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MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • For the past two years the Queen’s College of Guyana Alumni Association (NY) has been hosting its Queen’s College Summer Maths Institute, and Maplesoft has supported this initiative by giving students access to Maple.   With Dr. Terrence Blackman at its helm, the institute aims to sustainably implement a developmentally appropriate and culturally resonant middle school learning environment that engages Guyanese students in a cognitively rich mathematics learning experience.  The experience is intended to place them securely on pathways to STEM (Science, Technology, Engineering and Math) excellence.

    The program uses a developmentally appropriate approach that combines significant mathematical content with a setting that encourages a sense of discovery and excitement about math through problem solving and exploration. Program Manager Shindy Johnson, a former student of Queens College, noted that by the end of the first week math sceptics fall in love and gained confidence, and math lovers renew their passion.

    As avid Maple users, Dr. Terence Blackman and Cleveland Waddell, one of the main organizers and lecturer,  give the students the opportunity to use Maple. Last year, the students were amazed by Maple's computational power. “It was nothing like they have seen before.  Many students also wrote their first lines of computer code using Maple during the summer camp.  Maple is an invaluable resource for us during the camp,” said Waddell.

    Dr. Terence Blackman and Cleveland Waddell

    Students receive further enrichment through field trips to broaden their appreciation for education and industry in Guyana.  In addition, Guyanese professionals visit the Institute to share their expertise, career journeys and practical applications of math and other STEM disciplines in their professions.

    Field trip to Uitvlugt Sugar Estate

    Students who participate ranged from self-professed math lovers to teens who confessed to fearing and even loathing math. By the end of the first week, math lovers had discovered even greater “beauty in the mathematics” and those who quaked at the thought of math were beginning to commit – with confidence – to improving their math grades. This year’s Queen’s College Summer Maths Institute will take place July 26-August 3, 2018 in Georgetown, Guyana.   

    Recently I looked through an interesting book D. Wells "The Penquin book of Curious and Interesting Geometry" and came across this result, which I did not know about before: starting with a given quadrilateral , construct a square on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. See the picture below:

                                      

    It is interesting that this is true not only for a convex quadrilateral, but for arbitrary one and even self-intersecting one. This post is devoted to proving this result in Maple. The proof was very short and simple. Note that the coordinates of points are given not by lists, but by vectors. This is convenient when using  LinearAlgebra:-DotProduct  and  LinearAlgebra:-Norm  commands.

    The code of the proof (the notation of the points on the picture coincide with their names in the code):

    restart;
    with(LinearAlgebra):
    assign(seq(P||i=<x[i],y[i]>, i=1..4)):
    P||5:=P||1:
    assign(seq(Q||i=(P||i+P||(i+1))/2+<0,1; -1,0>.(P||(i+1)-P||i)/2, i=1..4)):
    expand(DotProduct((Q||3-Q||1),(Q||4-Q||2), conjugate=false));
    is(Norm(Q||3-Q||1, 2)=Norm(Q||4-Q||2, 2));
    
    

    The output:

                                                          0
                                                        true

     

    VA.mw

    The Maple command line interface (cmaple), often referred to as the "TTY interface" for its original use on Teletype terminals, is still the tool of choice for many Maple developers and power users. Maple 2018.1 introduces several new capabilities to this long-lived interface:

    This post, the first in a series of three, will address color syntax highlighting. We'll start with a very short sample session:

        |\^/|     Maple 2018.1 (X86 64 LINUX)
    ._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2018
     \  MAPLE  /  All rights reserved. Maple is a trademark of
     <____ ____>  Waterloo Maple Inc.
          |       Type ? for help.
    > piecewise(4 < x^2 and x < 8, f(x));
                                {                  2
                                { f(x)        4 < x  and x < 8
                                {
                                {  0             otherwise
    
    > p := unapply(%,x);
                                                   2
                          p := x -> piecewise(4 < x  and x < 8, f(x))
    
    > 1/p(1);
    Error, numeric exception: division by zero
    > quit
    memory used=5.3MB, alloc=41.3MB, time=0.07
    

    In the above example, you can see that general keywords are in bold blue, variables in italics (not supported by all terminals), error messages in bold red, control flow interrupting keywords in bold magenta, and memory usage messages in normal blue.

    Color syntax highlighting is turned on by default in cmaple for Linux and OS/X if the terminal you are using (as specified by the TERM environment variable) is known to support it. It is currently turned off by default under Windows. It can be explicitly turned on or off for 2D and message output using interface(ansi=) where is true or false (under Windows, you can put interface(ansi=true) in your maple.ini file to automatically turn it on). Likewise, interface(ansilprint=) controls highlighting for 1D output (such as that produced by lprint), and interface(ansiedit=) for input.

    Not all terminals support all possible highlighting modes. The following two commands show what colors your terminal can display, and how they are used by Maple's syntax highlighting:

    > interface(showtermcolors):
    
    ANSI X3.64 Standard Attributes
    
    Normal  Bold  Italic  Underlined  Reverse  
    
    System Colors (0-15) Using ANSI Escape Sequences
    
    Color00  Color01  Color02  Color03  Color04  Color05  Color06  Color07  
    Color08  Color09  Color10  Color11  Color12  Color13  Color14  Color15  
    
    System Colors (0-15) Using Extended Escape Sequences
    
      0    0    1    1    2    2    3    3    4    4    5    5    6    6    7    7  
      8    8    9    9   10   10   11   11   12   12   13   13   14   14   15   15  
    
    Extended 6x6x6 Color Cube (16-231)
    
     16   16   17   17   18   18   19   19   20   20   21   21  
     22   22   23   23   24   24   25   25   26   26   27   27  
     28   28   29   29   30   30   31   31   32   32   33   33  
     34   34   35   35   36   36   37   37   38   38   39   39  
     40   40   41   41   42   42   43   43   44   44   45   45  
     46   46   47   47   48   48   49   49   50   50   51   51  
    
