Given a compact n-dimensional immersed Riemannian manifold Mn we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then Mn is homeomorphic to the sphere Sn. A consequence of our main theorems is a conjecture which is equivalent to Poincaré Conjecture. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces are ...

We provide a new construction of the modes of the Poincaré dodecahedral space S3/I∗. The construction uses the Hopf map, Maxwell’s multipole vectors and orbifolds. In particular, the *235-orbifold serves as a parameter space for the modes of S3/I∗, shedding new light on the geometrical significance of the dimension of each space of k-modes, as well as on the modes themselves.

This is a preliminary announcement of a controlled algebraic surgery theory, of the type first proposed by Quinn [1]. We define and study the -controlled L-groups Ln(X, pX , ), extending to L-theory the controlled K-theory of Ranicki and Yamasaki [4]. The most immediate application of the algebra to controlled geometric surgery is the controlled surgery obstruction: a normal map (f, b) : K → L ...

Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical syste...

The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincaré inequality but not the log-Sobolev one.

It is shown that a periodically forced impact oscillator may exhibit chaotic dynamics on two symbols, as well as an infinity of periodic solutions. Two cases are considered, depending if the impact velocity is finite or infinite. In the second case, the Poincaré map is well defined by continuation of the energy. The proof combines the study of phase plane curves together with the “stretching al...

The control method of Ott, Grebogi, and Yorke (OGY) is used to stabilize the unstable periodic orbits of a chaotic relay system. Small variations in the height of the relay output are used as control input. The influence of the control activation bound is studied in detail via the one-dimensional Poincaré map of the controlled system. The reduced sensitivity of the multi-step OGY method for hig...

Nonlinear dynamic model of the coupled lateral-torsional vibrations of a gear transmission system is developed, in which time-varying gear meshing stiffness, errors, and gaps are considered. The fundamental rules of the system’s multi-harmonic dynamic response have been obtained based on the time domain diagram, frequency domain diagram, phase plot, and Poincaré map. Therefore, estimation of th...

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a C-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most not...

Abstract We study numerically the properties of entanglement two interacting, or noninteracting, particles evolving in a regime quantum chaos Chirikov standard map. Such pairs can be viewed as on Einstein–Podolsky–Rosen chaos. The analysis is done with such tools Loschmidt echo and Poincaré recurrences presence absorption. obtained results show unusual features entropy spectrum Schmidt decompos...