This was written as part of a reading group I organized at my university. These notes are meant to explain probability theory, Shannon entropy, and measured dynamical entropy, with one application. My sources are Kerr and Li’s Ergodic Theory and Claude Shannon’s paper A Mathematical Theory of Communication.
Category Archives: Math
Rotation Numbers and the Poincare Classification
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are meant to explain rotation numbers of circle homeomorphisms, and some general dynamical concepts like factor maps and homoclinicity. Denjoy’s theorem is also discussed.
Rotations of the Circle: Dynamical Properties
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are meant to explain the properties of minimality and topological transitivity.
Classifying 2-dimensional Linear Systems
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are meant to explain the phase portraits of 2-dimensional linear differential equations.
Vector Fields, Flows, and the Matrix Exponential
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are meant to explain concepts from Chapter 3.
Basic Hyperbolicity, the Differential, and Contraction Mapping Theorem
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are meant to explain concepts from Chapter 2.
Topology Crash Course Part 2
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are part of a series meant to provide the participants the necessary topological background.
Topology Crash Course, Part 1
This was written as part of a reading group I organized at my university. The text we are using is Katok and Hasselblatt’s First Course in Dynamics. These notes are part of a series meant to provide the participants the necessary topological background.
Lie Groups and their Lie Algebras
There is a saying that “groups, like men, will be known by their actions.” The groups we are interested in here are known by their actions as linear transformations on vector spaces (of finite dimension, over or ). Thus for every we must have for all scalars and . Since we require that , theContinue reading “Lie Groups and their Lie Algebras”
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