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  •   Elena, Liya

      "Researching turkish song: the selection of the main element and its graphic transformations",

       Russia, Kazan, school #57

    The setting and visualization of the melodic line of the song
    > restart:
    > with(plots):with(plottools):
    > p0:=plot([[0.5,9],[1,7],[2,9],[4,11],[6,9],[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9],[17,7],[18,9]],color=magenta):p1:=plot([[18,9],[20,11],[22,9],[23,11],[24,9],[26,11],[28,11],[29.5,8],[30,11],[32,9],[33.5,8],[34,9],[36,7],[37.5,5],[38,9],[40,7],[42,5],[44,5],[46,4],[47,5],[48,2],[50,4],[51,5],[51.5,4],[52,2],[54,4],[56,4],[56.5,5],[57,4],[58,5],[60,7],[62,5],[64,7],[66,5]],color=cyan):
    > p2:=plot([[66,5],[68,5],[69,5],[70,4],[71,5],[71.5,4],[72,2],[73,4],[74,5],[75,7],[76,5],[78,4],[78.5,7],[80,5],[82.5,4],[83.5,4],[84,2],[86,4],[88,4],[90.5,4],[91.5,4]],color=red):
    > p3:=plot([[91.5,4],[92,2],[94,4],[96,4],[96.5,9],[97,7],[98,9],[100,11],[100.5,9],[101,11],[102,9],[104,11],[106,9],[108,9],[109,9],[109.5,9],[110,7],[111,9],[112,7],[113,7],[114,9],[116,11],[116.5,9],[117,11],[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9],[126,11],[128,9],[129,7],[130,9],[132,11],[132.5,9],[133,11],[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9],[142,11],[143,7],[143.5,7],[144,9],[144.5,9],[145,7],[146,9],[148,11],[148.5,9],[149,11],[150,9],[151.5,11],[152,9],[154.5,9],[156,9],[156.5,9],[157,11],[157.5,9],[158,11],[160,9],[161,7],[162,9],[164,11],[164.5,9],[165,11],[166,9],[168,11],[168.5,9],[171.5,9],[172,9],[172.5,9],[173.5,11],[174,9],[174.5,11],[175,7],[175.5,7],[176,9],[176.5,9],[177,7],[178,9],[180,11],[180.5,9],[181,11],[182,9],[183.5,11],[184,9],[186.5,9],[188,9],[188.5,9],[189,11],[189.5,9],[190,11],[192,9],[192.5,9],[193,7],[194,9],[196,11],[196.5,9],[197,11],[198,9],[200,11],[201.5,9],[202,11],[203,9],[203.5,8],[204,9],[205,7],[205.5,9],[206,11],[207,9],[208,7],[209,8],[209.5,7],[210,9],[211,7],[212,5],[213,5],[213.5,5],[214,9],[215,7],[216,5],[217,5],[217.5,5],[218,7],[219,5],[220,4],[221,4],[221.5,4],[222,7],[223,5],[224,4],[225,4],[227,4],[227.5,4],[228,2],[230,4]],color=blue):
    > p4:=plot([[230,4],[232,4],[232.5,5],[233,4],[234,5],[236,7],[236.5,5],[237,5],[238,9],[240,7],[242.5,5],[244,5],[245,5],[246,4],[246.5,5],[247,4],[248,2],[250,4],[250.5,7],[251,5],[252,4],[254,4],[254.5,7],[255,5],[256,4],[258,4]],color=brown):
    > p5:=plot([[258,4],[259,4],[260,2]],color=green):
    > plots[display](p0,p1,p2,p3,p4,p5,thickness=2);



