Tetrahedral tellurate

A tetraethylammonium (TEA) salt of TeO 4 (2-) was synthesized by reacting Te(OH) 6 with tetraethylammonium hydroxide. X-ray structural analysis of (TEA) 2TeO 4.2H 2O confirmed that this new compound consists of discrete TeO 4 (2-) anions of distorted tetrahedral structure together with TEA cations and water molecules of crystallization [ a = 8.0820(9) A, b = 13.7730(12) A, c = 20.1590(18) A, V = 2244.0(4) A (3), Z = 4, and space group Pccb]. The (125)Te NMR spectrum indicated that water reacts with the TeO 4 (2-) anion to regenerate octahedral tellurate species in solution.     

Nonlinear dynamics and bifurcations in external feedback control of microcantilevers in atomic force microscopy

We study nonlinear dynamics and bifurcations in external feedback control of vibrating microcantilevers in atomic force microscopy. The efficiency and validity of the control methodology for microcantilevers was demonstrated and abundant nonlinear phenomena were observed in a previous numerical study. Using the averaging method and center manifold theory, we analyze a degenerate bifurcation of codimension two to explain a key feature of the previous numerical results. Numerical examples are also given, in which theoretical results are compared with numerical computations for bifurcations and unstable manifolds.     

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov?s method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.     

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.     

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose is a connected Riemann surface. Let denote the homeomorphism group of with the compact-open topology, and denote the subgroup of quasiconformal mappings of onto itself, and let and denote the identity components of and respectively. In this paper we show that the pair is an -manifold, and determine their topological types.

     

Icosaniobate: a new member of the isoniobate family

Icosaniobate [Nb20O54]8- was synthesized by reacting [H4Nb6O19]4- with NO in tetrahydrofuran or MeNO2. A single-crystal X-ray diffraction study of its n-tetrabutylammonium salt [a = 17.7284(18) A, b = 33.542(3) A, c = 34.316(2) A, Z = 4, and space group P22(1)2(1)] revealed a dimeric structure where two decaniobate ions are condensed sharing two terminal O atoms. Unlike that in [(NbW5O18)2O]4-, the Nb-O-Nb bridges in icosaniobate are bent. The nonlinear bridging reduces the maximum possible symmetry of the dimeric anion to mm2, which it closely approximates.     

A novel method for analyzing energy relaxation in condensed phases using nonequilibrium molecular dynamics simulations: application to the energy relaxation of intermolecular motions in liquid water

We present a novel method to investigate energy relaxation processes in condensed phases using nonequilibrium molecular dynamics simulations. This method can reveal details of the time evolution of energy relaxation like two-color third-order IR spectroscopy. Nonetheless, the computational cost of this method is significantly lower than that of third-order response functions. We apply this method to the energy relaxation of intermolecular motions in liquid water. We show that the intermolecular energy relaxation in water is characterized by four energy transfer processes. The structural changes of the liquid associated with the energy relaxation are also analyzed by the nonequilibrium molecular dynamics technique.     

Groups of measure-preserving homeomorphisms of noncompact 2-manifolds

Suppose M is a noncompact connected 2-manifold and μ is a good Radon measure of M with μ(∂M)=0. Let H(M) denote the group of homeomorphisms of M equipped with the compact-open topology and H₀(M) denote the identity component of H(M). Let H(M;μ) denote the subgroup of H(M) consisting of μ-preserving homeomorphisms of M and H₀(M;μ) denote the identity component of H(M;μ). We use results of A. Fathi and R. Berlanga to show that H₀(M;μ) is a strong deformation retract of H₀(M) and classify the topological type of H₀(M;μ).