     52   52   53   53   54   54   55   55   56   56   57   57  
     58   58   59   59   60   60   61   61   62   62   63   63  
     64   64   65   65   66   66   67   67   68   68   69   69  
     70   70   71   71   72   72   73   73   74   74   75   75  
     76   76   77   77   78   78   79   79   80   80   81   81  
     82   82   83   83   84   84   85   85   86   86   87   87  
    
     88   88   89   89   90   90   91   91   92   92   93   93  
     94   94   95   95   96   96   97   97   98   98   99   99  
    100  100  101  101  102  102  103  103  104  104  105  105  
    106  106  107  107  108  108  109  109  110  110  111  111  
    112  112  113  113  114  114  115  115  116  116  117  117  
    118  118  119  119  120  120  121  121  122  122  123  123  
    
    124  124  125  125  126  126  127  127  128  128  129  129  
    130  130  131  131  132  132  133  133  134  134  135  135  
    136  136  137  137  138  138  139  139  140  140  141  141  
    142  142  143  143  144  144  145  145  146  146  147  147  
    148  148  149  149  150  150  151  151  152  152  153  153  
    154  154  155  155  156  156  157  157  158  158  159  159  
    
    160  160  161  161  162  162  163  163  164  164  165  165  
    166  166  167  167  168  168  169  169  170  170  171  171  
    172  172  173  173  174  174  175  175  176  176  177  177  
    178  178  179  179  180  180  181  181  182  182  183  183  
    184  184  185  185  186  186  187  187  188  188  189  189  
    190  190  191  191  192  192  193  193  194  194  195  195  
    
    196  196  197  197  198  198  199  199  200  200  201  201  
    202  202  203  203  204  204  205  205  206  206  207  207  
    208  208  209  209  210  210  211  211  212  212  213  213  
    214  214  215  215  216  216  217  217  218  218  219  219  
    220  220  221  221  222  222  223  223  224  224  225  225  
    226  226  227  227  228  228  229  229  230  230  231  231  
    
    Extended 24-Level Grayscale (232-255)
    
    232  232  233  233  234  234  235  235  236  236  237  237  238  238  239  239  
    240  240  241  241  242  242  243  243  244  244  245  245  246  246  247  247  
    248  248  249  249  250  250  251  251  252  252  253  253  254  254  255  255  
    

    If your terminal does not support 256 color mode, then many of the colored blocks shown above will appear differently or not at all.

    > interface(showcolors):
    
     1 Normal output:           evalf(1/2) = 0.5
     2 Italics (variables):     x, y, z
     3 Symbol text (not used):  symbol
     4 Bold (fallback):         Begin, be bold, and venture to be wise.
     5 Underlined (fallback):   Morality, like art, means drawing a line someplace.
     6 Reversed (not used):      The reverse side also has a reverse side. 
     7 Input prompts:           >  DBG>
     8 User input:              1/(x^4+1);
     9 Userinfo output:         message, x, y
    10 Trace output:            {--> enter f, args = x, y
    11 Warning messages:        Warning, x is implicitly declared local
    12 Error messages:          Error, (in f) invalid subscript selector
    13 Debugger output:         No breakpoints set
    14 General Maple keywords:  for  from  to  while  do  until
    15 Declaration keywords:    local  option  description
    16 Flow interruptions:      break  return
    17 Exception keywords:      error  try  catch
    18 Subexpression labels:    %1  %2
    19 Special & quoted names:  thisproc  `diff/sin`
    20 String literals:         "Hello, world!"
    21 Maple startup message:   Maple 2019
    22 Output from printf:      x=1.234 y=5.678
    23 Status messages:         memory used=1.7MB, alloc=8.3MB, time=0.03
    24 System command output:   1466  4739  43140  myprog.mpl
    25 Maple comments:          # Comments are free but facts are sacred.
    

    The colors used for the different categories of output as listed by the command above are user selectable. The default is to use only the sixteen ANSI X3.64 standard colors (or Windows command prompt standard colors). These may appear differently than shown here depending on the color palette of your terminal window.

    The color settings can be queried or set as follows:

    > currentColors := interface(ansicolor);
    currentColors := [-1, -1, -1, -1, -1, -1, 2, -1, 2, 3, 11, 9, 6, 12, 10, 13, 9, 14, 6,
    
        5, 2, 136, 4, 134, 3]
    
    # Individual colours, as numbered in the output of interface(showcolors), can
    # be changed. Let's make keywords bright yellow:
    > myColors := subsop(14=226,currentColors);
    myColors := [-1, -1, -1, -1, -1, -1, 2, -1, 2, 3, 11, 9, 6, 226, 10, 13, 9, 14, 6, 5,
    
        2, 136, 4, 134, 3]
    
    > interface(ansicolor=myColors);
    [-1, -1, -1, -1, -1, -1, 2, -1, 2, 3, 11, 9, 6, 12, 10, 13, 9, 14, 6, 5, 2, 136, 4,
    
        134, 3]
    
    > piecewise(4 < x^2 and x < 8, f(x));
                                {                  2
                                { f(x)        4 < x  and x < 8
                                {
                                {  0             otherwise

    There are several predefined color schemes that can be selected using interface(ansicolor=), where is an integer from 0 to 6. Scheme 0, the default, should work on any terminal. Of the remaining schemes, the odd numbered ones are designed to look good on light backgrounds, and the even numbered ones on dark backgrounds.

