Multimodal - Packing a bike for a trip with several different transport stages
Packaging a bike for flying with a bike bag
List of lists changes reflected across sublists unexpectedly Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, we can take $\mu$ to be supported on a point, say $0$, then it is not necessarily true that  $$f'(0)g(0) = -g'(0)f(0).$$  If $\mu$ is absolute continuous w.r.t. Lebesgue measure, we have $\frac{d\mu}{dx} = h$ $$\int f'gd\mu = \int f'gh dx = -\int f(gh)'dx$$ where $(gh)'dx$ might be a measure. but we can not recover the form $\int g'fh dx$.  Thank you very much! Sending form POST request to PHP What is (meant by) a non-compact $U(1)$ Lie group? In John Preskill's  of monopoles he states on p. 471     Nowadays, we have another way of understanding why electric charge is quantized. Charge is quantized if the electromagnetic $U(l)_{\rm em}$ gauge group is compact. But $U(l)_{\rm em}$ is automatically compact in a unified gauge theory in which $U(l)_{\rm em}$ is embedded in a nonabelian semisimple   group. [Note that the standard Weinberg-Salam-Glashow (35) model is   not "unified" according to this criterion.]   The implication of the third sentence is that, in some circumstances, the $U(1)_{\rm em}$ gauge group may not be compact.  How could this be?  Since $U(1)$ as a differentiable manifold is diffeomorphic to $S^1$ isn't it automatically always compact?  The following paragraph:     In other words, in a unified gauge theory, the electric charge operator obeys nontrivial commutation relations with other operators in the theory. Just as the angular momentum algebra requires the eigenvalues of $J_z$ to be integer multiples of $\frac{\hbar}{2}$, the commutation relations satisfied by the electric charge operator require its   eigenvalues to be integer multiples of a fundamental unit. This   conclusion holds even if the symmetries generated by the charges that   fail to commute with electric charge are spontaneously broken.   is OK, but I don't follow what that has to do with the compactness of $U(1)$. In the middle of touching up a new texture, I pressed save and suddenly my textures all turned black on the model and UV texture - just vanished.  I tried reopening the file in another instance of Blender, I tried to CtrlZ too and it just did this weird patchy thing on my texture.   How do I recover it and what did I do wrong? How is Russell’s Paradox equal to the universal set? Is Ubuntu an open source software? Why have  on every line I've seen this convention pretty much everywhere, and, at times, it comes close to driving me nuts:  &lt;?php //The loop ?&gt; &lt;?php while ( have_posts() ) : the_post(); ?&gt;     &lt;?php the_content(); ?&gt; &lt;?php endwhile; // end of the loop. ?&gt;   Where the &lt;?php and closing ?&gt; are on every single line, even if there is no intervening HTML code.    My question is: Why?  Why include all these extra tags?      To me, it seems this convention adds a significant amount of clutter to code, is annoying to follow in the first place, and adds that many more places to accidentally leave out an opening or closing tag.  NOTE  This is code pulled from the Twenty-Twelve theme, the example given by WordPress. How do I simplify and evaluate the limit of $(\sqrt x - 1)/(\sqrt[3] x - 1)$ as $x\to 1$? Consider this limit:  $$ \lim_{x \to 1} \frac{\sqrt x -  1}{ \sqrt[3] x -  1}  $$  The answer is given to be 2 in the textbook.  Our math professor skipped this question telling us it is not in our syllabus, but how can it be solved? is it possible to redefine { and } in C? How can metagaming retries on poor skill checks be prevented? Proving with induction $(1-x)^n<\frac 1 {1+nx}$ Prove using induction that $\forall n\in\mathbb N, \forall x\in \mathbb R: 0&lt;x&lt;1: (1-x)^n&lt;\frac 1 {1+nx}$   My attempt:   Base: for $n=1: 1-x&lt;\frac 1 {1+x}\iff 1-x^2&lt;1$, true since $0&lt;x&lt;1$.  Suppose the statement is true for $n$, prove for $n+1$:  $(1-x)^{n+1}=(1-x)(1-x)^{n}\overset{i.h}&lt;\frac{(1-x)}{1+nx}$  Now I got stuck, maybe another induction to show that $1+nx+x&lt;1+nx$? Is there another way?  Moreover, I was told it's wrong to begin with $(1-x)^{n+1}$ and reach to $\frac 1 {1+(n+1)x}$ but why? Is it assuming what I need to prove? Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$ So My professor assigned this question and I am really stuck on part B of the question.  For $n \in \mathbb{N}$ let {$T_n$} = {$1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}$}.  (a) Prove that {$T_n$} is non-decreasing.  (b) Use {$S_n$} = {$(1+\frac{1}{n})^n$} to prove that {$T_n$} is bounded above and prove that $lim_n(T_n)\ge lim_n (S_n)$   Like I mentioned I can prove part (a) simply by supposing that $T_{n+1} \lt T_n$ then after some algebra concluding the statement is false and than with one other example proving that its not strictly increasing either.  