Using Maple to prove one of the axioms in mathematics.

The idea is to prove that (-1)*(-1)= 1.  The mere fact that 0's are involved in the proof makes it harder.  So it's easy enough on paper to solve, as I will show in the steps below.

It is known 0*x=0 for all values of x,
and 1-1=0,
If we let x=-1, then 0*(-1)=0,
sub in for zero   (1-1)(-1)=0,
expand (1)*(-1) + (-1)*(-1)=0, also we agree that 1*y=y for all values of y,
so ......  (-1) + (-1)*(-1)=0,
and therefore (-1)*(-1) = 1

Does anyone have an elegant way of showing that in Maple?

OneFrame := proc(k)
local Courbe, T, a, b, c, t, P, Q, NormM, F, Ell, sol, N1, N2, dr, tx;
a := 11; b := 7; c := sqrt(a^2 - b^2); t := 1/3*Pi;
Ell := x^2/a^2 + y^2/b^2 = 1;
geometry:-point(T, (a^2 - b^2)*cos(t)^3/a, -(a^2 - b^2)*sin(t)^3/b);
Courbe := plots:-implicitplot(Ell, x = -a - 10 .. a + 10, y = -b - 10 .. b + 10, scaling = constrained, color = blue);
NormM := plots:-implicitplot(y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), x = -a - 5 .. a + 10, y = -b - 10 .. b + 10, color = orange); geometry:-line(Per, y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), [x, y]);
geometry:-point(P, subs(y = 0
, geometry:-Equation(Per), 0));
geometry:-point(Q, 0, subs(x = 0, geometry:-Equation(Per)));
geometry:-point(M, a*cos(t), b*sin(t));
geometry:-point(N1, a*cos(k), b*sin(k));
geometry:-point(F, 2.329411765, -2.567510609);
geometry:-line(L, N1, F);
sol := solve({geometry:-Equation(L), Ell}, {x, y},explicit);
geometry:-point(N2, subs(sol[2], x), subs(sol[2], y));
geometry:-segment(sg, N1, N2);
tx := plots:-textplot([[geometry:-coordinates(M)[], "M"],
[geometry:-coordinates(N1)[], "N1"], [geometry:-coordinates(N2)[], "N2"],
[geometry:-coordinates(P)[], "P"],
[geometry:-coordinates(Q)[], "Q"],
[geometry:-coordinates(F)[], "F point de Frégier"],
[geometry:-coordinates(T)[], "T"]], font = [times, bold, 16], align = [above, left]);
dr := geometry:-draw([sg(color = magenta, linestyle = dash),
Per(color = black), P(color = red, symbol = solidcircle, symbolsize = 12),
Q(color = red, symbol = solidcircle, symbolsize = 12),
M(color = black, symbol = solidcircle, symbolsize = 12),
F(color = red, symbol = solidcircle, symbolsize = 12),
N1(color = black, symbol = solidcircle, symbolsize = 8),
N2(color = black, symbol = solidcircle, symbolsize = 8),
T(color = black, symbol = solidcircle, symbolsize = 8)]);
plots:-display(Courbe, tx, dr, scaling = constrained, axes = none); end proc;

plots:-animate(OneFrame, [k], k = Pi/3 .. Pi, frames = 50);
Error, (in plots/animate) wrong type of arguments
Why this animation does't work ? Thank you very much.
 

Dear Maple users

I am defining various geometrical items in the plottools package and finally plot them using the display command. Let's say I have defined:

Circle1:=circle(...):

Circle2:=circle(...):

Line1:=line(...):

etc.

and finally want to plot them with the display command:

display(Circle1,Circle2,Line1,...)

It works, however sometimes I only want some of these items to be plotted. Certain variables at the beginning of the document do control if those items should be plotted or not. Let's say a variable CircleOrNot can be set to have the values "no" or "yes", and if I set it to "yes", it should plot it and if the value is "no", it should not plot it. I know that I can make a conditional statement in the end when dealing with the display command, but it can be quite a mess (condition after condition ...).

What I would like is making a conditional assignment like the following:

If CircleOrNot="yes" then Circle1:=circle(...) else Circle1:=null end if

or something like that, and afterwards use the full display command including all objects:

display(Circle1,Circle2,Line1,...)

But it does not work. I receive an error. There doesn't seem to be an empty statement, which will just be ignored by the display command. How can I do it? Probably someone has a good way to handle this situation.

Best regards,

Erik

What is the correct way to find all terms of the form  _Cn^m where n is non negative integer and m is exponent which can be 1?

The problem is finding _Cn with no exponent. Maple's   _Cn^anything does not match _Cn but only when there is an actual exponent present.

It is easy to find _Cn^m with m present and easy to find _Cn with no exponent, but how to combine the two is the problem. Here is an example.

