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 EGSA  Differential Equations, Geometry, Control and Dynamical Systems, and Applications
 Department
 Department of Applied Mathematics I
 School
 Barcelona School of Industrial Engineering (ETSEIB)
 jesus.fernandez.sanchezupc.edu
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Testing of the three multiplicatively closed (Lie Markov) model heirarchies which respect purine/pyrimidine, WatsonCrick, and amino/keto nucleotide groupings
Woodhams, Michael D.; Fernández Sánchez, Jesús; Sumner, Jeremy
Date: 201403
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Read the abstract View Share Reference managersWe present three hierarchies of Lie Markov models of DNA sequence evolution. These models are (locally) ¿multiplicatively closed,¿ meaning that the composition of two Markov matrices in the model results, with some (rare) exceptions, in a third Markov matrix that is still in the model. Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines (RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat the four nucleotides without distinction. The multiplicative closure property allows mathematically consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such as the general timereversible model (GTR) and many of its submodels. We derive the nesting relationships of the three model hierarchies and present software implementing the models. For a diverse range of biological data sets, we perform Bayesian information criterion model comparision analogous to that of the ModelTest framework. We find that our models outperform the GTR model in some (but not all) cases.
We present three hierarchies of Lie Markov models of DNA sequence evolution. These models are (locally) “multiplicatively closed,” meaning that the composition of two Markov matrices in the model results, with some (rare) exceptions, in a third Markov matrix that is still in the model. Additionally, the models in each hierarchy respectively distinguish between (i) purines and pyrimadines (RY), (ii) WatsonCrick pairs (WS), and (iii) amino/keto pairs (MK), but otherwise treat the four nucleotides without distinction. The multiplicative closure property allows mathematically consistent modeling of timeinhomogeneous scenarios, unlike models that are not closed, such as the general timereversible model (GTR) and many of its submodels. We derive the nesting relationships of the three model hierarchies and present software implementing the models. For a diverse range of biological data sets, we perform Bayesian information criterion model comparision analogous to that of the ModelTest framework. We find that our models outperform the GTR model in some (but not all) cases. 
Lie Markov models with purine/pyrimidine symmetry
Fernández Sánchez, Jesús; Sumner, Jeremy; Jarvis, Peter; Woodhams, Michael D.
Date: 201306
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Read the abstract Access to the full text Share Reference managersContinuoustime Markov chains are a standard tool in phylogenetic inference. If homo geneity is assumed, the chain is formulated by specifying timeindependent rates of substitu tions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to gen eralise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the denition of \Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines { that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally dened over the complex eld, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra.
Continuoustime Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying timeindependent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with timedependent rates satisfying the same constraints. It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of "Lie Markov models", which are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines  that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra. 
Local description of phylogenetic groupbased models
Casanellas Rius, Marta; Fernández Sánchez, Jesús; Michalek, Mateusz
Date: 201312
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Read the abstract Access to the full text Share Reference managersMotivated by phylogenetics, our aim is to obtain a system of equations that dene a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties dened via groupbased models. For any nite abelian group G, we provide an explicit construction of codimX phylogenetic invariants (polynomial equations) of degree at most jGj that dene the variety X on a Zariski open set U. The set U contains all biologically meaningful points when G is the group of the Kimura 3parameter model. In particular, our main result conrms [Mic12, Conjecture 7.9] and, on the set U, Conjectures 29 and 30 of [SS05].
Motivated by phylogenetics, our aim is to obtain a system of equations that de ne a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties de ned via groupbased models. For any nite abelian group G, we provide an explicit construction of codimX phylogenetic invariants (polynomial equations) of degree at most jGj that de ne the variety X on a Zariski open set U. The set U contains all biologically meaningful points when G is the group of the Kimura 3parameter model. In particular, our main result con rms [Mic12, Conjecture 7.9] and, on the set U, Conjectures 29 and 30 of [SS05]. 
GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES
Alberich Carramiñana, Maria; Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Ventura González Alonso, Daniel; Barbieri Solha, Romero; Miranda Galceran, Eva; Pascual Gainza, Pedro; Roig Marti, Agustin; Roig Maranges, Abdo; Feliu Trijueque, Elsienda; Gálvez Carrillo, Maria Immaculada; Casanellas Rius, Marta
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Modelos de Lie Markov con simetría fijada
Fernández Sánchez, Jesús
Congreso de Jóvenes Investigadores Real Sociedad Matemática Española
Presentation's date: 20130916
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Lie Markov models with prescribed symmetry
Fernández Sánchez, Jesús
Applied Algebraic Geometry
Presentation's date: 20130803
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Is the general timereversible model bad for molecular phylogenetics?
Sumner, Jeremy; Jarvis, Peter; Fernández Sánchez, Jesús; Kaine, Bodie; Woodhams, Michael; Holland, Barbara
Systematic biology
Vol. 61, num. 6, p. 10691074
DOI: 10.1093/sysbio/sys042
Date of publication: 20120322
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Lie Markov models
Sumner, Jeremy; Fernández Sánchez, Jesús; Jarvis, Peter
Journal of theoretical biology
Vol. 298, num. 7, p. 1631
DOI: 10.1016/j.jtbi.2011.12.017
Date of publication: 20120407
Journal article
Read the abstract View Share Reference managersRecent work has discussed the importance of multiplicative closure for the Markov models used in phylogenetics. For continuoustime Markov chains, a sufficient condition for multiplicative closure of a model class is ensured by demanding that the set of ratematrices belonging to the model class form a Lie algebra. It is the case that some wellknown Markov models do form Lie algebras and we refer to such models as “Lie Markov models”. However it is also the case that some other wellknown Markov models unequivocally do not form Lie algebras (GTR being the most conspicuous example). In this paper, we will discuss how to generate Lie Markov models by demanding that the models have certain symmetries under nucleotide permutations. We show that the Lie Markov models include, and hence provide a unifying concept for, “groupbased” and “equivariant” models. For each of two and four character states, the full list of Lie Markov models with maximal symmetry is presented and shown to include interesting examples that are neither groupbased nor equivariant. We also argue that our scheme is pleasing in the context of applied phylogenetics, as, for a given symmetry of nucleotide substitution, it provides a natural hierarchy of models with increasing number of parameters. We also note that our methods are applicable to any application of continuoustime Markov chains beyond the initial motivations we take from phylogenetics. 
The space of phylogenetic mixtures for equivariant models
Casanellas Rius, Marta; Fernández Sánchez, Jesús; Kedzierska, A.M.
Algorithms for Molecular Biology
Vol. 7, num. 1, p. 33
DOI: 10.1186/17487188733
Date of publication: 20121128
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Read the abstract Access to the full text Share Reference managersThe selection of the most suitable evolutionary model to analyze the given molecular data is usually left to biologist's choice. In his famous book, J Felsenstein suggested that certain linear equations satisfied by the expected probabilities of patterns observed at the leaves of a phylogenetic tree could be used for model selection. It remained open the question regarding whether these equations were enough for characterizing the evolutionary model. Here we prove that, for equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees on a set of taxa. In other words, we prove that an alignment is produced from a mixture of phylogenetic trees under an equivariant evolutionary model if and only if its distribution of column patterns satisfies the linear equations mentioned above. Moreover, for each equivariant model and for any number of taxa, we provide a set of linearly independent equations defining this space of phylogenetic mixtures. This is a powerful tool that has already been successfully used in model selection. We also use the results obtained to study identifiability issues for phylogenetic mixtures. 
Lie Markov models
Fernández Sánchez, Jesús
Algebraic Statistics
Presentation's date: 20120612
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Phylogenetic mixtures for equivariant models
Fernández Sánchez, Jesús
Algebraic Statistics
Presentation's date: 20120612
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An approach to phylogenetic inference by means of group representation theory
Fernández Sánchez, Jesús
Congreso de Jóvenes Investigadores Real Sociedad Matemática Española
Presentation's date: 20110906
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Relevant phylogenetic invariants of evolutionary models
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Date: 201001
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Read the abstract Access to the full text Share Reference managersRecently there have been several attempts to provide a whole set of generators of the ideal of the algebraic variety associated to a phylogenetic tree evolving under an algebraic model. These algebraic varieties have been proven to be useful in phylogenetics. In this paper we prove that, for phylogenetic reconstruction purposes, it is enough to consider generators coming from the edges of the tree, the socalled edge invariants. This is the algebraic analogous to Buneman's Splits Equivalence Theorem. The interest of this result relies on its potential applications in phylogenetics for the widely used evolutionary models such as JukesCantor, Kimura 2 and 3 parameters, and General Markov models. 
