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

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  • The Mobius strip

    Variants :

    The line and the curve on the surface.


    Recently, I came across an addendum to a problem that appears in many calculus texts, an addendum I had never explored. It intrigued me, and I hope it will capture your attention too.

    The problem is that of girding the equator of the earth with a belt, then extending by one unit (here, taken as the foot) the radius of the circle so formed. The question is by how much does the circumference of the belt increase. This problem usually appears in the section of the calculus text dealing with linear approximations by the differential. It turns out that the circumference of the enlarged band is 2*Pi ft greater than the original band.

    (An alternate version of this has the circumference of the band increased by one foot, with the radius then being increased by 0.16 ft.)

    The addendum to the problem then asked how high would the enlarged band be over the surface of the earth if it were lifted at one point and drawn as tight as possible around the equator. At first, I didn't know what to think. Would the height be some surprisingly large number? And how would one go about calculating this height.

    It turns out that the enlarged and lifted band would be some 616.67 feet above the surface of the earth! This is significantly larger than the increase in the diameter of the original band. So, the result is a surprise, at least to me.

    This is the kind of amusement that retirement affords. I heartily recommend both the amusement and the retirement. The supporting calculations can be found in the attached worksheet:

    Let us consider 

    MultiSeries:-limit(sin(n)/n, n = infinity, complex);

    The answer is wrong: in view of the Casorati-Weierstrass theorem the limit does not exist. Let us try another limit command of Maple

    limit(sin(n)/n, n = infinity, complex);
    (lim) (sin(n))/(n)

    which fails. Therefore, Maple user does not obtain the correct answer. 

    Suppose we have some simple animations. Our goal - to build a more complex animation, combining the original animations in different ways.
    We show how to do it on the example of the three animations. The technique is general and can be applied to any number of animations.

    Here are the three simple animations:

    A:=animate(plot, [sin(x), x=-Pi..a, color=red, thickness=3], a=-Pi..Pi):
    B:=animate(plot, [x^2-1, x=-2..a, thickness=3, color=green], a=-2..2): 
    C:=animate(plot, [[4*cos(t),4*sin(t), t=0..a], color=blue, thickness=3], a=0..2*Pi):


    In Example 1 all three animation executed simultaneously:

    display([A, B, C], view=[-4..4,-4..4]);



    In Example 2, the same animation performed sequentially. Note that the previous animation disappears completely when the next one begins to execute:

    display([A, B, C], insequence);



    Below we show how to save the last frame of every previous animation into subsequent animations:

    display([A, display(op([1,-1,1],A),B), display(op([1,-1,1],A),op([1,-1,1],B),C)], insequence);



    Using this technique, we can anyhow combine the original animations. For example, in the following example at firstly animations   and  B  are executed simultaneously, afterwards C is executed:

    display([display(A, B), display(op([1,-1,1],A),op([1,-1,1],B),C)], insequence);



    The last example in 3D I have taken from here:

    A:=animate(plot3d,[[2*cos(phi),2*sin(phi),z], z =0..a, phi=0..2*Pi, style=surface, color=red], a=0..5):
    B:=animate(plot3d,[[(2+6/5*(z-5))*cos(phi), (2+6/5*(z-5))*sin(phi),z], z=5..a, phi=0..2*Pi, style=surface, color=blue], a=5..10):
    C:=animate(plot3d,[[8*cos(phi),8*sin(phi),z], z =10..a, phi=0..2*Pi, style=surface, color=green], a=10..20):
    display([A, display(op([1,-1,1],A),B), display(op([1,-1,1],A),op([1,-1,1],B),C)], insequence, scaling=constrained, axes=normal);



    We have just released an update to Maple.  It includes updates to the Maple Workbook, the video component, the Physics package, and many other small improvements throughout the product. It is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2016.2 download page.


    We have just released a major update to MapleSim and the MapleSim family of products. This update includes significant enhancements in the areas of model development and toolchain connectivity, including:

    • Live simulations let you see results as the simulation is running, so you can track progress and react to problems immediately.
    • A new 3-D overlay option lets you easily compare simulation visualizations by overlaying one visualization on top of another
    • Tools for revision control enable a structured approach to managing and tracking changes to your model, making it easier to manage projects when multiple engineers are working on the same model and reducing development risk.
    • MapleSim now supports direct import of models created in other FMI-compatible software, providing even greater cross-tool compatibility and opportunities for co-simulation.
    • The MapleSim Connector, for connectivity with Simulink®, and the MapleSim Connector for FMI, for exporting MapleSim models to other FMI-compatible tools, have been expanded to allow you to explore simulation results involving exported MapleSim models from within MapleSim, even though the simulation was done in the target tool.


    This update is being distributed through the automatic Check for Updates system, and is also available from our website. See the MapleSim 2016.2  downloads page for details on obtaining this update.



