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Hi everyone,

at the suggestion of Carl I am making my question a post.

History:

Newbies often get fascinated with the power tower: x^x^x^...

The generalized power tower z^z^z^... is a special case of the Euler sequence:

z_{n+1}=c^z_n, for c\in C.

Like the Mandelbrot set: z_{n+1}=z^2-c, the power tower, often called infinite exponential, also has a general periodic map. It is included below and is taken from Daniel Geisler's site:
www.tetration.org

Shel and Thron proved that the infinite exponential conveges whenever c belongs to the red region, called today Shell-Thron region.

Definition of Julia Sets for the iterated exponential map:

Also like the Mandelbrot set, the infinite exponential admits Juila Sets. The Julia Sets of the infinite exponential however, are defined differently from the Julia Sets of the Mandelbrot set. They are defined to be  the closure of the periodic repellers of the Euler sequence . They are Cantor Bouquets.  Geisler's colored map then is a general map of how the corresponding Julia set behaves roughly, with c taken from the map. 

We can then introduce small cuts which go from the interior of the Shell-Thron region to the exterior, crossing at various angles, and this will tells us how the infinite exponential evolves. Generally speaking, each time one crosses the Shell-Thron boundary, one wittnesses what's called a Knaster explosion, wherein the exponential explodes into p subregions, where p is the pre-period of the multiplier.


When the parameter c exits the Shell-Thron region at angles of 2*Pi and Pi from the real axis (cuts right and left, p=1, 2), the infinite exponential either transitions from converging to a single feature to exploding into multiple indecomposable contiua (p=1), or it breaks into a period 2 bifurcation (p=2), which itself, also may explode into continua.

When it exits at angles 2*Pi/p, where p>2 is the preperiod of the multiplier, then the infinite exponential evolves from converging to a single feature, to exploding into  p major regions, called Fatou regions, each one having its own attractor, displaying a p-furcation.


In all cases,  the Knaster explosions may introduce the presence of indecomposable continua, as some Fatou regions end up covering entire parts of the complex plane after each transition. In the animations, Knaster explosions occur whenever the background is red. There may be more than one explosion present in the evolution of the power tower. 

Cantor Bouquets are strange creatures. They are essentially quantum sets, and no point of them is actually visible. The probability a point is visible varies directly with the area of the corresponding bouquet "finger" which is rendered in the area of interest. Devaney uses these "fingers"  to obtain "iteneraries" of the iterated exponential map.

Points "close"  to the fingers-hairs o the Cantor Bouquets escape towards complex infinity at (final) angles 2*Pi/p.

The "hairs" of a Cantor Bouquet are C^\infty curves, hence they can be termed easthetically pretty. Only hairs from the main Cantor bouquet for c=e^{1/e} are globally convex/concave. Every other bouquet may contain hairs which change curvature "unpredictably".

Inlcuded files:

1) Euler.mw

2) EulerAnimation.mw

3) EulerAnimation2_noaxes.mw

 

Code for static fractal with given parameters in 1). Parameters can be changed in the constant section (Try p=2,4 or Pi)

Code for morphing animations through cuts in the Shell-Thron region in 2). Parameters d1-d2, N

Code for zooming animations in 3). Parameters M, M1-M2,

Limitations:

Will show attractors of broken Fatou basins only up to pre-period p=5. No big deal though. If you want to increase the preperiod past p=5, you need to add the relavnt tests in procedure ppc(c), otherwise the new attractors are not calculated and the plot ends up red.

Colors are assigned a bit more dispersed, using ln(m). You can also use m in that place. It depends on which area you are in.

Basins of attraction and Fatou regions change appearance under different epsilons. If you want different shapes for the basins, you can try using the Manhattan metric in proc Jhf, instead of |z-lim|.

Included below are the main map of tetration by Daniel Geisler, a short excerpt of the Shell-Thron region which shows pre-periodic cuts and  6 morhing transitions, for p\in {1,2,3,4,5} and one which exits at the angle of 2 radians (p=Pi).

References:

More than 300. To be published with my thesis. Patches, problems and code corrections in the original question page:

http://www.mapleprimes.com/questions/203593-Maple-13-Fast-Maple-18-Crawling

Hello

Is there any option to create a vector with logaritmically spaced? somthing like "logspace" in matlab. (logspace(0:10:20))

Thank you.

