LaTeX2HTML Demonstration LaTeX Doc
\documentclass[12pt, reqno, b5size]{amsart}
\usepackage{color}
\usepackage{bm}
\usepackage{hyperref}
\hypersetup{
pdfstartview=FitH,
bookmarks=true,
CJKbookmarks=true,
pdfstartview=FitH,
colorlinks=true,
linkcolor=blue
}
%new theorem
\newtheorem{defn}{Definiton}
\newtheorem{prop}[defn]{Proposition}
\newtheorem{lem}[defn]{Lemma}
\newtheorem{thm}[defn]{Theorem}
\newtheorem{cor}[defn]{Corollary}
\newtheorem{rem}{Remark}
\newtheorem{prob}{Problem}
\newtheorem{excs}{Exercise}
\newtheorem{answer}{Answer}[prob]
%newcommand
\newcommand{\eps}{\varepsilon}
\newcommand{\R}{\bm{R}}
\newcommand{\rd}{\operatorname{d}}
\newcommand{\set}[1]{\left\{#1\right\}}
%new environments
%the quote
\renewenvironment{quote}{%
\par\vspace{1.5\baselineskip}%
\hspace{\stretch{0.2}}\minipage[c]{.9\textwidth}\itshape}{%
\endminipage\hspace{\stretch{1}}\vspace{1.5\baselineskip}\par}
\begin{document}
\title{LaTeX2HTML Demonstration LaTeX Doc}
\author{van abel}
\email{[email protected]}
\address{Southwest University}
\date{\today}
\keywords{Latex, Html}
\begin{abstract}
This doc is the demonstration of LaTeX doc which can be directly use in a new post. You can copy and paste all the content of this document into a new post (in HTML mode), and see the demonstration.
\end{abstract}
\maketitle
\tableofcontents
\section{What did This Doc Do}
To demonstrate my plugins LaTeX2HTML, I write this demo-Doc of LaTeX, which can be compiled by LaTeX or PDFLaTeX on one hand, and can publish on your blog by directly copy and paste all the content into a new post in HTML mode.
\section{How to Get this Doc}
This doc has published with the LaTeX2HTML plugin with version higher than 1.1.0, the LaTeX2HTML plugin can be download at WordPress.com.
\section{Itemize and Enumerate: How to List Stuffs}
As you have already saw, we can list the element as
\begin{itemize}
\item Firstly
\item Secondly
\end{itemize}
What’s more you can nest them, for example
\begin{itemize}
\item Firstly
\item Secondly
\begin{itemize}
\item The first item of Second
\item The second item of Second
\begin{itemize}
\item The first item of the third item of second item
\end{itemize}
\end{itemize}
\end{itemize}
\emph{Enumerate} works almost the same as \emph{itemize}, except the style is different. For example replace \emph{enumerate} in the above example with \emph{enumerate}, then we get:
\begin{enumerate}
\item Firstly
\item Secondly
\begin{enumerate}
\item The first item of Second
\item The second item of Second
\begin{enumerate}
\item The first item of the third item of second item
\end{enumerate}
\end{enumerate}
\end{enumerate}
\section{The Color Scheme: How to Colorize You Content}
Firstly, you must decide which class your content belonged to, the \textcolor{green}{text} content or the \textcolor{green}{mathematics} content, since these two classes are proceeded by different scheme, \textcolor{green}{text} is by \emph{latex2html} and \textcolor{green}{mathematics} by \emph{mathjax}.
Just as in Latex, the you can set a color for your formula, for example
\[
\color{red}{a+b},\quad\color{blue}{a+b},\quad\color{green}{a+b}
\]
\[
\frac{\color{cyan}{a+b}}{c+d}, \quad
\frac{a}{\color{magenta}{a+b}},\quad
\frac{a}{a+\color{yellow}{b}}
\]
On the other hand, if you want to give color for your \textcolor{green}{text} content, then you should use something like this:
\textcolor{red}{red} and \textcolor{blue}{blue} and \textcolor{green}{green} and \textcolor{cyan}{cyan} and \textcolor{magenta}{magenta} and \textcolor{yellow}{yellow}.
