On a Class Almost Contact Manifolds with Norden Metric

Certain curvature properties and scalar invariants of the mani- folds belonging to one of the main classes almost contact manifolds with Norden metric are considered. An example illustrating the obtained results is given and studied.

On The Topology Of Manifolds With Positive Isotropic Curvature

We show that a closed orientable Riemannian n-manifold, n >= 5, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of Sn-1 x S-1.

Manifolds with nonnegative isotropic curvature

We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature,then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.

Isotropic isotopy and symplectic null sets

Capacity is an important numerical invariant of symplectic manifolds. This paper studies when a subset of a symplectic manifold is null, i.e., can be removed without affecting the ambient capacity. After examples of open null sets and codimension-2 non-null sets, geometric techniques are developed to perturb any isotopy of a loop to a hamiltonian flow; it follows that sets of dimension 0 and 1 are null. For isotropic sets of higher dimensions, obstructions to the perturbation are found in homotopy groups of the orthogonal groups.

A practical implementation of a continuous isotropic spherical omnidirectional drive

This paper presents a continuous isotropic spherical omnidirectional drive mechanism that is efficient in its mechanical simplicity and use of volume. Spherical omnidirectional mechanisms allow isotropic motion, although many are limited from achieving true isotropic motion by practical mechanical design considerations. The mechanism presented in this paper uses a single motor to drive a point on the great circle of the sphere parallel to the ground plane, and does not require a gearbox. Three mechanisms located 120 degrees apart provide a stable drive platform for a mobile robot. Results show the omnidirectional ability of the robot and demonstrate the performance of the spherical mechanism compared to a popular commercial omnidirectional wheel over edges of varying heights and gaps of varying widths.

Sequential nonlinear manifold learning

The computation of compact and meaningful representations of high dimensional sensor data has recently been addressed through the development of Nonlinear Dimensional Reduction (NLDR) algorithms. The numerical implementation of spectral NLDR techniques typically leads to a symmetric eigenvalue problem that is solved by traditional batch eigensolution algorithms. The application of such algorithms in real-time systems necessitates the development of sequential algorithms that perform feature extraction online. This paper presents an efficient online NLDR scheme, Sequential-Isomap, based on incremental singular value decomposition (SVD) and the Isomap method. Example simulations demonstrate the validity and significant potential of this technique in real-time applications such as autonomous systems.

Heteroscedastic probabilistic linear discriminant analysis for manifold learning in video-based face recognition

To recognize faces in video, face appearances have been widely modeled as piece-wise local linear models which linearly approximate the smooth yet non-linear low dimensional face appearance manifolds. The choice of representations of the local models is crucial. Most of the existing methods learn each local model individually meaning that they only anticipate variations within each class. In this work, we propose to represent local models as Gaussian distributions which are learned simultaneously using the heteroscedastic probabilistic linear discriminant analysis (PLDA). Each gallery video is therefore represented as a collection of such distributions. With the PLDA, not only the within-class variations are estimated during the training, the separability between classes is also maximized leading to an improved discrimination. The heteroscedastic PLDA itself is adapted from the standard PLDA to approximate face appearance manifolds more accurately. Instead of assuming a single global within-class covariance, the heteroscedastic PLDA learns different within-class covariances specific to each local model. In the recognition phase, a probe video is matched against gallery samples through the fusion of point-to-model distances. Experiments on the Honda and MoBo datasets have shown the merit of the proposed method which achieves better performance than the state-of-the-art technique.

Energy-based positioning control of underactuated vehicles using manifold regulation

In this paper, we consider the problem of position regulation of a class of underactuated rigid-body vehicles that operate within a gravitational field and have fully-actuated attitude. The control objective is to regulate the vehicle position to a manifold of dimension equal to the underactuation degree. We address the problem using Port-Hamiltonian theory, and reduce the associated matching PDEs to a set of algebraic equations using a kinematic identity. The resulting method for control design is constructive. The point within the manifold to which the position is regulated is determined by the action of the potential field and the geometry of the manifold. We illustrate the performance of the controller for an unmanned aerial vehicle with underactuation degree two-a quadrotor helicopter.

