Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.

/pdf/random-graph-dynamics-5durp7r63x.pdf

Random graph dynamics

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

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Random Graphs and Complex Networks

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

Considering an infinite string of iid random letters drawn from a finite alphabet we define the cover timeWn as the number of random letters needed until each pattern of lenghtn appears at least once as a substring Sharp weak and as limit results onWn are known in the symmetric case, ie, when the random letters are uniformly distributed over the alphabet In this paper we determine the limit distribution ofWn in the nonsymmetric case asn→∞ Generalizations in terms of point processes are also proved

Covering with blocks in the non-symmetric case

Let A,, A2, " , A, be events on a probability space. Denote by Sk the kth binomial moment of the number M, of those A 's which occur. Sharp lower and upper bounds of S,, will be given in terms of Sk and S,. The same method can be applied for proving Bonferroni-Galambos-type inequalities.

Bonferroni-galambos-type inequalities

The origin of eukaryotes and organellogenesis have been recognized as a major evolutionary transition and subject to in-depth studies. Acknowledging the fact that the initial interactions and conditions of cooperative behaviour between free-living single-celled organisms are widely debated, we narrow our scope to a single mechanism that could possibly have set-off multi-species associations. We hypothesize that the very first step in the evolution of such cooperative behaviour could be a single mutation in an ancestral symbiont genome that results in the formation of an ecto-commensalism with its obligate ancestral host. We investigate the ecological and evolutionary stability of inter-species microbial interactions with vertical transmissions as an association based on syntrophy (cross-feeding). To the best of our knowledge, this is the first time that a commensalistic model based on the syntrophy hypothesis is considered in the framework of coevolutionary dynamics and invadability by mutant phenotype into a monomorphic resident system.

https://www.biorxiv.org/content/biorxiv/early/2022/09/05/2022.09.04.506536.full.pdf

Tamás F. Móri

Papers

Covering with blocks in the non-symmetric case

Bonferroni-galambos-type inequalities

Genesis of Ecto-symbiotic features based on Commensalistic Syntrophy