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Ronan

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14 years, 125 days
East Grinstead, United Kingdom
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These are answers submitted by Ronan

Sum a finite amount of terms as an appro...

Ronan 1426
December 13 2020
1 1

Would this be acceptable as the sum converges
My internet is faulty at present so I can't get the document to display.

restart;


fd := j -> 256/3*j^5*(j - 1)^(2*j - 4)/(j + 1)^(2*j + 4);
S := x -> sum(fd(n)*ln(1 - 1/n^2), n = 2 .. x);
plot(S(x), x = 2 .. 20);
fd(10);
                             "(->)"
evalf(S(10));
for x from 2 by 5 to 200 do
    x, evalf(S(x));
end do;

 

Download MP_sum.mw

 

 

DirectSearch:-SolveEquations...

Ronan 1426
November 29 2020
0 0

Maybe I miss understand what you are after.
 

> 

restart; with(DirectSearch)
> 

E := (-7.29511346879067*10^9*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(2.02316788462733*10^17*b+4.107030812*10^12)*phi(x)+1.74615339888993*10^17*b

(-7295113469.*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(0.2023167885e18*b+0.4107030812e13)*phi(x)+0.1746153399e18*b

 (1)
> 

dsolve(E, phi(x))
> 

ivp := {E, phi(-0.406e-3) = 1, phi(0.406e-3) = 1, (D(phi))(-0.406e-3) = 0, (D(phi))(0.406e-3) = 0}

{(-7295113469.*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(0.2023167885e18*b+0.4107030812e13)*phi(x)+0.1746153399e18*b, phi(-0.406e-3) = 1, phi(0.406e-3) = 1, (D(phi))(-0.406e-3) = 0, (D(phi))(0.406e-3) = 0}

 (2)
> 

N := dsolve(ivp, phi(x))
> 

A := simplify(rhs(dsolve(ivp, phi(x))))
> 

W := evalf(subs(phi(x) = simplify(rhs(dsolve(ivp, phi(x)))), (2.74505742817409*10^8+3.18054074382591*10^8*phi(x))^2-(1.63833629390402*(2.74505742817409*10^8+3.18054074382591*10^8*phi(x)))*(diff(phi(x), x, x))+3*(6.98727355657655*10^7+32282.4885498329*(diff(phi(x), x)))^2+2.68414581287153*(diff(phi(x), x, x))^2 = 0.9e17))
> 

 

> 

with(DirectSearch)

[BoundedObjective, CompromiseProgramming, DataFit, ExponentialWeightedSum, GlobalOptima, GlobalSearch, Minimax, ModifiedTchebycheff, Search, SolveEquations, WeightedProduct, WeightedSum]

 (3)
> 

Digits := 30

30

 (4)
> 

SolveEquations(eval(W,{x=-0.0000039}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[0.147107922240261059617024e-2, Vector(1, {(1) = 0.383546505968e-1}), [b = 0.127348584174732672649668035297e-4], 661]

 (5)
> 

SolveEquations(eval(W, {x = -0.3e-5}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[0.419387893880979948510784e-2, Vector[column](%id = 18446745433166527174), [b = 0.127348584175363152130255238047e-4], 791]

 (6)
> 

SolveEquations(eval(W, {x = -0.2e-5}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[0.483612833457601000000e-5, Vector[column](%id = 18446745433826848886), [b = 0.127348584176092433330136395418e-4], 510]

 (7)
> 

SolveEquations(eval(W, {x = -0.1e-5}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[0.230450895918287510431716e-2, Vector[column](%id = 18446745433229447406), [b = 0.127348584176852498189365168024e-4], 666]

 (8)
> 

SolveEquations(eval(W, {x = 0}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[.19220551392974036042704144, Vector[column](%id = 18446745433149170022), [b = 0.127348584177643908884594179237e-4], 605]

 (9)
> 

SolveEquations(eval(W, {x = 0.39e-5}))

Warning, complex or non-numeric value encountered; trying to find a feasible point

  

[0.2802257031156986788225e-4, Vector[column](%id = 18446745433231180254), [b = 0.127348584181041268501301157811e-4], 763]

 (10)
> 

``


 

Download MZ2-1.mw

solve...

