MaplePrimes Questions

Dear Maple Community,

I come to you with a question about the reduced involutive form (rif) package. Namely, I decided to try the classic example from the "LONG GUIDE TO THE STANDARD FORM PACKAGE", which dates back to 1993. Here is the link to the complete documentation:

https://wayback.cecm.sfu.ca/~wittkopf/files/standard_manual.txt

So, the example is the following:

2.1 SIMPLE EXAMPLES

EXAMPLE A

Consider the system of nonlinear PDEs:       

y Zxxx - x Zxyy  =  Zyy - y Zy

                        2     2    2
2 y x Zxxx Zxyy + x Zxxx + x y Zxyy  =  0

                  2    2
y Zxyy - x W + 2 x  y Z  =  0

                 2    2
Zyy - y Zy  + 2 x  y W  =  x W

where the dependent variables W and Z are functions of the
independent variables x and y, and Zxxx denotes the third partial
derivative of Z with respect to x etc.

After making computations back in 1993 with Maple V, they obtain the following involutive form:

In our original notation the (considerably) simplified system is:
                                            2
  Zxxx = 0, Zxy = 0, Zyy = y Zy, W = 2 x y Z

So, I tried the same system of PDEs in the modern Maple and the modern rifsimp() command. You can find the result below:

demo_question.mw

The problem is that nowadays [Maple 2022.1] , I get only the trivial solution: z = 0 and w = 0.

Could someone clarify, please, where the truth is and what am I doing wrong?

Thanks a lot in advance for any help and clarification!

Best regards,

Dr. Denys D.
 

restart:

with(DETools):

PDE1 := y*diff(z(x,y), x$3) - x*diff(z(x,y),x,y$2) = diff(z(x,y),y$2) - y*diff(z(x,y), y);

y*(diff(diff(diff(z(x, y), x), x), x))-x*(diff(diff(diff(z(x, y), x), y), y)) = diff(diff(z(x, y), y), y)-y*(diff(z(x, y), y))

(1)

PDE2 := 2*x*y*diff(z(x,y),x$3)*diff(z(x,y),x,y$2) + x*(diff(z(x,y),x$3))^2 + x*y^2*(diff(z(x,y),x,y$2))^2 = 0;

2*x*y*(diff(diff(diff(z(x, y), x), x), x))*(diff(diff(diff(z(x, y), x), y), y))+x*(diff(diff(diff(z(x, y), x), x), x))^2+x*y^2*(diff(diff(diff(z(x, y), x), y), y))^2 = 0

(2)

PDE3 := y*diff(z(x,y),x,y$2) - x*w(x,y) + 2*x^2*y*z(x,y)^2 = 0;

y*(diff(diff(diff(z(x, y), x), y), y))-x*w(x, y)+2*x^2*y*z(x, y)^2 = 0

(3)

PDE4 := diff(z(x,y), y$2) - y*diff(z(x,y),y) + 2*x^2*y*w(x,y)^2 = x*w(x,y);

diff(diff(z(x, y), y), y)-y*(diff(z(x, y), y))+2*x^2*y*w(x, y)^2 = x*w(x, y)

(4)

sys := [PDE1, PDE2, PDE3, PDE4]:

rif := rifsimp(sys, [[w], [z]], indep = [x,y]);

table( [( Case ) = [[z(x, y)*(8*z(x, y)^2*y^2*x^2-1) = 0, diff(z(x, y), x), "false split"]], ( Solved ) = [w(x, y) = 0, z(x, y) = 0] ] )

(5)
 

 

I have one random variable as follows:  r := sqrt(RandomVariable(Uniform(0,1)))

I have another as follows:  x := cos(RandomVariable(Uniform(0, 2*Pi)));

The r variable returns a PDF of 2*t, where t<1; 0 otherwise.  The variable x returns a PDF of 1/(Pi*sqrt(1-t^2)), where |t|<1; 0 otherwise.

For both random variables, I have validated through simulated values.  I am convinced the above PDF functions are correct.  When I multiply r*x in Maple and then ask for a PDF of the resultant random variable, I fail.  Maple seems to not like this.

My question is then as follows...can anyone help me get the PDF of r*x?

Which version of Maple contains its most optimal and elegant coding, regardless of utility,  and was it the product of someone's direct creative input (not auto-coded with AI for example).

As an aside, there's a spelling mistake in the word "separate" below within the "Tags" instruction:

"Tags are words are used to describe and categorize your content. Combine multiple words with dashes(-), and seperate tags with spaces."

