Advanced Search

Kitonum

21006 Reputation

26 Badges

16 years, 185 days
Contact Kitonum

MaplePrimes Activity


These are answers submitted by Kitonum

Solution...

Kitonum 21006
April 03 2023
2 1

The problem is solved with good accuracy by reducing to a system of 4 equations with 4 unknowns:


 

> 

restart;
f:=(x1,x2)->4*(x1-0.25)^4-x1^2*x2^2+(x2-0.25)^4-1.21:
P:=plots:-implicitplot(f, -2..2, -2..2, color=blue, scaling=constrained):
S:=[seq([x0,y0]+~d*~[cos(t+Pi/2*k),sin(t+Pi/2*k)], k=0..3)];
fsolve({seq(f(S[i][]), i=1..4)}, {x0=-2..2,y0=-2..2,t=0..2*Pi,d=0..2});
S:=eval(S, %);
plots:-display(P, plottools:-polygon(S, style=line, color=red));
Dist:=(X,Y)->sqrt((X[1]-Y[1])^2+(X[2]-Y[2])^2):
seq(Dist(S[i],S[i+1]), i=1..3),Dist(S[1],S[4]); # Side lengths of the square

[[x0+d*cos(t), y0+d*sin(t)], [x0-d*sin(t), y0+d*cos(t)], [x0-d*cos(t), y0-d*sin(t)], [x0+d*sin(t), y0-d*cos(t)]]

  

{d = 1.044286758, t = 3.846075883, x0 = .3125573019, y0 = .4101664686}

  

[[-.4831331659, -.2661555792], [.9888793497, -.3855239992], [1.108247770, 1.086488516], [-.3637647459, 1.205856936]]

  

 

1.476844497, 1.476844496, 1.476844497, 1.476844496

 (1)
> 

 


 

Download Square.mw

convert...

Kitonum 21006
March 29 2023
1 2

I do not see any problems in this to waste my time on such manipulations. If anyone likes one form, then he can easily switch to it using the  convert  command.

Examples:

restart;
convert(sec(x), sincos); 
convert(csc(x), sincos);
convert(tan(x), sincos);
convert(sin(x)/cos(x), tan);

                                             

 

plots:-animate...

Kitonum 21006
March 26 2023
3 0 
P:=a->plots:-display(
	tubeplot([seq](L(s)), s=-2..2.5, radius=0.08, color="Red"),   # the line L
	tubeplot([seq](C(s)), s=-4*Pi..4*Pi, radius=0.08, color="Green", numpoints=200), # the curve C
	plot3d(S(s,t), s=-4*Pi..4*Pi, t=0..a, color="Gold", grid=[80,25]),
	scaling=constrained, style=patch, lightmodel=light4,
	orientation=[-120,75,0], labels=[x,y,z], axes=framed):

plots:-animate(P,[a], a=0..2*Pi, frames=90);

                           

Two surfaces...

Kitonum 21006
March 25 2023
1 4


 

 

> 

restart;
lambda1:=-sqrt(64-mu^2):
lambda2:=sqrt(64-mu^2):
plot3d([[lambda1-lambda1*t,mu*t,3-6*t],[lambda2-lambda2*t,mu*t,3-6*t]], t=-8..8, mu=-8..8, grid=[100,100]);
P1:=plots:-implicitplot3d(9*x^2*(3-z)^2+9*y*(3+z)^4 = 16*(-z^2+9)^2, x=-8..8, y=-3..3, z=-3..3, style=surface, grid=[50,50,50]):
P2:=plots:-spacecurve([[t,0,3], [0,t,-3]], t=-8..8, color=red, thickness=3):
plots:-display(P1, P2);

 

 

 

Equation of Generatrix :"`G__lambda,mu`≡{[[x=lambda-lambdat],[y=mu t],[z=3-6 t]](t in `ℝ`)" with lambda^2+mu^2 = 64

Cartesian equation of surface:  "S&equiv;{[[9 x^(2)*(3-z)^(2)+9 y^(2)(3+z)^(2)=16*(9-z^(2))^(2) if z<>3 and z<>-3],[[-8<=x<=8 and y=0] if z=3],[[ x=0 and -8<=y<=8] if z=-3]]"


The answer has been edited.
 

