Differential response of cholesterol based pyrimidine systems with oxyethylene type spacers to gelation and mesogen formation in the presence of alkali metal ions

A new series of lipophilic cholesteryl derivatives of 2,4,6-trichloro-pyrimidine-5-carbaldehyde has been synthesized. Oxyethylene spacers of variable lengths were inserted between the hydrogen bonding promoting pyrimidine core and the cholesteryl tail in order to understand their effect on the selfassembly of these compounds. Only compound 1a with the shortest spacer formed a gel in organic solvents such as n-butanol and n-dodecane. While other members (1b and c) having longer spacers led to sol formation and precipitation in n-butanol and n-dodecane respectively. The self-assembly phenomena associated with the gelation process were investigated using temperature-dependent UVVis and CD-spectroscopy. The morphological features of the freeze-dried gels obtained from different organic solvents were examined by scanning electron microscopy (SEM) and atomic force microscopy (AFM). The solid phase behaviours of these molecules and their associated alkali metal ion complexes were explored using polarized optical microscopy (POM) and differential scanning calorimetry (DSC). The molecular arrangements in the xerogel and in the solid state were further probed using a wide-angle Xray diffraction (WAXD) technique. Analysis of the wide-angle X-ray diffraction data reveals that this class of molecules adopts a hexagonal columnar organization in the gel and in the solid state. Each slice of these hexagonal columnar structures is composed of a dimeric molecular-assembly as a building block. Significant changes in the conformation of the oxyethylene chains could be triggered via the coordination of selected alkali metal ions. This led to the production of interesting metal ion promoted mesogenic behaviour.

Part II: Design of Well Doping Profile for Improved Breakdown and Mixed-Signal Performance of STI-Type DePMOS Device

Shallow-trench isolation drain extended pMOS (STI-DePMOS) devices show a distinct two-stage breakdown. The impact of p-well and deep-n-well doping profile on breakdown characteristics is investigated based on TCAD simulations. Design guidelines for p-well and deep-n-well doping profile are developed to shift the onset of the first-stage breakdown to a higher drain voltage and to avoid vertical punch-through leading to early breakdown. An optimal ratio between the OFF-state breakdown voltage and the ON-state resistance could be obtained. Furthermore, the impact of p-well/deep-n-well doping profile on the figure of merits of analog and digital performance is studied. This paper aids in the design of STI drain extended MOSFET devices for widest safe operating area and optimal mixed-signal performance in advanced system-on-chip input-output process technologies.

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Novel applications of BEM based Poisson level set approach

Accurate and efficient computation of the distance function d for a given domain is important for many areas of numerical modeling. Partial differential (e.g. HamiltonJacobi type) equation based distance function algorithms have desirable computational efficiency and accuracy. In this study, as an alternative, a Poisson equation based level set (distance function) is considered and solved using the meshless boundary element method (BEM). The application of this for shape topology analysis, including the medial axis for domain decomposition, geometric de-featuring and other aspects of numerical modeling is assessed. © 2011 Elsevier Ltd. All rights reserved.

Unified form language: A domain-specific language for weak formulations of partial differential equations

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies formultifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted. © 2014 ACM.

PREPARATION, CRYSTAL-STRUCTURE, AND VIBRATIONAL-SPECTRA OF PEROVSKITE-TYPE MIXED OXIDES LAM(Y)M'(1-Y)O(3) (M,M'=MN, FE, CO)

Three series of samples LaMnyCo1-yO3+/-lambda, LaFeyMn1-yO3+/-lambda, and LaFeyCo1-yO3+/-lambda (y = 0.0 to 1.0) with Perovskite structure were prepared by an explosion method different from the generally used ceramic techniques. The variation of crystal