DOCSLIB.ORG

Hopf Algebras and Markov Chains

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hopf Algebras and Markov Chains HOPF ALGEBRAS AND MARKOV CHAINS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Chung Yin Amy Pang July 2014 © 2014 by Chung Yin Amy Pang. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial-Share Alike 3.0 United States License. http://creativecommons.org/licenses/by-nc-sa/3.0/us/ This dissertation is online at: http://purl.stanford.edu/vy459jk2393 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Persi Diaconis, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Daniel Bump I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Eric Marberg Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract This thesis introduces a way to build Markov chains out of Hopf algebras. The transition matrix of a Hopf-power Markov chain is (the transpose of) the matrix of the coproduct- then-product operator on a combinatorial Hopf algebra with respect to a suitable basis. These chains describe the breaking-then-recombining of the combinatorial objects in the Hopf algebra. The motivating example is the famous Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, which arise in this manner from the shuffle algebra. The primary reason for constructing Hopf-power Markov chains, or for rephrasing fa- miliar chains through this lens, is that much information about them comes simply from translating well-known facts on the underlying Hopf algebra. For example, there is an explicit formula for the stationary distribution (Theorem 3.4.1), and constructing quotient algebras show that certain statistics on a Hopf-power Markov chain are themselves Markov chains (Theorem 3.6.1). Perhaps the pinnacle is Theorem 4.4.1, a collection of algorithms for a full left and right eigenbasis in many common cases where the underlying Hopf alge- bra is commutative or cocommutative. This arises from a cocktail of the Poincare-Birkhoff- Witt theorem, the Cartier-Milnor-Moore theorem, Reutenauer’s structure theory of the free Lie algebra, and Patras’s Eulerian idempotent theory. Since Hopf-power Markov chains can exhibit very different behaviour depending on the structure of the underlying Hopf algebra and its distinguished basis, one must restrict at- tention to certain styles of Hopf algebras in order to obtain stronger results. This thesis will focus respectively on a free-commutative basis, which produces "independent breaking" chains, and a cofree basis; there will be both general statements and in-depth examples. Remark. An earlier version of the Hopf-power Markov chain framework, restricted to free- commutative or free state space bases, appeared in [DPR14]. Table 1 pairs up the results iv [DPR14] thesis construction 3.2 3.2,3.3 stationary distribution 3.7.1 3.4 reversibility 3.5 projection 3.6 diagonalisation general 3.5 4 algorithm for free-commutative basis Th. 3.15 Th. 4.4.1.A algorithm for basis of primitives Th. 4.4.1.B algorithm for shuffle basis Th. 4.4.1.A0 algorithm for free basis Th. 3.16 Th. 4.4.1.B0 unidirectionality for free-commutative basis 3.3 5.1.2 right eigenfunctions for free-commutative basis 3.6 5.1.3 link to terminality of QSym 3.7.2 5.1.4 examples rock-breaking 4 5.2 tree-pruning 5.3 riffle-shuffling1 5 6.1 descent sets under riffle-shuffling 6.2 Table 1: Corresponding sections of [DPR14] and the present thesis and examples of that paper and their improvements in this thesis. (I plan to update this table on my website, as the theory advances and more examples are available.) In addition, a summary of Section 6.2, on the descent-set Markov chain under riffle-shuffling, appeared in [Pan13]. v Acknowledgement First, I must thank my advisor Persi Diaconis for his patient guidance throughout my time at Stanford. You let me roam free on a landscape of algebra, probability and combinatorics in whichever direction I choose, yet are always ready with a host of ideas the moment I feel lost. Thanks in addition for putting me in touch with many algebraic combinatorialists. Thanks to my coauthor Arun Ram for your observations about card-shuffling and the shuffle algebra, from which grew the theory in this thesis. I also greatly appreciate your help in improving my mathematical