MaplePrimes Posts

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.

Latest Post
  • Latest Posts Feed
  • A new code based on higher derivative method has been implemented in Maple. A sample code is given below and explained. Because of the symbolic nature of Maple, this method works very well for a wide range of BVP problems.

    The code solves BVPs written in the first order form dy/dx = f (Maple’s dsolve numeric converts general BVPs to this form and solves).

    The code can handle unknown parameters in the model if sufficient boundary conditions are provided.

    This code has been tested from Maple 8 to Maple 2017. For Digits:=15 or less, this code works in all of the Maple versions tested.

    Most problems can be solved with Digits:=15 with atol = 1e-10 or so. This code can be used to get a tolerance value of 1e-20 or any high precision as needed by changing the number of Digits accordingly. This may be needed if the original variables are not properly sacled. With arbitrarily high Digits, the code fails in Maple 18 or later version, etc because Maple does not support SparseDirect Solver at high precision in some of the versions (hopefuly this bug can be removed in the future versions).

    For simple problems, Maple’s dsolve/numeric is superior to the code developed as it is implemented in hardware floats. For large scale problems and stiff problems, the method developed is much more superior to Maple and comparable to (and often times better than) state of the art codes for BVPs - bvp4c (MATLAB), COLSYS,TWPBVP, etc.

    The code, as written, cannot be used for problems with a singularity at end points (doable in the future). In addition, mixed boundary conditions are not supported in this version of the code (for example, y1(1)=y2(0)). Future updates will include the application of this approach for DAE-BVPs, currently not supported by Maple’s dsolve/numeric command.

    A paper has been submitted to JCAM. I welcome feedback on the code and solicit input from Mapleprimes members if they are able to test (and break this code) for any BVP.


    PDF of the paper submitted, example maple code and the solver as a text file needed are uploaded here. Additional examples are hosted on my website at




    Troesch's problem
    This is an inherently unstable, difficult, nonlinear, two-point BVP formulated by Weibel and Troesch that describes the confinement of a place column by radiation pressure. Increasing epsilon increases the stiffness of the ODE.
    1. E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Phys. Fluids. 2 (1959) 52-56.
    2. B. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys. 21 (1976) 279-290.

    The package HDM solves boundary value problems (BVPs) using higher derivative methods (HDM) in Maple®. We explain how to solve BVPs using this package. HDM can numerically solve BVPs of ordinary differential equations (ODEs) of the form shown is the fowllowing example.




    Reset the program to clear the memory from previous execution command.



    Read the txt file which contains the HDM solver for BVPs.



    Declare the precision for the entire Maple® sheet.


    Digits := 15



    Enter the first-order ODEs into EqODEs list.


    EqODEs := [diff(y1(x), x) = y2(x), diff(y2(x), x) = epsilon*sinh(epsilon*y1(x))]



    Define the left boundary condition (bc1), and the right boundary condition (bc2). One should collect all the terms in one side.


    bc1 := [y1(x)]



    bc2 := [y1(x)-1.]



    Define the range (bc1 to bc2) of this BVP.


    Range := [0., 1.]



    List any known parameters in the list.


    pars := [epsilon = 2]



    List any unknown parameters in the list. When there is no unknown parameter, use [ ].


    unknownpars := []



    Define the initial derivative in nder (default is 5 for 10th order) and the number of the nodes in nele (default is 10 and distributed evenly across the range provided by the user). The code adapts to increase the order. For many problems, 10th order method with 10 elements are sufficient.


    nder := 5

    nele := 10



    Define the absolute and relative tolerance for the local error. The error calculation is done based on the norm of both the 9th and 10th order simulation results.


    atol := 0.1e-5

    rtol := 0.100000000000000e-7



    Call HDMadapt procedure, input all the information entered above and save the solution in sol. HDMadapt procedure does not need the initial guess for the mesh.

    sol:= HDMadapt(EqODEs,bc1,bc2,pars,unknownpars,nder,nele,Range,atol,rtol):


    Present some details of the solution.

    sol[4]; # final derivative



    sol[5]; # Maximum local RMSE




    Store the dimension of the solution (after adjusting the mesh) to NN.


