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A course Mathematics and Technology

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Title: A course Mathematics and Technology


1
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.

2
  • 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

3
  • 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.

4
Mathematics and technology
  • New course since winter 2001. Most students are
    preservice secondary school teachers.
  • Purpose Discover mathematics present in everyday
    technologies

5
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.

6
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

7
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

8
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

9
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

10
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)

11
  • Covering a territory with antennas for a mobile
    phone network (Euclidean geometry)

12
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)
13
  • 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

14
Voronoï diagrams (Euclidean geometry)
15
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)

16
  • 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

17
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

18
Benford law of significant digit
19
How to complete the hole in Eschers painting
Print Gallery
20
Polyhedra and fullerenes
Carbone 60 Truncated icosahedron
21
  • 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

22
  • 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

23
A remarkable property of the parabola
All rays parallel to the axis are reflected to a
single point.
24
  • Applications the shape of many objects among
    which
  • Telescope mirrors

25
  • Solar furnaces
  • Parabolic antennas
  • Radars

26
The corresponding property of the ellipse
Any ray issued from one focus is reflected to the
other focus.
27
Applications mirrors, accoustic phenomena
  • Elliptic mirrors for instance behind the lamp of
    a cinema projector
  • Accoustic phenomena for instance Paris subway

28
Google and the PageRank algorithm
  • A search engine that does not order entries
    properly is useless.

29
Where are we after two clicks?

30
Where are we after n clicks?
Why?
31
Order of pages
B, A, C, E, D
32
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?

33
  • 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
34
  • 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.

35
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.

36
  • 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

37
Principle for drawing the fern
  • The fern is the union
  • of a stalk
  • of three copies of the initial fern

38
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.

39
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.

40
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

41
(No Transcript)
42
This works with any initial set! Lets try
another one
43
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.

44
Example
Sixth iterate
First iterate
45
Some exercices
46
The GPS (Global positioning system)fully
operational since 1995
  • Network of orbiting satellites whose position is
    known

47
  • 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.

48
  • 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.

49
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.

50
4 unknowns
  • 4 measured times
  • The shift between clocks
  • The three coordinates of position

With this method we get a precision of 20 meters.
51
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

52
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.

53
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.

54
Signal of the GPS
  • Shift-register

55
  • 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?
56
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.

57
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.

58
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.

59
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.

60
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
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