@article{norris_2018, title={Gromov's theorem in n-symplectic geometry on LRn}, volume={59}, ISSN={["1089-7658"]}, DOI={10.1063/1.5029474}, abstractNote={If M is an n-dimensional manifold, then the associated bundle of linear frames LM of M supports the canonically defined Rn-valued soldering 1-form θ^. The pair (LM,dθ^) is an n-symplectic manifold, where dθ^ is the n-symplectic 2-form. We adapt the proofs of de Gosson and McDuff and Salamon of Gromov’s non-squeezing theorem on R2n to give a proof of Gromov’s theorem for affine n-symplectomorphisms on LRn.}, number={5}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Norris, L. K.}, year={2018}, month={May} }
@article{norris_2017, title={On the n-symplectic structure of faithful irreducible representations}, volume={58}, ISSN={["1089-7658"]}, DOI={10.1063/1.4979625}, abstractNote={Each faithful irreducible representation of an N-dimensional vector space V1 on an n-dimensional vector space V2 is shown to define a unique irreducible n-symplectic structure on the product manifold V1×V2. The basic details of the associated Poisson algebra are developed for the special case N = n2, and 2n-dimensional symplectic submanifolds are shown to exist.}, number={4}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Norris, L. K.}, year={2017}, month={Apr} }
@article{yi_krim_norris_2012, title={Human Activity as a Manifold-Valued Random Process}, volume={21}, ISSN={["1941-0042"]}, DOI={10.1109/tip.2012.2197008}, abstractNote={Most of the previous shape-based human activity models are built with either a linear assumption or an extrinsic interpretation of the nonlinear geometry of the shape space, both of which proved to be problematic on account of the nonlinear intrinsic geometry of the associated shape spaces. In this paper, we propose an intrinsic stochastic modeling of human activity on a shape manifold. More importantly, within an elegant and theoretically sound framework, our work effectively bridges the nonlinear modeling of human activity on a nonlinear space, with the classic stochastic modeling in a Euclidean space, and thereby provides a foundation for a more effective and accurate analysis of the nonlinear feature space of activity models. From a video sequence, human activity is extracted as a sequence of shapes. Such a sequence is considered as one realization of a random process on a shape manifold. Different activities are then modeled as manifold valued random processes with different distributions. To address the problem of stochastic modeling on a manifold, we first construct a nonlinear invertible map of a manifold valued process to a Euclidean process. The resulting process is then modeled as a global or piecewise Brownian motion. The mapping from a manifold to a Euclidean space is known as a stochastic development. The advantage of such a technique is that it yields a one-one correspondence, and the resulting Euclidean process intrinsically captures the curvature on the original manifold. The proposed algorithm is validated on two activity databases and compared with the related works on each of these. The substantiating results demonstrate the viability and high-accuracy of our modeling technique in characterizing and classifying different activities.}, number={8}, journal={IEEE TRANSACTIONS ON IMAGE PROCESSING}, author={Yi, Sheng and Krim, Hamid and Norris, Larry K.}, year={2012}, month={Aug}, pages={3416–3428} }
@inproceedings{yi_krim_norris_2012, title={Human activity modeling as Brownian motion on shape manifold}, volume={6667}, DOI={10.1007/978-3-642-24785-9_53}, abstractNote={In this paper we propose a stochastic modeling of human activity on a shape manifold. From a video sequence, human activity is extracted as a sequence of shape. Such a sequence is considered as one realization of a random process on shape manifold. Then Different activities are modeled by manifold valued random processes with different distributions. To solve the problem of stochastic modeling on a manifold, we first regress a manifold values process to a Euclidean process. The resulted process then could be modeled by linear models such as a stationary incremental process and a piecewise stationary incremental process. The mapping from manifold to Euclidean space is known as a stochastic development. The idea is to parallelly transport the tangent along curve on manifold to a single tangent space. The advantage of such technique is the one to one correspondence between the process in Euclidean space and the one on manifold. The proposed algorithm is tested on database [5] and compared with the related work in [5]. The result demonstrate the high accuracy of our modeling in characterizing different activities.}, booktitle={Scale space and variational methods in computer vision}, author={Yi, S. and Krim, H. and Norris, L. K.}, year={2012}, pages={628–639} }