Ronan 1426
September 24 2020
0 2

Use solve

T has 4 different value.

solve(T[1], Q); gives  -q*sigma^3/3

You get the same answer for T_2, T_3, T_4


                            

 

This Thread should help you....

Ronan 1426
September 10 2020
1 0

 

https://www.mapleprimes.com/posts/212577-Scalable-Subplots-Within-Figure

Possibly check save large calculations...

Ronan 1426
January 01 2020
1 1

 

Might be worth checking this setting.

Shift + F5...

Ronan 1426
September 06 2019
1 2

Shif +F5 stops executable math from running the blue box turns grey. You can use it inside test too. Also RMB to untick executible maths works. When in a line of test and you want to enter an equation click on math button enter equation then RMB untick executible math then click on text button again.
 

restart

``

P = 2*x+5

P = 2*x+5

 (1)

P = 2*x+5

``

Test section P = 2*x+5  now more text
Shift Return for this line.

``


 

Download Text_and_equations.mw

Here are a couple...

Ronan 1426
August 31 2019
1 1

https://www.mapleprimes.com/questions/201298-How-To-Animate-A-Bouncing-Ball-On-A-3d-Surface

or another

https://www.mapleprimes.com/questions/224449-How-To-Model-A-Bouncing-Ball-Through

 

Just Search for Ball.

 

 

Windows 10 File History...

Ronan 1426
June 01 2019
0 0

If you are using Windows 10 turn File History. That keeps a running backup of all files you work on. You coud set to save  once a hour. It has gotten me out of trouble a few times.

Evaluate - Remove output...

Ronan 1426
May 17 2019
1 1

 

They moved it to Evaluate Remove Output. Ctrl+Shift+R

This is one way of doing it...

Ronan 1426
April 26 2019
1 1

This works.
 

restart

with(plots)

[animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d, conformal, conformal3d, contourplot, contourplot3d, coordplot, coordplot3d, densityplot, display, dualaxisplot, fieldplot, fieldplot3d, gradplot, gradplot3d, implicitplot, implicitplot3d, inequal, interactive, interactiveparams, intersectplot, listcontplot, listcontplot3d, listdensityplot, listplot, listplot3d, loglogplot, logplot, matrixplot, multiple, odeplot, pareto, plotcompare, pointplot, pointplot3d, polarplot, polygonplot, polygonplot3d, polyhedra_supported, polyhedraplot, rootlocus, semilogplot, setcolors, setoptions, setoptions3d, shadebetween, spacecurve, sparsematrixplot, surfdata, textplot, textplot3d, tubeplot]

 (1)

c1 := x*y+x+y = 7

x*y+x+y = 7

 (2)

c2 := x^2+x*y+y^2 = 13

x^2+x*y+y^2 = 13

 (3)

implicitplot([c1, c2], x = -5 .. 5, y = -5 .. 5, colour = [blue, green])

  

sol := `assuming`([solve({c1, c2}, [x, y], useassumptions)], [0 < x and x < 4, 0 < y and y < 4])

[[x = 1, y = 3], [x = 3, y = 1]]

 (4)

``


 

Download This_works.mw

Using affine geometry...

