Thesis (Ph. D.)--University of Wisconsin--Madison, 1959. / Typescript with manuscript equations. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 60-61).

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Geometric modeling of biomolecules plays an important role in the study of biochemical processes. Many simulation methods depend heavily on the geometric models of biomolecules. Among various studies, shape analysis is one of the most important topics, which reveals the functionalities of biomolecules.

Motivated by the desire to understand classical-quantum correspondences, we study concentration phenomena of approximate eigenfunctions of a semiclassical pseudodifferential operator $P(h)$. Such eigenfunctions appear as steady state solutions of quantum systems. Here we think of $h$ as being a small parameter such that $h^{2}$ is inversely proportional to the energy of such a system. As we understand classical mechanics to be the high energy (or small $h$) limit of quantum mechanics we expect the behaviour of eigenfunctions $u(h)$ for small $h$ to be related to properties of the associated classical system. In particular we study the connection between the classical flow and the quantum concentration properties. The flow, $(x(t),\xi(t))$, of a classical system describes the system's motion through phase space where $x(t)$ is interpreted as position and $\xi(t)$ is interpreted as momentum. In the quantum regime we think of an eigenfunction as being composed of highly localised packets moving along bicharacteristics of the classical flow. With this intuition we relate concentration of eigenfunctions in a region to the time spent by projections of bicharacteristics there. We use the $L^{p}$ norm of $u$ when restricted to submanifolds as a measure of concentration. A high $L^{p}$ norm particularly for small $p$ is indicative of concentration near the submanifold. We reduce the estimates on eigenfunctions to operator norm estimates on associated evolution operators. Using the semiclassical analysis methods developed in Chapter 3 we express these evolution operators as oscillatory integral operators. Chapter 2 covers the technical background needed to work with such operators. In Chapter 4 we determine eigenfunction estimates for eigenfunctions restricted to a smooth embedded submanifold $Y$ of arbitrary dimension. If $Y$ is a hypersurface, the greatest concentration occurs when there are bicharacteristics of the classical flow embedded in $Y$. In Chapter 5 we assume that projections of such bicharacteristics can be at worst simply tangent to $Y$ and thereby obtain better results for small values of $p$.

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The behavior of many systems of practical interest in communications and other areas is well modeled by a single server exponential queueing system in which the arrival and service rates are dependent upon the state of a Markov chain, the dynamics of which are independent of the queue length. Formal solution to such models based on Neuts's matrix geometric approach have appeared frequently in the literature. A major problem in using the matrix geometric approach is the computation of the rate matrix, which requires the solution of a matrix polynomial. In particular, computational times appear to be unpredictable and excessive for many problems of practical interest. Alternative techniques which employ eigenanalysis have been developed. These techniques are polynomially bounded and yield results very quickly compared to iterative routines. On the other hand, the class of systems to which the eigenanalysis based techniques apply have been somewhat restricted. In this thesis, we modify the eigenanalysis approach initially presented in order to remove some of these restrictions. / Master of Science

The new modeling method presented here is classified as a substructure synthesis (SS) technique. The distinction between the new SS method and the component mode synthesis formulation is that no transformation between local coordinates and generalized coordinates occurs in the new SS method. The advantage of this is a retention of physical insight into the model and the ability to form equations of motions directly with generalized coordinates. The new formulation differs from other substructure synthesis formulation because it satisfies geometric, natural, displacement and force constraints between substructures into one mathematical process, instead of using both kinematic chains and boundary condition approximation methods. This has the advantage of reducing the complexity of the integrals that are required in the computation. The new formulation also results in global eigenfunction approximations and global generalized coordinates, which eventually satisfies the inclusion principle which means eigenvalue estimates converge from above their actual values. The analysis method also facilitates the examination of boundary conditions in a unique manner. The method is unique because constraints are explicitly examined and selectively satisfied. This allows the identification of extraneous constraints and provides guidance in the selection of admissible functions. The new SS formulation may be divided into two steps. The first step is to satisfy geometric boundary conditions of substructures with appropriate admissible functions. The second step is the modification of these admissible functions to minimally satisfy geometric constraints imposed by the interaction of substructures. Natural constraints can also be satisfied to improve convergence to the exact eigenvalues. The MAF-SS formulation results in explicit knowledge of the constraints coupling substructures. Changing these constraints with active feedback results in a modified structure. The effect of active feedback of terms proportional to the coupling constraints is to lower the stiffness of the structure. This increases the isolation between substructures. The ability to improve isolation using this unique type of feedback is demonstrated. The concept of structural modification through substructure constraint alteration 1s applied to systems using a multivariable feedback method. This is accomplished by combing the MAF-SS method with a standard eigenstructure assignment technique. This method uses the MAF-SS formulation to define a system with substructure constraint eigenstructure properties, the active feedback gain that realizes these systems is calculated with an eigenstructure assignment method. The MAF-SS has application to active control formulation, the result of this control can be an improvement in substructure isolation. / Ph. D.

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