@inproceedings{yi_krim_norris_2011, title={A invertible dimension reduction of curves on a manifold}, DOI={10.1109/iccvw.2011.6130412}, abstractNote={In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential geometry tools such as moving frame and parallel transportation are successfully adapted into the dimension reduction problem of high dimensional curves. Intuitively, instead of searching for a global flat subspace for curve embedding, we deployed a sequence of local flat subspaces adaptive to the geometry of both of the curve and the manifold it lies on. In practice, the experimental results of the dimension reduction and reconstruction algorithms well illustrate the advantages of the proposed theoretical innovation.}, booktitle={2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)}, author={Yi, S. and Krim, H. and Norris, L. K.}, year={2011} }
@article{norris_2001, title={n-symplectic algebra of observables in covariant Lagrangian field theory}, volume={42}, ISSN={["1089-7658"]}, DOI={10.1063/1.1396835}, abstractNote={n -symplectic geometry on the adapted frame bundle λ:LπE→E of an n=(m+k)-dimensional fiber bundle π:E→M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ:LπE→J1π we lift a Lagrangian L:J1π→R to a Lagrangian L≔ρ*(L):LπE→R, and then use L to define a “modified n-symplectic potential” θ̂L on LπE, the Cartan–Hamilton–Poincaré (CHP) Rn-valued 1-form. If the lifted Lagrangian is nonzero, then (LπE,dθ̂L) is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL≔φL(LπE) is a submanifold of LE, and (QL,d(θ̂|QL)) is shown to be an n-symplectic manifold. We prove the theorem that θ̂L=φL*(θ|QL), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE,dθ̂L). To find the reduced n-symplectic algebra on QL we set up the equations of n-symplectic reduction, and apply the general theory to the model of a k-tuple of massless scalar fields on Minkowski space–time. The formalism set forth in this paper lays the ground work for a geometric quantization theory of fields.}, number={10}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Norris, LK}, year={2001}, month={Oct}, pages={4827–4845} }
@article{mclean_norris_2000, title={Covariant field theory on frame bundles of fibered manifolds}, volume={41}, ISSN={["0022-2488"]}, DOI={10.1063/1.1288797}, abstractNote={We show that covariant field theory for sections of π : E→M lifts in a natural way to the bundle of vertically adapted linear frames LπE. Our analysis is based on the fact that LπE is a principal fiber bundle over the bundle of 1-jets J1π. On LπE the canonical soldering 1-forms play the role of the contact structure of J1π. A lifted Lagrangian ℒ: LπE→R is used to construct modified soldering 1-forms, which we refer to as the Cartan–Hamilton–Poincaré 1-forms. These 1-forms on LπE pass to the quotient to define the standard Cartan–Hamilton–Poincaré m-form on J1π. We derive generalized Hamilton–Jacobi and Hamilton equations on LπE, and show that the Hamilton–Jacobi and canonical equations of Carathéodory–Rund and de Donder–Weyl are obtained as special cases.}, number={10}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={McLean, M and Norris, LK}, year={2000}, month={Oct}, pages={6808–6823} }
@book{davis_chu_mcconnell_dolan_norris_ortiz_plemmon_ridgeway_scaife_stewart_et al._1998, title={Cornelius Lanczos: Collected published papers with commentaries}, ISBN={0929493003}, publisher={Raleigh, NC: College of Physical and Mathematical Sciences, North Carolina State University}, author={Davis, W. R. and Chu, M. T. and McConnell, J. R. and Dolan, P. and Norris, L. K. and Ortiz, E. and Plemmon, R. J. and Ridgeway, D. and Scaife, B.K.P. and Stewart, W. J. and et al.}, year={1998} }
@article{norris_1997, title={Schouten-Nijenhuis brackets}, volume={38}, ISSN={["0022-2488"]}, DOI={10.1063/1.531981}, abstractNote={The Poisson and graded Poisson Schouten–Nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields, respectively, on an n-dimensional manifold M are shown to be n-symplectic. This is accomplished by showing that both brackets may be defined in a unified way using the n-symplectic structure on the bundle of linear frames LM of M. New results in n-symplectic geometry are presented and then used to give globally defined representations of the Hamiltonian operators defined by the Schouten–Nijenhuis brackets.}, number={5}, journal={JOURNAL OF MATHEMATICAL PHYSICS}, author={Norris, LK}, year={1997}, month={May}, pages={2694–2709} }