\section{How to Write a Theorem}
There are two group of environments, which proceed the content such as \emph{Theorem}. One is used for a post, in which you mainly state something formally, just like you do in a research paper; The other is used for problem-discussion situation, in which you ask questions and hope for some examples and answers.
Of course the border is not so strict, for example the \emph{examp} environment can used in both cases.
Here are the complete list of environments you can use (The example will be pop up at some time later), I take first few words of a environment to represent it, for example \emph{thm} for \emph{Theorem}:
\begin{itemize}
\item First group: \emph{defn}(Definition), \emph{lem}(Lemma), \emph{prop}(Proposition), \emph{thm}(Theorem), \emph{cor}(Corollary), \emph{rem}(Remark), \emph{excs}(Exercise), \emph{proof}(Proof)
\item Second group: \emph{prob}(Problem), \emph{answer}(Answer)
\item Mixed: \emph{examp}(Example), \emph{quote}(Quotation)
\end{itemize}
I will go through the two group one by one, let’s begin with the first group.
\subsection{Definition, Lemma, Proposition, Theorem, Corollary…}
\subsubsection{Basic Usage of Environments}
Here is an example of \emph{definition} environment:
\begin{defn}
Suppose that $(X,\mathcal M)$ and $(Y,\mathcal N)$ are measurable spaces,
and $f:X\to Y$ is a map. We call $f$ is \emph{measurable} if for every $B\in\mathcal N$
the set $f^{-1}(B)$ is in $\mathcal M$.
\end{defn}
The other is similar, just replace \emph{defn} with any one of the above environment:
\begin{rem}
If $Y$ is a topological space, and $\mathcal N$ is the $\sigma$-algebra of Borel sets,
then $f$ is measurable if and only if the following condition satisfied:
\begin{itemize}
\item For every open set $V$ in $Y$, the inverse image $f^{-1}(V)$ is measurable.
\end{itemize}
\end{rem}
\subsubsection{Assign a Name for Your Theorem}
You can even assign a name to these environment, just as you did in latex, use \emph{[text]} just behind the environment, for example:
\begin{lem}[fundamental lemma of integration]
Let $\set{f_n}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise almost everywhere, and satisfies the additional property: given $\eps$ there exists a set $Z$ of measure $<\eps$ such that this subsequence converges absolutely and uniformly outside $Z$.
\end{lem}
\subsubsection{The Proof Environment}
The following are the \emph{proof} environment, and some more examples, if you are impatient to other contents, then just skip this subsection by click \ref{sec:5.2}.
\begin{proof}
For each integer $k$ there exists $N_k$ such that if $m,n\geq N_k$, then
\[
\|f_m-f_n\|_1< \frac{1}{2^{2k}}.
\]
We let our subsequence be $g_k=f_{N_k}$, taking the $N_k$ inductively to be strictly increasing. Then we have for all $m,n$:
\[
\|g_m-g_n\|_1 \]
We shall show that the series
\[
g_1(x)+\sum_{k=1}^\infty\left(g_{k+1}(x)-g_k(x)\right)
\]
converges absolutely for almost all $x$ to an element of $E$, and in fact we shall prove that this convergence is uniform except on a set of arbitrarily small measure.
Let $Y_n$ be the set of $x\in X$ such that
\[
|g_{n+1}(x)-g_n(x)|\geq\frac{1}{2^n}.
\]
Since $g_n$ and $g_{n+1}$ are step mappings, it follows that $Y_n$ has finite measure. On $Y_n$ we have the inequality
\[
\frac{1}{2^n}\leq|g_{n+1}-g_n|
\]
whence
\[
\frac{1}{2^n}\mu(Y)=\int_{Y_n}\frac{1}{2^n}\leq\int_X|g_{n+1}-g_{n}|\leq\frac{1}{2^{2n}}.
\]
Hence
\[
\mu(Y_n)\leq\frac{1}{2^n}.
\]
Let
\[
Z_n=Y_n\cup Y_{n+1}\cup\cdots.
\]
Then
\[
\mu(Z_n)\leq \frac{1}{2^{n-1}}.