    There is also a new character plot driver, selectable using interface(plotdevice=colorchar), which supports character plotting in color. Colors are mapped to the nearest color supported by the terminal:

    > interface(plotdevice=colorchar):
    > p1 := plot(sin(x),x=-Pi..Pi,thickness=1,color="DeepPink"):
    > p2 := plot(sin(x)+sin(3*x)/3,x=-Pi..Pi,thickness=2,color="LawnGreen"):
    > p3 := plot(sin(x)+sin(5*x)/5,x=-Pi..Pi,thickness=3,color="DodgerBlue"):
    > plots[display](p1,p2,p3);
    
                                                                                           
                                               |                                           
                                               |                  @@@@@                    
                                               |                 @@   @@                   
                                               |                 @     @                   
                                             1 |                *.......*                  
                                               |       *******.*@       @*..******         
                                               |      **    .**@         @**.    **        
                                               |     **   ..  ***       ***  ..   **       
                                               |    **   ..  @@ **** **** @@  ..   **      
                                               |    *   ..  @@     ***     @@  ..   *      
                                               |   *@@@**@@@                 @@@**@@@*     
                                           0.5 |  **@ ..                         .. @**    
                                               |  *@ ..                           .. @*    
                                               | ** .                               . **   
                                               | * .                                 . *   
                                               |**..                                 ..**  
                                               |*..                                   ..*  
                                               |*.                                     .*  
                                               *.                                       .* 
     **---------------------------------------**-----------------------------------------* 
      -3           -2            -1          0*|            1             2            3   
      *..                                   ..*|                                           
      **..                                 ..**|                                           
       * .                                 . * |                                           
       ** ..                              . ** |                                           
        *@ ..                           .. @*  |                                           
        **@ ..                         .. -0.5 |                                           
         *@@@**@@@                 @@@**@@@*   |                                           
          *   ..  @@     ***     @@  ..   *    |                                           
          **   ..  @@ **** **** @@  ..   **    |                                           
           **   ..  @**       **@  ..   **     |                                           
            **    .**@         @**.    **      |                                           
             *******.*@       @*.*******       |                                           
                      *.......*             -1 |                                           
                       @     @                 |                                           
                       @@   @@                 |                                           
                        @@@@@                  |                                           
                                               |                                           
                                                                                           
    
    > plot3d([1,x,y],x=0..2*Pi,y=0..2*Pi,coords=toroidal(10));
    
                                                                                           
                                      -------------------                                  
                                 --------\\\-|-|-|-///--------                             
                              ---------\\\-\-|-|-|-/-///---------                          
                            ----------\\\-\-||-|-||-/-///----------                        
                          /-/-/-/-----\\\\\-|||||-|-/////------------                      
                         -//-/-/-/---\-\\-\||-|||-||/-//-/---\-\-\-\--                     
                       --//-|||-/-/--\\\\\\|||||||-|//////--\-\-|||\|\-\                   
                      /---||-||||-|--\\-\\\|||||||||///-//--|-||||--|---\                  
                     /--|--\-\-\-\\--\\/\\|||||||||||//-//-|-///-/-/--|--\                 
                    //-//--\\-\-----\-\\\\|||||||||||////-///////-//--\\-\\                
                    |//\/\------------\\\|||||||||||||////-------//--/\\\\|                
                   /// / / -- /--- -----\|||||||||||||/---------\ --/\ \\\\\               
                   //\/ /-//-/-//-//-/--/-/--/-|-\--\-\--\-\\-\\-\-\\/\ \/\\               
                   ///\/ /--/ //-// /--/--/-|--|--|-\--\--\ \\-\\ \--\ \/\\\               
                   /||/-// /--/  /--/-//-|  |  |  |  |-\\-\--\  \\-\ \\/\||\               
                   || |-/ /  /--/  /  | -|--|--|--|--|- |  \ -\--\  \ \/| |||              
                  |||| | -/ /  ||--/-||- |  |  |  |  | -||--\  | \\-\- | |/|               
                   |||-| | -/--|  |  |  -|--|--|--|---|  |  |  |--\  | |/|||               
                   |\| |-|  | ||--|- |  |   |  |  |   |  | -|--|| |  |-- |/|               
                   |\||  |-|  |  |  -|--|---|--|---|--|--|-  |  |  |-|  ||/|               
                   \|-| |  |--|- |   |  |  |   |   |  |  |   | -|--| |  |/|/               
                    \||-|- |  |--|--|-  |  |   |   |  |  -|--|--|  | -|-||/                
                    \|-|||-|- |  |  | --|--|---|---|--|-- |  |  |  |-|||/|/                
                     \\|-| |--|--|  |   |  |   |   |  |   |  |--|--| |-|//                 
                      \\ |-|  |  |--|---|  |   |   |  |---|--|  |  |-| //                  
                       -\| |--|  |  |   |--|---|---|--|   |  |  |--| |//                   
                         \|-| |--|--|   |  |   |   |  |   |--|--| |-|/                     
                          \\|\|| |  |---|--|---|---|--|---|  | ||-|//                      
                            -\----|--|  |  |   |   |  |  |--|----/-                        
                              --\-\\ |--|--||--|---|--|--|  /-/--                          
                                 ----\\-||--|--|--|---|--///--                             
                                      \--\--|--|--|--/--/                                  
                                                                                           
    

    For more details, please refer to the help page, ?ansicolor.

    Hi MaplePrimes Users!

    It’s your friendly, neighborhood tech support team; here to share some tips and tricks from issues we help users with on a daily basis.

    A customer contacted us through a Help Page feedback form, asking how to add a row or column in a Matrix. The form came from the Row Operations help page, but the wording of the message suggested that the customer actually wanted to insert a new row or column altogether. Such manipulations can often be accomplished by a command in the ArrayTools package, but the only Insert command currently available is the one for Vectors and 1-D Arrays. Using the Concatenate command from that package, and various commands from the LinearAlgebra package (including the SubMatrix command), we were able to write two custom procedures to perform these manipulations:

    InsertRow := proc (A::rtable, n::integer, v::Vector[row])
        local R, C, top, bottom;
        uses LinearAlgebra;
        R := RowDimension(A); C := ColumnDimension(A);
        top := SubMatrix(A, [1 .. n-1], [1 .. C]);
        bottom := SubMatrix(A, [n .. R], [1 .. C]);
        return ArrayTools:-Concatenate(1, top, v, bottom);
    end proc:
    
    InsertColumn := proc (A::rtable, n::integer, v::Vector[column])
        local R, C, left, right;
        uses LinearAlgebra;
        R := RowDimension(A); C := ColumnDimension(A);
        left := SubMatrix(A, [1 .. R], [1 .. n-1]);
        right := SubMatrix(A, [1 .. R], [n .. C]);
        return ArrayTools:-Concatenate(2, left, v, right)
    end proc:
    
