Applications, Examples and Libraries

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Hare in the forest

The rocket flies

  

Быльнов_raketa_letit.mws

 

Plotting the function of a complex variable

Plotting_the_function_of_a_complex_variable.mws

 

Animated 3-D cascade of dolls

 

3d_matryoshkas_en.mws

 

With this application developed entirely in Maple using native syntax and embedded components for science and engineering students. Just replace your data and you're done.

Pearson_Coeficient.mw

Lenin Araujo Castillo

Ambassador of Maple

 

Foucault’s Pendulum Exploration Using MAPLE18

https://www.ias.ac.in/describe/article/reso/024/06/0653-0659

In this article, we develop the traditional differential equation for Foucault’s pendulum from physical situation and solve it from
standard form. The sublimation of boundary condition eliminates the constants and choice of the local parameters (latitude, pendulum specifications) offers an equation that can be used for a plot followed by animation using MAPLE. The fundamental conceptual components involved in preparing differential equation viz; (i) rotating coordinate system, (ii) rotation of the plane of oscillation and its dependence on the latitude, (iii) effective gravity with latitude, etc., are discussed in detail. The accurate calculations offer quantities up to the sixth decimal point which are used for plotting and animation. This study offers a hands-on experience. Present article offers a know-how to devise a Foucault’s pendulum just by plugging in the latitude of reader’s choice. Students can develop a miniature working model/project of the pendulum.

Exercises solved online with Maple exclusively in space. I attach the explanation links on my YouTube channel.

Part # 01

https://www.youtube.com/watch?v=8Aa2xzU8LwQ

Part # 02

https://www.youtube.com/watch?v=qyGT28CeSz4

Part # 03

https://www.youtube.com/watch?v=yf8rjSPbv5g

Part # 04

https://www.youtube.com/watch?v=FwHPW7ncZTg

Part # 05

https://www.youtube.com/watch?v=bm3frpukb0I

Link for download the file:

Vector_Exercises-Force_in_space.mw

Lenin AC

Ambassador of Maple

 

 

 


 

Solving a Numbrix Puzzle with Logic

Background

 

 

Parade magazine, a filler in the local Sunday newspaper, contains a Numbrix puzzle, the object of which is to find a serpentine path of consecutive integers, 1 through 81, through a nine by nine grid.  The puzzle typically specifies the content of every other border cell.

 

The Maple Logic  package has a procedure, Satisfy , that can be used to solve this puzzle.  Satisfy is a SAT-solver; given a boolean expression it attempts to find a set of equations of the form {x__1 = b__1, x__2 = b__2, () .. ()}, where x__i are the boolean variables in the given expression and b__i are boolean values (true or false) that satisfy the expression (cause it to evaluate true).

 

A general technique to solve this and other puzzles with Satisfy is to select boolean-values variables that encode the state of the puzzle (a trial solution, whether valid or not), generate a boolean-expression of these variables that is satisfied (true) if and only if the variables are given values that correspond to a solution, pass this expression to Satisfy, then translate the returned set of boolean values (if any) to the puzzle solution.

Setup

 

Assign a matrix that defines the grid and the initial position.  Use zeros to indicate the cells that need values. To make it easy to inspect the expressions, a small 2 x 3 matrix is used for this demo---a full size example is given at the end.

M := Matrix(2,3, {(1,1) = 1, (1,3) = 5});

Matrix(2, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 5, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0})

(2.1)

 

Extract the dimensions of the Matrix

(m,n) := upperbound(M);

2, 3

(2.2)

Boolean Variables

 

Let the boolean variable x[i,j,k] mean that cell (i,j) has value k. For example, x[2,3,6] is true when cell (2,3) contains 6, otherwise it is false. There are (m*n)^2 boolean variables.

Initial Position

 

The initial position can be expressed as the following and-clause.

initial := &and(seq(seq(ifelse(M[i,j] = 0, NULL, x[i,j,M[i,j]]), i = 1..m), j = 1..n));

`&and`(x[1, 1, 1], x[1, 3, 5])

(4.1)

Adjacent Cells

 

The requirement that an interior cell with value k is adjacent to the cell with value k+1 can be expressed as the implication
   

   x[i,j,k] &implies &or(x[i-1,j,k+1], x[i+1,j,k+1], x[i,j-1,k+1], x[i,j+1,k+1])

 

Extending this to handle all cells results in the following boolean expression.

adjacent := &and(seq(seq(seq(
         x[i,j,k] &implies &or(NULL
                               , ifelse(1<i, x[i-1, j, k+1], NULL)
                               , ifelse(i<m, x[i+1, j, k+1], NULL)
                               , ifelse(1<j, x[i, j-1, k+1], NULL)
                               , ifelse(j<n, x[i, j+1, k+1], NULL)
                               )
                            , i = 1..m)
                        , j = 1..n)
                    , k = 1 .. m*n-1));

