Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

Entanglement and quantum phase transitions

We examine several well known quantum spin models and categorize behavior of pairwise entanglement at quantum phase transitions. A unified picture on the connection between the entanglement and quantum phase transition is given.Comment: 4 pages, 3 figure

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria, are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that, in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non ideal equation of state.Comment: 10 pages, 1 figur

CONSTRUCTION OF AIRBORNE MOVEMENTS AND MODEL APPROACH FOR LEARNING

The purposes of this research were a) to identify differences in the biomechanical description of movements between the biomechanist (external view), the athlete (internal sight) and the coach system (internal sight from external view; Lippens, 1997) and b) to supply applicable and relevant information for learning sport skills. The research consists of biomechanical modelling, collection of anthropometric and kinematic data, analysis, construction of a learning model and its application to practice. Results of the research are: (a) The inertial and the non-inertial system as well as coupling of body segments establish the differences between the views 1 to 3. (b) Joint rotations are not identical with the muscular moments, passive rotations (McGeer. 1990) can occur. (c) Knowledge of muscular moments, "critical phases" and passive phases simplify learning of motor skills

Combining cluster observables and stacked weak lensing to probe dark energy: Self-calibration of systematic uncertainties

We develop a new method of combining cluster observables (number counts and cluster-cluster correlation functions) and stacked weak lensing signals of background galaxy shapes, both of which are available in a wide-field optical imaging survey. Assuming that the clusters have secure redshift estimates, we show that the joint experiment enables a self-calibration of important systematic errors including the source redshift uncertainty and the cluster mass-observable relation, by adopting a single population of background source galaxies for the lensing analysis. It allows us to use the relative strengths of stacked lensing signals at different cluster redshifts for calibrating the source redshift uncertainty, which in turn leads to accurate measurements of the mean cluster mass in each bin. In addition, our formulation of stacked lensing signals in Fourier space simplifies the Fisher matrix calculations, as well as the marginalization over the cluster off-centering effect, the most significant uncertainty in stacked lensing. We show that upcoming wide-field surveys yield stringent constraints on cosmological parameters including dark energy parameters, without any priors on nuisance parameters that model systematic uncertainties. Specifically, the stacked lensing information improves the dark energy FoM by a factor of 4, compared to that from the cluster observables alone. The primordial non-Gaussianity parameter can also be constrained with a level of f_NL~10. In this method, the mean source redshift is well calibrated to an accuracy of 0.1 in redshift, and the mean cluster mass in each bin to 5-10% accuracies, which demonstrates the success of the self-calibration of systematic uncertainties from the joint experiment. (Abridged)Comment: 29 pages, 17 figures, 6 tables, accepted for publication in Phys. Rev.

Complexity dimensions and learnability

A stochastic model of learning from examples has been introduced by Valiant [1984]. This PAC-learning model (PAC = probably approximately correct) reflects differences in complexity of concept classes, i.e. very complex classes are not efficiently PAC-learnable. Blumer et al. [1989] found, that efficient PAC-learnability depends on the size of the Vapnik Chervonenkis dimension [Vapnik & Chervonenkis, 1971] of a class. We will first discuss this dimension and give an algorithm to compute it, in order to provide the reader with the intuitive idea behind it. Natarajan [1987] defines a new, equivalent dimension is defined for well-ordered classes. These well-ordered classes happen to satisfy a general condition, that is sufficient for the possible construction of a number of equivalent dimensions. We will give this condition, as well as a generalized notion of an equivalent dimension. Also, a relatively efficient algorithm for the calculation of one such dimension for well-ordered classes is given