As a reminder, we regularly host live webinars on a variety of topics for our customers, and we wanted to make this information available to the MaplePrimes community as well.

This featured webinar for this month will outline the Finance Package in Maple 18 including capabilities like the mathematical, statistical, and connectivity tools required to analyse data, calculate forecasts, estimate risks, prototype and develop quantitative algorithms, and leverage parallel programming techniques.

Other topics include:

• Data feed connectivity and system integration

• Price equity and interest rate derivatives

• Populate reporting tools, deliver documents and share worksheets

• Optimize portfolios of financial instruments

• High-Performance Computing (HPC)

To join us for the live presentation, please click here to register.

In Maple 18, the Database package has been updated to include support for SQlite databases as well as a new option for plots to change the background images on plots.  To showcase both of these features, our engineering team put together an example that optimizes the flight path of a pan-US delivery drone.

This application extracts the latitude and longitude of those zip codes from an SQLlite database (the application includes the database, which cross-references US zip codes against their latitude, longitude, city and state). The application then performs a traveling salesman optimization and plots the shortest path on a map of the US.

To download the application click here: PanUSDeliveryDro.zip

This is a 5-days mini-course I gave in Brazil last week, at the CBPF (Brazilian Center for Physics Research). The material will still receive polishment and improvements, towards evolving into a sort of manual, but it is also interesting to see it exactly as it was presented to people during the course. This material uses the update of Physics available at the Maplesoft Physics R&D webpage.

BrasilComputacaoAlgebrica.mw.zip

BrasilComputacaoAlgebrica.pdf 

Edgardo S. Cheb-Terrab
Physics, Maplesoft

In this post we present the solution with Maple to the logical problem of "Gardens Puzzle"

http://www.mathsisfun.com/puzzles/gardens-solution.html

The Puzzle:

Five friends have their gardens next to one another, where they grow three kinds of crops: fruits (apple, pear, nut, cherry), vegetables (carrot, parsley, gourd, onion) and flowers (aster, rose, tulip, lily).

1. They grow 12 different varieties.
2. Everybody grows exactly 4 different varieties
3. Each variety is at least in one garden.
4. Only one variety is in 4 gardens.
5. Only in one garden are all 3 kinds of crops.
6. Only in one garden are all 4 varieties of one kind of crops.
7. Pear is only in the two border gardens.
8. Paul's garden is in the middle with no lily.
9. Aster grower doesn't grow vegetables.
10. Rose growers don't grow parsley.
11. Nuts grower has also gourd and parsley.
12. In the first garden are apples and cherries.
13. Only in two gardens are cherries.
14. Sam has onions and cherries.
15. Luke grows exactly two kinds of fruit.
16. Tulip is only in two gardens.
17. Apple is in a single garden.
18. Only in one garden next to Zick's is parsley.
19. Sam's garden is not on the border.
20. Hank grows neither vegetables nor asters.
21. Paul has exactly three kinds of vegetable.

Who has which garden and what is grown where?

About methods of solution. At first I just wanted to generate all variations and using conditions 1 .. 21 to find all solutions. But even if we use the condition that everybody grows exactly 4 different varieties then the total number variants equals  5!^2*binomial(12,4)^5=427945522455000000

So from the very beginning using some of the conditions 1 .. 21 we maximally reduce the number of possible variants. For example from the conditions 11, 18 and 6 implies that only in one garden are all 4 varieties of flowers. Next we pass through these variants and using conditions 1 .. 21 and finally come to a unique solution:

restart;

Fruits:={apple, pear, nut, cherry}:

Vegetables:={carrot, parsley, gourd, onion}:

Flowers:={aster, rose, tulip, lily}:

Set1:=Flowers:

Garden1:={Set1}:

Set2:=Fruits union Vegetables union Flowers minus {nut,gourd,parsley} minus {apple,cherry,rose}:

Garden2:={seq({nut,gourd,parsley} union {Set2[i]}, i=1..nops(Set2))}:

Set3:=Vegetables union Flowers minus {parsley}:

Garden3:={seq({apple,cherry,pear} union {Set3[i]}, i=1..nops(Set3))}:

Set4:=combinat[choose](Fruits union Vegetables union Flowers minus {onion,cherry} minus {apple,parsley,pear,nut}, 2):

Garden4:={seq({onion,cherry} union Set4[i], i=1..nops(Set4))}:

Set5:=Fruits union Vegetables union Flowers minus {apple, cherry,nut,parsley}:

Garden5:=combinat[choose](Set5, 4):

