Exact design manifold control of a class of nonlinear singularly perturbed systems

The integral manifold approach captures from a geometric point of view the intrinsic two-time-scale behavior of singularly perturbed systems. An important class of nonlinear singularly perturbed systems considered in this note are fast actuator-type systems. For a class of fast actuator-type systems, which includes many physical systems, an explicit corrected composite control, the sum of a slow control and a fast control, is derived. This corrected control will steer the system exactly to a required design manifold.

Exact design manifold control of a class of nonlinear singularly perturbed systems

The integral manifold approach captures from a geometric point of view the intrinsic two-time-scale behavior of singularly perturbed systems. An important class of nonlinear singularly perturbed systems considered in this note are fast actuator-type systems. For a class of fast actuator-type systems, which includes many physical systems, an explicit corrected composite control, the sum of a slow control and a fast control, is derived. This corrected control will steer the system exactly to a required design manifold.

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

The Functionality of the three-sited Ferroxidase center of E. coli Bacterial Ferritin (EcFtnA)

At least three ferritins are found in the bacterium Escherichia coli, the heme-containing bacterioferritin (EcBFR) and two non-heme bacterial ferritins (EcFtnA and EcFtnB). In addition to the conserved A- and B-sites of the diiron ferroxidase center, EcFtnA has a third iron-binding site (the C-site) of unknown function that is nearby the diiron site. In the present work, the complex chemistry of iron oxidation and deposition in EcFtnA has been further defined through a combination of oximetry, pH stat, stopped-flow and conventional kinetics, UV-visible, fluorescence and EPR spectroscopic measurements on the wildtype protein and site-directed variants of the A-, B- and C-sites. The data reveal that, while H2O2 is a product of dioxygen reduction in EcFtnA and oxidation occurs with a stoichiometry of Fe(II)/O2 ~ 3:1, most of the H2O2 produced is consumed in subsequent reactions with a 2:1 Fe(II)/H2O2 stoichiometry, thus suppressing hydroxyl radical formation. While the A- and B-sites are essential for rapid iron oxidation, the C-site slows oxidation and suppresses iron turnover at the ferroxidase center. A tyrosyl radical, assigned to Tyr24 near the ferroxidase center, is formed during iron oxidation and its possible significance to the function of the protein is discussed. Taken as a whole, the data indicate that there are multiple iron-oxidation pathways in EcFtnA with O2 and H2O2 as oxidants. Furthermore, the data are inconsistent with the C-site being a transit site, providing iron to the A- and B-sites, and does not support a universal mechanism for iron oxidation in all ferritins as recently proposed.

Sparse density estimation on the multinomial manifold

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion for the finite mixture model. Since the constraint on the mixing coefficients of the finite mixture model is on the multinomial manifold, we use the well-known Riemannian trust-region (RTR) algorithm for solving this problem. The first- and second-order Riemannian geometry of the multinomial manifold are derived and utilized in the RTR algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with an accuracy competitive with those of existing kernel density estimators.

Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Heterogeneous tensor decomposition for clustering via manifold optimization

Tensor clustering is an important tool that exploits intrinsically rich structures in real-world multiarray or Tensor datasets. Often in dealing with those datasets, standard practice is to use subspace clustering that is based on vectorizing multiarray data. However, vectorization of tensorial data does not exploit complete structure information. In this paper, we propose a subspace clustering algorithm without adopting any vectorization process. Our approach is based on a novel heterogeneous Tucker decomposition model taking into account cluster membership information. We propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model. All but the last mode have closed-form updates. Updating the last mode reduces to optimizing over the multinomial manifold for which we investigate second order Riemannian geometry and propose a trust-region algorithm. Numerical experiments show that our proposed algorithm compete effectively with state-of-the-art clustering algorithms that are based on tensor factorization.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Performance of the Quantitative Food Frequency Questionnaire Used in the Brazilian Center of the Prospective Study Natural History of Human Papillomavirus Infection in Men: The HIM Study