    # test cases:
    
    M := Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]):
    v := Vector[row]([2, 2, 2]):
    v2 := Vector[column]([2, 2, 2]):
    seq(InsertRow(M, i, v), i = 1 .. 4);
    seq(InsertColumn(M, i, v2), i = 1 .. 4);

    We then reworked this problem using some handy indexing and construction notation that allows our previous procedures to save on the variable constructions and syntax:

    InsertRow := proc( A :: rtable, V :: Vector[row], r :: posint )
        return < A[1..r-1,..]; V; A[r..-1,..] >:
    end proc:
    
    InsertColumn := proc( A :: rtable, V :: Vector[column], c :: posint )
        return < A[..,1..c-1] | V | A[..,c..-1] >:
    end proc:
    
    M := Matrix(3, 3, [seq(i, i = 1 .. 9)]);
    A := convert(M, Array);
    U := Vector[row]( [ a, b, c ] );
    V := convert( U, 'Vector[column]' );
    seq(InsertRow( A, U, i ), i=1..4);
    seq(InsertColumn( A, V, i ), i=1..4);
    seq(InsertRow( M, U, i ), i=1..4);
    seq(InsertColumn( M, V, i ), i=1..4);

    In order to explore the use of signal processing package in image processing, @Samir Khan and I created this application that performs discrete Haar wavelet transform on images to achieve both lossy (irreversible) and lossless (reversible) compression.

    Haar wavelet compression modifies the image matrix to increase the number of zero entries in the matrix, which results in a sparse matrix that can be stored more efficiently, thus reducing the file size. A Haar wavelet transform groups adjacent items in the matrix, stores the average and difference of the two numbers. Notice that this computation is reversible since knowing the values of a, b and given that x1-x2 = a; (x1+x2)/2 = b, we can solve for x1 and x2. Haar wavelet compression is taking advantage of the property that neighboring pixels in an image usually share very similar value; hence recursively applying Haar wavelet transform to the rows and columns of an image matrix significantly increases the number of zero entries. In the application we achieved a compression ratio of 1.296 (number of non-zero entries in original: number of non-zero entries in processed matrix).

    The fact that Haar wavelet transform is reversible means that we can use it to perform lossless image compression: the decompressed image will be exactly the same as the image before compression. Transmission and temporary storage of the data would be made more efficient without any loss of details in the image.

    But what if the file size is still too big or we simply don’t need that many details in the image? That is when lossy compression comes into use. By omitting some details/fidelity, lossy compression is able to achieve notably smaller file size. In this application, we apply a filter to the transformed image matrix, setting entries that are close to zeros to actual zeros (i.e. pick a positive number ϵ such that all x < ϵ are changed to 0 in the matrix). The value of ϵ directly impacts the quality of the decompressed image so should be chosen carefully in practice. In this application, we chose ϵ = 0.01, which results in a compression ratio of 19.38, but instead produces a very blurry image after reversing the compression.

    (left: Original image, right: lossy compression with ϵ = 0.01)

    The application can be accessed here for more details.


     

    For Maple 2018.1, there are improvements in pdsolve's ability to solve PDE with boundary and initial conditions. This is work done together with E.S. Cheb-Terrab. The improvements include an extended ability to solve problems involving non-homogeneous PDE and/or non-homogeneous boundary and initial conditions, as well as improved simplification of solutions and better handling of functions such as piecewise in the arguments and in the processing of solutions. This is also an ongoing project, with updates being distributed regularly within the Physics Updates.

    Solving more problems involving non-homogeneous PDE and/or non-homogeneous boundary and initial conditions

     

     

    Example 1: Pinchover and Rubinstein's exercise 6.17: we have a non-homogenous PDE and boundary and initial conditions that are also non-homogeneous:

    pde__1 := diff(u(x, t), t)-(diff(u(x, t), x, x)) = 1+x*cos(t)
    iv__1 := (D[1](u))(0, t) = sin(t), (D[1](u))(1, t) = sin(t), u(x, 0) = 1+cos(2*Pi*x)

    pdsolve([pde__1, iv__1])

    u(x, t) = 1+cos(2*Pi*x)*exp(-4*Pi^2*t)+t+x*sin(t)

    (1)

    How we solve the problem, step by step:

       

     

    Example 2: the PDE is homogeneous but the boundary conditions are not. We solve the problem through the same process, which means we end up solving a nonhomogeneous pde with homogeneous BC as an intermediate step:

    pde__2 := diff(u(x, t), t) = 13*(diff(u(x, t), x, x))
    iv__2 := (D[1](u))(0, t) = 0, (D[1](u))(1, t) = 1, u(x, 0) = (1/2)*x^2+x

    pdsolve([pde__2, iv__2])

    u(x, t) = 1/2+Sum(2*(-1+(-1)^n)*cos(n*Pi*x)*exp(-13*Pi^2*n^2*t)/(Pi^2*n^2), n = 1 .. infinity)+13*t+(1/2)*x^2

    (12)

    How we solve the problem, step by step:

       

     

    Example 3: a wave PDE with a source that does not depend on time:

    pde__3 := (diff(u(x, t), x, x))*a^2+1 = diff(u(x, t), t, t)
    iv__3 := u(0, t) = 0, u(L, t) = 0, u(x, 0) = f(x), (D[2](u))(x, 0) = g(x)

    `assuming`([pdsolve([pde__3, iv__3])], [L > 0])

    u(x, t) = (1/2)*(2*(Sum(sin(n*Pi*x/L)*(2*L*(Int(sin(n*Pi*x/L)*g(x), x = 0 .. L))*sin(a*Pi*t*n/L)*a-Pi*(Int(sin(n*Pi*x/L)*(-2*f(x)*a^2+L*x-x^2), x = 0 .. L))*cos(a*Pi*t*n/L)*n)/(Pi*n*a^2*L), n = 1 .. infinity))*a^2+L*x-x^2)/a^2

    (23)

    How we solve the problem, step by step:

       

     

