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Games with pseudo-fractals

by: Alsu 445 Maple
geometry fractal homothety
June 14 2018
4  0

Mukhametshina Liya

Games with pseudo-fractals
 

Homothety_Fractals.mw

 

  

  

Animated Picture on the Coordinate Plane

by: Alsu 445 Maple
animation
June 14 2018
7  0

Aleksandrov Denis,
7th grade
secondary school #57 of Kazan

_ANIMATED_PICTURE_ON_THE_COORDINATE_PLANE_Aleksandrov_D..mw

 

World Cup 2018 simulation

by: Christopher2222 6035
simulation
June 11 2018
0  19

vv if you could please help adjust your code.  I've adjusted the start of the eurocup code to match the world cup however I haven't decoded your coding and probably won't be able to have time before the world cup starts.  I've got as far as adding the teams, flags and ratings of each team.

Let me just say while copying and pasting the flag bytes to the code, Maple became a bitch to work worth (pardon my language) but I became so frustated because my laptop locked up twice.  The more I worked with Maple the slower it got, until it froze right up.  Copying and pasting large data in maple is almost to near IMPOSSIBLE.  .. perhaps this could be a side conversation.

Here's the world cup file so far.

2018_World_Cup.mw

**edit added**
Fixed flag sizes, couple of other fixes in other stats and added some additional stats
2018_World_Cup7.mw

Cascade of Opening Matryoshkas

by: Alsu 445 Maple
June 09 2018
5  2

 

ANIMATED image of cascade of opening matryoshkas

E.R. Ibragimova

 

 

ИбрагимоваЭ.Р_03_Казань_Матрёшки.mws

Flasc with bubbles

by: Alsu 445 Maple
animation
June 09 2018
5  1

Murtazin Shamil, 6th grade

Simulation of the animated image "Flask with bubbles"

 

FLASC_Murtazin_S.A..mw

A limit from an undergraduate competition

by: vv 14092 Maple
limit int
May 17 2018
1  8
 

At a recent undegraduate competition the students had to compute the following limit

 

> 

Limit( n * Diff( (exp(x)-1)/x, x$n), n=infinity ) assuming x<>0;

Limit(n*(Diff((exp(x)-1)/x, `$`(x, n))), n = infinity)

 (1)

 

Maple is able to compute the symbolic n-fold derivative and I hoped that the limit will be computed at once.

Unfortunately it is not so easy.
Maybe someone finds a more more straightforward way.

 

> 

restart;
> 

f := n * diff( (exp(x)-1)/x, x$n );

n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x

 (2)
> 

limit(%, n=infinity);

limit(n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x, n = infinity)

 (3)
> 

simplify(%) assuming x>0;

limit(-(-1)^n*x^(-n-1)*n*(GAMMA(n+1)-GAMMA(n+1, -x)), n = infinity)

 (4)

 

So, Maple cannot compute directly the limit.

 

> 

convert(f, Int) assuming n::posint;

-n*(-1/x)^n*(-x)^(n+1)*GAMMA(2+n)*(Int(exp(_t1*x)*_t1^n, _t1 = 0 .. 1))/((n+1)*(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))*x)

 (5)
> 

J:=simplify(%)  assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))*GAMMA(n+1)/(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))

 (6)
> 

L:=convert(J, Int) assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))

 (7)
> 

L:=subs(_k1=u, L);

n*(Int(exp(x*u)*u^n, u = 0 .. 1))

 (8)

 

Now it should be easy, but Maple needs help.

 

> 

with(IntegrationTools):
> 

L1:=Change(L, u^n = t, t) assuming n::posint;

Int(exp((x*t^(1/n)*n+ln(t))/n), t = 0 .. 1)

 (9)
> 

limit(L1, n=infinity);  # OK

exp(x)

 (10)
> 

####################################################################

Note that the limit can also be computed using an integration by parts, but Maple refuses to finalize:

> 

Parts(L, exp(u*x)) assuming n::posint;

n*(exp(x)/(n+1)-(Int(u^(n+1)*x*exp(x*u)/(n+1), u = 0 .. 1)))

 (11)
> 

simplify(%);

n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1)

 (12)
> 

limit(%, n=infinity);

limit(n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1), n = infinity)

 (13)
> 

value(%);  # we are almost back!

limit(n*((-x)^(-n)*(-(n+1)*n*GAMMA(n)/x-(-x)^n*(x-n-1)*exp(x)/x+(n+1)*n*GAMMA(n, -x)/x)+exp(x))/(n+1), n = infinity)

 (14)
> 

 

The use of manipulators as multi-axis CNC machines...