`&and`(`&implies`(x[1, 1, 1], `&or`(x[2, 1, 2], x[1, 2, 2])), `&implies`(x[2, 1, 1], `&or`(x[1, 1, 2], x[2, 2, 2])), `&implies`(x[1, 2, 1], `&or`(x[2, 2, 2], x[1, 1, 2], x[1, 3, 2])), `&implies`(x[2, 2, 1], `&or`(x[1, 2, 2], x[2, 1, 2], x[2, 3, 2])), `&implies`(x[1, 3, 1], `&or`(x[2, 3, 2], x[1, 2, 2])), `&implies`(x[2, 3, 1], `&or`(x[1, 3, 2], x[2, 2, 2])), `&implies`(x[1, 1, 2], `&or`(x[2, 1, 3], x[1, 2, 3])), `&implies`(x[2, 1, 2], `&or`(x[1, 1, 3], x[2, 2, 3])), `&implies`(x[1, 2, 2], `&or`(x[2, 2, 3], x[1, 1, 3], x[1, 3, 3])), `&implies`(x[2, 2, 2], `&or`(x[1, 2, 3], x[2, 1, 3], x[2, 3, 3])), `&implies`(x[1, 3, 2], `&or`(x[2, 3, 3], x[1, 2, 3])), `&implies`(x[2, 3, 2], `&or`(x[1, 3, 3], x[2, 2, 3])), `&implies`(x[1, 1, 3], `&or`(x[2, 1, 4], x[1, 2, 4])), `&implies`(x[2, 1, 3], `&or`(x[1, 1, 4], x[2, 2, 4])), `&implies`(x[1, 2, 3], `&or`(x[2, 2, 4], x[1, 1, 4], x[1, 3, 4])), `&implies`(x[2, 2, 3], `&or`(x[1, 2, 4], x[2, 1, 4], x[2, 3, 4])), `&implies`(x[1, 3, 3], `&or`(x[2, 3, 4], x[1, 2, 4])), `&implies`(x[2, 3, 3], `&or`(x[1, 3, 4], x[2, 2, 4])), `&implies`(x[1, 1, 4], `&or`(x[2, 1, 5], x[1, 2, 5])), `&implies`(x[2, 1, 4], `&or`(x[1, 1, 5], x[2, 2, 5])), `&implies`(x[1, 2, 4], `&or`(x[2, 2, 5], x[1, 1, 5], x[1, 3, 5])), `&implies`(x[2, 2, 4], `&or`(x[1, 2, 5], x[2, 1, 5], x[2, 3, 5])), `&implies`(x[1, 3, 4], `&or`(x[2, 3, 5], x[1, 2, 5])), `&implies`(x[2, 3, 4], `&or`(x[1, 3, 5], x[2, 2, 5])), `&implies`(x[1, 1, 5], `&or`(x[2, 1, 6], x[1, 2, 6])), `&implies`(x[2, 1, 5], `&or`(x[1, 1, 6], x[2, 2, 6])), `&implies`(x[1, 2, 5], `&or`(x[2, 2, 6], x[1, 1, 6], x[1, 3, 6])), `&implies`(x[2, 2, 5], `&or`(x[1, 2, 6], x[2, 1, 6], x[2, 3, 6])), `&implies`(x[1, 3, 5], `&or`(x[2, 3, 6], x[1, 2, 6])), `&implies`(x[2, 3, 5], `&or`(x[1, 3, 6], x[2, 2, 6])))

(5.1)

 

All Values Used

 

The following expression is true when each integer k, from 1 to m*n, is assigned to one or more cells.

allvals := &and(seq(seq(&or(seq(x[i,j,k], k=1..m*n)), i=1..m), j=1..n));

`&and`(`&or`(x[1, 1, 1], x[1, 1, 2], x[1, 1, 3], x[1, 1, 4], x[1, 1, 5], x[1, 1, 6]), `&or`(x[2, 1, 1], x[2, 1, 2], x[2, 1, 3], x[2, 1, 4], x[2, 1, 5], x[2, 1, 6]), `&or`(x[1, 2, 1], x[1, 2, 2], x[1, 2, 3], x[1, 2, 4], x[1, 2, 5], x[1, 2, 6]), `&or`(x[2, 2, 1], x[2, 2, 2], x[2, 2, 3], x[2, 2, 4], x[2, 2, 5], x[2, 2, 6]), `&or`(x[1, 3, 1], x[1, 3, 2], x[1, 3, 3], x[1, 3, 4], x[1, 3, 5], x[1, 3, 6]), `&or`(x[2, 3, 1], x[2, 3, 2], x[2, 3, 3], x[2, 3, 4], x[2, 3, 5], x[2, 3, 6]))

(6.1)

Single Value

 

The following expression is satisfied when each cell has no more than one value.

 single := &not &or(seq(seq(seq(seq(x[i,j,k] &and x[i,j,kk], kk = k+1..m*n), k = 1..m*n-1), i = 1..m), j = 1..n));