Dear Community Members,

 

We have problem with calculation in Maple v11 and v18. when we make a calculation by using maple v11 and v18, we was not able to get the solution as you see enclosed. when we clicked to "enter + ; ", programme does not run.

 

Hi

how can solve this system of equations:

-dp/dz=(0.855*(1+2*x))/p

dx/dz=((6.39*10^-3*p*(1-x))/(1+2*x))

that x=f(z) and p=f(z)

and at z=0     p=5 bar  & x=0

Third Party Package...

Yesterday at 12:23 AM mskalsi 5

Dear all

I want to know, how one can install third party package into Maple13, the package is "wkptest" i downloaded it from link http://cpc.cs.qub.ac.uk/summaries/ADTY. If anyone knows how to do this please help.

When I create a component, here a label component, and set a name property and then click OK, that name seems to go into some list never to be erased. If I then delete the first name and put in a new name and click OK. thus saving it, and the then try to use that name again on a different component, I am told that the name is already in use. Well, no existing component has the name, but it is still in Maple's list of components. If I go to edit an action for any component and input %+command completion, I get a list of components which includes the deleted name which does not belong to any existing component. Since the name is on this list, I cannot reuse the component name.

How do I clean up this erroneous list of component names, so that a mistake in naming one component doesn't forever prevent me from using it on the component for which it was intended?

 

Hello every one,

 

I am trying to solve a backward induction problem (game theoretic problem) in maple. Lets say we have S rounds. I start from bottom (S-th round) and solve a parametric equation. Then I put the solution in the upper level (S-1). Then, update the functions at level and solve the equation pertatining to this level (S-1). Again, I put the solution ofround S-1 into the equations of S-2 and update the equations and parametrically solve the equation belonging to level S-2 . This process repeats till the first level.

 

The issue is that the solution gradually becomes larger and larger. I guess that's why Maple is not able to solve it. 

 

Have any of you guys faced to a similar problem. Any suggestion?

 

I was thinking of asking maple to reduce its precision in computations. I mean there may not be necessary to store a number with 100 digits precision! However, I don't know how to do that (I don't know the command). Any suggestion?

 

Thank you in advance.

 

Ahmadreza

I've got

f(x,y)= a.exp(1+xy) +( a^2 )*sin(x)+1

for which I've shown that there exists an implicit function x=g(y). ( df/dx <>0)

and df/dx = a*exp(1+xy) +( a^2 )*cosx now in the neighborhood of P=(0,0) for the implicit function to exist I'd need a*exp(1+xy)*y <>0 but at P, wouldn't this be 0?

Given, g(y)=x, how do I find the max,min,saddle points?

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Just one example is improvements to Maple’s one-step app creation functionality, which instantly generates an interactive Math App with plots, sliders and dials as well as typeset math so you can interactively explore mathematical expressions and plots. These Apps are now more customizable than ever, and can be used not only in Maple but by sharing them in the MapleCloud, they can be used in the Maple Player, and online using only a web browser.

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Hello

1)I need some guides to solve my problem. I have attached my file. Could you please help me to understand what's wrong in my code?

2) Is there any option to pause "Loops" and watching results, something like break point.

THank you.

 

 

restart

 

with(ImageTools):``

with(FileTools):````

Input Section

 

NULL

filelocation := "C:\\Users\\Mohammad\\Desktop\\1.jpg":

k := 1:NULL

 

zimage := Read(filelocation):

zwidth := Width(zimage):NULL

kernel__length := VectorCalculus:-`+`(VectorCalculus:-`*`(2, k), 1):````

kernel__data := `~`[`*`](Matrix(1 .. kernel__length, 1 .. kernel__length, 1), 1/kernel__length^2):

imheight := Height(zimage):``

imwidth := Width(zimage):NULLNULL

Width(zimage, lower):

Width(zimage, upper):

View(zimage):``

new1zpic := Convolution(zimage, kernel__data):

View(new1zpic)``NULL

new2zpic := zimage:

dummy := 0:

