Computational & Applied Mathematics

Computational & Applied MathematicsUnderstanding our World

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Martin J. GanderDepartment of Mathematics and Statistics

McGill UniversityOn leave at the University of Geneva, 2002/2003

Computational and Applied Mathematics – p.1/31

Optimal Intercept Time

Suspect Vessel: speed v

Police Boat: speed u

y

xx0

y0

What direction does the police boat have to choose

to approach the suspect vessel up to a distance R as

quickly as possible ?Computational and Applied Mathematics – p.2/31

Geometric Solutiony

x0x

R

y0

Imagine the police boat going into all directions at thesame time =⇒ y2

0 + (vt − x0)2 = (ut + R)2.

In “Note on the Optimal Intercept Time of Vessels to a Nonzero Range”

G, SIAM Review Vol. 40, No. 3, 1998.


The Donut Problem

Top View

Side View

What is the maximum number of pieces one can getfrom a donut when cutting it with three planar cuts ?

Starting with an apple, how many pieces can we get ?


First Cut

Top view First cut

The most we can get with a single planar cut is twopieces.


Second Cut

Top view

First and second cut

We get six pieces in total, each of the first two is cut

twice.


Third Cut

How many pieces do we have now ?


The Total Number of Pieces isUpper right octant:

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15

3 pieces

Upper top octant:

Lower top octant:

1 piece

Lower left octant:

2 pieces

Upper left octant:

2 pieces 1 piece

Lower right octant:

1 piece

Lower bottom octant:

2 pieces

Upper bottom octant:

3 pieces

15 Pieces with 3 planar cuts.


Is this Solution Really Correct ?The answer is NO, the maximum number of piecesone can get is 13 ! How ?

An article by Martin Gardner in the “ScientificAmerican” contains the general result for a torus in ndimensions and m cuts along hyperplanes.

Suppose you are allowed to move the pieces betweenthe cuts, what is now the maximum number of piecesyou can get ?


Circular Billiard

In which direction do you have to push the red ball sothat it bounces of the rim exactly once and then hitsthe blue ball ?

Is there more than one solution ?Computational and Applied Mathematics – p.10/31

Circular Billiard: Algebraic Solution

PSfrag replacements γ

θx

y

−c a

b

1

f(θ) = (1 + c cos θ)√

(a − cos θ)2 + (b − sin θ)2

− (1 − a cos θ − b sin θ)√

(c + cos θ)2 + (sin θ)2


Solution for the Given Example

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θ

f(θ

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PSfrag replacements

θ

f(θ)

Are there always four solutions ?


The Geometric Point of View

PSfrag replacements

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x

y

e−e m1

m2

α

θ

We need to find an ellipse which touches the circle tan-

gentially.Computational and Applied Mathematics – p.13/31

Number of Solutions

An example with 4 solutions and one with 2 only.

Q(u) = (m2−m2m1)u4+(2m1−2m2

1+2e2+2m22)u

3+6u2m2m1

+ (−2m22 + 2m2

1 + 2m1 − 2e2)u − m2m1 − m2 = 0

where θ = 2 arctan(u). Q(u) is a 4th degreepolynomial, there can not be more than four solutions.


Dependence on the Ball PositionA numerical experiment: fixing the position of oneball and varying the position of the second ball.Counting the number of solutions gives the grayshade:


Dependence on the Ball Position

The separatrix (x,y) can be computed analytically (t parameter):

x(t)=−c

h

[

(1 + c)t6 + 3(1 + 3c)t4 + 3(1 − 3c)t2 + (1 − c)]

,

y(t) =16

hc2t3, h = (1 + 3c + 2c2)t6 + 3(1 + c + 2c2)t4+

+3(1 − c + 2c2)t2 + (1 − 3c + 2c2)

where c is the position of the fixed ball on the x axis.Computational and Applied Mathematics – p.16/31

Dependence on the Ball Position

Top view of a coffee mug with a point source of lightemulating the circular billiard game

(Drexler and G, SIAM review Vol. 40, No. 2, 1998)


Digital Signal ProcessingA periodic signal f(t) can be decomposed into itsFourier components:

f(t) =∞

∑

k=−∞

fkeikt.

Discrete version: for the vector f of length n, wherefj := f(tj), tj = j∆t, ∆t = 2π/n, we have

fj =n−1∑

k=0

fkeiktj .

What if the signal is an image, or a piece of music?Computational and Applied Mathematics – p.18/31

Population Dynamics

Joint work with Antonio Steiner, Il Volterriano1997-2003


The Lotka Volterra SystemWe consider rabbits and foxes living in a commonhabitat. If x denotes the rabbit population and y thefox population, the Lotka Volterra model states

x = x − xy

y = −y + xy

Approximating the derivative using its definition:

x(tn+1) − x(tn)

∆t= x(tn) − x(tn)y(tn)

y(tn+1) − y(tn)

∆t= −y(tn) + x(tn)y(tn)

We get a discrete dynamical system.Computational and Applied Mathematics – p.20/31

SolutionsThe exact solutions are cycles, but in general exact

solutions can not be found. Numerical solutionsspiral:

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A Symplectic MethodA very small change in the original method, instead of

x(tn+1) − x(tn)


y(tn+1) − y(tn)

∆t= −y(tn) + x(tn)y(tn)

changing in the second line tn to tn+1,

x(tn+1) − x(tn)


y(tn+1) − y(tn)

∆t= −y(tn) + x(tn+1)y(tn)

leads to a method for which one can prove that the

approximate solution is cyclic like the exact solution.Computational and Applied Mathematics – p.22/31

Success...

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The approximate solutions are also cycles, like the

mathematically exact or “biological” ones.Computational and Applied Mathematics – p.23/31

and Failure of the new method

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But here, in spite of the proof of cyclic behavior, the

method failed. On the right one can see why with the

zoom!


So When does the Method Work?

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The boundary beteween where the method works andwhere it does not is very complicated: it is a fractal.


Room Temperature in Montreal

∂u

∂t(x, t) =

∂2u

∂x2(x, t)+

∂2u

∂y2(x, t)+

∂2u

∂z2(x, t)+f(x, t)

Room temperaturein our living roomin Montreal (outsidetemperature up to -47degrees): the heat equa-tion.

Insulated walls, notwell insulated windowsand doors.


Why a Turntable in the Microwave ?The physical model is Maxwell’s equation

∇× E = −µH t,

∇× H = εEt + σE

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Noise Levels in a VOLVO S90

Noise simulation on a par-allel computer to improvepassenger comfort

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Radiation Therapy for Cancer

For cancer treatment it is important to have a very pre-

cise model of the body part that will be exposed to ra-

diation.Computational and Applied Mathematics – p.29/31

Aircraft Industry: B-747 in Flight

Simulation of a B-747 flying through athunderstorm, com-putation of the iceaccumulation on thewings and aroundthe engine intake.


The Fundamental Role of CAM

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− Fluid Dynamics

− Radiation− Particle Physics

− Aero Dynamics

Engineering:

− Active Noise Cancellation− Cellular Phone Systems

− Population Dynamics− Antibiotics− Micromachines

Physics:

Applied Mathematics

Pure Mathematics:− Geometry− Asymptotics− Existence

ANDComputational

Biology:

− Circuits Simulation

See also: www.math.mcgill.ca/mganderComputational and Applied Mathematics – p.31/31

Computational & Applied Mathematics

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