writing while we worked on our paper together. I’m grateful to Marcelo Aguiar for pointing me to Patras’s work, which underlies a key part of this thesis, and for introducing me to many of the combinatorial Hopf algebras I describe here. To the algebraic combinatorics community: thanks for taking me in as a part of your family at FPSAC and other conferences. Special mentions go to Sami Assaf and Aaron Lauve for spreading the word about my work; it makes a big difference to know that you are as enthusiastic as I am about my little project. I’d like to thank my friends, both at Stanford and around the world, for mathematical insights, practical tips, and just cheerful banter. You’ve very much brightened my journey in the last five years. Thanks especially to Agnes and Janet for taking time from your busy schedules to hear me out during my rough times; because of you, I now have a more positive look on life. Finally, my deepest thanks go to my parents. You are always ready to share my ex- citement at my latest result, and my frustrations at another failed attempt, even though I’m expressing it in way too much technical language. Thank you for always being there for me and supporting my every decision. vi Contents Abstract iv Acknowledgement vi 1 Introduction 1 1.1 Markov chains . 1 1.2 Hopf algebras . 6 1.3 Hopf-power Markov chains . 7 2 Markov chains from linear operators 9 2.1 Construction . 10 2.2 Diagonalisation . 12 2.3 Stationarity and Reversibility . 13 2.4 Projection . 16 3 Construction and Basic Properties of Hopf-power Markov Chains 19 3.1 Combinatorial Hopf algebras . 21 3.1.1 Species-with-Restrictions . 24 3.1.2 Representation rings of Towers of Algebras . 26 3.1.3 Subalgebras of Power Series . 28 3.2 First Definition of a Hopf-power Markov Chain . 32 3.3 General Definition of a Hopf-power Markov Chain . 35 3.4 Stationary Distributions . 44 3.5 Reversibility . 47 vii 3.6 Projection . 49 4 Diagonalisation of the Hopf-power operator 54 4.1 The Eulerian Idempotent . 54 4.2 Eigenvectors of Higher Eigenvalue . 57 4.3 Lyndon Words . 58 4.4 Algorithms for a Full Eigenbasis . 60 5 Hopf-power Markov chains on Free-Commutative Bases 69 5.1 General Results . 71 5.1.1 Independence . 71 5.1.2 Unidirectionality . 73 5.1.3 Probability Estimates from Eigenfunctions . 76 5.1.4 Probability Estimates from Quasisymmetric Functions . 87 5.2 Rock-Breaking . 90 5.2.1 Constructing the Chain . 91 5.2.2 Right Eigenfunctions . 92 5.2.3 Left Eigenfunctions . 95 5.2.4 Transition Matrix and Eigenfunctions when n = 4 . 98 5.3 Tree-Pruning . 99 5.3.1 The Connes-Kreimer Hopf algebra . 100 5.3.2 Constructing the Chain . 103 5.3.3 Right Eigenfunctions . 108 6 Hopf-power Markov Chains on Cofree Commutative Algebras 117 6.1 Riffle-Shuffling . 117 6.1.1 Right Eigenfunctions . 118 6.1.2 Left Eigenfunctions . 125 6.1.3 Duality of Eigenfunctions . 127 6.2 Descent Sets under Riffle-Shuffling . 129 6.2.1 Notation regarding compositions and descents . 132 viii 6.2.2 Quasisymmetric Functions and Noncommutative Symmetric Func- tions . 133 6.2.3 The Hopf-power Markov chain on QSym . 136 6.2.4 Right Eigenfunctions Corresponding to Partitions . 138 6.2.5 A full Basis of Right Eigenfunctions . 142 6.2.6 Left Eigenfunctions Corresponding to Partitions . 144 6.2.7 A full Basis of Left Eigenfunctions . 147 6.2.8 Duality of Eigenfunctions . 149 6.2.9 Transition Matrix and Eigenfunctions when n = 4 . 152 Bibliography 155 ix List of Tables 1 Corresponding sections of [DPR14] and the present thesis . v x List of Figures 3.1 An example coproduct calculation in G¯, the Hopf algebra of graphs . 25 5.1 The “two triangles with one common vertex” graph . 79 5.2 The tree [•Q3] .................................101 5.3 Coproduct of [•Q3] ..............................103 6.1 The trees T(13245) and T(1122) . 120 6.2 The Lyndon hedgerow T(35142) . 121 xi Chapter 1 Introduction This chapter briefly summarises the basics of the two worlds that this thesis bridges, namely Markov chains and Hopf algebras, and introduces the motivating example of riffle-shuffling of a deck of cards. 1.1 Markov chains A friendly introduction to this topic is Part I of the textbook [LPW09]. A (discrete time) Markov chain is a simple model of the evolution of an object over time.