Download SurfaceMP_new1.mw

 

combinat:-permute...

Kitonum 21006
March 25 2023
3 1
> 

restart;
with(combinat):
> 

permute([a,b],1);

[[a], [b]]

 (1)
> 

permute([p$3,q$3], 3);

[[p, p, p], [p, p, q], [p, q, p], [p, q, q], [q, p, p], [q, p, q], [q, q, p], [q, q, q]]

 (2)
> 

permute([5$3,7$3], 3);

[[5, 5, 5], [5, 5, 7], [5, 7, 5], [5, 7, 7], [7, 5, 5], [7, 5, 7], [7, 7, 5], [7, 7, 7]]

 (3)
> 

permute([5$3,7$3,9$3], 2);

[[5, 5], [5, 7], [5, 9], [7, 5], [7, 7], [7, 9], [9, 5], [9, 7], [9, 9]]

 (4)
> 

permute([5$3,7$3,9$3], 4);

[[5, 5, 5, 7], [5, 5, 5, 9], [5, 5, 7, 5], [5, 5, 7, 7], [5, 5, 7, 9], [5, 5, 9, 5], [5, 5, 9, 7], [5, 5, 9, 9], [5, 7, 5, 5], [5, 7, 5, 7], [5, 7, 5, 9], [5, 7, 7, 5], [5, 7, 7, 7], [5, 7, 7, 9], [5, 7, 9, 5], [5, 7, 9, 7], [5, 7, 9, 9], [5, 9, 5, 5], [5, 9, 5, 7], [5, 9, 5, 9], [5, 9, 7, 5], [5, 9, 7, 7], [5, 9, 7, 9], [5, 9, 9, 5], [5, 9, 9, 7], [5, 9, 9, 9], [7, 5, 5, 5], [7, 5, 5, 7], [7, 5, 5, 9], [7, 5, 7, 5], [7, 5, 7, 7], [7, 5, 7, 9], [7, 5, 9, 5], [7, 5, 9, 7], [7, 5, 9, 9], [7, 7, 5, 5], [7, 7, 5, 7], [7, 7, 5, 9], [7, 7, 7, 5], [7, 7, 7, 9], [7, 7, 9, 5], [7, 7, 9, 7], [7, 7, 9, 9], [7, 9, 5, 5], [7, 9, 5, 7], [7, 9, 5, 9], [7, 9, 7, 5], [7, 9, 7, 7], [7, 9, 7, 9], [7, 9, 9, 5], [7, 9, 9, 7], [7, 9, 9, 9], [9, 5, 5, 5], [9, 5, 5, 7], [9, 5, 5, 9], [9, 5, 7, 5], [9, 5, 7, 7], [9, 5, 7, 9], [9, 5, 9, 5], [9, 5, 9, 7], [9, 5, 9, 9], [9, 7, 5, 5], [9, 7, 5, 7], [9, 7, 5, 9], [9, 7, 7, 5], [9, 7, 7, 7], [9, 7, 7, 9], [9, 7, 9, 5], [9, 7, 9, 7], [9, 7, 9, 9], [9, 9, 5, 5], [9, 9, 5, 7], [9, 9, 5, 9], [9, 9, 7, 5], [9, 9, 7, 7], [9, 9, 7, 9], [9, 9, 9, 5], [9, 9, 9, 7]]

 (5)
> 

 

Download permutations.mw

The result is 2268...

