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infinitely many solutions...

vv 13805
December 05 2020
1 3

The system has infinitely many solutions:

Sys:=lhs~(sys):
Groebner:-IsZeroDimensional(Sys);  # ==> false

To see some of them:

solve(sys, maxsols=10);

 

local (escaped)...

vv 13805
December 04 2020
2 3

Somehow the names of the parameters returned by dsolve are local

 

> 

sol:=dsolve( {diff(x(t),t)=a, x(0)=1}, numeric, parameters=[a]): # x(t)=a*t+1
> 

type(a, `local`);

false

 (1)
> 

sol(parameters);

[a = undefined]

 (2)
> 

aa:=lhs(%[]);

a

 (3)
> 

type(aa,`local`); # !!

true

 (4)
> 

sol(parameters=[a=3]); # Works

[a = 3.]

 (5)
> 

sol(10)

[t = 10., x(t) = HFloat(31.000000000000007)]

 (6)
> 

sol(parameters=[aa=5]); # the locals not accepted

Error, (in dsolve/numeric/process_parameters) parameter(s) {a} are specified, but are not defined for the problem

  
> 

 

 

I don't see this as a real (annoying) bug, because sol(parameters)  and sol(parameters=...) is supposed to be used only for display purposes.

working solutions...

vv 13805
December 01 2020
2 0

Here are two working solutions.

1. Create a file A.mpl and use $include

module A()
$include "B.mpl";
export foo:=proc()
  0;
 end proc;
end module;

and in the worksheet use
read "A.mpl"

2. Remove the export from B.mpl, i.e. it will contain 

boo:=proc()
return 0;
end proc;

and use in the worksheet:

module A()
export boo,foo;
read "B.mpl";

foo:=proc()
  0;
 end proc;
end module;


 

constant...

vv 13805
December 01 2020
0 0

You cannot, because they differ by a constant.

conformal...

vv 13805
November 30 2020
1 4

The transform seems to be conformal. The picture was copied from https://en.wikipedia.org/wiki/Conformal_map  but the aspect ratio is not the same, so, the angles are not exactly preserved.

A projective transformation does not preserve the angles but preserves the lines.

And yes, of course, plottools:-transform command can be used.

 

> 

restart;
> 

N:=15:
> 

# f:=(x,y) -> (Re,Im)(sin(x+I*y):
> 

f:=(x,y) -> (Re,Im)(ln(1+I*(1+x+I*y)^2/9)):
> 

P:=plot([seq([x,y,x=0..1],y=0..1,1/N), seq([x,y,y=0..1],x=0..1,1/N)]);
plot([seq([f(x,y),x=0..1],y=0..1,1/N), seq([f(x,y),y=0..1],x=0..1,1/N)],scaling=constrained);

 

 
> 

# Using plottools:-transform
with(plottools):with(plots):
F := transform((x, y) -> [f(x,y)]):
> 

plots:-display(<P|F(P)>, scaling=constrained); # Not executed, to be accepted by mapleprimes.
> 

 

 

Opinion...

vv 13805
November 30 2020
1 0
The RootOf function is used to represent (not to compute) algebraic numbers (and other numbers). It will be used by many Maple commands, so, a compact and uniform representation is essential and the _Z notation seems to be convenient.
Commands like limit and sum are used in general to compute, not to represent expressions.
Anyway, a beginner seldom uses RootOf in input; he/she will see it only as a result of a computation (in solve, int etc) and any tutorial should mention this.
 
P.S. I was always puzzled by another unique Maple notation for a function name: GAMMA, instead of Gamma, the only one with all the letters capitalized.

int...

vv 13805
November 26 2020
3 0

Why dou you think that the integral can be expressed in terms of known functions?
Note that even int(sqrt(cos(t)^2+cos(2*t)^2),t)  has a complicated form and int(sqrt(cos(t)^2+cos(3*t)^2),t)  is already too much! 

Answer...

vv 13805
November 26 2020
0 0

1. The parantheses are not ignored; they are not displayed, being superfluous in this case, similar to a + (b*c), due to the order of precedence of the Maple operators.

2. Unfortunately Maple does not simplify the set operators; they are used mainly for explicit sets.
If you need to check or simplify expressions in set theory, you can use the Logic package because it works practically in a Boolean algebra.

Answer...

vv 13805
November 25 2020
1 7

If you want the general form of a vector in the kernel, try

B := <<6,3,0>|<4,2,0>|<10,5,0>>:
K := LinearAlgebra:-NullSpace(B):
add(alpha[i]*K[i],i=1..nops(K));

(alpha[i] being arbitrary scalars.)