    Example 4: Pinchover and Rubinstein's exercise 6.23 - we have a non-homogenous PDE and initial condition:

    pde__4 := diff(u(x, t), t)-(diff(u(x, t), x, x)) = g(x, t)
    iv__4 := (D[1](u))(0, t) = 0, (D[1](u))(1, t) = 0, u(x, 0) = f(x)

    pdsolve([pde__4, iv__4], u(x, t))

    u(x, t) = Int(f(tau1), tau1 = 0 .. 1)+Sum(2*(Int(f(tau1)*cos(n*Pi*tau1), tau1 = 0 .. 1))*cos(n*Pi*x)*exp(-Pi^2*n^2*t), n = 1 .. infinity)+Int(Int(g(x, tau1), x = 0 .. 1)+Sum(2*(Int(g(x, tau1)*cos(n1*Pi*x), x = 0 .. 1))*cos(n1*Pi*x)*exp(-Pi^2*n1^2*(t-tau1)), n1 = 1 .. infinity), tau1 = 0 .. t)

    (30)

    If we now make the functions f and g into specific mappings, we can compare pdsolve's solutions to the general and specific problems:

    f := proc (x) options operator, arrow; 3*cos(42*x*Pi) end proc
    g := proc (x, t) options operator, arrow; exp(3*t)*cos(17*x*Pi) end proc

     

    Here is what pdsolve's solution to the general problem looks like when taking into account the new values of f(x) and g(x,t):

    value(simplify(evalindets(u(x, t) = Int(f(tau1), tau1 = 0 .. 1)+Sum(2*(Int(f(tau1)*cos(n*Pi*tau1), tau1 = 0 .. 1))*cos(n*Pi*x)*exp(-Pi^2*n^2*t), n = 1 .. infinity)+Int(Int(g(x, tau1), x = 0 .. 1)+Sum(2*(Int(g(x, tau1)*cos(n1*Pi*x), x = 0 .. 1))*cos(n1*Pi*x)*exp(-Pi^2*n1^2*(t-tau1)), n1 = 1 .. infinity), tau1 = 0 .. t), specfunc(Int), proc (u) options operator, arrow; `PDEtools/int`(op(u), AllSolutions) end proc)))

    u(x, t) = 3*cos(42*Pi*x)*exp(-1764*Pi^2*t)+cos(Pi*x)*(65536*cos(Pi*x)^16-278528*cos(Pi*x)^14+487424*cos(Pi*x)^12-452608*cos(Pi*x)^10+239360*cos(Pi*x)^8-71808*cos(Pi*x)^6+11424*cos(Pi*x)^4-816*cos(Pi*x)^2+17)*(exp(289*Pi^2*t+3*t)-1)*exp(-289*Pi^2*t)/(289*Pi^2+3)

    (31)

     

    Here is pdsolve's solution to the specific problem:

    pdsolve([pde__4, iv__4], u(x, t))

    u(x, t) = ((867*Pi^2+9)*cos(42*Pi*x)*exp(-1764*Pi^2*t)+cos(17*Pi*x)*(exp(3*t)-exp(-289*Pi^2*t)))/(289*Pi^2+3)

    (32)

     

    And the two solutions are equal:

    simplify(combine((u(x, t) = 3*cos(42*x*Pi)*exp(-1764*Pi^2*t)+cos(x*Pi)*(65536*cos(x*Pi)^16-278528*cos(x*Pi)^14+487424*cos(x*Pi)^12-452608*cos(x*Pi)^10+239360*cos(x*Pi)^8-71808*cos(x*Pi)^6+11424*cos(x*Pi)^4-816*cos(x*Pi)^2+17)*(exp(289*Pi^2*t+3*t)-1)*exp(-289*Pi^2*t)/(289*Pi^2+3))-(u(x, t) = ((867*Pi^2+9)*cos(42*x*Pi)*exp(-1764*Pi^2*t)+cos(17*x*Pi)*(exp(3*t)-exp(-289*Pi^2*t)))/(289*Pi^2+3)), trig))

    0 = 0

    (33)

    f := 'f'; g := 'g'

     

    Improved simplification in integrals, piecewise functions, and sums in the solutions returned by pdsolve

     

     

    Example 1: exercise 6.21 from Pinchover and Rubinstein is a non-homogeneous heat problem. Its solution used to include unevaluated integrals and sums, but is now returned in a significantly simpler format.

    pde__5 := diff(u(x, t), t)-(diff(u(x, t), x, x)) = t*cos(2001*x)
    iv__5 := (D[1](u))(0, t) = 0, (D[1](u))(Pi, t) = 0, u(x, 0) = Pi*cos(2*x)

    pdsolve([pde__5, iv__5])

    u(x, t) = (1/16032024008001)*(4004001*t+exp(-4004001*t)-1)*cos(2001*x)+Pi*cos(2*x)*exp(-4*t)

    (34)

    pdetest(%, [pde__5, iv__5])

    [0, 0, 0, 0]

    (35)

     

    Example 2: example 6.46 from Pinchover and Rubinstein is a non-homogeneous heat equation with non-homogeneous boundary and initial conditions. Its solution used to involve two separate sums with unevaluated integrals, but is now returned with only one sum and unevaluated integral.

    pde__6 := diff(u(x, t), t)-(diff(u(x, t), x, x)) = exp(-t)*sin(3*x)
    iv__6 := u(0, t) = 0, u(Pi, t) = 1, u(x, 0) = phi(x)

    pdsolve([pde__6, iv__6], u(x, t))

    u(x, t) = (1/8)*(8*(Sum(2*(Int(-(-phi(x)*Pi+x)*sin(n*x), x = 0 .. Pi))*sin(n*x)*exp(-n^2*t)/Pi^2, n = 1 .. infinity))*Pi-Pi*(exp(-9*t)-exp(-t))*sin(3*x)+8*x)/Pi

    (36)

    pdetest(%, [pde__6, iv__6])

    [0, 0, 0, (-phi(x)*Pi^2+Pi*x+2*(Sum((Int(-(-phi(x)*Pi+x)*sin(n*x), x = 0 .. Pi))*sin(n*x), n = 1 .. infinity)))/Pi^2]