Load more

Recommended publications
  • On an Extension of Knuth's Rotation Correspondence to Reduced Planar Trees
    ON AN EXTENSION OF KNUTH’S ROTATION CORRESPONDENCE TO REDUCED PLANAR TREES KURUSCH EBRAHIMI-FARD AND DOMINIQUE MANCHON Abstract. We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth’s rotation correspondence between planar binary trees and planar rooted trees. The operadic coun- terpart of the new bijection is explained. Related to this, the space of planar reduced forests is endowed with a combinatorial Hopf algebra structure. The corresponding structure on the space of planar rooted hyperforests is also described. Contents 1. Introduction 1 2. Planar rooted trees and hypertrees 2 2.1. Planar trees 2 2.2. The Butcher product 3 2.3. Knuth’s correspondence between planar binary and planar rooted trees 3 2.4. Reduced planar rooted trees and planar rooted hypertrees 4 2.5. Adding decorations 6 2.6. Operadic structure 6 3. Hopf algebra structures on trees 7 3.1. Connected filtered bialgebras 7 3.2. The Butcher–Connes–Kreimer Hopf algebra of rooted forests 8 3.3. Two isomorphic Hopf algebras of rooted trees 8 3.4. A Hopf algebra structure on reduced planar forests 9 3.5. The associated pre-Lie structure 10 3.6. A Hopf algebra structure on planar rooted hyperforests 11 References 12 arXiv:1203.0425v1 [math.CO] 2 Mar 2012 1. Introduction Rooted trees have been extensively used in many branches of pure and applied mathematics. Es- pecially in the latter case they gained particular prominence due to the pioneering work on numerical integration methods by John Butcher in the 1960s [3, 14, 20].
    [Show full text]
  • New Combinatorial Computational Methods Arising from Pseudo-Singletons Gilbert Labelle
    New combinatorial computational methods arising from pseudo-singletons Gilbert Labelle To cite this version: Gilbert Labelle. New combinatorial computational methods arising from pseudo-singletons. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.247-258. hal-01185187 HAL Id: hal-01185187 https://hal.inria.fr/hal-01185187 Submitted on 19 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2008, Valparaiso-Vi~nadel Mar, Chile DMTCS proc. AJ, 2008, 247{258 New combinatorial computational methods arising from pseudo-singletons Gilbert Labelley Laboratoire de Combinatoire et d'Informatique Math´ematique,Universit´edu Qu´ebec `aMontr´eal,CP 8888, Succ. Centre-ville, Montr´eal(QC) Canada H3C3P8 Abstract. Since singletons are the connected sets, the species X of singletons can be considered as the combinatorial logarithm of the species E(X) of finite sets. In a previous work, we introduced the (rational) species Xb of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that E(X) = exp(Xb) in the context of rational species, where exp(T ) denotes the classical analytical power series for the exponential function in the variable T .
    [Show full text]
  • An Introduction to Combinatorial Species
    An Introduction to Combinatorial Species Ira M. Gessel Department of Mathematics Brandeis University Summer School on Algebraic Combinatorics Korea Institute for Advanced Study Seoul, Korea June 14, 2016 The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. What are combinatorial species? The theory of combinatorial species, introduced by André Joyal in 1980, is a method for counting labeled structures, such as graphs. The main reference for the theory of combinatorial species is the book Combinatorial Species and Tree-Like Structures by François Bergeron, Gilbert Labelle, and Pierre Leroux. If a structure has label set A and we have a bijection f : A B then we can replace each label a A with its image f (b) in!B. 2 1 c 1 c 7! 2 2 a a 7! 3 b 3 7! b More interestingly, it allows us to count unlabeled versions of labeled structures (unlabeled structures). If we have a bijection A A then we also get a bijection from the set of structures with! label set A to itself, so we have an action of the symmetric group on A acting on these structures. The orbits of these structures are the unlabeled structures. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions. What are species good for? The theory of species allows us to count labeled structures, using exponential generating functions.
