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

Dear all , I' would like to join a group to produce quantum information tools in Maple

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

We’re kicking off 2018 right, with another Meet Your Developers interview! This edition comes from Erik Postma, Manager of the Mathematical Software Group.

To catch up on previous interviews, search the “meet-your-developers” tag.

Without further ado…

1. What do you do at Maplesoft?
   I’m the manager of the mathematical software group, a team of 7 mathematicians and computer scientists working on the mathematical algorithms in Maple (including myself). So my work comes in two flavours: I do the typical managerial things, involving meetings to plan new features and solve my team’s day to day problems, and in the remaining time I do my own development work.
    
2. What did you study in school?
   I studied at Eindhoven University of Technology in the Netherlands. The first year, I took a combined program of mathematics and computer science; then for the rest of my undergrad, I studied mathematics. The program was called Applied Mathematics, but with the specialization I took it really wasn’t all that applied at all. Afterwards I continued in the PhD program at the same university, where my thesis was on a subject in abstract algebra (Lie algebras over finite fields).
    
3. What area(s) of Maple are you currently focusing on in your development?
   I’ve spent quite a bit of time over the past two years making the facilities for working with units of measurement in Maple easier to use. There is a very powerful package for doing this that has been part of Maple for many years, but we keep hearing from our users it’s difficult to use. So I’ve worked on keeping the power of the package but making it easier to use.
    
4. What’s the coolest feature of Maple that you’ve had a hand in developing?
   This was actually working on a problem in a part of the code that existed long before I started with Maplesoft. We have a very clever algorithm for drawing random numbers according to a custom, user-specified probability distribution. I wrote about it on MaplePrimes in a series of four blog posts, here. I’ve talked at various workshops and the like about this algorithm and how it is implemented in Maple.
    
5. What do you like most about working at Maplesoft? How long have you worked here?
   I love working at the crossroads of mathematics and computer science; there aren’t many places in the world where you can do that as much as at Maplesoft. But the best thing is the people I work with: us mathematicians are all crazy in slightly different ways, and that makes for a very interesting working environment.
    
6. Favourite hobby?
   Ultimate frisbee. I captain a mixed (i.e., coed) team called The Clockwork. (We play in orange jerseys – it references the book/movie A Clockwork Orange.) We play in a couple of local leagues, and some of the other members also work here. We don’t win much – but we work hard and have fun!
    
7. What do you like on your pizza?
   Mushrooms. Mushrooms on everything!
    
8. What’s your favourite movie?
   Probably Black Book, a dark movie about the Dutch resistance in the second world war from 2006, directed by Paul Verhoeven. I think what I like best about it is that it highlights the moral shades of grey in even so morally elevated a group as the resistance.
    
9. What skill would you love to learn? Why?
   I’d love to learn to speak Russian! I’m trying, but I have a very hard time with it. It would allow me to communicate with my in-laws more easily; they speak Russian.
    
10. Who’s your favourite mathematician?
    Oh, so many to choose from! I’m torn between:

• Ada Lovelace (1815-1852), known as the first programmer.
• Felix Klein (1849-1925), driving force behind a lot of research into geometries and their underlying symmetry groups.
• Wilhelm Killing (1847-1923), a secondary school teacher who made big contributions to the theory of Lie algebras.

Or wait, can I choose my wife?

Please take a look at the attached document, a partial design for a power supply I'm working on.  I find I am spending a lot of time reformatting results with units to look as nice as what you see here.  For every result, I need to do Units Formatting, change to a sensible unit like uH instead of 10^-6 H, and then do Numeric Formatting to change the number to show just three significant digits.  That requires from 0 to 2 decimals, in fixed point.

This is the way engineering documents should look.  You want to see a fixed point number from 1.00 to 999, with a certain number of significant digits (not decimal points), and have the unit scaled accordingly.  You want to see 12.3 uA, not 1.23402 x 10^-5 A.

I would like to see Maple add "N significant digits" to its Numeric Formatting options and auto-scale results with units to the appropriate multiplier.  If I could set that as my default result formatting it would save a huge amount of work.  Often as a design progresses the multiplier will change, also.  A result may initially come out in mA but later change to uA.  Not only do I have to do them all manually now, but I have to go back and change them.  Automating all that would be a great help.

(You may also notice that my vector results with units are not scaled like I describe here.  If anyone can tell me how to do that I would appreciate it.  Otherwise, it looks like a bug to me.)

Example_Document.zip

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.

Irrational numbers: numbers that cannot be represented as a ratio of integers. The decimal form of a rational number is non-repeating and non-terminating.

Change to:

Irrational numbers: numbers that cannot be represented as a ratio of integers. The decimal form of an irrational number is non-repeating and non-terminating. 

or change to

Irrational numbers can be represented by decimal fractions in which the digits go on forever without ever repeating a pattern.  See Downing, Douglas. Dictionary of Mathematics Terms. 2nd ed. Hauppauge, NY: Barron's Ed. Series, Inc., 1995, p. 176).