S:=[]:

for s1 in Garden1 do

for s2 in Garden2 do

for s3 in Garden3 do

for s4 in Garden4 do

for s5 in Garden5 do

s:=[s1,s2,s3,s4,s5]: s_4:=combinat[choose](s,4): m:=0: n:=0: k:=0: p:=0: q:=0:

for i in s do

if `intersect`(i,Fruits)<>{} and `intersect`(i,Vegetables)<>{} and `intersect`(i,Flowers)<>{} then m:=m+1: fi:

if pear in i then n:=n+1: fi:

if tulip in i then k:=k+1: fi:

if aster in i and `intersect`(i,Vegetables)<>{} then p:=p+1: fi:

if i=Fruits or i=Vegetables or i=Flowers then q:=q+1: fi:

od:

if nops(`union`(op(s)))=12 and nops(`union`(seq(`intersect`(op(s_4[j])), j=1..nops(s_4))))=1 and m=1 and n=2 and k=2 and p=0 and q=1 then S:=[op(S),[s1,s2,s3,s4,s5]]: fi:

od: od: od: od: od:

L1:=[seq([[3,Paul],[2,Sam],seq([combinat[permute]([1,4,5])[i,j],[Luke,Zick,Hank][j]],j=1..3)],i=1..6)]:

L2:=[seq([[3,Paul],[4,Sam],seq([combinat[permute]([1,2,5])[i,j],[Luke,Zick,Hank][j]],j=1..3)],i=1..6)]:

L0:=[op(L1),op(L2)]:

L:=[seq(op(combinat[permute](L0[i])),i=1..nops(L0))]:

Sol:=[]:

for l in L do

for s in S do

sol:=[seq([op(l[i]),s[i]], i=1..5)]:

gad1:=op(select(has,sol,parsley)): gad2:=op(select(has,sol,Zick)):

if abs(gad1[1]-gad2[1])=1 and convert([seq(((pear in sol[i][3] implies (sol[i][1]=1 or sol[i][1]=5)) and (sol[i][2]=Paul implies (not lily in sol[i][3])) and (sol[i][1]=1 implies (apple in sol[i][3] and cherry in sol[i][3])) and (sol[i][2]=Sam implies (onion in sol[i][3] and cherry in sol[i][3])) and (sol[i][2]=Luke implies nops(`intersect`(sol[i][3],Fruits))=2) and (sol[i][2]=Hank implies (`intersect`(sol[i][3],Vegetables)={} and not aster in sol[i][3])) and (sol[i][2]=Paul implies nops(`intersect`(sol[i][3],Vegetables))=3)), i=1..5)], `and`) then Sol:=[op(Sol), sol]: fi:

od: od:

for i in Sol do

Matrix(sort(i,(x,y)->x[1]<y[1]));

od;

In this post we present another compact proof of this remarkable theorem without using  geometry package.
The proof uses a procedure called  Cc , which for three points returns a list of the coordinates of the center and the radius of the circumscribed circle.

restart;

Cc:=proc(A,B,C)

local x1, y1, x2, y2, x3, y3, x, y;

x1,y1:=op(A);  x2,y2:=op(B);  x3,y3:=op(C);

solve({(x2-x1)*(x-(x1+x2)/2)+(y2-y1)*(y-(y1+y2)/2)=0, (x2-x3)*(x-(x2+x3)/2)+(y2-y3)*(y-(y2+y3)/2)=0},{x,y});

assign(%);

[simplify([x,y]), simplify(sqrt((x-x1)^2+(y-y1)^2))];

end proc:

Proof for arbitrary triangle:

A, B, C:=[x1,y1], [x2,y2], [x3,y3]:

A1, B1, C1, M:=(B+C)/2, (A+C)/2, (A+B)/2, (A+B+C)/3:

P1:=Cc(A,M,B1)[1]: P2:=Cc(B1,M,C)[1]: P3:=Cc(C,M,A1)[1]:

P4:=Cc(A1,M,B)[1]: P5:=Cc(B,M,C1)[1]: P6:=Cc(C1,M,A)[1]:

Cc1:=Cc(P1,P2,P3):  Cc2:=Cc(P4,P5,P6):

is(Cc1=Cc2);

                                                  true

Let  us consider the general case of symbolic values C(xC,yC). I make use of the idea suggested by edgar in http://www.mapleprimes.com/questions/97743-How-To-Prove-Morleys-Trisector-Theorem : no assumptions.