`&not`(`&or`(`&and`(x[1, 1, 1], x[1, 1, 2]), `&and`(x[1, 1, 1], x[1, 1, 3]), `&and`(x[1, 1, 1], x[1, 1, 4]), `&and`(x[1, 1, 1], x[1, 1, 5]), `&and`(x[1, 1, 1], x[1, 1, 6]), `&and`(x[1, 1, 2], x[1, 1, 3]), `&and`(x[1, 1, 2], x[1, 1, 4]), `&and`(x[1, 1, 2], x[1, 1, 5]), `&and`(x[1, 1, 2], x[1, 1, 6]), `&and`(x[1, 1, 3], x[1, 1, 4]), `&and`(x[1, 1, 3], x[1, 1, 5]), `&and`(x[1, 1, 3], x[1, 1, 6]), `&and`(x[1, 1, 4], x[1, 1, 5]), `&and`(x[1, 1, 4], x[1, 1, 6]), `&and`(x[1, 1, 5], x[1, 1, 6]), `&and`(x[2, 1, 1], x[2, 1, 2]), `&and`(x[2, 1, 1], x[2, 1, 3]), `&and`(x[2, 1, 1], x[2, 1, 4]), `&and`(x[2, 1, 1], x[2, 1, 5]), `&and`(x[2, 1, 1], x[2, 1, 6]), `&and`(x[2, 1, 2], x[2, 1, 3]), `&and`(x[2, 1, 2], x[2, 1, 4]), `&and`(x[2, 1, 2], x[2, 1, 5]), `&and`(x[2, 1, 2], x[2, 1, 6]), `&and`(x[2, 1, 3], x[2, 1, 4]), `&and`(x[2, 1, 3], x[2, 1, 5]), `&and`(x[2, 1, 3], x[2, 1, 6]), `&and`(x[2, 1, 4], x[2, 1, 5]), `&and`(x[2, 1, 4], x[2, 1, 6]), `&and`(x[2, 1, 5], x[2, 1, 6]), `&and`(x[1, 2, 1], x[1, 2, 2]), `&and`(x[1, 2, 1], x[1, 2, 3]), `&and`(x[1, 2, 1], x[1, 2, 4]), `&and`(x[1, 2, 1], x[1, 2, 5]), `&and`(x[1, 2, 1], x[1, 2, 6]), `&and`(x[1, 2, 2], x[1, 2, 3]), `&and`(x[1, 2, 2], x[1, 2, 4]), `&and`(x[1, 2, 2], x[1, 2, 5]), `&and`(x[1, 2, 2], x[1, 2, 6]), `&and`(x[1, 2, 3], x[1, 2, 4]), `&and`(x[1, 2, 3], x[1, 2, 5]), `&and`(x[1, 2, 3], x[1, 2, 6]), `&and`(x[1, 2, 4], x[1, 2, 5]), `&and`(x[1, 2, 4], x[1, 2, 6]), `&and`(x[1, 2, 5], x[1, 2, 6]), `&and`(x[2, 2, 1], x[2, 2, 2]), `&and`(x[2, 2, 1], x[2, 2, 3]), `&and`(x[2, 2, 1], x[2, 2, 4]), `&and`(x[2, 2, 1], x[2, 2, 5]), `&and`(x[2, 2, 1], x[2, 2, 6]), `&and`(x[2, 2, 2], x[2, 2, 3]), `&and`(x[2, 2, 2], x[2, 2, 4]), `&and`(x[2, 2, 2], x[2, 2, 5]), `&and`(x[2, 2, 2], x[2, 2, 6]), `&and`(x[2, 2, 3], x[2, 2, 4]), `&and`(x[2, 2, 3], x[2, 2, 5]), `&and`(x[2, 2, 3], x[2, 2, 6]), `&and`(x[2, 2, 4], x[2, 2, 5]), `&and`(x[2, 2, 4], x[2, 2, 6]), `&and`(x[2, 2, 5], x[2, 2, 6]), `&and`(x[1, 3, 1], x[1, 3, 2]), `&and`(x[1, 3, 1], x[1, 3, 3]), `&and`(x[1, 3, 1], x[1, 3, 4]), `&and`(x[1, 3, 1], x[1, 3, 5]), `&and`(x[1, 3, 1], x[1, 3, 6]), `&and`(x[1, 3, 2], x[1, 3, 3]), `&and`(x[1, 3, 2], x[1, 3, 4]), `&and`(x[1, 3, 2], x[1, 3, 5]), `&and`(x[1, 3, 2], x[1, 3, 6]), `&and`(x[1, 3, 3], x[1, 3, 4]), `&and`(x[1, 3, 3], x[1, 3, 5]), `&and`(x[1, 3, 3], x[1, 3, 6]), `&and`(x[1, 3, 4], x[1, 3, 5]), `&and`(x[1, 3, 4], x[1, 3, 6]), `&and`(x[1, 3, 5], x[1, 3, 6]), `&and`(x[2, 3, 1], x[2, 3, 2]), `&and`(x[2, 3, 1], x[2, 3, 3]), `&and`(x[2, 3, 1], x[2, 3, 4]), `&and`(x[2, 3, 1], x[2, 3, 5]), `&and`(x[2, 3, 1], x[2, 3, 6]), `&and`(x[2, 3, 2], x[2, 3, 3]), `&and`(x[2, 3, 2], x[2, 3, 4]), `&and`(x[2, 3, 2], x[2, 3, 5]), `&and`(x[2, 3, 2], x[2, 3, 6]), `&and`(x[2, 3, 3], x[2, 3, 4]), `&and`(x[2, 3, 3], x[2, 3, 5]), `&and`(x[2, 3, 3], x[2, 3, 6]), `&and`(x[2, 3, 4], x[2, 3, 5]), `&and`(x[2, 3, 4], x[2, 3, 6]), `&and`(x[2, 3, 5], x[2, 3, 6])))

(7.1)

Solution

 

Combine the boolean expressions into a a single and-clause and pass it to Satisfy.

sol := Logic:-Satisfy(&and(initial, adjacent, allvals, single));

{x[1, 1, 1] = true, x[1, 1, 2] = false, x[1, 1, 3] = false, x[1, 1, 4] = false, x[1, 1, 5] = false, x[1, 1, 6] = false, x[1, 2, 1] = false, x[1, 2, 2] = false, x[1, 2, 3] = false, x[1, 2, 4] = false, x[1, 2, 5] = false, x[1, 2, 6] = true, x[1, 3, 1] = false, x[1, 3, 2] = false, x[1, 3, 3] = false, x[1, 3, 4] = false, x[1, 3, 5] = true, x[1, 3, 6] = false, x[2, 1, 1] = false, x[2, 1, 2] = true, x[2, 1, 3] = false, x[2, 1, 4] = false, x[2, 1, 5] = false, x[2, 1, 6] = false, x[2, 2, 1] = false, x[2, 2, 2] = false, x[2, 2, 3] = true, x[2, 2, 4] = false, x[2, 2, 5] = false, x[2, 2, 6] = false, x[2, 3, 1] = false, x[2, 3, 2] = false, x[2, 3, 3] = false, x[2, 3, 4] = true, x[2, 3, 5] = false, x[2, 3, 6] = false}

(8.1)

Select the equations whose right size is true

sol := select(rhs, sol);

