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  • I don't stand by all I said here, so I'm deleting. I think there was valid criticism to be made, but I didn't make it any valid way.

    I found in the Application Center a quite old work (2010) titled Generation of correlated random numbers  (see here view.aspx).
    This work contains a few errors that I thought it was worth correcting. 

    Basically the works I refer to concern the sampling of linearly correlated random variables (or correlation in the Pearson sense). Classical textbooks about the subject generally discuss this topic by considering only gaussian random variables and present two methods to generate linearlycorrelated samples: one base on the Cholesky decomposition of the correlation matrix, the other based on its SVD decomposition.

    Now the question is: can we apply any of these two procedures to generate linearly correlated samples of arbitrary random variables?
    The answer is NO and the reason why it is so is strongly related to a fundamental property of gaussian random variables (GRV) that is that any linear combination of GRVs is still a GRV.
    But things are not that simple because even the multi gaussian case handmed with Cholesky's decomposition or SVD can lead to undesired results if no precautions are taken.

    The aim of this post is to show what are those wrong results we obtain by thoughtlessly applying these decompositiond and, of course, to show how we must proceed to avoid them.

    Let's start by a very simple point of natural good sense: suppose U1 and U2 are two independant identically distributed (iid) random variables and that we have some "function" F which, when applied to the couple (U1, U2) generate a couple (A1, A2) of linearly correlated random variables. Thus F(U1, U2) = (A1, A2).
    Let's suppose this same relation holds if we replace U1 and U2 by "a sample of U1" and "a sample of U2", and thus (A1, A2) by "a sample of the bivariate (A1, A2) whose components are linearly correlated". Let's call S the cloud one could obtain by using for instance the ScatterPlot(A1, A2) procedure of Maple.

    Let's suppose now that instead of computing F(U1, U2) I decide to compute (U2, U1). Let's call (A1' , A2') the corresponding joint sample and write S' := ScatterPlot(A2', A1').
    It seems natural (and it is!) to think that S and S' will be the same up to sampling artifacts. 

    Any correct method to generate samples from (linearly or not) correlated random variables must verify this similarity of patterns between S and S' S and S'. But this is not the case in this work view.aspx.

    The safer way to correlate, even in the Pearson's sense, random variables is to use the concept of COPULAS (there is a work on copulas in the Application Center, but for a quick overview see here Copula_(probability_theory)).
    For this special case of linear correlation on can use copulas without knowing it, and this is very simple: as soon as our  procedure F introduced above gives correct results if U1 and U2 are standard GRVs,

    • take any couple (R1, R2) of arbitrary random variables,
    • build a map M(R1, R2) --> (U1, U2),
    • generate (A1, A2) = F(U1, U2),
    • compute M^(-1)(A1, A2)


    What is the point of correcting a work that is 10 years old?
    A very simple answer is that the Cholesky's decomposition (or SVD) is still the emblematic method to use for linearly correlating random variable. This is the only one presented in scholar textbooks, the only one a lot of students have been taught about (unless they have  they have had an extensive background in probability or statistics), and thus a systematic source of wrong results users are not even aware of.


    Next point: it's well known that the Pearson's correlation cannot be lower than -1 or higher than +1, but this is common mistake to think any value between -1 and +1 can be reached.
    This is guaranteed for GRVs, but  not for some other random variables.
    For a classical counter-example see  04_correlation_2016_cost_symposium_fkuo_tagged.pdf 

    The notation used in the attached file are mainly those used in the initial work  view.aspx.

    restart:


    This work is aimed to correct the procedure used in  https://fr.maplesoft.com/applications/view.aspx?SID=99806
    to correlate arbitrary random variables in the (common) Pearson's sense.

    with(LinearAlgebra):
    with(plots):
    with(Statistics):

     

    GAUSSIAN RANDOM VARIABLES

     

    # First example: both A1 and A2 are centered gaussian random variables
    #                The order we use (A1 next A2 or A2 next A1) to define Ma doesn't matter

    Y   := RandomVariable(Normal(0, .25)):
    rho := .9:
    Q   := 10^4:
    A1  := Sample(Y, Q):
    A2  := Sample(Y, Q):
    Ma  := `<,>`(`<,>`(A1), `<,>`(A2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:
    A1A2 := ScatterPlot(Column(Rs2, 1), Column(Rs2, 2), title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(A2), `<,>`(A1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2), Column(Rs2, 1), title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

    # Second example : both A1 and A2 are non-centered gaussian random variables with equal standard deviations.
    #                  The order we use to define Ma does matter

    Y   := RandomVariable(Normal(1, .25)):
    rho := .9:
    Q   := 10^4:
    A1  := Sample(Y, Q):
    A2  := Sample(Y, Q):
    Ma  := `<,>`(`<,>`(A1), `<,>`(A2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:


    A1A2 := ScatterPlot(Column(Rs2, 1), Column(Rs2, 2), title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(A2), `<,>`(A1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2), Column(Rs2, 1), title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

     

    # Second example corrected: to avoid order's dependency proceed this way
    #    1/ center A1 and A2
    #    2/ correlate the now centered rvs
    #    3/ uncenter the couple of correlated rvs


    C1  := convert(Scale(A1, scale=Mean), Vector[row]):
    C2  := convert(Scale(A2, scale=Mean), Vector[row]):

    Ma  := `<,>`(`<,>`(C1), `<,>`(C2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:


    A1A2 := ScatterPlot(Column(Rs2, 1)+~Mean(A1), Column(Rs2, 2)+~Mean(A2), title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(C2), `<,>`(C1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2)+~Mean(A1), Column(Rs2, 1)+~Mean(A2), title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

    # Third example : both A1 and A2 are centered gaussian random variables with unequal standard deviations.
    #                 The order we use to define Ma does matter

    rho := .9:
    Q   := 10^4:
    A1  := Sample(Normal(0, 1), Q):
    A2  := Sample(Normal(0, 2), Q):
    Ma  := `<,>`(`<,>`(A1), `<,>`(A2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:


    A1A2 := ScatterPlot(Column(Rs2, 1), Column(Rs2, 2), title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(A2), `<,>`(A1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2), Column(Rs2, 1), title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

