Twisted K-homology,Geometric cycles and T-duality

Twisted $K$-homology corresponds to $D$-branes in string theory. In this paper we compare two different models of geometric twisted $K$-homology and get their equivalence. Moreover, we give another description of geometric twisted $K$-homology using bundle gerbes. We establish some properties of geometric twisted $K$-homology. In the last part we construct $T$-duality isomorphism for geometric twisted $K$-homology.Comment: We modify the statement about the six-term exact sequence of geometric twisted $K$-homology. Some Typos are corrected. Comments are welcome

On the Laplace-Beltrami operator on compact complex spaces

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.Comment: To appear on Transactions of the American Mathematical Society. Comments are welcome. arXiv admin note: text overlap with arXiv:1607.0028

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\Omega_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\overline{\mathfrak{d}}_{m,0}$, compactness of the inclusion $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})\hookrightarrow L^2\Omega^{m,0}(reg(X),h)$ where $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})$, the domain of $\overline{\mathfrak{d}}_{m,0}$, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$ with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces.Comment: Final version. To appear on Advances in Mathematic

Anisotropic Wetting Property of Superhydrophobic Surfaces and Electrokinetic Flow on Liquid-Filled Surfaces

Understanding the wetting property of rough surface is critical in guiding droplets and novel superhydrophobic surface design. The Cassie-Baxter model and Wenzel model are always used to describe the totally non-wetting and completely wetting states, however, there were few discussions about the intermediate state. Through measuring the contact angles of groove patterned surfaces in different groove orientations, the anisotropic wetting properties of groove patterned superhydrophobic surface were investigated. The degree of water penetration into the grooves was experimentally observed and it was found that the degree of water penetration was different with groove orientations, which would affect the corresponding contact angle. Besides guiding droplets, superhydrophobic surfaces are also very important in microfluidic due to their ability to generate fluid slip and flow enhancement. After a deeper understanding of the wetting property of groove patterned superhydrophobic surface, I further investigated its important role in microfluidics. In this dissertation, I mainly focus on electrokinetics on groove patterned surface and liquid-filled slippery surfaces, a new kind of surface by filling low surface tension oil into the interstices of groove patterned surfaces. I experimentally measured the streaming potential on flat parylene surface, air-filled groove patterned surface and liquid-filled surfaces and compared their effects in streaming potential enhancement. The liquid-filled surfaces were shown to be able to enhance the generated streaming potential due to its slippery property and liquid-oil interface charges. As the electrokinetic on liquid-filled surfaces is a new phenomenon, the underlying physics is still not clear. I further investigated the influences of filled oil properties and groove orientation on streaming potentials and fluid slip. Oils with different densities, viscosities, dielectric constant, conductivities and surface tensions were filled into the interstices of groove patterned surfaces to make different types of liquid-filled surfaces. The streaming potentials on liquid-filled surfaces with different oils were experimentally measured. An empirical relationship between streaming potential and oil properties was found and the effects of electrical properties, such as interface charge density and dielectric constant of filled oil, on fluid slip were also studied. Finally, the groove orientation was varied to study the tensorial effects on streaming potential. Through both streaming potential measurement and theoretical analysis, it was found that the streaming potential at 45° was always smaller than the arithmetic mean of those at 0° and 90°, and the pressure gradient in the transvers direction generated by tensorial effects was important in the streaming potential modification. My work will be important in guiding droplets, flow patterning, lab-on-chip devices and the development of electrokientic based power sources

A direct manipulation object-oriented environment to support methodology-independent CASE tools : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Computer Science at Massey University

The aim of the thesis is research into application of direct-manipulable OO graphical environments to the development of methodology-independent CASE tools. In this thesis, a Methodology-Independent Graphical OO CASE Environment (M1GOCE) is proposed. MIGOCE consists of three parts: OO Notation Workshop, OO Notation Repository and Universal OO Diagramming Tool. OO Notation Workshop is an OO graphical editor which is used to design existing and new notations; OO Notation Repository is a notation database that stores different notations designed by the notation workshop; Universal OO Diagramming Tool is an upper-CASE graphical environment, by which a user can draw arbitrary OO diagrams of different methodologies. The MIGOCE database management system provides OO notation sets management, OOA/OOD diagrams management and OO repository management for data integrity and sharing. MIGOCE has three outstanding characteristics: Methodology-independence, Directly-manipulable graphical environment and Easily-expanded program structure MIGOCE is completely methodology-independent. It not only supports existing OO methodologies, but also supports users' own notation designs. It provides support for mixing, updating existing methodologies or defining new ones. It typically allows the user to switch quickly different OO notation sets supported by corresponding methodologies for designing diagrams. Direct manipulation interfaces of MIGOCE enable it more flexible and distinctive. The user can easily add, delete, edit or show notation shapes, and get the system feedback very quick on the screen. The MIGOCE system itself is programmed using object-oriented programming language - C++. Its program structure enable the functions of itself easy to be modified and expanded. Although MIGOCE is a prototype, it provides a new way to develop the real methodology-independent OO CASE environment. So far, the way and style taken by MIGOCE have not been found in OO CASE literatures. This system gives a complete possibility of implementing a methodology-independent OO CASE tool and shows distinct effectiveness of such a tool in practice