Miscellaneous Mathematical Texts and Snippets

by Klaus Pommerening

Sets and Mappings

• Finding the Period of a Periodic Sequence (for example by Floyd's algorithm)

Number Theory

• The Euclidean Algorithm
• A Remark on Subset Sums
• A Remark on Subsemigroups (Dickson's lemma)
• The Indecomposable Solutions of Linear Diophantine Equations
• The Indecomposable Solutions of Linear Congruences

Groups

• The Number of Orbits of Finite Group Actions
• Permutations and Rejewski's Theorem

Fields and Polynomials

• Quadratic Equations in Finite Fields of Characteristic 2
• Polynomial Functions (interpolation from an algebraic view)
• Fourier Analysis of Boolean Maps

Algebraic Geometry and Invariant Theory

• Hilbert's Nullstellensatz over the complex numbers (a very short proof)
• Gordan's Finiteness Theorem (Hilbert's proof – slightly modernized)

Lie Theory and Algebraic Groups

• Richardson's Finiteness Theorem
• The Morozov-Jacobson Theorem (on 3-dimensional simple Lie subalgebras)

Analysis

• Stirling's Formula (with tight bounds)

Complex Analysis

• Extension of Conformal Maps to the Boundary

Probability

• Uniformly Distributed Random Variables in Groups (another view at the One Time Pad)