Higher-dimensional algebra

In mathematics, especially () , higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra. Higher-dimensional categories Category theory#Higher-dimensional categories A first step towards defining higher dimensional algebras is the concept of of , followed by the more 'geometric' concept of double category. A higher level concept is thus defined as a of categories, or super-category, which generalises to higher dimensions the notion of – regarded as any structure which is an interpretation of Lawvere's axioms of the elementary theory of abstract categories (ETAC). Ll. , Thus, a supercategory and also a , can be regarded as natural extensions of the concepts of , , and multi-graph, k-partite graph, or colored graph (see a color figure, and also its definition in graph theory). Supercategories were first introduced in 1970, and were subsequently developed for applica

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A functional approach to the numerical conformal bootstrap

Bernardo Zan

We apply recently constructed functional bases to the numerical conformal bootstrap for 1D CFTs. We argue and show that numerical results in this basis converge much faster than the traditional derivative basis. In particular, truncations of the crossing equation with even a handful of components can lead to extremely accurate results, in opposition to hundreds of components in the usual approach. We explain how this is a consequence of the functional basis correctly capturing the asymptotics of bound-saturating extremal solutions to crossing. We discuss how these methods can and should be implemented in higher dimensional applications.

SPRINGER2020

Extracting partial decay rates of helium from complex rotation: autoionizing resonances of the one-dimensional configurations

Pierre Lugan

Partial autoionization rates of doubly excited one-dimensional helium in the collinear Zee and eZe configuration are obtained by means of the complex rotation method. The approach presented here relies on a projection of back-rotated resonance wave functions onto singly ionized He+ channel wave functions and the computation of the corresponding particle fluxes. In spite of the long-range nature of the Coulomb potential between the electrons and the nucleus, an asymptotic region where the fluxes are stationary is clearly observed. Low-lying doubly excited states are found to decay predomintantly into the nearest single-ionization continuum. This approach paves the way for a systematic analysis of the decay rates observed in higher-dimensional models, and of the role of electronic correlations and atomic structure in recent photoionization experiments.

Iop Publishing Ltd2015

Velocity-Scheduling Control for a Unicycle Mobile Robot: Theory and Experiments

Dominique Bonvin, Davide Buccieri, Philippe Müllhaupt, Damien Perritaz

Improvement over classical dynamic feedback linearization for a unicycle mobile robots is proposed. Compared to classical extension, the technique uses a higher-dimensional state extension, which allows rejecting a constant disturbance on the robot rotational axis. The proposed dynamic extension acts as a velocity scheduler that specifies, at each time instant, the ideal translational velocity that the robot should have. By using a higher-order extension, both the magnitude and the orientation of the velocity vector can be generated, which introduces robustness in the control scheme. Stability for both asymptotic convergence to a point and trajectory tracking is proven. The theoretical results are illustrated first in simulation, and then experimentally on the autonomous mobile robot are given. The theory is illustrated through both simulation and experiments obtained on an autonomous mobile robot called Fouzy III.

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