Maple 2016 Questions and Posts

These are Posts and Questions associated with the product, Maple 2016

    

please where is the problem 

with(DEtools);

eq1 := (D(x))(t) = -y(t);

eq2 := (D(y))(t) = x(t)+2*x(t)^3-signum(z(t));

eq3 := (D(z))(t) = w(t);

eq4 := (D(w))(t) = -z(t)*(1+6*x(t)^2);

sys := eq1, eq2, eq3, eq4;

ic1 := [x(0) = 0, y(0) = 0, z(0) = cos(1), w(0) = sin(1)];

ic2 := [x(0) = 0, y(0) = 0, z(0) = cos(2.5), w(0) = sin(2.5)];

ic := ic1, ic2;

DEplot([sys], [x(t), y(t), z(t), w(t)], t = 0 .. 10, [ic], stepsize = 0.5e-1, scene = [x(t), y(t)], linecolor = [blue, red]);

sol1:=dsolve({sys,x(0)=0,y(0)=0,z(0)=cos(1),w(0)=sin(1)},{x(t),y(t),z(t),w(t)},type=numeric);

T := 10.0 ; N := 100 ; h := T/N;

xk := 0;

for k from 1 to N do

 solk := sol1(k*h);

  xknew := subs(solk,x(t));

 yknew := subs(solk,y(t));

 if xk*xknew<=0 and abs(yknew-6)<0.5 then break fi;

 xk := xknew;

od; sol1(k*h);

temps := proc(alpha,eps)

local sol,solk,T,N,h,k,xk,xknew,yknew,t0,t1,tm,x0,x1,xm;

sol := dsolve({sys,x(0)=0,y(0)=0,z(0)=cos(alpha),w(0)=sin(alpha)},{x(t),y(t),z(t),w(t)},type=numeric);

T := 10.0 ; N := 100 ; h := T/N;

xk := 0;

for k from 1 to N do

 solk := sol(k*h);

  xknew := subs(solk,x(t));

 yknew := subs(solk,y(t));

 if xk*xknew<=0 and abs(yknew-6)<0.5 then break fi;

 xk := xknew;

od; 

t0 := (k-1)*h ; t1 := k*h ;

x0 := subs(sol(t0),x(t)) ; x1 := subs(sol(t1),x(t)) ;

while abs(x0-x1)>eps do

 tm := (t0+t1)/2;

 xm := subs(sol(tm),x(t));

 if xm*x0<0 then x1 := xm; t1:=tm;

            else x0 := xm; t0:=tm;

 fi;

od;

RETURN(t0);

end;

 

dicho := proc(eps)

local a,b,m,sola,solb,solm,ta,tb,tm,ya,yb,ym;

a := 1 ; b := 2.5 ;

sola := dsolve({sys,x(0)=0,y(0)=0,z(0)=cos(a),w(0)=sin(a)},{x(t),y(t),z(t),w(t)},type=numeric);

solb := dsolve({sys,x(0)=0,y(0)=0,z(0)=cos(b),w(0)=sin(b)},{x(t),y(t),z(t),w(t)},type=numeric);

ta := temps(a,eps) ; tb := temps(b,eps) ;

ya := subs(sola(ta),y(t)) ; yb := subs(solb(tb),y(t)) ;

while abs(yb-ya)>eps do

m := evalf((a+b)/2);

solm := dsolve({sys,x(0)=0,y(0)=0,z(0)=cos(m),w(0)=sin(m)},{x(t),y(t),z(t),w(t)},type=numeric);

tm := temps(m,eps) ;

yb := subs(sol(tm),y(t));

 if (ym-6)*(ya-6)<0 then b := m; yb := ym;

            else a := m; ya := ym;

 fi;

od;

RETURN(a);

end;

dicho(0.01);

2.136718750

temps(2.136718750,0.01);

8.737500000

DEplot([sys], [x(t), y(t), z(t), w(t)], t = 0 .. 8.7375, [[x(0)=0,y(0)=0,z(0)=cos(2.136718750),w(0)=sin(2.136718750)]], stepsize = 0.5e-1, scene = [x(t), y(t)], linecolor = [blue]);

 

 

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

 Hi guys,
I am trying write a code for homotopy perturbation, i have already generated the polynomial as you can see, i have also  solve for concentration equation since is not couple. But i have a lot of error massages for temperature, velocity and induced magnetic field. can some one please go through the code?
 