    The selection of the main melodic element in graph of whole song. The whole song is divided into separate elements - results of transformationss0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > s1:=plot([[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9]],color=blue):
    > s2:=plot([[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9]],color=blue):
    > s3:=plot([[150,9],[151.5,11],[152,9],[154.5,9],[156,9],[156.5,9],[157,11],[157.5,9]],color=blue):
    > s4:=plot([[166,9],[168,11],[168.5,9],[171.5,9],[172,9],[172.5,9],[173.5,11],[174,9]],color=blue):
    > s5:=plot([[182,9],[183.5,11],[184,9],[186.5,9],[188,9],[188.5,9],[189,11],[189.5,9]],color=blue):
    > s6:=plot([[250,4],[250.5,7],[251,5],[252,4],[254,4],[254.5,7],[255,5],[256,4]],color=blue):
    > plots[display](s0,s1,s2,s3,s4,s5,s6);
    > s:=plots[display](s0,s1,s2,s3,s4,s5,s6):


    Animated display of grafical transformation of the basic element (to click on the picture - on the panel of instruments appears player - to play may step by step).m0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > pm:=plot([[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9]],color=red,style=line,thickness=4):
    > iop:=plots[display](m0,pm,insequence=true):
    > plots[display](iop,s0);

    > m0_t:=translate(m0,110,0):
    > m0_r:=reflect(m0_t,[[0,9],[24,9]]):
    > plots[display](m0,m0_r,insequence=true);
    > m0r:=plots[display](m0,m0_r,insequence=true):

    > pm0:=plots[display](pm,m0):
    > plots[display](pm0,m0r);

    > m0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > pn:=plot([[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9]],color=blue,thickness=3):
    > iop:=plots[display](m0,pn,insequence=true):
    > plots[display](iop,s0);

    > m0_t1:=translate(m0,126,0):
    > m0_r1:=reflect(m0_t1,[[0,9],[24,9]]):
    > plots[display](m0,m0_r1,insequence=true);
    > m0r1:=plots[display](m0,m0_r1,insequence=true):

    > pm01:=plots[display](pn,m0):
    > plots[display](pm01,m0r1);


    > pm2:=plots[display](pn,pm,m0):
    > plots[display](pm0,m0r,pm01,m0r1);

    > pt_i_1:=seq(translate(pm,5*11*i,0),i=0..4):
    > plots[display](pt_i_1);

    > pm_i:=seq(translate(pm,5*11*i,0),i=0..4):
    > plots[display](pm_i);
    > iop1:=plots[display](pm_i,insequence=true):
    > plots[display](iop1,s0);


    > pm_i_0:=seq(translate(m0_r,5*11*i,0),i=0..4):
    > plots[display](pm_i_0);
    > iop2:=plots[display](pm_i_0,insequence=true):
    > plots[display](iop2,s0);







    Construction of arabesques of melodic line BACH

    Elena, Liya "Construction of arabesques of melodic line BACH", Kazan, Russia, school#57
    > restart:
    > with(plots):with(plottools):

          The setting and visualization of line BACH: B - note b-flat, A - note la, C - note do, H - note si.
    > p0:=plot([[0,1],[2,0],[4,1.5],[6,1]],thickness=4,color=cyan,scaling=constrained);
    >   p0 := PLOT(
    >         CURVES([[0, 1.], [2., 0], [4., 1.500000000000000], [6., 1.]])
    >         COLOUR(RGB, 0, 1.00000000, 1.00000000),
    >         VIEW(DEFAULT, DEFAULT))
    > plots[display](p0);
    > r_i:=seq(rotate(p0,i*Pi/4),i=1..8):
    > p1:=display(r_i,p0):plots[display](p1,scaling=constrained);

    > c1:=circle([0,0],6,color=blue,thickness=2):
    > plots[display](c1,p1,scaling=constrained);
    > p_c:=plots[display](c1,p1,scaling=constrained):

    > pt_i_2:=seq(translate(p1,0,2*6*i),i=0..4):
    > plots[display](pt_i_2,scaling=constrained);
    > pt_i_22:=seq(translate(p1,0,6*i),i=0..4):
    > plots[display](pt_i_22,scaling=constrained);
    > pt_i_222:=seq(translate(p1,0,1/2*6*i),i=0..4):
    > plots[display](pt_i_222,scaling=constrained);

