A course Mathematics and Technology

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A course Mathematics and Technology 

• Mathematics is a living science, everywhere
  present in science and technology
• The teacher should have experienced how science
  develops in the real world. He(she) can then show
  it.

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• The scientist 
• He(she) asks questions
• He(she) dares to say I do not know. 
• He(she) has an open and critical mind.
• Helping discovering the power of the mathematical
  method
• Modelling
• Problem solving
• Mathematical sophistication
• Use of computer

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• Some messages through the course 
• Mathematics are useful and constantly developing
  around us.
• The questions Why? and What is it useful for?
  should be encouraged and deserve an answer.
• The beauty of mathematical constructions
• Mathematics are much more than numbers
• Most subjects treated are too advanced so that
  preservice teachers following the course can hope
  bring them directly to classroom. The purpose of
  the course is rather to teach them how to prepare
  such kind of material.

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Mathematics and technology

• New course since winter 2001. Most students are
  preservice secondary school teachers.
• Purpose Discover mathematics present in everyday
  technologies

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Joint creation with my colleague Yvan
Saint-Aubin

• Yvan is a physicist and I am a mathematician.
• We knew very little of the material of the course
  when it was created. We now have enough material
  for at least two courses.
• The game is to take some technologies into
  pieces, to dismantle them in order to discover
  and explain the mathematics that make them work.
• We play the game to prepare the course. We try to
  teach the students to do the same.

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Description

• Two formats
• Flashs-science (1 hour)
• More elaborate subject
• - 1 week 3 hours plus 2 hours of exercices
• - or two weeks on one subject
• The lectures are of two different types
• elementary parts (subject matter for exams)
• conference type lectures on advanced parts

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Evaluation 

• Two exams with open book and personal notes. Non
  cumulative contents
• A session project on an application of
  mathematics (by teams of two, if possible by
  larger teams (4-6) otherwise )
• A half-hour oral presentation of the project

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The exercices

• We have spent a lot of time writing interesting
  exercices that make the students practice
  modelling and review their elementary maths
• Finding appropriate exam questions is not a
  trivial task
• A few examples below

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A book for the course

• Mathematics and technology (september 2008)
• C. Rousseau and Y. Saint-Aubin, Springer-Verlag
• Mathématiques et technologie (october 2008)
• C. Rousseau and Y. Saint-Aubin, Springer-Verlag

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Flashs-science

• Antennas and radars are parabolic. Why?
  (Geometric definition of conics)
• Computer vision calculating the position of one
  object from its position on two photos (The
  parametric equations of lines in 3-dimensional
  space)

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• Covering a territory with antennas for a mobile
  phone network (Euclidean geometry)

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The corresponding exercice at the exam

• We fill a large planar region with nonoverlapping
  disks of radius r. We use two methods in the
  first method we place the centers of the disks on
  a square network and in the second method we
  place them on a regular triangular network of
  equilateral triangles.
• Which method gives the denser filling?
  Suggestion compute the proportion of each
  square covered by portions of disks in case (a)
  and the proportion of each triangle covered by
  portions of disks in case (b).

(b)
(a)
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• Physics  unifying the laws of reflection and
  refraction. The laws of nature follow
  optimization principles. Applications  short
  waves, optical fiber
• A short look in the architecture of computers
  describing logic circuits
• The regular tiling of the sphere with twelve
  spherical pentagons

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Voronoï diagrams (Euclidean geometry)
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More elaborate subjects

• Positioning in space  GPS, GPS signal,
  cartography, localization of thunderstorms
  (Geometric locus, differential geometry, theory
  of finite fields)
• How is a musical CD engraved why 44100 numbers
  per second? (Elementary Fourier analysis)
• Public key cryptography (Elementary number
  theory congruences)
• Error correcting codes  Hamming codes and
  Reed-Solomon codes (Linear algebra, finite
  fields)
• Image compression iterated function systems
  (Affine transformations of the plane)
• The JPEG format (.jpg) (Elementary Fourier
  analysis)

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• Robots (Rotations in 3-dimensional space, change
  of reference frame)
• Friezes and tilings (Symmetries linear algebra)
• Google and the Pagerank algorithm (Markov chains
  linear algebra)
• The skeleton and the gamma-knife surgery
  (Geometry)
• Turing machines and DNA computers (The hierarchy
  of functions starting from the basic ones)
• Random number generators (Finite fields)
• Calculus of variations (Multi-variable calculus)
• Sparing and borrowing money

