Miscellaneous Mathematical Texts and Snippets

by Klaus Pommerening

Elementary Trigonometry

• Wie hoch muß ein Turm sein, damit ich ihn sehe? (Z. B. beim Blick von Konstanz nach Bregenz) [März 2020]

Mathematical Models of Reality

• Exponentielles Wachstum (Ein ganz einfaches Beispiel) [Juni 2021]
• Die mediane Überlebenszeit [April 2017]
• Das Fehlalarm-Paradoxon (Warum produziert ein guter Test so miserable Ergebnisse?) [Mai 2020, erweitert November 2020 – alte Version hier]
• Anzahlschätzung anhand von Seriennummern (Wieviele Kraftfahrzeuge gibt es im Kanton Schaffhausen?) [Juni 2020]
• Der Gini-Koeffizient [Oktober 2021]
• Wer ist arm? (Wie der Kampf gegen die Armut durch eine schlechte Definition vernebelt wird.) [Februar 2022]
• Computersimulation dynamischer Systeme (dargestellt am Beispiel der Räuber-Beute-Systeme und anderer Wachstumsmodelle aus der Ökologie) [März 1987]

Sets and Mappings

• Finding the period of a periodic sequence (for example by Floyd's algorithm) [February 2009]
• The period of a random recursive sequence [May 1989/ english version May 2020]
• Recursive and periodic sequences (How to find the shortest feedback shift register that generates a given finite sequence) [April 2021]

Number Theory

• The euclidean algorithm [January 2000]
• Identities for binomial coefficients and Pascal tableaus [July 2020]
• Orbits of the multiplictive group mod m [February 2018]
• Remarks on subset sums [August 1987/ extended english version March 2018]
• A remark on subsemigroups (Dickson's lemma) [April 1975]
• Linear congruences with two unknowns [October 1986/ extended english version February 2018]
• The indecomposable solutions of linear congruences [October 1986/ extended english version March 2018]
• The indecomposable solutions of linear Diophantine equations [February 1986/ extended english version December 2017]
• Zero-sum multisets (a survey) [November 2017]
• On sums of two squares (Zagier's one-sentence proof, slightly blown-up) [April 2020]

Groups

• The number of orbits of finite group actions (»Burnside's lemma«) [February 2009]
• Permutations and Rejewski's theorem [January 2008]

Fields and Polynomials

• Konstruierbarkeit mit Zirkel und Lineal – ein Vorspiel zur Galois-Theorie [Oktober 1979]
• Bemerkungen zur Auflösung von Polynomgleichungen durch Radikale [Oktober 1979]
• Quadratic equations in finite fields of characteristic 2 [May 2000]
• Polynomial functions (interpolation from an algebraic view) [April 1997]
• Fourier analysis of Boolean maps [May 2000]

Algebraic Geometry and Invariant Theory

• Hilbert's Nullstellensatz over the complex numbers (a very short proof) [April 1982]
• Gordan's finiteness theorem (Hilbert's proof – slightly modernized) [April 1975]
• On systems of generators of graded algebras [July 2016]
• The Hilbert-Mumford criterion [January 1987/ April 2017]
• Some remarks on the complexity of invariant algebras [February 1986/ September 2016]

Lie Theory and Algebraic Groups

• Richardson's finiteness theorem [July 1979]
• The Morozov-Jacobson theorem (on 3-dimensional simple Lie subalgebras) [July 1979]
• Fixed binary forms [April 2017]
• Linear factors and stabilizers of binary forms [August 2017]
• The Clebsch-Gordan isomorphism [August 2017]
• On stabilizers and orbits of binary forms [May 2017]
• Closed orbits of reductive groups [September 2016]

Analysis

• Stirling's formula (with tight bounds) [September 1999]

Complex Analysis

• Extension of conformal maps to the boundary [April 1979]

Probability and Statistics

• The mean value of a sample [September 2020]
• Der Median (auch für beliebige Verteilungen) [Februar 2022]
• Uniformly distributed random variables in groups (another view at the One Time Pad) [June 1994]