Lecture December 22 and 29th

We advance through the chapter 12: Image compression and JPEG standard.
The main ideas of the compression are the following two: first, the pictures form a linear space, namely the space of functions defined on the pixels of the camera sensor, with values in real numbers. Second, the basis of this space used by camera (just one pixel is white, all other pixels are black) is not natural for humans, and the natural basis for humans is what we called the Fourier basis.

The later consists of oscillating pictures, and pictures with higher oscillations tend to be less important. JPEG compression of a picture throws away the less important components unless they are REALLY significant.

The third exercise is due on January 12.

Correction (Jan 19): Due date is moved to January 26. The exercise was corrected.

Lecture December 15

We finished the Chapter 9, with proof of Frobenius theorem.
Here is the second exercise, due on December 27.
Next topic will be the Chapter 12, about the JPEG compression.
Happy Hanuka!

Lecture of november 24 (second hour)

We started to read Chapter 9 of "Mathematics and Techonology" by Ch. Rousseau and Y. Saint-Aubin .
The main idea of the PageRank algorithm is not to judge a website by its content, but rather by how often a randomly clicking websurfer will return to it. If we represent a collection of websites by a directed graph (edges of the graph represent links from one website to another), then we have to discuss random walks on graphs. We discussed Markov property of the random walk, and defined transition matrix P. We finished by noticing that a limiting probability vector p, if exists, should satisfy p=Pp, i.e. to be an eigenvector with eigenvalue equal to 1.

Lecture of November 24 (first hour)

The first hour was devoted to Euler angles. There are many variations of Euler angles, but the main idea is the same: you can represent any rotation of three-dimensional space as a composition of three consecutive rotations around COORDINATE AXIS. So, as soon as you chose the axis of rotation for each step, any rotation will be defined by three numbers, namely by the three angles of these three rotations. For the choice of X,Z and again X axes we built Euler angles explicitly, essentially proving this representability.


Euler angles are widely used everywhere where you have a moving solid body, e.g. aeronautics, cinematography (camera movements) etc. If you google "Euler angles", many sites you'll be referenced to will try to explain in layman terms that you should keep right ORDER of these rotations (i.e. non-commutativity of group of space rotations). In aeronautics rotation around each coordinate axis has its own name: if you choose X axis going from the tail to the nose of the airplane, and Z axis going vertically up, then rotation around Z axis is the "heading change", around Y axis is the "pitch change" and around X axis is the "roll change", see e.g.