Hello everyone,
I hope this message finds you well. I am trying to plot a function f(x, y) and overlay its contour on a quarter ellipse using Maple 2015. However, I’ve encountered some difficulties and have not been successful so far. I would greatly appreciate any assistance in resolving this issue. Thank you!

Plotting in 2D

restart:with(plots):  aa := 4: bb := 2:  
f := -((x^(2))/(aa^(2))+(y^(2))/(bb^(2))-1)*((aa^(2)*bb^(2))/(aa^(2)+bb^(2))):  
plot3d(f,x = 0 .. aa/(2),y = 0 .. bb/(2),region = (x, y) -> ((2 x)/(aa))^(2) + ((2 y)/(bb))^(2)<= 1,axes = boxed,style = patchcontour, grid = [50, 50],orientation = [-120, 45],shading = zhue,title = "f(x,y) over quarter ellipse domain");

Contour plotting
xrange := 0 .. aa/(2): yrange := 0 .. bb/(2):  
nx := 100:   ny := 100:  
dx := (rhs(xrange) - lhs(xrange))/(nx-1):dy := (rhs(yrange) - lhs(yrange))/(ny-1):  
Z := Matrix(nx, ny, (i, j) -> local x, y, inside;x := lhs(xrange) + (i-1)*dx;y := lhs(yrange) + (j-1)*dy;inside := (((2 x)/aa)^2 + ((2 y)/bb)^2 <= 1);if inside then f(x, y) else NULL end if):  
contourplot(Z, xrange, yrange,contours = 15, filled = true, coloring = [blue, green, yellow, red], axes = boxed, title = "Contour plot over quarter ellipse", grid = [nx, ny]);  

Hi

I hope everyone is fine and doing well. I want to constrcut the set of monomials {p[0],p[1],...,p[m-1]} for any value of m for example, for m=6 the monoial is define as:

p[0]:=1;

p[1]:=x;

p[2]:=y;

p[3]:=x^2;

p[4]:=x*y;

p[5]:=y^2;

and similarly for m=10 the monomials should be given as:

p[0]:=1;

p[1]:=x;

p[2]:=y;

p[3]:=x^2;

p[4]:=x*y;

p[5]:=y^2;

p[6]:=x^3;

p[7]:=x^2*y;

p[8]:=x*y^2;

p[9]:=y^3;

I am waiting for your positive response. Please take care and thanks

A classical probability result says that if G₁ and G₂ are two independent Gamma random variables with same scale parameter (let's say 1 to simplify) and shape parameters a₁ and a₂ respectively, then G_k / (G₁ + G₂ ) is a Beta random variable with parameters (a_k , a_3-k ) (k=1..2).

In the attached file it is shown that (Maple 2015) function Statistics:-PDF fails in computing the PDF of G_k.
Noting strange here if you observe that even in the extremely simple case Z = X / (X+Y), where both X and Y are independent Uniform random variable with support [0, 1), Maple 2015 already fails in computing PDF(Z).

An alternative to Statistics:-PDF is to write explicitely the double integration which defines CDF(Z) (to begin with, and later PDF(G_k)) and ask Maple to do the integrations.
This approach works for Z but requires helping Maple when X and Y are still independent Uniform random variables but with respective non instanciated supports [0, a₁) and [0, a₂).

Applying to the Gamma-Gamma case the recipies I introduced in the Uniform-Uniform case does not give any result, unless in the very particular case where the shape parameters a₁ and a₂ are (strictly) positive integers.

All the details are in X_over_(X_plus_Y).mw

Do you have any idea how to prove with Maple the probability result mentioned at the head of this question?

PS: The "classical method" to do compute PDF(Z) consists in changing the integration variables < x₁, x₂ > into < x₁ = v₁v₂, x₂ = v₂ (1-v₁) > (see for instance Stack exchange)... but even after having dome it I still cannot get the desired result.

Thanks in advance.

As I was comparing visually the first terms of a priori identical sums produced by add , I was surprised to find them different.
So I suspected some error in what I have done, until I realized that add randomly permuted the terms.
Each term is of the form (R + P)² where R is a random number and P a polynomial.

This behaviour is illustrated in worksheet add_changes_ranks.mw and appears only when random numbers are used (provided the seed is not forced to some constant value)

Does someone ever onserved that or have any idea of what happens here (maybe this behaviour no longer happens in recent versions?) ?

Thanks in advance

I use  Maple 2015 and I try to understand how the simplification rules apply in the case of the expression 

f := n -> (ln(x)^n)^(1/n)

Here n is assumed to be a strictly positive and I consider only the cases "n is an integer" or "1/n is an integer".

