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Unable to Use Array without Issue(s) bar turing red

asked 2017-05-27 03:26:21 +0100

Krishnan Arbuja gravatar image

In my Sage/CoCalc article (Sage has recently merged with CoCalc), I am unable to create an Array without a red issue bar popping up. Morover, I am unable to cross out element in my array. The issue(s) is PackageArrayError; illegal pream-token (1): PackageArrayError; illegal pream-token (2): ...... and so on.

The whole Document is below.

     \documentclass{article}
        % set font encoding for PDFLaTeX or XeLaTeX
        \usepackage{ifxetex}
        \ifxetex
          \usepackage{fontspec}
        \else
          \usepackage[T1]{fontenc}
          \usepackage[utf8]{inputenc}
          \usepackage{lmodern}
        \fi
        \usepackage{amssymb}
        \usepackage{amsmath}
        \usepackage{ragged2e}
        \usepackage{graphicx}
        \usepackage{breqn}
        \usepackage{fixltx2e}
        \usepackage{hyperref}
        \usepackage{titlesec}
        \usepackage{amsopn}
        \usepackage{array}
        \newcommand{\dd}[1]{\mathrm{d}#1}

        % used in maketitle
        \title{Regarding Your Version of Natural Density to Rationals}
        \author{Aiyappan Nair}


        \begin{document}
        \maketitle
        Your version of asymptotic density to rationals differs from my own version. It is not a simplification but rather a change in definition. And although it is much more simple there are flaws that I would like to mention.

        \subsection{Counter-Example}
        As mentioned earlier, if sets $T_1,T_2,T_p$ are subsets of $\mathbb{Q}$, a density $D$ should have the following qualities.

        (1) If $T_1=T_2$ then $D(T_1)=D(T_2)$

        (2) If $T_1\subset{T_2}$ then $D(T_1)\le D(T_2)$

        In your new version of the density

        $$D=\lim_{r\to\infty}\frac{\left|T_1\cap V(r)\right|}{|V(r)|}$$

        Where

        $$V(r)=\left\{\left.\frac{m}{n}\right|1<m<r,1<n<r,\gcd(m,n)=1\right\}$$

        and $m,n,r\in\mathbb{Z}$ 

        If we set $T_1=\left\{\left.\frac{m}{2n+1}\right|m,n\in\mathbb{Z}\right\}$ and $T_2=\left\{\left.\frac{m}{4n+2}\right|m,n\in\mathbb{Z}\right\}$, I found that $D(T_1)>D(T_2)$ even though $T_1\subset T_2$. Which breaks requirment (2).

        \subsection{Proof of Counter Example}
        Imagine $V(r)$ as an array of values, restricted by $r$. The numerator is from listed as columns each labelled from $[1,r]$ and the denominator is listed as row labelled from $[1,r]$.

        \begin{array}{|l|col1 col2 col3 col4 col5 col6 col7 col8}
  & 1 & 2 & 3 & 4 & 5 & .. & r\\
\hline
1 & 1/1 & 2/1 & 3/1 & 4/1 & 5/1 & ..& r/1\\
2 & 1/2 & \st{2/2} & 3/2 & 4/2 & 5/2 & ..& r/2\\
3 & 1/3 & 2/3 & 3/3 & 4/3 & 5/3 & ..& r/3\\
4 & 1/4 & 2/4 & 3/4 & 4/4 & 5/4 & ..& r/4\\
5 & 1/5 & 2/5 & 3/5 & 4/5 & 5/5 & ..& r/5\\
..&  .. & ..  & ..  &  .. & ..  & ..& .. \\
r &  1/r & 2/r & 3/r  & 4/r & 5/r  & .. & r/r\\
\end{array}




    \end{document}

I have never had this problem with SageMath before it merged with CoCalc. Please help.

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Comments

This has nothing to do with sagemath the free software. You should ask cocalc's customer support instead.

nbruin gravatar imagenbruin ( 2017-05-27 04:57:56 +0100 )edit

1 Answer

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answered 2017-05-28 02:51:43 +0100

calc314 gravatar image

This is a LaTeX issue. The array environment must be inside a math environment. Try the following:

\[ 
\begin{array}{|l|cccccccc}
  & 1 & 2 & 3 & 4 & 5 & .. & r\\
\hline
1 & 1/1 & 2/1 & 3/1 & 4/1 & 5/1 & ..& r/1\\
2 & 1/2 & {2/2} & 3/2 & 4/2 & 5/2 & ..& r/2\\
3 & 1/3 & 2/3 & 3/3 & 4/3 & 5/3 & ..& r/3\\
4 & 1/4 & 2/4 & 3/4 & 4/4 & 5/4 & ..& r/4\\
5 & 1/5 & 2/5 & 3/5 & 4/5 & 5/5 & ..& r/5\\
..&  .. & ..  & ..  &  .. & ..  & ..& .. \\
r &  1/r & 2/r & 3/r  & 4/r & 5/r  & .. & r/r\\
\end{array}
\]

Also, you had a \st command in one line that was not defined.

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Asked: 2017-05-27 03:26:21 +0100

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Last updated: May 28 '17