{x[1, 1, 1] = true, x[1, 2, 6] = true, x[1, 3, 5] = true, x[2, 1, 2] = true, x[2, 2, 3] = true, x[2, 3, 4] = true}

(8.2)

Extract the lhs of the true equations

vars := map(lhs, sol);

{x[1, 1, 1], x[1, 2, 6], x[1, 3, 5], x[2, 1, 2], x[2, 2, 3], x[2, 3, 4]}

(8.3)

Extract the result from the indices of the vars and assign to a new Matrix

S := Matrix(m,n):

for v in vars do S[op(1..2,v)] := op(3,v); end do:

S;

Matrix(2, 3, {(1, 1) = 1, (1, 2) = 6, (1, 3) = 5, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4})

(8.4)

Procedure

 

We can now combine the manual steps into a procedure that takes an initialized Matrix and fills in a solution.

Numbrix := proc( M :: ~Matrix, { inline :: truefalse := false } )

Example

 

Create the initial position for a 9 x 9 Numbrix and solve it.

P := Matrix(9, {(1,1)=11, (1,3)=7, (1,5)=3, (1,7)=81, (1,9)=77, (3,9)=75, (5,9)=65, (7,9)=55, (9,9)=53
               , (9,7)=47, (9,5)=41, (9,3)=39, (9,1)=37, (7,1)=21, (5,1)=17, (3,1)=13});

Matrix(9, 9, {(1, 1) = 11, (1, 2) = 0, (1, 3) = 7, (1, 4) = 0, (1, 5) = 3, (1, 6) = 0, (1, 7) = 81, (1, 8) = 0, (1, 9) = 77, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (2, 6) = 0, (2, 7) = 0, (2, 8) = 0, (2, 9) = 0, (3, 1) = 13, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (3, 5) = 0, (3, 6) = 0, (3, 7) = 0, (3, 8) = 0, (3, 9) = 75, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 0, (4, 5) = 0, (4, 6) = 0, (4, 7) = 0, (4, 8) = 0, (4, 9) = 0, (5, 1) = 17, (5, 2) = 0, (5, 3) = 0, (5, 4) = 0, (5, 5) = 0, (5, 6) = 0, (5, 7) = 0, (5, 8) = 0, (5, 9) = 65, (6, 1) = 0, (6, 2) = 0, (6, 3) = 0, (6, 4) = 0, (6, 5) = 0, (6, 6) = 0, (6, 7) = 0, (6, 8) = 0, (6, 9) = 0, (7, 1) = 21, (7, 2) = 0, (7, 3) = 0, (7, 4) = 0, (7, 5) = 0, (7, 6) = 0, (7, 7) = 0, (7, 8) = 0, (7, 9) = 55, (8, 1) = 0, (8, 2) = 0, (8, 3) = 0, (8, 4) = 0, (8, 5) = 0, (8, 6) = 0, (8, 7) = 0, (8, 8) = 0, (8, 9) = 0, (9, 1) = 37, (9, 2) = 0, (9, 3) = 39, (9, 4) = 0, (9, 5) = 41, (9, 6) = 0, (9, 7) = 47, (9, 8) = 0, (9, 9) = 53})

(10.1)

CodeTools:-Usage(Numbrix(P));

memory used=0.77GiB, alloc change=220.03MiB, cpu time=15.55s, real time=12.78s, gc time=3.85s

 

Matrix(9, 9, {(1, 1) = 11, (1, 2) = 10, (1, 3) = 7, (1, 4) = 81, (1, 5) = 3, (1, 6) = 4, (1, 7) = 81, (1, 8) = 78, (1, 9) = 77, (2, 1) = 12, (2, 2) = 9, (2, 3) = 8, (2, 4) = 7, (2, 5) = 6, (2, 6) = 5, (2, 7) = 80, (2, 8) = 79, (2, 9) = 76, (3, 1) = 13, (3, 2) = 14, (3, 3) = 27, (3, 4) = 28, (3, 5) = 71, (3, 6) = 72, (3, 7) = 73, (3, 8) = 74, (3, 9) = 75, (4, 1) = 16, (4, 2) = 15, (4, 3) = 26, (4, 4) = 29, (4, 5) = 70, (4, 6) = 69, (4, 7) = 68, (4, 8) = 67, (4, 9) = 66, (5, 1) = 17, (5, 2) = 18, (5, 3) = 25, (5, 4) = 30, (5, 5) = 61, (5, 6) = 62, (5, 7) = 63, (5, 8) = 64, (5, 9) = 65, (6, 1) = 20, (6, 2) = 19, (6, 3) = 24, (6, 4) = 31, (6, 5) = 60, (6, 6) = 59, (6, 7) = 58, (6, 8) = 57, (6, 9) = 56, (7, 1) = 21, (7, 2) = 22, (7, 3) = 23, (7, 4) = 32, (7, 5) = 43, (7, 6) = 44, (7, 7) = 49, (7, 8) = 50, (7, 9) = 55, (8, 1) = 36, (8, 2) = 35, (8, 3) = 34, (8, 4) = 33, (8, 5) = 42, (8, 6) = 45, (8, 7) = 48, (8, 8) = 51, (8, 9) = 54, (9, 1) = 37, (9, 2) = 38, (9, 3) = 39, (9, 4) = 40, (9, 5) = 41, (9, 6) = 46, (9, 7) = 47, (9, 8) = 52, (9, 9) = 53})

(10.2)

 

numbrix.mw

I describe here a finite difference scheme for solving the boundary value
problem for the heat equation

"(&PartialD; u)/(&PartialD; t)= ((&PartialD;)^)/((&PartialD;)^( )x^)(c(x)(&PartialD; u)/(&PartialD; x)) + f(x,t)   a<x<b,   t>0"

for the unknown temperature u(x, t)subject to the boundary conditions

u(a, t) = alpha(t), u(b, t) = beta(t), t > 0

and the initial condition

"u(x,0)=`u__0`(x),    a < x < b."