    I'd like to pay attention of Maple community to the recent work by Alex Degtyarev in algebraic geometry done with Maple.

    I am pleased to announce that we have just released a significant update to Maple T.A. 2016, our online assessment system.

    Maple T.A. 2016.1 includes a wide range of features and improvements that have been requested by customers, including new options for questions and assignments, improved content management, and enhanced integration with course management systems. It also includes a substantial number of small enhancements and corrections across all areas of the product, providing improved responsiveness, more efficient load handling, and smoother workflow for instructors and students.

    For more information, visit What’s New in Maple T.A.

    Jonny Zivku
    Product Manager, Online Education Products

    Let us consider 

    U := RandomVariable(DiscreteUniform(-10, 10)):
    V := RandomVariable(DiscreteUniform(-10, 10)):
    Probability(U^2-V^2 <= 1/9, numeric);

    , whereas a positive number greater than 1/21 is expected. 


    Let us consider the example from Maple help to ?ProbabilityFunction (also see ?Geometric)

    ProbabilityFunction(Geometric(1/3), 5);
                                  32 /729

    Let us continue the investigation

    ProbabilityFunction(Geometric(1/3), 5.1);
    ProbabilityFunction(Geometric(1/3), 5.12);
    ProbabilityFunction(Geometric(1/3), 51/10)

    whereas the result 0 is expected in all the three cases up to Wiki. I am aware of the line

    "t-algebraic; point (assumed to be an integer)"

    in the help. However, 

    ProbabilityFunction(Geometric(1/3), -.5);

    The same issue with the DiscreteUniform distribution. This bug lasts from  at least Maple 16. The question arises: may we trust Maple?

    HI and other watchers,

    Please enjoy the attaced files about combinatorics.
    You may already know what '4 choose 3' is.


    Hopefully this can be useful to the casual mathematical observer.




    Graphical Programming with MapleSim in Vector Mechanics to Structures 2D

    At the present time before constructing or starting to develop a mechanical structures project it is necessary to model it using graphic programming; In this opportunity and used MapleSim as a computational tool belonging to the company Maplesoft. The modern approach to modeling and simulation makes the fabrication of complex designs easy to solve. We will cover some examples taken from the engineering being implemented in Maplesim with insertion of physical objects; To be seen in real time through video output; Then integrates with Maple to analyze the equations and data through the static and dynamic behavior of the fabricated. Solved methods of physical block components include functionality for many domains: rotational and translational mechanics, multi-body dynamics, logic, and structural blocks; With techniques like: Drag-and-Drop Physical Modeling Environment and Create Custom Components Directly From Their Equations, thus the systems that would take hours or days to build from equations; In principle they can be created in a fraction of time using MapleSim, so it can incorporate significantly more complex graphical algorithms. In MapleSim, I use the revolutionary multibody technology that perfectly combines advanced multi-domain modeling tools to provide all the functionality you need in one environment.


    Lenin Araujo Castillo

    Ambassador Maple - Perú



    Everything is simple, until you go underwater – This is what the University of Waterloo Submarine Racing team, or in short ‘WatSub’ coined as their motto. Never mind learning to scuba dive, and dealing with such things as rust, this newly formed team would have to compete against university teams with a decade or more of experience.

    But that did not deter the team, and they started work on Ontario’s first submarine racing project. The team approached Maplesoft to be a sponsor and we are proud to have supported this ingenious venture. The team has used Maplesoft technology in the design and testing of the submarine.

    “Maple has been our go-to calculations and analysis tool throughout the development of Amy (2015-2016 season), and we will continue using it throughout the development of Bolt (2016-2017 season),” said Gonzalo Espinoza Graham, President of the WatSub Team. “Its familiar interface and computing environment allowed us to set design benchmark targets from early on the design process and follow through with them on the later stage.”

    What started as an engineering project in December 2014, becoming officially the first submarine racing team in Ontario. The team soon grew to over 130 general members and a tight core-team, who were eager to tackle new challenges.  The team resides inside the Sedra Student Design Centre, University of Waterloo’s state of the art facility that houses over 25 student teams, the largest of its kind in North America.  

    WatSub made its first appearance on the European International Submarine Races (eISR) back in July 2016, with its 1st submarine ‘Amy’, where a single scuba diver piloted the submarine and propelled it through an unforgiving winding course marked by obstacles and turns 10 meters underwater. The team has since then participated in other competitions and is constantly improving the design and performance of the submarine, learning from each competition they participate in.  Next year Amy will participate in the 14th edition of the eISR international competition. “I think the greatest thing we learned is never to give up,” said Ana Krstanovic, a third-year political science student who manages communications for the team. “We’re more motivated now than ever.”


