The investigation of this thesis is introduced by finding all the relationships existing among the various types of set functions. Characterizations of the different set functions will be formulated in order to facilitate the construction and identification of each type of set function considered. This problem has important applications to measure theory and integration.Ball State UniversityMuncie, IN 47306

In this thesis we study the problem of stabilizing smooth embedded submanifolds in the state space of smooth, nonlinear, autonomous, deterministic control-affine systems. Our motivation stems from a realization that important applications, such as path following and synchronization, are best understood in the set stabilization framework. Instead of directly attacking the above set stabilization problem, we seek feedback equivalence of the given control system to a normal form that facilitates control design. The process of putting a control system into the normal form of this thesis is called transverse feedback linearization. When feasible, transverse feedback linearization allows for a decomposition of the nonlinear system into a “transverse” and a “tangential” subsystem relative to the goal submanifold. The dynamics of the transverse subsystem determine whether or not the system’s state approaches the submanifold. To ease controller design, we ask that the transverse subsystem be linear time-invariant and controllable. The dynamics of the tangential subsystem determine the motion on the submanifold. The main problem considered in this work, the local transverse feedback linearization problem (LTFLP), asks: when is such a decomposition possible near a point of the goal submanifold? This problem can equivalently be viewed as that of finding a system output with a well-defined relative degree, whose zero dynamics manifold coincides with the goal submanifold. As such, LTFLP can be thought of as the inverse problem to input-output feedback linearization. We present checkable, necessary and sufficient conditions for the existence of a local coordinate and feedback transformation that puts the given system into the desired normal form. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set. When the goal submanifold is diffeomorphic to Euclidean space, we present sufficient conditions for feedback equivalence in a tubular neighbourhood of it. These results are used to develop a technique for solving the path following problem. When applied to this problem, transverse feedback linearization decomposes controller design into two separate stages: transversal control design and tangential control design. The transversal control inputs are used to stabilize the path, and effectively generate virtual constraints forcing the system’s output to move along the path. The tangential inputs are used to control the motion along the path. A useful feature of this twostage approach is that the motion on the set can be controlled independently of the set stabilizing control law. The effectiveness of the proposed approach is demonstrated experimentally on a magnetically levitated positioning system. Furthermore, the first satisfactory solution to a problem of longstanding interest, path following for the planar/vertical take-off and landing aircraft model to the unit circle, is presented. This solution, developed in collaboration with Luca Consolini and Mario Tosques at the University of Parma, is made possible by taking a set stabilization point of view.

The study of combinatorial patterns of words has raised great interest since the early 20th century. In this master's thesis presentation we study two combinatorial patterns. The first pattern is “abelian k-th power free” and the second one is “representability of sets of words of equal length”. For the first pattern we study the context-freeness of non-abelian k-th powers. A word is a non-abelian k-th power if it cannot be factorized in the form w1w2...wk where the wi are permutations of w1 for 2 ≤ i ≤ k. We show that neither the language of non-abelian squares nor the language of non- abelian cubes is context-free. For the second pattern we study the representability of a set of words of fixed length. A set S of words of length n is representable if there exists some word w such that the set of length-n factors of w equals S. We will give lower and upper bounds for the number of such representable sets. Furthermore, we study a variation of the problem: we fix a length t, and try to evaluate the number of sets of words of length n such that there exists some word w of length t such that the set of length-n factors of w equals S. We give a closed-form formula in the case where n ≤ t < 2n. In particular, we give a characterization on two distinct words having the same subset of length-n factors.

This thesis discusses how to use topology and shape optimization, specifically the level set method, for innovative design. The level set method is a numerical algorithm that simulates the expansion of dynamic implicit surfaces. In this research, the equations for manufacturability are generated and solved through use of the level set method joined with the COMSOL multi-physics package. Specific constraints are added to make the optimization practical for engineering design. The resulting method was applied to design the best underlying support structure, conforming to both curvature and manufacturability constraints, for the longerons used with the International Space Station solar panels.

In this thesis, we study the behaviour of Lovasz and Schrijver's lift-and-project operators N and N_0 while being applied recursively to the fractional stable set polytope of a graph. We focus on two related conjectures proposed by Liptak and Tuncel: the N-N_0 Conjecture and Rank Conjecture. First, we look at the algebraic derivation of new valid inequalities by the operators N and N_0. We then present algebraic characterizations of these valid inequalities. Tightly based on our algebraic characterizations, we give an alternate proof of a result of Lovasz and Schrijver, establishing the equivalence of N and N_0 operators on the fractional stable set polytope. Since the above mentioned conjectures involve also the recursive applications of N and N_0 operators, we also study the valid inequalities obtained by these lift-and-project operators after two applications. We show that the N-N_0 Conjecture is false, while the Rank Conjecture is true for all graphs with no more than 8 nodes.