    NN := 11



    Plot the interested variable (the ath ODE variable will be sol[1][i+NN*(a-1)] )


    node := 2

    odevars := [y1(x), y2(x)]




    for i from 1 to nops(sol[3]) do xx[i+1]:=xx[i]+sol[3][i]: od:

    for j from 1 to node do
    end do;




    Download Example_3_Troesch.mws


    Was just pondering this idea and posted this in the post topic for discussion. 

    Each Maple finished version of Maple may still have certain bugs that will not be updated for that version, so I am suggesting (I think anyone could implement it) that if there is a workaround, one could wrap it up in something I would call a patch package updateable by us users we could update here on mapleprimes.  It would be good for people who haven't upgraded or can't upgrade due to costs etc...

    For example, there was recent issue with pdsolve that was fixed quite quickly in the seperate updateable Physics package.  Things could be done similarily that might work with other workarounds using this patch package idea. 

    If anyone thinks this is good or even viable idea then lets implement it.  I envisioned it with just this one rule to follow - the name of the patch package would reflect the version we are patching (ie. with(patch12) or with(patch2016) for Maple 12 and Maple 2016 respectively etc...)  We could make these patch packages available in this post or start another.

    As I said, I'm just throwing the idea out there.  Thoughts?

    There seems to be a bug with improper integration:




    Substituting any number for x, or assuming x >= 0  (or x<=0) does give the correct result,

    The problem also persists when assuming x>-1 (or x>-Maple_floats(MIN_FLOAT))

    Hi MaplePrimes,


    These two files have the same content.  One is a .pdf and the other is a Maple Worksheet.  I explore integer sequences of the form - 

    a(r) = c*a(r-1)+d*a(r-2) with a(1) and a(2) given.

    Some of these sequences are in (the Online Encyclopedia of Integer Sequences) and some are not.  If we restrict c to 1 and assume that a(1)=1 and a(2) = 2 we have the parameter d remaining.  See additional webpage -

    Let me know if you like the code.





    Books free. Like!!!

    Lenin Araujo Castillo

    As you can see this app performs the trace of a given path r (t), then locate the position vector in a specific time. It also graphs the velocity vector, acceleration, Tangential and Normal unit vectors, along with the Binormal. Very good app developed entirely in Maple for our engineering students.

    Lenin Araujo Castillo

    Ambassador of Maple

    It seems a large number of people, when initially using maple, wrongly deduce that for example sin(60) is the sin of 60 degrees and not the sin of 1/3 Pi radians.  I believe mathematica's default is degrees.  When a student compares an expression to another but forgets to realize a value is read as radians and not degrees they are perplexed when Maple returns false and Mathematica returns true.

    As a suggestion, under tools->options allow a user to be able to change how maple reads values within trigonometric funtions as either radians or degrees.

    Most times when someone computes the sin(60) what they really mean in Maple..
    is sin(convert(60 degrees, radians))


    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fifth presentation, about "The Appell doubly hypergeometric functions", describes a very recent project I've been working at Maple, i.e. the very first complete computational implementation of the Appell doubly hypergeometric functions. This work appeared in Maple 2017. These functions have a tremendous potential in that, at the same time, they have a myriad of properties, and include as particular cases most of the existing mathematical language, and so they have obvious applications in integration, differential equations, and applied mathematics all around. I think these will be the functions of this XXI century, analogously to what happened with hypergeometric functions in the previous century.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

    The four double-hypergeometric Appell functions,

    a complete implementation in a computer algebra system


    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft


    The four multi-parameter Appell functions, AppellF1 , AppellF2 , AppellF3  and AppellF4  are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric  and some cases of the MeijerG  function, and with them most of the known functions of mathematical physics. Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity. In this talk, a full implementation of these functions in the Maple computer algebra system, including, for the first time, their numerical evaluation over the whole complex plane, is presented, with details about the symbolic and numerical strategies used.

