I am trying to find a cubic polynomial over GF(5) that evaluates to zero at the points (1,2,0),(2,0,0),(1,4,1),(-2,-1,-1),(-3,2,1),(1,-1,0),(2,-3,1),(-3,-2,2),(1,-2,1).  Clearly the zero polynomial would satisfy the condition, thus I'm looking for a proper cubic polynomial. I note that the generic form of such a polynomial is $f= ax^3+ by^3+ cz^3+ dx^2y + ex^2z+ fy^2x+ iy^2z+ gz^2x+ kz^2y+ lxyz$, that is I need to solve for the coefficients $a,b,\ldots,l$.  I would prefer to do this calculation in GAP as I am using the software already for other calculations.
given a polynomial $p(x,y)$ from $\mathbb Z[x,y]$. I want to substitute $x=1$ leaving $y$ as it is. The command Value(p,[1,y]) does not work. Can you give me a hint?  After that it gets a little more complicated. Let  $$p(x,y)=x^6+y^6$$ After substituting $x=1$ we have $p(1,y)=1+y^6$. Given a natural number $j$. I want to find all expressions $y^j$ in $p$ and substitute these $y^j$ by $1$ and the others by $0$. In this case (and for $j=6$) we would get $1+1=2$. If $$p(x,y)=x^4y^2+x^2y^6$$ we should get $0+1=1$. Is there any way to do this?  Thanks for reading! edit: Well, it turns out I used the function Value wrong. The right code for my problem would be Value(p,[x,y],[1,y]). So my first question is solved.
I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a Riemannian Manifold are very similar except for the detail that makes them Riemannian (having a inner product on the tangent space of every element which varies smoothly with respect to this element of the R. Manifold).  So my question is: is there any example of a Smooth Manifold that is NOT a Riemannian Manifold?  Thanks. Explaining to an 8-year-old why nothing can travel faster than light Finding local maximum and minimum I've got a few macros that perform floating point lookups and calculations for me and I'm interesting in plotting their results. They're the kind of calculation which requires if/else, so they can't be easily written as a simple math formula for \addplot.  Thanks to a previous question (), I now know to use \pgfmathdeclarefunction, but after trying it I've found that addplot seems to prepend a "0Y" or "1Y" to values before passing them as function arguments. My external macros aren't prepared to handle this extra text, and I'm not sure why it's there.   In the course of testing things out, I've also found that calling a function defined with pgfmathdeclarefunction can also change the positioning of the graph...  \documentclass{article}  \usepackage{xparse}  \usepackage{pgfplots} \pgfplotsset{compat=1.10}  \begin{document}  % Example of a macro which does math outside of pgf     \ExplSyntaxOn     \DeclareExpandableDocumentCommand \doubleMacro { m }%      {%       \fp_eval:n { #1*2 }%Math performed by a package other than pgfplots      }%     \ExplSyntaxOff    %Demo: macro does work in regular text \doubleMacro{1.1}.  % Defining custom pgf functions for plotting      \pgfmathdeclarefunction{mathDoubleMacro}{1}{%         \pgfmathparse{\doubleMacro{#1}} %Calls macro which performs math done by another package     }      \pgfmathdeclarefunction{mathDoublePgf}{1}{%         \pgfmathparse{#1*2} %Uses pgf to do the same thing as an external macro could     }  %Generic plot with function given directly to \addplot \begin{center} \begin{tikzpicture}     \begin{axis}         \addplot[domain=0:1] {2*x};     \end{axis} \end{tikzpicture} \end{center}  %Plot using first macro: complains about 'Unknown fp word Y.' \begin{center} \begin{tikzpicture}     \begin{axis}         %\addplot[domain=0:1] {mathDoubleMacro(x)};%Commented out because it gives problems     \end{axis} \end{tikzpicture} \end{center}  %Plot using second macro: works, but graph is displaced to the right of the expected position \begin{center} \begin{tikzpicture}     \begin{axis}         \addplot[domain=0:1] {mathDoublePgf(x)};     \end{axis} \end{tikzpicture} \end{center}  \end{document}   If I define mathDoubleMacro as calling a non-existent macro, then the errors show which x-values which the addplot is trying to evaluate y at:    0Y0.0e0  1Y4.1666e-2 1Y8.3332e-2 ...  1Y9.9998401e-1   What are these nY's all about, and why does \addplot[domain=0:1] {mathDoublePgf(x)} give a graph with different positioning? Can toddlers be taught to safely navigate stairs? I have    iPhone 5 (with iOS 8.3) MacBook Pro Early 2013 (with OS X Yosemite 10.10.3) iPad Air (with iOS 8.3)   My iPhone is paired with the MacBook, so e.g when my phone rings, I can see it in my computer. But when I want to AirDrop photos from my iPhone the MacBook, then I can't see the MacBook in my iPhone AirDrop targets.  I can see the iPad, I can see my wife's iPhone, but not the MacBook.  On MacBook, I can see the iPad, my wife's phone and even my iPhone and I can drop to there, but not the other way around.  Before I started troubleshooting, the iPhone and MacBook were mutually invisible (I only saw iPad from both). After a while of trying things and sending files back and forth, my MacBook started to see the iPhone, but I did not manage to make it work from the other way. Still, the iPhone seems to be much "less" visible than the iPad - which I see almost instantly and always, the iPhone only seems to appear after a while.  