/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Recently introduced cost-effective depth sensors coupled with the real-time skeleton estimation algorithm of Shotton et al. [16] have generated a renewed interest in skeleton-based human action recognition. Most of the existing skeleton-based approaches use either the joint locations or the joint angles to represent a human skeleton. In this paper, we propose a new skeletal representation that explicitly models the 3D geometric relationships between various body parts using rotations and translations in 3D space. Since 3D rigid body motions are members of the special Euclidean group SE(3), the proposed skeletal representation lies in the Lie group SE(3)×…×SE(3), which is a curved manifold. Using the proposed representation, human actions can be modeled as curves in this Lie group. Since classification of curves in this Lie group is not an easy task, we map the action curves from the Lie group to its Lie algebra, which is a vector space. We then perform classification using a combination of dynamic time warping, Fourier temporal pyramid representation and linear SVM. Experimental results on three action datasets show that the proposed representation performs better than many existing skeletal representations. The proposed approach also outperforms various state-of-the-art skeleton-based human action recognition approaches.

http://ravitejav.weebly.com/uploads/2/4/7/2/24725306/action_recognition_presentation.pdf

Human Action Recognition by Representing 3D Skeletons as Points in a Lie Group

where dEλ is the projection valued measure associated with /^Δ\". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of various geometric quantities on M. As a particularly important example we have the heat kernel E(xl9 x2, t) — ke-\\2t. By use of the local parametrix and the standard elliptic estimates, one can show that for / > 0, E(xλ9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say) %2> ι s * the domain of all positive powers of Δ as a function of xλ; see e.g. [9]. In works of Garding [19] and Donnelly [16], upper estimates for E(xu x2, t) (and its derivatives) were given under the assumption that M has bounded geometry. They showed that as x2 -> oo, the behavior of E{xλ, x2, t) is roughly similar to that of the e-p 2(xx,x2)/4 Euclidean heat kernel, — (p(xx, x2) denotes distance). Recall that (4ττ/) M is said to have bounded geometry if the injectivity radius i(x) of the

/pdf/finite-propagation-speed-kernel-estimates-for-functions-of-4p1qmao5ak.pdf

Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds

I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

Lectures on Spaces of Nonpositive Curvature

It is known by classification theory of complex surfaces that CP 2 # nCp2(o <_ n < 8) and CP t • CP 1 are only compact differential 4-manifolds on which there is a complex structure with positive first Chern class. In [TY], the authors proved that for any n between 3 and 8, there is a compact complex surface M diffeomorphic to CP 2 # nCP 2 such that CI(M ) > 0 and M admits a K~ihlerEinstein metric. This paper is the continuation of my joint work with professor S.T. Yau [TY]. The main result of this paper is the following.

/pdf/on-calabi-s-conjecture-for-complex-surfaces-with-positive-55ntsylqys.pdf

On Calabi's conjecture for complex surfaces with positive first Chern class.

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/pdf/some-isoperimetric-inequalities-and-eigenvalue-estimates-2cxi3gqpq5.pdf

Some isoperimetric inequalities and eigenvalue estimates

/pdf/rigidity-and-the-distance-between-boundary-points-4q62yuxxwd.pdf

Rigidity and the distance between boundary points

Spaces with nonpositive curvature and their ideal boundaries

In this paper we consider the question of boundary rigidity for surfaces of nonpositive curvature. Given a compact manifold, M, with smooth boundary N, a riemannian metric go on M induces a nonnegative real valued function, do, on N x N where do(p, q) is the distance in (M, go) between p and q. A riemannian manifold (M, go) is called boundary rigid if for any riemannian manifold (M~, g~) with the same boundary, N, if d t = do then gt is isometric to go. This question was recently considered by the author in [C] where one was led to the quesiton: \"Are all SGM manifolds boundary rigid?\". The condition SGM is a condition on the boundary distance function do which roughly speaking is equivalent to the condition that all geodesic segments in M are the unique minimizing paths between the endpoints (see [C] for a precise definition.) By geodesic segments we mean geodesics that intersect the boundary at most at the boundary points (i.e. they do not \"graze\" the boundary at interior points of the segment.) Any compact subdomain in the interior of a convex manifold with possibly empty boundary (i.e. between any two points there is a unique geodesic) will be SGM. Hence, in particular, any subdomain of a complete simply connected manifold of nonpositive curvature will be SGM. Also any disk of nonpositive curvature will be SGM. In this paper we show:

Christopher B. Croke

Papers

Some isoperimetric inequalities and eigenvalue estimates

Rigidity and the distance between boundary points

Spaces with nonpositive curvature and their ideal boundaries

Rigidity for surfaces of non-positive curvature

A sharp four dimensional isoperimetric inequality.