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

A simple proof that comonotonic risks have the convex-largest sum.

In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.Risk; Actuarial; Distribution;

Risk measurement with the equivalent utility principles.

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable at- tention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker. In incomplete financial markets, prices are no more unique but depend on the agents' attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities.Actuarial; Coherence; Decision; Expected; Market; Markets; Measurement; Preference; Premium; Prices; Pricing; Principles; Random variables; Research; Risk; Risk measure; Risk measurement; Space; Studies; Theory; Uncertainty; Utilities; Variables;

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

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

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

The cleavage surface of the BaFe_(2-x)Co_(x)As_(2) and Fe_(y)Se_(1-x)Te_(x) superconductors: from diversity to simplicity

We elucidate the termination surface of cleaved single crystals of the BaFe_(2-x)Co_(x)As_(2) and Fe_(y)Se_(1-x)Te_(x) families of the high temperature iron based superconductors. By combining scanning tunneling microscopic data with low energy electron diffraction we prove that the termination layer of the Ba122 systems is a remnant of the Ba layer, which exhibits a complex diversity of ordered and disordered structures. The observed surface topographies and their accompanying superstructure reflections in electron diffraction depend on the cleavage temperature. In stark contrast, Fe_(y)Se_(1-x)Te_(x) possesses only a single termination structure - that of the tetragonally ordered Se_(1-x)Te_(x) layer.Comment: 4 pages, 4 figure

Shape analysis on homogeneous spaces: a generalised SRVT framework

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and Control". v3: amended the text to improve readability and clarify some points; updated and added some references; added pseudocode for the dynamic programming algorithm used. The main results remain unchange