Given e:=_C1^2+_C1+_C2^3+a+b; I want to obtain the set {_C1^2,_C1,_C2^3}  but I get instead {_C1, _C2, _C1^2, _C2^3} using ther following code

e:=_C1^2+_C1+_C2^3+a+b;
indets(e,   Or(  And(symbol, suffixed(_C, nonnegint)),
                 And(symbol, suffixed(_C, nonnegint))^anything  ));

If instead I do this

e:=_C1^2+_C1+_C2^3+a+b;
indets(e, And(symbol, suffixed(_C, nonnegint))^anything);

This gives {_C1^2, _C2^3} , so it did not pick the _C1 term since exponent is missing. If I do this instead

e:=_C1^2+_C1+_C2^3+a+b;
indets(e, And(symbol, suffixed(_C, nonnegint)));

This it gives {_C1, _C2} and if I do 

e:=_C1^2+_C1+_C2^3+a+b;
indets(e,   And(symbol, suffixed(_C, nonnegint))^Or(anything,identical(1)))

it gives {_C1^2, _C2^3}

What to do to obtain {_C1^2,_C1,_C2^3} 

I am trying to find all _Cn^m in an expression where m can be anything including not being there (or 1 basically) but in Maple, exponent 1 is not counted as exponent.

Update

Another input example is e:=_C1^2+_C1+ _C1*_C2^3+a+b;  This should be produce the set {_C1^2,_C1,_C2^3} 

Any trick to use for this?

Maple 2024.1

What is the relationship between font size and screen pixels on plots? My screen are 2560 x1440 and 1920 x1200. Is for exanple size 12 font 12 pixels high?

Also when a diplay plot is say 800 x 600 pixels in size does the plot  does it use the max horizontal or vertical number of pixels? I know it may not fill the plot area in both directions. Or are some pixels trimmed of around the sides. Is it that the bounding box is the 800 x 600? I know when a caption is added the plot area is reduced in size vertically. 

Download Heat_Eq_2D.mw

Hi. 

I am trying to solve a polynomial equation but the structure leads Maple to return a trivial solution and the other solutions are given as a RootOf expression. The equation involves a single variable, x, that is raised to a power, b and a multiplier, a (both are positive-valued). Please see attached worksheet.

I have not encountered this before and I cannot find a way to get to an explicit solution. Perhaps it is not possible (?).

Does anybody know how to deal with this? 

Thanks in advance ...

Roots_of_a_Polynomial_MaplePrimes.mw 

Any ideas, guys?

i use all code really sometime this is happen i can find each term seperatly without any factoring and collecting term i want all of them term by term i did all simplify code but i did't get result

Download simplify.mw

While Maple provides built-in functions for handling continued fractions, such as NumberTheory[ContinuedFraction], I employ custom implementations to better suit educational purposes. The custom code allows for step-by-step demonstrations, making it easier for students to understand how each convergent is calculated. Additionally, it offers more flexibility in modifying
the process of exploring generalized or specific continued fractions, such as those used in solving
quadratic equations or demonstrating the convergence properties of irrational numbers.

If the logarithmic error plot is approaching zero, the errors are decreasing as the number of iterations increases. This behavior shows that the continued fraction is indeed converging to the actual value.

However, a logarithmic scale that approaches zero might be a bit misleading. Given that the current code produces a logarithmic error plot that may not appear to go sufficiently negative, it might indicate that the approximations are not yet close enough.

The issue I face with the error plot stopping after 12 terms might be due to the logarithmic transformation attempting to take the logarithm of very small or zero errors, which can cause issues when plotting. However, I don't know how to fix this problem

Download continued-fraction.mw

Can we find the conditions on A1, A2, B1, B2, beta1, and beta2 for which numerator of Eq. (1) is zero? ss.mw

I am trying to create a ring of difference operators with customly defined actions. I have managed to get the code to work to an extent, but I am having trouble creating the inverse operators. For example, if Sn sends n to n+2, I want to create Sn^(-1) that sends n to n-2. When I tried to input this into the algebra itself, it gave me the following error: "Error, (in Ore_algebra:-skew_algebra) indeterminate n may appear in a single commutation only". Below is the code that creates the difference algebra. Apologies for the bad formatting.

> with(Ore_algebra);

> CreateDifferenceAlgebra := proc (vars::(list(symbol)), offsets::(list(numeric)))

local actions, i, var, diff_op,
shift_offset, algebra; actions := table();
for i to numelems(vars) do
   var := vars[i];
   diff_op := convert(cat("S",\var), symbol);
   shift_offset := offsets[i];
   actions[diff_op] := (proc (values) options operator, arrow; proc (u, order)
      local res; res := u;
      to order do res := subs(values[1] = values[1]+values[3], res*values[2]) end do;
      res
      end proc

   end proc)([var, diff_op, shift_offset]) end do;

algebra := Ore_algebra:-skew_algebra(seq(shift = [convert(cat("S",

vars[i]), symbol), vars[i]], i = 1 .. numelems(vars)),

action = {seq(convert(cat("S", vars[i]), symbol) =
actions[convert(cat("S", vars[i]), symbol)], i = 1 ..
numelems(vars))}); return algebra end proc


;
 

I have angled some test to be parallel to the slope of a line. It tuns out that the text is only parallel if scaling = constrained is used. That is not necasserily pratical for many plots. In prcttice I have a nice proedure for this. Is there any way to pickup the plot x-y scaling factor that maple decides on so I could built that into the procedure. I have manually rescaled her to demonstrate.

mmm...I see the website viewer does not rotate the text.

Download 2024-10-15_text_not_always_parallel.mw