 

This finite difference scheme is designed expressly with the goal of avoiding

differentiating the conductivity c(x), therefore c(x) is allowed to be

nonsmooth or even discontinuous. A discontinuous c(x) is particularly
important in applications where the heat conduction takes place through layers
of distinct types of materials.

 

The animation below, extracted from the worksheet, demonstrates a solution 

corresponding to a discontinuous c(x).  The limit of that solution as time goes to

infinity, which may be calculated independently and exactly, is shown as a gray
line.

Download worksheet: heat-finite-difference.mw

 

 

 

 

Maple 2019 has a new add-on package Maple Quantum Chemistry Toolbox from RDMChem for computing the energies and properties of molecules.  As a member of the team at RDMChem that developed the package, I would like to tell the story of its origins and provide a brief demonstration of the package.  

 

Thinking about Quantum Chemistry at Harvard

 

The story of the Maple Quantum Chemistry Toolbox begins with my graduate studies in Chemical Physics at Harvard University in the late 1990s.  Even in 1998 programs for computing the energies and properties of molecules were extremely complicated and nonintuitive.  Many of the existing programs had begun in the 1970s on computers whose programs would be recorded on punchcards.

Fig. 1: Used Punchcard by Pete Birkinshaw from Manchester, UK CC BY 2.0

 

Even today some of these programs have remnants of their early versions such as input files that must start on the second column to account for the margin of the now non-existent punchcards.  As a student, I made a bound copy of one of these manuals at a local Kinkos photocopy shop and later found myself in Harvard Yard, thinking that there must be a better way to present quantum chemistry computations.  The idea for a Maple-like package for quantum chemistry was born in that moment.

 

At the same time I was learning about something called the two-electron reduced density matrix (2-RDM).  The basic variable in quantum chemistry is the wave function which is the probability amplitude for finding each of the electrons in a molecule.  Because electrons are indistinguishable with pairwise interactions, the wave function contains much more information than is needed for computing the energies and electronic properties of molecules.  The energies and properties of any molecule with any number of electrons can be expressed as a function of a 2 electron matrix, the 2-RDM [1-3].  A quantum chemistry based on the 2-RDM, it was known, would have potentially significant advantages over wave function calculations in terms of accuracy and computational cost, especially for molecules far from the mean-field limit.  A 2-RDM approach to quantum chemistry became the focus of my Ph.D. thesis.

 

Representing Many Electrons with Only Two Electrons

 

The idea of using the 2-RDM in quantum chemistry can be attributed to four scientists: two physicists Kodi Husimi and Joseph Mayer, a chemist Per-Olov Lowdin, and a mathematician John Coleman [1-3].  In the early 1940s Husimi first published the idea in a Japanese physics journal, but in the midst of World War II the paper was not widely disseminated in the West.  In the summer of 1951 John Coleman, which attending a physics conference at Chalk River, realized that the ground-state energy of any atom or molecule could be expressed as functional of the 2-RDM, and similar ideas later occurred to Per-Olov Lowdin and Joseph Mayer who published their ideas in Physical Review in 1955.  It was soon recognized that computing the ground-state energy of an atom or molecule with the 2-RDM was potentially difficult because not every two-electron density matrix corresponds to an N-electron density matrix or wave function.  The search for the appropriate constraints on the 2-RDM, known as N-representability conditions, became known as the N-representability problem [1-3].  

 

Beginning in the late 1990s and early 2000s, Carmela Valdemoro and Diego Alcoba at the Consejo Superior de Investigaciones Científicas (Madrid, Spain), Hiroshi Nakatsuji, Koji Yasuda, and Maho Nakata at Kyoto University (Kyoto, Japan), Jerome Percus and Bastiaan Braams at the Courant Institute (New York, USA), John Coleman and Robert Erdahl at Queens University (Kingston, Canada), and my research group and I at The University of Chicago (Chicago, USA) began to make significant progress in the computation of the 2-RDM without computing the many-electron wave function [1-3].  Further contributions were made by Eric Cances and Claude Le Bris at CERMICS, Ecole Nationale des Ponts et Chaussées (Marne-la-Vallée, France), Paul Ayers at McMaster University (Hamilton, Canada), and Dimitri Van Neck at the University of Ghent (Ghent, Belgium) and their research groups.  By 2014 several powerful 2-RDM methods had emerged for the computation of molecules.  The Army Research Office (ARO) issued a proposal call for a company to develop a modern, built-from-scratch package for quantum chemistry that would contain two newly developed 2-RDM-based methods from our group: the parametric 2-RDM method [1] and the variational 2-RDM method with a fast algorithm for solving the semidefinite program [4,5,6].   The company RDMChem LLC was founded to work with the ARO to develop such a package built around RDMs, and hence, the name of the company RDMChem was selected as a hybrid of the RDM abbreviation for Reduced Density Matrices and the Chem colloquialism for Chemistry.  To achieve a really new design for an electronic structure package with access to numeric and symbolic computations as well as advanced visualizations, the team at RDMChem and I developed a partnership with Maplesoft to build something new that became the Maple Quantum Chemistry Package (or Toolbox), which was released with Maple 2019 on Pi Day.