    # Third example corrected: to avoid order's dependency proceed this way
    #    1/ scale A1 and A2
    #    2/ correlate the now scaled rvs
    #    3/ unscale the couple of correlated rvs


    C1  := A1 /~ StandardDeviation(A1):
    C2  := A2 /~ StandardDeviation(A2):

    Ma  := `<,>`(`<,>`(C1), `<,>`(C2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:


    A1A2 := ScatterPlot(Column(Rs2, 1)*~StandardDeviation(A1), Column(Rs2, 2)*~StandardDeviation(A2), title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(C2), `<,>`(C1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2)*~StandardDeviation(A1), Column(Rs2, 1)*~StandardDeviation(A2), title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

    # More generally: to avoid order's dependency proceed this way
    #    1/ transform A1 and A2 into standard gaussian random variables (mean and standard deviation scalings)
    #    2/ correlate the now scaled rvs
    #    3/ unscale the couple of correlated rvs

     

    A MORE COMPLEX EXAMPLE:

    NON GAUSSIAN RANDOM VARIABLES
    (here two LogNormal rvs)

     

     

    # Preliminary
    #   the expectation (mean) of a LogNormal rv cannot be 0;
    #   as a consequence it is expected that the order used to buid Ma will matter
    #
    # Proceed as Igor Hlivka did

     

    Y   := RandomVariable(LogNormal(.5, .25)):
    rho := .9:
    Q   := 1000:
    A1  := Sample(Y, Q):
    A2  := Sample(Y, Q):
    Ma  := `<,>`(`<,>`(A1), `<,>`(A2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    ScatterPlot(A1, A2, color = red, title = ["Raw LogNormal RV", font = [TIMES, BOLD, 12]]):
    A1A2 := ScatterPlot(Column(Rs2, 1), Column(Rs2, 2), title = "Correlated LogNormal RV", opts, color=blue):

    # And now change, as usual, the order in Ma

    Ma  := `<,>`(`<,>`(A2), `<,>`(A1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    A2A1 := ScatterPlot(Column(Rs2, 2), Column(Rs2, 1), title = "Correlated LogNormal RV", opts, color=red):

    display(A1A2, A2A1);

     

    # How can we avoid that the order used to assemble Ma do matter?
    #
    # A close examination of what was done with gaussiann rvs show that in all the cases we
    # went back to standard gaussian rvs before correlating them.
    # So let's just do the same thing here.
    #
    # Of course it's not as immediate as previously...
    # (please do not focus on the slowness of the code, it is written to clearly explain 
    # what is done, not to be fast)



    #-------------------------------------- from Y to standard gaussian
    G  := RandomVariable(Normal(0, 1)):
    G1 := Vector[row](Q, q -> Quantile(G, Probability(Y > A1[q], numeric), numeric)):
    G2 := Vector[row](Q, q -> Quantile(G, Probability(Y > A2[q], numeric), numeric)):
    # could be replaced by this faster code
    #   cdf_Y := unapply(CDF(Y, z), z) assuming z > 0;
    #   cdf_G := unapply(CDF(G, z), z);
    #   S1    := sort(A1):
    #   ini   := -10:
    #   V     := Vector[row](Q):
    #   for q from 1 to Q do
    #     V[q] := fsolve(cdf_G(z)=cdf_Y(S1[q]), z=ini);
    #     ini  := V[q]:
    #   end do:
    #------------------------------------------------------------------

    Ma  := `<,>`(`<,>`(G1), `<,>`(G2)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:


    opts := titlefont = [TIMES, BOLD, 12], symbol=point, transparency=0.5:


    #-------------------------------------- from standard gaussian to Y
    C1 := Vector[row](Q, q -> Quantile(Y, Probability(G > Rs2[q, 1], numeric), numeric)):
    C2 := Vector[row](Q, q -> Quantile(Y, Probability(G > Rs2[q, 2], numeric), numeric)):
    #------------------------------------------------------------------
    A1A2 := ScatterPlot(C1, C2, title = "Correlated Normal RV", opts, color=blue):



    Ma  := `<,>`(`<,>`(G2), `<,>`(G1)):
    MA  := Transpose(Ma):
    Cor := Matrix([[1, rho], [rho, 1]]):
    Cd2 := LUDecomposition(Cor, method = 'Cholesky', output = ['U']):
    Rs2 := MA . Cd2:

    #-------------------------------------- from standard gaussian to Y
    C1 := Vector[row](Q, q -> Quantile(Y, Probability(G > Rs2[q, 1], numeric), numeric)):
    C2 := Vector[row](Q, q -> Quantile(Y, Probability(G > Rs2[q, 2], numeric), numeric)):
    #------------------------------------------------------------------

    A2A1 := ScatterPlot(C2, C1, title = "Correlated Normal RV", opts, color=red):

    display(A1A2, A2A1);

     

     

    CONCLUSION: Be extremely careful when correlating non standard gaussian random variables,
                                 and more generally non gaussian random variables.


    Correlating rvs the way Igor Hlivka did can be replaced in the more general framework of COPULA THEORY.

    Mathematically a bidimensional copula C is a function from [0, 1] x [0, 1] to [0, 1] if C is joint CDF of a bivariate random variable
    both with uniform marginals on [0, 1].
    See for instanc here  https://en.wikipedia.org/wiki/Copula_(probability_theory)

    What I did here to "correlate" A1 and A2 was nothing but to apply in a step-by-step way a GAUSSIAN COPULA to the bivariate
    (A1, A2) random variable.
    In  Quantile(G, Probability(Y > A1[q], numeric), numeric) the blue expression maps A1 onto [0, 1] (as it is needed
    in the definition of a copula), while the brown sequence is the copula itself (when the same operation on A2 has been done).

     

     


     

    Download LInear_Correlated_Random_Variables.mw

    This year, the International Mathematics Competition for University Students  (IMC) took place online (due to Coronavirus), https://www.imc-math.org.uk/?year=2020

    One of the sponsors was Maplesoft.


    Here is a Maple solution for one of the most difficult problems.

     

    Problem 4, Day 1.

    A polynomial p with real coeffcients satisfies the equation

    p(x+1)-p(x) = x^100, for all real x.