    > pr:=rotate(p1,Pi/8):
    > plots[display](pr,scaling=constrained);
    > plots[display](p1,pr,scaling=constrained);
    > pr_i:=seq(rotate(p1,Pi/16*i),i=0..8):
    > plots[display](pr_i,scaling=constrained);

    > pt_1:=translate(p1,0,2*6):
    > pr_1_i:=seq(rotate(pt_1,Pi/3.5*i),i=0..6):
    > plots[display](pr_1_i,scaling=constrained);
    > pr_11_i:=seq(rotate(pt_1,Pi/5*i),i=0..10):
    > plots[display](pr_11_i,scaling=constrained);
    > pr_111_i:=seq(rotate(pt_1,Pi/6.5*i),i=0..12):
    > plots[display](pr_111_i,scaling=constrained);

    Elena, Liya "Designing of islamic arabesques", Kazan, Russia, school #57

    > restart:
          At the theorem of cosines  c^2 = a^2+b^2-2*a*b*cos(phi);
          In our case  c=a0 ,  a=1 ,  a=b , phi; - acute angle of a rhombus (the tip of the kalam).
          s0 calculated at theorem of  Pythagoras.
         (а0 - horizontal diagonal of a  rhombus, s0 - vertical diagonal of a  rhombus)
    > a:=1:phi:=Pi/4:
    > a0:=sqrt(a^2+a^2-2*a^2*cos(phi));

                           a0 := sqrt(2 - sqrt(2))

    > solve((s0^2)/4=a^2-(a0^2)/4,s0);

                    sqrt(2 + sqrt(2)), -sqrt(2 + sqrt(2))

          The setting of initial parameters : the size of the tip of the pen-kalam and  depending on its - the main module size - point
           (а0 - horizontal diagonal of a  rhombus, s0 - vertical diagonal of a  rhombus)
    > a0:=sqrt(2-sqrt(2)):
    > s0:=sqrt(2+sqrt(2)):
          Connection the graphical libraries Maple
    > with(plots):with(plottools):
          Construction of unit of measure (point) - rhombus - the tip of the kalam
    > p0:=plot([[0,0],[a0/2,s0/2],[0,s0],[-a0/2,s0/2],[0,0]],scaling=constrained,color=gold,thickness=3):
    > plots[display](p0);

    The setting and construction of altitude of alif - the basis of the rules compilation of the proportions      Example, on style naskh altitude of alif amount five points
    > p_i:=seq(plot([[0,0+s0*i],[a0/2,s0/2+s0*i],[0,s0+s0*i],[-a0/2,s0/2+s0*i],[0,0+s0*i]],scaling=constrained,color=black),i=0..4):
    > pi:=display(p_i):
    > plots[display](p_i);
    The setting of appropriate circle of diameter, amount altitude of alifd0:=s0+s0*i:
    > i:=4:
    > d0:=d0:
    > c0:=circle([0,d0/2],d0/2,color=blue):
    > plots[display](p_i,c0);

    Construction of flower by turning "point"r_i:=seq(rotate(p0,i*Pi/4),i=1..8):
    > p1:=display(r_i,p0):plots[display](p1,scaling=constrained);

     The setting of circumscribed circlec1:=circle([0,0],s0,color=blue,thickness=2):
          Construction and the setting of flower inscribed in a circle
    > plots[display](c1,p1,scaling=constrained);
    > p_c:=plots[display](c1,p1,scaling=constrained):

    The setting and construction of arabesque by horizontal parallel transport original flower with different stepspt_i_1:=seq(translate(p1,5*a0*i,0),i=0..4):
    > plots[display](pt_i_1);
    > pt_i_11:=seq(translate(p1,2*a0*i,0),i=0..4):
    > plots[display](pt_i_11);
    > pt_i_111:=seq(translate(p1,a0*7*i,0),i=0..4):
    > plots[display](pt_i_111);