restart; with(geometry); point(A, 0, 0);
point(B, 1, 0);
point(C, xC, yC);
point(MA, (xC+1)*(1/2), (1/2)*yC);
point(MC, 1/2, 0);
point(MB, (1/2)*xC, (1/2)*yC);
point(E, (0+1+xC)*(1/3), (0+0+yC)*(1/3));# the center of mass
line(l1, x = 1/4, [x, y]);
The coordinates of the center of the first described circle are found as the solutions of the system of the equations of midperpendiculars.

midpoint(ae, A, E); coordinates(ae);


S1 := solve({x = 1/4, ((xC+1)*(1/3))*(x-(xC+1)*(1/6))+(1/3)*yC*(y-(1/6)*yC) = 0}, {x, y});

BTW, Maple can't create the midperpendiculars in this case.

point(O1, op(map(rhs, S1)));
                               O1

Simple details are omitted in the above. The coordinates of the centers of the two next described circles are found similarly.
coordinates(midpoint(mce, MC, E));

S2 := solve({x = 3/4, ((-1/2+xC)*(1/3))*(x-5/12-(1/6)*xC)+(1/3)*yC*(y-(1/6)*yC) = 0}, {x, y});

point(O2, op(map(rhs, S2)));

                               O2
coordinates(midpoint(bma, B, MA)); coordinates(midpoint(be, B, E));
  

                

S3 := solve({(xC-1)*(x-(xC+3)*(1/4))+yC*(y-(1/4)*yC) = 0, ((-2+xC)*(1/3))*(x-(4+xC)*(1/6))+(1/3)*yC*(y-(1/6)*yC) = 0}, {x, y});

point(O3, op(map(rhs, S3)));

                               O3

Now we find the equation of the circumference which passes through O1, O2, and O3.

eq := a*x+b*y+x^2+y^2+c = 0:
sol := solve({eval(eq, S1), eval(eq, S2), eval(eq, S3)}, {a, b, c});

A long output can be seen in the attached .mw file.

eq1 := eval(eq, sol);

  Now we find (in suspense)  the coordinates of the next center and verify whether it belongs to the sircumference O1O2O3.

coordinates(midpoint(mac, C, MA)); coordinates(midpoint(ec, E, C)); S4 := solve({(xC-1)*(x-(3*xC+1)*(1/4))+yC*(y-3*yC*(1/4)) = 0, ((2*xC-1)*(1/3))*(x-(4*xC+1)*(1/6))+(2*yC*(1/3))*(y-4*yC*(1/6)) = 0}, {x, y});

 point(O4, op(map(rhs, S4)));

                               O4
simplify(eval(eq1, S4));

                             0 = 0

Hope the reader will have a real pleasure to find the two residuary centers and to verify these on his/her own.

geom2.mw

It is well known that the medians of a triangle divide it into 6 triangles.
It is less known that the centers of their circumscribed circles belong to one circumference as drawn below

This remarkable theorem  was proved in the 21st century! Unfortunately, I lost its source.
I can't prove this difficult  theorem by hand. However, I can prove it with Maple.
The aim of this post is to expose these proofs. Everybody knows that it is scarcely possible
to construct a general triangle with help of the geometry package of Maple.
Without loss of generality one may assume that the vertex A is placed at the origin,
the vertex B is placed at (1,0), and the vertex C(xC,yC). We firstly consider the theorem
in the case of concrete values of xC and yC.

restart; with(geometry):with(plots):
point(A, 0, 0);
point(B, 1, 0);
xC := 15*(1/10); yC := sqrt(3); point(C, xC, yC);
triangle(T, [A, B, C]);
median(mA, A, T, MA);
median(mB, B, T, MB);
median(mC, C, T, MC);
line(m1, [A, MA]);
line(m2, [B, MB]);
intersection(E, m1, m2);
triangle(AEMB, [A, E, MB]);
circumcircle(c1, AEMB, 'centername' = C1);
circumcircle(c2, triangle(CEMB, [C, E, MB]), 'centername' = C2);
circumcircle(c3, triangle(CEMA, [C, E, MA]), 'centername' = C3);
circumcircle(c4, triangle(BEMA, [B, E, MA]), 'centername' = C4);
circumcircle(c5, triangle(BEMC, [B, E, MC]), 'centername' = C5);
circumcircle(c6, triangle(AEMC, [A, E, MC]), 'centername' = C6);
circle(CC, [C1, C2, C3]);
IsOnCircle(C4, CC);
                              true

