1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Evolution of Protein Molecules

Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.

Statistical mechanics of cellular automata

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

http://materiais.dbio.uevora.pt/Evo/A_review_of_long-branch_attraction.pdf

A review of long-branch attraction

Evolutionary algorithms use crossover to combine information from pairs of solutions and use selection to retain the best solutions. Ideally, crossover takes distinct good features from each of the two structures involved. This process creates a conflict: progress results from crossing over structures with different features, but crossover produces new structures that are like their parents and so reduces the diversity on which it depends. As evolution continues, the algorithm searches a smaller and smaller portion of the search space. Mutation can help maintain diversity but is not a panacea for diversity loss. This paper explores evolutionary algorithms that use combinatorial graphs to limit possible crossover partners. These graphs limit the speed and manner in which information can spread giving competing solutions time to mature. This use of graphs is a computationally inexpensive method of picking a global level of tradeoff between exploration and exploitation. The results of using 26 graphs with a diverse collection of graphical properties are presented. The test problems used are: one-max, the De Jong functions, the Griewangk function in three to seven dimensions, the self-avoiding random walk problem in 9, 12, 16, 20, 25, 30, and 36 dimensions, the plus-one-recall-store (PORS) problem with n=15,16, and 17, location of length-six one-error-correcting DNA barcodes, and solving a simple differential equation semi-symbolically. The choice of combinatorial graph has a significant effect on the time-to-solution. In the cases studied, the optimal choice of graph improved solution time as much as 63-fold with typical impact being in the range of 15% to 100% variation. The graph yielding superior performance is found to be problem dependent. In general, the optimal graph diameter increases and the optimal average degree decreases with the complexity and difficulty of the fitness landscape. The use of diverse graphs as population structures for a collection of problems also permits a classification of the problems. A phylogenetic analysis of the problems using normalized time to solution on each graph groups the numerical problems as a clade together with one-max; self-avoiding walks form a clade with the semisymbolic differential equation solution; and the PORS and DNA barcode problems form a superclade with the numerical problems but are substantially distinct from them. This novel form of analysis has the potential to aid researchers choosing problems for a test suite

Graph-based evolutionary algorithms

Cellular automata can generate fractals

Computing fractal dimensions for additive cellular automata

Properties of normal phylogenetic networks.

A rooted acyclic digraph N with labeled leaves displays a tree T when there exists a way to select a unique parent of each hybrid vertex resulting in the tree T. Let Tr(N) denote the set of all trees displayed by the network N. In general, there may be many other networks M, such that Tr(M) = Tr(N). A network is regular if it is isomorphic with its cover digraph. If N is regular and {\cal D} is a collection of trees displayed by N, this paper studies some procedures to try to reconstruct N given {\cal D}. If the input is {\cal D}=Tr(N), one procedure is described, which will reconstruct N. Hence, if N and M are regular networks and Tr(N) = Tr(M), it follows that N = M, proving that a regular network is uniquely determined by its displayed trees. If {\cal D} is a (usually very much smaller) collection of displayed trees that satisfies certain hypotheses, modifications of the procedure will still reconstruct N given {\cal D}.

Stephen J. Willson

Papers

Graph-based evolutionary algorithms

Cellular automata can generate fractals

Computing fractal dimensions for additive cellular automata

Properties of normal phylogenetic networks.

Regular Networks Can be Uniquely Constructed from Their Trees