Ronan 1426
January 19 2019
1 0


Hi,

This really is just a more expanded version of vv 5208  answer.

restart

P := [x3, y3, z3]

[x3, y3, z3]

 (1)

A := [x1, y1, z1]

[x1, y1, z1]

 (2)

B := [x2, y2, z2]

[x2, y2, z2]

 (3)

H lies on the line connecting AB, H can be defined as an affine combination of A and B. -∞ < λ < ∞. If 0 < λ < 1 then λ lies between A and B

 

H := simplify(expand((1-lambda)*A+lambda*B))

[(-x1+x2)*lambda+x1, (-y1+y2)*lambda+y1, (-z1+z2)*lambda+z1]

 (4)

``

Get distance between P and H

distPH := sqrt((H[1]-P[1])^2+(H[2]-P[2])^2+(H[3]-P[3])^2)

(((-x1+x2)*lambda+x1-x3)^2+((-y1+y2)*lambda+y1-y3)^2+((-z1+z2)*lambda+z1-z3)^2)^(1/2)

 (5)

Miniminist the distance by differentiating wrt λ. Then solve for λ

mdist := diff(distPH, lambda)

(1/2)*(2*((-x1+x2)*lambda+x1-x3)*(-x1+x2)+2*((-y1+y2)*lambda+y1-y3)*(-y1+y2)+2*((-z1+z2)*lambda+z1-z3)*(-z1+z2))/(((-x1+x2)*lambda+x1-x3)^2+((-y1+y2)*lambda+y1-y3)^2+((-z1+z2)*lambda+z1-z3)^2)^(1/2)

 (6)

lambda := solve(mdist, lambda)

(x1^2-x1*x2-x1*x3+x2*x3+y1^2-y1*y2-y1*y3+y2*y3+z1^2-z1*z2-z1*z3+z2*z3)/(x1^2-2*x1*x2+x2^2+y1^2-2*y1*y2+y2^2+z1^2-2*z1*z2+z2^2)

 (7)

Back substitute. This is the shortest possible distance  between P and H

simplify(distPH)

((x1^2*y2^2-2*x1^2*y2*y3+x1^2*y3^2+x1^2*z2^2-2*x1^2*z2*z3+x1^2*z3^2-2*x1*x2*y1*y2+2*x1*x2*y1*y3+2*x1*x2*y2*y3-2*x1*x2*y3^2-2*x1*x2*z1*z2+2*x1*x2*z1*z3+2*x1*x2*z2*z3-2*x1*x2*z3^2+2*x1*x3*y1*y2-2*x1*x3*y1*y3-2*x1*x3*y2^2+2*x1*x3*y2*y3+2*x1*x3*z1*z2-2*x1*x3*z1*z3-2*x1*x3*z2^2+2*x1*x3*z2*z3+x2^2*y1^2-2*x2^2*y1*y3+x2^2*y3^2+x2^2*z1^2-2*x2^2*z1*z3+x2^2*z3^2-2*x2*x3*y1^2+2*x2*x3*y1*y2+2*x2*x3*y1*y3-2*x2*x3*y2*y3-2*x2*x3*z1^2+2*x2*x3*z1*z2+2*x2*x3*z1*z3-2*x2*x3*z2*z3+x3^2*y1^2-2*x3^2*y1*y2+x3^2*y2^2+x3^2*z1^2-2*x3^2*z1*z2+x3^2*z2^2+y1^2*z2^2-2*y1^2*z2*z3+y1^2*z3^2-2*y1*y2*z1*z2+2*y1*y2*z1*z3+2*y1*y2*z2*z3-2*y1*y2*z3^2+2*y1*y3*z1*z2-2*y1*y3*z1*z3-2*y1*y3*z2^2+2*y1*y3*z2*z3+y2^2*z1^2-2*y2^2*z1*z3+y2^2*z3^2-2*y2*y3*z1^2+2*y2*y3*z1*z2+2*y2*y3*z1*z3-2*y2*y3*z2*z3+y3^2*z1^2-2*y3^2*z1*z2+y3^2*z2^2)/(x1^2-2*x1*x2+x2^2+y1^2-2*y1*y2+y2^2+z1^2-2*z1*z2+z2^2))^(1/2)

 (8)

