\documentclass[a4paper,12pt, twoside]{article} \usepackage{enumerate} \renewcommand{\baselinestretch}{1.5} \usepackage{amsmath,  amsthm,  amsfonts, mathrsfs, latexsym, amssymb, pstricks,graphicx,setspace,textcomp} \usepackage{pdfpages,caption} \usepackage[round]{natbib}  \bibliographystyle{apa} \usepackage{cite} \usepackage{nomencl} \usepackage{graphicx} \usepackage{float} \usepackage{underlin} \usepackage{url} \usepackage{longtable} \usepackage{titletoc} \usepackage{titlesec} \usepackage[normalem]{ulem} \usepackage{hyperref} \usepackage{caption} \usepackage{parskip} \setlength{\parindent}{20pt} \usepackage{setspace}  \usepackage{lipsum}  \usepackage{pdflscape}  \usepackage{multirow} \usepackage[utf8x]{inputenc} \begin document \begin{landscape} \chapter*{Plan of work} \begin{tabular}{|c|*7{p{32mm}|}} \hline \multirow{2}{*}{Task}  &amp;  \multicolumn{6}{c|}{Duration}\\ \cline{2-7}  &amp; Jan-15 to May-15 &amp; June-15 to Dec-15 &amp; Jan-16 to May-16 &amp; June-16 to Dec-16 &amp; Jan-17 to May-17 &amp; June-17 to Dec-17\\ \hline \multirow{2}{*}{Course Work} &amp;   &amp;  &amp;&amp;&amp;&amp;\\ \cline{2-7} &amp;  &amp;  &amp;&amp;&amp;&amp;\\ \hline \multirow{2}{*}{Literature Survey} &amp;&amp;&amp;&amp;&amp;&amp;\\ \cline{2-7} &amp;&amp;&amp;&amp;&amp;&amp;\\ \hline \multirow{2}{*}{Develop CZM }   &amp;&amp;&amp;&amp;&amp;&amp;\\ \cline{2-7} &amp;&amp;&amp;&amp;&amp;&amp;\\ \hline \multirow{2}{*}{Fracture Toughness}  &amp;&amp;&amp;&amp;&amp;&amp;\\  \cline{2-7}  for pure modes &amp;&amp;&amp;&amp;&amp;&amp;\\ \hline \multirow{2}{*}{Applying to mixed }  &amp;&amp;&amp;&amp;&amp;&amp;\\ \cline{2-7}   mode cases &amp;&amp;&amp;&amp;&amp;&amp;\\ \hline \multirow{2}{*}{Publications and }  &amp;&amp;&amp;&amp;&amp;&amp;\\ \cline{2-7}  Thesis Preparation &amp;&amp;&amp;&amp;&amp;&amp;\\ \hline \end{tabular} \newpage \end{landscape} \end document
I am looking for a convenient way to set the background of tabular cells. I'd like to do two things:  (1) Set the background shaded in a color (2) Set the background hatched  I learned that the TikZ package provides means to do this but I am unable to come up with a solution that does not use matrices (such as here: ) but instead uses the tabular package.  Here's what I'd like to do:  \documentclass{article} \begin{document} \begin{tabular}{|l|l|l|} \hline   1 &amp; \setBGGray 2 &amp; 3 \\\hline   4 &amp; 5 &amp; 6 \\\hline   7 &amp; \setBGHatched 8 &amp; 9 \\\hline \end{tabular} \end{document}   The output should look like this:    Can someone please help me achieving this using tabular (or tabularx) and TikZ?
