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 www.maplesoft.com/jmm 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!

Daniel

Maple Product Manager

The code for the animation:

L:=[[-0.12,2],[-0.14,0],[0.14,0],[0.12,2]]:
L1:=[[0.05,2],[4,1],[2,4],[3.5,3.5],[1,7],[2,6.5],[0,10]]:
A:=plot(L, color=brown, thickness=10):
B:=plot([op(L1),op(map(t->[-t[1],t[2]],ListTools:-Reverse(L1)))], color="Green", thickness=10):
C:=plottools:-polygon([op(L1),op(map(t->[-t[1],t[2]],ListTools:-Reverse(L1)))], color=green):
Tree:=plots:-display([A, B, C], scaling=constrained, axes=none):
T:=[[-3.2,-2, Happy, color=blue, font=[times,bold,30]], [0,-2,New, color=blue, font=[times,bold,30]], [2.5,-2,Year, color=blue, font=[times,bold,30]], [-5,-3.5, "&", color=yellow, font=[times,bold,30]],[-2.5,-3.5, Merry, color=red, font=[times,bold,30]], [2.3,-3.5, Christmas!, color=red, font=[times,bold,30]], [0,-5, "2017", color=cyan, font=[times,bold,36]]$5]:
F:=k->plottools:-homothety(Tree, k, [0,5]):
A:=plots:-animate(plots:-display, ['F'(k)], k=0..1, frames=60, paraminfo=false):
B:=plots:-animate(plots:-textplot,[T[1..round(i)]], i=0..nops(T), frames=60, paraminfo=false):
plots:-display(A, B, size=[500,550], scaling=constrained);

Christmas_Tree.mw

 Edit.

Parametric equation of second-order curve in 3d. Draghilev method.
PLAN_CURVE_3d_1.mw
Examples:
x1^2+x1*x3+13*x2^2+x3-1=0;
x1+x2+x3=0;

 x1^2+0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

 x1^2-0.1*x2^2+x3^2-9=0;
 x1+3*x3+1=0;

Parametric equation of a circle in 3d by three points. Draghilev method.

CIRCLE_3_POINTS_geom3d_2.mw

In this post I want to present an easy method to obtain a discrete parametrization of a surface S defined implicitly (f(x,y,z)=0).
This problem was discussed here several times, the most recent post is
http://www.mapleprimes.com/posts/207661-Isolation-Of-Sides-Of-The-Surface-On-The-Graph

S is supposed to be the boundary of a convex body having (x0,y0,z0) an interior point and contained in a ball of radius R centered at (x0,y0,z0).
Actually, the procedure also works if the body is only star-shaped with respect to the interior point, and it is also possible to plot only a part of the surface
inside a solid angle centered at (x0,y0,z0).

Usage:
Par3d(f, x=x0, y=y0, z=z0, R, m, n,  theta1 .. theta2,  phi1 .. phi2)

f           is an expression depending on the variables x, y, z
x0, y0, z0  are the coordinates of the interior point
R           is the radius of the ball which contains the surface,
m, n        are the numbers of the grid lines which will be generated
The last two parameters are optional and are used when only a part of S will be parametrized.

The procedure Par3d returns a MESH structure M, which can be plotted with PLOT3D(M).

Par3d :=proc(f,x::`=`,y::`=`,z::`=`,R,m,n,th:=0..2*Pi,ph:=0..Pi)
    local A,i,j, rij,fij,Cth,Sth,Cph,Sph, theta,phi, r;
    A:=Array(1..m+1,1..n+1,1..3,datatype=float[8]);
    for i from 0 to m do for j from 0 to n do
      theta:=op(1,th)+i/m*(op(2,th)-op(1,th));
      phi:=op(1,ph)+j/n*(op(2,ph)-op(1,ph));
      Cth:=evalf(cos(theta)); Sth:=evalf(sin(theta));
      Cph:=evalf(cos(phi));   Sph:=evalf(sin(phi));
      fij:= eval(f, [lhs(x)=rhs(x)+r*Sph*Cth, lhs(y)=rhs(y)+r*Sph*Sth, lhs(z)=rhs(z)+r*Cph]);
      rij:=fsolve(fij,r=0..R);  if [rij]::list(numeric) then rij:=min(rij) fi; 
      if [rij]=[] or not(type(rij,numeric)) then print(['i'=i,'j'=j], fij); rij:=undefined fi; 
      A[i+1,j+1,1]:=evalf(rhs(x)+rij*Sph*Cth);
      A[i+1,j+1,2]:=evalf(rhs(y)+rij*Sph*Sth);
      A[i+1,j+1,3]:=evalf(rhs(z)+rij*Cph);
    od;od:
    MESH(A);
end:

The procedure is not optimized, e.g.
- Cth, etc could be Vectors computed outside the loops
- Some small changes to use evalhf.

###### EXAMPLES ######

f1 := x^2+3*y^2+4*z^2 - x*y - 2*y*z - 10:
plots:-implicitplot3d(f1, x=-5..5, y=-5..5, z=-2..2);

M:=Par3d(f1, x=0,y=0,z=0,5,40,40):
PLOT3D(M);

f2 := x^4+y^4+z^4-1:
M:=Par3d(f2, x=0,y=0,z=0,5,40,40):
PLOT3D(M);

M:=Par3d(f2, x=0,y=0,z=0, 5,40,40, 0..Pi, 0 .. Pi/3): #Plot half of the top only
plots:-display(PLOT3D(M), scaling=constrained);

M:=Par3d(f2,      x=0,y=0,z=0, 5,30,30, 0..Pi, 0 .. Pi):
N:=Par3d(f2+0.01, x=0,y=0,z=0, 5,30,30, 0..Pi, 0 .. Pi):
plots:-display(PLOT3D(M), color=red):
plots:-display(PLOT3D(N), color=green):
plots:-display(%,%%, orientation=[-40,65,10]);

f3 := (x^2+y^2-1)^2+(z+sin(x*y+z))^4-120:
plots:-implicitplot3d(f3, x=-4..4,y=-4..4,z=-5..5, numpoints=10000);

Par3d(f3, x=0,y=0,z=0,5, 30,30):
PLOT3D(%);

Note.
The procedure could be used to plot locally around a point (x0,y0,z0)
One may use the spherical coordinates (theta0,phi0) and then call the procedure taking theta0-a .. theta0+a,  phi0-b, .. phi0+b  for the trailing parameters
The spherical coordonates can be computed using:

ThetaPhi :=proc(x,y,z, X,Y,Z)
    local r:=sqrt((X-x)^2+(Y-y)^2+(Z-z)^2);
    ['theta'=arctan(Y-y,X-x), 'phi'=arccos((Z-z)/r)]
end:

ThetaPhi(10,20,30, 11,21,28);evalf(%);

At 3:00 PM EST on Thursday, December 15, Maplesoft hosted a momentous hour in my life, my "retirement party" ending my career at Maplesoft. It was a day I had planned some four years ago when I dropped to a lighter schedule, and a day my wife has been awaiting for six years.

Jim Cooper, CEO at Maplesoft, presented a very brief sketch of some milestones in my life, including my high school graduation in 1958, BA in 1963, MS in 1966, PhD in 1970, jobs at the University of Nebraska-Lincoln, Memorial University of Newfoundland, and the Rose-Hulman Institute of Technology. There was a picture of me taken from my high school graduation yearbook. There was a cake. There were kind words about my contributions to Maple, including "Clickable Calculus," the term and its meaning.

I was handed the microphone - I knew what I wanted to say. My wife was present in the gathering. I pointed to her and said that all the congratulations should go to her who had waited so patiently for my retirement for six years. I thanked Maplesoft and all its employees for nearly 14 of the best years of my life, for I have thoroughly enjoyed my return to Canada and my work (more like play) at Maplesoft. 

It's been a great opportunity to be part of the Maple experience, and now it's time for new ones. There'll be more woodworking in my basement woodshop where I make mostly noise and sawdust, some extra travel, more exercise and fresh air, long-delayed household projects, and whatever else my mate of 49 years asks.

But the best part of all is that I'll still have a connection to Maplesoft - I'll continue doing two webinars a month, will maintain and update much of the content I've created for Maple while at Maplesoft, and contribute additional content of relevance to the Maple community. 

