Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.

Material Synthesis and Characterization on Low-Dimensional Cobaltates

In this thesis, results of the investigation of a new low-dimensional cobaltates Ba2-xSrxCoO 4 are presented. The synthesis of both polycrystalline and single crystalline compounds using the methods of conventional solid state chemical reaction and floating-zone optical furnace is first introduced. Besides making polycrystalline powders, we successfully, for the first time, synthesized large single crystals of Ba2CoO4. Single crystals were also obtained for Sr doped Ba2-xSrxCoO 4. Powder and single crystal x-ray diffraction results indicate that pure Ba2CoO4 has a monoclinic structure at room temperature. With Sr doping, the lattice structure changes to orthorhombic when x ≥ 0.5 and to tetragonal when x = 2.0. In addition, Ba2CoO4 and Sr2CoO4, have completely different basic building blocks in the structure. One is CoO4 tetrahedron and the later is CoO6 octahedron, respectively. Electronic and magnetic properties were characterized and discussed. The magnetic susceptibility, specific heat and thermal conductivity show that Ba2CoO4 has an antiferromagnetic (AF) ground state with an AF ordering temperature TN = 25 K. However, the magnitude of the Néel temperature TN is significantly lower than the Curie-Weiss temperature (:&thetas;: ∼ 110 K), suggesting either reduced-dimensional magnetic interactions and/or the existence of magnetic frustration. The AF interaction persists in all the samples with different doping concentrations. The Néel temperature doesn't vary much in the monoclinic structure regime but decreases when the system enters orthorhombic. Magnetically, Ba2CoO4 has an AF insulating ground state while Sr2CoO4 has a ferromagnetic (FM) metallic ground state. Neutron powder refinement results indicate a magnetic structure with the spin mostly aligned along the a-axis. The result from a μ-spin rotation/relaxation (μ+SR) experiment agrees with our refinement. It confirms the AF order in the ab -plane. We also studied the spin dynamics and its anisotropy in the AF phase. The results from inelastic neutron scattering show that spin waves have a clear dispersion along a-axis but not along c-axis, indicating spin anisotropy. This work finds the strong spin-lattice coupling in this novel complex material. The interplay between the two degrees of freedom results an interesting phase diagram. Further research is needed when large single crystal samples are available.

Fibrations and contact structures

We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms.

Development and Application of a Multi-dimensional Strategy for Measuring Fidelity of Implementation of a Districtwide Intervention

This symposium describes a multi-dimensional strategy to examine fidelity of implementation in an authentic school district context. An existing large-district peer mentoring program provides an example. The presentation will address development of a logic model to articulate a theory of change; collaborative creation of a data set aligned with essential concepts and research questions; identification of independent, dependent, and covariate variables; issues related to use of big data that include conditioning and transformation of data prior to analysis; operationalization of a strategy to capture fidelity of implementation data from all stakeholders; and ways in which fidelity indicators might be used.