    Appell Functions (symbolic)



    The main references:


    P. Appel, J.Kamke de Feriet, "Fonctions hypergeometriques et Hyperspheriques", 1926


    H. Srivastava, P.W. Karlsson, "Multiple Gaussian Hypergeometric Series", 1985


    24 papers in the literature, ranging from 1882 to 2015


    Definition and Symmetries


    Polynomial and Singular Cases


    Single Power Series with Hypergeometric Coefficients


    Analytic Extension from the Appell Series to the Appell Functions


    Euler-Type and Contiguity Identities


    Appell Differential Equations


    Putting all together


    Problem: some formulas in the literature are wrong or miss the conditions indicating when are they valid (exchange with the Mathematics director of the DLMF - NIST)


    Appell Functions (numeric)






    Compute these Appell functions over the whole complex plane


    Considering that this is a research problem, implement different methods and flexible optional arguments to allow for:

    a) comparison between methods (both performance and correctness),

    b) investigation of a single method in different circumstances.


    Develop a computational structure that can be reused with other special functions (abstract code and provide the main options), and that could also be translated to C (so: only one numerical implementation, not 100 special function numerical implementations)

    Limitation: the Maple original evalf command does not accept optional arguments


    The cost of numerically evaluating an Appell function



    If it is a special hypergeometric case, then between 1 to 2 hypergeometric functions


    Next simplest case (series/recurrence below) 3 to 4 hypergeometric functions plus adding somewhat large formulas that involve only arithmetic operations up to 20,000 times (frequently less than 100 times)


    Next simplest case: the formulas themselves are power series with hypergeometric function coefficients; these cases frequently converge rapidly but may involve the numerical evaluation of up to hundreds of hypergeometric functions to get the value of a single Appell function.


    Strategy for the numerical evaluation of Appell functions (or other functions ...)



    The numerical evaluation flows orderly according to:

    1) check whether it is a singular case

    2) check whether it is a special value

    3) compute the value using a series derived from a recurrence related to the underlying ODE

    4) perform an sum using an infinite sum formula, checking for convergence

    5) perform the numerical integration of the ODE underlying the given Appell function

    6) perform a sequence of concatenated Taylor series expansions





    Numerical integration of an underlying differential equation (ODEs and dsolve/numeric)


    Concatenated Taylor series expansions covering the whole complex plane




    Improvements in the numerical evaluation of hypergeometric functions


    Evalf: an organized structure to implement the numerical evaluation of special functions in general


    To be done



    Download Appell_Functions.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This fourth presentation, about "The FunctionAdvisor: extending information on mathematical functions with computer algebra algorithms", describes the FunctionAdvisor project at Maple, a project I started working during 1998, where the key idea I am trying to explore is that we do not need to collect a gazillion of formulas but just core blocks of mathematical information surrounded by clouds of algorithms able to derive extended information from them. In this sense this is also unique piece of software: it can derive properties for rather general algebraic expressions, not just well known tabulated functions. The examples illustrate the idea.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

    The FunctionAdvisor: extending information on mathematical functions

    with computer algebra algorithms


    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft



    A shift in paradigm is happening, from: encoding information into a database, to: encoding essential blocks of information together with algorithms within a computer algebra system. Then, the information is not only searchable but can also be recreated in many different ways and actually used to compute. This talk focuses on this shift in paradigm over a real case example: the digitizing of information regarding mathematical functions as the FunctionAdvisor project of the Maple computer algebra system.

    The FunctionAdvisor (basic)


    Beyond the concept of a database


    " Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

    Formal power series for algebraic expressions


    Differential polynomial forms for algebraic expressions


    Branch cuts for algebraic expressions


    The nth derivative problem for algebraic expressions


    Conversion network for mathematical and algebraic expressions






    Download FunctionAdvisor.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra -2017" . It was a very interesting event. This third presentation, about "Computer Algebra in Theoretical Physics", describes the Physics project at Maplesoft, also my first research project at University, that evolved into the now well-known Maple Physics package. This is a unique piece of software and perhaps the project I most enjoy working.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.



    Computer Algebra in Theoretical Physics


    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft




    Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system have implemented most of the mathematical objects and mathematics used in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum.


