Markiyan Hirnyk

Markiyan Hirnyk
10 years, 316 days


These are questions asked by Markiyan Hirnyk

I mean the root of the equation

GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1

belonging to RealRange(Open(1),4). It should be noticed there are solutions outside this interval. Here is my try.

 

``

solve({n > 1, GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n < 4}, [n])``

[]

(1)

`in`(which*is*wrong, view*of)

simplify(eval(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)), n = (1/2)*sqrt(5)+1/2))

1

(2)

Also

Student[Calculus1]:-Roots(A = 1, n = 1 .. 4)

[1.618033989]

(3)

There is a substitute

fsolve(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n = 1 .. 4)

1.618033989

(4)

NULL

identify(%)

(1/2)*5^(1/2)+1/2

(5)

``

There is a shade of hope that GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n))  can be simplified.

Download solution.mw

 PS. An SCR was submitted by me.

of the implicit function sin(x+y)+sin(x) = y at x = Pi , y=0 of order 15? Here is one of the difficulties

restart; eval(implicitdiff(sin(x+y)+sin(x) = y, y, x$15), [x = Pi, y = 0]);
Error, (in expand/bigprod) Maple was unable to allocate enough memory to complete this computation. Please see ?alloc

Let us denote the cardinality of the subsets of {1,..,n} without two consequent numbers
(e.g. {..,4,5,..} is not allowed) by A[n]. What is the asymptotics of A[n] as n approaches infinity?
The same question for the case of three consequent numbers.
Here is my math experiment.
restart; L := combinat:-powerset({seq(i, i = 1 .. 11)}):#n = 11
nops(%);
2048
M := selectremove(c-> min([seq(c[k+1]-c[k], k = 1 .. nops(c)-1)]) = 1, L)[2]:
nops(M);
233
The other results are [11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368].
These points are very close to some straight line in logarithmic scale as
plot([[11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368]], axis[2] = [mode = log]);
shows. However, the ones do not exactly belong to a straight line:
evalf(ln(46368)-ln(28657), 15);
0.4812118247230
evalf(ln(28657)-ln(17711), 15);
0.48121182594077
eval(exp(.4812118247230*n), n = 15);
1364.000725  .
These results suggest that A[n] is asymptotically equal to exp(c*n) with c about 0.481.
I have not succeeded to find out the nature of the constant c.

question_on_asymptotics.mw

How to prove or disprove the flatness of the surface x = (u-v)^2, y =  u^2-3*v^2, z = (1/2)*v*(u-2*v), where u and v are real-valued parameters? Here is my try:

 

plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame);plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame)

 

eliminate([x = (u-v)^2, y = u^2-3*v^2, z = (1/2)*v*(u-2*v)], [u, v])

[{u = -2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = -2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}]

(1)

NULL

I think Gaussian curvature should be used to this end. Dr. Robert J. Lopez is my hope.

Download flat.mw

Let a non-planar non-self-intersecting closed polygon in three dimensions P  be given, say
with(plots): with(plottools):
P := polygon([[0, 1, 1], [1, -1, 2], [3, 0, 5], [1, 1, 1]]):
How to find the minimal surface  with the boundary P?
There is no chance to find the solution as a closed-form expression.
Thus, the numerical solution (or/and a triangulation which approximates the minimal surface up to the given accuracy) is required.

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