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Some students projects(a list on my webpage)

• Rollercoasters
• The search of boundaries in a photo
• Morphing IMAGES
• Text compression
• Mathematical morphology in treating images

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Benford law of significant digit
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How to complete the hole in Eschers painting
Print Gallery
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Polyhedra and fullerenes
Carbone 60 Truncated icosahedron
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• Voronoï diagrams and Delaunay triangulation in
  image analysis
• Sphere packings and honeycombs
• The best skateboard track
• Other cryptographic methods
• Reed-Müller error-correcting codes
• Knots and the action of enzymes on DNA
• Digital fingerprintING
• Image compression  from fractals to practical
  applications

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• Penrose tilings
• The seasons, the locus of the sunrise and sunset
  at a given date, the length of day at a given
  date,
• Calculation of astronomic distances, from the
  ancient Greeks to now
• The eclipses
• The shape of sand dunes
• Phyllotaxy (how to explain spirals in sunflowers,
  etc.)
• Population growth under constraints
• Mathematical modelling of epidemics
• Chaos

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A remarkable property of the parabola
All rays parallel to the axis are reflected to a
single point.
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• Applications the shape of many objects among
  which
• Telescope mirrors

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• Solar furnaces
• Parabolic antennas
• Radars

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The corresponding property of the ellipse
Any ray issued from one focus is reflected to the
other focus.
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Applications mirrors, accoustic phenomena

• Elliptic mirrors for instance behind the lamp of
  a cinema projector

• Accoustic phenomena for instance Paris subway

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Google and the PageRank algorithm

• A search engine that does not order entries
  properly is useless.

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Where are we after two clicks?

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Where are we after n clicks?
Why?
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Order of pages
B, A, C, E, D
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Image compression

• The easiest way to store an image inside the
  memory of a computer is to store the color of
  each pixel.
• This requires an enormous quantity of memory!
• Can we do better?

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• Lets suppose we have drawn a city

We store in memory the line segments, circle
arcs, etc, which approximate our image.
We approximate our image by known geometric
objects
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• To store a line segment in memory it is
  sufficient to store
• the two endpoints of the line segment
• a program explaining to the computer how to draw
  a line segment with given endpoints.
• The geometric objects are our alphabet.

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How to store more complex images, for instance
landscapes?

• We use the same principle but we enlarge our
  alphabet
• We approximate our landscape by fractals, for
  instance the fern.

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• We store in memory a program to draw the fern.
  Such a program on Mathematica
• m15000
• Ln_If1ltnlt87,2,n
• Hn_If86ltnlt94,3,Ln
• Kn_Ifngt93,4,Hn
• RTableKRandomInteger,1,100,m
• F1,x_,y_0
• G1,x_,y_0.16y
• F2,x_,y_x0.85y0.04
• G2,x_,y_-x0.04y0.851.6
• F3,x_,y_x0.2-y0.26
• G3,x_,y_0.23x0.22y1.6
• F4,x_,y_-x0.15y0.28
• G4,x_,y_x0.26y0.240.44
• x10
• y10
• Doxn1,yn1FRn,xn,yn,GRn,x
  n,yn,n,1,m
• TTablexn,yn,n,m

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Principle for drawing the fern

• The fern is the union
• of a stalk
• of three copies of the initial fern

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We can reconstruct the fern from 4 affine
transformations

• the transformation T1 which sends the large
  fern to the fern minus two branches,
• the transformation T2 which sends the large fern
  on the left branch,
• the transformation T3 which sends the large fern
  on the right branch,
• the transformation T4 which sends the large fern
  on the stalk.

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In order to reconstruct the fern, it suffices to
store in memory this information!

• Algorithm
• We take a point P on the fern.
• We choose at random i in 1,2,3,4 and we plot
  
  P1 Ti(P).
• We choose at random i in 1,2,3,4 and we plot
  P2 Ti(P).
• Etc...
• This method is called Iterated function systems
  . It works because the fern is self similar.

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Why does it work?

• Lets look at the Sierpinski carpet
• It is a union of three Sierpinski carpets.
• Let us start with a square and iterate a
  construction algorithm

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(No Transcript)
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This works with any initial set! Lets try
another one
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In practice

• Coding

• We replace any small square by the image of a
  similar larger square under a homothety of ratio
  ½ composed with one of 8 transformations
• Identity plus 3 rotations
• 4 symetries
• We adjust contrast.
• We make a translation of the level of grey.

