restart:with(LinearAlgebra):with(plots):with(geometry):with(plottools):
On appelle alpha la moitié de l'angle de rotation de la roue menée par tour de roue menante.
alpha=Pi/n en radians?
soit Pi/8 pour 8 rainures..
On a alors les relations suivantes entre l'entaxe E, le rayon de la roue ùenante R1 et le rayon de la roue menée R2 :
R1=E.sin(alpha), R2=E*cos(alpha)
Intersection du cercle (O,R2) avec la droite tan(phi)x-r/cos(phi), on obtient les coordonnées de P3
sol:=allvalues(solve([tan(phi)*x-r/cos(phi)=y,y^2+x^2=R2^2],[x,y])):
Intersection de 2 cercles
sol1:=allvalues(solve([(x-E)^2+y^2=(R-a)^2,y^2+x^2=R2^2],[x,y])):
Coordonnées des points du pourtour de l'élément de croix
Oo:=point([0,0]):
phi:=Pi/8:R2:=5:r:=1/4:E:=R2/cos(phi):evalf(%):R:=R2*tan(phi):evalf(%):a:=0.5:
P1:=point([(R2/2-r)*cos(phi),(R2/2-r)*sin(phi)]):
P2:=point([(R2/2)*cos(phi)+r*sin(phi),(R2/2)*sin(phi)-r*cos(phi)]):
xP2:=(R2/2)*cos(phi)+r*sin(phi):yP2:=(R2/2)*sin(phi)-r*cos(phi):
xP1:=(R2/2-r)*cos(phi):yP1:=(R2/2-r)*sin(phi):
Equation paramétrique du segment OP1 (t varie de 0 à 1) ;
x1:=t*(0-xP1)+xP1:
y1:=t*(0-yP1)+yP1:
n1:=5:
dt:=1/(n1-1):#t varie entre 0 et 1
for i to n1 do
tau:=(i-1)*dt:
xx[i]:=evalf(subs(t=tau,x1)):
yy[i]:=evalf(subs(t=tau,y1)):
#print(i,xx[i],yy[i]);
od:
Equation paramétrique du quart de cercle P1P2 de la rainure (t varie de 0 à 1)
x2:=R2/2*cos(phi)+r*cos(t):#attention au sens de rotation du parcours de l'objet
y2:=R2/2*sin(phi)+r*sin(t):
n2:=6:
dt:=Pi/2/(n2-1):#arc de Pi/2
for i to n2 do
tau:=phi-Pi+(i-1)*dt:
xx[i]:=evalf(subs(t=tau,x2)):
yy[i]:=evalf(subs(t=tau,y2)):
od:
for i to n2 do Vector[row]([i,xx[i],yy[i]]) od:
droite:=plot((tan(phi)*x-r/cos(phi),x=0..3),linestyle=dot,color=blue):
sol[1]:
xP3:=evalf(subs(op(1,sol[1]),x)):yP3:=evalf(subs(op(1,sol[1]),y)):
xP2:yP2:
xP4:=evalf(subs(op(1,sol1[1]),x)):yP4:=evalf(subs(op(1,sol1[1]),y)):
xP5:=E-(R-a):yP5:=0:
x3:=t*(xP3-xP2)+xP2:
y3:=t*(yP3-yP2)+yP2:
n3:=10:
dt:=1/(n3-1):#t varie entre 0 et 1
for i to n3 do
tau:=(i-1)*dt:
xx[i+n2]:=evalf(subs(t=tau,x3)):
yy[i+n2]:=evalf(subs(t=tau,y3)):
od:
for i to n3 do Vector[row]([i,xx[i],yy[i]]) od:
x4:=xP5+R-a+(R-a)*cos(t):#attention au sens de rotation du parcours de l'objet
y4:=(R-a)*sin(t):
n4:=11:
eta:=arcsin(yP4/(R-a)):
dt:=(-eta)/(n2-1)/2:#arc de Pi/2
for i to n4 do
tau:=(Pi+eta)+(i-1)*dt:#recherche de tau ?
xx[i+n2+n3]:=evalf(subs(t=-tau,x4)):
