Curvature of the Gauss map for normally flat submanifolds in space forms

For a submanifold with flat normal bundle in a space form there is a normal orthonormal basis that simultaneously diagonalizes the corresponding Weingarten operators, and at which these operators satisfy a simple Codazzi symmetry. When the second fundamental form has zero index of relative nullity, the image of the Gauss map as defined by Obata is a submanifold of a generalized Grassmannian manifold, with induced metric given by the third fundamental form, corresponding to the sum of the squares of these operators. In this case, we show that the Riemann curvature tensor of the Gauss image is completely determined by the curvature and Weingarten operators of the original submanifold, in a form analogous to a theorema egregium that relates the intrinsic geometry of the Gauss map with the extrinsic geometry of the original embedding. For this, we derive the difference vector of the induced Levi-Civita connections and express the Riemann curvature tensor with a generalized Kulkarni-Nomizu product

Covariance integral invariants of embedded Riemannian manifolds for manifold learning

2018 Summer.Includes bibliographical references.This thesis develops an effective theoretical foundation for the integral invariant approach to study submanifold geometry via the statistics of the underlying point-set, i.e., Manifold Learning from covariance analysis. We perform Principal Component Analysis over a domain determined by the intersection of an embedded Riemannian manifold with spheres or cylinders of varying scale in ambient space, in order to generalize to arbitrary dimension the relationship between curvature and the eigenvalue decomposition of covariance matrices. In the case of regular curves in general dimension, the covariance eigenvectors converge to the Frenet-Serret frame and the corresponding eigenvalues have ratios that asymptotically determine the generalized curvatures completely, up to a constant that we determine by proving a recursion relation for a certain sequence of Hankel determinants. For hypersurfaces, the eigenvalue decomposition has series expansion given in terms of the dimension and the principal curvatures, where the eigenvectors converge to the Darboux frame of principal and normal directions. In the most general case of embedded Riemannian manifolds, the eigenvalues and limit eigenvectors of the covariance matrices are found to have asymptotic behavior given in terms of the curvature information encoded by the third fundamental form of the manifold, a classical tensor that we generalize to arbitrary dimension, and which is related to the Weingarten map and Ricci operator. These results provide descriptors at scale for the principal curvatures and, in turn, for the second fundamental form and the Riemann curvature tensor of a submanifold, which can serve to perform multi-scale Geometry Processing and Manifold Learning, making use of the advantages of the integral invariant viewpoint when only a discrete sample of points is available

Union bound minimization approach for designing grassmannian constellations

In this paper, we propose an algorithm for designing unstructured Grassmannian constellations for noncoherent multiple-input multiple-output (MIMO) communications over Rayleigh block-fading channels. Unlike the majority of existing unitary space-time or Grassmannian constellations, which are typically designed to maximize the minimum distance between codewords, in this work we employ the asymptotic pairwise error probability (PEP) union bound (UB) of the constellation as the design criterion. In addition, the proposed criterion allows the design of MIMO Grassmannian constellations specifically optimized for a given number of receiving antennas. A rigorous derivation of the gradient of the asymptotic UB on a Cartesian product of Grassmann manifolds, is the main technical ingredient of the proposed gradient descent algorithm. A simple modification of the proposed cost function, which weighs each pairwise error term in the UB according to the Hamming distance between the binary labels assigned to the respective codewords, allows us to jointly solve the constellation design and the bit labeling problem. Our simulation results show that the constellations designed with the proposed method outperform other structured and unstructured Grassmannian designs in terms of symbol error rate (SER) and bit error rate (BER), for a wide range of scenarios.This work was supported by Huawei Technologies, Sweden under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

A measure preserving mapping for structured Grassmannian constellations in SIMO channels

In this paper, we propose a new structured Grassmannian constellation for noncoherent communications over single-input multiple-output (SIMO) Rayleigh block-fading channels. The constellation, which we call Grass-Lattice, is based on a measure preserving mapping from the unit hypercube to the Grassmannian of lines. The constellation structure allows for on-the-fly symbol generation, low-complexity decoding, and simple bit-to-symbol Gray coding. Simulation results show that Grass-Lattice has symbol error rate performance close to that of a numerically optimized unstructured constellation, and is more power efficient than other structured constellations proposed in the literature.This work was supported by Huawei Technologies Sweden, under the project GRASSCOM. The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltr´an was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/AEI /10.13039/501100011033. The work ofI. Santamaria was also partly supported under grant PID2019-104958RB-C43(ADELE) funded by MCIN/ AEI /10.13039/501100011033

