This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA∈KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of f(n) K (v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

Population dynamics is the dominant branch of mathematical biology. The first model for population dynamics was developed by Thomas Malthus. A more complicated model was developed by Pierre Fran&ccedil / ois Verhulst and it is called the logistic equation. Our aim in this thesis is to extend the models using piecewise constant arguments and to find the conditions when the models have fixed points, periodic solutions and chaos with investigation of stability of periodic solutions.

This dissertation addresses the new models in mathematical neuroscience: artificial neural networks, which have many similarities with the structure of human brain and the functions of cells by electronic circuits. The networks have been investigated due to their extensive applications in classification of patterns, associative memories, image processing, artificial intelligence, signal processing and optimization problems. These applications depend crucially on the dynamical behaviors of the networks. In this thesis the dynamics are presented by differential equations with discontinuities: differential equations with piecewise constant argument of generalized type, and both impulses at fixed moments and piecewise constant argument. A discussion of the models, which are appropriate for the proposed applications, are also provided. Qualitative analysis of existence and uniqueness of solutions, global asymptotic stability, uniform asymptotic stability and global exponential stability of equilibria, existence of periodic solutions and their global asymptotic stability for these networks are obtained. Examples with numerical simulations are given to validate the theoretical results. All the properties are rigorously approved by using methods for differential equations with discontinuities: existence and uniqueness theorems / stability analysis through the Second Lyapunov method and linearization. It is the first time that the problem of stability with the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type is investigated. Despite the fact that these equations are with deviating argument, stability criteria are merely found in terms of Lyapunov functions.

Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA[set]KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of fK(n)(v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

自民國91年，國內法令准許私募，私募是指公司的內部人，透過協商，尋求願意投資公司的投資人，把公司的股票，售給少數的投資人，不需經過承銷商，揭露也比較少。 國外的文獻，有些學者實證的結果，私募宣告時有正的異常報酬，但有些學者實證的結果發現有負的異常報酬，本篇主要探討國內宣告私募時，對公司價值的影響，以及國內公司大多屬於家族企業，在這麼高的控制權之下，股權集中度對於私募發行公司的價值又有什麼影響？ 實證的結果發現，國內公司宣告私募時有正的異常報酬，除了資訊不對稱會影響公司價值之外，本文還發現，公司在私募時，家族集團的控股如果低於30%時，增加公司持股，可以增加公司的價值，但當家族集團的股權超過30%時，就不明顯了。 此外，當股權集中度在10%~30%時，公司價值的變化與股權集中度成反比，顯示國內股權集中度在10%~30%時，符合利益掠奪假（Entrenchment Hypothesis），增加公司的股權集中度反而會減少公司的價值。

In this document we review the status of existing techniques for nonlinear model order reduction by investigating how well these techniques perform for typical industrial needs. In particular the TPWL-method (Trajectory Piecewise Linear-method) and the POD-approach (Proper Orthogonal Decomposion) is taken under consideration. We address several questions that are (closely) related to both the theory and application of nonlinear model order reduction techniques. The goal of this document is to provide an overview of available methods together with a classification of nonlinear problems that in principle could be handled by these methods.

Το θέμα της διπλωματικής μεταπτυχιακής εργασίας είναι οι συνήθεις μη γραμμικότητες και οι εφαρμογές τους. Σχεδιάζονται, εξομοιώνονται και υλοποιούνται στην πράξη απλοί και σύνθετοι μη γραμμικοί αντιστάτες τμηματικής γραμμικότητας (PieceWise-Linear – PWL). Ιδιαίτερη προσοχή δίνεται στην υλοποίηση της διόδου Chua καθώς και στο σχεδιασμό και υλοποίηση του ίδιου του χαοτικού κυκλώματος Chua. Ταυτόχρονα δίνονται συνοπτικά στοιχεία θεωρίας χάους. Επιπλέον, μελετάται και υλοποιείται υπερχαοτικός ταλαντωτής κατάλληλος για συγχρονισμό και εφαρμογή σε ασφαλείς επικοινωνίες. Αυτός ο υπερχαοτικός ταλαντωτής βασίζεται σε έναν LC ταλαντωτή και το γνωστό Deliyannis SAB συζευγμένα μέσω μιας διόδου. Σε όλα τα παραπάνω πρέπει να προστεθεί η συγκέντρωση όλης της σχετικής βιβλιογραφίας. / The subject of this diploma thesis is to study usual nonlinearities and their applications. Simple and composite nonlinear piecewise-linear resistors have been designed, simulated and implemented. A great care is shown towards implementing the Chua’s diode and designing and implementing the chaotic Chua’s circuit itself. At the same time some basics of chaos theory are given. Moreover, a hyperchaotic oscillator is studied which is suitable for synchronization and application in secure communications. This hyperchaotic oscillator is based on a LC oscillator and the well-known Deliyannis SAB coupled by means of a diode. To all above, one should add the very rich bibliography, which has now been accumulated for the benefit of all concerned in the Electronics Laboratory.

Good quality management decisions often rely on the evaluation and interpretation of data. One of the most popular ways to investigate possible relationships in a given data set is to follow a process of fitting models to the data. Regression models are often employed to assist with decision making. In addition to decision making, regression models can also be used for the optimization and prediction of data. The success of a regression model, however, relies heavily on assumptions made by the model builder. In addition, the model may also be influenced by the presence of outliers; a more robust model, which is not as easily affected by outliers, is necessary in making more accurate interpretations about the data. In this research study robust techniques for regression models with minimal assumptions are explored. Mathematical programming techniques such as linear programming, mixed integer linear programming, and piecewise linear regression are used to formulate a nonlinear regression model. Outlier detection and smoothing techniques are included to address the robustness of the model and to improve predictive accuracy. The performance of the model is tested by applying it to a variety of data sets and comparing the results to those of other models. The results of the empirical experiments are also presented in this study. / Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2011.

Good quality management decisions often rely on the evaluation and interpretation of data. One of the most popular ways to investigate possible relationships in a given data set is to follow a process of fitting models to the data. Regression models are often employed to assist with decision making. In addition to decision making, regression models can also be used for the optimization and prediction of data. The success of a regression model, however, relies heavily on assumptions made by the model builder. In addition, the model may also be influenced by the presence of outliers; a more robust model, which is not as easily affected by outliers, is necessary in making more accurate interpretations about the data. In this research study robust techniques for regression models with minimal assumptions are explored. Mathematical programming techniques such as linear programming, mixed integer linear programming, and piecewise linear regression are used to formulate a nonlinear regression model. Outlier detection and smoothing techniques are included to address the robustness of the model and to improve predictive accuracy. The performance of the model is tested by applying it to a variety of data sets and comparing the results to those of other models. The results of the empirical experiments are also presented in this study. / Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2011.