    Ojaswi Tagore, Gonzalo Espinoza Graham, and Janna Henzl represented WatSub at the European International Submarine Race in Gosport, UK.


    Another example of an innovative project that Maplesoft supported in 2016 is Waterloop: The Canadian SpaceX Hyperloop Competition Team, Canada's only SpaceX Hyperloop Pod Competition team. This project, which could change the way we travel in the future, is driven by a group of dedicated University of Waterloo students who have taken on the challenge to design and build a functional prototype Hyperloop pod. They will test it on a one-mile test track in Hawthorne, California in January 2017, pitting it against 22 of the 1200+ teams who originally entered the competition.

    The Hyperloop is a conceptual next generation high-speed transit system that will take commuters between cities at speeds over 1,000 km/h. The technology will differ from previous rail transit by having pods ride on a cushion of air in a reduced pressure tube in order to reach greater speeds with a smoother ride, and is powered entirely by renewable energy.

     The Hyperloop Pod Competition was launched by Elon Musk, the billionaire engineer and founder of SpaceX and Tesla Motors.  The competition is separated into 3 rounds. The first one was held in late December, where selected teams sent in their initial designs to be reviewed. From there, 180 teams were chosen to compete at Texas A&M University. Each team set up a booth and a panel of judges critiqued them and chose 31 teams to move onto the final, build and test stage.

    Waterloop Goose I

    Waterloop Goose X

    The GOOSE I is Waterloop’s half-scale, functional prototype vehicle pod, which will be the one in the competition.  The GOOSE X pod is a conceptual full size Hyperloop vehicle inspired by the prototype they are building. The full size pod will have a capacity of 26 passengers per pod.

    "Our prototype has been designed to be as simple and economical as possible, while still performing all necessary functions for the full size Hyperloop. If it is successful, it has the potential to revolutionize the transit industry in the same manner the train and airplane has before it," said Montgomery de Luna, architectural design lead for Waterloop. “We would like to thank Maplesoft for their generous support.  Without sponsors like Maplesoft supporting our vision and encouraging innovative student projects, we wouldn’t be able to achieve our goal.”

    Revolutionizing the transportation industry isn’t easy and is at times frustrating and time consuming for these teams, but having the best tools and resources will ensure that the teams have a good chance at excelling in competitions and creating innovative models that could change our future.

    The Joint Mathematics Meetings are taking place this week (January 4 – 7) in Atlanta, Georgia, U.S.A. This will be the 100th annual winter meeting of the Mathematical Association of America (MAA) and the 123nd annual meeting of the American Mathematical Society (AMS).

    Maplesoft will be exhibiting at booth #118 as well as in the networking area. Please stop by our booth or the networking area to chat with me and other members of the Maplesoft team, as well as to pick up some free Maplesoft swag or win some prizes.

    There are also several interesting Maple-related talks and events happening this week:


    Teaching Cryptology to Increase Interest in Mathematics for Students Majoring in Non-Technical Disciplines and High School Students

    Wednesday, January 4, 0820, L401 & L402, Lobby Level, Marriott Marquis

    Neil Sigmon, Radford University


    Enigma: A Combinatorial Analysis and Maple Simulator

    Wednesday, January 4, 0900, L401 & L402, Lobby Level, Marriott Marquis

    Rick Klima, Appalachian State University


    MYMathApps Calculus - Building on Maplets for Calculus

    Thursday, January 5, 0800, Courtland, Conference Level, Hyatt Regency

    Philip B. Yasskin, Texas A&M University 
    Douglas B. Meade, University of South Carolina 
    Andrew Crenwelge, Texas A&M University


    Maple Software Technology as a Stimulant Tool for Dynamic Interactive Calculus Teaching and Learning

    Thursday, January 5, 1000, Courtland, Conference Level, Hyatt Regency

    Lina Wu, Borough of Manhattan Community College-The City University of New York 


    Collaborative Research: Maplets for Calculus

    Thursday, January 5, 1400, Marquis Ballroom, Marquis Level, Marriott Marquis

    Philip Yasskin, Texas A&M University 
    Douglas Meade, U of South Carolina


    Digital Graphic Calculus Art Design in Maple Software

    Thursday, January 5, 1420, International 7, International Level, Marriott Marquis

    Lina Wu, Borough of Manhattan Community College-The City University of New York 


    Maplesoft will also be hosting a catered reception and brief presentation on Teaching STEM Online: Challenges and Solutions, Thursday January 5th, from 6:00pm – 7:30pm, at the Hyatt Regency, Hanover AB, on the exhibitor level. Please RSVP at or at Maplesoft booth #118.


    If you are attending the Joint Math meetings this week and plan on presenting anything on Maple, please feel free to let me know and I'll update this list accordingly.

    See you in Atlanta!


    Maple Product Manager

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