Lower bounds on Ricci flow invariant curvatures and geometric applications

We consider Ricci flow invariant cones C in the space of curvature operators lying between the cones ``nonnegative Ricci curvature'' and ``nonnegative curvature operator''. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies R + epsilon I is an element of C at the initial time, then it satisfies R + epsilon I is an element of C on some time interval depending only on the scalar curvature control. This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I is an element of C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C. Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allows us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.

NON-COERCIVE RICCI FLOW INVARIANT CURVATURE CONES

This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness. We show, in dimensions greater than 4, that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

A Generalization of Gravity

I consider theories of gravity built not just from the metric and affine connection, but also other (possibly higher rank) symmetric tensor(s). The Lagrangian densities are scalars built from them, and the volume forms are related to Cayley's hyperdeterminants. The resulting diff-invariant actions give rise to geometric theories that go beyond the metric paradigm (even metric-less theories are possible), and contain Einstein gravity as a special case. Examples contain theories with generalizeations of Riemannian geometry. The 0-tensor case is related to dilaton gravity. These theories can give rise to new types of spontaneous Lorentz breaking and might be relevant for ``dark'' sector cosmology.

PLUTO plus : Near-Complete Modeling of Affine Transformations for Parallelism and Locality

Affine transformations have proven to be very powerful for loop restructuring due to their ability to model a very wide range of transformations. A single multi-dimensional affine function can represent a long and complex sequence of simpler transformations. Existing affine transformation frameworks like the Pluto algorithm, that include a cost function for modern multicore architectures where coarse-grained parallelism and locality are crucial, consider only a sub-space of transformations to avoid a combinatorial explosion in finding the transformations. The ensuing practical tradeoffs lead to the exclusion of certain useful transformations, in particular, transformation compositions involving loop reversals and loop skewing by negative factors. In this paper, we propose an approach to address this limitation by modeling a much larger space of affine transformations in conjunction with the Pluto algorithm's cost function. We perform an experimental evaluation of both, the effect on compilation time, and performance of generated codes. The evaluation shows that our new framework, Pluto+, provides no degradation in performance in any of the Polybench benchmarks. For Lattice Boltzmann Method (LBM) codes with periodic boundary conditions, it provides a mean speedup of 1.33x over Pluto. We also show that Pluto+ does not increase compile times significantly. Experimental results on Polybench show that Pluto+ increases overall polyhedral source-to-source optimization time only by 15%. In cases where it improves execution time significantly, it increased polyhedral optimization time only by 2.04x.

Exact Phase Retrieval in Principal Shift-Invariant Spaces

We address the problem of phase retrieval from Fourier transform magnitude spectrum for continuous-time signals that lie in a shift-invariant space spanned by integer shifts of a generator kernel. The phase retrieval problem for such signals is formulated as one of reconstructing the combining coefficients in the shift-invariant basis expansion. We develop sufficient conditions on the coefficients and the bases to guarantee exact phase retrieval, by which we mean reconstruction up to a global phase factor. We present a new class of discrete-domain signals that are not necessarily minimum-phase, but allow for exact phase retrieval from their Fourier magnitude spectra. We also establish Hilbert transform relations between log-magnitude and phase spectra for this class of discrete signals. It turns out that the corresponding continuous-domain counterparts need not satisfy a Hilbert transform relation; notwithstanding, the continuous-domain signals can be reconstructed from their Fourier magnitude spectra. We validate the reconstruction guarantees through simulations for some important classes of signals such as bandlimited signals and piecewise-smooth signals. We also present an application of the proposed phase retrieval technique for artifact-free signal reconstruction in frequency-domain optical-coherence tomography (FDOCT).

Coarse-to-fine vision-based localization by indexing scale-invariant features

This paper presents a novel coarse-to-fine global localization approach inspired by object recognition and text retrieval techniques. Harris-Laplace interest points characterized by scale-invariant transformation feature descriptors are used as natural landmarks. They are indexed into two databases: a location vector space model (LVSM) and a location database. The localization process consists of two stages: coarse localization and fine localization. Coarse localization from the LVSM is fast, but not accurate enough, whereas localization from the location database using a voting algorithm is relatively slow, but more accurate. The integration of coarse and fine stages makes fast and reliable localization possible. If necessary, the localization result can be verified by epipolar geometry between the representative view in the database and the view to be localized. In addition, the localization system recovers the position of the camera by essential matrix decomposition. The localization system has been tested in indoor and outdoor environments. The results show that our approach is efficient and reliable. © 2006 IEEE.

Image-based localization and pose recovery using scale invariant features

In this paper, we propose a vision based mobile robot localization strategy. Local scale-invariant features are used as natural landmarks in unstructured and unmodified environment. The local characteristics of the features we use prove to be robust to occlusion and outliers. In addition, the invariance of the features to viewpoint change makes them suitable landmarks for mobile robot localization. Scale-invariant features detected in the first exploration are indexed into a location database. Indexing and voting allow efficient recognition of global localization. The localization result is verified by epipolar geometry between the representative view in database and the view to be localized, thus the probability of false localization will be decreased. The localization system can recover the pose of the camera mounted on the robot by essential matrix decomposition. Then the position of the robot can be computed easily. Both calibrated and un-calibrated cases are discussed and relative position estimation based on calibrated camera turns out to be the better choice. Experimental results show that our approach is effective and reliable in the case of illumination changes, similarity transformations and extraneous features. © 2004 IEEE.

Rotation Numbers Of Invariant Manifolds Around Unstable Periodic Orbits For The Diamagnetic Kepler Problem

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincare section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincare section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.