     The setting and construction of arabesque by vertical parallel transport original flower with different stepspt_i_2:=seq(translate(p1,0,2*s0*i),i=0..4):
    > plots[display](pt_i_2);
    > pt_i_22:=seq(translate(p1,0,s0*i),i=0..4):
    > plots[display](pt_i_22);
    > pt_i_222:=seq(translate(p1,0,1/2*s0*i),i=0..4):
    > plots[display](pt_i_222);
     Getting arabesques by turning original flower on different anglespr:=rotate(p1,Pi/8):
    > plots[display](pr);
    > plots[display](p1,pr);

    > pr_i:=seq(rotate(p1,Pi/16*i),i=0..8):
    > plots[display](pr_i);

    > pt_1:=translate(p1,0,2*s0):
    > pr_1_i:=seq(rotate(pt_1,Pi/3.5*i),i=0..6):
    > plots[display](pr_1_i);
    > pr_11_i:=seq(rotate(pt_1,Pi/5*i),i=0..10):
    > plots[display](pr_11_i);
    > pr_111_i:=seq(rotate(pt_1,Pi/6.5*i),i=0..12):
    > plots[display](pr_111_i);

    Construction of standard quadrilaterals

          Muchametshina Liya,  8th class,  school № 57, Kazan, Russia




    Construction of square

    > restart:
    > with(plottools):
           Сoordinates (x;y) of the lower left corner of the square and the side "а"
    > x:=0;y:=3;a:=6;

                                    x := 0

                                    y := 3

                                    a := 6

          Construction of the square
    > P1:=plot([[x,y],[x,y+a],[x+a,y+a],[x+a,y],[x,y]],color=green,thickness=4):
    > plots[display](P1,scaling=CONSTRAINED);

    The setting of the second square wich moved relative to the first on the vector (2;-3) (vector can be changed) and with side "а-1" (the length of a side can be changed)P2:=plot([[x+2,y-3],[x+2,y-3+a-1],[x+2+a-1,y-3+a-1],[x+2+a-1,y-3],[x+2,y-3]],color=black,thickness=4):
    > plots[display](P1,P2,scaling=CONSTRAINED);

    Construction of rectangle

    > restart:
    > with(plottools):
            Сoordinates (x;y) of the lower left corner of the square and the "а" and "b" sides
    > x:=0;y:=2;a:=3;b:=9;

                                    x := 0

                                    y := 2

                                    a := 3

                                    b := 9

           The rectangle is specified by the sequence of vertices with given the lengths "a" and "b"
    > l:=plot([[x,y],[x,y+a],[x+b,y+a],[x+b,y],[x,y]]):
    > plots[display](l,scaling=CONSTRAINED,thickness=4);
    Construction of rhombus

    > restart:
    > with(plottools):
          The coordinates (x;y) of the initial vertex of the rhombus and the half of the diagonals "a" and "b"
    > x:=0;y:=2;a:=3;b:=4;

                                    x := 0

                                    y := 2

                                    a := 3

                                    b := 4

           Rhombus is specified by the sequence of vertices with the values "a" and "b"
    > ll:=plot([[x,y],[x+a,y+b],[x+a+a,y],[x+a,y-b],[x,y]]):
    > plots[display](ll,scaling=CONSTRAINED,thickness=4);

    Construction of parallelogram

    > restart:
    > with(plottools):
          (х;у) - the starting point, (i;j) - the displacement vector of starting point, "а" - the base of the parallelogram
    > x:=0;y:=0;i:=4;j:=5;a:=10;

                                    x := 0

                                    y := 0

                                    i := 4

                                    j := 5

                                   a := 10

         The parallelogram is defined by the sequence of vertices
    > P1:=plot([[x,y],[x+i,y+j],[x+i+a,y+j],[x+a,y],[x,y]]):
    > plots[display](P1,scaling=CONSTRAINED,thickness=4);
     If  i= 0  it turns out the rectangleget.
           If  j= а  it turns out the  square.
           If  a := sqrt(i^2+j^2) it turns out the rhombus. a:=sqrt(i^2+j^2):