    (37)

     

    More accuracy when returning series solutions that have exceptions for certain values of the summation index or a parameter

     

     

    Example 1: the answer to this problem was previously given with n = 0 .. infinity instead of n = 1 .. infinity as it should be:

    pde__7 := diff(v(x, t), t, t)-(diff(v(x, t), x, x))

    iv__7 := v(0, t) = 0, v(x, 0) = -(exp(2)*x-exp(x+1)-x+exp(1-x))/(exp(2)-1), (D[2](v))(x, 0) = 1+(exp(2)*x-exp(x+1)-x+exp(1-x))/(exp(2)-1), v(1, t) = 0

    pdsolve([pde__7, iv__7])

    v(x, t) = Sum(-2*sin(n*Pi*x)*((Pi^2*(-1)^n*n^2-Pi^2*n^2+2*(-1)^n-1)*sin(Pi*t*n)-(-1)^n*cos(Pi*t*n)*Pi*n)/(Pi^2*n^2*(Pi^2*n^2+1)), n = 1 .. infinity)

    (38)

     

    Example 2: the answer to exercise 6.25 from Pinchover and Rubinstein is now given in a much simpler format, with the special limit case for w = 0 calculated separately:

    pde__8 := diff(u(x, t), t) = k*(diff(u(x, t), x, x))+cos(w*t)
    iv__8 := (D[1](u))(L, t) = 0, (D[1](u))(0, t) = 0, u(x, 0) = x

    `assuming`([pdsolve([pde__8, iv__8], u(x, t))], [L > 0])

    u(x, t) = piecewise(w = 0, (1/2)*L+Sum(2*L*(-1+(-1)^n)*cos(n*Pi*x/L)*exp(-Pi^2*n^2*k*t/L^2)/(n^2*Pi^2), n = 1 .. infinity)+t, (1/2)*(L*w+2*(Sum(2*L*(-1+(-1)^n)*cos(n*Pi*x/L)*exp(-Pi^2*n^2*k*t/L^2)/(n^2*Pi^2), n = 1 .. infinity))*w+2*sin(w*t))/w)

    (39)

     

    Improved handling of piecewise, eval/diff in the given problem

     

     

    Example 1: this problem, which contains a piecewise function in the initial condition, can now be solved:

    pde__9 := diff(f(x, t), t) = diff(f(x, t), x, x)
    iv__9 := f(0, t) = 0, f(1, t) = 1, f(x, 0) = piecewise(x = 0, 0, 1)

    pdsolve([pde__9, iv__9])

    f(x, t) = Sum(2*sin(n*Pi*x)*exp(-Pi^2*n^2*t)/(n*Pi), n = 1 .. infinity)+x

    (40)

     

    Example 2: this problem, which contains a derivative written using eval/diff, can now be solved:

    pde__10 := -(diff(u(x, t), t, t))-(diff(u(x, t), x, x))+u(x, t) = 2*exp(-t)*(x-(1/2)*x^2+(1/2)*t-1)

    iv__10 := u(x, 0) = x^2-2*x, u(x, 1) = u(x, 1/2)+((1/2)*x^2-x)*exp(-1)-((3/4)*x^2-(3/2)*x)*exp(-1/2), u(0, t) = 0, eval(diff(u(x, t), x), {x = 1}) = 0

    pdsolve([pde__10, iv__10], u(x, t))

    u(x, t) = -(1/2)*exp(-t)*x*(x-2)*(t-2)

    (41)

     

    References:

     

    Pinchover, Y. and Rubinstein, J.. An Introduction to Partial Differential Equations. Cambridge UP, 2005.


     

    Download What_is_New_after_Maple_2018.mw

    Katherina von Bülow

    In an attempt to explore the field of image processing, @Samir Khan and I created an application (download here) that demonstrates the removal of two types of noises from an image through frequency and spatial filtering.

    Periodic noises and salt & pepper noises are two common types of image noises, usually caused by errors during the image capturing or data transmission process. Periodic noises result in repetitive patterns being added onto the original image, while salt & pepper noises are the irregular appearance of dark pixels in the bright area and bright pixels in the dark area of the image. In this application, we artificially generate these noises and pollute a clean picture in order to demonstrate the removal techniques.

    (Fig 1: Picture of Waterloo Office taken by Sophie Tan            Fig 2: Converted to greyscale for processing, added two noises)

    In order to remove periodic noises from the image, we apply a 2D Fourier Transform to convert the image from spatial domain to frequency domain, where periodic noises can be visually detected as separate, discrete spikes and therefore easily removed.

    (Fig 3 Frequency domain of the magnitude of the image)

    One way to remove salt and pepper noises is to apply a median filter to the image. In this application, we run a 3 by 3 kernel across the image matrix that sorts and places the median among the 9 elements as the new matrix entry, thus resulting in the whole image being median-filtered.

    Comparison of the image before and after noise removal:

    Please refer to the application for more details on the implementation of the two removal techniques.

     

    We have released an update to Maple, Maple 2018.1. This release provides enhancements to the mathematical computation engine, including physics and DEs.  It also provides substantial improvements to the command line version, easier access to group management tools in the MapleCloud, and a few other interface improvements.

    This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2018.1 download page, where you can also find more details.

    Hello, everyone! My name’s Sophie and I’m an intern at Maplesoft. @Samir Khan asked me to develop a couple of demonstration applications using the DeepLearning package - my work is featured on the Application Center

    I thought I’d describe two critical commands used in the applications – DNNClassifier() and DNNRegressor().

    The DNNClassifier calls tf.estimator.DNNClassifier from the Tensorflow Python API. This command builds a feedforward multilayer neural network that is trained with a set of labeled data in order to perform classification on similar, unlabeled data.

    Dataset used for training and validating the classifier has the type DataFrame in Maple. In the Prediction of malignant/benign of breast mass example, the training set is a DataFrame with 32 columns in total, with column labels: “ID Number”, “Diagnosis”, “radius”, “texture”, etc. Note that labeling the columns of the dataset is mandatory, as later the neural network needs to identify which feature column corresponds to which list of values.