by: one man 1370 Maple 17
animation manipulator inverse
May 10 2018
5  5

It post can be called a continuation of the theme “Determination of the angles of the manipulator with the help of its mathematical model. Inverse  problem”.
Consider  the use of manipulators as multi-axis CNC machines.
Three-link manipulator with 5 degrees of freedom. In these examples  one of the restrictions on the movement of the manipulator links is that the position of the last link coincides with the normal to the surface along the entire trajectory of the working point movement.
That is, we, as it were, mathematically transform a system with many degrees of freedom to an analog of a lever mechanism with one degree of freedom, so that we can do the necessary work in a convenient to us way.
It seems that this approach is fully applicable directly to multi-axis CNC machines.

(In the texts of the programs, the normalization is carried out with respect to the coordinates of the last point, in order that the lengths of the integration interval coincide with the path length.)
MAN_3_5_for_MP.mw

MAN_3_5_for_MP_TR.mw

add, floats, and Kahan sum

by: vv 14092 Maple
evalf add
April 27 2018
3  2

add, floats, and Kahan sum

 

I found an intresting fact about the Maple command add for floating point values.
It seems that add in this case uses a summation algorithm in order to reduce the numerical error.
It is probably the Kahan summation algorithm (see wiki), but I wonder why this fact is not documented.

Here is a simple Maple procedure describing and implementing the algorithm.

 

 

> 

restart;
> 

Digits:=15;

15

 (1)
> 

KahanSum := proc(f::procedure, ab::range)  
local S,c,y,t, i;      # https://en.wikipedia.org/wiki/Kahan_summation_algorithm
S := 0.0;              # S = result (final sum: add(f(n), n=a..b))
c := 0.0;              # c = compensation for lost low-order bits.
for i from lhs(ab) to rhs(ab) do
    y := f(i) - c;     
    t := S + y;              
    c := (t - S) - y;        
    S := t;                  
od;                         
return S
end proc:

 

Now, a numerical example.

 

 

> 

f:= n ->  evalf(1/(n+1/n^3+1) - 1/(n+1+1/(n+1)^3+1));

proc (n) options operator, arrow; evalf(1/(n+1/n^3+1)-1/(n+2+1/(n+1)^3)) end proc

 (2)
> 

n := 50000;
K := KahanSum(f, 1..n);

50000

  

.333313334133301

 (3)
> 

A := add(f(k),k=1..n);

.333313334133302

 (4)
> 

s:=0.0:  for i to n do s:=s+f(i) od:
's' = s;

s = .333313334133413

 (5)
> 

exact:=( 1/3 - 1/(n+1+1/(n+1)^3+1) );

6250249999999900000/18751875067501050009

 (6)
> 

evalf( [errK = K-exact, errA = A-exact, err_for=s-exact] );

[errK = 0., errA = 0.1e-14, err_for = 0.112e-12]

 (7)
> 

evalf[20]( [errK = K-exact, errA = A-exact, err_for=s-exact] );

[errK = -0.33461e-15, errA = 0.66539e-15, err_for = 0.11166539e-12]

 (8)
> 

 


Download KahanSum.mw

STEM Education: Application generator in Maple

by: Lenin Araujo Castillo 1790 Maple 2018
April 18 2018
2  0

Due to the mechanistic process of our students and little creativity in analysis in schools and universities to be professionally trained is that STEM education appears (science, technology, engineering and mathematics) is a new model that is being considered in other countries and with very slow step in our city. In this work the methods with STEM will be visualized but using computational tools provided by Maplesoft which is a company that leads online education for adolescents and adults in the current market. In Spanish.

ECI_UNT_2018.pdf

ECI_UNT_2018.mw

Lenin Araujo Castillo

Ambassador of Maple

Bug corrected in inttrans[hilbert]

by: vv 14092 Maple
inttrans hilbert
April 10 2018
2  2

There is a bug in inttrans:-hilbert:

> 

restart;
> 

inttrans:-hilbert(sin(a)*sin(t+b), t, s);
# should be:
sin(a)*cos(s+b);   expand(%);

sin(a)*cos(s)

  

sin(a)*cos(s+b)

  

sin(a)*cos(s)*cos(b)-sin(a)*sin(s)*sin(b)