    Construction of trapeze

    Trapeze general form
    > restart:
    > with(plottools):
            (х;у) - the starting point, (i;j) - the displacement vector of starting point, а - the larger base of the trapezoid
    > x:=0;y:=2;i:=1;j:=5;a:=11;

                                    x := 0

                                    y := 2

                                    i := 1

                                    j := 5

                                   a := 11

             The trapez is defined by the sequence of vertices     
    > P1:=plot([[x,y],[x+i,y+j],[x+i+j,y+j],[x+i+a,y],[x,y]]):
    > plots[display](P1,scaling=CONSTRAINED,thickness=4);
    Rectangular trapezoid
    > restsrt:
    > with(plottools):
    > x:=0;y:=2;i:=0;j:=6;a:=11;

                                    x := 0

                                    y := 2

                                    i := 0

                                    j := 6

                                   a := 11

    > P1:=plot([[x,y],[x,y+j],[x+j,y+j],[x+a,y]]):
    > plots[display](P1,scaling=CONSTRAINED,thickness=4);
    Isosceles trapezoid
    > restart:
    > with(plottools):
    > x:=0;y:=2;i:=4;j:=6;a:=15;

                                    x := 0

                                    y := 2

                                    i := 4

                                    j := 6

                                   a := 15

    > P1:=plot([[x,y],[x+i,y+j],[x+j+i,y+j],[x+a,y],[x,y]]):
    > plots[display](P1,scaling=CONSTRAINED,thickness=4);




    Anyone interested in modifying and expanding the Maple FIFA simulation for the Eurocup to include the knockout round?  Any interest?

    > restart;
    > a := -10; b := 10; ps := seq(plot([i, t, t = -20 .. 20], x = -10 .. 10, y = -20 .. 20, color = red, style = point), i = a .. b);

    plots[display](ps, insequence = true); p := plots[display](ps, insequence = true);

















    Post gialid_GEODROMchik - what is this?

    Pilot project of Secondary school # 57 of Kazan, Russia

    Use of Maple

    in Mathematics Education by mathematics teacher Alsu Gibadullina

    and in scientific work of schoolchildren


    Examples made using the Maple

    the 6th class


                  Arina                         Elza                             David    


           Book.mws              Kolobok.mws               sn_angl.mws





    A few people have asked me how I created the sections in the Maple application in this video:

    Here's the worksheet (Maple 2016 only). As you can see, the “sections” look different what you would normally expect (I often like to experiment with small changes in presentation!)

    These aren't, however, sections in the traditional Maple sense; they're a demonstration of Maple 2016's new tools for programmatically changing the properties of a table (including the visibility of its rows and columns). @dskoog gets the credit for showing me the technique.

    Each "section" consists of a table with two rows.

    • The table has a name, specified in its properties.
    • The first row (colored blue) contains (1) a toggle button and (2) the title of each section (with the text in white)
    • The second row (colored white) is visible or invisible based upon the state of the toggle button, and contains the content of my section.

    Each toggle button has

    • a name, specified in its properties
    • + and - images associated with its on and off states (with the image background color matching the color of the first table row)
    • Click action code that enables or disables the visibility of the second row

    The Click action code for the toggle button in the "Pure Fluid Properties" section is, for example,

    if DocumentTools:-GetProperty(buttonName, 'value') = "false" then   
         DocumentTools:-SetProperty([tableName, 'visible[2..]', true]);
         DocumentTools:-SetProperty([tableName, 'visible[2..]', false]);
    end if;

    As I said at the start, I often try to make worksheets look different to the out-of-the-box defaults. Programmatic table properties have simply given me one more option to play about with.

    Disclaimer: This blog post has been contributed by Prof. Nicola Wilkin, Head of Teaching Innovation (Science), College of Engineering and Physical Sciences and Jonathan Watkins from the University of Birmingham Maple T.A. user group*. 

    Written for Maple T.A. 2016. For Maple T.A. 10 users, this question can be written using the queston designer.


    This is the second of three blog posts about working with data sets in Maple.

