in this case $A=0$. It follows that $\left \| A \right \|=\left \| A \right \|_{HS}=0$ If $rank(A)=1$, than: $ A=\begin{pmatrix} A_{1}\\  \alpha _{2}A_{1}\\  ...\\  \alpha _{n}A_{1} \end{pmatrix}$ where $A_{1}$ is the first row of $A$   and in this case: $\left \| A \right \|=\left \| A \right \|_{HS}=\left \| A_{1} \right \|\sqrt{1+\alpha _{1}^{2}+...+\alpha _{n}^{2}}$  Proof of the implication $\Rightarrow $   We know that $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|$  On the other hand: $ A=\begin{pmatrix} A_{1}\\  A_{2}\\  ... \\ A_{n}  \end{pmatrix}$. So, $$\left \| Ax \right \|=\left \| \begin{pmatrix} \left \langle A_{1},x \right \rangle\\  \left \langle A_{2},x \right \rangle\\  ...\\   \left \langle A_{n},x \right \rangle\end{pmatrix} \right \|=\sqrt{\left \langle A_{1},x \right \rangle^{2}+\left \langle A_{2},x \right \rangle^{2}+...+\left \langle A_{n},x \right \rangle^{2}} $$  Where $ \left \langle A_{i},x \right \rangle$ is the inner product of $A_{i}$ and $x$  Using the Cauchy-Schwarz inequality: $\left \langle A_{i},x \right \rangle^{2}\leq \left \| A_{i} \right \|^{2}\left \| x \right \|^{2}$, we get: $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|\leq max_{\left \| x \right \|=1}\left \| x \right \|\left \| A \right \|_{HS}=\left \| A \right \|_{HS}$. In order to have the equality: $\left \| A \right \|=\left \| A \right \|_{HS}$, we should have: $\left \langle A_{i},x \right \rangle^{2}=\left \| A_{i} \right \|^{2}\left \| x \right \|^{2} $ which occurs only if $A_{i}=\lambda _{i} x$. So, the rows of A are dependent, which implies that $rank(A)=1$. Note that the case $rank(A)=0$ happens when $A=0$  Please let me know if my solution makes sense. error: file '/grub/i386-pc/normal.mod' not found. grub rescue&gt;   What can I do? I just sit and stare at it.   I found my old netbook (Dell Inspiron 1010) which I have not used for about four years. I replaced Windows XP with Ubuntu 12.10. I used my bootable USB drive. I installed and rebooted. I got the message that normal.mod is not found.   What should I do? Type exit, reboot, or quit? Should I re-install? Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$ Let $l \in\mathbb{R} $ and $ f:(0,\infty) \to \Bbb R $ be a function such that $ \lim_{x\to \infty} xf(x)=l $ .  Prove that  $ \lim_{x\to \infty} f(x)=0 $.  Any help would be appreciated - I found this difficult to prove Change bend angle and swap node position on arrow