The present thesis investigated how words are processed within the context of visual search. Both explicit and implicit measures were used to assess whether spatial attention is a prerequisite for words to undergo processing. In the explicit search task, subjects searched a display and indicated whether a word was present or absent among nonword distractors. In the implicit task, priming was employed to index word processing. Subjects viewed the same search displays that were used in the explicit task, however, the displays were presented briefly and were followed by a single target letter string to which subjects performed a lexical decision. In Experiments 3 through 6, in which the target was always presented at fixation, no priming was evident. In Experiments 7 and 8 when the location of the target moved from trial to trial, priming was observed. It is argued that attentional resources are narrowly allocated to a location in visual space when target location is certain but diffusely allocated when target location is uncertain. Furthermore, processing only occurs for words that fall within the suffusion of this strategically pliable attentional beam. The results are also interpreted within the domains of perceptual cuing and attentional capture.

This thesis presents the theory of affine algebraic sets defined over an arbitrary field K. We define basic concepts such as the Zariski topology, coordinate ring of functions, regular functions, and dimension. We are interested in the relationship between the geometry of an affine algebraic set over a field K and its geometry induced by the algebraic closure of K. Various versions of Hilbert-Nullstellensatz are presented, introducing a new variant over finite fields. Examples are provided throughout the paper and a question on the dimension of irreducible affine algebraic sets is formulated.

In this thesis, we study the behaviour of Lovasz and Schrijver's lift-and-project operators N and N_0 while being applied recursively to the fractional stable set polytope of a graph. We focus on two related conjectures proposed by Liptak and Tuncel: the N-N_0 Conjecture and Rank Conjecture. First, we look at the algebraic derivation of new valid inequalities by the operators N and N_0. We then present algebraic characterizations of these valid inequalities. Tightly based on our algebraic characterizations, we give an alternate proof of a result of Lovasz and Schrijver, establishing the equivalence of N and N_0 operators on the fractional stable set polytope. Since the above mentioned conjectures involve also the recursive applications of N and N_0 operators, we also study the valid inequalities obtained by these lift-and-project operators after two applications. We show that the N-N_0 Conjecture is false, while the Rank Conjecture is true for all graphs with no more than 8 nodes.

The present thesis investigated how words are processed within the context of visual search. Both explicit and implicit measures were used to assess whether spatial attention is a prerequisite for words to undergo processing. In the explicit search task, subjects searched a display and indicated whether a word was present or absent among nonword distractors. In the implicit task, priming was employed to index word processing. Subjects viewed the same search displays that were used in the explicit task, however, the displays were presented briefly and were followed by a single target letter string to which subjects performed a lexical decision. In Experiments 3 through 6, in which the target was always presented at fixation, no priming was evident. In Experiments 7 and 8 when the location of the target moved from trial to trial, priming was observed. It is argued that attentional resources are narrowly allocated to a location in visual space when target location is certain but diffusely allocated when target location is uncertain. Furthermore, processing only occurs for words that fall within the suffusion of this strategically pliable attentional beam. The results are also interpreted within the domains of perceptual cuing and attentional capture.

Sequence alignment has become one of the most common tasks in bioinformatics. Most of the existing sequence alignment methods use general scoring schemes. But these alignments are sometimes not completely relevant because they do not necessarily provide the desired information. It would be extremely difficult, if not impossible, to include any possible objective into an algorithm. Our goal is to allow a working biologist to augment a given alignment with additional information based on their knowledge and objectives.<p></p>In this thesis, we will formally define constraints and compatible constraint sets for an alignment which require some positions of the sequences to be aligned together. Using this approach, one can align some specific segments such as domains within protein sequences by inputting constraints (the positions of the segments on the sequences), and the algorithm will automatically find an optimal alignment in which the segments are aligned together.<p></p>A necessary prerequisite of calculating an alignment is that the constraints inputted be compatible with each other, and we will develop algorithms to check this condition for both pairwise and multiple sequence alignments. The algorithms are based on a depth-first search on a graph that is converted from the constraints and the alignment. We then develop algorithms to perform pairwise and multiple sequence alignments satisfying these compatible constraints.<p></p>Using straightforward dynamic programming for pairwise sequence alignment satisfying a compatible constraint set, an optimal alignment corresponds to a path going through the dynamic programming matrix, and as we are only using single-position constraints, a constraint can be represented as a point on the matrix, so a compatible constraint set is a set of points. We try to determine a new path, rather than the original path, that achieves the highest score which goes through all the compatible constraint set points. The path is a concatenation of sub-paths, so that only the scores in the sub-matrices need to be calculated. This means the time required to get the new path decreases as the number of constraints increases, and it also varies as the positions of the points change. It can be further reduced by using the information from the original alignment, which can offer a significant speed gain.<p></p>We then use exact and progressive algorithms to find multiple sequence alignments satisfying a compatible constraint set, which are extensions of pairwise sequence alignments. With exact algorithms for three sequences, where constraints are represented as lines, we discuss a method to force the optimal path to cross the constraint lines. And with progressive algorithms, we use a set of pairwise alignments satisfying compatible constraints to construct multiple sequence alignments progressively. Because they are more complex, we leave some extensions as future work.