Christian Wolinski

MaplePrimes Activity


These are answers submitted by Christian Wolinski

Try:

printlevel := 3;
for x in [ -1, 1 ] do for y in [ -1, 1 ] do 'x*y'=x*y; end do; end do;

printlevel := 2;
for x from 1 to 2 do for y from 1 to 2 do 'x*y'=x*y; end do; end do;

Include sol:=radnormal(sol); in your code before ode test and everything turns fine. That radical of yours is very poorly presented. Maple will not waste any effort doing this for you. In general fidling radicals is an expensive and unproductive affair.

That last one should have been:

eqs := [a = b, c = d, e = f]:
map2(map, `^`, eqs, 2); #or replace map2 with map[2]

Is this the answer:
[x, y] in {[-2, 2], [0, 0], [1, 1], [2, 2]};

There are 98 subcases in DNF form. All of them point to empty set.

Solving H1..H6 we get:

{-1.960600500 <= tau2, 0.2072815534e-2 <= tau1, tau1 <= 0.2365533980e-2, tau2 <= 0.3994999999e-3}

to satisfy objective 0<=H<=1.

It is as reliable as solve command, which in this example fails.

{solve}(radnormal(diff(e, x), rationalized), {x});

The problem is e represents a root of quadratic, rather than an apparent higher degree root:

e := sqrt(2)*sqrt((3*x^3 + 5*sqrt(x^2 + 8)*x^2 + 24*x + 36*sqrt(x^2 + 8))/sqrt(x^2 + 8))/8 + x/8 + (3*x^2)/(8*sqrt(x^2 + 8)) + 3/sqrt(x^2 + 8);
e2 := (1/4)*x+3*sqrt(x^2+8)*(1/4);
(evala@Norm)(e2-e);
f := lambda -> -lambda*(3*(x^2+8)^(1/2)-4*lambda+x);
radnormal(f(e));

 

The result of:
solve({x*(1-y)/y*(1-x) < t,  0 < x, x < 1, 0 < y, y < 1, t > 0}, {y}, parameters = {x, t});
is quite self explanatory. So any of these should suffice:
plt1 := t -> plot([-x*(x-1)/(-x^2+t+x), 1], x = 0 .. 1, color = [red$2], thickness=[2$2]);
plt1(0.3);
plt2 := t -> plots[inequal]([-x*(x-1)/(-x^2+t+x) < y, y < 1], x = 0 .. 1, y = 0 .. 1);
plt2(0.3);

 

Not much to say here. Would the solution solf := ln(y-1)+2*arctanh(y^(1/2))-I*Pi; be better?
evalc(simplify(Im(solf)));
plot(%, y = -4 .. 4);

Complex analisys is the Maple way.

Hope this helps (edited):

restart;
A[1]:=alpha=1-(1/2)/(1-(RootOf(16*_Z*(_Z*(2*_Z*(_Z*(8*_Z*(_Z*(_Z*(_Z*(32*_Z*(8*_Z-33)+1513)-812)-13)+267)-1469)-330)+811)+279)+345,index=2)-1/2)**2);
A[2]:=expr=(1+alpha)*sqrt(1-alpha**2)+(3+4*alpha)/12*sqrt(3-4*alpha**2)+2*(1+alpha)/3*sqrt(2*(1+alpha)*(1-2*alpha))+(1+2*alpha)/6*sqrt(2*((1-alpha)**2-3*alpha**2));
alias(beta=indets(A[1],RootOf)[1]);
F := lambda^2+(8159503/855552)*beta-(236960/1671)*beta^9+(2085340/5013)*beta^8-(13276063/40104)*beta^7+(414451/20052)*beta^6+(10073755/160416)*beta^5-(13077487/160416)*beta^4+(1471997/213888)*beta^3+(5380211/160416)*beta^2+1770613/2566656;
evalf(subs(lambda = expr, evalf(subs(A[1], A[2]), 40), F), 40);
F:=(evala@Norm)(F);
evalf(subs(lambda = expr, evalf(subs(A[1], A[2]), 40), F), 40);


Also, the file you've submitted appears corrupted, so says Maple 2017.
Sorry, my previous posting was based on a typo. This one is corrected.

restart;
f1 := x1 - x1*sin(x1 + 5*x2) - x2*cos(5*x1 - x2);
f2 := x2 - x2*sin(5*x1 - 3*x2) + x1*cos(3*x1 + 5*x2);
ds := {d1 = 3/13, d2 = 2/13};
rgs := subs(ds, [seq(seq([x1 = 2*Pi*i*d1 .. 2*Pi*(i + 1)*d1, x2 = 2*Pi*j*d2 .. 2*Pi*(j + 1)*d2], i=(floor(subs(ds, -1/d1)) .. ceil(subs(ds, 1/d1))-1)), j = (floor(subs(ds, -1/d2)) .. ceil(subs(ds, 1/d2)) - 1))]):
t0 := time():
Digits := 32:
S := NULL:
for r in rgs do
#print(r); 
S0 := NULL;
for i from 1 to 20 do
q := fsolve({f1, f2}, {x1, x2}, avoid = {S0}, op(r)):
if type(q, specfunc(anything, fsolve)) then break else S0 := S0, q fi; od:
S := S, S0;
od:
t1 := time():
print(t1 - t0);
print(nops([S]));
C := plots[pointplot](map(subs, [S], [x1, x2]), symbol = solidcircle, color = violet):
Digits := 12:
plots[display](C, seq(plots[implicitplot](L[1], x1 = -2*Pi .. 2*Pi, x2 = -2*Pi .. 2*Pi, grid = [300, 300], color = L[2]), L=[[f1, red], [f2, blue]]), size = [1200, 1200], scaling = constrained);

Also replace color specification in plot C: violet for COLOUR(RGB, .723921568, .673725492, .723921568). It looks better.

Notice that x1=0 and x2=0 cases are not in the solution set. That is probably because these are boundary cases in the relevant sub ranges. Easily solved using solve.

Moderator edit: Here's the plot, showing 190 solutions.

map2(map, convert, A, D);

with(PolynomialIdeals);
R := PolynomialRing([x, y, z, u, v], 'lex');
P := PolynomialIdeal([x^7*v, y^8*v, z^9*v, u*v]);
decomposition := PrimaryDecomposition(P);
htP := nops([decomposition]);

f := x -> x^3:
g := x -> x^(1/3):


#For complex domain these answers are correct:   
f(g(x));
g(f(x));

#For real domain these answers are correct:
A := [x < 0, x >= 0]:
f(g(x)) assuming x, real;
proc(E) global A; local a; local S := proc(E, a) a, simplify(E) assuming a end; 'piecewise'(seq(S(E, a), a = A)); end(g(f(x)));
F2 := %:


plot([Re, Im](F2), x = -2 .. 2, color = [red, blue], axes = framed);
plot([Re, Im](g(f(x))), x = -2 .. 2, color = [red, blue], axes = framed);

After simplifying for real t, res appears as:

-abs(t)^3*csgn(2*sqrt(2)+3*ln(t))+t^3;


 

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