Kitonum

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11 years, 361 days

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These are answers submitted by Kitonum

The original code has been executed in Maple 12 (classical worksheet). Found mistake in Dist2 (bracket in the wrong place).
  
Here's another variant of the code:

M1 := [-2, 3, -4]: M2 := [4, -1, 3]:

A := [-t+3, 2*t-4, t+2]:

B := [s+4, 2*s, 2*s-1]:

Dist1 := sqrt((M1[1]-A[1])^2+(M1[2]-A[2])^2+(M1[3]-A[3])^2)+sqrt((B[1]-A[1])^2+(B[2]-A[2])^2+(B[3]-A[3])^2)+sqrt((B[1]-M2[1])^2+(B[2]-M2[2])^2+(B[3]-M2[3])^2):

Dist2 := sqrt((M1[1]-B[1])^2+(M1[2]-B[2])^2+(M1[3]-B[3])^2)+sqrt((B[1]-A[1])^2+(B[2]-A[2])^2+(B[3]-A[3])^2)+sqrt((A[1]-M2[1])^2+(A[2]-M2[2])^2+(A[3]-M2[3])^2):

Optimization[Minimize](Dist1); Optimization[Minimize](Dist2);

min(%[1], %%[1]);


[17.8369577415605676,

[s = 0.602330835119798569, t = 1.52243814242405584]]


[14.4132655077310190,

[s = -0.189757117490280680, t = 0.845165089700380400]]


14.4132655077310190

 

 

This is quite simply:

 

M1:=[-2,3,-4]:  M2:=[4,-1,3]:

A:=[-t+3,2*t-4,t+2]:  B:=[s+4,2*s,2*s-1]:

Dist1:=sqrt((M1[1]-A[1])^2+(M1[2]-A[2])^2+(M1[3]-A[3])^2)+sqrt((B[1]-A[1])^2+(B[2]-A[2])^2+(B[3]-A[3])^2)+sqrt((B[1]-M2[1])^2+(B[2]-M2[2])^2+(B[3]-M2[3])^2):

Dist2:=sqrt((M1[1]-B[1])^2+(M1[2]-B[2])^2+(M1[3]-B[3])^2)+sqrt((B[1]-A[1])^2+(B[2]-A[2])^2)+(B[3]-A[3])^2+sqrt((A[1]-M2[1])^2+(A[2]-M2[2])^2+(A[3]-M2[3])^2):

evalf(minimize(Dist1));  evalf(minimize(Dist2));

min(%, %%); 


                                                                         21.59170144


                                                                         16.24881940


                                                                         16.24881940

1) If you consider the function y=psi(x)  in the real range, then you should write surd(1-x, 3)  instead  (1-x)^(1/3). Else for  (1-x)^(1/3)  if  x>1 , you get complex values.

2) By the code  q:=y->fsolve(y=psi(x), x)  you are looking the function inverse to the function y=psi(x) . For correct solution of the problem the function y=psi(x) should be monotonic in the corresponding range. Such ranges simpliest can be found using the plot of the function. Function  y=psi(x)  in the entire domain does not have an inverse function.

V:=1.8e-5:  R:=8.314:  T:=298:  k:=0.841:  G:=1e6:

psi:=unapply((ln(x)-x+k*(1-x)^2+(V*G)/(R*T)*(surd(1-x,3)-(1-x)/2))*(R*T/V), x);

plot(psi(x), x=0..3, title="The plot of function y=psi(x)");

q:=y->fsolve(psi(x)=y, x):

plot('q'(y), y=-.9e9..-.1.4e9);

 

If this is important to you, then you can specify  the corresponding function, called P

P:=unapply(n!/(n-r)!, n,r);
 P(10,6);  P(n,r);

                        n!
P := (n, r) -> --------
                     (n - r)!


151200


   n!
--------
(n - r)!

It is easy to see that the product of the members of which are equidistant from the ends is 4, for example

simplify((sqrt(3)+tan(Pi/180))*(sqrt(3)+tan(29*Pi/180)));
                                           4

So, we obtain

mul(simplify((sqrt(3)+tan(Pi*k/180))*(sqrt(3)+tan((30-k)*Pi/180))), k=1..14)*simplify(sqrt(3)+tan(15*Pi/180));
                                     536870912

 

z:=(x,y)->5/(1+x^2+y^2):

A:=plot3d(z(x,y), x=-4..7, y=-4..4, numpoints=3000):

x:=t->2+3*cos(t): y:=t->-1+2*sin(t):

N:=100:

B:=seq(plots[spacecurve]([x(t),y(t),z(x(t), y(t))], t=0..2*Pi*k/N, color=black, thickness=3), k=0..N):

bug:=seq(plottools[sphere]([x(2*Pi*k/N),y(2*Pi*k/N),z(x(2*Pi*k/N), y(2*Pi*k/N))], 0.15, style=surface, color=red),  k=0..N):

C:=seq(plots[display](A, B[k], bug[k]), k=1..N+1):

plots[display](C, insequence=true, axes=normal, view=[-4.7..7.7, -4.7..4.7, -1..5.7]);

 

Your task is similar to the problem of wandering drunk sailor. Do a keyword search drunken sailor.

Vector(10,[seq(sin(i), i=-0.5..-0.1, 0.1), seq(sin(i), i=0.1..0.5, 0.1)]);

or

Vector(10, [seq(sin(i), i in {seq(-0.5..0.5, 0.1)} minus {0.})]);

subs(a=`3`, sqrt(a)/a);

 

 

To find the center of rotation is sufficient to use two points:

restart; with(geometry):

point(X, x0, y0), point(A, 4, -2), point(B, 5, -4):   #  X - the center of rotation

L:=coordinates(rotation(A1, A, alpha, 'counterclockwise', X)):

M:=coordinates(rotation(B1, B, alpha, 'counterclockwise', X)):

solve({L[1]=4, L[2]=2, M[1]=6, M[2]=3});

 

 

 

You can use  seq  command.

Example:

X:=[seq(i, i=0..20)]:
Y:=[seq((-1)^i*x, i=0..19)]:
y:=piecewise(seq(op([X[i]<x and x<=X[i+1], Y[i]]), i=1..20)):
plot(y, x=0..20);

 

 

 

The last line should be

dsolve({eq, bcs}, v(x));

Matrix([seq([seq(a[k], k = x+m .. y+m)], m = 0 .. 2)]);

a:=1:  c:=2:

fsolve(5*b^5+(60-5*a)*b^4+(125+50*c-80*a)*b^3+(594*c-445*a-775)*b^2+(2324*c-1005*a-3270)*b+3000*c-750*a-3000=0, b);

                                          -5.950891678, -4.378215842, -3.371729567

 

PS. We find the real roots. If also complex roots are needed, then

fsolve(5*b^5+(60-5*a)*b^4+(125+50*c-80*a)*b^3+(594*c-445*a-775)*b^2+(2324*c-1005*a-3270)*b+3000*c-750*a-3000=0, b, complex); 

                                          -5.950891678, -4.378215842, -3.371729567,
                              1.350418543 - 1.816274549 I, 1.350418543 + 1.816274549 I

Consider the function  f:=x->sqrt(a* x + b) +  sqrt(c*x + d) . If  a*c>=0  then  f   is a monotonic function. Therefore, the equation   sqrt(a* x + b) +  sqrt(c*x + d) = m  can not have two solutions. So the necessary condition is   a*c<0 . For definiteness, let  a>0 . From these conditions and from the conditions a*x+b>=0,  c*x+d>=0  we get  -b/a<=x<=-d/c . Therefore, in the limited ranges of parameters  a, b, c, d  all the solutions can be found by the usual brute force.

The following code finds all equations with integers  a=1 .. 10,  b=-10 .. 10,  c=-10 .. -1,  d=-10 .. 10, each of which has exactly two integer solutions:

L:=[]:

for a to 10 do

for b from -10 to 10 do

for c from -10 to -1 do

for d from -10 to 10 do

if -b/a<=-d/c then s:=floor(-b/a): t:=ceil(-d/c):

M:=[]:

for x from s to t do

u:=sqrt(a*x+b): v:=sqrt(c*x+d):

if type(u, integer) and type(v, integer) then M:=[op(M), [x, u+v]]: fi:

od:

if nops(M)=2 and M[1,2]=M[2,2] then L:=[op(L), [a,b,c,d,M[2,2],[M[1,1],M[2,1]]]]: fi: fi:

od: od: od: od:

nops(L);

L[1..50];

ListTools[Search]([1, 5, -1, 8, 5, [-1,4]], L); 

  

 

For example, the list  [1, -9, -1, 10, 1, [9, 10]]  corresponds to the equation  sqrt(x-9)+sqrt(-x+10)=1  with the roots 9 and 10. The list  L  contains all the solutions in the specified ranges. Received 319 solutions. Displayed first 50 solutions. The original equation is also in the list  L  at position 82.

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