Download MinimizeTensorComponents.mw

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

Hello, 

I study mainly subjects that fall under umbrella of number theory, but i have specified a little further in the worksheet. This is really a request for assistance, because in as much as i have met so many brilliant people online via social media etc,  I would always love to meet more, and especially ones who are more experienced in this field. 

Basically i am too cheap and old to think about going to a good university, so I am trying to get free advice from the people who have probably completed doctorates in the relevant field. Got to be honest I say.

Anyway my contact email is at the top of the attached worksheet.

First thing that stood out to me about the distributions produced in this worksheet is how sparse the number of points is for N=17 relative to all the other values of N.

EXAMPLE_FOR_MAPLE3.mw

Edit: Another example worksheet added.

MAPLE_EXAMPLE_13.mw

I wanted to use MAPLE to preform symbolic quantum computations. The role
of quantum operators and their tensor product is very important in simplying
the understating of such new calculus at least for the beginners. For instance,
(using "o" for the tensor product and "." for the scalar product, H being the Hadamard
operator on a qubit, I the identity operator, and CNOT the 2 qubit controled not
operator)
1) generating the Bells states |Bxy> two stages of operators are needed
     (CNOT) .  (H o  I)  . |x> o |y>

2) performing quantum teleportation of |psi>
     (H o I o I) . (CNOT o I ) . |psi>o |B00>
    followed by a measurements on the first two qubits for driving the application of
    quantum gates to the third qubit.

All these tensor products of operators can be easily written in MAPLE.

Here is a first version of the ExpandQop procedure that will be usefull the purpose of
expanding correctly the tensor product of two quantum operator expressed in Dirac notation.

I hope this is usefull.

LL 

Download ExpandQop.mw

Lesson_on_functions.mws

As the title says, a lesson on functions:  eg the -> operator, f(2), eval, evalf etc 

Using the learning sequence as an alternative to learn problems related to "balance of a body" is shown in this video; thanks to the kindness that Maple offers us in its fundamental programming syntax.

Balance_of_a_body_with_learning_sequence.mw

Lenin Araujo

Ambassador Of Maple

One case where an "expansion beyond all orders" may be needed is investigating the asymptotic behavior of the difference of two functions with coinciding dominant series.

Download steep.mw

Just a simple little worksheet to see if I have enough propane to heat my house for the rest of the winter.

Download Propane_usage-.mw

On the example of a manipulator with three degrees of freedom.
A mathematical model is created that takes into account degrees of freedom of the manipulator and the trajectory of the movement from the initial point to the final one (in the figure, the ends of the red curve). In the text of the program, these are the equations fi, i = 1..5.
Obviously, the straight line could be the simplest trajectory, but we will consider a slightly different variant. The solution of the system of equations is the coordinates of the points of the manipulator (x1, x2, x3) and (x4, x5, x6) in all trajectory. After that, knowing the lengths of the links and the coordinates of the points at each moment of time, any angles of the manipulator are calculated. The same selected trajectory is reproduced from these angles. The possible angles are displayed by black color.
All the work on creating a mathematical model and calculating the angles can be done without the manipulator itself, is sufficient to have only the instruction with technical characteristics.
To display some angles, the procedure created by vv is used.
MAN_2.mw

This post is devoted to the rigorous proof of Miquel's five circles theorem, which I learned about from this question. The proof is essentially very simple and takes only 15 lines of code. The figure below, in which all the labels coincide with the corresponding names in the code, illustrates the basic ideas of the code. First, we symbolically define common points of intersection of blue circles with a red unit circle  (these parameters  s1 .. s5  are the polar coordinates of these points). All other parameters of this configuration can be expressed through them. Then we find the centers  M  and  N  of two circles. Then we find the coordinates of the point  K  from the condition that  CK  is perpendicular to  MN . Then we find the point  E  and using the result obtained, we easily find the coordinates  of all the points  A1 .. A5. Then we find the coordinates of the point   P  as the point of intersection of the lines  A1A2  and  A3A4 . Finally, we verify that the point  P  lies on a circle with center at the point  N , which completes the proof.

