DOCSLIB.ORG

Butcher Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Butcher Series Asia Pacific Mathematics Newsletter 1 ButcherButcher Series Series AA Story Story of RootedRooted Trees Trees and and Numerical Numerical Methods Methods for forEvolution Evolution Equations Equations Robert I McLachlan, Klas Modin, Hans Munthe-Kaas and Olivier Verdier Robert I McLachlan, Klas Modin, Hans Munthe-Kaas and Oliver Verdier Abstract. Butcher series appear when Runge–Kutta fields on vector spaces. Indeed, let f be a smooth methods for ordinary differential equations are ex- vector field on a vector space V, defining the panded in power series of the step size parameter. Each ordinary differential equation (ODE) term in a Butcher series consists of a weighted elemen- tary differential, and the set of all such differentials x˙ = f (x), (1.1) is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher where x˙ = dx denotes the derivative with respect discovered that rooted trees can also be used to obtain dt the order conditions of Runge–Kutta methods, and he to time t. One way to study (1.1) is to develop found a natural group structure, today known as the the Taylor series of its solutions. Let x(h) be the Butcher group. It is now known that many numerical solution to (1.1) at time t = h subject to the initial methods can also be expanded in Butcher series; these condition x(0) = x . The Taylor series of x(h) in h is are called B-series methods. A long-standing problem 0 has been to characterize, in terms of qualitative fea- 1 x(h) = x(0) + hx˙(0) + h2x¨(0) + . (1.2) tures, all B-series methods. Here we tell the story of 2 ··· Butcher series, stretching from the early work of Cayley, We already know that x(0) = x and x˙(0) = f (x ). to modern developments and connections to abstract 0 0 algebra, and finally to the resolution of the character- The additional terms can be found by repeat- isation problem. This resolution introduces geometric edly applying the chain and product rules. For tools and perspectives to an area traditionally explored example, using analysis and combinatorics. d d x¨ = x˙ = f (x) = f (x)x˙ = f (x) f (x), dt dt 1. From Cayley to Butcher or, relative to a basis in which x = x1e + + xne , 1 ··· n n ∂f i Butcher series are mathematical objects that were x¨i = (x) f j(x), introduced by the New Zealand mathematician ∂xj j=1 John Butcher in the 1960s. He introduced them 1 n where f (x) = f (x)e1 + + f (x)e . Continuing in as part of his study of Runge–Kutta methods, a ··· n this way gives popular class of numerical methods for evolu- tion equations such as initial-value problems for x˙ = f (x), ordinary differential equations, and they remain x¨ = f (x) f (x), ... indispensable in the numerical analysis of differ- x = f (x) f (x) f (x) + f (x)( f (x), f (x)), ential equations. In this article we provide a brief .... x = f (x) f (x) f (x) f (x) + f (x) f (x)( f (x), f (x)) introduction to Butcher series, survey their early + 3f (x)( f (x) f (x), f (x)) + f (x)( f (x), f (x), f (x)), history up to their introduction by John Butcher, . and relate the story of the many connections that . (1.3) have recently been discovered between Butcher Here the kth derivative f (k)(x) of the vector field series and other parts of mathematics, notably f is regarded as a multilinear map Vk V. For algebra and geometry.1 We begin, however, with → example, f ( f , f ) is the vector field on V whose the traditional definition. ith coordinate is Butcher series are intimately associated with n ∂2f i the set of smooth (infinitely differentiable) vector (x) f j(x) f k(x). ∂xj∂xk j,k=1 1This article is not a comprehensive review and is focussed on our own interests. Useful companions to this article are A vector field of the form appearing in (1.3), com- the detailed mathematical review of Butcher series by Sanz- bining f and its derivatives, is called an elementary Serna and Murua [35] and the textbook treatments of Hairer et al. [21, 23]. differential. Using (1.3), the Taylor series (1.2) for December 2017, Volume 7 No 1 1 Asia Pacific Mathematics Newsletter 2 the solution of (1.1) can be written as in Hairer, Lubich and Wanner [21], and a detailed history in Butcher and Wanner [9]. 1 2 1 3 1 3 x(h) = x + hf + h f f + h f f f + h f ( f , f ) + 0 2 6 6 ··· The first breakthrough paper dates from 1963 (1.4) [5]. Here Butcher found for the first time the where each elementary differential is evaluated coefficients ci of the B-series (1.5) of xn+1 of the at x . Notice that the power of h in each term is 0 Taylor expansion in h of an arbitrary Runge– determined by the multiplicity of f in the elemen- Kutta method. This gave the order conditions tary differential. However, the coefficients 1, 1, for Runge–Kutta methods in complete generality. 