"for i from (k+1)  to (Width(zimage,upper)-k) do     for j from (k+1)  to (Height(zimage,upper)-k) do           for m from 1 to kernel[length]  do               for n from 1 to kernel[length]  do                         dummy:=dummy+kernel[data] (m, n)*new2zpic(i+m-2, j+n-2)                                         end do;              end do;                      new2zpic(i,j):=dummy:                   dummy:=0:   end do; end do;"

Error, final value in for loop must be numeric or character

 

 

 

 

NULL

 

Download Exercise_II-1.0.mw

restart

 

with(ImageTools):``

with(FileTools):````

Input Section

 

NULL

filelocation := "C:\\Users\\Mohammad\\Desktop\\1.jpg":

k := 1:NULL

 

zimage := Read(filelocation):

zwidth := Width(zimage):NULL

kernel__length := VectorCalculus:-`+`(VectorCalculus:-`*`(2, k), 1):````

kernel__data := `~`[`*`](Matrix(1 .. kernel__length, 1 .. kernel__length, 1), 1/kernel__length^2):

imheight := Height(zimage):``

imwidth := Width(zimage):NULLNULL

Width(zimage, lower):

Width(zimage, upper):

View(zimage):``

new1zpic := Convolution(zimage, kernel__data):

View(new1zpic)``NULL

new2zpic := zimage:

dummy := 0:

"for i from (k+1)  to (Width(zimage,upper)-k) do     for j from (k+1)  to (Height(zimage,upper)-k) do           for m from 1 to kernel[length]  do               for n from 1 to kernel[length]  do                         dummy:=dummy+kernel[data] (m, n)*new2zpic(i+m-2, j+n-2)                                         end do;              end do;                      new2zpic(i,j):=dummy:                   dummy:=0:   end do; end do;"

Error, final value in for loop must be numeric or character

 

 

 

 

NULL

 

Download Exercise_II-1.0.mw

f:=sin((x-1)*(y+3))/(exp((x-1)^2+(y+3)^2)-1);
limit(f,{x=1,y=-3});


On running the above code, I don't get a result. I don't understand why, and what are some of the underlying math principles that I would have to use to evaluate the limit of f at (1,-3) if I were to solve this question on paper?

I need to find where the limit of the function:

{1-cos(x*y^3)}/(x^2+y^6)^(1+a) exists/does not exists for different values of a, given 0<=a<=1 and as (x,y)->(0,0).

I've tried writing a procedure that changes a in particular increments but that is clearly not the most efficient way. Are there any rules of limits that I should be using?

 

I've got a set

E:={(x,y,z): x^2+y^2=-2*z-x, z^2+y^2=1} and need to find points of E which have minimal or maximal distance from (0,0,0). I've set up the Lagrangian as F:=sqrt(x^2+y^2+z^2) + L1(x^2+y^2+2z+x)+L2(z^2+y^2-1)

and consequently obtained the equations:

x/sqrt(x^2+y^2+z^2) + 2*x*L1+L1=0

y/sqrt(x^2+y^2+z^2) + 2*y*L1+2*y*L2=0

z/sqrt(x^2+y^2+z^2)+2*L1+2*L2*z=0

for which I've set up
eqn1,eqn2,eqn3 as the three equations and vars:=x,y,z

and used solve() but I'm not getting the right answer( I need to first express x,y,z in terms of L1, L2 and then get values for L1 and L2 by substituting in the constraints and eventually get values of x,y,z.)

How should I implement that?

For example, given a 3d point p(x,y,z), with (x,y,z) as its coordinates. Then it is transformed by rotation and translation, as 

p'=R(p)*p+t(p), where R(p) is a 3x3 rotation matrix that is a matrix of functions of p, and t(p) is a 3x1 vector function of p. 

My question is how to derive dp'/d(as a 3x3 matrix) using maple? 

To make it clear,I want to do it in a way that dp'/dp = ∂p'/∂p + ∂p'/∂R*∂R/∂p +  ∂p'/∂t*∂t/∂p

And I'd like to know each intermediate quantity, such as p'/∂R, R/∂p.


Anyone can help?

Thanks a lot. 

What Maple15 commands will display a Rouleaux tetrahedron or a Meissner tetrahedron?

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