On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

On algebraic structures of numerical integration on vector spaces and manifolds

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.Comment: 42 pages, final versio

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

Topography of supplementary eye field afferents to frontal eye field in macaque: Implications for mapping between saccade coordinate systems

Two discrete areas in frontal cortex are involved in generating saccadic eye movements—the frontal eye field (FEF) and the supplementary eye field (SEF). Whereas FEF represents saccades in a topographic retinotopic map, recent evidence indicates that saccades may be represented craniotopically in SEF. To further investigate the relationship between these areas, the topographic organization of afferents to FEF from SEF in Macaco mulatto was examined by placing injections of distinct retrograde tracers into different parts of FEF that represented saccades of different amplitudes. Central FEF (lateral area 8A), which represents saccades of intermediate amplitudes, received afferents from a larger portion of SEF than did lateral FEF (area 45), which represents shorter saccades, or medial FEF (medial area 8A), which represents the longest saccades in addition to pinna movements. Moreover, in every case the zone in SEF that innervated lateral FEF (area 45) also projected to medial FEF (area 8A). In one case, a zone in rostral SEF projected to both lateral area 8A from which eye movements were evoked by microstimulation as well as medial area 8A from which pinna movements were elicited by microstimulation. This pattern of afferent convergence and divergence from SEF onto the retinotopic saccade map in FEF is indicative of some sort of map transformation between SEF and FEF. Such a transformation would be necessary to interconnect a topographic craniotopic saccade representation in SEF with a topographic retinotopic saccade representation in FE

One-site density matrix renormalization group and alternating minimum energy algorithm

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT) / matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201

Opgørelse af regularitet på jernbaner

Regularitet på jernbaner opgøres forskelligt, hvorfor det er svært at sammenligne regularitet mellem forskellige selskaber og lande. Opgørelse af regularitet vanskeliggøres endvidere af, at regulariteten afhænger af både kapacitetsudnyttelsen, pålideligheden, køreplanstillægget og rettidigheden. Erfaringer viser, at fokus på regularitet kan forbedre regulariteten. Fokus på regularitet kan opnås gennem fx kampagner overfor passagerer og togpersonale. Imidlertid kan ”overdrevent” fokus på togenes regularitet være skadeligt for passagerernes regularitet. Dette skyldes, at fokus på togregularitet kan resultere i, at et forsinket korresponderende tog ikke afventes, at rejsetiden forlænges for at have mere luft i køreplanerne, og at forsinkede tog aflyses for ikke at generere følgeforsinkelser. Det er muligt at opgøre regulariteten for passagerer i stedet for tog. Opgørelse af passagerre-gularitet i stedet for togregularitet betyder, at passagererne ikke stilles ringere ved ”overdrevent” fokus på regularitet. Brug af passagerregularitet i bod-bonus-kontrakter betyder endvi-dere, at der vil være fokus på at sikre så gode forhold for passagererne som muligt. Passagerernes forhold kan endvidere forbedres, ved at sikre det rette tillæg i køreplanerne i forhold til risikoen for forsinkelser og det samlede tidsforbrug, hvilket kan sikres ved simulering og vurdering af passagerforsinkelser

The Organization and Evolution of Dorsal Stream Multisensory Motor Pathways in Primates

In Prosimian primates, New World monkeys, and Old World monkeys microstimulation with half second trains of electrical pulses identifies separate zones in posterior parietal cortex (PPC) where reaching, defensive, grasping, and other complex movements can be evoked. Each functional zone receives a different pattern of visual and somatosensory inputs, and projects preferentially to functionally matched parts of motor and premotor cortex. As PPC is a relatively small portion of cortex in most mammals, including the close relatives of primates, we suggest that a larger, more significant PPC emerged with the first primates as a region where several ethologically relevant behaviors could be initiated by sensory and intrinsic signals, and mediated via connections with premotor and motor cortex. While several classes of PPC modules appear to be retained by all primates, elaboration and differentiation of these modules likely occurred in some primates, especially humans