50 research outputs found

    A novel algebraic approach to time-reversible evolutionary models

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    In the last years algebraic tools have been proven to be useful in phylogenetic reconstruction and model selection by means of the study of phylogenetic invariants. However, up to now, the models studied from an algebraic viewpoint are either too general or too restrictive (as group-based models with a uniform stationary distribution) to be used in practice. In this paper we provide a new framework to work with time-reversible models, which are the most widely used by biologists. In our approach we consider algebraic time-reversible models on phylogenetic trees (as defined by Allman and Rhodes) and introduce a new inner product to make all transition matrices of the process diagonalizable through the same orthogonal eigenbasis. This framework generalizes the Fourier transform widely used to work with group-based models and recovers some of the well known results. As illustration, we exploit the combination of our technique with algebraic geometry tools to provide relevant phylogenetic invariants for trees evolving under the Tamura-Nei model of nucleotide substitution

    Computing Implicitizations of Multi-Graded Polynomial Maps

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    In this paper, we focus on computing the kernel of a map of polynomial rings φ\varphi. This core problem in symbolic computation is known as implicitization. While there are extremely effective Gr\"obner basis methods used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the map φ\varphi is multigraded, we consider an alternative approach. We demonstrate how to quickly compute a matrix of maximal rank for which φ\varphi has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gr\"obner techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is \emph{embarassingly parallel} and a fully parallelized version of our algorithm will be forthcoming in OSCAR.Comment: 16 pages, 2 figures. An implementation of our main algorithm can be found on our MathRepo page as well as our GitHu

    New Directions for Contact Integrators

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    Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    Algebraic Statistics in Practice: Applications to Networks

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    Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra and computational algebra), geometry and combinatorics to provide insight into knotty problems in mathematical statistics. In this survey we illustrate this on three problems related to networks, namely network models for relational data, causal structure discovery and phylogenetics. For each problem we give an overview of recent results in algebraic statistics with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

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    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

    Non-acyclicity of coset lattices and generation of finite groups

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