``

The coordinates to H

H

[(-x1+x2)*(x1^2-x1*x2-x1*x3+x2*x3+y1^2-y1*y2-y1*y3+y2*y3+z1^2-z1*z2-z1*z3+z2*z3)/(x1^2-2*x1*x2+x2^2+y1^2-2*y1*y2+y2^2+z1^2-2*z1*z2+z2^2)+x1, (-y1+y2)*(x1^2-x1*x2-x1*x3+x2*x3+y1^2-y1*y2-y1*y3+y2*y3+z1^2-z1*z2-z1*z3+z2*z3)/(x1^2-2*x1*x2+x2^2+y1^2-2*y1*y2+y2^2+z1^2-2*z1*z2+z2^2)+y1, (-z1+z2)*(x1^2-x1*x2-x1*x3+x2*x3+y1^2-y1*y2-y1*y3+y2*y3+z1^2-z1*z2-z1*z3+z2*z3)/(x1^2-2*x1*x2+x2^2+y1^2-2*y1*y2+y2^2+z1^2-2*z1*z2+z2^2)+z1]

 (9)

``


 

Download Affine_Geometry.mw

Union...

Ronan 1426
December 29 2018
0 0

Try this, I extended your list,

L1 := [{3, 5}, {4, 5}, {4, 8, 9}, {-7, 2, 3}]:
      
L2 := {};  #empty set
                            L2 := {}
for i to nops(L1) do
L2 := L1[i] union L2
end do;
                          L2 := {3, 5}
                        L2 := {3, 4, 5}
                     L2 := {3, 4, 5, 8, 9}
                  L2 := {-7, 2, 3, 4, 5, 8, 9}
 

gridrefine...

Ronan 1426
November 29 2018
1 1

Try   implicitplot(cos(x)*cosh(y) = 1, x = -3 .. 3, y = -5 .. 5, gridrefine=2).

You can set higher values for gridrefine   3,4,5

You can download a numerical solution fr...

Ronan 1426
October 24 2018
1 0

http://www.yorku.ca/marko/ComPhys/Euler/Euler.html

 

you would need to do your own animation though.

Also here in a question, I posted at the time @Rouben Rostamian    gave a quaternion solution.

https://www.mapleprimes.com/questions/221298-I-Am-Looking-For-A-rotate-Type-Command#answer236768

This is an animation I did from following the blog I mentioned in the previous reply. It shows the momentum vectors.

Error appears to be at c:=...

Ronan 1426
August 26 2018
0 0


I changed your c:= to an "if then else end if".That stops producing the Error  But I could have intrepited your statement  incorrectly.

 

> 

restart;

NULL

> 

fdsolve := proc({gamma:=NULL, rho:=NULL, mu:=NULL, omega:=NULL, t0:=NULL,
                 t1:=NULL, x0:=NULL, y0:=NULL,
                 N:=NULL}, params)
    local t, h, c, b, x, y, L, n, l, X, Y, f, g, s1, s2, s3, s4, s5, s6, s7, s8, s9;
    eval(F(t,x,y), params);
    f := unapply(%, [t,x,y]);
    eval(G(t,x,y), params);
    g := unapply(%, [t,x,y]);
    L := floor(1/mu);
    h := (t1 - t0)/N;
s1:= sum(((omega*(h*(n-1))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s2:= sum(((omega*(h*n)^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s3:= sum(((omega*(h*n)^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s4:= sum(((omega*(h*(n-i+1))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s5:= sum(((omega*(h*(n-i-1))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s6:= sum(((omega*(h*(n-i))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s7:= sum(((omega*(h*(n-i))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s8:= sum(((omega*(h*(n-i-1))^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);
s9:=sum(((omega*h^rho)^(j)*GAMMA(gamma+j))/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100);    
    c := (i,n) ->
        if i=0 then
              (h^mu)*(  ( ((n-1)^(mu+1))*s1 )-((n^(mu+1))*s2)+((n^(mu))*s3)  )else
              (h^mu)*( (((n-i+1)^(mu+1))*s4)+(((n-i-1)^(mu+1))*s5)-(2*((n-i)^(mu+1))*s6) )
        end if;