1/2, 1/6, 1/6, and so on are not determined by their As previous studies had laboriously expanded corresponding elementary differentials. A Butcher the solutions of particular (e.g. explicit) meth- series, shortly denoted B-series, is a generalisation ods by hand, this was an enormously important of (1.4) allowing arbitrary coefficients, i.e. a formal development. series of the form Butcher did have, however, some precursors. 2 3 B(c, f ):= c0x0 + c1hf + c2h f ( f ) + c3h f ( f ( f )) The most notable example is the paper of Mer- son [32] from 1957. Robert Henry ‘Robin’ Merson 3 + c h f ( f , f ) + (1.5) 4 ··· (1921–1992) was a scientist at the Royal Aircraft where c R. Although presented here in coor- Establishment, Farnborough, UK, who was in- i ∈ dinates, we shall see that Butcher series do not vited along with more senior numerical analysts depend on the choice of basis. to a conference on Data Processing and Automatic Computing Machines at Australia’s Weapons Re- search Establishment in Salisbury, South Aus- 2. Early History tralia.2 It seems like a long way to go for a Butcher series are named in honour of the New conference in 1957. However, the UK was still per- Zealand mathematician John Butcher. In a pub- forming above-ground atomic bomb tests in South lication career spanning (so far) 60 years he has Australia at that time and the Australian govern- written 167 papers and books, all but 18 of them ment was very keen to be a part of the emerging concerned with Runge–Kutta methods and their era. Merson’s work is bound up with one of generalisations. Most of them involve in some the most significant events of 1957, the launch way the fundamental structure that bears his of Sputnik 1 on 4 October 1957, and the tale of name. Butcher series were introduced in a remark- Farnborough’s involvement is told in detail by able series of ten sole-authored papers in the years one of the key participants, Desmond King-Hele, 1963–1972. in his book A Tapestry of Orbits [28]. The short version is that with the aid of a large radio an- A Runge–Kutta method is a numerical ap- tenna hastily erected in a nearby field, and some proximation xn xn+1 of the exact flow of (1.1) → calculations of Robin Merson, within two weeks defined by the following equations in xn, xn+1, they had an accurate orbit for Sputnik 1. This X1, ..., Xν V: ∈ allowed them to estimate the density of the upper ν atmosphere and (after Sputnik 2) the shape of the Xi = xn + h aij f (Xj), = earth. Robin Merson became an expert in prac- j 1 (2.1) ν tical numerical analysis and orbit determination. xn+1 = xn + h bj f (Xj). Merson’s paper explains clearly the struc- j=1 ture of the elementary differentials f ( f ), f ( f , f ), Here ν is the number of stages of the method etcetera, and, crucially, shows how they are in one-to-one correspondence with rooted trees. He and aij, bj are real numbers parameterising the Runge–Kutta method. Associated with the ab- stract Runge–Kutta method (2.1) are its order con- 2Flight-related research at Farnborough began with the Army Balloon Factory in 1904, which became the Royal Aircraft ditions, polynomials equations in aij and bj — one Factory in 1912, the Royal Aircraft Establishment in 1918, equation per elementary differential — that deter- and then the Royal Aerospace Establishment in 1988. It was merged into the Defence Research Agency in 1991 and then mine the order of convergence of the method and into the Defence Evaluation and Research Agency in 1995. its local error. Their derivation has been simplified This was split up in 2001, with Farnborough becoming part of the private company Qinetiq. Desmond King-Hele’s version over the years; a modern exposition can be found of these later developments is recorded at [29]. 2 December 2017, Volume 7 No 1 4 McLachlan, Modin, Munthe-Kaas, and Verdier 4 McLachlan, Modin, Munthe-Kaas, and Verdier The first breakthrough paper dates from 1963 [5]. Here Butcher found for the The first breakthrough paper dates from 1963 [5]. Here Butcher found for the first time the coefficients ci of the B-series (1.5) of xn+1 of the Taylor expansion in first time the coefficients ci of the B-series (1.5) of xn+1 of the Taylor expansion in 4 McLachlan, Modin, Munthe-Kaas,h of an and arbitrary Verdier Runge–Kutta method.