\]
If $x\not\in Z_n$, then for $k\geq n$ we have
\[
|g_{k+1}(x)-g_k(x)| \]
and from this we conclude that our series
\[
\sum_{k=n}^\infty\left(g_{k+1}(x)-g_{k}(x)\right)
\]
is absolutely and uniformly convergent, for $x\not\in Z_n$. This proves the statement concerning the uniform convergence. If we let $Z$ be the intersection of all $Z_n$, then $Z$ has measure $0$, and if $x\not\in Z$, then $x\not\in Z_n$ for some $n$, whence our series converges for this $x$. This proves the lemma.
\end{proof}
\subsubsection{More Examples of Environments}
\begin{thm}
Let $f_n$ be a Cauchy sequence in $\mathcal L^1$ which is $L^1$-convergent to an element $f$ in $\mathcal L^1$. Then there exists a subsequence which converges to $f$ almost everywhere, and also such that given $\eps$, there exists a set $Z$ of measure $< \eps$ such that the convergence is uniform on the complement of $Z$.
\end{thm}
\begin{cor}
An element $f\in\mathcal L^1$ has seminorm $\|f\|_1=\int_X|f|\rd\mu=0$ if and only if $f$ is equal to $0$ almost everywhere.
\end{cor}
\begin{prop}[Monotone Convergence Theorem]
Let $\set{f_n}$ be an increasing (resp. decreasing) sequence of real valued functions in $\mathcal L^1$ such that the integrals
\[
\int_X f_n\rd\mu
\]
are bounded. Then $\set{f_n}$ is a Cauchy sequence, and is both $\mathcal L^1$ and almost everywhere convergent to some function $f\in\mathcal L^1$.
\end{prop}
\begin{prop}[Fatou's Lemma]
Let $\set{f_n}$ be a sequence of real valued non-negative functions in $\mathcal L^1$. Assume that
\[
\liminf\|f_n\|_1
\]
is exists (so is a real number $\geq0$). Then $\liminf f_n(x)$ exists for almost all $x$, the function $\liminf f_n$ is in $\mathcal L^1$, and we have
\[
\int_X\liminf f_n\rd \mu\leq\liminf\int_X f_n\rd\mu=\liminf\|f_n\|_1.
\]
\end{prop}
\subsection{The Problem, Answer Environments}\label{sec:5.2}
The second group of environments are provided for discussion, after all, this is a discussion platform. They are: \emph{prob} for Problem, \emph{examp} for Example, and \emph{answer} for Answer.
It almost works the same as the first group, for example
\begin{prob}[Egoroff's theorem]
Assume that $\mu$ is $\sigma$-finite. Let $f:X\to E$ be a map and assume that $f$ is the pointwise limit of a sequence of simple maps $\set{\varphi_n}$. Given $\eps$, show that there exists a set $Z$ with $\mu(Z)< \eps$ such that the convergence of $\set{\varphi_n}$ is uniform on the complement of $Z$.
\end{prob}
But, the differences between them is that, the \emph{answer} is numbered with \emph{prob} (the \emph{examp}, \emph{excs}, \emph{rem} will numbered independently), to see this, for example:
\begin{answer}
Assume first that $\mu(X)$ is finite. Let $A_k$ be the set where $|f|\geq k$. The intersection of all $A_k$ is empty so their measures tend to $0$. Excluding a set of small measure, you can assume that $f$ is bounded, in which case $f$ is in $\mathcal L^1(\mu)$ and you can use the fundamental lemma of integration.
\end{answer}
\begin{answer}
This is another answer for the problem.
\end{answer}
You should note that the number of answer is reset to 1 by \emph{prob}, of course, more sensible. For example:
\begin{prob}
Why we should firstly process the positive measurable functions, then the real measurable functions and at last the complex measurable functions for the integral of measurable functions?
\end{prob}
There are the answers for this problem:
\begin{answer}
In fact, you can define the integral of complex function directly.
\end{answer}
\begin{answer}
There is another more instructive answer...
\end{answer}
\subsection{The Exercise Environment}
Maybe, at somewhere, you want the reader consider about something, then you can use \emph{excs} environment for Exercise. \textcolor{red}{Please keep in mind that it will have independent numbering, just as \emph{prob}, but will \emph{not reset} the number of answer}. Here is an example:
\begin{excs}
Suppose $(X,\mu)$ is a measure space, and that $f$ is measurable, then $\int_X f\rd \mu=0$ if and only if $f\equiv0$ almost everywhere.
