2017-08-20 10:43:48 -0600 received badge ● Popular Question (source) 2017-05-26 20:26:21 -0600 asked a question Unable to Use Array without Issue(s) bar turing red 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|1D(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. 2017-02-16 05:55:12 -0600 received badge ● Popular Question (source) 2015-12-28 02:19:08 -0600 received badge ● Student (source) 2015-12-28 01:59:41 -0600 received badge ● Famous Question (source) 2015-12-24 12:34:02 -0600 commented answer Maximize the integral of an implicit relation with two parameters @Slelievere How did you figure this out by using sage software. Can you show me? 2015-12-24 11:51:14 -0600 received badge ● Commentator 2015-12-24 11:50:42 -0600 received badge ● Scholar (source) 2015-01-27 16:07:07 -0600 received badge ● Famous Question (source) 2015-01-27 16:07:07 -0600 received badge ● Notable Question (source) 2015-01-27 16:07:07 -0600 received badge ● Popular Question (source) 2015-01-08 18:38:12 -0600 commented question Maximize the integral of an implicit relation with two parameters 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. 2015-01-08 18:33:20 -0600 received badge ● Notable Question (source) 2014-12-29 08:21:44 -0600 received badge ● Popular Question (source) 2014-12-21 15:48:10 -0600 commented question Maximize the integral of an implicit relation with two parameters Is there anyway I can solve this problem? 2014-12-17 11:52:53 -0600 commented question Maximize the integral of an implicit relation with two parameters Here I made some edits. Hopefully, this could help. 2014-12-14 13:33:04 -0600 asked a question Maximize the integral of an implicit relation with two parameters 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)$$. Note: The blue stripes should stop at 2\pi, and if the equation has regions that are "UNDEFINED", try to ignore it. I MADE EDITS! 2014-11-24 21:00:32 -0600 asked a question How does one graph this? I am trying to parametrize x^2+y^2+sin(4x)+sin(4y)=4. I need to find a way of taking the intersections between x^2+y^2+\sin(4x)+\sin(4y)=4, and \tan(nx). As n increases from 0\le{n}\le{2\pi}, I can take the following in coordinate-form....$$(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. 2014-09-14 17:04:59 -0600 commented question Need help with an error? 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! 2014-09-14 14:47:53 -0600 asked a question Need help with an error? 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 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? 2014-08-12 19:27:21 -0600 commented answer Finding the maximum of a parameter 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? 2014-08-12 08:57:30 -0600 commented question Finding the maximum of a parameter 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. 2014-08-11 15:30:17 -0600 commented question Finding the maximum of a parameter Is it possible to answer my question now? 2014-08-09 17:22:29 -0600 commented question Finding the maximum of a parameter If there's something that doesn't make sense, please say so. 2014-08-08 14:23:25 -0600 asked a question Finding the maximum of a parameter 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=uy=vz=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 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  2014-07-28 09:54:45 -0600 commented question Need help finding maximum values over 3-d parameters? If my question is not clear please be free to say it. 2014-07-27 15:40:32 -0600 received badge ● Editor (source) 2014-07-27 15:24:10 -0600 asked a question Need help finding maximum values over 3-d parameters? 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 $q$.