NULL

restart

PDEtools[declare](f(x),theta(x),u(x),w(x), prime=x):

f(x)*`will now be displayed as`*f

 

theta(x)*`will now be displayed as`*theta

 

u(x)*`will now be displayed as`*u

 

w(x)*`will now be displayed as`*w

 

`derivatives with respect to`*x*`of functions of one variable will now be displayed with '`

(1)

N := 4:

NULL

NULL

f(x):=sum((p^(i))*f[i](x),i=0..N);

f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)

(2)

theta(x) := sum(p^i*theta[i](x), i = 0 .. N);

theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x)

(3)

``

u(x) := sum(p^i*u[i](x), i = 0 .. N);

u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x)

(4)

``

w(x) := sum(p^i*w[i](x), i = 0 .. N);

w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x)

(5)

HPMEq := (1-p)*(diff(f(x), `$`(x, 2)))+p*(diff(f(x), `$`(x, 2))-k1*(diff(f(x), x))-k2*f(x));

(1-p)*(diff(diff(f[0](x), x), x)+p*(diff(diff(f[1](x), x), x))+p^2*(diff(diff(f[2](x), x), x))+p^3*(diff(diff(f[3](x), x), x))+p^4*(diff(diff(f[4](x), x), x)))+p*(diff(diff(f[0](x), x), x)+p*(diff(diff(f[1](x), x), x))+p^2*(diff(diff(f[2](x), x), x))+p^3*(diff(diff(f[3](x), x), x))+p^4*(diff(diff(f[4](x), x), x))-k1*(diff(f[0](x), x)+p*(diff(f[1](x), x))+p^2*(diff(f[2](x), x))+p^3*(diff(f[3](x), x))+p^4*(diff(f[4](x), x)))-k2*(f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)))

(6)

HPMEr := (1-p)*(diff(theta(x), `$`(x, 2)))+p*(diff(theta(x), `$`(x, 2))-k11*(diff(theta(x), x))+k12*(diff(u(x), x))^2+k13*(diff(w(x), x))^2+k14*theta(x));

(1-p)*(diff(diff(theta[0](x), x), x)+p*(diff(diff(theta[1](x), x), x))+p^2*(diff(diff(theta[2](x), x), x))+p^3*(diff(diff(theta[3](x), x), x))+p^4*(diff(diff(theta[4](x), x), x)))+p*(diff(diff(theta[0](x), x), x)+p*(diff(diff(theta[1](x), x), x))+p^2*(diff(diff(theta[2](x), x), x))+p^3*(diff(diff(theta[3](x), x), x))+p^4*(diff(diff(theta[4](x), x), x))-k11*(diff(theta[0](x), x)+p*(diff(theta[1](x), x))+p^2*(diff(theta[2](x), x))+p^3*(diff(theta[3](x), x))+p^4*(diff(theta[4](x), x)))+k12*(diff(u[0](x), x)+p*(diff(u[1](x), x))+p^2*(diff(u[2](x), x))+p^3*(diff(u[3](x), x))+p^4*(diff(u[4](x), x)))^2+k13*(diff(w[0](x), x)+p*(diff(w[1](x), x))+p^2*(diff(w[2](x), x))+p^3*(diff(w[3](x), x))+p^4*(diff(w[4](x), x)))^2+k14*(theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x)))

(7)

HPMEs := (1-p)*(diff(u(x), `$`(x, 2)))+p*(diff(u(x), `$`(x, 2))-R*(diff(u(x), x))-A-k8*w(x)-k7*u(x)+k5*theta(x)+k6*f(x));