    Prove that p(x) <= p(1-x) for   0 <= x and x <= 1/2.

     

    A Maple solution.

    Obviously, the degree of the polynomial must be 101.

    We shall find effectively p(x).

     

    restart;

    n:=100;

    100

    (1)

    p:= x -> add(a[k]*x^k, k=0..n+1):

    collect(expand( p(x+1) - p(x) - x^n ), x):

    S:=solve([coeffs(%,x)]):

    f:=unapply(expand(eval(p(1-x)-p(x), S)), x);

    proc (x) options operator, arrow; (94598037819122125295227433069493721872702841533066936133385696204311395415197247711/16665)*x-37349543370098022593228114650521983084038207650677468129990678687496120882031450*x^3-1185090416633200*x^87+5974737180020*x^89-(86465082200/3)*x^91+133396340*x^93-597520*x^95+2695*x^97-(50/3)*x^99+x^100-(2/101)*x^101+(16293234618989521508515025064456465992824384487957638029599182473343901462949018943/221)*x^5-69298763242215246970576715450882718421982355083931952097853888722419955069286800*x^7+(113991896447569512043394769396957538374962221763587431560580742819193991151970540/3)*x^9-(450021969146981792096716260960657763583495746057337083106755737535521294639081800/33)*x^11+3451079104335626303615205945922095523722898887765464179344409464422173275181060*x^13-648776866983969889704838151840901241863730925272452260127881376737469460326640*x^15+(1224135636503373678241493336115166408006020118605202014423201964267584789018590/13)*x^17-(32609269812588448517851078111423700053874956628293000710950261666057691492700/3)*x^19+(17369174852688147212979009419766100341356836811271344020859968314555332168046/17)*x^21-79714896335448291043424751268405443765709493999285019374276097663327217200*x^23+(26225149723490747954239730131127580683873943002539194987613420614551124468/5)*x^25-294965074792241210541282428184641838437329968596736990461830398732050600*x^27+(186430797065926226062569133543332579493666384095775768758650822594552980/13)*x^29-608766986011732859031810279841713016991034114339196337222615083429200*x^31+22758671683254934243234770245768111655371809025564559292966948184145*x^33-755022138514287934394628273773230341731572817528392747252537299270*x^35+(380420681562789081339436627697748498619486609696130138245054547645/17)*x^37-596110444235534895977389751553577405150617862905657345084592800*x^39+(186546013247587274869312959605954587283787420112828231587660264/13)*x^41-313678397368440441190125909536848768199325715147747522784400*x^43+6254306446857003025144445909566034709396500424382183891144*x^45-114204496639521606716779723226539643746613722246036949600*x^47+1916927215404111401325904884511116319416726263341690260*x^49-29677354167404548158728688629916697559643435320275800*x^51+(93950257927474972838978328999588595121346462082404180/221)*x^53-5650787690628744633775927032927548604440367748960*x^55+69888520126633344286255800412032531913013033640*x^57-806279422358340503473340514496960223283853200*x^59+8696895011389170857678332370276446830499368*x^61-87900576836101226420991143179656778525600*x^63+(10844299000116828980379757772973769420469/13)*x^65-7447304814595165455238549781183862150*x^67+(1065245686771269279784908613651828005/17)*x^69-497741911503981694520541768814800*x^71+3738596479537236832468307626580*x^73-26593490941061853727808593704*x^75+179403449737703736809514420*x^77-1149393958953185579079600*x^79+(21007540356807993839074/3)*x^81-(121855249152521399900/3)*x^83+(3818021878637120462/17)*x^85 end proc

    (2)

    plot(f, 0..1); # Visual check: f(x)>0 for 0<x<1/2

     

    f(0), f(1/4), f(1/2);

    0, 2903528346661097497054603834764435875077553006646158945080492319146997643370625023889353447129967354174648294748510553528692457632980625125/3213876088517980551083924184682325205044405987565585670602752, 0

    (3)

    sturm(f(x), x, 0, 1/2);

    1

    (4)

    So, the polynomial f has a unique zero in the interval (0, 1/2]. Since f(1/2) = 0  and f(1/4) > 0, it results that  f > 0 in the interval  (0, 1/2). Q.E.D.

     

    Download imc2020-1-4.mw

    One way to find the equation of an ellipse circumscribed around a triangle. In this case, we solve a linear system of equations, which is obtained after fixing the values of two variables ( t1 and t2). These are five equations: three equations of the second-order curve at three vertices of the triangle and two equations of a linear combination of the coordinates of the gradient of the curve equation.
    The solving of system takes place in the ELS procedure. When solving, hyperboles appear, so the program has a filter. The filter passes the equations of ellipses based on by checking the values of the invariants of the second-order curves.
    FOR_ELL_ТR_OUT_PROCE_F.mw  ( Fixed comments in the text  01, 08, 2020)

    A lot of scientific software propose packages enabling drawing figures in XKCD style/
    Up to now I thought this was restricted to open products (R, Python, ...) but I recently discovered Matlab and even Mathematica were doing same.

    Layton S (2012). “XKCDIFY! Adding flair to boring Matlab Axes one plot at a time.” Last accessed on December 08, 2014, URL https://github.com/slayton/matlab-xkcdify.

    Woods S (2012). “xkcd-style graphs.” Last accessed on December 08, 2014, URL http://mathematica.stackexchange.com/questions/11350/xkcd-style-graphs/ 11355#11355.

     

    So why not Maple?

    As a regular user of R, I could have visualize the body of the corresponding procedures to see how these drawings were made and just translate theminto Maple.
    But copying for the sake of copying is not of much interest.
    So I started to develop some primitives for "XKCD-drawing" lines, polygons, circles and even histograms.
    My goal is not to write an XKCD package (I don't have the skills for that) but just to arouse the interest of (maybe) a few people here who could continue this preliminary work


    A main problem is the one of the XKCD fonts: no question to redefine them in Maple and I guess using them in a commercial code is not legal (?). So no XKCD font in this first work, nor even the funny guy who appears recurently on the drawings (but it could be easily constructed in Maple).