 

Maple Quantum Chemistry Toolbox

 The Maple Quantum Chemistry Toolbox provides a powerful, parallel platform for quantum chemistry calculations that is directly integrated into the Maple 2019 environment.  It is optimized for both cutting-edge research as well as chemistry education.  The Toolbox can be used from the worksheet, document, or command-line interfaces.  Plus there is a Maplet interface for rapid exploration of molecules and their properties.  Figure 2 shows the Maplet interface being applied to compute the ground-state energy of 1,3-dibromobenzene by density functional theory (DFT) in a 6-31g basis set.           

Fig. 2: Maplet interface to the Quantum Chemistry Toolbox 2019, showing a density functional theory (DFT) calculation         

After entering a name into the text box labeled Name, the user can click on: (1) the button Web to import the geometry from an online database containing more than 96 million molecules,  (2) the button File to read the geometry from a standard XYZ file, or (3) the button Input to enter the geometry.  As soon the geometry is entered, the Maplet displays a 3D picture of the molecule in the window on the right of the options.  Dropdown menus allow the user to select the basis set, the electronic structure method, and a boolean for geometry optimization.  The user can click on the Compute button to perform the computation.  When the quantum computation completes, the total energy appears in the box labeled Total Energy.  The dropdown menu Analyze contains a list of data tables, plots, and animations that can be selected and then displayed by clicking the Analyze button.  The Maplet interface contains nearly all of the options available in the worksheet interface.   The Help Pages of the Toolbox include extensive curricula and lessons that can be used in undergraduate, graduate, and even high school chemistry courses.  Next we look at some sample calculations in the worksheet interface.     

 

Reproducing an Early 2-RDM Calculation

 

One of the earliest variational calculations of the 2-RDM was performed in 1975 by Garrod, Mihailović,  and  Rosina [1-3].  They minimized the electronic ground state of the 4-electron atom beryllium as a functional of only two electrons, the 2-RDM.  They imposed semidefinite constraints on the particle-particle (D), hole-hole (Q), and particle-hole (G) metric matrices.  They solved the resulting optimization problem of minimizing the energy as a linear function of the 2-RDM subject to the semidefinite constraints, known as a semidefinite program, by a cutting-plane algorithm.  Due to limitations of the cutting-plane algorithm and computers circa 1975, the calculation was a difficult one, likely taking a significant amount of computer time and memory.

 

With the Quantum Chemistry Toolbox we can use the command Variational2RDM to reproduce the calculation on a Windows laptop.  First, in a Maple 2019 worksheet we load the commands of the Add-on Quantum Chemistry Toolbox:

with(QuantumChemistry);

[AOLabels, ActiveSpaceCI, ActiveSpaceSCF, AtomicData, BondAngles, BondDistances, Charges, ChargesPlot, CorrelationEnergy, CoupledCluster, DensityFunctional, DensityPlot3D, Dipole, DipolePlot, Energy, FullCI, GeometryOptimization, HartreeFock, Interactive, Isotopes, MOCoefficients, MODiagram, MOEnergies, MOIntegrals, MOOccupations, MOOccupationsPlot, MOSymmetries, MP2, MolecularData, MolecularGeometry, NuclearEnergy, NuclearGradient, Parametric2RDM, PlotMolecule, Populations, RDM1, RDM2, ReadXYZ, SaveXYZ, SearchBasisSets, SearchFunctionals, SkeletalStructure, Thermodynamics, Variational2RDM, VibrationalModeAnimation, VibrationalModes, Video]

(1.1)

Then we define the atom (or molecule) using a Maple list of lists that we assign to the variable atom:

atom := [["Be",0,0,0]];

[["Be", 0, 0, 0]]

(1.2)

 

We can then perform the variational 2-RDM method with the Variational2RDM command to compute the ground-state energy and properties of beryllium in a minimal basis set like the one used by Rosina and his collaborators.  By default the method uses the D, Q, and G N-representability conditions and the minimal "sto-3g" basis set.  The calculation, which completes in seconds, contains a wealth of information in the form of a convenient Maple table that we assign to the variable data.

data := Variational2RDM(atom);

table(%id = 18446744313704784158)

(1.3)

 

The table contains the total ground-state energy of the beryllium atom in the atomic unit of energy (hartrees)

data[e_tot];

HFloat(-14.40370016681039)

(1.4)

 

We also have the atomic orbitals (AOs) employed in the calculation

data[aolabels];

Vector(5, {(1) = "0 Be 1s", (2) = "0 Be 2s", (3) = "0 Be 2px", (4) = "0 Be 2py", (5) = "0 Be 2pz"})

(1.5)

 

as well as the Mulliken populations of these orbitals

data[populations];

Vector(5, {(1) = 1.9995807710723152, (2) = 1.7913484714571852, (3) = 0.6969023822632789e-1, (4) = 0.6969026475511847e-1, (5) = 0.6969029119010149e-1})

(1.6)

 

We see that 2 electrons are located in the 1s orbital, 1.8 electrons in the 2s orbital, and about 0.2 electrons in the 2p orbitals.  By default the calculation also returns the 1-RDM

data[rdm1];

Matrix(5, 5, {(1, 1) = 1.9999258249189755, (1, 2) = -0.37784860208539793e-2, (1, 3) = 0., (1, 4) = 0., (1, 5) = 0., (2, 1) = -0.37784860208539793e-2, (2, 2) = 1.7910034176105256, (2, 3) = 0., (2, 4) = 0., (2, 5) = 0., (3, 1) = 0., (3, 2) = 0., (3, 3) = 0.6969023822632789e-1, (3, 4) = 0., (3, 5) = 0., (4, 1) = 0., (4, 2) = 0., (4, 3) = 0., (4, 4) = 0.6969026475511847e-1, (4, 5) = 0., (5, 1) = 0., (5, 2) = 0., (5, 3) = 0., (5, 4) = 0., (5, 5) = 0.6969029119010149e-1})

(1.7)

 

