Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c${}^c$ manifolds; and conversely, in the presence of a spin^c${}^c$ structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

用迈克耳逊干涉仪观察多种不同形式空气劈尖形成的牛顿环

本通过将迈克耳逊干涉仪两臂上的平面镜分别用凹面镜和凸面镜代换,可观察不同形式的空气劈尖形成的牛顿环,对动镜移动时干涉条纹的变化规律作了观察、分析和讨论,此改进后的装置可用于测量曲面镜曲率半径和比较两曲面镜曲率半径的大小。     

Four complementary theoretical approaches for the analysis of NMR paramagnetic relaxation

Four theoretical and computational approaches used at the University of Michigan to analyze NMR paramagnetic relaxation enhancement (NMR-PRE) are described. The primary objective of the theory is to describe the relationship of the NMR-PRE phenomenon to the electron spin hamiltonian and the spin energy level structure when zero field splitting interactions are significant. Four formulations of theory are discussed: (1) spin dynamics simulation; (2) the laboratory frame "constant H(S)" formulation; (3) the Molecular Frame "constant H(S)" formulation; and (4) the zfs-limit "constant H(S)" formulation. No single theoretical approach describes all important aspects of the relaxation mechanism in a fully satisfactory way. We use the four formulations in a complementary manner to provide as complete a picture of the relaxation mechanism as possible. We also discuss the integration of NMR-PRE theory and recently developed theory of electron spin relaxation which accounts for effects of the permanent zfs hamiltonian.     

Liouville—Ttype Theorems for Conformal Gaussian Curvature Equations in R^2

In this paper,we derive an elementary identity for smooth solutions of the following equation:△u(x) K(x)e^2u(x)=0 in R^2 and use it to get some global properties of the solutions.     

关于Cp,q圆型域的酉曲率

先用级数法得到圆型域Cp，q=E｜a｜z1｜^2/p＋b｜z2｜^2/q〈1，0〈a，b≤1，p，q为正整数｜的Bergman核函数显表达式，然后计算了它的Bergman度量，酉曲率和Ricci曲率。     

D = 5, N = 2 Geometric Higher Curvature Supergravity in the Second-Order Canonical Theory

The supersymmetric extension of the five-dimensional Chern–Simons gravity is studied from the Hamiltonian point of view. This model containing the Gauss–Bonnet term quadratic in the Riemann curvature is the gauge theory of the supergroup SU(2,2/1). In the first order, the theory has a polynomial structure, but the second-order leads to a nonpolynomial structure for both the Hamiltonian and the supersymmetry transformation rules of the fields. The second-order theory has the advantage that the apparent gauge degrees of freedom are unambiguously removed leaving only the physical ones. This important feature is analyzed by constructing the second-order Hamiltonian theory. The gauge invariances of the model and the generator of time evolution are found.     

Letter: Active Gravitational Mass and the Invariant Characterization of Reissner–Nordström Spacetime

We analyse the concept of active gravitational mass for Reissner-Nordström spacetime in terms of scalar polynomial invariants and the Karlhede classification. We show that while the Kretschmann scalar does not produce the expected expression for the active gravitational mass, both scalar polynomial invariants formed from the Weyl tensor, and the Cartan scalars, do.