 (1)
> 

########## correction ##############
> 

`inttrans/expandc` := proc(expr, t)
local xpr, j, econst, op1, op2;
      xpr := expr;      
      for j in indets(xpr,specfunc(`+`,exp)) do
          econst := select(type,op(j),('freeof')(t));
          if 0 < nops(econst) and econst <> 0 then
              xpr := subs(j = ('exp')(econst)*combine(j/('exp')(econst),exp),xpr)
          end if
      end do;
      for j in indets(xpr,{('cos')(linear(t)), ('sin')(linear(t))}) do
          if type(op(j),`+`) then
              op1:=select(has, op(j),t); ##
              op2:=op(j)-op1;            ##
              #op1 := op(1,op(j));
              #op2 := op(2,op(j));
              if op(0,j) = sin then
                  xpr := subs(j = cos(op2)*sin(op1)+sin(op2)*cos(op1),xpr)
              else
                  xpr := subs(j = cos(op1)*cos(op2)-sin(op1)*sin(op2),xpr)
              end if
          end if
      end do;
      return xpr
end proc:
> 

#######################################
> 

inttrans:-hilbert(sin(a)*sin(t+b), t, s); expand(%);

-(1/2)*cos(a-b)*sin(s)+(1/2)*sin(a-b)*cos(s)+(1/2)*cos(a+b)*sin(s)+(1/2)*sin(a+b)*cos(s)

  

sin(a)*cos(s)*cos(b)-sin(a)*sin(s)*sin(b)

 (2)
> 

 


Download hilbert.mw

 

Recognizing Handwritten Digits with Machine Learni...

by: Samir Khan 2181 Maple 2018
April 09 2018
10  5

To demonstrate Maple 2018’s new Python connectivity, we wanted to integrate a large Python library. The result is the DeepLearning package - this offers an interface to a subset of the Tensorflow framework for machine learning.

I thought I’d share an application that demonstrates how the DeepLearning package can be used to recognize the numbers in images of handwritten digits.

The application employs a very small subset of the MNIST database of handwritten digits. Here’s a sample image for the digit 0.

This image can be represented as a matrix of pixel intensities.        

The application generates weights for each digit by training a two-layer neural network using multinomial logistic regression. When visualized, the weights for each digit might look like this.

Let’s say that we’re comparing an image of a handwritten digit to the weights for the digit 0. If a pixel with a high intensity lands in

  • an intensely red area, the evidence is high that the number in the image is 0
  • an intensely blue area, the evidence is low that the number in the image is 0

While this explanation is technically simplistic, the application offers more detail.

Get the application here

Calculating components of acceleration in Maple

by: Lenin Araujo Castillo 1790 Maple 18
engineering embedded-components normal
April 08 2018
2  2

Using Maple's native syntax, we can calculate the components of acceleration. That is, the tangent and normal scalar component with its respective units of measure. Now the difficult calculations were in the past because with Maple we solved it and we concentrated on the interpretation of the results for engineering. In spanish.

Calculo_Componentes_Aceleracion_Curvilínea.mw

Uso_de_comandos_y_operadores_para_calculos_de_componentes_de_la_aceleración.mw

Lenin Araujo Castillo

Ambassador of Maple

 

 

Maple 2018: Updates to Physics Differential Equations...

by: ecterrab 14670 Maple
maplecloud
March 28 2018
7  0

The first update to the Maple 2018 Physics, Differential Equations and Mathematical Functions packages is available. As has been the case since 2013, this update contains fixes, enhancements to existing functionality, and new developments in the three areas. 

The webpage for these updates will continue being the Maplesoft R&D Physics webpage. Starting with Maple 2018, however, this update is also available from the MapleCloud.

To install the update: open Maple and click the Cloud icon (upper-right corner), select "Packages" and search for "Physics Updates". Then, in the corresponding "Actions" column, click the third icon (install pop-up).

NOTE May/1: the "Updates" icon of the MapleCloud toolbar (that opens when you click the upper-right icon within a Maple document / worksheet), now works fine, after having installed the Physics Updates version 32 or higher.

These first updates include:

  • New Physics functionality regarding Tensor Products of Quantum States; and Coherent States.
  • Updates to pdsolve regarding PDE & Boundary Conditions (exact solutions);
  • A change in notation: d_(x), the differential of a coordinate in the Physics package, is now displayed as shown in this Mapleprimes post.


Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Tensor Product of Quantum State Spaces

by: ecterrab 14670 Maple
quantum
March 23 2018
4  0

This is about the recent implementation of tensor products of quantum state spaces in the Physics package, in connection with an exchange with the Physics of Information Lab of the University of Waterloo. As usual this development is available to everybody from the Maplesoft R&D Physics webpage. This is the last update for Maple 2017. The updates for Maple 2018, starting with this same material, will begin being distributed through the MapleCloud next week.