Just to make it clear:   both iPhone and MBP have BT turned on both have AirDrop visibility set to Everyone   I would be glad for any suggestions to fix this - what settings to check, logs, whatever. How to convert a std::string to const char* or char* Aborting a shell script if any command returns a non-zero value An English book tells me that the following sentence is correct:     This publication is made from paper from sustainable forests.   Shouldn't it be of paper instead? A topological space $X$ is said to be irreducible if it cannot be written as a union $X = A∪B$, where $A$ and $B$ are proper closed subspaces of $X$  I want to try the following:  Show:  $(i)$ That every set with the indiscrete topology is irreducible, and every infinite set with the finite complement topology is irreducible.  $(ii)$ That an irreducible Hausdorff space contains at most one point.  Can you give me any suggestions on how to try this? How to make a pattern follow a stroke, path or line? Here is my MS Paint job on a real bad attempt at what I'm trying to achieve:    I have access to Photoshop and Illustrator CS6, and I want to be able to draw a thick line (that is not straight) and have a pattern within the line show arrowheads within the line to indicate the direction of the line. What is the best way to add options to a select from a JavaScript object with jQuery? What is the best method for adding options to a &lt;select&gt; from a JavaScript object using jQuery?  I'm looking for something that I don't need a plugin to do, but I would also be interested in the plugins that are out there.  This is what I did:  selectValues = { "1": "test 1", "2": "test 2" };  for (key in selectValues) {   if (typeof (selectValues[key] == 'string') {     $('#mySelect').append('&lt;option value="' + key + '"&gt;' + selectValues[key] + '&lt;/option&gt;');   } }   A clean/simple solution:  This is a cleaned up and simplified :  $.each(selectValues, function(key, value) {      $('#mySelect')           .append($('&lt;option&gt;', { value : key })           .text(value)); });   Changes from matdumsa's: (1) removed the close tag for the option inside append() and (2) moved the properties/attributes into an map as the second parameter of append(). If just a single photon hit a single slit interferometer, what would happen? Would you just see a dot on the screen, or would there be a diffraction pattern? Furthermore, if you had a double slit interferometer but also had which-path information would there still be a diffraction pattern even though there is no interference? GPU compute grayed out Is there a way to reorganize the fields/attributes in a data extension?  I've added some extra fields that relate to other fields in that extension.   I haven't found a way to regroup those fields. For what values of $\theta$ does $\sum_{1}^{\infty} \frac{e^{i n\theta}}{n}$ Converge? I just came across the problem of what happens in the boundary cases of .  More concisely, for what $\theta$ will the following sum converge? $$\sum_{n=1}^{\infty} \frac{e^{ni\theta}}{n}$$ It is well known that this sum will diverge for $\theta=0$ and will converge for $\theta=\pi$. In fact, for any odd multiple of an even root of unity, since we have symmetry, for each line through the origin of the complex plane, we can use the alternating series theorem to show that they converge.  Another idea I had was to split  the sum into imaginary and real parts, and if we can prove each of these converges, then we are all set. So now we must prove that both of the following converge $$\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}\space, \space \sum_{n=1}^{\infty}\frac{\sin(n\theta)}{n}$$ Now these can be shown to converge easily by Dirichlet's test for any rational multiple of pi, but what about irrational multiples of pi? I know it has been shown to converge with $\theta=1$, but I don't see how to generalize this. Calculating average raster value for area in ArcGIS Desktop? I have Raster data for a country which has Evaporation Data in its pixels. I also have Vector layer which has the 30 cities of the country.   What I need is the average Evaporation value of each city from the raster data using ArcGIS?  I have access to the spatial analyst extension. Can I configure Ubuntu to never install a specific package even if it is required by another package I install? I have a Win 7/11.10 dual boot system with Grub 2 as the default bootloader. However, I want to make Windows bootloader the default bootloader (grub 'dos' like interface scrares the hell out of my sister) so when I start the PC I will have the following boot options: 1. Windows 7 2. Ubuntu  NB: A process to reverse this operation so Grub can be the default bootloader again will also be appreciated Science fiction short story: beam transporting people into the distant future