Load more

Recommended publications
  • Vita Walter Gautschi
    VITA WALTER GAUTSCHI April 1, 2021 EDUCATION Ph.D. University of Basel, Switzerland 1953 (Thesis advisor: A. M. Ostrowski) PROFESSIONAL EXPERIENCE Research Fellow Istituto Nazionale per le Applicazioni del Calcolo, Rome 1954–55 Research Fellow Harvard Computation Lab. 1955–56 Research Mathematician Natl. Bureau of Standards 1956–59 Professor. Lecturer American U., Washington, D.C. Mathematician Oak Ridge National Lab. 1959–63 Professor of Math. & Computer Science Purdue University 1963–2000 Professor Emeritus Purdue University 2000– Visiting Professor Technical Univ. of Munich, Germany 1970–71 Visiting Professor Mathematics Res. Center, Univ. of WI 1976–77 Visiting Professor ETH Zurich 1996–2001 Visiting Professor University of Padova 1997 Visiting Professor University of Basel 2000 1 PROFESSIONAL SOCIETIES AND HONORS Schweizerische Mathematische Gesellschaft American Mathematical Society Mathematical Association of America Society for Industrial and Applied Mathematics Corresponding Member, Bavarian Academy of Sciences, Munich, 2001– Foreign Member, Academy of Sciences, Turin, 2001– SIAM Fellow, Class 2012 Member, Council of the American Mathematical Society, 1975–80, 1984–95 Fulbright Research Scholar, Munich, 1970–71 Listed in Who is Who in the World Listed in the International Biographical Centre’s Top 100 Educators 2009 and in 2000 Outstanding Intellectuals of the 21st Century Listed in S. Gottwald, H.-J. Ilgauds, and K.-H. Schlote, Lexikon bedeutender Mathematiker, 2d ed., Verlag Programm Mathematik, Leipzig, in preparation. RESEARCH INTERESTS Numerical Analysis Constructive Approximation Theory Special Functions Orthogonal Polynomials GENERAL INFORMATION Birthdate: December 11, 1927 Birthplace: Basel, Switzerland Marital Status: Married – Erika Children: 4 Citizenship: USA PUBLICATIONS Books B1. (with H. Bavinck and G. M. Willems) Colloquium approximatietheorie, MC Syllabus 14, Mathematisch Centrum Amsterdam, 1971.
    [Show full text]
  • 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]
  • 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]
  • HA-LU 2019 International Conference in Honor of Ernst Hairer and Christian Lubich
    HA-LU 2019 International conference in honor of Ernst Hairer and Christian Lubich Gran Sasso Science Institute, L'Aquila 17{21 June 2019 ii Program at a glance June 17 June 18 June 19 June 20 June 21 9.15 Opening Sanz-Serna Overton Hochbruck Zennaro 10.00 Wanner Gander Yserentant Chartier Vilmart 10.45 Coffee break Coffee break Coffee break Coffee break Coffee break 11.15 Ascher Quarteroni Vandereycken Jahnke Cohen 12.00 Deuflhard Banjai Palencia Li Lasser 12.45 Lunch Lunch Lunch Lunch Lunch 14.30 Ostermann Akrivis Group photo Hairer 15.15 Gonzalez-Pinto Calvo 16.00 Coffee break Coffee break 16.30 Photo exhibition iii iv List of abstracts Monday, 17 June 2019 1 Zigzags with B¨urgi,Bernoulli, Euler and the Seidel-Entringer-Arnol'd tri- angle (Gerhard Wanner)......................... 1 Different faces of stiffness (Uri Ascher).................... 1 Convergence results for collocation methods different from the Bible (Peter Deuflhard)................................. 2 Low-rank splitting integrators for stiff differential equations (Alexander Ostermann)................................ 2 On the convergence in `p norms of a MoL approach based on AMF-W- methods for m-dimensional linear parabolic problems of diffusion- reaction type (Severiano Gonzalez-Pinto)................ 3 Tuesday, 18 June 2019 5 Numerical integrators for the Hamiltonian Monte Carlo method (Jesus Maria Sanz Serna)............................ 5 The Method of Reflections (Martin Gander)................. 5 Modeling the heart function (Alfio Quarteroni) ............... 6 Fast and oblivious quadrature for the Schr¨odingerequation (Lehel Banjai) 6 Energy-decaying Runge-Kutta methods for phase field equations (Georgios Akrivis) .................................. 6 High-order stroboscopic averaging methods for highly oscillatory delay problems (Mari Paz Calvo) ......................