(1-p)*(diff(diff(u[0](x), x), x)+p*(diff(diff(u[1](x), x), x))+p^2*(diff(diff(u[2](x), x), x))+p^3*(diff(diff(u[3](x), x), x))+p^4*(diff(diff(u[4](x), x), x)))+p*(diff(diff(u[0](x), x), x)+p*(diff(diff(u[1](x), x), x))+p^2*(diff(diff(u[2](x), x), x))+p^3*(diff(diff(u[3](x), x), x))+p^4*(diff(diff(u[4](x), x), x))-R*(diff(u[0](x), x)+p*(diff(u[1](x), x))+p^2*(diff(u[2](x), x))+p^3*(diff(u[3](x), x))+p^4*(diff(u[4](x), x)))-A-k8*(w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x))-k7*(u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x))+k5*(theta[0](x)+p*theta[1](x)+p^2*theta[2](x)+p^3*theta[3](x)+p^4*theta[4](x))+k6*(f[0](x)+p*f[1](x)+p^2*f[2](x)+p^3*f[3](x)+p^4*f[4](x)))

(8)

HPMEt := (1-p)*(diff(w(x), `$`(x, 2)))+p*(diff(w(x), `$`(x, 2))-R*(diff(w(x), x))+k9*u(x)-k10*w(x))

(1-p)*(diff(diff(w[0](x), x), x)+p*(diff(diff(w[1](x), x), x))+p^2*(diff(diff(w[2](x), x), x))+p^3*(diff(diff(w[3](x), x), x))+p^4*(diff(diff(w[4](x), x), x)))+p*(diff(diff(w[0](x), x), x)+p*(diff(diff(w[1](x), x), x))+p^2*(diff(diff(w[2](x), x), x))+p^3*(diff(diff(w[3](x), x), x))+p^4*(diff(diff(w[4](x), x), x))-R*(diff(w[0](x), x)+p*(diff(w[1](x), x))+p^2*(diff(w[2](x), x))+p^3*(diff(w[3](x), x))+p^4*(diff(w[4](x), x)))+k9*(u[0](x)+p*u[1](x)+p^2*u[2](x)+p^3*u[3](x)+p^4*u[4](x))-k10*(w[0](x)+p*w[1](x)+p^2*w[2](x)+p^3*w[3](x)+p^4*w[4](x)))

(9)

for i from 0 to N do equ[1][i] := coeff(HPMEq, p, i) = 0 end do;

diff(diff(f[0](x), x), x) = 0

 

diff(diff(f[1](x), x), x)-k1*(diff(f[0](x), x))-k2*f[0](x) = 0

 

diff(diff(f[2](x), x), x)-k2*f[1](x)-k1*(diff(f[1](x), x)) = 0

 

diff(diff(f[3](x), x), x)-k2*f[2](x)-k1*(diff(f[2](x), x)) = 0

 

diff(diff(f[4](x), x), x)-k1*(diff(f[3](x), x))-k2*f[3](x) = 0

(10)

for i from 0 to N do equa[1][i] := coeff(HPMEr, p, i) = 0 end do;

diff(diff(theta[0](x), x), x) = 0

 

diff(diff(theta[1](x), x), x)-k11*(diff(theta[0](x), x))+k12*(diff(u[0](x), x))^2+k13*(diff(w[0](x), x))^2+k14*theta[0](x) = 0

 

diff(diff(theta[2](x), x), x)+2*k13*(diff(w[0](x), x))*(diff(w[1](x), x))-k11*(diff(theta[1](x), x))+2*k12*(diff(u[0](x), x))*(diff(u[1](x), x))+k14*theta[1](x) = 0

 

diff(diff(theta[3](x), x), x)+k12*(2*(diff(u[0](x), x))*(diff(u[2](x), x))+(diff(u[1](x), x))^2)+k14*theta[2](x)+k13*(2*(diff(w[0](x), x))*(diff(w[2](x), x))+(diff(w[1](x), x))^2)-k11*(diff(theta[2](x), x)) = 0

