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Rouben Rostamian

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These are questions asked by Rouben Rostamian

What is the asymptotic value of the root...

Rouben Rostamian 8259 Maple
roots asymptotic
June 16 2019
2 6

I am interested in the root of the equation ((2*n-1)*x + 2*n + 1)*x^n = 1 - x, where 0 < x < 1 and n is a large positive integer.  I believe that the root converges to 1 as n goes to infinity.  How does one obtain an asymptotic estimate of the root for large n?

 

How to simplify this expression?...

Rouben Rostamian 8259 Maple
complex simplify
February 10 2019
2 1

I don't know how to simplify the expression A/B in the worksheet below.  I know that it should simplify to exp(I*Pi/3).  How do I lead Maple to discover that result without telling it the solution ahead of the time?  All variables other than A and B are real.

> 

restart;
> 

local gamma:
> 

A := I*sqrt(3) + 2*c*exp(I*(gamma+(1/3)*Pi)) + 2*a*exp(I*alpha) - 2*a*exp(I*(alpha+(1/3)*Pi)) - 2*b*exp(I*beta) - 1;

I*3^(1/2)+2*c*exp(I*(gamma+(1/3)*Pi))+2*a*exp(I*alpha)-2*a*exp(I*(alpha+(1/3)*Pi))-2*b*exp(I*beta)-1

> 

B := I*sqrt(3) - 2*a*exp(I*(alpha+(1/3)*Pi)) - 2*b*exp(I*beta) + 2*b*exp(I*(beta+(1/3)*Pi)) + 2*c*exp(I*gamma) + 1;

I*3^(1/2)-2*a*exp(I*(alpha+(1/3)*Pi))-2*b*exp(I*beta)+2*b*exp(I*(beta+(1/3)*Pi))+2*c*exp(I*gamma)+1

How to show that A/B = e^((1/3)*i*Pi)?

 

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How to specify multiple lights in plot3d...

Rouben Rostamian 8259 Maple 2018
plot3d lightning plotting
February 01 2019
2 2

This is from the documentation of ?plot3d,light:

 

light=[phi, theta, r, g, b]

    This option adds [my emphasis] a directed light source

    from the direction phi, theta in spherical coordinates.

 

The "adds" in that sentence seems to imply that

multiple light sources are possible, but it does not seem

to work that way.  Each subsequent light option overrides

the previous one as we see in the following illustration

where we render a ball three times by illuminating it

(a) from the north, (b) from the south, and (c) from

both the north and south.  In (c) Maple accounts for only

the south light.

> 

restart;
> 

with(plots): with(plottools):
> 

ball := sphere(1, style=surface, color=gold):
> 

display(<
    display(ball,light=[0,0,1,1,1])   |
    display(ball,light=[180,0,1,1,1]) |
    display(ball,light=[0,0,1,1,1], light=[180,0,1,1,1])
 >);

 

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How to produce animation in Threads?...

Rouben Rostamian 8259 Maple 2018
animation threads sequence
January 20 2019
2 4

In the worksheet below I produce a sequence of frames for an animation by distributing the task into several threads.  The result has strange artifacts as we see in the sample.  The artifacts vary randomly from run to run.  Am I doing something wrong?

> 

restart;
> 

kernelopts(version);

`Maple 2018.2, X86 64 LINUX, Oct 23 2018, Build ID 1356656`

> 

kernelopts(numcpus);

4

Draw a simple picture:

> 

frame := proc(t)
  plot(x^2 + t, x=-1..1, title=typeset("t = %1", t));
end proc:

Generate a sequence of pictures:

> 

frames := Threads:-Seq(frame(t), t=0..1, 0.05):

Animate the sequence:

> 

plots:-display([frames], insequence);

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How does PDEtoos[declare] work?...

Rouben Rostamian 8259 Maple 2018
pdetools documentation declare
December 25 2018
3 9

I don't quite understand the behavior of PDEtools[declare].  My reading of the documentation is that PDEtools[declare](y(t)) tells Maple that y is a function of t, and therefore y(t) is displayed as y and the derivative of y is displayed as yt.  I did not expect other variables to be similarly affected but apparently they are.  For instance, in the worksheet below, why is the derivative of p displayed as ps?

> 

restart;

The normal display of derivatives:

> 

diff(y(t),t);
diff(p(s),s);

diff(y(t), t)

diff(p(s), s)

Declare y as a function of t:

> 

PDEtools[declare](y(t));

` y`(t)*`will now be displayed as`*y

> 

diff(y(t),t);    # this is displayed in subscript notation, as expected
diff(p(s),s);    # why is this displayed in subscript notation?

diff(y(t), t)

diff(p(s), s)

> 

 

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