Should I put the verbs in the infinitive form in the following sentence:     As the federal auctions provide for a limited time to elaborate the   technical studies, to develop the project and to apply for the   necessary licenses   or should I omit the preposition "to" and instead write:     ... to elaborate the technical studies, develop the project, and apply   for the necessary licenses
I googled it but I couldn't find the answer... maybe I didn't seek it in the proper way. My question is if &quot;to&quot; must be repeated in sentences such as the following ones:   In the past, women used to do the housework and to take care of children.  The aim of this meeting is to control their progress and to discuss the new working plan.    If I had to bet, I would omit the second to... But I am not sure about it.
How to use downloaded fonts with pdfLaTeX on Ubuntu I want to use the  font in my pdfLatex document.  What steps should I do for using a downloaded font   Should I download the TTF or the OpenType version? Where should I store the font? Should any font processing be done? How can pdfLatex find this font?   I am using TexLive 2011 installed in a custom location (/somepath/texlive/2011) on Ubuntu Egyptian fractions with prime power denominators summing to 1? Inspired by        Can we solve the equation $$1=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots +\frac{1}{a_n}$$ where $a_1,a_2,a_3,\cdots ,a_n$ are distinct prime powers (primes are allowed) ?   I tried a modified greedy-algorithm with different starting vectors and brute force with $6$ entris with limit $200$, but did not find a representation. Let $A$ be a square matrix of order $n$ such that $A^2 = I$.  Prove that if $1$ is the only eigenvalue of $A$, then $A = I$. Prove that $A$ is diagonalizable.   For  (1), I know that there are two eigenvalues which are $1$ and $-1$, how do I go about proving what the question asks me? Can I use two different cards with the same name in Commander? It's using Euclidian inverse. Can you show it step by step?   $$N=7, q=11, a = 3+2X^2 -3X^4 +X^6$$  the inverse of $a \pmod {11}$ is  $$A=-2+4X+2X^2 +4X^3 -4X^4 +2X^5 -2X^6$$  How do we compute $A$? Simple enough question: What math should all game programmers have a firm grasp of in order to be successful?  I'm not specifically talking about rendering math or anything in the niche areas of game programming, more specifically just things that even game programmers should know about, and if they don't they'll probably find it useful.  Note: as there is no one correct answer, this question (and its answers) is a community wiki.  Also, if you would like fancy latex math equations, feel free to use . ActionBarSherlock with ViewPager not calling ViewPager fragments lifecycle methods when going to and returning from backstack To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$ if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here $L^p$ spaces and counting measure currently I am working on the following two exercises as a revision for my exam.  Let $\mu$ be the counting measure on $\mathbb N$.   Show that if $1 \le p &lt; s &lt; \infty$ then $f \in L^p$ implies $f \in L^s$ and $\|f\|_s \le \|f\|_p$. Show that if $f \in L^p$ for some $p \ge 1$ then $\lim_{s \to \infty} \|f\|_s = \|f\|_\infty$.   My attempts so far:   It is $f \in L^s$ iff $\sum_{k = 1}^\infty |f(k)|^s$ converges. Since $f \in L^p$ we know that $\sum_{k = 1}^\infty |f(k)|^p$ converges. Thus for any $\varepsilon &gt; 0$ there is $N \in \mathbb N$ s.t. \begin{align*} k \ge N \Longrightarrow |f(k)|^p &lt; \varepsilon \Longrightarrow |f(k)| &lt; \varepsilon^{1/p} \end{align*} We have $\forall k \ge N$ that $|f(k)|^s &lt; \varepsilon^{s/p}$. Originally I wanted to use the comparison test now, showing that \begin{align*} \sum_{k = N}^\infty \varepsilon^{s/p} &lt; \infty, \end{align*} but that does not work since \begin{align*} \sum_{k = N}^\infty \varepsilon^{s/p} = \varepsilon^{s/p} \underbrace{\sum_{k \ge N} 1}_{= \infty},  \end{align*} so I don't know how to go on here. Let $K := \|f\|_\infty$. We need to show that $\forall \varepsilon &gt; 0$ $\exists M \in \mathbb N$ s.t. \begin{align*} s \ge M \Longrightarrow \left|\left(\sum_{k = 1}^\infty |f(k)|^s\right)^{1/s} - K\right| &lt; \varepsilon. \end{align*} Choose $M$ s.t. $M &gt; p$. Then $s &gt; p$ and from part (i) we know that \begin{align*} \left(\sum_{k = 1}^\infty |f(k)|^s\right)^{1/s} \le \left(\sum_{k = 1}^\infty |f(k)|^p\right)^{1/p}. \end{align*} But how can I use this? Conditional Number Formatting In Java Is $(x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}}$ Differentiable in $(0,0,0)$? Book, it follows a guy in a dystopian future What is a rocket engine thrusting against in space? I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? What is the best way to find out a researcher's current affiliation? Let's say I know a name and a field, but not much more.   A google search seems like the most obvious approach, but unfortunately university personnel or research group pages often seem to be quite poorly maintained, and it is not uncommon that someone is listed even if they left more than a year ago.  Early-career researchers are particularly difficult to track down since they often have less of a "presence" on institution webpages and are likely to be moving relatively frequently. Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks! This is, how I download various master branches from GitHub, and I aim to have a prettier script (and maybe more reliable?).  wget -P ~/ https://github.com/user/repository/archive/master.zip unzip ~/master.zip mv ~/*-master ~/dir-name   Can this be shorten to one line somehow, maybe with tar and pipe?  Please address issues of downloading directly to the home directory ~/ and having a certain name for the directory (mv really needed?). I am typing to get a sale amount (by input) to be multiplied by a defined sales tax (0.08) and then have it print the total amount (sales tax times sale amount).  I run into this error. Anyone know what could be wrong or have any suggestions?   salesAmount = raw_input (["Insert sale amount here \n"]) ['Insert sale amount here \n']20.99 &gt;&gt;&gt; salesTax = 0.08 &gt;&gt;&gt; totalAmount = salesAmount * salesTax  Traceback (most recent call last):   File "&lt;pyshell#57&gt;", line 1, in &lt;module&gt;     totalAmount = salesAmount * salesTax TypeError: can't multiply sequence by non-int of type 'float' CSS3 selector to find the 2nd div of the same class I need a CSS selector that can find the 2nd div of 2 that has the same class. I've looked at nth-child() but it's not what I want since I can't see a way to further clarify what class I want. These 2 divs will be siblings in the document if that helps.  My HTML looks something like this:  &lt;div class="foo"&gt;...&lt;/div&gt; &lt;div class="bar"&gt;...&lt;/div&gt; &lt;div class="baz"&gt;...&lt;/div&gt; &lt;div class="bar"&gt;...&lt;/div&gt;   And I want the 2nd div.bar (or the last div.bar would work too). What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists? Uninstall GRUB and use Windows bootloader