Discrete Mathematics I - Lecture 13

Heutige Themen:

• Einheiten in kommutativen Ringen,
• Fundamentalsatz der Arithmetik,
• Lineare Gleichungen in Restklassenringen (Z mod n und Q[x] mod f),
• Berechnung von Inversen
• Nullteiler in kommutativen Ringen,
• Integritätsbereiche und Körper

Computational Biology - Lecture 13

Today, the following topics were on the roster:

• Closure theorem,
• parametric representation,
• implicitization theorem,
• model invariants for algebraic statistical models.

Introduction to Bioinformatics - Lecture 13

Today, statistical sampling methods were considered. Here are the topics:

• Statistical mechanics,
• canonical and micro-canonical ensemble,
• observables and partition function,
• molecular dynamics simulations (verlet and velocity verlet algorithm),
• consideration of time step,
• improvement of simulation (cutoff, distance, and multipole schemes),
• standard Monte Carlo method,
• calculation of partition function,
• importance sampling,
• sampling of protein structures.

Discrete Mathematics I - Lecture 12

Heutige Themen:

• Hauptsatz der Arithmetik,
• Restklassenringe modulo Z bzw. Q[x].

Computational Biology - Lecture 12

Yesterday, we resumed with the introduction into algebraic geometry:

• ideal-variety correspondence,
• elimination theorem,
• extension theorem.

Introduction to Bioinformatics - Lecture 12

Today, we will give an introduction to 3D structure prediction of proteins:

• Force fields (CHARMM, Oobatake-Crippen),
• rigid geometry models,
• buildup method,
• basic heuristic methods,
• conformational space annealing,
• HP model.

Discrete Mathematics I - Lecture 11

Kurze Einführung in die Teilbarkeitslehre (Ring der ganzen Zahlen, Polynomring):

• Division mit Rest
• ggT und kgV
• Euklidischer Algorithmus
• Satz von Bezout, erweiterter Euklidischer Algorithmus.

In der Vorlesung gab es Fragen nach der Definition des ggT, d.h., gemeinsamer Teiler und größter unter allen gemeinsamen Teilern. Mathematisch handelt es sich um das Infimum der beteiligten Zahlen. Die gemeinsamen Teiler bilden die unteren Schranken und das Infimum ist die größte untere Schranke. Für das kgV gilt die duale Aussage.

Computational Biology - Lecture 11

Today, we will give an introduction to affine algebraic sets:

• Hilbert's Nullstellensatz (weak and strong version),
• correspondence between affine algebraic sets and ideals.

Introduction to Bioinformatics - Lecture 11

Today, we finished the considerations about 2D structure prediction:

• Nearest neighbor classification (intrinsic dimension, Bhattacharyya distance).
• Consensus prediction
• Neural network classification (Rost-Sander approach).

Exams in the Winter Term

Written exams:

• Bioinformatics, Feb. 8, 2011.
• Discrete Mathematics I and II, March 21, 2011.

Oral exams do not follow a fixed schedule, day and time can be negotiated on a case-by-case basis.

Discrete Mathematics I - Lecture 10

In der heutigen Vorlesung wurden folgende Themen behandelt:

• Aufbau der ganzen Zahlen 
• Definition von Ringen und ihre Grundeigenschaften
• Polynomringe
• Binomialsatz

Computational Biology - Lecture 10

Today, we will finish the topic on Groebner bases:

• Minimal Groebner bases,
• Reduced Groebner bases,
• Buchberger's S-criterion,
• Buchberger's algorithm.

Introduction to Bioinformatics - Lecture 10

Today, an introduction to the prediction of secondary protein structures was given:

• GOR method,
• Chow-Fasman method,
• Generation of sample sets,
• Nearest neighbor classification.

Reed-Muller Codes - Revisited

Recently, we published the article:

• From Ideals in Polynomial Rings to Linear Codes Using Groebner Bases, Int. J. Pure Applied Math., vol. 65, no. 1, pp. 41-53, 2010.

This paper studies linear codes as ideals in the group algebra over an elementary abelian p-group. These codes are described in terms of Groebner bases and corresponding encoding and decoding procedures are given. It is shown that the class of codes studied contains an interesting family of linear codes. These codes have a designed Hamming distance and turn out to be superior to the primitive Reed-Muller codes in the non-binary case.

The Reluctant Mr. Darwin

The other day, I read the book The Reluctant Mr. Darwin from David Quammen, 2006. The book provides insight into the course of Charles Darwin's life from his return from the voyage on the Beagle until his death. Darwin spent years to catalogue the vast collection of specimens he brought back. He was a self-taught scientist and eventually came to a theory of evolution. He was reluctant to publish but first choose to share his ideas with colleagues. He was afraid of a public backlash and sometimes diverted his attention and energies elsewhere. In the meantime, another scientist, Alfred R. Wallace, independently came to the same ideas about evolution. Then Darwin's colleagues eventually convinced him to publish his theory. His book, The Origin of the Species, found both, acclaim and dislike.

German translation:

• David Quammen: Charles Darwin - Der große Forscher und seine Theorie der Evolution, Piper Verlag, 2010, 9,95 €.

Happy New Year!