By changing width task environment is not working properly I am working in extbookdocument class. I prepared my book with setting width size 7 inches by using the command  \usepackage[paperheight=9in,paperwidth=7in,top=1in,bottom=1in,right=1in]{geometry}   But if I change my width size to 6 inches then task environment is NOT giving required output.  My code is      ...     \usepackage[paperheight=9in,paperwidth=7in,top=1in,bottom=1in,right=1in]{geometry}     ...      \NewTasks[counter-format = {\bf tsk[1].},label-width=24pt]{numbering}[\op]      ...      \section{Answers}     \begin{numbering}(5)     \op (C)     \op (A)     \op (C)     \op (A)     \op (B)     \op (D)     \op (D)     \op (C)     \op (A)     \op (D)     \op (C)     \op (A)     \op (A)     \op (C)     \op (B)     \op (D)     \op $\pm2$, $\pm\sqrt2$     \op*(3) $(q-s)^2-rqp-rsp+sp^2+qr^2$     \op\SetTaskNumber{21} (D)     \op (B)     \op (A)     \op (C)     \op (C)     \end{numbering} ...   The ouput of this code is perfecly alright but if I change the width size to 6 inches the output is not good. I am sharing both outputs. Please tell me what alterations do I need to do if I want the same as it is in the width of 7 inches.  I use font size 10 pt for 7 inches width and 9 pt for 6 inches width by using the command \documentclass[10pt,a4paper]{extbook}.
tasks package: items are in two line with 4 columns and 4 items I write some choice questions into beamer frame, after some research, here is my code:  \documentclass[aspectratio=169]{beamer} \usetheme{Berkeley} \usepackage{tasks,CJK} \begin{document} \begin{CJK*}{utf8}{bsmi} \begin{frame} \transwipe 1.Which one you want to choose? \begin{tasks}[counter-format=tsk[A].,debug=true](4) \task A \task B \task C \task D \end{tasks} \end{frame} \end{CJK*} \end{document}   output with pdflatex:  But I don't want the D choice goto the second line, I want them stay in one line, what should I do?
Generate random numbers uniformly over an entire range If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges? Let $a_n$ be a sequence of complex numbers that converge to zero. Can we always find $s_n \in \{-1,1\}$ such that $\sum_{n=1}^{\infty} s_n a_n$ converges?  If the $a_n$ are real numbers, we can find such a sequence $s_n$. If the partial sum of the first $N$ terms is positive we make sure the following terms are negative until the sum becomes less than zero. Then we switch to making the terms positive until the partial sum becomes greater than zero, and so on. It is easy to see that the partial sums will either tend monotonically towards zero, or oscillate around zero with decreasing amplitude.   Edit: Actually there is no reason that the partials sums should go to zero in the monotone case. They will still converge however. What is the ultimate postal code and zip regex? I'm looking for the ultimate postal code and zip code regex. I'm looking for something that will cover most (hopefully all) of the world. Is there a way in photoshop to convert k values to alpha values? Basically, I would like to make it so that the lighter a pixel is (the more white it has) the more transparent it becomes. Black pixels would be opaque, white pixels would be transparent.  I currently use the slider in blending options on the first tab to get rid of the white and make it transparent but find it works slightly better if I use the magic wand tool with delete.  I figure that there is some blending option out there but there are so many and I'm not sure which to use to achieve my goal.  (Using Photoshop CS6) Does white whole wheat flour need to be refrigerated? I know that regular whole wheat flour needs to be refrigerated, but does white whole wheat flour also need refrigeration? I hate to use up the space in the refrigerator if I don't have to.  I have 5 lb. (2.3 kg), and it would probably take me a month or so to use it all. How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k$ for some positive integer $k$? Prove or disprove: if $\sum\limits_{n=0}^∞a_n$ converges and $\lim\limits_{n\to\infty}b_n =0$, then $\sum\limits_{n=0}^{\infty}a_n b_n$ converges. Hydraulic Disk brake piston not fully retracting My front disk brake's piston doesn't fully retract.  While I can ride the bike, the brake is slightly rubbing.  (IE, if I just spin the wheel with my hand, it'll go around about one and a half times and then stop.) It's definitely slowing me down.   If this fixable?  The advice I got was that it's not worth the trouble and to buy a replacement set.  The current brakes are Avid Juicy 7s that are about 7 years old I am trying to make an explosion in Blender.  I am having a problem with the smoke simulation with the Cycles rendering engine. More specifically, I have been able to render smoke, but flames do not show up. In the viewport however, everything looks perfectly fine.  The render looks like this:    But in the viewport, it is clear that there are flames. Does anyone have any idea what is going on?     This is NOT a duplicate of  because I have Flame Rate set to 1.0 I'm trying to show that for $b&gt;1$, $x&gt;0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0&lt;p&lt;x\} = \inf\{b^q:q\in Q,\;x&lt;q\}$$ I know this follows immediately if we define $b^x=\exp(x\log(b))$ and use the fact that the exponential function is strictly increasing and continuous over $\mathbb R$, but I'm not allowed to do that. In fact, I only have that $b^{m/n}=(b^{1/n})^m$, where $b^{1/n}$ means the positive $n^{th}$ root of $b$; $b^x$ is not yet defined for irrational $x$.  I know that the supremum and infimum both exist because if $b&gt;1$ and $p&lt;q$, then $b^p&lt;b^q$. Also, by assuming that the inf $&lt;$ sup, I got a contradiction. The part I'm stuck on is deriving a contradiction from assuming that sup $&lt;$ inf. Does anybody have any ideas?  Thank you. How to turn a ruled pattern into a multivariable function Prove that $\sqrt 2 + \sqrt 3$ is irrational I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct? Let $K$ be any field. Is it true that:   If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, then is it prime? In particular, is it true that $K[x_1,\dots,x_n]/I$ is isomorphic to a polynomial ring (in the case that $I$ is proper and hence prime by the statement above)?   If they are true, how can i prove these statements in an elementary but rigorous way?  P.S.: Birth of my question: I found on my book that "affine varieties given by linear equations are irreducibile", so I translated (I don't know if I have made it right) this geometric statement into the analogous algebraic version. My Bridgestone MB-5 currently has a 7-speed rear wheel. I'd like to convert it to a 10-speed. Is this possible? Is it practical? How do I get from   $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1}  (1-t)^{ \beta -1} \,dt}$$ to:   $$2^{2- \alpha - \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha - \beta )x)}{( \cosh(x))^{ \alpha + \beta }} \,dx}$$  ? I have tried different change of variables such as $t=1- \frac{1}{ \cosh (x)}$ and got $\int_{0}^{ \infty}{ \frac{ \sinh (x)  ( \cosh (x)-1)^{ \alpha -1} }{( \cosh(x))^{ \alpha + \beta }} \,dx}$ but without success. Which is correct if I want to request for a pen?        Can I have your pen please?   May I have your pen please? I see people say statement as question while speaking by raising their voice at the of it to mean it's a question.   For example consider followings.         You want to stay here? ==> Is used to mean "Do you want to stay here?"   You guys spoke to anyone? ==> Is used to mean "Did you guys speak to anyone?"      Are these really used by native speaker in casual speech? And can I write them in the same way while writing them as dialogue by putting them between quotes? Are there any special grammar rules for these kind of questions? Will C++ exceptions safely propagate through C code? Node.js on multi-core machines Laravel loading error in Ubuntu 14.04