IsOnCircle(C5, CC);
                              true
IsOnCircle(C6, CC);
                              true
display([draw([T(color = black), mA(color = black), mB(color = black), mC(color = black), C1(color = blue), C2(color = blue), C3(color = blue), C4(color = blue), C5(color = blue), C6(color = blue), CC(color = red)], symbol = solidcircle, symbolsize = 15, thickness = 2, scaling = constrained), textplot({[-0.5e-1, 0.5e-1, "A"], [.95, 0.5e-1, "B"], [xC-0.5e-1, yC+0.5e-1, "C"]})], axes = frame, view = [-.1 .. max(1, xC)+.1, 0 .. yC+.1]);

This can be done as a procedure in such a way.

restart; SixPoints := proc (xC, yC) geometry:-point(A, 0, 0); geometry:-point(B, 1, 0); geometry:-point(C, xC, yC); geometry:-triangle(T, [A, B, C]); geometry:-median(mA, A, T, MA); geometry:-median(mB, B, T, MB); geometry:-median(mC, C, T, MC); geometry:-line(m1, [A, MA]); geometry:-line(m2, [B, MB]); geometry:-intersection(E, m1, m2); geometry:-triangle(AEMB, [A, E, MB]); geometry:-circumcircle(c1, AEMB, 'centername' = C1); geometry:-circumcircle(c2, geometry:-triangle(CEMB, [C, E, MB]), 'centername' = C2); geometry:-circumcircle(c3, geometry:-triangle(CEMA, [C, E, MA]), 'centername' = C3); geometry:-circumcircle(c4, geometry:-triangle(BEMA, [B, E, MA]), 'centername' = C4); geometry:-circumcircle(c5, geometry:-triangle(BEMC, [B, E, MC]), 'centername' = C5); geometry:-circumcircle(c6, geometry:-triangle(AEMC, [A, E, MC]), 'centername' = C6); geometry:-circle(CC, [C1, C2, C3]); return geometry:-IsOnCircle(C4, CC), geometry:-IsOnCircle(C5, CC), geometry:-IsOnCircle(C6, CC), geometry:-draw([CC(color = blue), C1(color = red), C2(color = red), C3(color = red), C4(color = red), C5(color = red), C6(color = red), T(color = black), mA(color = black), mB(color = black), mC(color = black), c1(color = green), c4(color = green), c2(color = green), c3(color = green), c5(color = green), c6(color = green)], symbol = solidcircle, symbolsize = 15, thickness = 2) end proc;
SixPoints(1.5, 1.2);

true, true, true, PLOT(...)
 SixPoints(1.5, 1.2)[4];

See geom1.mw

To be continued (The general case will be considered in  part 2http://www.mapleprimes.com/posts/200210-Six-Points-On-Circumference-2 .).

I am using "Maple 15" and "Windows 7" and I want to share the tip for using "Classic worksheet" and "GR Tensor". Windows 7 is 64-bit but you can use Classic worksheet and GR Tensor, which work basically in 32-bit, by simply downlording "32-bit Maple 15" !! I haven't tried other more current versions of Maple but Maple 15 works well: you can choose 32-bit Maple in the download options.

Enjoy the Classic worksheet and GR Tensor. Good luck !  

Code of the animation:

restart;

N := 192:

A := seq(plot([[.85*sin(t)^3-2+1.25*i/N, .85*(13*cos(t)*(1/15)-(1/3)*cos(2*t)-2*cos(3*t)*(1/15)-(1/15)*cos(4*t)), t = 0 .. Pi*i/N], [-.85*sin(t)^3-2+1.25*i/N, .85*(13*cos(t)*(1/15)-(1/3)*cos(2*t)-2*cos(3*t)*(1/15)-(1/15)*cos(4*t)), t = 0 .. Pi*i/N], [sin(t)^3+2-1.25*i/N, 13*cos(t)*(1/15)-(1/3)*cos(2*t)-2*cos(3*t)*(1/15)-(1/15)*cos(4*t), t = 0 .. Pi*i/N], [-sin(t)^3+2-1.25*i/N, 13*cos(t)*(1/15)-(1/3)*cos(2*t)-2*cos(3*t)*(1/15)-(1/15)*cos(4*t), t = 0 .. Pi*i/N]], color = red, thickness = 5, view = [-3 .. 3, -1.2 .. .9]), i = 1 .. N):

plots[display](A, insequence = true, scaling = constrained, axes = none);

Congratulations to Andriy Andrusyk, from the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, who won last quarter’s Möbius Challenge with his application Heat Equation.  Dr. Andrusyk won a DSLR Camera Prize Pack.

Remember that you have until Mar. 31 to enter your Möbius Apps for a chance to win the next prize, an Xbox One!   Visit Möbius App Challenge for full contest details.

eithne

I thought it would be interesting to review what happened with Physics in Maple during 2013. The proposed theme for the Physics project was the consolidation and integration of the package with the rest of the Maple library. There were more than 500 changes, enhancements in most of the Physics commands, plus 17 new Physics:-Library commands. The impact of these changes is across the board, from Vector Analysis to Quantum Mechanics, Relativity and Field Theory.