    Feature columns are what come between the raw input data and the classifier model; they are required by Tensorflow to specify how the input data should be transformed before given to the model. Maple now supports three types of Feature Columns, including:

    • NumericColumn that represents real, numerical figure,
    • CategoricalColumn that denotes categorical(ordinal) data
    • BucketizedColumn that organizes continuous data into a discrete number buckets with specified boundaries.

    In this application, the input data consists of 30 real, numeric values that represents physical traits of a cell nucleus computed from a digitized image of the breast mass. We create a list of NumericColumns by calling

    with(DeepLearning):
    fc := [seq(NumericColumn(u,shape=[1]), u in cols[3..])]:

    where cols is a list of column labels and shape[1] indicates that each data input is just a single numeric value.

    When we create a DNNClassifier, we need to specify the feature columns (input layer), the architecture of the neural network (hidden layers) and the number of classes (output layer). Recall that the DNNClassifier builds a feedforward multilayer neural network, hence when we call the function, we need to indicate how many hidden layers we want and how many nodes there should be on each of the layer. This is done by passing a list of non-negative integers as the parameter hidden_units when we call the function. In the example, we did:

    classifier := DNNClassifier(fc, hidden_units=[20,40,20],num_classes=2):

    where we set 3 hidden layer each with 20, 40, 20 nodes respectively. In addition, there are 30 input nodes (i.e. the number of feature columns) and 1 output node (i.e. binary classification). The diagram below illustrates a simpler example with an input layer with 3 nodes, 2 hidden layers with 7, 5 nodes and an output layer with 1 node.

    (Created using NN-SVG by https://github.com/zfrenchee/NN-SVG)

    After we built the model, we can train it by calling

    classifier:-Train(train_data[3..32], train_data[2], steps = 256, num_epochs = 3, shuffle = true):

    where we

    1. Give the training data (train_data[3..32]) and the corresponding labels (train_data[2]) to the model.
    2. Specified that the entire dataset will be passed to the model for three times and each iteration has 256 steps.
    3. Specified that data batches for training will be created by randomly shuffling the tensors.

    Now the training process is complete, we can use the validation set to evaluate the effectiveness of our model.

    classifier:-Evaluate(test_data[3..32],test_data[2], steps = 32);

    The output indicates an accuracy of ~92.11% in this case. There are more indices like accuracy_basline, auc, average_loss that help us decide if we need to modify the architecture for better performance.

    We then build a predictor function that takes an arbitrary set of measurements as a DataSeries and returns a prediction generated by the trained DNN classifier.

    predictor := proc (ds) classifier:-Predict(Transpose(DataFrame(ds)), num_epochs = 1, shuffle = false)[1] end proc;

    Now we can pass a DataSeries with 30 labeled rows to the predictor: (Recall the cols is a list of the column names)

    ds := DataSeries([11.49, 14.59, 73.99, 404.9, 0.1046, 8.23E-02, 5.31E-02, 1.97E-02, 0.1779, 6.57E-02, 0.2034, 1.166, 1.567, 14.34, 4.96E-03, 2.11E-02, 4.16E-02, 8.04E-03, 1.84E-02, 3.61E-03, 12.4, 21.9, 82.04, 467.6, 0.1352, 0.201, 0.2596, 7.43E-02, 0.2941, 9.18E-02], labels = cols[3..]); 
    predictor(ds);
    

    The output indicates that the probability of this data being a class _id [0] is ~90.79%. In other words, according to our model, the probability of this breast mass cell being benign is ~90.79%.

    The use of the DNNRegressor is very similar (almost identical) to that of the Classifier, the only significant difference is that while the Classifier predicts discrete labels as classes, the Regressor predicts a continuous qualitative result with the provided data (Note that CategoricalColumn is still applicable). For more details about the basic usage of the DNNRegressor, please refer to Predicting the burnt area of a forest fires with DNN Regressor.

     

    Mukhametshina Liya

    Games with pseudo-fractals
     

    Homothety_Fractals.mw

     

      

      

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

    _ANIMATED_PICTURE_ON_THE_COORDINATE_PLANE_Aleksandrov_D..mw

     

    We have just released a new version of MapleSim.  The MapleSim 2018 family of products offers new tools for developing digital twins, greater connectivity with other modeling tools, and expanded modeling scope. Improvements include:

    • New tools for creating motion profiles
    • FMI  import for FMI 2.0 Fixed-Step Co-Simulation
    • Optimized handling of large models
    • Inclusion of temperature effects in the MapleSim Hydraulics Library from Modelon and MapleSim Pneumatics Library from Modelon
    • Heat transfer through air and water with the MapleSim Heat Transfer Library from CYBERNET

    See What’s New in MapleSim 2018 for more information about these and other improvements.

    Who should be considered an 'expert'? How does one achieve expert status? In this guest MaplePrimes blog post, 'Understanding Maple' author Ian Thompson discusses his view of what makes an expert, his journey of becoming an expert in Maple, and the process of putting together and perfecting this resource for Maple users.

     

    In days of 8-bit computers, one would sometimes encounter individuals who knew everything about a particular device or piece of software. Single programmers wrote entire applications or games, and some could debug their work by looking directly at a core dump (a printout of the numbers stored in the computer’s memory). Some even managed to take computers beyond their specifications by exploiting design loopholes that the manufacturers hadn’t foreseen or intended. It would be fair to classify such individuals as ‘experts’.

    Fast forward twenty five years, and the picture is far less clear. The complexity of computers and software has grown to such an extent that even relatively small smartphone applications are created by teams of developers, and nobody understands every aspect of a CPU chip, much less an entire PC or tablet. Who now should be classified as an expert? One possibility is that an expert is a person who may sometimes need to look up the details of a rarely used command or feature, but who is never confused or frustrated by the behavior of the system or software in question (except where there is a bug), and never needs help from anyone, except perhaps on rare occasions from its creators.