    In this talk, the Physics project at Maplesoft is presented and the resulting Physics package is illustrated by tackling problems in classical and quantum mechanics, using tensor and Dirac's Bra-Ket notation, general relativity, including the equivalence problem, and classical field theory, deriving field equations using variational principles.





    ... and why computer algebra?


    We can concentrate more on the ideas instead of on the algebraic manipulations


    We can extend results with ease


    We can explore the mathematics surrounding a problem


    We can share results in a reproducible way


    Representation issues that were preventing the use of computer algebra in Physics


    Classical Mechanics


    *Inertia tensor for a triatomic molecule


    Quantum mechanics


    *The quantum operator components of  `#mover(mi("L",mathcolor = "olive"),mo("&rarr;",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `&epsilon;`[j,k,m] L[m]"


    *Unitary Operators in Quantum Mechanics


    *Eigenvalues of an unitary operator and exponential of Hermitian operators


    *Properties of unitary operators



    Consider two set of kets " | a[n] >" and "| b[n] >", each of them constituting a complete orthonormal basis of the same space.

    *Verify that "U=(&sum;) | b[k] >< a[k] |" , maps one basis to the other, i.e.: "| b[n] >=U | a[n] >"


    *Show that "U=(&sum;) | b[k] > < a[k] | "is unitary


    *Show that the matrix elements of U in the "| a[n] >" and  "| b[n] >" basis are equal


    Show that A and `&Ascr;` = U*A*`#msup(mi("U"),mo("&dagger;"))`have the same spectrum (eigenvalues)


    Schrödinger equation and unitary transform


    Translation operators using Dirac notation


    *Quantization of the energy of a particle in a magnetic field


    Classical Field Theory


    The field equations for the lambda*Phi^4 model


    *Maxwell equations departing from the 4-dimensional Action for Electrodynamics


    *The Gross-Pitaevskii field equations for a quantum system of identical particles


    General Relativity


    Exact Solutions to Einstein's Equations  Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]


    *"Physical Review D" 87, 044053 (2013)


    The Equivalence problem between two metrics


    *On the 3+1 split of the 4D Einstein equations


    Tetrads and Weyl scalars in canonical form






    Download Physics.pdf

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

    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This second presentation, about "Differential algebra with mathematical functions, symbolic powers and anticommutative variables", describes a project I started working in 1997 and that is at the root of Maple's dsolve and pdsolve performance with systems of equations. It is a unique approach. Not yet emulated in any other computer algebra system.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.

    Differential algebra with mathematical functions,

    symbolic powers and anticommutative variables


    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft


    Computer algebra implementations of Differential Algebra typically require that the systems of equations to be tackled be rational in the independent and dependent variables and their partial derivatives, and of course that A*B = A*B, everything is commutative.


    It is possible, however, to extend this computational domain and apply Differential Algebra techniques to systems of equations that involve arbitrary compositions of mathematical functions (elementary or special), fractional and symbolic powers, as well as anticommutative variables and functions. This is the subject of this presentation, with examples of the implementation of these ideas in the Maple computer algebra system and its ODE and PDE solvers.



    restartwith(PDEtools); interface(imaginaryunit = i)

    sys := [diff(xi(x, y), y, y) = 0, -6*(diff(xi(x, y), y))*y+diff(eta(x, y), y, y)-2*(diff(xi(x, y), x, y)) = 0, -12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(eta(x, y), x, y))-(diff(xi(x, y), x, x)) = 0, -8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(eta(x, y), x, x) = 0]


    declare((xi, eta)(x, y))

    xi(x, y)*`will now be displayed as`*xi


    eta(x, y)*`will now be displayed as`*eta


    for eq in sys do eq end do

    diff(diff(xi(x, y), y), y) = 0


    -6*(diff(xi(x, y), y))*y+diff(diff(eta(x, y), y), y)-2*(diff(diff(xi(x, y), x), y)) = 0


    -12*(diff(xi(x, y), y))*a^2*y-9*(diff(xi(x, y), y))*a*y^2-3*(diff(xi(x, y), y))*b-3*(diff(xi(x, y), x))*y-3*eta(x, y)+2*(diff(diff(eta(x, y), x), y))-(diff(diff(xi(x, y), x), x)) = 0