Consolidation of the Physics package is about making it robust and versatile in real case scenarios. With the launch of the Physics: Research and Development updates webpage, Maplesoft has pioneered feedback, adjustments in the package and new developments provided around the clock for all of its users. The result of this accelerated exchange with people around the world is what you find in Maple's updated Physics today.

In addition to changes improving the functionality in mathematical-physics, changes were introduced towards making the computational experience as natural as possible, now including textbook-like typesetting of inert forms for the whole mathematical language and vectorial differential operators.

Physics doubled in size in Maple 16, almost doubled again in Maple 17, and during 2013 Physics received the largest number of changes ever in the package in one year. We are aiming for real to provide a state-of-the-art environment for algebraic computations in Physics. The links at the end show the same but with the Examples sections expanded.

Physics2013.pdf    Physics2013.mw

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Maplesoft is holding its first ever Virtual User Summit on Feb. 27.  You’ll be able to watch presentations by both Maplesoft and Maplesoft customers, ask questions, have discussions in the lounge with other attendees, and even enter a draw, all from the comfort of your own home or office.

Here’s the agenda.  We’ll release more detailed information on speakers and session times in the next couple of weeks.

For more information and to register:  Maplesoft Virtual User Conference

We're looking forward to seeing you there. (Well, "seeing you" :-))

eithne

I'd like to pay attention to an application "Periodicity of Sunspots " by Samir Khan, where a real data is analysed. That application can be used in teaching statistics.

PS. The code by Samir Khan works well for me.

Voting is open for the next individual prize to be awarded as part of the Möbius App Challenge.  The winner will receive a DSLR Camera Prize Pack! 

Here are the finalist Apps:

• Interactive Beam Analysis
• Heat Equation
• Ellipse
• Solving Systems of Equations

If you do not have the latest version of Java installed, released last week, you may have problems viewing these apps, as older versions of Java seem to be quietly disabling themselves.  I’ve attached a zip file of the finalist Apps in case it is helpful. finalists.zip

Note that, if you ever have any problems viewing Apps in your browser, or simply want to work offline, you can always download a Möbius App and view it in Maple or the free Maple Player. To download a Möbius App, follow the link to the App and then click on the Download button near the top left of the page.

You can vote for your favorite through our Facebook page or, if you’re not on Facebook, send an email with your vote to Mobius-Project@maplesoft.com.

And remember, we are now accepting entries for the next quarterly prize. You could win an Xbox One!  See the Möbuis App Challenge for details.

Voting closes Jan. 30.

eithne

Greetings to all.

It is a new year (for some time now) and I am writing to indicate that the mathematical adventures with cycle index computations and Maple continue!

Here are the previous installments:

• Power Group Enumeration
• Fascinating cycle indices with Maple
• The enumeration of non-isomorphic graphs
• Automorphisms of the Petersen Graph.

My purpose this time is to alert readers who might be interested to a new cycle index computation that is neither an application of the classical form of the Polya Enumeration Theorem (PET) nor of Power Group Enumeration. The former counts objects being distributed into slots with a group acting on the slots and the latter objects going into slots with a second group which permutes the objects in addition to the slots being permuted. What I am about to present treats a third possible case: when the slot permutation group and the object permutation group are one and the same and act simultaneously (not exactly the same but induced by the action of a single group).

This requires quite radical proceedings in the etymological sense of the word, which is to go back to the roots of a problem. It seems that after working with the PET sooner or later one is confronted with enumeration problems that demand the original unmitigated power of Burnside's lemma, sometimes called the lemma that is not Burnside's. This is the case with the following problem. Suppose you have an N-by-N matrix whose entries are values from 1 to N, with all assignments allowed and the symmetric group on N elements acts on the row and column indices permuting rows and columns as well as the entries simultaneously. We ask how many such matrices there are taking these double symmetries into account. This also counts the number of closed binary operations on a set of N elemnents and there is a discussion as well as the Maple code (quite simple in my opinion and no more than a few lines) that solves this problem at the following Math Stackexchange link, which uses Lovasz Formula for the cycle index of the symmetric group which some readers may remember.

In continuing the saga of Polya and Burnside exploration I have often reflected on how best to encapsulate these techniques in a Maple package. With this latest installment it would appear that a command to do Burnside enumeration probably ought to be part of such a package.

Best regards,

Marko Riedel