Shooting...

vv 13805
November 24 2020
1 1

> 

xodephi := {diff(x(t), t) = 16250.25391*(1 - (487*x(t))/168 + 4*Pi*x(t)^(3/2) + (274229*x(t)^2)/72576 - (254*Pi*x(t)^(5/2))/21 + (119.6109573 - (856*ln(16*x(t)))/105)*x(t)^3 + (30295*Pi*x(t)^(7/2))/1728 + (7.617741607 - 23.53000000*ln(x(t)))*x(t)^4 + (535.2001594 - 102.446*ln(x(t)))*x(t)^(9/2) + (413.8828821 + 323.5521650*ln(x(t)))*x(t)^5 + (1533.899179 - 390.2690000*ln(x(t)))*x(t)^(11/2) + (2082.250556 + 423.6762500*ln(x(t)) + 33.2307*ln(x(t)^2))*x(t)^6)*x(t)^5, diff(xphi(t), t) = 5078.204347*x(t)^(3/2), x(0) = 0.03369973351, xphi(0) = a}:  #xphi(10.92469316) = 0}:
> 

sol := dsolve(xodephi, parameters=[a], numeric):
> 

ff:=proc(A)
  sol(parameters=[A]);
  eval(xphi(t), sol(10.92469316))
end:
> 

A:=fsolve(ff);
sol(parameters=[A]);
sol(10.92469316);

-467.1843838

  

[a = -467.1843838]

  

[t = 10.92469316, x(t) = HFloat(0.061040830524973895), xphi(t) = HFloat(1.6909911249030074e-9)]

 (1)
> 

plots:-odeplot(sol, [t,xphi(t)], t=0..11);

 
> 

plots:-odeplot(sol, [t,x(t)], t=0..11);

 
> 

 

 

Generalizations...

vv 13805
November 24 2020
1 0

The term "generalization"  is not always the right one. This happens in maths too. E.g. the Lebesgue integral is a "generalization" of the Riemann integral. However, int(sin(1/x)/x, x=0..1) does not exist  in the Lebesgue sense, but it exists as a Riemann (improper)  integral, and Maple computes it.

Inscribed ellipses of a quadrilateral ...

vv 13805
November 22 2020
3 4

> 

restart;

f maps the unit square onto the quadrilateral; ek = ellipses tangent to the unit square.

> 

Ellipses:=proc(P::listlist(realcons), t::name, k::name)
local a1,b1,c1,a2,b2,c2,b0,c0,ek,
      f:=(x,y)->[a1*x+b1*y+c1,a2*x+b2*y+c2]/~(x+b0*y+c0);
ek:=1/2+k/2*cos(t)-sqrt(1-k^2)/2*sin(t), 1/2+sqrt(1-k^2)/2*sin(t)+k/2*cos(t);
solve(op~([f(0,0)=~P[1], f(1,0)=~P[2], f(1,1)=~P[3], f(0,1)=~P[4]]));
eval([f(ek)[], t=0..2*Pi], %)
end proc:
> 

# Example
> 

P:=[[0,0], [5,0], [4,3], [1,2], [0,0]]: # quadrilateral
> 

Ell:=Ellipses(P, t, k);

[(-4-4*k*cos(t)+(-k^2+1)^(1/2)*sin(t))/(-2-(-k^2+1)^(1/2)*sin(t)), (-3-3*(-k^2+1)^(1/2)*sin(t)-3*k*cos(t))/(-2-(-k^2+1)^(1/2)*sin(t)), t = 0 .. 2*Pi]

 (1)
> 

plots:-display(plot(P,color=blue, thickness=4),
seq(plot(Ell, color=COLOR(HUE,k)), k=0..1,0.05), scaling=constrained, size=[800,800]);

 
> 

plots:-animate(plot, [Ell], k=0..1, background=plot(P,color=blue, thickness=4), scaling=constrained);

 
> 

 

 

Bug in series for JacobiTheta3&4...

vv 13805
November 18 2020
2 3

 

Yes, in the implementation of series for JacobiTheta3 and JacobiTheta4, the term 1 is missing.

 

> 

restart;
> 

FunctionAdvisor(definition, JacobiTheta3);

[JacobiTheta3(z, q) = Sum(2*q^(_k1^2)*cos(2*_k1*z), _k1 = 1 .. infinity)+1, abs(q) < 1]

 (1)
> 

FunctionAdvisor(series, JacobiTheta3);

series(JacobiTheta3(z, q), z, 4) = series(2*(Sum(q^(_k1^2), _k1 = 1 .. infinity))-(4*(Sum(q^(_k1^2)*_k1^2, _k1 = 1 .. infinity)))*z^2+O(z^4),z,4), series(JacobiTheta3(z, q), q, 4) = series(1+(2*cos(2*z))*q+O(q^4),q,4)

 (2)
> 

 

 

Type less...

vv 13805
November 18 2020
1 0

I always prefer to change myself the coordinates. This way I avoid the two distinct conventions used by Maple for spherical.
As a bonus, I type less!

int(eval((x^2 + y^2)*z, [x=sin(u)*cos(v), y=sin(u)*sin(v), z=cos(u)]) * sin(u), u = 0 .. Pi/2, v = 0 .. 2*Pi);

        Pi/2

 

select...

vv 13805
November 17 2020
2 1 
if type(lhs(ode),`*`) and rhs(ode)=0 then 
  ODE:=select(has, lhs(ode), diff) = 0
fi;

If lhs has a factor depending on y, but has not diff, you may keep it, or solve it for y.

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