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1

Simulating an infinite mean waiting time

Bartoszek Krzysztof

Mathematica Applicanda

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2019

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Vol. 47, no. 1

93--102

EN

We consider a hybrid method to simulate the return time to the initial state in a critical-case birth-death process. The expected value of this return time is infinite, but its distribution asymptotically follows a power-law. Hence, the simulation approach is to directly simulate the process, unless the simulated time exceeds some threshold and if it does, draw the return time from the tail of the power law.

PL

W pracy rozważany jest mieszany sposób symulowania czasu powrotu do stanu początkowego określonego krytycznego procesu narodzin i śmierci. Ten czas powrotu ma nieskończoną wartość oczekiwaną przy czym jego asymptotyczny rozkład jest potęgowy. Zatem dopóki symulowany czas nie przekroczy pewnej granicznej wartości proces jest symulowany bezpośrednio. W chwili przekroczenia tej wartości granicznej czas powrotu jest losowany z ogona tego rozkładu potęgowego.

2

Critical case stochastic phylogenetic tree model via the Laplace transform

Bartoszek K., Krzemiński M.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

474--481

EN

Birth–and–death models are now a common mathematical tool to describe branching patterns observed in real–world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth–and–death model that is compatible with phylogenetic trees of both influenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non–trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this, we correct the proof of Liggett and Schinazi (2009) in the critical case.

3

On a matching distance between rooted phylogenetic trees

Bogdanowicz D., Giaro K.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 3

669--684

EN

The Robinson–Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values of similarity between clusters are transformed to the final MC-score of the dissimilarity of trees. The analyzed properties give insight into the structure of the metric space generated by MC, its relations with the Matching Split (MS) distance of unrooted trees and asymptotic behavior of the expected distance between binary n-leaf trees selected uniformly in both MC and MS [...].

4

A Method for Nucleotide Sequence Analysis

Kozarzewski B.

Computational Methods in Science and Technology

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2012

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Vol. 18, No. 1

5-10

EN

Symbolic sequence decomposition into a set of consecutive, distinct subsequences (mers) is presented. Several statistical distributions of nucleotide subsequences are defined and analysed. Sequence entropy and similarity between sequences in terms of mer lengths distribution are defined. An alignment-free method of phylogenetic tree construction is proposed.

5

H-trees: a Model of Evolutionary Scenarios with Horizontal Gene Transfer

Górecki P.

Fundamenta Informaticae

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2010

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Vol. 103, nr 1-4

105-128

EN

In this paper, we present a model of evolution of genes in the context of evolution of species. The concept is based on reconciliation models. We assume that the gene evolution is modeled by macro-evolutionary events like gene duplications, losses and horizontal gene transfers (HGTs) while the evolution of species is shaped by speciation events. We define an evolutionary scenario (called an H-tree) which will represent the common evolution of genes and species. We propose a rewrite system for transforming the scenarios. We prove that the system is confluent, sound and strongly normalizing. We show that a scenario in a normal form (that is, non-reducible) is unique and minimal in the sense of the cost computed as the total number of gene duplications, losses and HGTs (mutation cost). We present a classification of the scenarios and analyze their hierarchies.