    In my previous post, I discussed how to use Maple to access a large number of data sets from Quandl, an online data aggregator. In this post, I’ll focus on exploring built-in data sets in Maple.

    Data is being generated at an ever increasing rate. New data is generated every minute, adding to an expanding network of online information. Navigating through this information can be daunting. Simply preparing a tabular data set that collects information from several sources is often a difficult and time consuming effort. For example, even though the example in my previous post only required a couple of lines of Maple code to merge 540 different data sets from various sources, the effort to manually search for and select sources for data took significantly more time.

    In an attempt to make the process of finding data easier, Maple’s built-in country data set collects information on country-specific variables including financial and economic data, as well as information on country codes, population, area, and more.

    The built-in database for Country data can be accessed programmatically by creating a new DataSets Reference:

    CountryData := DataSets:-Reference( "builtin", "country" );

    This returns a Reference object, which can be further interrogated. There are several commands that are applicable to a DataSets Reference, including the following exports for the Reference object:

    exports( CountryData, static );

    The list of available countries in this data set is given using the following:

    GetElementNames( CountryData );

    The available data for each of these countries can be found using:

    GetHeaders( CountryData );

    There are many different data sets available for country data, 126 different variables to be exact. Similar to Maple’s DataFrame, the columns of information in the built-in data set can be accessed used the labelled name.

    For example, the three-letter country codes for each country can be returned using:

    CountryData[.., "3 Letter Country Code"];

    The three-letter country code for Denmark is:

    CountryData["Denmark", "3 Letter Country Code"];

    Built-in data can also be queried in a similar manner to DataFrames. For example, to return the countries with a population density less than 3%:

    pop_density := CountryData[ .., "Population Density" ]:
    pop_density[ `Population Density` < 3 ];

    At this time, Maple’s built-in country data collection contains 126 data sets for 185 countries. When I built the example from my first post, I knew exactly the data sets that I wanted to use and I built a script to collect these into a larger data container. Attempting a similar task using Maple’s built-in data left me with the difficult decision of choosing which data sets to use in my next example.

    So rather than choose between these available options, I built a user interface that lets you quickly browse through all of Maple’s collection of built-in data.

    Using a couple of tricks that I found in the pages for Programmatic Content Generation, I built the interface pictured above. (I’ll give more details on the method that I used to construct the interface in my next post.)

    This interface allows you to select from a list of countries, and visualize up to three variables of the country data with a BubblePlot. Using the preassigned defaults, you can select several countries and then visualize how their overall number of internet users has changed along with their gross domestic product. The BubblePlot visualization also adds a third dimension of information by adjusting the bubble size according to the relative population compared with the other selected countries.

    Now you may notice that the list of available data sets is longer than the list of available options in each of the selection boxes. In order to be able to generate BubblePlot animations, I made an arbitrary choice to filter out any of the built-in data sets that were not of type TimeSeries. This is something that could easily be changed in the code. The choice of a BubblePlot could also be updated to be any other type of Statistical visualization with some additional modifications.

    You can download a copy of this application here:

    You can also interact with it via the MapleCloud:

    I’ll be following up this post with an in-depth post on how I authored the country selector interface using programmatic content generation.

    Since it's not every day we receive submission to the Maple Application Center that have words like "quantum entanglement" (and "teleportation"!) in the title, I thought I'd share this one:

    Matrix Representation of Quantum Entangled States: Understanding Bell's Inequality and Teleportation



    I'm an educator (physicist) who has migrated to Maple because of the lower "activation barrier" to get something of interest produced by the student. The students in my courses are exposed to several language (Python, C++, Java) and mathematical systems (Mathematica, Maple, MATLAB.) Many claim that unless forced to used a particular language or system, their first choice is Python and Maple for the reason I cite. 

    As a consequence, it is my experience that students truly perfer the math-like appearance of the 2-D Math notation as opposed to the Maple notation. They see it as more natural - again with a lower activation barrier. Hence I see no reason to change. However, I would be interested in reasons why it might be beneficial.

