Trees and tensors on Kähler manifolds

We present an organized method to convert between partial derivatives of metrics (functions) and covariant derivatives of curvature tensors (functions) on Kähler manifolds. Basically, it reduces the highly recursive computation in tensor calculus to the enumeration of certain trees with external legs.     

Generalized projective chaos synchronization of gyroscope systems subjected to dead-zone nonlinear inputs

This Letter presents a robust control scheme to generalized projective synchronization between two identical two-degrees-of-freedom heavy symmetric gyroscopes with dead zone nonlinear inputs. Because of the nonlinear terms of the gyroscope system, the system exhibits complex and chaotic motions. By the Lyapunov stability theory with control terms, two suitable sliding surfaces are proposed to ensure the stability of the controlled closed-loop system in sliding mode. Then, two sliding mode controllers (SMC) are designed to guarantee the hitting of the sliding surfaces even when the control inputs contain dead-zone nonlinearity. This method allows us to arbitrarily direct the scaling factor onto a desired value. Numerical simulations show that this method works very well for the proposed controller.     

Maximal degeneracy points of GKZ systems

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel´fand-Kapranov-Zelevinsky (GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exist certain special boundary points, which we called maximal degeneracy points, at which all solutions but one become singular.

     

Explicit construction of graph invariant for strongly pseudoconvex compact 3-dimensional rational CR manifolds

Let X be a strongly pseudoconvex compact 3-dimensional CR manifolds which bounds a complex variety with isolated singularities in some C^N. The weighted dual graph of the exceptional set of the minimal good resolution of V is a CR invariant of X; in case X has a tranversal holomorphic S¹ action, we show that it is a complete topological invariant of except for two special cases. When X is a rational CR manifolds, we give explicit algebraic algorithms to compute the graph invariant in terms of the ring structure of _k=0 m^k/m^k+1, where m is the maximal ideal of each singularity. An example is computed explicitly to demonstrate how the algorithms work.     

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

We consider random Schrödinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ _t the solution with initial data ψ₀. The space and time variables scale as \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) with 0 < κ <  κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ _t converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.     

Large Matrix Computations on Vector Computers

Preconditionings have proved to be a powerful technique for accelerating the rate of convergence of an iterative method. This paper, which is concerned with the conjugate gradient algorithm for large matrix computations, investigates an approximate polynomial preconditioning strategy. The method is particularly attractive for implementation on vector computers.     

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr‐Newman black hole geometry, the Dirac equation has no normalizable, time‐periodic solutions. A key tool is Chan‐drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.     

Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass

The definition of quasi-local mass for a bounded space-like region Ω in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type of flux integral on the boundary two-surface \({\Sigma=\partial \Omega}\) and should be independent of whichever space-like region \({\Sigma}\) bounds. An important idea which is related to the Hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi-local mass from the difference of the extrinsic geometries. This approach has been taken by Brown-York [4,5] and Liu-Yau [16,17] (see also related works [3,6,9,12,14,15,28,32]) to define such notions using the isometric embedding theorem into the Euclidean three space. However, there exist surfaces in the Minkowski space whose quasilocal mass is strictly positive [19]. It appears that the momentum information needs to be accounted for to reconcile the difference. In order to fully capture this information, we use isometric embeddings into the Minkowski space as references. In this article, we first prove an existence and uniqueness theorem for such isometric embeddings. We then solve the boundary value problem for Jang’s [13] equation as a procedure to recognize such a surface in the Minkowski space. In doing so, we discover a new expression of quasi-local mass for a large class of “admissible” surfaces (see Theorem A and Remark 1.1). The new mass is positive when the ambient space-time satisfies the dominant energy condition and vanishes on surfaces in the Minkowski space. It also has the nice asymptotic behavior at spatial infinity and null infinity. Some of these results were announced in [29].