    In a recent post (Plot styling - experimenting with Maple's plotting...) Samir Khan proposed a few styles made of several plotting options,  some of which he named "Excel style" or "Oscilloscope style"... maybe a future "XKCD xtyle" in Maple ?


    This work has been done with Maple 2015 and reuses an old version of a 1D-Kriging procedure 

     

    restart:

    with(LinearAlgebra):
    with(plots):
    with(Statistics):

     

    The principle is always the same:
        1/   Let L a straight line which is either defined by its two ending points (xkvd_hline) or taken as the default [0, 0], [1, 0] line.
              For xkvd_hline the given line L is firstly rotate to be aligned with the horizontal axis.

        2/   Let P1, ..., PN N points on L. Each Pn writes [xn, yn]

        3/   A random perturbation rn is added yo the values y1, ..., yN

        4/   A stationnary random process RP, with gaussian correlation function is used to build a smooth curve passing through the points
              (x1, y1+r1), ..., (xN, yN+rN) (procedure KG where "KG" stands for "Kriging")

        5/   The result is drawn or mapped to some predefined shape :
                      xkcd_hist,
                      xkcd_polyline,
                      xkcd_circle

        6/   A procedure xkcd_func is also provided to draw functions defined by an explicit relation.
     

    KG := proc(X, Y, psi, sigma)
      local NX, DX, K, mu, k, y:
      NX := numelems(X);
      DX := < seq(Vector[row](NX, X), k=1..NX) >^+:
      K  := (sigma, psi) -> evalf( sigma^2 *~ exp~(-((DX - DX^+) /~ psi)^~2) ):
      mu := add(Y) / NX;
      k  := (x, sigma, psi) -> evalf( convert(sigma^2 *~ exp~(-((x -~ X ) /~ psi)^~2), Vector[row]) ):
      y  := mu + k(x, sigma, psi) . (K(sigma, psi))^(-1) . convert((Y -~ mu), Vector[column]):
      return y
    end proc:


    xkcd_hline := proc(p1::list, p2::list, a::nonnegative, lc::positive, col)
      # p1 : first ending point
      # p2 : second ending point
      # a  : amplitude of the random perturbations
      # lc : correlation length
      # col: color
      local roll, NX, LX, X, Z:
      roll := rand(-1.0 .. 1.0):
      NX   := 10:
      LX   := p2[1]-p1[1]:
      X    := [seq(p1[1]..p2[1], LX/(NX-1))]:
      Z    := [p1[2], seq(p1[2]+a*roll(), k=1..NX-1)]:
      return plot(KG(X, Z, lc*LX, 1), x=min(X)..max(X), color=col, scaling=constrained):
    end proc:


    xkcd_line := proc(L::list, a::nonnegative, lc::positive, col, {lsty::integer:=1})
      # L  : list which contains the two ending point
      # a  : amplitude of the random perturbations
      # lc : correlation length
      # col: color
      local T, roll, NX, DX, DY, LX, A, m, M, X, Z, P:
      T    := (a, x0, y0, l) ->
                 plottools:-transform(
                   (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
                 ):
      roll := rand(-1.0 .. 1.0):
      NX   := 5:
      DX   := L[2][1]-L[1][1]:
      DY   := L[2][2]-L[1][2]:
      LX := sqrt(DX^2+DY^2):
      if DX <> 0 then
         A := arcsin(DY/LX):
      else
         A:= Pi/2:
      end if:
      X := [seq(0..1, 1/(NX-1))]:
      Z := [ seq(a*roll(), k=1..NX)]:
      P := plot(KG(X, Z, lc, 1), x=0..1, color=col, scaling=constrained, linestyle=lsty):
      return T(A, op(L[1]), LX)(P)
    end proc:


    xkcd_func := proc(f, r::list, NX::posint, a::positive, lc::positive, col)
      # f  : function to draw
      # r  : plot range
      # NX : number of equidistant "nodes" in the range r (boundaries included)
      # a  : amplitude of the random perturbations
      # lc : correlation length
      # col: color
      local roll, F, LX, Pf, Xf, Zf:
      roll := rand(-1.0 .. 1.0):
      F    := unapply(f, indets(f, name)[1]);
      LX   := r[2]-r[1]:
      Pf   := [seq(r[1]..r[2], LX/(NX-1))]:
      Xf   := Pf +~ [seq(a*roll(), k=1..numelems(Pf))]:
      Zf   := F~(Pf) +~ [seq(a*roll(), k=1..numelems(Pf))]:
      return plot(KG(Xf, Zf, lc*LX, 1), x=min(Xf)..max(Xf), color=col):
    end proc:




    xkcd_hist := proc(H, ah, av, ax, ay, lch, lcv, lcx, lcy, colh, colxy)
      # H   : Histogram
      # ah  : amplitude of the random perturbations on the horizontal boundaries of the bins
      # av  : amplitude of the random perturbations on the vertical boundaries of the bins
      # ax  : amplitude of the random perturbations on the horizontal axis
      # ay  : amplitude of the random perturbations on the vertical axis
      # lch : correlation length on the horizontal boundaries of the bins
      # lcv : correlation length on the vertical boundaries of the bins
      # lcx : correlation length on the horizontal axis
      # lcy : correlation length on the vertical axis
      # colh: color of the histogram
      # col : color of the axes
      local data, horiz, verti, horizontal_lines, vertical_lines, po, rpo, p1, p2:
      data  := op(1..-2, op(1, H)):
      verti := sort( [seq(data[n][3..4][], n=1..numelems([data]))] , key=(x->x[1]) );
      verti := verti[1],
               map(
                    n -> if verti[n][2] > verti[n+1][2] then
                            verti[n]
                          else
                            verti[n+1]
                          end if,
                    [seq(2..numelems(verti)-2,2)]
               )[],
               verti[-1];
      horiz := seq(data[n][[4, 3]], n=1..numelems([data])):

      horizontal_lines := NULL:
      for po in horiz do
        horizontal_lines := horizontal_lines, xkcd_hline(po[1], po[2], ah, lch, colh):
      end do:

      vertical_lines := NULL:
      for po in [verti] do
        rpo := po[[2, 1]]:
        vertical_lines := vertical_lines, xkcd_hline([0, rpo[2]], rpo, av, lcv, colh):
      end do:

      p1 := [2*verti[1][1]-verti[2][1], 0]:
      p2 := [2*verti[-1][1]-verti[-2][1], 0]:

      return
        display(
          horizontal_lines,
          T~([vertical_lines]),
          xkcd_hline(p1, p2, ax, lcx, colxy),
          T(xkcd_hline([0, 0], [1.2*max(op~(2, [verti])), 0], ay, lcy, colxy)),
          axes=none,
          scaling=unconstrained
        );
    end proc:


    xkcd_polyline := proc(L::list, a::nonnegative, lc::positive, col)
      # xkcd_polyline reduces to xkcd_line whebn L has 2 elements
      # L  : List of points
      # a  : amplitude of the random perturbations
      # lc : correlation length
      # col: color
      local T, roll, NX, n, DX, DY, LX, A, m, M, X, Z, P:
      T    := (a, x0, y0, l) ->
                 plottools:-transform(
                   (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
                 ):
      roll := rand(-1.0 .. 1.0):
      NX   := 5:
      for n from 1 to numelems(L)-1 do
        DX   := L[n+1][1]-L[n][1]:
        DY   := L[n+1][2]-L[n][2]:
        LX := sqrt(DX^2+DY^2):
        if DX <> 0 then
          A := evalf(arcsin(abs(DY)/LX)):
          if DX >= 0 and DY <= 0 then A := -A end if:
          if DX <= 0 and DY >  0 then A := Pi-A end if:
          if DX <= 0 and DY <= 0 then A := Pi+A end if:
        else
          A:= Pi/2:
          if DY < 0 then A := 3*Pi/2 end if:
        end if:
        if n=1 then
          X := [seq(0..1, 1/(NX-1))]:
          Z := [seq(a*roll(), k=1..NX)]:
        else
          X := [0    , seq(1/(NX-1)..1, 1/(NX-1))]:
          Z := [Z[NX], seq(a*roll(), k=1..NX-1)]:
        end if:
        P    := plot(KG(X, Z, lc, 1), x=0..1, color=col, scaling=constrained):
        P||n := T(A, op(L[n]), LX)(P):
      end do;
      return seq(P||n, n=1..numelems(L)-1)
    end proc:


    xkcd_circle := proc(a::nonnegative, lc::positive, r::positive, cent::list, col)
      # a   : amplitude of the random perturbations
      # lc  : correlation length
      # r   : redius of the circle
      # cent: center of the circle
      # col : color
      local roll, NX, LX, X, Z, xkg, A:
      roll := rand(-1.0 .. 1.0):
      NX   := 10:
      X    := [seq(0..1, 1/(NX-1))]:
      Z    := [0, seq(a*roll(), k=1..NX-1)]:
      xkg  := KG(X, Z, lc, 1):
      A    := Pi*roll():
      return plot([cent[1]+r*(1+xkg)*cos(2*Pi*x+A), cent[2]+r*(1+xkg)*sin(2*Pi*x+A), x=0..1], color=col)
    end proc:

    T := plottools:-transform((x,y) -> [y, x]):
     

    # Axes plot

    x_axis := xkcd_hline([0, 0], [10, 0], 0.03, 0.5, black):
    y_axis := xkcd_hline([0, 0], [10, 0], 0.03, 0.5, black):
    display(
      x_axis,
      T(y_axis),
      axes=none,
      scaling=constrained
    )

     

    # A simple function

    f := 1+10*(x/5-1)^2:
    F := xkcd_func(f, [0.5, 9.5], 6, 0.3, 0.4, red):

    display(
      x_axis,
      T(y_axis),
      F,
      axes=none,
      scaling=constrained
    )

     

    # An histogram

    S := Sample(Normal(0,1),100):
    H := Histogram(S, maxbins=6):
    xkcd_hist(H,   0, 0.02, 0.001, 0.01,   1, 0.1, 0.01, 1,   red, black)

     

    # Axes plus grid with two red straight lines

    r := rand(-0.1 .. 0.1):

    x_axis := xkcd_line([[-2, 0], [10, 0]], 0.01, 0.2, black):
    y_axis := xkcd_line([[0, -2], [0, 10]], 0.01, 0.2, black):
    d1     := xkcd_line([[-1, 1], [9, 9]] , 0.01, 0.2, red):
    d2     := xkcd_line([[-1, 9], [9, -1]], 0.01, 0.2, red):
    display(
      x_axis, y_axis,
      seq( xkcd_line([[-2+0.3*r(), u+0.3*r()], [10+0.3*r(), u+0.3*r()]], 0.005, 0.5, gray), u in [seq(-1..9, 2)]),
      seq( xkcd_line([[u+0.3*r(), -2+0.3*r()], [u+0.3*r(), 10+0.3*r()]], 0.005, 0.5, gray), u in [seq(-1..9, 2)]),
      d1, d2,
      axes=none,
      scaling=constrained
    )

     

    # Axes and a couple of polygonal lines

    d1 := xkcd_polyline([[0, 0], [1, 3], [3, 5], [7, 1], [9, 7]], 0.01, 1, red):
    d2 := xkcd_polyline([[0, 9], [2, 8], [5, 2], [8, 3], [10, -1]], 0.01, 1, blue):

    display(
      x_axis, y_axis,
      d1, d2,
      axes=none,
      scaling=constrained
    )

     

    # A few polygonal shapes

    display(
      xkcd_polyline([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]], 0.01, 1, red),
      xkcd_polyline([[1/3, 1/3], [2/3, 1/3], [2/3, 4/3], [-1, 4/3], [1/3, 1/3]], 0.01, 1, blue),
      xkcd_polyline([[-1/3, -1/3], [4/3, 1/2], [1/2, 1/2], [1/2,-1], [-1/3, -1/3]], 0.01, 1, green),
      axes=none,
      scaling=constrained
    )

     

    # A few circles

    cols  := [red, green, blue, gold, black]:                                # colors
    cents := convert( Statistics:-Sample(Uniform(-1, 3), [5, 2]), listlist): # centers
    radii := Statistics:-Sample(Uniform(1/2, 2), 5):                         # radii
    lcs   := Statistics:-Sample(Uniform(0.2, 0.7), 5):                       # correlations lengths

    display(
      seq(
        xkcd_circle(0.02, lcs[n], radii[n], cents[n], cols[n]),
        n=1..5
      ),
      axes=none,
      scaling=constrained
    )