Is this a bug?

hypergeom([1, -1, 1/2], [-12,-3], 1);
Error, (in hypergeom/check_parameters) function doesn't exist: missing appropriate negative integers in the first list of parameters to compensate the negatives integer(s): [-3], found in the second list.
 

Yet this hypergeometric series terminates and Maple should be able to handle it, at least according to the Maple help page (the second rule below applies, yet the numerator has a smaller absolute value, so the first rule below applies).

If some   n[i] is a non-positive integer, the series is finite (that is,   F(n, d, z)  is a polynomial in    z).
If some  d[j]  is a non-positive integer, the function is undefined for all non-zero  z, unless there is also a negative upper parameter of smaller absolute value, in which case the previous rule applies.
 

Interestingly, the Wolfram Mathematica app can evaluate this to 311/312.

One way is coloring a surface on both sides. We build equidistant surface with very small radius and stain the equidistant surface in color different from the color of the original surface.
Examples coloring of surfaces on both sides.  Radius equal to abs (0.0001).
x3-0.5*exp(sin(x1+2.5*x2+x3))=0;
(x1^2+x2^2-0.4)^2+(x3+sin(x1*x2+x3))^4-0.1=0;
2_COLORS.mw

Hi. Wanting a procedure, but not wanting to reinvent the wheel, I did a search on mapleprimes for "Shoelace" (formula). I got the response below indicating a hit. But clicking on the hypertext, i get to page 41 of Kitonums replies, but not the actual thing i wanted. I go to "find on this page" in my browser firefox and input "Shoelace"...nothing. enter "thU"...nothing. Do I really have to go through all of his replies on the page?

Let us consider

restart; Digits := 20; evalf(Int(abs(cos(1/t)), t = 0 .. 0.1e-1), 3);
   -0.639e-2

Pay your attention to the minus sign. Simply no words. Mma produces 0.006377.

evalf@Int.mw

Ian Thompson has written a new book, Understanding Maple.

I've been browsing through the book and am quite pleased with what I've read so far. As a small format paperback of just over 200 pages it packs in a considerable amount of useful information aimed at the new Maple user. It says, "At the time of writing the current version is Maple 2016."

The general scope and approach of the book is explained in its introduction, which can currently be previewed from the book's page on amazon.com. (Click on the image of the book's cover, to "Look inside", and then select "First Pages" in the "Book sections" tab in the left-panel.)

While not intended as a substitute for the Maple manuals (which, together, are naturally larger and more comprehensive) the book describes some of the big landscape of Maple, which I expect to help the new user. But it also explains how Maple is working at a lower level. Here are two phrases that stuck out: "This book takes a command driven, or programmatic, approach to Maple, with the focus on the language rather than the interface", followed closely by, "...the simple building blocks that make up the Maple language can be assembled to solve complex problems in an efficient way."

A population  governed by the logistic equation with a constant rate of harvesting satisfies the initial value problem . This model is typically analyzed by setting the derivative equal to zero and finding the two equilibrium solutions . A sketch of solutions  for different values of a suggests that the larger equilibrium is stable; the smaller, unstable.

When a is less that the unstable equilibrium,  becomes zero at a time , and the population becomes extinct. If  is not interpreted as pertaining to a population, its graph exists beyond , and actually has a vertical asymptote between the two branches of its graph.

In the worksheet "Logistic Model with Harvesting", two questions are investigated, namely,

1. How does the location of this vertical asymptote depend on on a and h?
2. How does the extinction time , the time at which , depend on a and h?

To answer the second question, an explicit solution , readily provided by Maple, is set equal to zero and solved for . It turns out to be difficult both to graph the surface  and to obtain a contour map of the level sets of this function. Instead, we solve for  and obtain a graph of  with  as a slider-controlled parameter.

To answer the first question, the explicit solution, which has the form , exhibits its vertical asymptote when . Solving this equation for  gives the time at which the vertical asymptote is located, a function that is as difficult to graph as . Again the remedy is to solve for, and graph, , with  as a slider-controlled parameter.

Download the worksheet: Logistic_with_Harvesting.mw

Ever needed to measure something and all you had was a piece of paper?  This leads us to how we can use maple to figure out what we can measure using a sheet of standard 8-1/2" x 11" paper.