    My ultimate question is: do I start them with worksheet mode or documents mode? I'm use to worksheet mode and have found the call and response method easy for them to understand. But document mode has many valuable benefits. Is it worth the increase in learning (and frustration) for the benefits if the students use the software only a few times per semester? Or for some, every week?

    I would be interested in hearing about the experiences of other educators.


    Greetings to all. I am writing today to share a personal story / exploration using Maple of an algorithm from the history of combinatorics. The problem here is to count the number of strings over a certain alphabet which consist of some number of letters and avoid a set of patterns (these patterns are strings as opposed to regular expressions.) This counting operation is carried out using rational generating functions that encode the number of admissible strings of length n in the coefficients of their series expansions. The modern approach to this problem uses the Goulden-Jackson method which is discussed, including a landmark Maple implementation from a paper by D. Zeilberger and J. Noonan, at the following link at (Goulden-Jackson has its own website, all the remaining software described in the following discussion is available at the MSE link.) The motivation for this work was a question at the MSE link about the number of strings over a two-letter alphabet that avoid the pattern ABBA.

    As far as I know before Goulden-Jackson was invented there was the DFA-Method (Deterministic Finite Automaton also known as FSM, Finite State Machine.) My goal in this contribution was to study and implement this algorithm in order to gain insight about its features and how it influenced its powerful successor. It goes as follows for the case of a single pattern string: compute a DFA whose states represent the longest prefix of the pattern seen at the current position in the string as it is being scanned by the DFA, with the state for the complete pattern doubling as a final absorbing state, since the pattern has been seen. Translate the transitions of the DFA into a system of equations in the generating functions representing strings ending with a given maximal prefix of the pattern, very much like Markov chains. Finally solve the system of equations for the generating functions and thus obtain the sequence of values of strings of length n over the given alphabet that avoid the given pattern.

    I have also implemented the DFA method for sets of patterns as opposed to just one pattern. The algorithm is the same except that the DFA does not consist of a chain with backlinks as in the case of a single pattern but a tree of prefixes with backlinks to nodes higher up in the tree. The nodes in the tree represent all prefixes that need to be tracked where obviously a common prefix between two or more patterns is shared i.e. only represented once. The DFA transitions emanating from nodes that are leaves represent absorbing states indicating that one of the patterns has been seen. We run this algorithm once it has been verified that the set of patterns does not contain pairs of patterns where one pattern is contained in another, which causes the longer pattern to be eliminated at the start. (Obviously if the shorter pattern is forbidden the so is the longer.) The number of states of the DFA here is bounded above by the sum of the lengths of the patterns with subpatterns eliminated. The uniqueness property of shared common prefixes holds for subtrees of the main tree i.e. recursively. (The DFA method also copes easily with patterns that have to occur in a certain order.)

    I believe the Maple code that I provide here showcases many useful tricks and techniques and can help the reader advance in their Maple studies, which is why I am alerting you to the web link at MSE. I have deliberately aimed to keep it compatible with older versions of Maple as many of these are still in use in various places. The algorithm really showcases the power of Maple in combinatorics computing and exploits many different aspects of the software from the solution of systems of equations in rational generating functions to the implementation of data structures from computer science like trees. Did you know that Maple permits nested procedures as known to those who have met Lisp and Scheme during their studies? The program also illustrates the use of unit testing to detect newly introduced flaws in the code as it evolves in the software life cycle.

    Enjoy and may your Maple skills profit from the experience!

    Best regards,

    Marko Riedel

    The software is also available here: dfam-mult.txt

         Example of the equidistant surface at a distance of 0.25 to the surface
    -0.1 * (sin (4 * x1) + sin (3 * x2 + x3) + sin (2 * x2)) = 0
    Constructed on the basis of universal parameterization of surfaces. 

    Hi there, fellow primers, it's good to be back after almost 5 years! I just want to share a worksheet on Numerov's algorithm in Maple using procedures as I've recently found out that google could not find any Maple procedure that implements Numerov's algorithm to solve ODEs.   Reference.pdf 

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