Below - the code of the proof. Note that the code does not use any special (in particular geometric) packages, only commands from the Maple kernel. I usually try any geometric problems to solve in this style, it is more reliable,  and often shorter.

restart;
t1:=s1/2+s2/2: t2:=s2/2+s3/2:
M:=[cos(t1),sin(t1)]: N:=[cos(t2),sin(t2)]:
C:=[cos(s2),sin(s2)]: K:=(1-t)*~M+t*~N:
CK:=K-C: MN:=M-N:
t0:=simplify(solve(CK[1]*MN[1]+CK[2]*MN[2]=0, t)):
E:=combine(simplify(C+2*eval(CK,t=t0))):
s0:=s5-2*Pi: s6:=s1+2*Pi:
assign(seq(A||i=eval(E,[s2=s||i,s1=s||(i-1),s3=s||(i+1)]), i=1..5)):
Dist:=(p,q)->sqrt((p[1]-q[1])^2+(p[2]-q[2])^2):
LineEq:=(P,Q)->(y-P[2])*(Q[1]-P[1])=(x-P[1])*(Q[2]-P[2]):
Line1:=LineEq(A1,A2):
Line2:=LineEq(A3,A4):
P:=combine(simplify(solve({Line1,Line2},[x,y])))[]:
Circle:=(x-N[1])^2+(y-N[2])^2-Dist(N,C)^2:
is(eval(Circle, P)=0);  # The final result

                                                                    true

It may seem that this proof is easy to repeat manually. But this is not so. Maple brilliantly coped with very cumbersome trigonometric transformations. Look at the coordinates of point  P , expressed through the initial parameters  s1 .. s5 :

simplify(eval([x,y], P));  # The coordinates of the point  P

ProofMiquel.mw

My September 9, 2016, blog post ("Next Number" Puzzles) pointed out the meaninglessness of the typical "next-number" puzzle. It did this by showing that two such puzzles in the STICKELERS column by Terry Stickels had more than one solution. In addition to the solution proposed in the column, another was found in a polynomial that interpolated the given members of the sequence. Of course, the very nature of the question "What is the next number?" is absurd because the next number could be anything. At best, such puzzles should require finding a pattern for the given sequence, admitting that there need not be a unique pattern.

The STICKELERS column continued to publish additional "next-number" puzzles, now no longer of interest. However, the remarkable puzzle of December 30, 2017, caused me to pull from the debris on my retirement desk the puzzle of July 15, 2017, a puzzle I had relegated to the accumulating dust thereon.

The members of the given sequence appear across the top of the following table that reproduces the graphic used to provide the solution.

It turns out that the pattern in the graphic can be expressed as 100 – (-1)^k k(k+1)/2, k=0,…, a pattern Maple helped find. By the techniques in my earlier blog, an alternate pattern is expressed by the polynomial

which interpolates the nodes (1, 100), (2, 101) ... so that f(8) = -992.

The most recent puzzle consists of the sequence members 0, 1, 8, 11, 69, 88; the next number is given as 96 because these are strobogrammatic numbers, numbers that read the same upside down. Wow! A sequence with apparently no mathematical structure! Is the pattern unique? Well, it yields to the polynomial

which can also be expressed as

Hence, g(x) is an integer for any nonnegative integer x, and g(6) = -401, definitely not a strobogrammatic number. However, I do have a faint recollection that one of Terry's "next-number" puzzles had a pattern that did not yield to interpolation. Unfortunately, the dust on my desk has not yielded it up.
 

A few days ago, I drew attention to the question in which OP talked about the generation of triangles in a plane, for which the lengths of all sides, the area and radius of the inscribed circle are integers. In addition, all vertices must have different integer coordinates (6 different integers), the lengths of all sides are different and the triangles should not be rectangular. I prepared the answer to this question, but the question disappeared somewhere, so I designed my answer as a separate post.

The triangles in the plane, for which the lengths of all sides and the area  are integers, are called as Heronian triangles. See this very interesting article in the wiki about such triangles
https://en.wikipedia.org/wiki/Integer_triangle#Heronian_triangles

The procedure finds all triangles (with the fulfillment of all conditions above), for which the lengths of the two sides are in the range  N1 .. N2 . The left side of the range is an optional parameter (by default  N1=5). It is not recommended to take the length of the range more than 100, otherwise the operating time of the procedure will greatly increase. The procedure returns the list in which each triangle is represented by a list of  [list of coordinates of the vertices, area, radius of the inscribed circle, list of lengths of the sides]. Without loss of generality, one vertex coincides with the origin (obviously, by a shift it is easy to place it at any point). 

The procedure works as follows: one vertex at the origin, then the other two must lie on circles with integer and different radii  x^2+y^2=r^2. Using  isolve  command, we find all integer points on these circles, and then in the for loops we select the necessary triangles.