 

diff(diff(theta[4](x), x), x)+k12*(2*(diff(u[0](x), x))*(diff(u[3](x), x))+2*(diff(u[1](x), x))*(diff(u[2](x), x)))-k11*(diff(theta[3](x), x))+k14*theta[3](x)+k13*(2*(diff(w[0](x), x))*(diff(w[3](x), x))+2*(diff(w[1](x), x))*(diff(w[2](x), x))) = 0

(11)

for i from 0 to N do equat[1][i] := coeff(HPMEs, p, i) = 0 end do;

diff(diff(u[0](x), x), x) = 0

 

diff(diff(u[1](x), x), x)-R*(diff(u[0](x), x))-A-k7*u[0](x)+k5*theta[0](x)+k6*f[0](x)-k8*w[0](x) = 0

 

diff(diff(u[2](x), x), x)-R*(diff(u[1](x), x))-k7*u[1](x)+k6*f[1](x)-k8*w[1](x)+k5*theta[1](x) = 0

 

diff(diff(u[3](x), x), x)-R*(diff(u[2](x), x))+k6*f[2](x)-k7*u[2](x)+k5*theta[2](x)-k8*w[2](x) = 0

 

diff(diff(u[4](x), x), x)-R*(diff(u[3](x), x))+k5*theta[3](x)+k6*f[3](x)-k7*u[3](x)-k8*w[3](x) = 0

(12)

``

for i from 0 to N do equati[1][i] := coeff(HPMEt, p, i) = 0 end do;

diff(diff(w[0](x), x), x) = 0

 

diff(diff(w[1](x), x), x)-R*(diff(w[0](x), x))-k10*w[0](x)+k9*u[0](x) = 0

 

diff(diff(w[2](x), x), x)-k10*w[1](x)+k9*u[1](x)-R*(diff(w[1](x), x)) = 0

 

diff(diff(w[3](x), x), x)-k10*w[2](x)+k9*u[2](x)-R*(diff(w[2](x), x)) = 0

 

diff(diff(w[4](x), x), x)+k9*u[3](x)-R*(diff(w[3](x), x))-k10*w[3](x) = 0

(13)

con[1][0] := f[0](-1) = 1, f[0](1) = 1:

-.5000000000*k2+0.3435019841e-1*k2^4+.5000000000*k2*x^2-.2500000000*k2^2*x^2+.2083333333*k2^2+0.4166666667e-1*k2^2*x^4-0.2083333333e-1*k2^3*x^4+.1041666667*k2^3*x^2+0.4166666667e-1*k1^2*k2+0.5952380952e-3*k2^3*k1*x^7-0.8472222222e-1*k2^3+0.2480158730e-4*k2^4*x^8+0.4166666667e-2*x^6*k1^2*k2^2+0.8333333333e-2*k2*x^5*k1^3-0.9722222222e-2*k1*k2^3*x^5-0.3472222222e-1*k1^2*k2^2*x^4-0.2777777778e-1*k2*x^3*k1^3+0.5046296296e-1*k1*k2^3*x^3+0.6805555556e-1*k2^2*k1^2*x^2-0.4133597884e-1*k1*k2^3*x+0.1944444444e-1*k2*k1^3*x+1.+0.1388888889e-2*k2^3*x^6+0.1666666667e-1*k1*k2^2*x^5+0.4166666667e-1*k2*x^4*k1^2-.1111111111*k2^2*k1*x^3+0.9444444444e-1*k1*k2^2*x-0.8333333333e-1*k2*k1^2*x^2+.1666666667*k2*k1*x^3-0.3750000000e-1*k1^2*k2^2-0.6944444444e-3*k2^4*x^6+0.8680555556e-2*k2^4*x^4-0.4236111111e-1*k2^4*x^2-.1666666667*k1*k2*x

 