    -8*(diff(xi(x, y), x))*a^2*y-6*(diff(xi(x, y), x))*a*y^2+4*(diff(eta(x, y), y))*a^2*y+3*(diff(eta(x, y), y))*a*y^2-4*eta(x, y)*a^2-6*eta(x, y)*a*y-2*(diff(xi(x, y), x))*b+(diff(eta(x, y), y))*b-3*(diff(eta(x, y), x))*y+diff(diff(eta(x, y), x), x) = 0



    `casesplit/ans`([eta(x, y) = 0, diff(xi(x, y), x) = 0, diff(xi(x, y), y) = 0], [])



    Differential polynomial forms for mathematical functions (basic)


    Differential polynomial forms for compositions of mathematical functions


    Generalization to many variables


    Arbitrary functions of algebraic expressions


    Examples of the use of this extension to include mathematical functions


    Differential Algebra with anticommutative variables




    Download DifferentialAlgebra.pdf

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


    I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This first presentation, about "Active Learning in High-School Mathematics using Interactive Interfaces", describes a project I started working 23 years ago, which I believe will be part of the future in one or another form. This is work actually not related to my work at Maplesoft.

    At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.


    Active learning in High-School mathematics using Interactive Interfaces


    Edgardo S. Cheb-Terrab

    Physics, Differential Equations and Mathematical Functions, Maplesoft



    The key idea in this project is to learn through exploration using a web of user-friendly Highly Interactive Graphical Interfaces (HIGI). The HIGIs, structured as trees of interlinked windows, present concepts using a minimal amount of text while maximizing the possibility of visual and analytic exploration. These interfaces run computer algebra software in the background. Assessment tools are integrated into the learning experience within the general conceptual map, the Navigator. This Navigator offers students self-assessment tools and full access to the logical sequencing of course concepts, helping them to identify any gaps in their knowledge and to launch the corresponding learning interfaces. An interactive online set of HIGIS of this kind can be used at school, at home, in distance education, and both individually and in a group.



    Computer algebra interfaces for High-School students of "Colegio de Aplicação"  (UERJ/1994)





    When we are the average high-school student facing mathematics, we tend to feel



    Bored, fragmentarily taking notes, listening to a teacher for 50 or more minutes


    Anguished because we do not understand some math topics (too many gaps accumulated)


    Powerless because we don't know what to do to understand (don't have any instant-tutor to ask questions and without being judged for having accumulated gaps)


    Stressed by the upcoming exams where the lack of understanding may become evident


    Computer algebra environments can help in addressing these issues.




    Be as active as it can get while learning at our own pace.


    Explore at high speed and without feeling judged. There is space for curiosity with no computational cost.


    Feel empowered by success. That leads to understanding.


    Possibility for making of learning a social experience.


    Interactive interfaces




    Interactive interfaces do not replace the teacher - human learning is an emotional process. A good teacher leading good active learning is a positive experience a student will never forget



    Not every computer interface is a valuable resource, at all. It is the set of pedagogical ideas implemented that makes an interface valuable (the same happens with textbooks)



    A course on high school mathematics using interactive interfaces - the Edukanet project




    Brazilian and Canadian students/programmers were invited to participate - 7 people worked in the project.



    Some funding provided by the Brazilian Research agency CNPq.


    -Develop a framework to develop the interfaces covering the last 3 years of high school mathematics (following the main math textbook used in public schools in Brazil)

    - Design documents for the interfaces according to given pedagogical guidelines.

    - Create prototypes of Interactive interfaces, running Maple on background, according to design document and specified layout (allow for everybody's input/changes).


    The pedagogical guidelines for interactive interfaces


    The Math-contents design documents for each chapter


    Example: complex numbers


    Each math topic:  a interactive interrelated interfaces (windows)



    For each topic of high-school mathematics (chapter of a textbook), develop a tree of interactive interfaces (applets) related to the topic (main) and subtopics


    Example: Functions



    Main window



    Analysis window




    Parity window


    Visualization of function's parity


    Step-by-Step solution window

    The Navigator: a window with a tile per math topic





    Click the topic-tile to launch a smaller window, topic-specific, map of interrelated sub-topic tiles, that indicates the logical sequence for the sub-topics, and from where one could launch the corresponding sub-topic interactive interface.