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Example
Sixth iterate
First iterate
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Some exercices
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The GPS (Global positioning system)fully
operational since 1995

• Network of orbiting satellites whose position is
  known

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• The receptor measures the travelling time t of a
  signal from one satellite to the receptor.
• The distance from the satellite to the receptor
  is d ct
• c speed of light
• The points located at a distance d from a
  satellite are on a sphere of radius d, with
  center at the satellite.

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• The intersection of two spheres is a circle

• The intersection of three spheres is two points.
  One of them is excluded because it is non
  realistic.

• Hence, if we know the travelling time of the
  signals of three satellites to the receptor we
  know the position of the receptor.

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This is the theory

• In practice the satellites have atomic clocks
  perfectly synchronized.
• The receptor has a cheap clock.
• We have a fourth unknown the shift between the
  clock of the receptor and the clocks of the
  satellites.
• We then need to measure the travelling time of
  a signal from a fourth satellite.

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4 unknowns

• 4 measured times

• The shift between clocks
• The three coordinates of position

With this method we get a precision of 20 meters.
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Applications of the GPS

• Finding ones way in nature
• Drawing a map
• Managing a fleet of vehicles
• Measuring Mount Everest and observing its growth
• Helping blind people
• Find ones way on the road
• Landing a plane in the fog

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GPS are a reference of time!

• Electronic equipments can be synchronized with
  the help of GPS.
• Hydro-Québec uses this method to synchronize its
  lightnings detectors. Once thunderstorms are
  localized, one can reduce the current through
  lines passing through zones of thunderstorms so
  as to minimize the risk of breakdown of the
  electrical network, in case one transit line
  receives a lightning.

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A related exam question

• Meteorites regularly enter the atmosphere,
  rapidly heat up, disintegrate, and finally
  explode before hitting the surface of the Earth.
  This explosion generates a shock wave that
  travels in all directions at the speed of sound
  v. The shock wave is detected by seismographs
  installed at various locations on the surface of
  the Earth.
• If four stations (equipped with perfectly
  synchronized clocks) note the moment that the
  shock wave arrives, explain how to calculate both
  the position and time of the explosion.

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Signal of the GPS

• Shift-register

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• Example we take (q0, q1, q2, q3 )(1,1,0,0)
• 000100110101111
• 001001101011110
• 010011010111100
• 100110101111000
• 001101011110001
• 011010111100010
• 110101111000100
• 101011110001001
• 010111100010011
• 101111000100110
• 011110001001101
• 111100010011010
• 111000100110101
• 110001001101011
• 100010011010111

Why?
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Random number generators

• Consider sequences of 0 and 1
• 0 and 1 must each appear with probabilty ½.
• All sequences of length 2 must each appear with
  probability ¼.
• All sequences of length n must each appear with
  probability 1/2n.

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Theorem in the sequences of period 2n 1
generated by the shift-register

• 1 appears 2n-1 times and 0 appears 2n-1 1
  times,
• Each sequence of length 2 appears 2n-2 times
  except 00 which appears 2n-2 1 times
• Each sequence of length r appears 2n-r times
  except 00 which appears 2n-r 1 times, for r
  lt n.

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Error correcting codes

• Principle we lengthen the message in a redundant
  way. This allows to correct some errors.
• Example We repeat each bit three times. We want
  to send 0.
• We send 000.
• If we receive 000 we decode 0
• 100 we decode 0
• 010 we decode 0
• 001 we decode 0
• We have corrected 0 or 1 error.

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However

• If we receive 110 we decode 1
• 101 we decode 1
• 011 we decode 1
• 111 we decode 1
• And the transmission is erroneous.
• An error correcting code is efficient if there
  are few errors.
• This code is not economical a word of 4 bits is
  lengthened to 12 bits and we may only be able to
  correct one error.

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We can do much better

• Hamming code
• We want to send a 4 bits word u1, u2, u3, u4
• We send a 7 bits word. We add (mod 2)
• u5 u1 u2 u3
• u6 u2 u3 u4
• u7 u1 u2 u4
• This code can correct one error.
• u1 erroneous u5 and u7 incompatibles
• u2 erroneous u5, u6 and u7 incompatibles
• u3 erroneous u5 and u6 incompatibles
• u4 erroneous u6 and u7 incompatibles
• u5 erroneous u5 incompatible
• u6 erroneous u6 incompatible
• u7 erroneous u7 incompatible