Tensor Product of Quantum State Spaces

 

Basic ideas and design

 

 

Suppose A and B are quantum operators and Ket(A, n), et(B, m) are, respectively, their eigenkets. The following works since the introduction of the Physics package in Maple

> 

with(Physics)

> 

Setup(op = {A, B})

`* Partial match of 'op' against keyword 'quantumoperators'`

 

[quantumoperators = {A, B}]

(1)
> 

A*Ket(A, alpha) = A.Ket(A, alpha)

Physics:-`*`(A, Physics:-Ket(A, alpha)) = alpha*Physics:-Ket(A, alpha)

(2)
> 

B*Ket(B, beta) = B.Ket(B, beta)

Physics:-`*`(B, Physics:-Ket(B, beta)) = beta*Physics:-Ket(B, beta)

(3)

where on the left-hand sides the product operator `*` is used as a sort of inert form (it has all the correct mathematical properties but does not perform the contraction) of the dot product operator `.`, used on the right-hand sides.

 

Suppose now that A and B act on different, disjointed, Hilbert spaces.

 

1) To represent that, a new keyword in Setup , is introduced, to indicate which spaces are disjointed, say as in disjointedhilbertspaces = {A, B}.  We want this notation to pop up at some point as {`&Hscr;`[A], `&Hscr;`[B]} where the indexation indicates all the operators acting on that Hilbert space. The disjointedspaces keyword has for synonyms disjointedhilbertspaces and hilbertspaces. The display `&Hscr;`[A] is not yet implemented.

 

NOTE: noncommutative quantum operators acting on disjointed spaces commute between themselves, so after setting  - for instance - disjointedspaces = {A, B}, automatically, A, B become quantum operators satisfying (see comment (ii) on page 156 of ref. [1])

 

"[A,B][-]=0"

 

2) Product of Kets and Bras (KK, BB, KB and BK) where K and B belong to disjointed spaces, are understood as tensor products satisfying, for instance with disjointed spaces A and B (see footnote on page 154 of ref. [1]),

 

`&otimes;`(Ket(A, alpha), Ket(B, beta)) = `&otimes;`(Ket(B, beta), Ket(A, alpha)) 

 

`&otimes;`(Bra(A, alpha), Ket(B, beta)) = `&otimes;`(Ket(B, beta), Bra(A, alpha)) 

 

while of course

Bra(A, alpha)*Ket(A, alpha) <> Bra(A, alpha)*Ket(A, alpha)

 

Details

   

 

3) All the operators of one disjointed space act transparently over operators, Bras and Kets of the other disjointed spaces, for example

 

A*Ket(B, n) = A*Ket(B, n)

and the same for the Dagger of this equation, that is

Bra(B, n)*Dagger(A) = Bra(B, n)*Dagger(A)

 

And this happens automatically. Hence, when we write the left-hand sides and press enter, they are automatically rewritten (returned) as the right-hand sides.

 

Note that for the product of an operator times a Bra or a Ket we are not using the notation that expresses the product with the symbol 5.

 

Regarding the display of Bras and Kets and their tensor products, two enhancements are happening:

 

• 

A new Setup option hideketlabel makes all the labels in Kets and Bras to be hidden when displaying Kets, Bras and Bracket(s), with the indices presented one level up, as if they were a sequence of labels, so that:

 "Ket(A,m,n,l"  

is displayed as

Ket(A, m, n, l)

 

  

This is the notation used more frequently when working in quantum information. This hideketlabel option is already implemented entering Setup(hideketlabel = true)
• 

Tensor products formed with operators, or Bras and Kets, that belong to disjointed spaces (set as such using Setup ), are displayed with the symbol 5 in between, as in Ket(A, n)*Ket(B, n) instead of Ket(A, n)*Ket(B, n), and `&otimes;`(A, B) instead of A*B.

Tensor product notation and the hideketlabel option

   

The implementation of tensor products using `*` and `.`

   

Basic exercising with the new functionality

   

Related functionality already in place before these changes

   

Reference

 

[1] Cohen-Tannoudji, Diue, Laloe, Quantum Mechanics, Chapter 2, section F.

[2] Griffiths Robert B., Hilbert Space Quantum Mechanics, Quantum Computation and Quantum Information Theory Course, Physics Department, Carnegie Mellon University, 2014.

See also

   

 


 

Download TensorProductDesign.mw
 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Carmichael's lambda (n) function (reduced Euler...

by: kenrubenstein 30
animation totient
March 22 2018
1  0

Carmichael's lambda(n) function (as it relates to Euler's Totient Function).....this is just one of 8 stages of animation. 

 

Here's the complete animation with supporting music by the mighty Stormtroopers Of Death....

https://youtu.be/QN-s3EpZICs

 

 

 

 

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