     

    # A 3D drawing

    x_axis := xkcd_line([[0, 0], [5, 0]], 0.01, 0.2, black):
    y_axis := xkcd_line([[0, 0], [4, 2]], 0.01, 0.2, black):
    z_axis := xkcd_line([[0, 0], [0, 5]], 0.01, 0.2, black):

    f1 := 4*cos(x/6)-1:
    F1 := xkcd_func(f1, [0.5, 5], 6, 0.001, 0.8, red):
    F2 := xkcd_line([[0.5, eval(f1, x=0.5)], [3, 4]], 0.01, 0.2, red):
    f3 := 4*cos((x-2)/6):
    F3 := xkcd_func(f3, [3, 7], 6, 0.001, 0.8, red):
    F4 := xkcd_line([[5, eval(f1, x=5)], [7, eval(f3, x=7)]], 0.01, 0.2, red):

    dx := xkcd_line([[2, 1], [4, 1]], 0.01, 0.2, gray, lsty=3):
    dy := xkcd_line([[2, 0], [4, 1]], 0.01, 0.2, gray, lsty=3):
    dz := xkcd_line([[4, 1], [4, 3]], 0.01, 0.2, gray, lsty=3):

    po := xkcd_circle(0.02, 0.3, 0.1, [4, 3], blue):

    # Numerical value come from "probe info + copy/paste"

    nvect   := xkcd_polyline([[4, 3], [4.57, 4.26], [4.35, 4.1], [4.57, 4.26], [4.58, 4.02]], 0.01, 1, blue):
    tg1vect := xkcd_polyline([[4, 3], [4.78, 2.59], [4.49, 2.87], [4.78, 2.59], [4.46, 2.57]], 0.01, 1, blue):
    tg2vect := xkcd_polyline([[4, 3], [4.79, 3.35], [4.70, 3.13], [4.79, 3.35], [4.46, 3.35]], 0.01, 1, blue):
    rec1    := xkcd_polyline([[4.118, 3.286], [4.365, 3.396], [4.257, 3.108]], 0.01, 1, blue):
    rec2    := xkcd_polyline([[4.257, 3.108], [4.476, 2.985], [4.259, 2.876]], 0.01, 1, blue):



    display(
      x_axis, y_axis, z_axis,
      F1, F2, F3, F4,
      dx, dy, dz,
      po,
      nvect, tg1vect, tg2vect, rec1, rec2,
      axes=none,
      scaling=constrained
    )

     

    # Arrow

    d1 := xkcd_polyline([[0, 0], [1, 0], [0.9, 0.05], [1, 0], [0.9, -0.05]], 0.01, 1, red):


    T := (a, x0, y0, l) ->
                 plottools:-transform(
                   (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
                 ):


    display(
      seq( T(2*Pi*n/10, 0.5, 0, 1/2)(
               display(
                  xkcd_polyline(
                      [[0, 0], [1, 0], [0.9, 0.05], [1, 0], [0.9, -0.05]],
                      0.01,
                      1,
                      ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])
                   )
               )
            ),
           n=1..10
      ),
      axes=none,
      scaling=constrained
    )

     

     


     

    Download XKCD.mw

     

    An attempt to find the equation of an ellipse inscribed in a given triangle. 
    The program works on the basis of the ELS procedure.  After the procedure works, the  solutions are filtered.
    ELS procedure solves the system of equations f1, f2, f3, f4, f5 for the coefficients of the second-order curve.
    The equation f1 corresponds to the condition that the side of the triangle intersects t a curve of the second order at one point.
    The equation f2 corresponds to the condition that the point x1,x2  belongs to a curve of the second order.
    Equation f3 corresponds to the condition that the side of the triangle is tangent to the second order curve at the point x1,x2.
    The equation f4 is similar to the equation f2, and the equation f5 is similar to the equation f3.
    FOR_ELL_ТR_PROCE.mw
    For example

    @ianmccr posted here about making help for a package using makehelp. Here I show how to do this with the HelpTools package.

    The attached worksheet shows how to create the help database for the Orbitals package available at the Applications Center or in the Maple Cloud. The help pages were created as worksheets - start using an existing help page as a model - use View/Open Page As Worksheet and then save from there. The topics and other information are entered by adding Attributes under File/Document Properties - for example for the realY help page these are:

    Active=false means a regular help page; Active=true means an example worksheet.

    There may be several aliases, for example the cartesion help page also describes the fullcartesian command and so Alias is: Orbitals[fullcartesian],cartesian,fullcartesian and Keyword is: Orbitals,cartesian,fullcartesian

    Once a worksheet is created for each help page they are assembled into the help database with the attached file. More information is in the attached file

    Orbitals_Make_help_database.mw

    MacDude posted a worksheet for creating help files using the makehelp command that I found very useful.  However, his worksheet created a table of contents that organized the command help pages under a single folder.  I wanted to create a help file table of contents with the commands organized into sub-folders under the main application folder. ie. Package Name,Folder Name, command. The help page for makehelp is not very informative; in particular the example that shows (purportedly) how to override the existing help pages with your own help file is very misleading.  Also, the description of the parameters to the browser option of the makehelp command is too vague. I needed an error message to tell me that the browser option expects parameters in the form List(Name,String).  In the end, I was able to get a folder/sub-folder structure using a structure [`Folder Name`,`Subfolder Name`,"Command"].  Sub-folders are sorted alphabetically. If anyone has a need, the attached worksheet shows how I created the table of contents structure. 

    helpcode.mw

    I like tweaking plots to get the look and feel I want, and luckily Maple has many plotting options that I often play with. Here, I visualize the same data several times, but each time with different styling.