Download Integer_Triangle1.mw

Edit.

Implementation of Maple apps for the creation of mathematical exercises in
engineering

In this research work has allowed to show the implementation of applications developed in the Maple software for the creation of mathematical exercises given the different levels of education whether basic or higher.
For the majority of teachers in this area, it seems very difficult to implement apps in Maple; that is why we show the creation of exercises easily and permanently. The purpose is to get teachers from our institutions to use applications ready to be evaluated in the classroom. The results of these apps (applications with components made in Maple) are supported on mobile devices such as tablets and / or laptops and taken to the cloud to be executed online from any computer. The generation of patterns is a very important alternative leaving aside random numbers, which would allow us to lose results
onscreen. With this; Our teachers in schools or universities would evaluate their students in parallel on the blackboard without losing the results of any student and thus achieve the competencies proposed in the learning sessions.
In these apps would be the algorithms for future research updates and integrated with systems in content management. Therefore what we show here is extremely important for the evaluation on the blackboard in bulk to students without losing any scientific criteria.

FAST_UNT_2018.mw

FAST_UNT_2018.pdf

Lenin Araujo Castillo

Ambasador of Maple

In the beginning, Maple had indexed names, entered as x[abc]; as early as Maple V Release 4 (mid 1990s), this would display as x_abc. So, x[1] could be used as x₁, a subscripted variable; but assigning a value to x₁ created a table whose name was x. This had, and still has, undesirable side effects. See Table 0 for an illustration in which an indexed variable is assigned a value, and then the name of the concomitant table is also assigned a value. The original indexed name is destroyed by these steps.
 

At one time this quirk could break commands such as dsolve. I don't know if it still does, but it's a usage that those "in the know" avoid. For other users, this was a problem that cried out for a solution. And Maplesoft did provide such a solution by going nuclear - it invented the Atomic Variable, which subsumed the subscript issue by solving a larger problem.

The larger problem is this: Arbitrary collections of symbols are not necessarily valid Maple names. For example, the expression  is not a valid name, and cannot appear on the left of an assignment operator. Values cannot be assigned to it. The Atomic solution locks such symbols together into a valid name originally called an Atomic Identifier but now called an Atomic Variable. Ah, so then x₁ can be either an indexed name (table entry) or a non-indexed literal name (Atomic Variable). By solving the bigger problem of creating assignable names, Maplesoft solved the smaller problem of subscripts by allowing literal subscripts to be Atomic Variables.

It is only in Maple 2017 that all vestiges of "Identifier" have disappeared, replaced by "Variable" throughout. The earliest appearance I can trace for the Atomic Identifier is in Maple 11, but it might have existed in Maple 10. Since Maple 11, help for the Atomic Identifier is found on the page 

In Maple 17 this help could be obtained by executing help("AtomicIdentifier"). In Maple 2017, a help page for AtomicVariables exists.

In Maple 17, construction of these Atomic things changed, and a setting was introduced to make writing literal subscripts "simpler." With two settings and two outcomes for a "subscripted variable" (either indexed or non-indexed), it might be useful to see the meaning of "simpler," as detailed in the worksheet AfterMath.mw.

Hi,

2 examples to explore components to illustrated Fractals
 

Download AppliFloconVonKoch.mw

It passed through my mind it would be interesting to collect the links to the most relevant Mapleprimes posts about Quantum Mechanics using the Physics package of the last couple of years, to have them all accessible from one place. These posts give an idea of what kind of computation is already doable in quantum mechanics, how close is the worksheet input to what we write with paper and pencil, and how close is the typesetting of the output to what we see in textbooks.

At the end of each page linked below, you will see another link to download the corresponding worksheet, that you can open using Maple (say the current version or the version 1 or 2 years ago).

• Quantum Commutation Rules Basics
• Quantum Mechanics: Schrödinger vs Heisenberg picture
• Quantization of the Lorentz Force
• Magnetic traps in cold-atom physics
• The hidden SO(4) symmetry of the hydrogen atom

This other set of three consecutive posts develops one problem split into three parts:

• (I) Ground state of a quantum system of identical boson particles
• (II) The Gross-Pitaevskii equation and Bogoliubov spectrum
• (III) The Landau criterion for Superfluidity

This other link is interesting as a quick and compact entry point to the use of the Physics package:

• Mini-Course: Computer Algebra for Physicists

There is an equivalent set of Mapleprimes posts illustrating the Physics package tackling problems in General Relativity, collecting them is for one other time.

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