All the questions are orange written in the attached file and resumed below:

1. Why simplify(f(2)) simplifies f(2) whereas simplify(f(n)) doesn't simplifies f(n) for any integer n > 2?
    
2. Why simplify(f(1/n)) simplifies f(1/n)?
    
3. Why simplify(f(3)) with adhoc assumptions returns a simplified expression of some form whereas, for any integer n > 3,  simplify(f(n)) with (the same corresponding) adhoc assumptions returns a simplified expression of a complete different form than with n=3?

Can you please have a look to it and give me some clarifications?
Simplification_rules.mw

Thanks in advance

As I was numerically investigating this recent question I incidentally discovered a strange behaviour of Maple 2015 (which maybe exists in more recent versions?)

The attached worksheet presents an erratic behaviour (plus a remanance isssue because saving it, and opening it again changes the displays).
Note that this strange behaviour seems to occur only when tickmarks use the atomic name `#mo("2")`.

display_issue.mw

Here is a pdf print of this same worksheet: as I hope you will see (because I don't know what you are about to see when opening the attached worksheet) its content differs from the worksheet's. 

display_issue.pdf

Here are 3 screen captures which show what MY worksheet looks like

PAGE 1


PAGE 2
There is a typo in the comment below: read "void" instead of "coid", sorry for the mistake.


PAGE 3


Is this a Maple 2015 issue which has been fixed in earlier versions?
Is there a way to fix these issues?

( squircle is the humoristic name for the 2D open ball of center 0 and radius 1 in L_n norm ).
The equation of the squircle in L_n norm writes  |x|ⁿ+|y|ⁿ = 1

The attached file gives the exact values of the areas of squircles in norms L₂, L₄, L₁₀₀, L₁. 
Unless for n=2 the results are dramatically poor (evalf/Int gives the same wrong results).

The function a(n) gives the exact expression of the squircle area in  L_n norm.

squircle.mw

I have some cubic and quartic equations with complex cofficients. Maple 2015 is able to solve these and returns the roots as labelled sets, so I can do things like "plot S[1]". I want to vary some parameters in the coefficients, and see what happens to the roots.
My problem is when I log out and then rerun the code, the labels 1,2,3,(4) are frequently attached to different roots than they were the first time. This is both unexpected and inconvenient. Is there any way to ensure that the same roots are always given the same labels?

[moderator: see also this Question from 2023]

Can anyone explain me the reason of the last result?
Thanks in advance

Download What-does-happen-here.mw

Hi!

I am using a proceure to conpute de integral of a function by he Simpson's rule. My function is defined from a function and a procedure, but I am getting the error  "Error, (in w) invalid input: hfun2 expects its 1st argument, t, to be of type numeric, but received (1/10)*i+1/20"

As you can see in the attaxhed file, I have tried several ways to compute the integral but always returns the above error. Please, can yo help me?

Thanks

forum.mw

Hi Dear,

I hope everyone is fine here. In the attached file, I have generated a square matrix "Q" using two-dimensional polynomials. The dimension of the square matrix "Q" depends on M1 and M2 parameters. In my simulation, sometimes I need this matrix of 1000 by 1000 dimensions. Using the attached method, it took a lot of time to compute two-dimensional polynomials and then to compute the general square matrix "Q." I wanted to write this matrix using proc (procedures). Maybe by using this way, I don't need to compute the polynomials, and it took less time to compute the square matrix "Q." I know how to generate a matrix using proc when its dimension depends on one parameter. However, here, the dimension of matrix "Q" depends on two parameters, M1 and M2. So, I am a little bit confused about how to adjust them in proc. Please see the attached file and share your useful ideas. 

help.mw

Thanks in advance

Hi
I hope you are doing well. I have plotted (in the attached file) the contour plot of the function and its density plot; both have the same behavior but different appearances (error in direction may be rotation needs to apply). I don't know why it happens because this code works well for other solutions. Kindly have a look and fix the issue. I shall be waiting for your positive response. Please take care.
Help.mw

It seems that Maple needs more help than necessary:

Download limits.mw

Why don't the commands labelled "Shouldn't this return.." do the job?

TIA

I have an expression equal to the sum of N terms of the form Int(f_n=1..N(x), x) and I want to replace each f_n(x) by its Taylor (or series) expansion.

When the integrals are definite, like J1 below, I can easily obtain a new expression (K1) where the integrand has been replaced by some expansion.
But when the integral is indefinite, like J2, I get an evaluated expression for K2.

It seems I have to do some gymnastic (J3 --> K3) to get what I want

Download Integration.mw

Why do I get this unevaluatedform for K2?
Do I have to use Intat to get K3?

Thanks in advance

For years I observe that package orthopoly is not considered as a package within a procedure.
Finally I have decided to ask for clarifications: Can someone explain me why procedure f generates an error?
 

Download meaning.mw

TIA