I am making changes in my code in order to make it work with Grid. This involves making many modules I had export on them as local.

I have lots of code of this form

use toX=A:-some_very_long_module_name:-toX in
local B := module() 
       export foo:=proc()
            toX(...);
       end proc;
end module;
end use;

This is done so I do not have to type A:-some_very_long_module_name:-toX() all the time as toX() is a very common call I make all over the place in many different modules.

This was working fine, except when I now changed some_very_long_module_name module to be local.

I really do not want to change all my code and change toX(...) to the explicit fully qualified name  some_very_long_module_name:-toX(....)

I could probably look into using alias instead, but I do not like aliases.

But before doing this, any one knows why this now makes Maple not happy? And if it possible to make use....end use while keep this long named module local?

Below is work sheet showing 4 examples. The first one is how I had things before, where the module was export.

The second and third examples showing the problems that show up when changing the module to local.

The last one showing it works if I remove use...end use and just type the full long name.

my goal is to keep this long named module local, but still use  use...end use. around other modules which needs to make calls to it. Just to safe typing, that is all. 

Any ideas to try are welcome. 

 

restart;

 

Original code work, but module has to be export

 

A := module()

  export main_entry:=proc()
     B:-foo();
  end proc;

  export some_very_long_name:= module() #NOTICE, export
        export toX:=proc(n::integer)::integer;
              n+1;
        end proc;
  end module;

  use toX=A:-some_very_long_name:-toX in  #works now
  local B:= module()
     export foo:=proc()
         toX(1);
     end proc;
  end module;
  end use;

end module;

_m1906974412096

A:-main_entry()

2

 

Example 1 where it now fails, since changed to local

 

A := module()

  export main_entry:=proc()
     B:-foo();
  end proc;

  local some_very_long_name:= module() #NOTICE, now local
        export toX:=proc(n::integer)::integer;
              n+1;
        end proc;
  end module;

  use toX=A:-some_very_long_name:-toX in
  local B:= module()
     export foo:=proc()
         toX(1);
     end proc;
  end module;
  end use;

end module;

_m1907025773216

A:-main_entry()

Error, (in foo) module does not export `some_very_long_name`

 

Example 2 remove A:- in the use call

 

A := module()

  export main_entry:=proc()
     B:-foo();
  end proc;

  local some_very_long_name:= module() #NOTICE, now local
        export toX:=proc(n::integer)::integer;
              n+1;
        end proc;
  end module;

  use toX= some_very_long_name:-toX in #notice, removed A:-
  local B:= module()
     export foo:=proc()
         toX(1);
     end proc;
  end module;
  end use;

end module;

Error, (in anonymous procedure created in anonymous module instantiated by anonymous module) nameless local variable in procedure

 

 

 

 

Example 3. One solution is to type the full name and remove use....end use; But I am trying to avoid this.

 

A := module()

  export main_entry:=proc()
     B:-foo();
  end proc;

  local some_very_long_name:= module() #NOTICE, now local
        export toX:=proc(n::integer)::integer;
              n+1;
        end proc;
  end module;
  
  local B:= module()
     export foo:=proc()
         some_very_long_name:-toX(1); #this works
     end proc;
  end module;


end module;

_m1907023851968

A:-main_entry()

2

 


 

Download trouble_with_use.mw

The following page of the documentation cites an article from the newsletter which I would be very interested in reading: 

- https://www.maplesoft.com/support/help/view.aspx?path=assume

The full citation for the article I'm interested in is as follows:
 

  

Corless, Robert, and Monagan, Michael. "Simplification and the Assume Facility." Maple Technical Newsletter, Vol. 1 No. 1. Birkhauser, 1994.

Hello,

I have updated to maple 2024 on both my desktop and my laptop, and now I am missing the feature "Convert Output Units:" in the context tab on my windows 11 laptop. It's still available on my windows 10 desktop.

I have tried reinstalling maple and installing java, but it unfortunately did not help. Due to limited school licences, I am unable to test with maple 2023.

Is this an issue you have heard of?

Thank you in advance,

Daniel

 

And here is the context menu on windows 11. Also nothing happens when I click "Format -> Convert Output Units" in the top menu. 

 

Hello, i have been drawing some cool 3d plots for my assignment, but when i use the export button and export it as pdf the plots turn out very low quality. 

See the image below is using the export function

Then i tried something different i tried using the print button and printing to a pdf.

