Krishnan Arbuja's profile - activity

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.

@Slelievere How did you figure this out by using sage software. Can you show me?

I doubt my question became clearer. It's been months. If u and v has values that make the function discontinuous isn't part of the answer I'm looking for.

Is there anyway I can solve this problem?

Here I made some edits. Hopefully, this could help.

Suppose we have an equation: $${(r\cos{(t)}+u})^{2}+{(-(r\sin{(t)}+v)^3+1)}^{\frac{2}{3}}=1$$

Where $x=r\cos{t}$ and $y=r\sin{t}$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, between the x-values ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis, so that u and v as point (u,v) is still inside the implicit relation $$(1-x^2)^3=(1-y^3)$$.

Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.

I MADE EDITS!

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$.

I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$. As n increases from $0\le{n}\le{2\pi}$, I can take the following in coordinate-form....

$$(n,\text{The x-intersection value})$$ $$(n,\text{The y-intersection value})$$

Finally I need to take the following to graph its parametric derivative. Which is...

$$\frac{({\text{The x-intersection value}})^2+4\cos(4(\text{The x-intersection value}))}{-(\text{The y-intersection value})^2-4\cos(4(\text{The y-intersection}))}$$

I have little knowledge with how to use sage. If someone can help I'll be thankful.

I want to plot a 3-d parameter. Try and think of it this way. I'm trying to find the area under a specific curve. That is A. Then if you see u, and v in inside find_root(find_root(((y*cos(x))^2+u)/4+((y*sin(x))^2+v)/9-1,0,4)), I need highest value of A, at u, and v...... x=u y=v z=A Just look above, and it will make sense. If not I'll figure it out myself. This is as much as I can clarify it. You'll only understand if you were literally with me. This is clearly not the case!

In sage math I put in the following software

var('n a b x y s u v A')
n = 5100
a = 0
b = 2*pi
h = (b-a)/n
s = 0
for i in range(n):
    x = a + i*h
    s = s + find_root(((y*cos(x))^2+u)/4+((y*sin(x))^2+v)/9-1,0,4)
A = float((b-a)/n*s)
var('u v A')
f_x = u
f_y = v
f_z = A
parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))

However, I was given the following error...

Error in lines 7-9
Traceback (most recent call last):
  File "/projects/180e8f3c-9dc5-424f-abcc-5267257c0d31/.sagemathcloud/sage_server.py", line 736, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 3, in <module>
  File "/usr/local/sage/sage-6.3.beta6/local/lib/python2.7/site-packages/sage/numerical/optimize.py", line 77, in find_root
    return f.find_root(a=a,b=b,xtol=xtol,rtol=rtol,maxiter=maxiter,full_output=full_output)
  File "expression.pyx", line 9683, in sage.symbolic.expression.Expression.find_root (build/cythonized/sage/symbolic/expression.cpp:42560)
NotImplementedError: root finding currently only implemented in 1 dimension.

Is there a way I can graph this? If so is there also a way to find the maximum values of u and v, from this parametric function?

Let's say I used the variables u, v, A, from the first code-block in my question, could I take those and make them a 3-d parameter. If there's something wrong with what is on the top how can I get the 3-d parameter I need?

I want too be able to know to fix the error message and make a function, out of u,v, and A. Then use the program for solving a maximum and minimum of that function.

Is it possible to answer my question now?

If there's something that doesn't make sense, please say so.

I was trying to use integrals, in order to find the average radius. What I did was I took a function, like $$f'(x,y)$$, and then I replaced f'(rcos(theta),rsin(theta)). Then I replaced r with y, and theta with x. This was divided into the positives above the x-axis, and the negatives below. Using computer programming I got used it all inorder to find the area under the positive curve.

var('n a b x y s u v A') 
n = 5100 
a = 0 
b =2*pi
h =(b-a)/n
s =0
for i in range(n):
  x = a + i*h
  s = s +find_root((y*cos(x)+u)^2)/4+((y*sin(x)+v)^2)/9-1,1.9,3.1)
A= float((b-a)/n*s)

Where a is the start of the integral, b is the end of it, h is the intervals for the integral, s is the root to identify the region for integration, the float is the summation,A is the area, and u and v are parameters.. Note: This is an 2-D implicit function, you can't use a direct integral, but you can use the definition. I'm trying to use the co-ordinates to find the maximum of u, and v in terms of A, or the area. I would like to find the co-ordinates of the maximum that comes out of this function. You can think of it as

$$x=u$$ $$y=v$$ $$z=A$$.

I tried this (at the very top of this post) on sage, but it gave...

 Error in lines 7-9
Traceback (most recent call last):
  File "/projects/180e8f3c-9dc5-424f-abcc-5267257c0d31/.sagemathcloud/sage_server.py", line 736, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 3, in <module>
  File "/usr/local/sage/sage-6.3.beta6/local/lib/python2.7/site-packages/sage/numerical/optimize.py", line 77, in find_root
    return f.find_root(a=a,b=b,xtol=xtol,rtol=rtol,maxiter=maxiter,full_output=full_output)
  File "expression.pyx", line 9685, in sage.symbolic.expression.Expression.find_root (build/cythonized/sage/symbolic/expression.cpp:42561)
NotImplementedError: root finding currently only implemented in 1 dimension.
19

If my question is not clear please be free to say it.

If you look into my work so far I was trying to solve under a specific section of a function using the left-endpoint rule, since it can't be computed explicitly.

In this case e is the change of the function by x, and f is the change by y. And z is equal to the area under an equation from $a=0$, to $b=2\pi$, where the area is positive. You can see here: https://www.desmos.com/calculator/kv4...

I tried to make a 3-d parameter by making $m(x)=e$, $m(y)=f$, and $m(z)=q$, and tried to find the maximum values of e, and f. I've tried using sage's programming, but there is something wrong with what I did as seen here: https://cloud.sagemath.com/projects/1...

Is there a way of finding the maximum value of e, and f values? If it is done correctly both of them should be calculated as $e=0$, and $f=0$, since this should have the maximum value of $q$.