    [Show full text]
  • Algebraic Structures of B-Series Philippe Chartier, Ernst Hairer, Gilles Vilmart
    Algebraic Structures of B-series Philippe Chartier, Ernst Hairer, Gilles Vilmart To cite this version: Philippe Chartier, Ernst Hairer, Gilles Vilmart. Algebraic Structures of B-series. Foundations of Computational Mathematics, Springer Verlag, 2010, 10 (4), pp.407-420. 10.1007/s10208-010-9065-1. inria-00598369 HAL Id: inria-00598369 https://hal.inria.fr/inria-00598369 Submitted on 6 Jun 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Found Comput Math (2010) 10: 407–427 DOI 10.1007/s10208-010-9065-1 Algebraic Structures of B-series Philippe Chartier · Ernst Hairer · Gilles Vilmart Received: 10 February 2009 / Revised: 10 November 2009 / Accepted: 16 February 2010 / Published online: 24 March 2010 © SFoCM 2010 Abstract B-series are a fundamental tool in practical and theoretical aspects of nu- merical integrators for ordinary differential equations. A composition law for B-series permits an elegant derivation of order conditions, and a substitution law gives much insight into modified differential equations of backward error analysis. These two laws give rise to algebraic structures (groups and Hopf algebras of trees) that have recently received much attention also in the non-numerical literature.
    [Show full text]
  • Lie-Butcher Series, Geometry, Algebra and Computation
    Lie–Butcher series, Geometry, Algebra and Computation Hans Z. Munthe-Kaas and Kristoffer K. Føllesdal Abstract Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie groups and homogeneous manifolds. On the algebraic side, B-series are based on pre-Lie algebras and the Butcher-Connes-Kreimer Hopf algebra. The LB-series are instead based on post-Lie algebras and their enveloping algebras. Over the last decade the algebraic theory of LB-series has matured. The purpose of this paper is twofold. First, we aim at presenting the algebraic structures underlying LB series in a concise and self contained manner. Secondly, we review a number of algebraic operations on LB-series found in the literature, and reformulate these as recursive formulae. This is part of an ongoing effort to create an extensive software library for computations in LB-series and B-series in the programming language Haskell. 1 Introduction Classical B-series are formal power series expressed in terms of rooted trees (con- nected graphs without any cycle and a designated node called the root). The theory has its origins back to the work of Arthur Cayley [5] in the 1850s, where he realized that trees could be used to encode information about differential operators. Being forgotten for a century, the theory was revived through the efforts of understanding numerical integration algorithms by John Butcher in the 1960s and ’70s [2, 3]. Ernst Hairer and Gerhard Wanner [15] coined the term B-series for an infinite formal se- arXiv:1701.03654v2 [math.NA] 27 Oct 2017 ries of the form H.
    [Show full text]
  • The Lie Group Structure of the Butcher Group
    The Lie group structure of the Butcher group Geir Bogfjellmo∗ and Alexander Schmeding† July 16, 2018 The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge–Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker–Campbell–Hausdorff Lie group modelled on a Fréchet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated to the Butcher group by Connes and Kreimer. Keywords: Butcher group, infinite-dimensional Lie group, Hopf algebra of rooted trees, regularity of Lie groups, symplectic methods MSC2010: 22E65 (primary); 65L06, 58A07, 16T05 (secondary) Contents 1 PreliminariesontheButchergroupandcalculus 5 2 AnaturalLiegroupstructurefortheButchergroup 13 3 The Lie algebra of the Butcher group 17 4 RegularitypropertiesoftheButchergroup 20 arXiv:1410.4761v3 [math.GR] 6 May 2015 5 TheButchergroupasanexponentialLiegroup 25 ∗NTNU Trondheim [email protected], Phone: +47 73591753, Fax: +47 73593524 (Corresponding author) †NTNU Trondheim [email protected] 1 6 The subgroup of symplectic tree maps 28 References 31 Introduction and statement of results In his seminal work [But72] J.C. Butcher introduced the Butcher group as a tool to study order conditions for a class of integration methods. Butcher’s idea was to build a group structure for mappings on rooted trees.