Kitonum 21006
March 24 2023
2 0 
restart;
with(algcurves):

f:=2*z^6 + z^7/2 - (5*z^11)/4 + 4*z^22 + (29*z^34)/10 - z^40 - (13*z^43)/2 + w^38*(z^2 - z^7/4) + 
 w^49*(-z^9 + z^13/4 + 2*z^14) + w^34*((7*z^14)/3 - (3*z^18)/2) + w^47*(z^10/3 + (7*z^11)/4 + (8*z^21)/5) + 
 w^24*(4*z^8 + (4*z^25)/5 - (3*z^27)/2) + w^9*((-6*z^2)/5 - z^6/2 + (7*z^31)/3) + 
 w^16*((7*z^21)/3 + (4*z^27)/5 + (4*z^32)/3) + w^18*(-6*z^14 - 2*z^31 - z^33) + w^3*(2*z^17 + (7*z^34)/2) + 
 w^16*((-3*z^5)/4 - 2*z^36 + z^39/3) + w^50*(-1/3*z^23 - (7*z^40)/2 + z^42) + w^4*((-3*z^30)/2 + (4*z^38)/3 + (8*z^42)/5) + 
 w^33*(-3*z^4 + (8*z^22)/3 - (8*z^43)/5) + w^16*(-1/4*z^26 - (3*z^41)/4 - z^43) + w^48*((2*z^2)/3 + 6*z^26 + (3*z^43)/5) + 
 w^49*(2*z^18 + z^36 - 2*z^44) + w^10*((-2*z^11)/5 - (3*z^26)/2 + z^45) + w^40*(-1/2*z^20 - z^29 + z^46) + 
 w^36*(-4 + 8*z^13 - (7*z^47)/4) + w^14*((7*z^24)/5 - 6*z^32 - 6*z^49) + w^22*(-2*z^27 - (8*z^50)/3) + 
 w^2*((3*z^10)/5 + (7*z^24)/4 - z^50/4);

genus(f,z,w);

                                                         2268

ListTools:-Collect...

Kitonum 21006
March 08 2023
0 1 
restart;
L:=[k$1,y$23,f$25];
L0:=sort(ListTools:-Collect(L), key=(t->t[2]));
map(t->t[1], L0); 
map(t->t[2], L0);

    

 

Check...

Kitonum 21006
February 27 2023
0 0

Yes, this is indeed an ellipse, most of which is below the Ox axis and it is strongly elongated along this axis. To see it, you need to lengthen the axes and use the scaling=constrained option to correctly show its shape:

plots:-implicitplot(1/1350000000000000000*x^2-1/14580000000000000000000000000000000*x*y+1/5400000000000000*y^2-1/2250000000*x+173/5400000000*y-1=0, x=-10^10..10^11, y=-3*10^8..3*10^7, gridrefine=3, scaling=constrained, size=[1000,100]);

                               

A way...

Kitonum 21006
February 26 2023
2 1

Use the inert multiplication sign  %.  and  InertForm:-Display( ... , inert=false)  command so that the multiplication sign is not gray, but the usual blue color:

restart;
U := <u1, u2>;
K := Matrix(2, 2, symbol = k);
K%.U;
InertForm:-Display(%, inert=false);

                                   

A way...

Kitonum 21006
February 24 2023
1 0

The task is easily solved in Maple without any financial packages. In the code below, q means how many times the debt increases monthly, x means the monthly payment, and  p[n]  means debt at the end of the n-th month:

restart;
q:=1+9/12/100:
p[1]:=120000*q-x:
for n from 2 to 30*12 do
p[n]:=p[n-1]*q-x;
od:
fsolve(p[30*12]=0);

                                                           965.5471403

identify...

Kitonum 21006
February 23 2023
1 1

We can find all the solutions of this system with rational coefficients, depending on 6 parameters:

restart;
eqs:=eval~(k[1]*x^3+k[2]*x^2*y+k[5]*x^2*z+k[3]*x*y^2+k[6]*x*y*z+k[8]*x*z^2+k[4]*y^3+k[7]*y^2*z+k[9]*y*z^2+k[10]*z^3,{{x=1,y=1,z=1},{x=RootOf(_Z^3-4*_Z^2+_Z+1,index=1),y=RootOf(_Z^3-4*_Z^2+_Z+1,index=2),z=RootOf(_Z^3-4*_Z^2+_Z+1,index=3)},{x=RootOf(_Z^3-4*_Z^2+_Z+1,index=2),y=RootOf(_Z^3-4*_Z^2+_Z+1,index=3),z=RootOf(_Z^3-4*_Z^2+_Z+1,index=1)},{x=RootOf(_Z^3-4*_Z^2+_Z+1,index=3),y=RootOf(_Z^3-4*_Z^2+_Z+1,index=1),z=RootOf(_Z^3-4*_Z^2+_Z+1,index=2)}}):
sol:=solve(eqs):
evalf[20](sol):
Sol:=identify(evalf[15](%));

       


Edit. To obtain integer solutions, you can do

isolve(Sol);

         

 

expand...