The eigenvalues of the 1-RDM are the natural orbital occupations

LinearAlgebra:-Eigenvalues(data[rdm1]);

Vector(5, {(1) = 1.9999941387490443+0.*I, (2) = 1.7909351037804568+0.*I, (3) = 0.6969023822632789e-1+0.*I, (4) = 0.6969026475511847e-1+0.*I, (5) = 0.6969029119010149e-1+0.*I})

(1.8)

 

We can display the density of the 2s-like 2nd natural orbital using the DensityPlot3D command providing the atom, the data, and the orbitalindex keyword

DensityPlot3D(atom,data,orbitalindex=2);

 

 

Similarly,  using the DensityPlot3D command, we can readily display the 2p-like 3rd natural orbital

DensityPlot3D(atom,data,orbitalindex=3);

 

 

By using Maple keyword arguments in the Variational2RDM command, we can readily change the basis set, use point-group symmetry, add active orbitals with or without self-consistent-field, change the N-representability conditions, as well as explore many other options.  Having reenacted one of the first variational 2-RDM calculations ever, let's examine a more complicated molecule.

 

Explosive TNT

 

We consider the molecule TNT that is used as an explosive. Using the command MolecularGeometry, we can import the experimental geometry of TNT from the online PubChem database.

mol := MolecularGeometry("TNT");

[["O", .5454, -3.514, 0.12e-2], ["O", .5495, 3.5137, 0.8e-3], ["O", 2.4677, -2.4539, -0.5e-3], ["O", 2.4705, 2.4513, 0.3e-3], ["O", -3.5931, -1.0959, 0.4e-3], ["O", -3.5922, 1.0993, 0.6e-3], ["N", 1.2142, -2.454, 0.2e-3], ["N", 1.217, 2.4527, 0], ["N", -2.9846, 0.15e-2, 0.1e-3], ["C", 1.2253, -0.6e-3, -0.9e-3], ["C", .5271, -1.2082, -0.8e-3], ["C", .5284, 1.2078, -0.8e-3], ["C", -1.5646, 0.8e-3, -0.4e-3], ["C", -.8678, -1.2074, -0.6e-3], ["C", -.8666, 1.2084, -0.6e-3], ["C", 2.7239, -0.16e-2, 0.11e-2], ["H", -1.4159, -2.1468, -0.3e-3], ["H", -1.4137, 2.1483, -0.3e-3], ["H", 3.1226, .2418, -.9891], ["H", 3.0863, .6934, .7662], ["H", 3.3154, -.8111, .4109]]

(1.9)

 

The command PlotMolecule generates a 3D ball-and-stick plot of the molecule

PlotMolecule(mol);

 

 

We perform a variational calculation of the 2-RDM of TNT in an active space of 10 electrons and 10 orbitals by setting the keyword active to the list [10,10].  The keyword casscf is set to true to optimize the active orbitals during the calculation.  The keyword basis is used to set the basis set to a minimal basis set sto-3g for illustration.   

data := Variational2RDM(mol, active=[10,10], casscf=true, basis="sto-3g");

table(%id = 18446744493271367454)

(1.10)

 

The ground-state energy of TNT in hartrees is

data[e_tot];

HFloat(-868.8629631593426)

(1.11)

 

Unlike beryllium, the electric dipole moment of TNT in debyes is nonzero

data[dipole];

Vector(3, {(1) = .5158925019252739, (2) = -0.5985274393363119e-1, (3) = .1277528280025474})

(1.12)

 

We can easily visualize the dipole moment relative to the molecule's ball-and-stick model with the DipolePlot command

DipolePlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

 

 

The 1-RDM is returned by default

data[rdm1];

_rtable[18446744313709602566]

(1.13)

 

The natural molecular-orbital (MO) occupations are the eigenvalues of the 1-RDM

data[mo_occ];

_rtable[18446744313709600150]

(1.14)

 

All of the occupations can be viewed at once by converting the Vector to a list

convert(data[mo_occ], list);

[HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(2.0), HFloat(1.9110133620349001), HFloat(1.8984139688344246), HFloat(1.6231436866358906), HFloat(1.6158489471020905), HFloat(1.6145310163161273), HFloat(0.38920731792133734), HFloat(0.387039366894289), HFloat(0.37786347287813526), HFloat(0.09734187094597906), HFloat(0.08559699476985069), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0), HFloat(0.0)]

(1.15)

 

We can visualize these occupations with the MOOccupationsPlot command

MOOccupationsPlot(mol,method=Variational2RDM, active=[10,10], casscf=true, basis="sto-3g");

 

 

The occupations, we observe, show significant deviations from 0 and 2, indicating that the electrons have substantial correlation beyond the mean-field (Hartree-Fock) limit.  The blue lines indicate the first N/2 spatial orbitals where N is the total number of electrons while the red lines indicate the remaining spatial orbitals.  We can visualize the highest "occupied" molecular orbital (58) with the DensityPlot3D command

DensityPlot3D(mol,data, orbitalindex=58);

 

 

Similarly, we can visualize the lowest "unoccupied" molecular orbital (59) with the DensityPlot3D command

DensityPlot3D(mol,data, orbitalindex=59);

 

 

Comparison of orbitals 58 and 59 reveals an increase in the number of nodes (changes in the phase of the orbitals denoted by green and purple), which reflects an increase in the energy of the orbital.

 

Looking Ahead

 

The Maple Quantum Chemistry Toolbox 2019, an new Add-on for Maple 2019 from RDMChem, provides a easy-to-use, research-grade environment for the computation of the energies and properties of atoms and molecules.  In this blog we discussed its origins in graduate research at Harvard, its reproduction of an early 2-RDM calculation of beryllium, and its application to the explosive molecule TNT.  We have illustrated only some of the many features and electronic structure methods of the Maple Quantum Chemistry package.  There is much more chemistry and physics to explore.  Enjoy!    