On the existence of shear-current effects in magnetized burgulence

| openaire: EC/H2020/818665/EU//UniSDynThe possibility of explaining shear flow dynamos by a magnetic shear-current (MSC) effect is examined via numerical simulations. Our primary diagnostics is the determination of the turbulent magnetic diffusivity tensor η. In our setup, a negative sign of its component η yx is necessary for coherent dynamo action by the SC effect. To be able to measure turbulent transport coefficients from systems with magnetic background turbulence, we present an extension of the test-field method (TFM) applicable to our setup where the pressure gradient is dropped from the momentum equation: the nonlinear TFM (NLTFM). Our momentum equation is related to Burgers' equation and the resulting flows are referred to as magnetized burgulence. We use both stochastic kinetic and magnetic forcings to mimic cases without and with simultaneous small-scale dynamo action. When we force only kinetically, negative η yx are obtained with exponential growth in both the radial and azimuthal mean magnetic field components. Using magnetokinetic forcing, the field growth is no longer exponential, while NLTFM yields positive η yx . By employing an alternative forcing from which wavevectors whose components correspond to the largest scales are removed, the exponential growth is recovered, but the NLTFM results do not change significantly. Analyzing the dynamo excitation conditions for the coherent SC and incoherent α and SC effects shows that the incoherent effects are the main drivers of the dynamo in the majority of cases. We find no evidence for MSC-effect-driven dynamos in our simulations.Peer reviewe

Coherence-based subspace packings for MIMO noncoherent communications

In this paper, we propose a new algorithm for designing unstructured Grassmannian constellations for noncoherent MIMO communications over Rayleigh block-fading channels. The algorithm minimizes the maximum coherence between subspaces, which is shown to be equivalent to the diversity product previously proposed in the literature. The coherence criterion is optimized by means of a gradient ascent algorithm on the Grassmann manifold. The method is generalized to optimize a weighted cost function that takes into account several neighboring codewords. Simulation results suggest that the constellations designed with the proposed algorithm achieve better SER performance than existing algorithms for unstructured Grassmannian constellation designs.The work of D. Cuevas was also partly supported under grant FPU20/03563 funded by Ministerio de Universidades (MIU), Spain. The work of Carlos Beltran was also partly supported under grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The work of I. Santamaria was also partly supported under grant PID2019-104958RB-C43 (ADELE) funded by MCIN/ AEI /10.13039/501100011033

SOCIB: The Balearic Islands Coastal Ocean Observing and Forecasting System Responding to Science, Technology and Society Needs

Publicado

VIII Jornadas de Intercambio de Experiencias Educativas : Avilés, 16, 17 y 18 de Noviembre de 2009

Comunicaciones presentadas en las VIII Jornadas de Intercambio de Experiencias Educativas, celebradas en el Centro del Profesorado y de Recursos de Avilés en noviembre de 2009, acompañadas en cada caso de las correspondientes presentaciones y aplicaciones empleadas en la exposición y de fragmentos videográficos relativos a cada intervención. En la línea de ediciones anteriores de las Jornadas, continúan su apuesta por dar a conocer experiencias educativas de centro y de aula. La selección de experiencias se ha llevado a cabo utilizando el mismo protocolo que en años anteriores. A partir de la valoración de las memorias presentadas por la totalidad de Seminarios y Grupos de Trabajo desarrollados en este C.P.R. durante el curso escolar 2008-2009, se ha efectuado la correspondiente selección teniendo en cuenta criterios de calidad y proyección en las actividades de aula. Se han incluido en la presente edición experiencias procedente de ámbitos geográficos próximos, en función de la valía y oportunidad de la temática abordada, pudiendo diferenciarse en conjunto tres grandes temáticas: el tratamiento de la diversidad desde la dimensión de las altas capacidades, las bibliotecas escolares en distintos tipos de centros y el tratamiento de las Tecnologías de la Información y la Comunicación desde la doble perspectiva de un modelo de integración de las TIC en un centro educativo y la elaboración de aplicaciones para la pizarra digital interactiva.AsturiasUniversidad de Oviedo. Facultad de Formación del Profesorado y Educación; Campus de Llamaquique, Calle Aniceto Sela, s. n.; 33005 Oviedo; +34985103215; +34985103214; [email protected]