1-(1/2)*k2+(277/8064)*k2^4+(1/2)*k2*x^2-(1/4)*k2^2*x^2+(5/24)*k2^2+(1/24)*k2^2*x^4-(1/48)*k2^3*x^4+(5/48)*k2^3*x^2+(1/24)*k1^2*k2+(1/1680)*k2^3*k1*x^7-(61/720)*k2^3+(1/40320)*k2^4*x^8+(1/240)*x^6*k1^2*k2^2+(1/120)*k2*x^5*k1^3-(7/720)*k1*k2^3*x^5-(5/144)*k1^2*k2^2*x^4-(1/36)*k2*x^3*k1^3+(109/2160)*k1*k2^3*x^3+(49/720)*k2^2*k1^2*x^2-(125/3024)*k1*k2^3*x+(7/360)*k2*k1^3*x+(1/720)*k2^3*x^6+(1/60)*k1*k2^2*x^5+(1/24)*k2*x^4*k1^2-(1/9)*k2^2*k1*x^3+(17/180)*k1*k2^2*x-(1/12)*k2*k1^2*x^2+(1/6)*k2*k1*x^3-(3/80)*k1^2*k2^2-(1/1440)*k2^4*x^6+(5/576)*k2^4*x^4-(61/1440)*k2^4*x^2-(1/6)*k1*k2*x

 

2.400000000*k2+0.3589208394e-1*k2^4+1.104000000*k2^2+2.713333334*k1*k2+1.904000000*k1^2*k2+0.115520003e-1*k2^3+.939244445*k1*k2^2+.3973226666*k1^2*k2^2+0.1412642116e-1*k1*k2^3+.9218444444*k1^3*k2

(14)

NULL

"cond[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0:  for j from 1 to N do:  cond[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0:  end do:    for i from 0 to N do:  dsolve({equa[1][i],cond[1][i]},theta[i](x));  theta[i](x):=rhs(`%`):    end do:    theta(x):=evalf(simplify(sum(theta[n](x),n=0..N)));  convert(theta(x),'rational'); "

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {u[0](-1) = 0, u[0](1) = 0, w[0](-1) = 0, w[0](1) = 0}

 

theta[0](x)+theta[1](x)+theta[2](x)+theta[3](x)+theta[4](x)

 

theta[0](x)+theta[1](x)+theta[2](x)+theta[3](x)+theta[4](x)

(15)

``

"condi[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0,f[0](-1)=1, f[0](1)=1:  for j from 1 to N do:  condi[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0, f[j](-1)=0, f[j](1)=0:  end do:    for i from 0 to N do:  dsolve({equat[1][i],condi[1][i]},u[i](x));  u[i](x):=rhs(`%`):    end do:    u(x):=evalf(simplify(sum(u[n](x),n=0..N)))"

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {f[0](-1) = 1, f[0](1) = 1, w[0](-1) = 0, w[0](1) = 0, theta[0](-1) = 1/10, theta[0](1) = 1}

 

u[0](x)+u[1](x)+u[2](x)+u[3](x)+u[4](x)

(16)

``

"condit[1][0]:=theta[0](-1)=0.1, theta[0](1)=1,w[0](-1)=0, w[0](1)=0,u[0](-1)=0, u[0](1)=0,f[0](-1)=1, f[0](1)=1:  for j from 1 to N do:  condit[1][j]:=theta[j](-1)=0, theta[j](1)=0,w[j](-1)=0, w[j](1)=0,u[j](-1)=0, u[j](1)=0, f[j](-1)=0, f[j](1)=0:  end do:    for i from 0 to N do:  dsolve({equati[1][i],condit[1][i]},w[i](x));  w[i](x):=rhs(`%`):    end do:    w(x):=evalf(simplify(sum(w[n](x),n=0..N)))"

Error, (in dsolve) found the following equations not depending on the unknowns of the input system: {f[0](-1) = 1, f[0](1) = 1, u[0](-1) = 0, u[0](1) = 0, theta[0](-1) = 1/10, theta[0](1) = 1}

 

w[0](x)+w[1](x)+w[2](x)+w[3](x)+w[4](x)

(17)

NULL

``

``


 

Download completecode.mw

Hi everyone,

I was wondering how I could modify the thickness of edges in a graph displayed as a Maple plot through DrawGraph. The point is, the graph comprises 100 vertices and 1000 edges. By default, edge thickness is set as 2 but due to the high number of edges I would like to set edge thickness to1 or even 0. How can I do so?