That resulted in a different looking plot

This plot using the print to pdf feature looks much nicer, but the 3d text plot has become impossible to read.

 

Is there a way to fix that? Or to make the export to pdf feature export at higher quality? 

Best Regards

I have a list of values in the range from 1 to 6 inclusive:

    [> y := [ seq(rand(1..6)(), i = 1..10) ] : # step 1

I would like to replace all elements <= 3 with -1 and all elements > 3 with 1

    [> y := subs(1=-1, 2=-1, 3=-1, 4=1, 5=1, 6=1, y) : # step 2

Then calculate the cumulative sum

    [> y := [ seq(add(y[ .. i]), i = 1..n) ] : # step 3

And plot the results

    [> plot([ seq(i, i = 1..n) ], y) # step 4
 

This all works, but step 2 is downright ugly. Is there a more elegant expression available?

And step 4 requires I generate an extra data set for the x-axis. Is there a way to specify just the the y-values and have Maple assume the x-values are in the range  1..nops(y)? (Something akin to just  plot(y), which produces output I don't understand.)

By way of example, I am trying to replicate this MATLAB expression:

    > y = cumsum(2*(randi(6, 10, 1) <= 3) - 1); plot(y)

 

 

restart;
Fig:=proc(t)
local a,b,c,A,B,C,Oo,P,NorA,NorB,NorC,lieu,Lieu,dr,tx:
uses plots, geometry;
a := 11:b := 7:
c := sqrt(a^2 - b^2):

point(A, a*cos(t), b*sin(t)):
point(B, a*cos(t + 2/3*Pi), b*sin(t + 2/3*Pi)):
point(C, a*cos(t + 4/3*Pi), b*sin(t + 4/3*Pi)):
point(Oo,0,0):
lieu:=a^2*x^2+b^2*y^2-c^4/4=0:
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green):

line(NorA, y-coordinates(A)[2] =((a^2*coordinates(A)[2])/(b^2*coordinates(A)[1]))*(x-coordinates(A)[1]),[x, y]):
line(NorB, y-coordinates(B)[2] =((a^2*coordinates(B)[2])/(b^2*coordinates(B)[1]))*(x-coordinates(B)[1]), [x, y]):
line(NorC, y-coordinates(C)[2] =((a^2*coordinates(C)[2])/(b^2*coordinates(C)[1]))*(x-coordinates(C)[1]),[x, y]):
intersection(P,NorA,NorB):

ellipse(p, x^2/a^2 + y^2/b^2 - 1, [x, y]);

tx := textplot([[coordinates(A1)[], "A"],[coordinates(A2)[], "B"], [coordinates(C)[], "C"], [coordinates(Oo)[], "O"],#[coordinates(P)[], "P"]], font = [times, bold, 16], align = [above, left]):
dr := draw([p(color = blue),NorA(color=red,NorB(color=red),NorC(color=red),p(color=blue),
Oo(color = black, symbol = solidcircle, symbolsize = 8), P(color = black, symbol = solidcircle, symbolsize = 8)]):
display(dr,tx,Lieu,scaling=constrained, axes=none,title = "Les triangles inscrits dans une ellipse ont leur centre de gravité en son centre . Lieu du point de concours des perpendicalaires issues des sommets", titlefont = [HELVETICA, 14]);
end:

Error, `:=` unexpected
plots:-animate(Fig, [t], t=0.1..2*Pi, frames=150);
 

Let X1, ...XN  names of RandomVariable type and R a random variable defined as a function of  X1, ...XN.
(In Maple type(R, RandomVariable) returns false as R doesn't inherit the RandomVariable type.)

Given an expression of R I would like to determine if R is :

  • either a linear combination of X1, ...XN ,
    example R = a*X+ X+ 1
    (I know how to handle this situation)
     
  • or a linear combination of products of the X1, ...XN ,
    example R = a*X1*X+ b*X3
    (I feel I know how to handle this situation too)
     
  • or a linear combination of functions, or products of functions, of  X1, ...XN .
    example R = a*f(X1, X2) + b*X3*g(X4) + c*X1 + d*X1*X2
    (here I've no idea at all)

In the command hastype(R, some_type), how can I construct type some_type in order it returns an answer which matches the reqierements above?

RV_Type.mw

Thanks in advance for any suggestion

page 37 of book Symmetry Methods for Differential Equations by Hydon gives this example

When I wanted to verify it using Maple., symgen did not find these symmetries. Only when I give it using HINT the exact form it find them.  