The Natural History of Human Papillomavirus (HPV) Infection in Men: The HIM Study is a prospective multi-center cohort study that, among other factors, analyzes participants` diet. A parallel cross-sectional study was designed to evaluate the validity and reproducibility of the quantitative food frequency questionnaire (QFFQ) used in the Brazilian center from the HIM Study. For this, a convenience subsample of 98 men aged 18 to 70 years from the HIM Study in Brazil answered three 54-item QFFQ and three 24-hour recall interviews, with 6-month intervals between them (data collection January to September 2007). A Bland-Altman analysis indicated that the difference between instruments was dependent on the magnitude of the intake for energy and most nutrients included in the validity analysis, with the exception of carbohydrates, fiber, polyunsaturated fat, vitamin C, and vitamin E. The correlation between the QFFQ and the 24-hour recall for the deattenuated and energy-adjusted data ranged from 0.05 (total fat) to 0.57 (calcium). For the energy and nutrients consumption included in the validity analysis, 33.5% of participants on average were correctly classified into quartiles, and the average value of 0.26 for weighted kappa shows a reasonable agreement. The intraclass correlation coefficients for all nutrients were greater than 0.40 in the reproducibility analysis. The QFFQ demonstrated good reproducibility and acceptable validity. The results support the use of this instrument in the HIM Study. J Am Diet Assoc. 2011;111:1045-1051.

A Longitudinal Three-Center Study of Dental Arch Relationship in Patients With Bilateral Cleft Lip and Palate

Objective: To compare and evaluate longitudinally the dental arch relationships from 4.5 to 13.5 years of age with the Bauru-BCLP Yardstick in a large sample of patients with bilateral cleft lip and palate (BCLP). Design: Retrospective longitudinal intercenter outcome study. Patients: Dental casts of 204 consecutive patients with complete BCLP were evaluated at 6, 9, and 12 years of age. All models were identified only by random identification numbers. Setting: Three cleft palate centers with different treatment protocols. Main Outcome Measures: Dental arch relationships were categorized with the Bauru-BCLP yardstick. Increments for each interval (from 6 to 9 years, 6 to 12 years, and 9 to 12 years) were analyzed by logistic and linear regression models. Results: There were no significant differences in outcome measures between the centers at age 12 or at age 9. At age 6, center B showed significantly better results (p = .027), but this difference diminished as the yardstick score for this group increased over time (linear regression analysis), the difference with the reference category (center C, boys) for the intervals 6 to 12 and 9 to 12 years being 10.4% (p = .041) and 12.9% (p = .009), respectively. Conclusions: Despite different treatment protocols, dental arch relationships in the three centers were comparable in final scores at age 9 and 12 years. Delaying hard palate closure and employing infant orthopedics did not appear to be advantageous in the long run. Premaxillary osteotomy employed in center B appeared to be associated with less favorable development of the dental arch relationship between 9 and 12 years.

Activation of Leishmania (Leishmania) amazonensis arginase at low temperature by binuclear Mn(2+) center formation of the immobilized enzyme on a Ni(2+) resin

In Leishmania, arginase is responsible for the production of ornithine, a precursor of polyamines required for proliferation of the parasite. In this work, the activation kinetics of immobilized arginase enzyme from L. (L.) amazonensis were studied by varying the concentration of Mn(2+) applied to the nickel column at 23 degrees C. The intensity of the binding of the enzyme to the Ni(2+) resin was directly proportional to the concentration of Mn(2+). Conformational changes of the enzyme may occur when the enzyme interacts with immobilized Ni(2+), allowing the following to occur: (1) entrance of Mn(2+) and formation of the metal bridge; (2) stabilization and activation of the enzyme at 23 degrees C; and (3) an increase in the affinity of the enzyme to Ni(2+) after the Mn(2+) activation step. The conformational alterations can be summarized as follows: the interaction with the Ni(2+) simulates thermal heating in the artificial activation by opening a channel for Mn(2+) to enter. (C) 2010 Elsevier Inc. All rights reserved.

The analytic torsion of a cone over an odd dimensional manifold

We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Muller theorem [3, 2] for a manifold with boundary, according to Bruning and Ma (2006) [5]. We also prove Poincare duality for the analytic torsion of a cone. (C) 2010 Elsevier B.V. All rights reserved.