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The aim of our work is to investigate how a relatively small set of clauses can be transformed into a running program capable of solving a number of problems. The problems are chosen from the domain of simple arithmetic, algebra and letter series completion. We describe how the problems are solved, how errors are detected and corrected by modification of the existing clauses. Various techniques useful in the process of error detection and correction are described in detail. Two types of errors are dealt with: selection errors arising due to incorrect selection of clauses, and instantiation errors arising when the partial results (subgoals) are not specific enough. The system described was implemented in Prolog.

A novel nanolithography technique is developed for nanoscale patterning of polymers: atomic force microscope assisted electrostatic nanolithography (AFMEN). In AFMEN, nanostructures are generated by mass transport of polymeric material within an initially uniform, planar film. The combination of localized softening of attolitres of polymer, a strongly non-uniform electric field to polarize and manipulate the soften dielectric, and single-step process methodology using conventional atomic force microscopy (AFM), establishes a new paradigm for polymer nanolithography. We develop a basic theoretical understanding of the processes associated with AFMEN and present modeling of the relevant phenomena. The analysis of polymer heating indicates that the AFMEN resolution is not directly limited by the radius of the AFM tip, which is distinctly different from alternative AFM–based lithographic techniques. Instead, the feature size depends critically on the thermal characteristics of the polymer, such as its thermal conductivity. The dielectric properties of the polymer play a secondary role, affecting the magnitude of the electric field but do not directly impact the feature size. The nanostructure shape is determined by the competition between the ponderomotive forces acting on the dielectric material in non–uniform electric field, and the polymer surface tension. An exact analytical solution, as well as numerical solutions, are determined for the electric field distribution in the tip–sample junction, which allow prediction of the nanostructure geometry. The response of the AFM tip during the nanolithography process is investigated. An analysis of the free energy of the system, comprising AFM tip, sample surface and water meniscus, shows that the tip is spontaneously lifted away from the polymer surface. The mechanical work required to lift the tip is drawn from the energy of electric field. In addition, water condensation in the proximity of nanoscale asperities such as an AFM tip is studied. A general framework based on the density functional theory (DFT) is developed and applied to describe the liquid meniscus condensation, providing the first (to the best of our knowledge) molecular–level theory of water condensation on the nanoscale in the presence of external electric fields.

The central goal of this work is to define and analyze least squares finite element methods for the Stokes and Navier-Stokes equations that are practical and optimal in a systematic and rigorous way. To accomplish this task we begin by developing the least squares theory for the linear Stokes equations. We introduce least squares methods based on the minimization of functionals that involve residuals of the equations of an equivalent first order formulation for the Stokes problem. We show that for the Stokes equations there are two general types of boundary conditions. For the first type, practical least squares methods can be defined and analyzed in a fairly standard way, based on application of the Agmon, Douglis and Nirenberg a priori estimates. For the second type of boundary conditions this task is more difficult and involves mesh dependent (weighted) least squares functionals. Among the main results are the optimal error estimates for the weighted least squares method in two and three space dimensions. Then, we formulate two least squares methods for the nonlinear Navier-Stokes equations written as a first order system. We consider the first method as a conforming discretization of an abstract nonlinear problem and the second weighted one, which is more practical, as a nonconforming discretization of the same abstract problem. As a result, the analysis of the first method fits into the framework of the approximation theory of Brezzi, Rappaz and Raviart and the analysis of the second does not. Thus, we develop an abstract approximation theory that is suitable for nonconforming discretizations of the abstract problem. The central result is based on the application of our abstract theory to the weighted least squares method. We prove that this method results in optimally accurate approximations for the Navier-Stokes equations. We believe that these error analyses of Chapter are the first treatment of a least squares formulation for a nonlinear problem in the current literature. We then discuss various implementation issues, including theoretical and numerical estimates of condition numbers and the presentation of numerical examples. In particular, we study the numerical convergence rates of various implementations of least squares methods and demonstrate that the weights are necessary for the optimal rates to hold. Finally, we compare numerical results for the driven cavity flow problem with some benchmark results reported in the literature.