    [Show full text]
  • Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected]
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 1-1-2014 Combinatorial Species and Labelled Structures Brent Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: http://repository.upenn.edu/edissertations Part of the Computer Sciences Commons, and the Mathematics Commons Recommended Citation Yorgey, Brent, "Combinatorial Species and Labelled Structures" (2014). Publicly Accessible Penn Dissertations. 1512. http://repository.upenn.edu/edissertations/1512 This paper is posted at ScholarlyCommons. http://repository.upenn.edu/edissertations/1512 For more information, please contact [email protected]. Combinatorial Species and Labelled Structures Abstract The theory of combinatorial species was developed in the 1980s as part of the mathematical subfield of enumerative combinatorics, unifying and putting on a firmer theoretical basis a collection of techniques centered around generating functions. The theory of algebraic data types was developed, around the same time, in functional programming languages such as Hope and Miranda, and is still used today in languages such as Haskell, the ML family, and Scala. Despite their disparate origins, the two theories have striking similarities. In particular, both constitute algebraic frameworks in which to construct structures of interest. Though the similarity has not gone unnoticed, a link between combinatorial species and algebraic data types has never been systematically explored. This dissertation lays the theoretical groundwork for a precise—and, hopefully, useful—bridge bewteen the two theories. One of the key contributions is to port the theory of species from a classical, untyped set theory to a constructive type theory. This porting process is nontrivial, and involves fundamental issues related to equality and finiteness; the recently developed homotopy type theory is put to good use formalizing these issues in a satisfactory way.
    [Show full text]
  • A COMBINATORIAL THEORY of FORMAL SERIES Preface in His
    A COMBINATORIAL THEORY OF FORMAL SERIES ANDRÉ JOYAL Translation and commentary by BRENT A. YORGEY This is an unofficial translation of an article that appeared in an Elsevier publication. Elsevier has not endorsed this translation. See https://www.elsevier.com/about/our-business/policies/ open-access-licenses/elsevier-user-license. André Joyal. Une théorie combinatoire des séries formelles. Advances in mathematics, 42 (1):1–82, 1981 Preface In his classic 1981 paper Une théorie combinatoire des séries formelles (A com- binatorial theory of formal series), Joyal introduces the notion of (combinatorial) species, which has had a wide-ranging influence on combinatorics. During the course of researching my PhD thesis on the intersection of combinatorial species and pro- gramming languages, I read a lot of secondary literature on species, but not Joyal’s original paper—it is written in French, which I do not read. I didn’t think this would be a great loss, since I supposed the material in his paper would be well- covered elsewhere, for example in the textbook by Bergeron et al. [1998] (which thankfully has been translated into English). However, I eventually came across some questions to which I could not find answers, only to be told that the answers were already in Joyal’s paper [Trimble, 2014]. Somewhat reluctantly, I found a copy and began trying to read it, whereupon I discovered two surprising things. First, armed with a dictionary and Google Translate, reading mathematical French is not too difficult (even for someone who does not know any French!)— though it certainly helps if one already understands the mathematics.
    [Show full text]
  • Analytic Combinatorics for Computing Seeding Probabilities
    algorithms Article Analytic Combinatorics for Computing Seeding Probabilities Guillaume J. Filion 1,2 ID 1 Center for Genomic Regulation (CRG), The Barcelona Institute of Science and Technology, Dr. Aiguader 88, Barcelona 08003, Spain; guillaume.fi[email protected] or guillaume.fi[email protected]; Tel.: +34-933-160-142 2 University Pompeu Fabra, Dr. Aiguader 80, Barcelona 08003, Spain Received: 12 November 2017; Accepted: 8 January 2018; Published: 10 January 2018 Abstract: Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here, we present an approach to estimate such quantities based on the theory of analytic combinatorics. The strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. The generating function can also be used to set up a simple recurrence to compute the probabilities with greater precision. We use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors, and we show that the estimates are accurate in practical situations. More generally, this work shows novel strategies based on analytic combinatorics to compute probabilities of interest in bioinformatics.