Identify GoogleTv from Android app change caption in list of tables I would like to change the caption in the List of Tables in a way it is different from the caption in the corresponding table. How can I do that? Does there exist a command to do that? Can a creature's sacrifice benefit be taken when it is sacrificed to Eldrazi Monument? Windows 10 64-bit requirements: Does my CPU support CMPXCHG16b, PrefetchW and LAHF/SAHF? I'm currently trying to find out whether or not it would be a good idea to update my slightly dated notebook (Windows 7, 64-bit) to Windows 10. The problem is that Microsoft states in their  that to use the 64-bit version of Windows 10, a processor which supports CMPXCHG16b, PrefetchW and LAHF/SAHF is needed. I know that my processor is 64-bit (Intel Core i5-2430M @ 2.40 GHz) but I don't know how to find out whether it supports CMPXCHG16b, PrefetchW and LAHF/SAHF, and I have no idea what these mean.  I tried using Google and got no results that made sense to me, only some comparison of my i5 and a similar AMD processor. I also checked Intel's ARK website. If somebody could point me to where I can get more info I would be really grateful. How do you handle aspect ratio differences with Unity 2D? I've gotten a lot of answers to this question, but they are all generic and generally not very useful.  None of the tutorials talk about aspect ratio and dealing with mobile devices and there are a zillion ways to do it, all seem to have gotcha's and flaws.  I really would love to know what successful games have used to handle different aspect ratios on iOS and Android without making a zillion different sized assets.  I am strictly speaking mobile, not desktop, specifically with Unity and I don't care about the UI, I only care about the gameplay canvas.  Issues I have in mind is when there are key things that have to be in certain places and cannot fall off the screen.  Using black bars on top or bottom is unacceptable these days. Title vanishes when editing a question As a non-native speaker, I cannot grasp any difference between the expressions "here we go" and "there we go": both expressions seem to underline an event that is going to happen immediately.  Is there any subtle usage detail that I am missing? Tikz shortandoff-on newcommand problem Usually I have no problem with babel and !:, and I don't use shortandoff-on switch. But since I made a command using !#1 parameter, the problem comes up again.  The same synthax used outside the command work fine, but in the command I get the error message :  ! Package tikz Error: + or - expected.  How to solve the problem without using the shortandoff-on switch ?  \documentclass[french,a4paper]{article} \RequirePackage{etex} \RequirePackage[utf8]{inputenc} \RequirePackage[T1]{fontenc} \RequirePackage{fourier}  \RequirePackage{tikz} \usetikzlibrary{calc,intersections}  %%%%%%%%%%%%%%%%%%%%%% %%%% Francisation %%%% %%%%%%%%%%%%%%%%%%%%%%  \RequirePackage{babel}  %%%%%%%%%%%%%%%%%%% ---------------------------------------- %%%% Géométrie %%%% %%%%%%%%%%%%%%%%%%%-----------------------------------------      % #1 option du path     % #2 premier point     % #3 second point     % #4 longueur dans un sens     % #5 longueur dans l'autre     % défini Mil#1#2 \newcommand{\Med}[5][]{%     \path[#1,name path=Med#2#3] ($(#2)!.5!(#3)$) node (Mil#2#3) {}      ($(Mil#2#3)!#4!90:(#3)$) -- ($(Mil#2#3)!#5!270:(#3)$) ; }  \tikzset{small dot/.style={fill=black,circle,scale=0.3}}  \begin{document}  %\shorthandoff{:!} \begin{tikzpicture}  \coordinate[label=below left:$A$] (A) at (0,0) ; \coordinate[label=below right:$B$] (B) at (4,0) ;  \draw (A)--(B) ;  %\Med[draw]{A}{B}{1}{1} ;  \draw($(A)!.5!(B)$) node (MilAB) {}      ($(MilAB)!1!90:(B)$) -- ($(MilAB)!1!270:(B)$) ;   \end{tikzpicture} %\shorthandon{:!}  \end{document} Finding Maximum value of consecutive positive integers with constraints Let, $x,y,z$ be consecutive positive integers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} &gt; \frac{1}{10}$ I need to find the maximum value of $x+y+z$ ?  My attempt:  I tried guessing first, To maximise $x+y+z$ it is necessary to have high value for $x$, But not too high which will contradict the constrain of the problem. As we can roughly split the $0.1$ into three parts so I started with numbers closer to $30$ with trial an error I got that for $x=29$ we get the optimum and final result would be 90 (=max. value of $x+y+z$).   Is there any way to arrive at this conclusion with using inequalities? Any hints will be appreciated... The Duals of $l^\infty$ and $L^{\infty}$ Disable the guest user completely in El Capitan without having to disable 'Find My Mac' I know it should be straight forward but unfortunately, it  isn't.  I did disable the guest user in the System Preferences by unchecking the checkbox for Allow guests to login in to this computer, but it still shows up on the login screen.  I have heard that disabling 'Find My Mac' would fix it for OS X Lion, but I want to keep it.  Is there a way to do this? Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$ Prove by mathematical  induction that:   $$\forall n \in \mathbb{N}: 3^{n} &gt; n^{3}$$  Step 1: Show that the statement is true for $n = 1$:  $$3^{1} &gt; 1^{3} \Rightarrow 3 &gt; 1$$  Step 2: Show that if the statement is true for $n = p$, it is true for $n = p + 1$  The general idea I had was to start with $(p+1)^{3}$ and during the process substitute in $3^{p}$ for $p^{3}$ as an inequality.  $$(p+1)^{3} = p^{3} + 3p^{2} + 3p + 1 &lt; 3^{p} + 3p^{2} + 3p + 1$$  Now, if it can be shown that:  $$\forall p \in \mathbb{N}: 3p^{2} + 3p + 1 \leq 2 \cdot 3^{p}$$  ...the proof is complete. This is because $3^{p+1} = 3 \cdot 3^{p}$ and one of those three have already been used.   We do this by mathematical induction. First, the base case of $n = 1$:  $$3\cdot 1^{2} + 3 \cdot 1 + 1 \not \leq 2 \cdot 3^{1}$$  ..which turns out to be false.  What are some more productive approaches to this step? Why are orbits elliptical? Many introductory quantum mechanics textbooks include simple exercises on computing the de Broglie wavelength of macroscopic objects, often contrasting the results with that of a proton, etc.  For instance, this example, taken from a textbook:     Calculate the de Broglie wavelength for      (a) a proton of kinetic energy 70 MeV      (b) a 100 g bullet moving at 900 m/s   The pedagogical motivation behind these questions is obvious: when the de Broglie wavelength is small compared to the size of an object, the wave behaviour of the object is undetectable.  But what is the validity of the actual approach of applying the formula naively? Of course, a 100g bullet is not a fundamental object of 100g but a lattice of some $10^{23}$ atoms bound by the electromagnetic force. But is the naive answer close to the actual one (i.e. within an order of magnitude or two)? How does one even calculate the de Broglie wavelength for a many body system accurately? How to participate in clan war instead of being a spectator I joined a new clan and now that the clan war has started I tried to attack but it says "Spectator can't attack". How do I get out of being a spectator (I haven't used either of my 2 attacks in the war). I went to the clan castle and the clan option and I selected "Yes I'll participate in this war" but nothing happens. I'm looking for a way to insert continuous colour legend for a raster layer.  I managed to create discrete symbols with values along the bar, but I would like to have a continuous bar with some values on the bar, similar to the old plugin "One-Band Raster", that it doesn't work with the QGIS version higher than 1.7  I tried to set in properties->style, where I selected the  Render Type "Singleband pseudocolor" and see the "generate new color ramp" box.  I tried both "Continuous" or "Equal interval". However, in the Composer the legend always is displayed with discrete symbols.  Is there any way that can make QGIS Composer to display a nice continuous band for a raster layer? I am wondering how to show that $\mathbb{Z}[\sqrt {10}]$ is not a UFD.    My only idea is to show that there are two factorizations of $10$, say, $ab, uv$ such that $a$ is not a unit times $u$ or $v$. In this ring $10=2\cdot5=\sqrt {10}\cdot \sqrt {10}$, so it suffices to show $2$ is not a unit times $\sqrt {10}$. Suppose $2=\sqrt {10}(a+b\sqrt{10})=a\sqrt{10}+10b$. Then $a=0$ since $\sqrt{10}$ is not rational. So $10b=2$, which has no integer solutions. So $\mathbb{Z}[\sqrt {10}]$ is not a UFD.  Is my reasoning correct? What are the flaws? Normal subgroup of a normal subgroup Let $F,G,H$ be groups such that $F\trianglelefteq G \trianglelefteq H$.  I am asked whether we necessarily have $F\trianglelefteq H$. I think the answer is no but I cannot find any counterexample with usual groups. Is there a simple case where this property is not true? I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$.  One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th edi, p442) is as following: $$ \langle x|[x,p]|y \rangle = \langle x|xp-px|y \rangle = (x-y) \langle x|p|y \rangle. $$ On the other hand,  $ \langle x|[x,p]|y \rangle = i \langle x|y \rangle = i \delta (x-y)$. Thus $$ (x-y) \langle x|p|y \rangle = i \delta(x-y). \tag{1} $$  We use $(x-y) \delta(x-y) = 0$. Take the derivative with respect to $x$; we have $\delta(x-y) + (x-y) \delta'(x-y) = 0$. Thus $$ (x-y) \delta'(x-y) = - \delta(x-y). \tag{2} $$  Comparing Eqs. (1) and (2), we identify $$ \langle x|p|y \rangle = -i \delta'(x-y). \tag{3} $$  In addition, we can add $\alpha \delta(x-y)$ on the right-hand side of Eq. (3), i.e. $$ \langle x|p|y \rangle = -i \delta'(x-y)  + \alpha \delta(x-y), $$ and $[x,p] = i$ is still satisfied. We can also add $$ \frac{\beta}{\sqrt{|x-y|}}\delta(x-y)$$ on the RHS of Eq. (3).  Here $\alpha$ and $\beta$ are two real numbers.  My question is, what is the most general expression of $\langle x|p|y \rangle$? Can we always absorb the additional term into a phase factor like Dirac's quantum mechanics book did?  Thank you very much in advance. sum of noncentral Chi-square random variables I need to find the distribution of the random variable  $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions for $X_i$s, and then transform back to obtain $Y$'s distribution. However, I wonder whether there is a general form for $Y$ like the Gaussian case: we know the sum of independent Gaussian is still a Gaussian, and thus we only need to know the summed mean and summed variance.   How about all $\sigma^2_i=\sigma^2$? Will this condition make a general solution?