 

interface(version);

`Standard Worksheet Interface, Maple 2024.2, Windows 10, October 29 2024 Build ID 1872373`

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1843 and is the same as the version installed in this computer, created 2025, January 25, 22:5 hours Pacific Time.`

libname;

"C:\Users\Owner\maple\toolbox\2024\Physics Updates\lib", "C:\Program Files\Maple 2024\lib"

restart;

ode:=diff(y(x),x) = (y(x)^3+y(x)-3*x^2*y(x))/(3*x*y(x)^2+x-x^3)

diff(y(x), x) = (y(x)^3+y(x)-3*x^2*y(x))/(3*x*y(x)^2+x-x^3)

the_syms:=DEtools:-symgen(ode)

the_syms:=DEtools:-symgen(ode,'way'='all')

the_syms:=DEtools:-symgen(ode,'way'='formal')

#only when I give it the exact general form, it finds them !
the_syms:=DEtools:-symgen(ode,'HINT'=[a*y^3+b*y-c*x^2*y,d*x^3-e*x-f*x*y^2])

[_xi = -3*x^2*y+y^3+y, _eta = x*(x^2-3*y^2-1)]

 


 

Download why_no_syms_feb_10_2025.mw

Is this expected? Should it not have found them on its own?

Is it possible Maple internally does not automatically try Ansatz  of polynomials higher than quadratic to keep computation time low?   If so, I wonder if there is a way to tell it to try cubic or higher orders (like tryhard) option but for symgen?  I will search help more to see if there is a way to do this...

In trying to evaluate the accuracy of an asymptotic approximation, I asked evalf to return the numerical value of the difference of two expressions.  It evaluated each expression but would not take their difference (see line 6).  Any idea what's going on?

restart

g := exp(-r*cosh(x))*x^(2*n); G := Int(g, x = 0 .. infinity)

Int(exp(-r*cosh(x))*x^(2*n), x = 0 .. infinity)

(1)

f0 := arccosh(1+y); df0 := diff(f0, y)

1/(y^(1/2)*(2+y)^(1/2))

(2)

f := f0^(2*n)*df0

arccosh(1+y)^(2*n)/(y^(1/2)*(2+y)^(1/2))

(3)

n := 2; series(f, y)

2*2^(1/2)*y^(3/2)-(7/6)*2^(1/2)*y^(5/2)+(47/80)*2^(1/2)*y^(7/2)-(17281/60480)*2^(1/2)*y^(9/2)+O(y^(11/2))

(4)

r := 6; evalf(G)

0.8560816424e-4

(5)

Ghat := exp(-r)*(2*sqrt(2)*GAMMA(5/2)*`-`*(7*sqrt(2)*(1/6))/r^(5/2)*(GAMMA(7/2)/r^(7/2))); Ghat1 := evalf(Ghat)

0.1056905544e-3-0.2568867641e-4*``

(6)

evalf(Ghat1)

0.1056905544e-3-0.2568867641e-4*``

(7)

 

Download asyapprox.mw

I am working with an expression in Maple that involves complex terms and an integral. After applying the simplify command, some terms remain unsimplified, even though they seem reducible (see (7)). Additionally, an integral in my expression remains unevaluated (see (9)).
 

restart;

kernelopts(version);

`Maple 2022.0, X86 64 WINDOWS, Mar 8 2022, Build ID 1599809`

(1)

with(plots)

interface(showassumed=0):

assume(x::real);assume(t::real);assume(lambda1::complex);assume(b::real);

alias(psi1 = psi1(x,t), psi2 = psi2(x,t), phi1 = phi1(x,t), phi2 = phi2(x,t), beta = beta(t), alpha =alpha(t));

psi1, psi2, phi1, phi2, beta, alpha

(2)

rel := {psi1 = exp((-I*lambda1)*x - (1/(4*I*lambda1))*int((alpha + b*beta),t)), psi2 = exp((I*lambda1)*x + (1/(4*I*lambda1))*int((alpha + b*beta),t)), phi1= exp((-I*conjugate(lambda1))*x - (1/(4*I*conjugate(lambda1)))*int((alpha + b*beta),t)), phi2 = exp((I*conjugate(lambda1))*x + (1/(4*I*conjugate(lambda1)))*int((alpha + b*beta),t))}