yy[i+n2+n3]:=evalf(subs(t=-tau,y4)):
od:
for i to n4 do Vector[row]([i,xx[i],yy[i]]) od:
n:=n2+n3+n4;
n := 27
for i to n do Vector[row]([i,xx[i],yy[i]]) od:
figure:=NULL:
for i from 0 to n do
xx[0]:=0:yy[0]:=0:
figure:=figure,[xx[i],yy[i]]:
#print(i,xx[i],yy[i]);
od:
polygonplot([figure],scaling=constrained,color=yellow,view=[-0.1..5,-0.1..3]):
for i to n do X[i]:=xx[i]: Y[i]:=yy[i] od:
d1:=plottools[disk]([xP1,yP1],0.05,color=blue):
d2:=plottools[disk]([xP2,yP2],0.05,color=red):
d3:=plottools[disk]([xP3,yP3],0.05,color=green):
d4:=plottools[disk]([xP4,yP4],0.05,color=green):
d5:=plottools[disk]([xP5,yP5],0.05,color=green):
fig:=pointplot([figure],scaling=constrained):
Po:=pointplot([[xP1,yP1],[xP2,yP2],[xP3,yP3]],color = blue, symbol = asterisk):
Cir:=plot([R2*cos(t),R2*sin(t),t=0..Pi/2],color=black):
Arc:=plot([E+(R-a)*cos(t),(R-a)*sin(t),t=3*Pi/4..Pi],linestyle=dot,color=blue):
textplot({[1, 2, "one point in 2-D"], [3, 2, "second point in 2-D"]}):
texte:=textplot([[xP1-0.2,yP1,"P1"],[xP2,yP2-0.3,"P2"],[xP3+0.2,yP3+0.2,"P3"],
[xP4+0.2,yP4+0.1,"P4"],[xP5-0.2,yP5+0.2,"P5"]]):
display({Arc,Cir,d1,d2,d3,d4,d5,Po,fig,droite,texte},scaling=constrained,view=[-1..7,-1..6]):
with(plottools):
printlevel:=3:
Miroir : symétrie par rapport à l'axes des x
for i from 0 to n/2 do
tt:=yy[i]:
yy[i]:=yy[n-i+1]:
yy[n-i+1]:=tt:
tt:=xx[i]:
xx[i]:=xx[n-i+1]:
xx[n-i+1]:=tt
od:
for i from 0 to n-1 do
xx[2*n-i]:=xx[i]:
yy[2*n-i]:=-yy[i]:
#print(i,xx[i],yy[i])
od:
Poly:=NULL:
for i from 0 to 2*n-1 do
xx[0]:=0:yy[0]:=0:
Poly:=Poly,[xx[i],yy[i]]:od:
polygonplot([Poly],color=yellow,scaling=constrained):
pointplot([Poly],color = blue, scaling=constrained,symbol = asterisk,view=[-1..5,-3..3]):
Rotation
unassign('xt','yt'):
#gc():
zt:=8:#8 rainures ou faisceaux
xt:=Vector(63,[]):
yt:=Vector(63,[]):
xt:=Vector((2*n-1),zt,[]):
yt:=Vector((2*n-1),zt,[]):
j:=0:
for k from 0 to zt-1 do
j:=0:
phi:=2*Pi*k/zt:
cs:=cos(phi):
sn:=sin(phi):
for kk from 1 to 2*n-1 do
j:=j+1:
xt[j][k]:=evalf(xx[kk]*cs-yy[kk]*sn):
yt[j][k]:=evalf(xx[kk]*sn+yy[kk]*cs):
od:
od:
N1:=j:
points:=seq(seq([xt[i][j], yt[i][j]], j=0..zt-1), i=1..2*n-1):
p_cross:= pointplot([points], scaling = constrained, color = black,linestyle=solid, filled=[yellow]):
polygonplot([points], color = yellow, scaling = constrained);
NULL;
display([p_cross]);#How to draw this cross with a line without points. Thank you.

