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Tags
 CayleyHamilton Theorem
 Chinese remainder theorem
 Compression
 Diophantine approximation
 Dirichlet's approximation theorem
 Dirichlet's unit theorem
 Dynamics
 eigenvalues
 Entropy
 Generic Matrix
 Geometry
 Geometry of numbers
 Homogeneous spaces
 Hyperbolic space
 Lattices
 Minkowski's Theorem
 Number Theory
 Pell's equations
 radar
 random walk
 self similar sets
 Sierpinski carpet
 Sierpinski triangle
 SL_2(Z)
 stochastic matrices
 Weak Law of Large Numbers
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Author Archives: eofirdavid
Improper integrals and periodic functions
The idea for this post came from a question I saw in a math help forum about improper integrals. While this problem has a very simple solution using basic tools in integral calculus, I want to show a more geometric … Continue reading
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Random walks on graphs
Imagine going to a new amusement park for the first time. Once you get there, you go to the first ride that you see. Once you finish it, you randomly choose one of the roads leaving it and follow it … Continue reading
Radars and the Chinese Remainder Theorem
The radar is a detection system that was developed before and during World War II for military uses, though by today it has many other applications including, for example, astronomical and geological research. The name radar is an acronym for … Continue reading
Billiard tables – and what is mathematical research
Mathematical research is something that most people don’t really understand. They can imagine someone in a lab mixing chemicals or doing experiments with some scientific machinery, but mathematical research? The goal of this post is to share with you a … Continue reading
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Random walks and self similar sets
1. Introduction In this post we consider an interesting mathematical process which can be easily simulated by a computer, and generates interesting pictures. A video version of this post can be seen in here (for now in Hebrew). Let and … Continue reading
How to measure information using entropy
Which text gives us more information – the full body of work of Shakespeare, or 884,647 random words written by 1000 monkeys? To answer this, we consider the following problem: Alice has a message over some finite alphabet and she … Continue reading
Lattice parametrization
We came to the point where we have already seen how lattices appear naturally in problems arising from number theory. In this post we construct a nice space which parametrize the set of all lattices of a certain dimension, with … Continue reading
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Tagged Homogeneous spaces, Hyperbolic space, Lattices, SL_2(Z)
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The rise of algebraic extensions
In the post about number theory and lattices, we tried to determine when is the Euclidean distance in is actually a Euclidean norm and we were led to study the embeddings of rings such as as lattices in . As mentioned … Continue reading
Minkowski’s theorem
In the post about Diophantine approximation, we saw that in order to find “good” rational approximations to a real number, it is enough to prove that given a lattice and a “big enough” box around the origin, the box must … Continue reading
From number theory to geometry of lattices
Number theory can mean a lot of thing to a lot of people. This is a very big part of mathematics, and it contains many areas starting with the elementary number theory (“simple” congruence like arguments), algebraic number theory (e.g. … Continue reading