The purpose of this laboratory exercise is to give you experience with the use of topology optimization as a first design tool in doing a construction. A 99 line topology optimization code written in Matlab will be used. The code is available at

http://www.topopt.dtu.dk

The details are explained in the paper “A 99 line topology optimization code writter in Matlab” which is published in the journal “Structual and Multidisciplinary Optimization” by Ole Sigmund, dept. of solid mechanics at Technical University of Denmark. The matlab code is also included as an appendix.

This matlab code should give understanding of the basics of a topology optimization program, and the user can easily modify the code

The laboratory exercise is divided into two parts. The first part is about the design of bicycles and is intended to make you familiar with the methodology and the matlab code. The second part consists of a bunch of projects that can be treated in the same way as the bicycle. It is advisable to choose one of the projects in the second part.

We give a simple and self-contained proof of the Parisi and Coppersmith-Sorkin formulas for the expected value of the minimum assignment in a matrix of independent exponential random variables.

The game of two-person single-suit whist is played with a deck consisting of a single totally ordered suit of 2n cards. Each of the two players receives n cards. Hence both players have complete information about the distribution of the cards. One of the players is said to be on lead. Play proceeds in rounds called tricks. The player on lead plays one of his cards, and with knowledge of this card, the other player plays one of his cards. The player with the higher card wins the trick, and obtains the lead. The cards that are played are then removed. Play continues until all cards are exhausted. Each player tries to win as many tricks as possible

We give a complete solution to this game, that is, an efficient algorithm for calculating the game theoretical value of any distribution of the cards.

We give an elementary proof of the Wallis product formula for pi. The proof does not require any integration or trigonometric functions

We complete a result of Svante Janson on the asymptotic normal distribution of the cost Wn of the minimum spanning tree in a complete graph on n vertices with independent uniform (0,1) edge costs. By evaluating a triple sum given by Janson, we show that the variance of Wn is asymptotically n-1(6ζ(4)-4(ζ(3)), where ζ is the Riemann zeta function.

We obtain an exact formula for the expected value of the optimum for a certain class of random combinatorial optimization problems on bipartite graphs. By applying this formula, we compute the limit as n → ∞ of the expected value of the minimum 2-factor in the complete bipartite graph Kn,n with independent exp(1) edge costs. This limit, approximately 4.0831, is conjectured to be twice the value obtained non-rigorously by W. Krauth, M. M´ezard and G. Parisi for the minimum travelling salesman tour on the complete graph.

We obtain exact formulas for all moments of the cost Cn of the inimum assignment in the complete bipartite graph Kn,n with independent exp(1) edge costs. In particular we prove that as n tends to infinity, the variance of Cn is c/n+O(1/n2), where c = 4ζ(2)-4ζ(3), or approximately 1.7715.

The game of two-person whist is played with a deck of cards. Each card belongs to a suit, and each suit is totally ordered. The cards are distributed between the two players so that each player receives the same number of cards. Hence both players have complete information about the deal. Play proceeds in tricks, with the obligation to follow suit, as in many real-world card games.

We study the symmetric case of this game, that is, we assume that in each suit, the two players have the same number of cards. Under this assumption, the second player in each trick will always be able to follow suit.

We show how to assign a value from a certain semigroup to each single-suit card distribution in such a way that the outcome of a multisuit deal under optimal play is determined by the sum of the values of the individual suit. This allows us to play a deal perfectly, provided that we can compute the values of its single-suit components. Although we do not have an efficient algorithm for doing this in general, we give methods that will allow us to find the value of a suit quickly in most cases.

We also establish certain general facts about the game, for instance the nontrivial fact that a higher card is always at least as good as a smaller card in the same suit.

The core of these lecture notes corresponds to the contents of a series of seminars held during November-December, 2003, at the Division of Biometry of the Rolf Nevanlinna Institute (RNI) (a research institute of mathematics, computer science and statistics, University of Helsinki). The seminars were organized within the “Centre of Population Genetic Analyses” at RNI. The centre is funded by a grant from the Academy of Finland.

The author thanks Professor Elja Arjas, head of the Division of Biometry, for the invitation to give these lectures and for several comments that have improved both the contents and of the presentation.

The original impetus for the studies underlying these notes was to understand certain issues in bacterial taxonomy (see e.g., Busse et.al. 1996, Gaunt et.al. 2001, Mougel et.al. 2002, Van de Peer et.al. 1993) especially when construction hidden Markov models using Hobohnm algorithm for preprocessing of sequence data.

The benefit of these lectures for their audience might be to have been exposed to many of the concepts and models that are being mentioned in the section on methods of a biological research paper like, e.g., (Gaunt et.al. 2001), and applied there, or in documents like PAML Manual (Yang 2002), MrByes Manual (Huelsenbeck and Ronquist 2001), and others.

The current version of the notes is not final; several additional sections and chapters are under construction. One obvious shortcoming of this version is that every item in the Bibliography is not necessarily referred to in the text. Some of the figures in the text have a sketchy standard.