    First, some data.

    restart:
    data_1 := [[0,0],[1,2],[2,1.3],[3,6]]:
    data_2 := [[0.5,3],[1,1],[2,5],[3,2]]:
    data_3 := [[-0.5,3],[1.3,1],[2.5,5],[4.5,2]]:

    This is the default look.

    plot([data_1, data_2, data_3])

    I think the darker background on this plot makes it easier to look at.

    gray_grid :=
     background      = "LightGrey"
    ,color           = [ ColorTools:-Color("RGB",[150/255, 40 /255, 27 /255])
                        ,ColorTools:-Color("RGB",[0  /255, 0  /255, 0  /255])
                        ,ColorTools:-Color("RGB",[68 /255, 108/255, 179/255]) ]
    ,axes            = frame
    ,axis[2]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
    ,axis[1]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
    ,axesfont        = [Arial]
    ,labelfont       = [Arial]
    ,size            = [400*1.78, 400]
    ,labeldirections = [horizontal, vertical]
    ,filled          = false
    ,transparency    = 0
    ,thickness       = 5
    ,style           = line:
    
    plot([data_1, data_2, data_3], gray_grid);

    I call the next style Excel, for obvious reasons.

    excel :=
     background      = white
    ,color           = [ ColorTools:-Color("RGB",[79/255,  129/255, 189/255])
                        ,ColorTools:-Color("RGB",[192/255, 80/255,   77/255])
                        ,ColorTools:-Color("RGB",[155/255, 187/255,  89/255])]
    ,axes            = frame
    ,axis[2]         = [gridlines = [10, thickness = 0, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
    ,font            = [Calibri]
    ,labelfont       = [Calibri]
    ,size            = [400*1.78, 400]
    ,labeldirections = [horizontal, vertical]
    ,transparency    = 0
    ,thickness       = 3
    ,style           = point
    ,symbol          = [soliddiamond, solidbox, solidcircle]
    ,symbolsize      = 15:
    
    plot([data_1, data_2, data_3], excel)

    This style makes the plot look a bit like the oscilloscope I have in my garage.

    dark_gridlines :=
     background      = ColorTools:-Color("RGB",[0,0,0])
    ,color           = white
    ,axes            = frame
    ,linestyle       = [solid, dash, dashdot]
    ,axis            = [gridlines = [10, linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
    ,font            = [Arial]
    ,size            = [400*1.78, 400]:
    
    plot([data_1, data_2, data_3], dark_gridlines);

    The colors in the next style remind me of an Autumn morning.

    autumnal :=
     background      =  ColorTools:-Color("RGB",[236/255, 240/255, 241/255])
    ,color           = [  ColorTools:-Color("RGB",[144/255, 54/255, 24/255])
                         ,ColorTools:-Color("RGB",[105/255, 108/255, 51/255])
                         ,ColorTools:-Color("RGB",[131/255, 112/255, 82/255]) ]
    ,axes            = frame
    ,font            = [Arial]
    ,size            = [400*1.78, 400]
    ,filled          = true
    ,axis[2]         = [gridlines = [10, thickness = 1, color = white]]
    ,axis[1]         = [gridlines = [10, thickness = 1, color = white]]
    ,symbol          = solidcircle
    ,style           = point
    ,transparency    = [0.6, 0.4, 0.2]:
    
    plot([data_1, data_2, data_3], autumnal);

    In honor of a friend and ex-colleague, I call this style "The Swedish".

    swedish :=
     background      = ColorTools:-Color("RGB", [0/255, 107/255, 168/255])
    ,color           = [ ColorTools:-Color("RGB",[169/255, 158/255, 112/255])
                        ,ColorTools:-Color("RGB",[126/255,  24/255,   9/255])
                        ,ColorTools:-Color("RGB",[254/255, 205/255,   0/255])]
    ,axes            = frame
    ,axis            = [gridlines = [10, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
    ,font            = [Arial]
    ,size            = [400*1.78, 400]
    ,labeldirections = [horizontal, vertical]
    ,filled          = false
    ,thickness       = 10:
    
    plot([data_1, data_2, data_3], swedish);

    This looks like a plot from a journal article.

    experimental_data_mono :=
    
    background       = white
    ,color           = black
    ,axes            = box
    ,axis            = [gridlines = [linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
    ,font            = [Arial, 11]
    ,legendstyle     = [font = [Arial, 11]]
    ,size            = [400, 400]
    ,labeldirections = [horizontal, vertical]
    ,style           = point
    ,symbol          = [solidcircle, solidbox, soliddiamond]
    ,symbolsize      = [15,15,20]:
    
    plot([data_1, data_2, data_3], experimental_data_mono, legend = ["Annihilation", "Authority", "Acceptance"]);

    If you're willing to tinker a little bit, you can add some real character and personality to your visualizations. Try it!

    I'd also be very interested to learn what you find attractive in a plot - please do let me know.

    I'm really fed up with installing the Physics package!

    My install directory of Maple is C:\Maths\Maple

    I don't want to do it manually each time a new Physics Package is installed.

    libname;
    "C:\Maths\Maple\lib", "C:\Users\jm\maple\toolbox\GRTensorIII\lib"

    Physics:-Version();
    The "Physics Updates" version "717" is installed but is not

       active. The active version of Physics is within the library

       C:\Maths\Maple\lib\maple.mla, created 2020, March 5, 2:36 hours

    Please software developpers at Maplesoft do something...Before rel 713 everything was OK.

    Thank you and kind regards to each and every one.

     

    Hi, 

    I would like to share this work I've done. 
    No big math here, just a demonstrator of Maple's capabilities in 3D visualization.

    All the plots in the file have been discarded to reduce the size of this post. Here is a screen capture to give you an idea of what is inside the file.

    Download 3D_Visualization.mw

    Hi, 

    In a recent post  (Monte Carlo Integration) Radaar shared its work about the numerical integration, with the Monte Carlo method, of a function defined in polar coordinates.
    Radaar used a raw strategy based on a sampling in cartesian coordinates plus an ad hoc transformation.
    Radaar obtained reasonably good results, but I posted a comment to show how Monte Carlo summation in polar coordinates can be done in a much simpler way. Behind this is the choice of a "good" sampling distribution which makes the integration problem as simple as Monte Carlo integration over a 2D rectangle with sides parallel to the co-ordinate axis.