    [Show full text]
  • Norges Teknisk-Naturvitenskapelige Universitet
    NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Algebraic Structures on Ordered Rooted Trees and Their Significance to Lie Group Integrators by H˚avard Berland and Brynjulf Owren PREPRINT NUMERICS NO. 3/2003 NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY TRONDHEIM, NORWAY This report has URL http://www.math.ntnu.no/preprint/numerics/2003/N3-2003.ps Address: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Abstract Most Lie group integrators can be expanded in series indexed by the set of ordered rooted trees. To each tree one can associate two distinct higher order derivation operators, which we call frozen and unfrozen operators. Composition of frozen operators induces a concatenation product on the trees, whereas composition of unfrozen operators in- duces a somewhat more complicated product known as the Grossman{ Larson product. Both of these algebra structures can be supplemented by the same coalgebra structure and an antipode, the result being two distinct cocommutative graded Hopf algebras. We discuss the use of these structures and characterize subsets of the Hopf algebras corre- sponding to vector fields and mappings on manifolds. This is further relevant for deriving order conditions for a general class of Lie group integrators and for deriving the modified vector field in backward error analysis for these integrators. 1 Introduction The derivation of high order Runge{Kutta methods was revolutionized by Butcher's discovery of the beautiful connection between their series expan- sion in terms of the stepsize and the set T of rooted trees [2]. Virtually overnight, long and tedious calculations were replaced by elegant recursion formulas expressed in terms of trees.
    [Show full text]
  • On the Hopf Algebraic Structure of Lie Group Integrators
    Foundations of Computational Mathematics manuscript No. (will be inserted by the editor) H. Z. Munthe-Kaas · W. M. Wright On the Hopf Algebraic Structure of Lie Group Integrators Received: date / Accepted: date Abstract A commutative but not cocommutative graded Hopf algebra HN , based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees HC , developed by Butcher in his study of Runge–Kutta methods and later rediscovered by Connes and Moscovici in the context of non-commutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that HN is naturally obtained from a universal object in a category of non- commutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on non-commutative Lie group actions on a manifold. Recursive and non-recursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of HN is a Hopf al- gebra of Grossman and Larson. HN contains two well-known Hopf algebras as special cases: The Hopf algebra HC of Butcher–Connes–Kreimer is identi- fied as a proper subalgebra of HN using the image of a tree symmetrization operator. The Hopf algebra HF of the Free Associative Algebra is obtained from HN by a quotient construction. Keywords Hopf algebra · ordered rooted trees · Lie group integrators · Lie–Butcher series · Butcher group · Connes–Kreimer Hopf algebra · Grossman–Larson Hopf algebra arXiv:math/0603023v1 [math.AC] 1 Mar 2006 Mathematics Subject Classification (2000) 16W25 · 16W30 · 22E60 · 37M99 · 65L05 · 65L06 · 81R60 · 81T15 H.
    [Show full text]
  • Species and Functors and Types, Oh My!
    University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 9-2010 Species and Functors and Types, Oh My! Brent A. Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Discrete Mathematics and Combinatorics Commons, and the Programming Languages and Compilers Commons Recommended Citation Brent A. Yorgey, "Species and Functors and Types, Oh My!", . September 2010. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/761 For more information, please contact [email protected]. Species and Functors and Types, Oh My! Abstract The theory of combinatorial species, although invented as a purely mathematical formalism to unify much of combinatorics, can also serve as a powerful and expressive language for talking about data types. With potential applications to automatic test generation, generic programming, and language design, the theory deserves to be much better known in the functional programming community. This paper aims to teach the basic theory of combinatorial species using motivation and examples from the world of functional pro- gramming. It also introduces the species library, available on Hack- age, which is used to illustrate the concepts introduced and can serve as a platform for continued study and research. Keywords combinatorial species, algebraic data types Disciplines Discrete Mathematics and Combinatorics | Programming Languages and Compilers This conference paper is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/761 Species and Functors and Types, Oh My! Brent A. Yorgey University of Pennsylvania [email protected] Abstract a certain size, or randomly generate family structures.
    [Show full text]