Kitonum 21006
February 21 2023
1 4

This can be done using one command  expand:

x^(n-1)*(expand((n*x^n - 2*n*x^(n - 1) + x^n)/x^(n-1)));

                                           

A way...

Kitonum 21006
February 19 2023
1 0

This can be done in many ways. Here is one:

restart;
L:=[["O",3.85090000,0.45160000,0.00120000],
["O",-2.59990000,1.40410000,-0.00180000],
["N",-1.57050000,-0.71710000,0.00010000],
["C",-0.20660000,-0.42310000,-0.00020000],
["C",0.22050000,0.90470000,0.00040000],
["C",0.72980000,-1.45700000,-0.00070000],
["C",1.58410000,1.19860000,0.00020000],
["C",2.09330000,-1.16290000,-0.00070000],
["C",2.52040000,0.16480000,-0.00030000],
["C",-2.64850000,0.17820000,0.00090000],
["C",-3.97350000,-0.54200000,0.00100000],
["H",-0.44360000,1.75770000,0.00120000],
["H",0.41130000,-2.49630000,-0.00100000],
["H",-1.80100000,-1.70860000,0.00010000],
["H",1.90530000,2.23700000,0.00090000],
["H",2.81800000,-1.97260000,-0.00080000],
["H",-4.06550000,-1.14630000,-0.90580000],
["H",-4.79040000,0.18440000,0.02880000],
["H",-4.04450000,-1.18860000,0.88020000],
["H",3.96500000,1.41760000,0.00170000]]:

k:=0: n:=nops(L):
for i from 1 to n do
if L[i,1]="H" then k:=k+1; S[i]:=i fi;
od:
S:=convert(S, list);

                                            S := [12, 13, 14, 15, 16, 17, 18, 19, 20]

Workaround...

Kitonum 21006
February 16 2023
1 0 
expr := ``(a*f(x) + b*f(x) + a*g(x)) + 1/(a*f(x) + b*f(x) + a*g(x));
applyop(expand,1,simplify(expr));

 

select...

Kitonum 21006
February 14 2023
1 2 
restart;
T:=table([1 = NULL, 2 = NULL, 3 = NULL, 4 = NULL, 5 = NULL, 6 = NULL, 7 = 5, 9 = NULL, 8 = NULL, 11 = 4, 10 = NULL, 13 = NULL, 12 = NULL, 15 = 9, 14 = NULL, 18 = NULL, 19 = 8, 16 = 9, 17 = NULL, 22 = 9, 23 = NULL, 20 = NULL, 21 = 8, 27 = NULL, 26 = 8, 25 = 4, 24 = NULL, 31 = NULL, 30 = 9, 29 = NULL, 28 = 9, 36 = NULL, 37 = 9, 38 = 9, 39 = NULL, 32 = 5, 33 = NULL, 34 = NULL, 35 = NULL, 45 = NULL, 44 = NULL, 47 = NULL, 46 = NULL, 41 = 8, 40 = NULL, 43 = NULL, 42 = NULL, 54 = NULL, 55 = NULL, 52 = NULL, 53 = NULL, 50 = NULL, 51 = NULL, 48 = 5, 49 = 9, 60 = 8, 59 = NULL, 58 = 7, 57 = 7, 56 = NULL]):
T1:=table(select(t->rhs(t)<>NULL, op(op(T))));
op~([indices(T1)]);
First 10 11 12 13 14 15 16 Last Page 12 of 286