    This comment I sent pushed me to share the present work on Monte Carlo integration over simple polygons ("simple" means that two sides do not intersect).
    Here again one can use raw Monte Carlo integration on the rectangle this polygon is inscribed in. But as in Radaar's post, a specific sampling distribution can be used that makes the summation method more elegant.

    This work relies on three main ingredients:

    1. The Dirichlet distribution, whose one form enables sampling the 2D simplex in a uniform way.
    2. The construction of a 1-to-1 mapping from this simplex into any non degenerated triangle (a mapping whose jacobian is a constant equal to the ratio of the areas of the two triangles).
    3. A tesselation into triangles of the polygon to integrate over.


    This work has been carried out in Maple 2015, which required the development of a module to do the tesselation. Maybe more recent Maple's versions contain internal procedures to do that.
     

    Monte_Carlo_Integration.mw

     

    This is an animation of the spread of the COVID-19 over the U.S. in the first 150 days.  It was created in Maple 2020, making extensive use of DataFrames. 

     

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

     

    The animation of 150 Day history includes COVID-19 data published by the NY Times and geographic data assembled from other sources. Each cylinder represents a county or in two special cases New York City and Kansas City. The cross-sectional area of each cylinder is the area in square miles of the corresponding county. The height of each cylinder is on a logarithmic scale (in particular it is 100*log base 2 of the case count for the county. The argument of the logarithm function is the number of cases per county divided by the are in square miles-so an areal density.  Using a logarithmic scale facilitates showing super high density areas (e.g., NYC) along with lower density areas.  The heights are scaled by a prefactor of 100 for visualization.

    Hi. My name is Eugenio and I’m a Professor at the Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas at the Facultad de Educación of the Universidad Complutense de Madrid (UCM) and a member of the Instituto de Matemática Interdisciplinar (IMI) of the UCM.

    I have a 14-year-old son. In the beginning of the pandemic, a confinement was ordered in Spain. It is not easy to make a kid understand that we shouldn't meet our friends and relatives for some time and that we should all stay at home in the first stage. So, I developed a simplified explanation of virus propagation for kids, firstly in Scratch and later in Maple, the latter using an implementation of turtle geometry that we developed long ago and has a much better graphic resolution (E. Roanes-Lozano and E. Roanes-Macías. An Implementation of “Turtle Graphics” in Maple V. MapleTech. Special Issue, 1994, 82-85). A video (in Spanish) of the Scratch version is available from the Instituto de Matemática Interdisciplinar (IMI) web page: https://www.ucm.es/imi/other-activities

    Introduction

    Surely you are uncomfortable being locked up at home, so I will try to justify that, although we are all looking forward going out, it is good not to meet your friends and family with whom you do not live.

    I firstly need to mention a fractal is. A fractal is a geometric object whose structure is repeated at any scale. An example in nature is Romanesco broccoli, that you perhaps have eaten (you can search for images on the Internet). You can find a simple fractal in the following image (drawn with Maple):

    Notice that each branch is divided into two branches, always forming the same angle and decreasing in size in the same proportion.

    We can say that the tree in the previous image is of “depth 7” because there are 7 levels of branches.

    It is quite easy to create this kind of drawing with the so called “turtle geometry” (with a recursive procedure, that is, a procedure that calls itself). Perhaps you have used Scratch programming language at school (or Logo, if you are older), which graphics are based in turtle geometry.

    All drawings along these pages have been created with Maple. We can easily reform the code that generated the previous tree so that it has three, four, five,… branches at each level, instead of two.

    But let’s begin with a tale that explains in a much simplified way how the spread of a disease works.

    - o O o -

    Let's suppose that a cat returns sick to Catland suffering from a very contagious disease and he meets his friends and family, since he has missed them so much.

    We do not know very well how many cats each sick cat infects in average (before the order to STAY AT HOME arrives, as cats in Catland are very obedient and obey right away). Therefore, we’ll analyze different scenarios:

    1. Each sick cat infects two other cats.
    2. Each sick cat infects three other cats.
    3. Each sick cat infects five other cats

     

    1. Each Sick Cat Infects Two Cats

    In all the figures that follow, the cat initially sick is in the center of the image. The infected cats are represented by a red square.

    · Before everyone gets confined at home, it only takes time for that first sick cat to see his friends, but then confinement is ordered (depth 1)

    As you can see, with the cat meeting his friends and family, we already have 3 sick cats.

    · Before all cats confine themselves at home, the first cat meets his friends, and these in turn have time to meet their friends (depth 2)

    In this case, the number of sick cats is 7.

    · Before every cat is confined at home, there is time for the initially sick cat to meet his friends, for these to meet their friends, and for the latter (friends of the friends of the first sick cat) to meet their friends (depth 3).

    There are already 15 sick cats...

    · Depth 4: 31 sick cats.

    · Depth 5: 63 sick cats.

    Next we’ll see what would happen if each sick cat infected three cats, instead of two.

     

    2. Every Sick Cat Infects Three Cats

    · Now we speed up, as you’ve got the idea.

    The first sick cat has infected three friends or family before confining himself at home. So there are 4 infected cats.

    · If each of the recently infected cats in the previous figure have in turn contact with their friends and family, we move on to the following situation, with 13 sick cats:

    · And if each of these 13 infected cats lives a normal life, the disease spreads even more, and we already have 40!

    · At the next step we have 121 sick cats:

    · And, if they keep seeing friends and family, there will be 364 sick cats (the image reminds of what is called a Sierpinski triangle):

     

    4. Every Sick Cat Infects Five Cats

    · In this case already at depth 2 we already have 31 sick cats.

     

    5. Conclusion

    This is an example of exponential growth. And the higher the number of cats infected by each sick cat, the worse the situation is.

    Therefore, avoiding meeting friends and relatives that do not live with you is hard, but good for stopping the infection. So, it is hard, but I stay at home at the first stage too!

    We have just released an update to Maple, Maple 2020.1.

    Maple 2020.1 includes corrections and improvement to the mathematics engine, export to PDF, MATLAB connectivity, support for Ubuntu 20.04, and more.  We recommend that all Maple 2020 users install these updates.

    This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2020.1 download page, where you can also find more details.

    In particular, please note that this update includes a fix to the SMTLIB problem reported on MaplePrimes. Thanks for the feedback!

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