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

Discovery of X-ray emission from the proto-stellar jet L1551 IRS5 (HH 154)

We have for the first time detected X-ray emission associated with a proto-stellar jet, on the jet emanating from L1551 IRS5. The IRS5 proto-star is hidden beyond a very large absorbing column density, making the direct observation of the jet's emission possible. The observed X-ray emission is likely associated with the shock ``working surface'', i.e. the interface between the jet and the circumstellar medium. The X-ray luminosity emanating from the jet is moderate, at LX ~ 3 times 10^29 erg/s, a significant fraction of the luminosity normally associated with the coronal emission from young stars. The spectrum of the X-ray emission is compatible with thermal emission from a hot plasma, with T ~ 0.5 MK, fully compatible with the temperature expected (on the basis of the jet's velocity) for the shock front produced by the jet hitting the circumstellar medium.Comment: To appear in "Stellar Coronae in the Chandra and XMM Era", ASP Conference Series in press, F. Favata & J. Drake ed

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

Information Optimization in Coupled Audio-Visual Cortical Maps

Barn owls hunt in the dark by using cues from both sight and sound to locate their prey. This task is facilitated by topographic maps of the external space formed by neurons (e.g., in the optic tectum) that respond to visual or aural signals from a specific direction. Plasticity of these maps has been studied in owls forced to wear prismatic spectacles that shift their visual field. Adaptive behavior in young owls is accompanied by a compensating shift in the response of (mapped) neurons to auditory signals. We model the receptive fields of such neurons by linear filters that sample correlated audio-visual signals, and search for filters that maximize the gathered information, while subject to the costs of rewiring neurons. Assuming a higher fidelity of visual information, we find that the corresponding receptive fields are robust and unchanged by artificial shifts. The shape of the aural receptive field, however, is controlled by correlations between sight and sound. In response to prismatic glasses, the aural receptive fields shift in the compensating direction, although their shape is modified due to the costs of rewiring.Comment: 7 pages, 1 figur

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

On the distribution of cash-flows using Esscher transforms.

In their seminal paper, Gerber and Shiu (1994) introduced the concept of the Esscher transform for option pricing. As examples they considered the shifted Poisson process, the random walk, a shifted gamma process and a shifted inverse Gaussian process to describe the logarithm of the stock price. In the present paper it is shown how upper and lower bounds in convex order can be obtained when we use these types of models to describe the financial stochasticity for a given cash-flow.Cash flow; Pricing; Processes; Models; Model;

Bounds for present value functions with stochastic interest rates and stochastic volatility.

The distribution of the present value of a series of cash flows under stochastic interest rates has been investigated by many researchers. One of the main problems in this context is the fact that the calculation of exact analytical results for this type of distributions turns out to be rather complicated, and is known only for special cases. An interesting solution to this difficulty consists of determining computable upper bounds, as close as possible to the real distribution.In the present contribution, we want to show how it is possible to compute such bounds for the present value of cash flows when not only the interest rates but also volatilities are stochastic. We derive results for the stop loss premium and distribution of these bounds.Distribution; Value; Cash flow; Interest rates; Researchers; Problems;

Congenital radial head dislocation with a progressive cubitus valgus: a case report

Congenital dislocation of the radial head is rare, although it is the most common congenital anomaly of the elbow. A concomitant progressive cubitus valgus of the elbow has not previously been described in literature. We describe a case of an 8-year-old girl with an unilateral congenital radial head dislocation with a progressive cubitus valgus of 35°, caused by a prematurely closing physis of the lateral humeral condyle. This might be caused by an increased pressure on the lateral physis by the anteriorly dislocated radial head. As no complaints or limitations were present, treatment was non-operative with clinical observation, with satisfactory results after a follow-up of 18 months. A concomitant progressive cubitus valgus can be present in patients with a congenital radial head dislocation. Non-operative treatment can provide satisfactory results

Land Collateral and Labor Market Dynamics in France

The value of land in the balance sheet of French firms correlates positively with their hiring and investment flows. To explore the relationship between these variables, we develop a macroeconomic model with firms that are subject to both credit and labor market frictions. The value of collateral is driven by the forward-looking dynamics of the land price, which reacts endogenously to fundamental and non-fundamental (sunspot) shocks. We calibrate the model to French data and find that land price shocks give rise to significant amplification and hump-shaped responses of investment, vacancies and unemployment that are in line with the data. We show that both the endogenous movements in the firm’s discount factor and the sluggish response of the land price are key elements that drive the results