    This topic-specific smaller window allows for identifying the pre-requisites and gaps in understanding, launching the corresponding interfaces to fill the gaps, and tracking the level of familiarity with a topic.






    The framework to create the interfaces: a version of NetBeans on steroids ...


    Complementary classroom activity on a computer algebra worksheet



    This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do with paper and pencil in high school and 1st year of undergraduate science courses.


    Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the learning of mathematics into interesting understanding, success and fun.

    1. Arithmetic operations and elementary functions


    2. Algebraic Expressions, Equations and Functions


    3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations


    4. Algebraic manipulation: simplify, factorize, expand


    5. Matrices (Linear Algebra)



    Advanced students: guiding them to program mathematical concepts on a computer algebra worksheet


    Status of the project



    Prototypes of interfaces built cover:



    Natural numbers




    Integer numbers


    Rational numbers


    Absolute value




    Numerical sequences








    Linear systems






    Derivative of the inverse function


    The point in Cartesian coordinates


    The line


    The circle


    The ellipse


    The parabole


    The hyperbole


    The conics

    More recent computer algebra frameworks: Maple Mobius for online courses and automated evaluation





    Download Computer_Algebra_in_Education.pdf

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

    We have just released an update to Maple. Maple 2017.2 includes updated translations for Japanese, Traditional Chinese, Simplified Chinese, Brazilian Portuguese, French, and Spanish. It also contains improvements to the MapleCloud, physics, limits, and PDEs. This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2017.2 download page.



    A geometric construction for the Summer Holiday

    Does every plane simple closed curve contain all four vertices of some square?

     This is an old classical conjecture. See:

    Maybe someone finds a counterexample (for non-analytic curves) using the next procedure and becomes famous!


    SQ:=proc(X::procedure, Y::procedure, rng::range(realcons), r:=0.49)
    local t1:=lhs(rng), t2:=rhs(rng), a,b,c,d,s;
    s:=fsolve({ X(a)+X(c) = X(b)+X(d),
                Y(a)+Y(c) = Y(b)+Y(d),
                (X(a)-X(c))^2+(Y(a)-Y(c))^2 = (X(b)-X(d))^2+(Y(b)-Y(d))^2,
                (X(a)-X(c))*(X(b)-X(d)) + (Y(a)-Y(c))*(Y(b)-Y(d)) = 0},
              {a=t1..t1+r*(t2-t1),b=rng,c=rng,d=t2-r*(t2-t1)..t2});  #lprint(s);
    if type(s,set) then s:=rhs~(s)[];[s,s[1]] else WARNING("No solution found"); {} fi;




    X := t->(10-sin(7*t)*exp(-t))*cos(t);
    Y := t->(10+sin(6*t))*sin(t);
    rng := 0..2*Pi;

    proc (t) options operator, arrow; (10-sin(7*t)*exp(-t))*cos(t) end proc


    proc (t) options operator, arrow; (10+sin(6*t))*sin(t) end proc


    0 .. 2*Pi


    s:=SQ(X, Y, rng):
       plot([X,Y,rng], scaling=constrained),
       plot([seq( eval([X(t),Y(t)],t=u),u=s)], color=blue, thickness=2));


    It appears google doesn't know about the haversine formula.  Huh?  Well at least google can't draw the proper path for it.  I typed in google "distance from Pyongyang to NewYork city"  and got 10,916km.  Ok that's fine but then it drew a map

    The map path definitely did not look right.  Pulled out my globe traced a rough path of the one google showed and I got 13 inches (where 1 inch=660miles) -> 8580 miles = 13808 km .. clearly looks like google goofed. 

    So we need Maple to show us the proper path.

    m := WorldMap();
    AddPath(m, [-74.0059, 40.7128], [125.7625, 39.0392]):

    Ok so you say that really doesn't look like the shortest path.  Well, lets visualize that on the globe projection

    Display(m, projection = Globe, orientation = [-180, 0, 0])