b := (i,n) -> (h^mu)*( ((n-i)^(mu)*s7)-((n-i-1)^(mu)*s8) );

    t := Array(0..N, i-> (1-i/N)*t0 + i/N*t1, datatype=float[8]);
    x[0], y[0] := x0, y0;                                           
    for n from 0 to N-1 do
        X[0], Y[0] :=
            x[0] + add(b(i,n+1)*f(t[i],x[i],y[i]), i=0..n),
            y[0] + add(b(i,n+1)*g(t[i],x[i],y[i]), i=0..n);
        for l from 1 to L do
            X[l], Y[l] :=
                x[0] + add(c(i,n+1)*f(t[i],x[i],y[i]), i=0..n)
                     + (h^mu)*s9*f(t[n+1], X[l-1], Y[l-1]),
                y[0] + add(c(i,n+1)*g(t[i],x[i],y[i]), i=0..n)
                     + (h^mu)*s9*g(t[n+1], X[l-1], Y[l-1]);
        end do;
        x[n+1], y[n+1] := X[L], Y[L];
        #printf("y[%d]=%a\n", n+1, y[n+1]);
    end do;
    return Array(0..N, i -> [t[i], x[i], y[i]]);
end proc

proc (params, { N := NULL, gamma := NULL, mu := NULL, omega := NULL, rho := NULL, t0 := NULL, t1 := NULL, x0 := NULL, y0 := NULL }) local t, h, c, b, x, y, L, n, l, X, Y, f, g, s1, s2, s3, s4, s5, s6, s7, s8, s9; eval(F(t, x, y), params); f := unapply(%, [t, x, y]); eval(G(t, x, y), params); g := unapply(%, [t, x, y]); L := floor(1/mu); h := (t1-t0)/N; s1 := sum((omega*(h*(n-1))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s2 := sum((omega*(h*n)^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s3 := sum((omega*(h*n)^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s4 := sum((omega*(h*(n-i+1))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s5 := sum((omega*(h*(n-i-1))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s6 := sum((omega*(h*(n-i))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s7 := sum((omega*(h*(n-i))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s8 := sum((omega*(h*(n-i-1))^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+1)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); s9 := sum((omega*h^rho)^j*GAMMA(gamma+j)/(GAMMA(j*rho+mu+2)*factorial(j)*GAMMA(gamma)), j = 0 .. 100); c := proc (i, n) options operator, arrow; if i = 0 then h^mu*((n-1)^(mu+1)*s1-n^(mu+1)*s2+n^mu*s3) else h^mu*((n-i+1)^(mu+1)*s4+(n-i-1)^(mu+1)*s5-2*(n-i)^(mu+1)*s6) end if end proc; b := proc (i, n) options operator, arrow; h^mu*((n-i)^mu*s7-(n-i-1)^mu*s8) end proc; t := Array(0 .. N, proc (i) options operator, arrow; (1-i/N)*t0+i*t1/N end proc, datatype = float[8]); x[0], y[0] := x0, y0; for n from 0 to N-1 do X[0], Y[0] := x[0]+add(b(i, n+1)*f(t[i], x[i], y[i]), i = 0 .. n), y[0]+add(b(i, n+1)*g(t[i], x[i], y[i]), i = 0 .. n); for l to L do X[l], Y[l] := x[0]+add(c(i, n+1)*f(t[i], x[i], y[i]), i = 0 .. n)+h^mu*s9*f(t[n+1], X[l-1], Y[l-1]), y[0]+add(c(i, n+1)*g(t[i], x[i], y[i]), i = 0 .. n)+h^mu*s9*g(t[n+1], X[l-1], Y[l-1]) end do; x[n+1], y[n+1] := X[L], Y[L] end do; return Array(0 .. N, proc (i) options operator, arrow; [t[i], x[i], y[i]] end proc) end proc

 (1)
> 

 

> 

 


 

Download _Prab-1.mw

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