3,049 research outputs found
Resonance Identities for Closed Characteristics on Compact Star-shaped Hypersurfaces in
Resonance relations among periodic orbits on given energy hypersurfaces are
very important for getting deeper understanding of the dynamics of the
corresponding Hamiltonian systems. In this paper, we establish two new
resonance identities for closed characteristics on every compact star-shaped
hypersurface in when the number of geometrically
distinct closed characteristics on is finite, which extend those
identities established by C. Viterbo in 1989 for star-shaped hypersurfaces
assuming in addition that all the closed characteristics and their iterates are
non-degenerate, and that by W. Wang, X. Hu and Y. Long in 2007 for strictly
convex hypersurfaces in .Comment: arXiv admin note: substantial text overlap with arXiv:math/0701608.
To appear in Journal of Functional Analysis, this is the final versio
The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
Let and be a nontrivial element of finite order in
, where the integer , is a finite group which acts
freely and isometrically on the -sphere and therefore is diffeomorphic
to a compact space form. In this paper, we establish first the resonance
identity for non-contractible homologically visible minimal closed geodesics of
the class on every Finsler compact space form when there exist
only finitely many distinct non-contractible closed geodesics of the class
on . Then as an application of this resonance identity, we prove
the existence of at least two distinct non-contractible closed geodesics of the
class on with a bumpy Finsler metric, which improves a result of
Taimanov in [Taimanov 2016] by removing some additional conditions. Also our
results extend the resonance identity and multiplicity results on
in [arXiv:1607.02746] to general compact space forms.Comment: 33 pages, All comments are welcome. arXiv admin note: substantial
text overlap with arXiv:1607.0274
Multiplicity of closed characteristics on symmetric convex hypersurfaces in
Let be a compact hypersurface in bounding a convex
set with non-empty interior. In this paper it is proved that there always exist
at least geometrically distinct closed characteristics on if
is symmetric with respect to the origin.Comment: 16 page
Hierarchical and Adaptive Filter and Refinement Algorithms for Geometric Intersection Computations on GPU
Geometric intersection algorithms are fundamental in spatial analysis in Geographic Information System (GIS). This dissertation explores high performance computing solution for geometric intersection on a huge amount of spatial data using Graphics Processing Unit (GPU). We have developed a hierarchical filter and refinement system for parallel geometric intersection operations involving large polygons and polylines by extending the classical filter and refine algorithm using efficient filters that leverage GPU computing. The inputs are two layers of large polygonal datasets and the computations are spatial intersection on pairs of cross-layer polygons. These intersections are the compute-intensive spatial data analytic kernels in spatial join and map overlay operations in spatial databases and GIS. Efficient filters, such as PolySketch, PolySketch++ and Point-in-polygon filters have been developed to reduce refinement workload on GPUs. We also showed the application of such filters in speeding-up line segment intersections and point-in-polygon tests. Programming models like CUDA and OpenACC have been used to implement the different versions of the Hierarchical Filter and Refine (HiFiRe) system. Experimental results show good performance of our filter and refinement algorithms. Compared to standard R-tree filter, on average, our filter technique can still discard 76% of polygon pairs which do not have segment intersection points. PolySketch filter reduces on average 99.77% of the workload of finding line segment intersections. Compared to existing Common Minimum Bounding Rectangle (CMBR) filter that is applied on each cross-layer candidate pair, the workload after using PolySketch-based CMBR filter is on average 98% smaller. The execution time of our HiFiRe system on two shapefiles, namely USA Water Bodies (contains 464K polygons) and USA Block Group Boundaries (contains 220K polygons), is about 3.38 seconds using NVidia Titan V GPU
- …