Hello, I'm using Maple 2021.2 to approximate the solution to a differential equation using a pseudospectral expansion in the basis of Hermite polynomials. After setting up some equations, I get a non-linear system of equations that I try to solve using the Maple's function solve, but Maple suddenly crashes during the execution of this function.

My code is the following:

HN := (n, y) -> exp(-1/2*y^2)*simplify(HermiteH(n, y))/sqrt(sqrt(Pi)*2^n*n!):
u7 := a0*HN(0, y) + a1*HN(1, y) + a2*HN(2, y) + a3*HN(3, y) + a4*HN(4, y) + a5*HN(5, y) + a6*HN(6, y):
resid := diff(u7, y $ 2) + 4*y*diff(u7, y)*u7 + 2*u7^2:
r := evalf(allvalues(RootOf(HermiteH(7, x), x)));
equations := {evalf(eval(u7, y = 0) - 1)};
for root_r in r do
if root_r != 0 then
equations := {op(equations), evalf(eval(resid, y = root_r))};
end if:
end do:
solutions := solve(equations, {a0, a1, a2, a3, a4, a5, a6}):

Does anyone have any idea about what is happening? Is there any better way of solving a non-linear system of equations?

I am attempting to use maplev mode for emacs. Ubuntu 20.04.3, Emacs 26.3, maplev-mode from git, release-2.36-59-g142c03. maple 2021.

I copied mload from an older computer and added a line to define maple, after the original maple definition line.

MAPLE=${MAPLE:-maple} # Maple script

MAPLE=/usr/local/Maple2021/bin/maple <=== added

I compiled and installed maplev-mode. I had to touch doc/version.texi because the file did not exist. But, otherwise, I saw no errors.

I added lines to my emacs init.el as per the README.md.

maplev-mode seems to load when I edit a .mpl file.

However, mint is not defined. Where should I change to define the location of mint?

/usr/local/Maple2021/bin/mint

Tom Dean

In this example, i have a top module A. Inside it, I have module named **my_RECORD** (which happened to be option object) that I want to return back to caller when they call a proc.

Maple allows me to declare the proc to return **my_RECORD**. No problem., But when making a local variable inside the proc and using **o::my_RECORD** it complains that my_RECORD does not exist.

The solution is to change **o::my_RECORD **to **o::A:-my_RECORD**

My question why it did not complain the same way on the return value on the proc? for me, they are both semantically the same. One says the proc returns this type, and the other says the local variable is this type. So why had to do **A:-my_RECORD **on one but not the other?

Here is an example. Maple 2021.2 on windows 10.

Download A_no_lib.mw

**Correction**

Please ignore this question. dsolve does hang, but I had typo in the timelimit command itself when I wrote the test. Fixing this, now it timesout OK.

Maybe someone can look why dsolve hangs on this ode. But since timelimit does work, there is a workaround.

**Original question**

I was checking Maple's dsolve on this textbook problem

The book gives the answer in the back as

When using Maple's dsolve, I found it hangs. The stange thing, is that adding timelimit() also hangs. I can understand dsolve() hanging sometimes. But what I do not understand is why with timelimit it also hangs?

I've waited 20 minutes and then gave up. As you see, the timelimit is 20 seconds. May be if I wait 2 hrs or 20 hrs or 20 days, it will finally timeout. I do not know but can't wait that long.

Do others see same problem on this ode? Does it hang for you? How about on the mac or Linux?

During this time, I see mserver.exe running at very high CPU. I restarted Maple few times, but this did not help.

Maple 2021.2 on windows 10. May be one day Maplesoft will fix timelimit so it works as expected.

Download test_ode_hang.mw

on DLMF page, they show this transformation on independent variable for second order ode

https://dlmf.nist.gov/1.13#Px7

About half way down the page, under **Elimination of First Derivative by Change of Independent Variable **section.

I tried to verify it using Maple dchange. But the problem it looks like dchange wants the old variable to be on the left side (z in this example) and the new variable (eta in this example) to be on the right side in the transformation. But on the above web page, it is the other way around.

Here are my attempts

restart;
ode:=diff(w(z),z$2)+f(z)*diff(w(z),z)+g(z)*w(z)=0;
tranformation:=eta=int(exp(-int(f(z),z)),z);
PDEtools:-dchange({tranformation},ode,known={z},unknown={eta});
PDEtools:-dchange({tranformation},ode,{eta},known={z});
PDEtools:-dchange({tranformation},ode,{eta});

All give errors

Error, (in dchange/info) missing a list with the new variables

Error, (in dchange/info) the new variables are not contained in the rhs of the direct transformation equations

Error, (in dchange/info) the new variables are not contained in the rhs of the direct transformation equations

The problem it does not seem possible to invert the transformation shown on the webpage, so that the old variable z show on the left side and the new variable (eta) on the right side.

Why is this restriction on dchange Since one tells it which is the new variable and which is the old variable? May be I am not using dchange correctly in this example.

Any suggestion for a workaround to use dchange to verify the above result?

Here is my hand derivation (pdf file attached also)

Can the above be done using dchange?

Maple 2021.2 on windows 10

sol.pdf