\end{excs}
\subsection{The Quote Environment}
Sometimes, there are some words or comments on the content, it is like a remark, but it is not so formal. And, if you are write a lecture notes, these words may be the lecturer said before or after an important thing, such as theorems. I have defined a new environment \emph{quote} to deal with these stuff. For example:
Before the theorem
\begin{thm}
Let $\Omega\subset\R^n$ and $u:\Omega\to\R$, then
\begin{enumerate}
\item If $u\in C^2(\Omega)$ is harmonic in $\Omega$, then $u$ satisfies MVP;
\item If $u\in C(\Omega)$ satisfies MVP, then $u$ is smooth and harmonic.
\end{enumerate}
\end{thm}
we want to add a comment on it, then you can use
\begin{quote}
A function satisfying mean-value properties is only required to be continuous. However, a harmonic function is required to be $C^2$. Thus, the equivalence of this two kind of functions will be significant.
\end{quote}
\section{Auto Numbering and Referring Back}
\subsection{Auto Numbering and Referring Back to Equations}
\subsubsection{Basic Usage of Mathematical Environments}
All the \emph{mathematical} environments: \emph{equation}, \emph{align}, \emph{multline}, \emph{gather} will auto-numbering. For example
\begin{equation}\begin{cases}
3=2x+y\\
3=y+2x\end{cases}
\end{equation}
An example of \emph{multline}, which will make the last line flush right:
\begin{multline}
\int_a^b \biggl\{ \int_a^b [ f(x)^2 g(y)^2 + f(y)^2 g(x)^2 ]
-2f(x) g(x) f(y) g(y) \,dx \biggr\} \,dy \\
=\int_a^b \biggl\{ g(y)^2 \int_a^b f^2 + f(y)^2
\int_a^b g^2 - 2f(y) g(y) \int_a^b fg \biggr\} \,dy
\end{multline}
The next example will show how to numbered the equation at a given line:
\begin{gather}
\begin{split}
\varphi(x,z)
&= z – \gamma_{10} x – \sum_{m+n\ge2} \gamma_{mn} x^m z^n\\
&= z – M r^{-1} x – \sum_{m+n\ge2} M r^{-(m+n)} x^m z^n
\end{split}\tag{4}\\
\begin{split}
\zeta^0 &= (\xi^0)^2, \\
\zeta^1 &= \xi^0 \xi^1
\end{split}\notag
\end{gather}
\subsubsection{Auto Numbering Scheme of Equations}
I hope you have already noticed that the above equations are numbered automatically, in fact, this is my first goal to write a plugin to proceed the latex code. Mathjax proceed mathematical perfectly, the only non-advantage is that it can't automatically add number for equations. When I got a way to add number for the equations automatically, I found that I can go a litter far
\begin{enumerate}
\item automatically numbering theorems, which can be realize by set \emph{CSS}'s \emph{before} element with a \emph{counter}, but it may not work for IE6;
\item automatically numbering sections, and which can produce a table of contents, as you have already see.
\end{enumerate}
\subsubsection{How to Numbered Equations by 3.a and 3.b}
Here is an example with varied number:
\begin{align}
x^2+y^2&=z^2,\quad{x,y,z\in\mathbb{Z}}\tag{5.a}\\
x^3+y^3&=z^3,\quad{x,y,z\in\mathbb{Z}}\tag{5.b}
\end{align}
\begin{equation}
x^4+y^4=z^4,\quad{x,y,z\in\mathbb{Z}}\tag{7.a.b}
\end{equation}
For example, if you write another equation, which you would like to numbered automatically, then this equation will be numbered as \emph{(8)} in the present situation, since we already used \emph{7} in \emph{7.a.b}. For example,
\begin{gather}
\int_a^b \biggl\{ \int_a^b [ f(x)^2 g(y)^2 + f(y)^2 g(x)^2 ]
-2f(x) g(x) f(y) g(y) \,dx \biggr\} \,dy \\
=\int_a^b \biggl\{ g(y)^2 \int_a^b f^2 + f(y)^2
\int_a^b g^2 - 2f(y) g(y) \int_a^b fg \biggr\} \,dy
\end{gather}
\end{document}
Excellent! I’m looking for this quite a long time.