 

Selected References

 

[1] D. A. Mazziotti, Chem. Rev. 112, 244 (2012). "Two-electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics"

[2]  Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules (Adv. Chem. Phys.) ; D. A. Mazziotti, Ed.; Wiley: New York, 2007; Vol. 134.

[3] A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: Coulson’s Challenge (Springer-Verlag,  New York, 2000).

[4] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). "Large-scale Semidefinite Programming for Many-electron Quantum Mechanics"

[5] A. W. Schlimgen, C. W. Heaps, and D. A. Mazziotti, J. Phys. Chem. Lett. 7, 627-631 (2016). "Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex"

[6] J. M. Montgomery and D. A. Mazziotti, J. Phys. Chem. A 122, 4988-4996 (2018). "Strong Electron Correlation in Nitrogenase Cofactor, FeMoco"

 

Download QCT2019_PrimesV17_05.05.19.mw

In this Post I derive the differential equations of motion of a homogeneous elliptic lamina of mass m and the major and minor axes of lengths of a and b which rolls without slipping along the horizontal x axis within the vertical xy plane.

If the initial angular velocity is large enough, the ellipse will roll forever and go to ±∞ in the x direction, otherwise it will just rock.

I have attached two files:

 rolling-ellipse.mw
        Worksheet to solve the differential equations and animate the motion

rolling-ellipse.pdf
         Documentation containing the derivation of the differential equations

And here are two animations extracted from the worksheet.

While googling around for Season 8 spoilers, I found data sets that can be used to create a character interaction network for the books in the A Song of Ice and Fire series, and the TV show they inspired, Game of Thrones.

The data sets are the work of Dr Andrew Beveridge, an associate professor at Macalaster College (check out his Network of Thrones blog).

You can create an undirected, weighted graph using this data and Maple's GraphTheory package.

Then, you can ask yourself really pressing questions like

  • Who is the most influential person in Westeros? How has their influence changed over each season (or indeed, book)?
  • How are Eddard Stark and Randyll Tarly connected?
  • What do eigenvectors have to do with the battle for the Iron Throne, anyway?

These two applications (one for the TV show, and another for the novels) have the answers, and more.

The graphs for the books tend to be more interesting than those for the TV show, simply because of the far broader range of characters and the intricacy of the interweaving plot lines.

Let’s look at some of the results.

This a small section of the character interaction network for the first book in the A Song of Ice and Fire series (this is the entire visualization - it's big, simply because of the shear number of characters)

The graph was generated by GraphTheory:-DrawGraph (with method = spring, which models the graph as a system of protons repelling each other, connected by springs).

The highlighted vertices are the most influential characters, as determined by their Eigenvector centrality (more on this later).

 

The importance of a vertex can be described by its centrality, of which there are several variants.

Eigenvector centrality, for example, is the dominant eigenvector of the adjacency matrix, and uses the number and importance of neighboring vertices to quantify influence.

This plot shows the 15 most influential characters in Season 7 of the TV show Game of Thrones. Jon Snow is the clear leader.

Here’s how the Eigenvector centrality of several characters change over the books in the A Song of Ice and Fire series.

A clique is a group of vertices that are all connected to every other vertex in the group. Here’s the largest clique in Season 7 of the TV show.

Game of Thrones has certainly motivated me to learn more about graph theory (yes, seriously, it has). It's such a wide, open field with many interesting real-world applications.

Enjoy tinkering!

Hi

The Physics Updates for Maple 2019 (current v.331 or higher) is already available for installation via MapleCloud. This version contains further improvements to the Maple 2019 capabilities for solving PDE & BC as well as to the tensor simplifier. To install these Updates,

  • Open Maple,
  • Click the MapleCloud icon in the upper-right corner to open the MapleCloud toolbar 
  • In the MapleCloud toolbar, open Packages
  • Find the Physics Updates package and click the install button, it is the last one under Actions
  • To check for new versions of Physics Updates, click the MapleCloud icon. If the Updates icon has a red dot, click it to install the new version

Note that the first time you install the Updates in Maple 2019 you need to install them from Packages, even if in your copy of Maple 2018 you had already installed these Updates.

Also, at this moment you cannot use the MapleCloud to install the Physics Updates for Maple 2018. So, to install the last version of the Updates for Maple 2018, open Maple 2018 and enter PackageTools:-Install("5137472255164416", version = 329, overwrite)

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

This application solves a set of compatible equations of two variables. It also graphs the intersection point of the variable "x" and "y". If we want to observe the intersection point closer we will use the zoom button that is activated when manipulating the graph. If we want to change the variable ("x" and "y") we enter the code of the button that solves and graphs. In spanish.

System_of_Equations_Determined_Compatible_2x2_and_3x3.mw

Lenin Araujo Castillo

Ambassador of Maple

It is my pleasure to announce the return of the Maple Conference! On October 15-17th, in Waterloo, Ontario, Canada, we will gather a group of Maple enthusiasts, product experts, and customers, to explore and celebrate the different aspects of Maple.

Specifically, this conference will be dedicated to exploring Maple’s impact on education, new symbolic computation algorithms and techniques, and the wide range of Maple applications. Attendees will have the opportunity to learn about the latest research, share experiences, and interact with Maple developers.

In preparation for the conference we are welcoming paper and extended abstract submissions. We are looking for presentations which fall into the broad categories of “Maple in Education”, “Algorithms and Software”, and “Applications of Maple” (a more extensive list of topics can be found here).

You can learn more about the event, plus find our call-for-papers and abstracts, here: https://www.maplesoft.com/mapleconference/

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