    [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]
  • 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]
  • Curriculum Vitæ
    Curriculum Vitæ LAURENT O. JAY September 2011 Position: Professor of Mathematics Associate Chair Director of Graduate Studies Address: Department of Mathematics 14 MacLean Hall The University of Iowa Iowa City, IA 52242-1419 USA Phone: +1 319 335 0898 Fax: +1 319 335 0627 E-mail: [email protected] Homepage: http://www.math.uiowa.edu/~ljay/ Higher Education • University of Geneva, Switzerland, Mathematics, Ph.D. (”Doctorat `es Sciences Mention Math´ematiques”), 1994. Advisor: Professor Ernst Hairer. • University of Geneva, Switzerland, Computer Science, M.Sc. (”Diplˆome d’Infor- maticien”), 1990. • University of Geneva, Switzerland, Mathematics, M.Sc. (”Diplˆome de Math´ema- ticien”), 1990. • University of Geneva, Switzerland, Computer Science, B.Sc. (”Licence `es Sci- ences Informatique”), 1989. Academic Positions Long Term Positions • Associate Chair and Director of Graduate Studies in Mathematics since July 2011. • Full Professor with tenure since July 2009, Department of Mathematics, Uni- versity of Iowa, Iowa City, USA. Associate Professor with tenure July 2003-June 2009. Assistant Professor, August 1998-June 2003. Affiliated faculty member of the interdisciplinary Program in Applied Mathematical and Computational 1 Sciences, July 2002-present. Courses taught so far: Graduate courses: Numeri- cal analysis I; Numerical analysis II; Nonlinear dynamics and chaos; Nonlinear dynamics with numerical methods; Optimization techniques; Ordinary differen- tial equations I; Ordinary differential equations II; Topics in numerical analy- sis; Topics in applied mathematics, Quantum mechanics (foundations, practical models, and algorithms). Undergraduate courses: Calculus I; Calculus II; Dif- ferential equations for engineers; Elementary numerical analysis; Engineering calculus I; Matrix algebra. • Postdoctoral Associate, February 1997-January 1998, Minnesota Supercom- puter Institute and Computer Science Department, University of Minnesota, Minneapolis, USA.
    [Show full text]
  • Internationale Mathematische Nachrichten International Mathematical News Nouvelles Mathematiques´ Internationales
    Internationale Mathematische Nachrichten International Mathematical News Nouvelles Mathematiques´ Internationales Die IMN wurden 1947 von R. Inzin- Bezug: ger als Nachrichten der Mathematischen ” Gesellschaft in Wien“ gegrundet.¨ 1952 Die IMN erscheinen dreimal jahrlich¨ und ¨ wurde die Zeitschrift in Internationale werden von den Mitgliedern der Oster- ” Mathematische Nachrichten“ umbenannt reichischen Mathematischen Gesellschaft und war bis 1971 offizielles Publikati- bezogen. onsorgan der Internationalen Mathema- Jahresbeitrag: 25,– ” tischen Union“. Bankverbindung: Konto Nr. AT83-1200- Von 1953 bis 1977 betreute W. Wunder- 0229-1038-9200, bei der Bank Austria- lich, der bereits seit der Grundung¨ als Re- Creditanstalt (BIC-Code BKAUATWW). dakteur mitwirkte, als Herausgeber die IMN. Die weiteren Herausgeber waren H. Vogler (1978–79), U. Dieter (1980– 81, 1984–85), L. Reich (1982–83), P. Flor (1986–99) und M. Drmota (2000–2007). Herausgeber: Osterreichische¨ Mathematische Gesell- schaft, Wiedner Hauptstraße 8–10/104, A-1040 Wien. email [email protected], http://www.oemg.ac.at/ Redaktion: J. Wallner (TU Graz, Herausgeber) H. Humenberger (Univ. Wien) Eigentumer,¨ Herausgeber und Verleger: R. Tichy (TU Graz) Osterr.¨ Math. Gesellschaft. Satz: Osterr.¨ R. Winkler (TU Wien) Math. Gesellschaft. Druck: Weinitzen- druck, 8044 Weinitzen. Standige¨ Mitarbeiter der Redaktion: ¨ B. Gittenberger (TU Wien) c 2015 Osterreichische Mathematische G. Eigenthaler (TU Wien) Gesellschaft, Wien. K. Sigmund (Univ. Wien) ISSN 0020-7926 Osterreichische¨ Mathematische Gesellschaft Gegrundet¨ 1903 P. M. Gruber (TU Wien) http://www.oemg.ac.at/ G. Helmberg (Univ. Innsbruck) email: [email protected] H. Heugl (Wien) W. Imrich (MU Leoben) Sekretariat: M. Koth (Univ. Wien) C. Krattenthaler (Univ. Wien) TU Wien, Institut 104, W. Kuich (TU Wien) Wiedner Hauptstr.
