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. The scientist : He(she) asks questions;
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.
Helping discovering the power of the mathematical method:
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.
- 1 week = 3 hours plus 2 hours of exercices
- or two weeks on one subject
Mathematics and technology, SUMAT, Springer
C. Rousseau and Y. Saint-Aubin,
Mathématiques et technologie, SUMAT, Springer
C. Rousseau and Y. Saint-Aubin,
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).
All rays parallel to the axis are reflected to a single point.
Applications: the shape of many objects among which The laws of nature follow optimization principles. Applications : short waves, optical fiber
Any ray issued from one focus is reflected to the other focus.
A search engine that does not order entries properly is useless.
B, A, C, E, D
The easiest way to store an image inside the memory of a computer is to store the color of each pixel.
This requiresan enormous quantity of memory!
Can we do better?
Let’s suppose we have drawn a city: The laws of nature follow optimization principles. Applications : short waves, optical fiber
We store in memory the line segments, circle arcs, etc…, which approximate our image.
We approximate our image by known
To store a line segment in memory it is sufficient to store: The laws of nature follow optimization principles. Applications : short waves, optical fiber
The geometric objects are ouralphabet.
We store in memory a program to draw the fern. Such a program on Mathematica
ListPlot[T, AspectRatio->1, Axes-> False]
Let’s look at the Sierpinski carpet:
It is a union of three Sierpinski carpets.
Let us start with a square and iterate a construction algorithm
c:speed of light
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.
4 unknowns is excluded because it is non realistic.
4 measured times
With this method we get a precision of 20 meters.
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.
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 receive000we decode0
We have corrected 0or 1error.
If we receive110we decode1
011we 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.
We want to send a 4 bits word:u1, u2, u3, u4
We send a 7 bits word. We add:
u5 = u1+ u2 + u3
u6 = u2+ u3 + u4
u7 = u1+ u2 + u4
This code can correct one error.
u2 erroneous:u5, u6andu7incompatibles
u3 erroneous:u5and u6incompatibles