Now for part (b) I have not a slightest clue about how to solve, I assume I will have to use the expanded form of $S_n$ but thats the best I have. So please give me some hints or nudges in the right direction. A function that is continuous at the origin but not differentiable there Show that $f(x,y)=\dfrac{xy^2}{x^2+y^2}$ (with $(x,y)\not=(0,0)$ and $f(0,0)=0$) is continuous but not differentiable at $(0,0)$.  I tried to show continuity with an $\epsilon -\delta$ argument but I don't know how to factorize the expression so that I can have something useful.   For differentiability I think I should show that the partials are not continuous at $(0,0)$. But finding the partials is also painful. Functional equations leading to sine and cosine drupal 7 + services 3 - inserting nodes with select fields I'm using Drupal 7 + Services 3 to create nodes with POST and JSON. Everything works fine, except field types of select - entity reference select or plain text.  The entity reference uses the autocomplete widget, while the text select is a plain radio button.  This is the format I'm using:  "field_urgency": {     "und": [  {  "value": 1    }   ] } Package incompatibilites: etoolbox, hyperref, and bm, standalone? I am trying to transition to using the etoolbox package, but seem to have run into some difficulties and they appear related to the hyperref and bm packages which to me seem totally unrelated.  The MWE example compiles since \RemoveEToolbox is defined and shows that there is no apparent problem before etoolbox was used.    Commenting out the \def\RemoveEToolbox{} line produces two problems:   With both the \renewcommand, and the usepackage{hyperref} commented, this results in "Runaway argument? Paragraph ended before \bm@test@token was complete". Including the usepackage{hyperref} (with \renewcommand commented) results in "Undefined control  sequence. \Call@AtVeryEndDocument ...cumentHook \@undefined"   But if uncomment the \renewcommand{\bm}{#1} line (effectively doing without any \bm), then this seems to compile.  I have been using this for a while now and things appeared to have been working fine, but now that I attempt to use etoolbox I seem to run into problems.  \documentclass{standalone} \def\RemoveEToolbox{}% Shows that this works fine without {etoolbox}  \usepackage{amsmath} \usepackage{xcolor} \usepackage{bm}  \ifdefined\RemoveEToolbox     \newcommand*{\iftoggle}[3]{#2}% Default to true value of "if" \else     \usepackage{etoolbox}     \newtoggle{paper}     \toggletrue{paper}      %\renewcommand{\bm}[1]{#1}% Why can't I have a normal \bm with {etoolbox}? \fi  % \usepackage{hyperref}  \begin{document}  This document is intended for  \iftoggle{paper}{paper}{electronic} distribution.  \bm{\textcolor{blue}{Solve $x^2-1=0$}}  \begin{align*}     a &amp;= b\\     \iftoggle{paper}{         c &amp;= d\\     }{}     e &amp;= f\\ \end{align*} \end{document} Stored Procedures, MySQL and PHP The question is a fairly open one. I've been using Stored Procs with MS SQLServer for some time with classic ASP and ASP.net and love them, lots.  I have a small hobby project I'm working on and for various reasons have gone the LAMP route. Any hints/tricks/traps or good starting points to get into using stored procedures with MySQL and PHP5? My version of MySQL supports Stored Procedures. Is it possible to change the color of a single bar when the bar plot is based on symbolic values? I am beginner in LaTeX, trying to use it for industrial documents. I wonder if there is a way to change the color of only one bar of the plot below, ie "TestDetails", to illustrate it is a local macro and not a LaTeX keyword. I have been trying to experiment with point meta, but could not find a way.  \documentclass{article} \usepackage{filecontents}  \usepackage{pgfplots}  \usepackage{pgfplotstable} \pgfplotsset{width=.95\textwidth, compat=newest}   \begin{document}  \begin{filecontents}{data.csv} {Occurences}, {keyword} 215, item 159, SI 134, subsubsection 106, TestDetails 58, caption \end{filecontents}  \pgfplotstableread[col sep=comma]{data.csv}{\datatable}  \begin{tikzpicture} \begin{axis}[ /pgf/number format/.cd,1000 sep={}, width=0.8\linewidth,height=6cm, xbar, xmin=30, xmax=250, xtick=\empty, enlarge x limits={value=0.1, upper}, enlarge y limits=0.1, ytick=data,  y dir=reverse, xlabel= {}, y tick label style={major tick length=0pt}, yticklabels from table={\datatable}{[index]1}, nodes near coords, nodes near coords align=horizontal, ] \addplot [draw,fill=blue!50]table[ y expr=\coordindex, x index=0]{\datatable}; \end{axis} \end{tikzpicture}  \end{document} Is it possible for a familiar to attune to and then wear or wield a magic item?