Here is an example with a random graph:

with(GraphTheory) : with(RandomGraphs) : G := RandomGraph(100, 1000) : DrawGraph(G)

 

Thank you very much

This worksheet is a modification to Kitonum's excellent http://www.mapleprimes.com/posts/202222-Contour-Curves-With-Labels.

The mod adds the ability to display labelled contours for expressions in x and y defined parametrically.

Your comments are welcome.

Contourplot_with_labels.mw

How to improve menu speed in Maple 2016?

I am using Maple 2016, when I right click the equation, it really takes a long time to show the menu. It seems that the CPU usage and Memory usage is low, it should be fast as it is only a showing menu action, what's wrong with that?

The worksheet below rolls an ellipse along the y axis with constant energy.

How can the physics be enhanced to roll the ellipse along a non-linear curve (e.g. a sine curve) with constant energy?

EllipseRoll.mw

Help create file Excel in ExcelTools, but error row 564?

thu_file.mw

Please help me? 

Hello

I am looking for the table containing the value m,vv and mm. and these values I can easily copy and paste in excel sheet. 

Plz suggest commands 

 

Thanks Dummy_5.mw

The conditional trigger in the code below does not cause a halt although odeplot shows x dipping to a value below 2.

What is the correct coding of this event to cause a halt when x becomes less than 2?

Does any Maple documentation show examples of working discrete events with conditional triggers?

restart;

DE1 := diff(x(t), t, t) = -x(t)-(diff(x(t), t))+y(t);

DE2 := diff(y(t), t, t) = y(t)-(diff(y(t), t))+x(t);

ICs := x(0) = 2, (D(x))(0) = 1, y(0) = 0, (D(y))(0) = 1;

sol1 := dsolve({DE1, DE2, ICs}, {x(t), y(t)}, numeric, event_initial = true, events = [[[0, x(t) < 2], halt]]);

plots:-odeplot(sol1, [t, x(t)], t = 0 .. 3);
 

The animation in this worksheet seems realistic but I wonder if the physics which is implemented is correct since there is no explicit mention of rotational energy.

ThrowBola.mw

Hello,

this is my code:

######################

restart;
g(x,y):=min((3+((x-y)^(2))/(10)+(|x+y|)/(sqrt(2))),(-|x-y|+(7)/(sqrt(2))));


q:=0; h(x,y):=Heaviside(g(x,y)-q);

                               0
(x, y) -> Heaviside(g(x, y))

p := int(int(h(x, y)*exp((-x^2-y^2)*(1/2))/(2*Pi), y = -infinity .. infinity), x = -infinity .. infinity);

Error, (in unknown) too many levels of recursion

 

######################

I keep having this error message, is this integral impossible to process or is there someting wrong with the code?

 

 

 

 

with(plots);
for x to 10 do Disp[x] := plot([x+cos(a), sin(a), a = 0 .. 2*Pi], colorscheme = ["Red", "Green"]) end do;

display(seq(Disp[i], i = 1 .. x-1), insequence = true, scaling = constrained);
 

What coding in the color or colorscheme parameter in the plot command above would rotate the circle's colors in conjunction with the circle's movement along the x axis to give the appearance of a rolling wheel? 

how can I get a positive output in the following example

solve((hv*hw+lv*lw)/(lv+hv)*(lv+hv) = dw*(lv+hv), hv)

instead of -lv*(dw-lw)/(dw-hw) I would like lv*(dw-lw)/(hw-dw)

How can type 

[
 

"seq('op'('S['i'])',i=1..5)"

op(S[1]), op(S[2]), op(S[3]), op(S[4]), op(S[5])

(1)

into

into

(2)

op(S[1]), op(S[2]), op(S[3]), op(S[4]), op(S[5])

``

(3)

``


 

Download op_seq.mw

seq('op'('S['i'])',i=1..5)]

export

[op(S[1]), op(S[2],op(S[3],op(S[4])]

Please help me

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