Principality of curves on sandwiched singularities
Fernández Sánchez, Jesús
Mathematical research letters
Vol. 17, p. 1126
Date of publication: 2010
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Phylogenetic reconstruction using algebraic geometry
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Arbor (Madrid)
Vol. 186, num. 746, p. 10231033
DOI: 10.3989/arbor.2010.746n1251
Date of publication: 20101231
Journal article
Read the abstract View Share Reference managersA new approach to phylogenetic reconstruction has been emerging in the last years. Given an evolutionary model, the joint probability distribution of the nucleotides for these species satisfy some algebraic constraints called invariants. These invariants have theoretical and practical interest, since they can be used to infer phylogenies. In this paper, we explain how to use these invariants to design algorithms for phylogenetic reconstruction and we show how the application of tools and theoretical results coming from commutative algebra and algebraic geometry can improve the performance and the efficiency of these algorithms. 
Invariants, una aproximació a la filogenètica des de l'àlgebra
Fernández Sánchez, Jesús
Butlletí de la Societat Catalana de Matemàtiques
Vol. 25, num. 2, p. 145169
DOI: 10.2436/20.2002.01.30
Date of publication: 2010
Journal article
Read the abstract Access to the full text Share Reference managersEn els darrers anys, una nova aproximació a la reconstrucció filogenètica basada en tècniques provinents de l’àlgebra s’ha estat consolidant. Fixat un model evolutiu per a un conjunt donat d’espècies, els invariants són relacions algebraiques satisfetes per les distribucions teòriques de nucleòtids d’aquestes espècies. En aquest article s’exposa com es poden fer servir els invariants per implementar algoritmes de reconstrucció filogenètica i s’explica com l’eficiència d’aquests algoritmes es veu beneficiada per resultats teòrics provinents de la geometria algebraica i la representació de grups. 
On adjacent complete ideals above a given complete ideal
Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
Proceedings of the Royal Society of Edinburgh. Section A, mathematics
Vol. 140, p. 225239
DOI: 10.1017/S030821050800067X
Date of publication: 2010
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Relevant phylogenetic invariants of evolutionary models
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Journal de mathématiques pures et appliquées
Vol. 96, num. 3, p. 207229
DOI: 10.1016/j.matpur.2010.11.002
Date of publication: 20101119
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GEOMETRÍA DE VARIEDADES ALGEBRAICAS Y APLICACIONES
Barja Yañez, Miguel Angel; Fernández Sánchez, Jesús; Amoros Torrent, Jaume; Alberich Carramiñana, Maria; Elgueta Montó, Josep; Gonzalez Alonso, Victor; Berna Sepulveda, Isabel Silvana; Ferragut Amengual, Antoni Manel; Casanellas Rius, Marta
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On the exceptional locus of the birational projections of a normal surface singularity into a plane
Fernández Sánchez, Jesús
Journal of algebra
Vol. 312, p. 24612473
Date of publication: 200901
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Classication of the birational projections of a sandwiched surface singularity to a plane
Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
Date: 200809
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Clases de equisingularidad de proyecciones birracionales de singularidades normales de superficie en el plano
Fernández Sánchez, Jesús
Date of publication: 200801
Book chapter
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Geometry of the Kimura 3parameter model
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Advances in applied mathematics
Vol. 41, num. 3, p. 265292
Date of publication: 200803
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Performance of a new invariants method for phylogenetic reconstruction
Fernández Sánchez, Jesús
Society for Molecular Biology and Evolution. Meeting
Presentation's date: 20080101
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Classification of the birational projections of a sandwiched surface singularity to a plane
Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
Seminar on Singularities: algebraic methods
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Factorization of adjacent complete ideals in dimension 2
Fernández Sánchez, Jesús
Proceedings of the Royal Society of Edinburgh. Section A, mathematics
Vol. 137A, num. 5, p. 913921
Date of publication: 200710
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Factorization of complete ideals in normal birational extensions in dimension two
Fernández Sánchez, Jesús
Journal of algebra
Vol. 314, num. 1, p. 344361
Date of publication: 200703