    [Show full text]
  • The Geometry of Characters of Hopf Algebras 3
    The geometry of characters of Hopf algebras Geir Bogfjellmo and Alexander Schmeding Abstract Character groups of Hopf algebras can be used to conveniently describe several species of “series expansions” such as ordinary Taylor series, B-series, aris- ing in the study of ordinary differential equations, Fliess series, arising from con- trol theory and rough paths. These ideas are a fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups. Our main result is the construction and study of Lie group struc- tures for Hopf algebras which are graded but not necessarily connected (in the sense that the degree zero subalgebra is finite-dimensional). Key words: infinite-dimensional Lie group, pro-Lie group, character group of a Hopf algebra, regularity of Lie groups, continuous inverse algebra MSC2010: 22E65 (primary); 16T05, 43A40, 58B25 (Secondary) 1 Foundations: Character groups and graded algebra . ...................... 4 2 Geometry of groups of characters . ..................... 7 3 Appendix: Infinite-dimensional calculus . ....................... 18 4 Appendix: Characters and the exponential map . ..................... 21 References ......................................... ............................ 23 arXiv:1704.01099v4 [math.GR] 27 Sep 2017 Geir Bogfjellmo Chalmers tekniska h¨ogskola och G¨oteborgs universitet, Matematiska vetenskaper, 412 96 G¨oteborg, Sweden e-mail: [email protected] Alexander Schmeding NTNU Trondheim, Institutt for matematiske fag, 7491 Trondheim, Norway e-mail: [email protected] 1 2 Geir Bogfjellmo and Alexander Schmeding Character groups of Hopf algebras appear in a variety of mathematical contexts.
    [Show full text]
  • On Enumeration Problems in Lie–Butcher Theory H
    Future Generation Computer Systems 19 (2003) 1197–1205 On enumeration problems in Lie–Butcher theory H. Munthe-Kaas∗, S. Krogstad Department of Computer Science, University of Bergen, N-5020 Bergen, Norway Abstract The algebraic structure underlying non-commutative Lie–Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie–Butcher series. © 2003 Published by Elsevier Science B.V. Keywords: Butcher theory; Trees; Enumeration; Lie group integrators; Runge–Kutta methods 1. Summary 2. Introduction to Lie–Butcher theory Let g be the free Lie algebra over ordered trees, Lie series, dating back to Sophus Lie (1842–1899), with a grading induced by the number of nodes in the is a version of Taylor series adapted to general man- trees. In the first part of this paper we prove that gn, the ifolds. Butcher series, invented by Butcher [4,5] homogeneous component of degree n, has dimension: and developed further in [8] (see also [9])isan adaption of Taylor series to the study of order con- n d (g ) = 1 µ 2 . ditions of Runge–Kutta methods. In recent papers dim n n d d 2 d|n [6,10,12,14,15,17,19], efforts have been made to ex- tend the theory of Runge–Kutta methods from Rn For n = 1, 2, 3,... these numbers are 1, 1, 3, 8, 25, to Lie groups and homogeneous spaces. Some very 75, 245, 800, 2700, 9225, 32 065,..
    [Show full text]