Manoeuvring control of underactuated surface vessels using manifold regulation for Port-Hamiltonian systems

This paper considers the manoeuvring of underactuated surface vessels. The control objective is to steer the vessel to reach a manifold which encloses a waypoint. A transformation of configuration variables and a potential field are used in a Port-Hamiltonian framework to design an energy-based controller. With the proposed controller, the geometric task associated with the manoeuvring problem depends on the desired potential energy (closed-loop) and the dynamic task depends on the total energy and damping. Therefore, guidance and motion control are addressed jointly, leading to model-energy-based trajectory generation.

Generalized azimuthal shear deformations in compressible isotropic elastic materials

In this article we study the azimuthal shear deformations in a compressible Isotropic elastic material. This class of deformations involves an azimuthal displacement as a function of the radial and axial coordinates. The equilibrium equations are formulated in terms of the Cauchy-Green strain tensors, which form an overdetermined system of partial differential equations for which solutions do not exist in general. By means of a Legendre transformation, necessary and sufficient conditions for the material to support this deformation are obtained explicitly, in the sense that every solution to the azimuthal equilibrium equation will satisfy the remaining two equations. Additionally, we show how these conditions are sufficient to support all currently known deformations that locally reduce to simple shear. These conditions are then expressed both in terms of the invariants of the Cauchy-Green strain and stretch tensors. Several classes of strain energy functions for which this deformation can be supported are studied. For certain boundary conditions, exact solutions to the equilibrium equations are obtained. © 2005 Society for Industrial and Applied Mathematics.

Swelling induced finite strain flexure in a rectangular block of an isotropic elastic material

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.

Some solutions for a compressible isotropic elastic material

In this article we obtain closed-form solutions for the combined inflation and axial shear of an elastic tube in respect of the compressible Isotropic elastic material introduced by Levinson and Burgess. Several other boundary-value problems are also examined, including the bending of a rectangular block and straightening of a cylindrical sector, both coupled with stretching and shearing, and an axially varying twist deformation. Some of the solutions appear in closed form, others are expressible in terms of elliptic functions.

Cidade é uma palavra plural [City is a manifold word]

This text elaborates on the city as cultural construct and representation and Lisbocópio, the installation by Pancho Guedes and Ricardo Jacinto in the context of the Official Representation of Portugal at the 10. Mostra Internazionale di Architettura-La Biennale di Venezia.

Robust clustering of multi-type relational data via a heterogeneous manifold ensemble

High-Order Co-Clustering (HOCC) methods have attracted high attention in recent years because of their ability to cluster multiple types of objects simultaneously using all available information. During the clustering process, HOCC methods exploit object co-occurrence information, i.e., inter-type relationships amongst different types of objects as well as object affinity information, i.e., intra-type relationships amongst the same types of objects. However, it is difficult to learn accurate intra-type relationships in the presence of noise and outliers. Existing HOCC methods consider the p nearest neighbours based on Euclidean distance for the intra-type relationships, which leads to incomplete and inaccurate intra-type relationships. In this paper, we propose a novel HOCC method that incorporates multiple subspace learning with a heterogeneous manifold ensemble to learn complete and accurate intra-type relationships. Multiple subspace learning reconstructs the similarity between any pair of objects that belong to the same subspace. The heterogeneous manifold ensemble is created based on two-types of intra-type relationships learnt using p-nearest-neighbour graph and multiple subspaces learning. Moreover, in order to make sure the robustness of clustering process, we introduce a sparse error matrix into matrix decomposition and develop a novel iterative algorithm. Empirical experiments show that the proposed method achieves improved results over the state-of-art HOCC methods for FScore and NMI.

Universality in the fast orientational relaxation near isotropic-nematic transition

Detailed molecular dynamics simulations of Lennard-Jones ellipsoids have been carried out to investigate the emergence of criticality in the single-particle orientational relaxation near the isotropic-nematic (IN) phase transition. The simulations show a sudden appearance of a power-law behavior in the decay of the second-rank orientational relaxation as the IN transition is approached. The simulated value of the power-law exponent is 0.56, which is larger than the mean-field value (0.5) but less than the observed value (0.63) and may be due to the finite size of the simulated system. The decay of the first-rank orientational time correlation function, on the other hand, is nearly exponential but its decay becomes very slow near the isotropic-nematic transition, The zero-frequency rotational friction, calculated from the simulated angular Velocity correlation function, shows a marked increase near the IN transition.