{phi1 = exp(-I*conjugate(lambda1)*x+((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1)), phi2 = exp(I*conjugate(lambda1)*x-((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1)), psi1 = exp(-I*lambda1*x+((1/4)*I)*(int(b*beta+alpha, t))/lambda1), psi2 = exp(I*lambda1*x-((1/4)*I)*(int(b*beta+alpha, t))/lambda1)}

(3)

Bnum := psi2*phi1*conjugate(lambda1) + psi1*lambda1*phi2;

psi2*phi1*conjugate(lambda1)+psi1*lambda1*phi2

(4)

Bnumexp := subs(rel,Bnum):

Den := -phi1*psi2 - phi2*psi1;

-phi1*psi2-phi2*psi1

(5)

expDen := subs(rel, Den)

-exp(-I*conjugate(lambda1)*x+((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1))*exp(I*lambda1*x-((1/4)*I)*(int(b*beta+alpha, t))/lambda1)-exp(I*conjugate(lambda1)*x-((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1))*exp(-I*lambda1*x+((1/4)*I)*(int(b*beta+alpha, t))/lambda1)

(6)

sr := Bnumexp/expDen: ratiosr := simplify(diff(sr,t), complex):

B := b - (4*I/beta(t))*ratiosr

b+2*(b*beta+alpha)*exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)*(-conjugate(lambda1)+lambda1)^2*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)/(beta(t)*(exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)+exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1))^2*lambda1*conjugate(lambda1))

(7)

p := {alpha(t) = t^2, beta = exp(-t)}

{beta = exp(-t), alpha(t) = t^2}

(8)

B1 := eval(subs(p, B))

b+2*(b*exp(-t)+alpha)*exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)*(-conjugate(lambda1)+lambda1)^2*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)/((exp(-t))(t)*(exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)+exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1))^2*lambda1*conjugate(lambda1))

(9)

NULL


 

Download simplify.mw

is(cos(x)=sqrt(1-sin(x)^2) or cos(x)=-sqrt(1-sin(x)^2)) assuming real;

                                          false  

In a simpler situation the result is correct

is(x=0 or x<>0) assuming real;

                                           true         

SymPy can handle the following query to its assumption system:

>>> from sympy import Q, ask
>>> from sympy.abc import x, y, z
>>> ask(Q.zero(x*y*z), Q.zero(x*z) & Q.real(y))
True
>>> ask(Q.zero(x*y*z), (Q.zero(x) | Q.zero(z)) & Q.real(y))
True
>>> ask(Q.zero(x*y) | Q.zero(y*z), Q.zero(x*z) & Q.real(y))
True

Mathematica can handle an equivalent query:

In[21]:= Assuming[{x*z == 0}, Refine[x*y*z == 0]]
Out[21]= True

However, it can't handle a similar type of query:

In[25]:= Assuming[{x == 0 || z == 0}, Simplify[x*y*z == 0]]
Out[25]= x y z == 0

In[28]:= Assuming[{x*y == 0}, Refine[x*z == 0 || z*y == 0]]
Out[28]= x z == 0 || y z == 0  

It seems that Maple can't handle this type of query at all:

> with(RealDomain);
> is(x*y*z = 0) assuming (x*z = 0, y::real);
FAIL   

However, Maple can handle simpler assumptions like x = 0:

> is(x*y*z = 0) assuming (x = 0, y::real, z::real);
true

Is there some way to use Maple's assumption system to reason about assumptions like x*y=0? I believe that's not possible. If Maple can infer either x=0 or y=0 from x*z=0, its assumption system doesn't seem to be capable of dealing with this sort of assumption.

Based on a paper by Weibel and Gonnet (1993), it doesn't seem like Maple can reason about assumptions that are disjunctions of properties for more than one variable, such as "property p holds for x OR property p holds for y."

SymPy can handle these sorts of queries because it uses a SAT solver as part of its assumption system. However, there is no mention of a SAT solver—or any method of checking multiple possibilities—in the referenced paper or related papers on Maple's assumption system.

Am I correct that Maple's assumption system is not capable of handling this sort of query?

Weibel, T., Gonnet, G.H. (1993). An assume facility for CAS, with a sample implementation for Maple. In: Fitch, J. (ed) Design and Implementation of Symbolic Computation Systems. DISCO 1992. Lecture Notes in Computer Science, vol 721. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57272-4_27

By the way, is there some way to use markdown for these questions? I made the above post by asking chatgpt to convert markdown into html. 

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