    This rather stringent definition makes me an expert in only two areas of computing: the Fortran programming language, and the mathematical computation system Maple. An argument could be made for the typesetting system LATEX, but whilst this has a large number of expert users, there is also a much smaller group of more exalted experts, who maintain the system and develop new packages and extensions. It would be fair to say that I fall into the first category, but not the second.*

    How does one achieve expert status? Some software actively prevents this, by hiding its workings to such an extent that fully understanding its behavior is impossible. Where it is possible to gain expert status, I have experienced two very different routes, both starting during my time as a research student, when it became clear that Fortran and Maple would be useful in my work. There were several parallels. I knew a little about both, having used them for basic tasks as an undergraduate. However, working out why things went wrong and how to fix them was time-consuming and unrewarding, since it often relied on magic recipes obtained from unreliable sources, and in many cases I didn’t really understand why these worked, any more than I understood why my own attempts had not. I realized then that knowing a little was at the root of these problems. Partial knowledge, supplemented by contradictory, outdated and even downright bad advice from websites and well-meaning individuals (some of whom invariably labor under false pretences of their own expert status) is not an efficient way to approach scientific computing. In fact it’s just a recipe for frustration. In the case of Fortran, fixing this turned out to be easy, because there are lots of good books on the subject. Reading one of these eliminated all of my problems with the language at a stroke. I can’t claim that I remembered every command and its syntax, nor do I know them all now. This is hardly surprising — the Fortran Language Standard (a very terse document that sets out everything the language provides) now extends to more than 600 pages. Instead, the book provided a general picture of how things work in Fortran, and showed the right way to go about tackling a problem. This investment in time has since paid itself back hundreds of times over.

    The route to expert status in Maple was far more challenging. Its own help pages give a very comprehensive description of individual commands, but they are intended as a reference guide, and if it’s possible to become an expert using these alone, then I never discovered the correct order in which to read them. I found a number of books on Maple in the university library, but most were too basic to be useful, and others focused on particular applications. None seemed likely to give me the general picture — the feel for how things work — that would make Maple into the time-saving resource it was intended to be.

    The picture became clearer after I taught Maple to students in three different courses. Nothing encourages learning better than the necessity to teach someone else! Investigating the problems that students experienced gave me new opportunities to properly understand Maple, and eventually the few remaining gaps were filled in by the Programming Guide. This is a complex document, similar in length to the Fortran Language Standard, but with more examples. Personally I would only recommend it to readers with experience of programming language specifications. Students now started to ask how I came to know so much about Maple, and whether there was a book that would teach them the same. Since no such book existed, I decided to write one myself. As the old adage goes, if you want something doing properly, do it yourself. The project soon began to evolve as I tried to set down everything that the majority of Maple users need to know. I’ve always hated books that skirt around important but difficult topics, so where before I might have used a dirty trick to circumnavigate a problem, now I felt compelled to research exactly what was going on, and to try to explain it in a simple, concise way. When the first draft was complete, I approached Cambridge University Press (CUP). The editor arranged for reviews by four anonymous referees**, and by Maplesoft’s own programming team. This led to several major improvements. My colleague, Dr Martyn Hughes, also deserves a mention for his efforts in reading and commenting on four different drafts. Meanwhile, Maplesoft continued to release new editions of their software, and the drafts had to be revised to keep up with these. The cover was created by one of CUP’s designers, with instructions that it should not look too ‘treeish’ — one might be surprised by the number of books that have been written about Maple syrup, and it would be a shame for Understanding Maple to be mixed up with these by potential readers browsing the internet. Then there were the minor details: how wide should the pages be? What font should be used? Should disk be spelled with a ‘c’ or a ‘k’? Could quotes from other sources be used without the threat of legal action over copyright infringement? One rights holder laughably tried to charge $200 for a fragment of text from one of their books. Needless to say, no greenbacks were forthcoming.

    The resulting book is concise, with all the key concepts needed to gain an understanding of Maple, alongside numerous examples, packed into a mere 228 pages. It gives new users a solid introduction, and doesn’t avoid difficult topics. It isn’t perfect (in fact I have already started to list revisions that will be made if a second edition is published in the future) but I’ve seen very few problems that can’t be solved with the material it contains. Only time will tell if Understanding Maple will it create new experts. At the very least, I would certainly like to think it will make Maple far easier to grasp, and help new users to avoid some of the traps that caught me out many years ago.

     

    Learn more about Understanding Maple, which is published by Cambridge University Press.

    There are many questions that complain about Latex conversion in Maple.

    I'd like to again request that Maplesoft improves Latex output of its expressions. If Maple can just fix how it generates fractions, that will good enough for now.

    I am willing to send Maplesoft a personal check of the amount of one month salary for one of your developers to do this fix if you are willing to do it. It should not take more than one month to do this simple fix in your code. It might even take one day if someone knows the code.

    The problem comes when there is a fraction in the expression. the Latex output instead of using proper latex code using "\frac{}{}", it instead uses "/" which makes the output terrible.

    Another case, where Maple generate (expression)^{-1} instead of \frac{1}{expression}.

    It can't be that hard to fix these 2 issues, which can go a long way towards making the latex generated by Maple much better. Here is an example

    eq:=-(1/2)*1/y = (1/3)*x^3+z:
    sol:=solve(eq,y);

    latex(sol);
    -3/2\, \left( {x}^{3}+3\,z \right) ^{-1}

    Which renders as

    Which is terrible. The screen output is much better.

    Compare this to Mathematica

    eq = -(1/2)*(1/y) == (1/3)*x^3 + z;
    sol = y /. First@Solve[eq, y];
    TeXForm[sol]
    
       -\frac{3}{2 \left(x^3+3 z\right)}

    Which renders in Latex as

    If Maplesoft does not think Latex is improtant, then they are completely wrong. CAS support in Latex is very important. Ignoring Latex means you will lose customers who want good Latex support of the math output of Maple. After all, Math and Latex go togother. And Maple is supposed to be all about Mathematics.

    Any chance of Maplesoft taking some time to fix these issues in Latex? Maple has not had any improvement in Latex for years and years. I keep buying Maple each year, and nothing changes in its Latex export.

    thank you

    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

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