Can we measure 6" with a sheet of paper?

> eq := (17/2)*x+11*y = 6;
                                  17             
                            eq := -- x + 11 y = 6
                                  2              
> eq2 := isolve(eq, a);
                   eq2 := {x = -20 - 22 a, y = 16 + 17 a}
> subs(a = 0, b = 0, eq2);
                              {x = -20, y = 16}

So that is the simplest case, stacking up 16 pieces on the long side and subtracting 20 on the short side.  A total of toppling the piece of paper over 36 times.  That's a high percentage of of error. 

But wait!  haha.   Wouldn't a fold make it simpler?  Sure!  Fold the 8.5" across and we now have 2.5" to work with.

> eq := (17/2)*x+11*y+(5/2)*z = 6;
                               17            5      
                         eq := -- x + 11 y + - z = 6
                               2             2      
> eq2 := isolve(eq, {a, b});
           eq2 := {x = a, y = 1 + 4 a + 5 b, z = -2 - 21 a - 22 b}
> subs(a = 0, b = 0, eq2);
                           {x = 0, y = 1, z = -2}

Less toppling of pieces of paper and much less error. 

Last week Michael Pisapia, Maplesoft European VP, attended the opening reception of Mathematics: The Winton Gallery at the Science Museum in London. Ahead of being open to the public on 8^th December, contributors and donors were invited to take a look behind the scenes of the new gallery, which explores how mathematicians, their tools and ideas have helped to shape the modern world over the last four hundred years.

The gallery is a spectacular space, designed by the world-renowned Zaha Hadid Architects, housing over a hundred artefacts of mathematical origin or significance. It is divided up into disciplines ranging from navigation to risk assessment, and gambling to architecture. Inspired by the Handley Page aircraft, the largest object on display, and suspended as the centrepiece, the gallery is laid out using principles of mathematics and physics. It follows the lines of airflow around it in a stunning display of imagined aerodynamics, brought to life using light and sculpture. You can learn more about its design in this video.

Guests at the reception enjoyed a specially commissioned piece of music from the Royal College of Music titled ‘Gugnunc’, named after the aircraft and inspired by the rhythms of Morse code and mathematical and mechanical processes, and performed at the centre of the gallery.

Of course any exhibit celebrating all things maths is of great interest to us here at Maplesoft, but this one especially so, since Mathematics: The Winton Gallery showcases the earliest available version of Maple.

A copy of Maple V, from 1997, sits in ‘The Power of Computers’ section of the Winton Gallery, in an exhibit which tells the story of the significant role played by mathematical software in improving the quality of mathematics education and research. Other objects in the section include a Calculating Machine from the Scientific Service circa 1939, a PDP-8 minicomputer from the 1960s, and part of Charles Babbage’s mid-19^th century analytical engine, intended as a high-powered mathematical calculator.

As many of you will remember, Maple V was a major milestone in the history of Maple, providing unparalleled interactivity, powerful symbolics and creative visualization in mathematical computation and modeling. For a walk down memory lane, check out Maple V: The Future of Mathematics (ca. 1994) on YouTube.

Seeing this copy of Maple finally in place in the exhibit marks the end of a long journey – and not just in the miles it travelled to arrive at the museum from its home in Canada. When we were first approached by the Science Museum for a donation of Maple, we launched a hunt to find not just the right copy of Maple with its box and manuals, but also artefacts that showcased the origin and history of Maple. It was a journey down memory lane for the inventors of Maple as well as the first few employees as they dug out old correspondences, photos, posters and other memorabilia that could be showcased. Today they can be proud of their contribution to this display at the Science Museum. 

Although the case of historic software packages is visually less impressive than many of the other items in the gallery, it certainly attracted plenty of attention as guests made their way in for the first time. 

For fans of Maple V - and there are many - it’s reassuring that the Science Museum are now entrusted with preserving not only the iconic packaging, but with telling the story of Maple’s history and marking its place in the evolution of mathematics and technology.

To learn more about Mathematics: The Winton Gallery, its highlights and architecture, visit http://www.sciencemuseum.org.uk/mathematics

To see the timeline of Maple’s evolution over the years, visit:  http://www.maplesoft.com/25anniversary/