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Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Molecular biology and evolution
Vol. 24, num. 1, p. 288293
Date of publication: 200701
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Equisingularity classes of birational projections of normal singularities to a plane
Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
Advances in mathematics
Vol. 216, num. 2, p. 753770
Date of publication: 200712
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Geometry of the Kimura 3parameter model
Casanellas Rius, Marta; Fernández Sánchez, Jesús
Future Directions in Phylogenetic Methods and Models
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Equisingularity classes of birational projections of normal singularities to a plane
Fernández Sánchez, Jesús; Alberich Carramiñana, Maria
I Congreso HispanoFrancés de Matemàticas
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Equisingularity classes of birational projections of normal singularities to a plane
Fernández Sánchez, Jesús
Approaching Mathematics through Algebra
p. 1
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Performance of a new invariants method for phylogenetic reconstruction
Fernández Sánchez, Jesús
Future Directions in Phylogenetic Methods and Models
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Equisingularity classes of birational projections of normal singularities to a plane
Fernández Sánchez, Jesús
I Congreso HispanoFrancés de Matemàticas
Presentation's date: 20070709
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Clases de equisingularidad de proyecciones birracionales de singularidades normales de superficie
Fernández Sánchez, Jesús
Approaching Mathematics through Algebra
p. 145159
Presentation's date: 20070625
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Singularidades de morfismos y variedades II
Casas ., E.; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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Ingenio Mathematica (iMATH)
Alberich Carramiñana, Maria; Zuazua ., Enrique; Fernández Sánchez, Jesús
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Geometría y Topología de Variedades Algebraicas y Simplécticas
Barja Yañez, Miguel Angel; Casanellas Rius, Marta; Fernández Sánchez, Jesús
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Some results relating to adjacent ideals in dimension two
Fernández Sánchez, Jesús
Journal of pure and applied algebra
Vol. 207, num. 2, p. 387395
Date of publication: 200608
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RTACA  Red Temática de Álgebra Conmutativa y Aplicaciones
Casanellas Rius, Marta; Elias ., Joan; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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Equivalence of the Nash conjecture for primitive and sandwiched singularities
Fernández Sánchez, Jesús
Proceedings of the American Mathematical Society
Vol. 133, num. 3, p. 677679
Date of publication: 200503
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Nash families of smooth arcs on a sandwiched singularity
Fernández Sánchez, Jesús
Mathematical proceedings of the Cambridge Philosophical Society
Vol. 138, num. 1, p. 117128
Date of publication: 200501
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Factoritación de ideales completos y singularidades sandwiched
Fernández Sánchez, Jesús
Congreso Conjunto de Matemáticas RSMESCMSEIOSEMA
Presentation's date: 20050202
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Geometría y topología de las variedades algebraicas
Roig Marti, Agustin; Barja Yañez, Miguel Angel; Pascual Gainza, Pedro; Elgueta Montó, Josep; Abío Roig, Ignasi; Amoros Torrent, Jaume; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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Singularidades de morfismos y variedades
Alberich Carramiñana, Maria; Casas ., E.; Fernández Sánchez, Jesús
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On sandwiched singularities and complete ideals
Fernández Sánchez, Jesús
Journal of pure and applied algebra
Vol. 185, num. 13, p. 165175
Date of publication: 200312
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Geometria Algebraica.
Casas ., E.; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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Geometria Algebraica.
Casanellas Rius, Marta; Casas ., E.; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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EAGEREuropean Algebraic Geometry Research training network
Conte ., Alberto; Amoros Torrent, Jaume; Casanellas Rius, Marta; Alberich Carramiñana, Maria; Fernández Sánchez, Jesús
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On polar germs of planet curves and related in variants
Fernández Sánchez, Jesús
Géométrie algébrique en Liberté VIII
Presentation's date: 20000328
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