This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex systems. This includes the spectral properties of such systems and in particular their influence on the systems dynamics. In the second part of this dissertation, or Chapter IV, we consider a new class of one-dimensional dynamical systems or functions with an eventual negative Schwarzian derivative motivated by some maps arising in neuroscience. To aid in understanding the interplay between the graph structure of a network and its dynamics we first introduce the concept of an isospectral graph reduction in Chapter I. Mathematically, an isospectral graph transformation is a graph operation (equivalently matrix operation) that modifies the structure of a graph while preserving the eigenvalues of the graphs weighted adjacency matrix. Because of their properties such reductions can be used to study graphs (networks) modulo any specific graph structure e.g. cycles of length n, cliques of size k, nodes of minimal/maximal degree, centrality, betweenness, etc. The theory of isospectral graph reductions has also lead to improvements in the general theory of eigenvalue approximation. Specifically, such reductions can be used to improved the classical eigenvalue estimates of Gershgorin, Brauer, Brualdi, and Varga for a complex valued matrix. The details of these specific results are found in Chapter II. The theory of isospectral graph transformations is then used in Chapter III to study time-delayed dynamical systems and develop the notion of a dynamical network expansion and reduction which can be used to determine whether a network of interacting dynamical systems has a unique global attractor. In Chapter IV we consider one-dimensional dynamical systems of an interval. In the study of such systems it is often assumed that the functions involved have a negative Schwarzian derivative. Here we consider a generalization of this condition. Specifically, we consider the functions which have some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. This includes both systems with regular as well as chaotic dynamic properties.

We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle and matrix, or on the nature of misfit. The particle shape is parameterized using a set of design variables which are the magnitudes of vectors from a reference point inside the particle to points on the particle/matrix interface. We use a sequential quadratic programming approach to carry out the optimization. Although this approach can be used to find equilibrium shapes of particles in three dimensional systems, we have presented the details of our formulation for two dimensional systems under plane strain conditions. Using our formulation, we have studied the equilibrium shapes in two dimensional systems with cubic anisotropy; the precipitate and matrix phases may have different elastic moduli, and the misfit may be dilatational or non-dilatational. The equilibrium shapes and their size dependence are analysed within the framework of symmetry-breaking shape transitions. These transitions are further characterized in terms their dependence on the cubic elastic anisotropy parameter, defined by A = 2C44/(C11 – C12), and on the modulus mismatch, defined by δ=μp/μm, where /μp and μm are the effective shear moduli of the precipitate and matrix phases, respectively. Depending on the type of misfit, the systems may be classified into the following four cases: Case A: For dilatational misfit, the equilibrium shapes in isotropic systems are circular (with an isotropic or I symmetry) at small sizes and undergo a transition at a critical size to become ellipse-like (with an orthorhombic or O symmetry). This I --O transition is continuous and is obtained only when the precipitate phase is softer than the matrix. These results are in good agreement with the analytical results of Johnson and Cahn. In cubic systems with dilatational misfit, the particles exhibit a transition from square-like shapes (with a tetragonal or T symmetry) at small sizes to rectangle-like shapes (with an O symmetry) at large sizes. This T -- O transition is continuous. It occurs even in systems with stiffer precipitates; however, it is forbidden for systems with δ >δC, where δ C represents a critical modulus mismatch. The critical size decreases with increasing cubic anisotropy (i.e., with increasing values of (A-1)/(A+1). The sides of the square-like and rectangle-like shapes are along the elastically soft directions. Case B: In these systems, the principal misfits e*xx and e*yy differ in magnitude but have the same sign. The precipitates at small sizes become elongated along the direction of lower misfit; this shape has an O symmetry. In systems with A > 1, they continue to become more elongated along the same direction, exhibiting no symmetry-breaking transition. However, in systems with A < 1, particles at large sizes are elongated along an intermediate direction between the direction of lower misfit and one of the elastically soft <11> directions; this shape has only a monoclinic or M symmetry. This O - M transition, in which the mirror symmetries normal to the x and y axes are lost, may be discontinuous or continuous. The critical size increases with δ (in the range 0.8 < δ <1.25), indicating that this transition would also be forbidden for systems with δ > δC. In systems with A < 1, the critical size decreases with increasing values of A-1/ A+1 Case C: In these systems, the principal misfits differ in both magnitude and sign, and the misfit strain tensor allows an invariant line along which the normal strain is zero. The precipitates at small sizes are elongated along the direction of lower absolute misfit, and possess an 0 symmetry. At large sizes, the mirror symmetries normal to the x and y axes are broken to yield shapes which are elongated along a direction between that of lower misfit and the invariant line. This 0 -> M transition is continuous and occurs in all the systems irrespective of the value of A The critical size increases with A and decreases with δ. Case D; The misfit in this case is a special form of that in Case C; the principal misfits have the same magnitude but opposite signs. The precipitates at small sizes have a square-like shape with its sides normal to the < 11 > axes, irrespective of the type of cubic anisotropy. At large sizes, they become rectangle-like with the long axis oriented along one of the <11> directions. Similar to Case C, this T - 0 transition is continuous and occurs in all the systems irrespective of the values of A. The critical size increases with A and decreases with δ. Thus, we have identified all the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two dimensional systems. We have identified their origin and nature, and characterized them in terms of their dependence on the anisotropy parameter and modulus mismatch.

Thèse de doctorat--Sciences de la terre--Rennes 1, 2004. / Contient aussi des textes en anglais. Bibliogr., 6 p. Notes bibliogr. Résumé en français et en anglais.

Thèse doctorat : Mathématiques : Strasbourg 1 : 2005. / Titre provenant de l'écran-titre. Bibliogr. 3 p.

Thèse doctorat : Mécanique : Villeurbanne, INSA : 2007. / Thèse rédigée en anglais. Titre provenant de l'écran-titre. Bibliogr. p. 145-155.

Reproduction de : Thèse de doctorat : Informatique : Paris 8 : 2007. / Titre provenant de l'écran-titre. Bibliogr. p. 133-137.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 99-114).

Abstract This study analyzed treatment of geometric transformations in presently available middle grades (6, 7, 8) student mathematics textbooks. Fourteen textbooks from four widely used textbook series were evaluated: two mainline publisher series, Pearson (Prentice Hall) and Glencoe (Math Connects); one National Science Foundation (NSF) funded curriculum project textbook series, Connected Mathematics 2; and one non-NSF funded curriculum project, the University of Chicago School Mathematics Project (UCSMP). A framework was developed to distinguish the characteristics in the treatment of geometric transformations and to determine the potential opportunity to learn transformation concepts as measured by textbook physical characteristics, lesson narratives, and analysis of student exercises with level of cognitive demand. Results indicated no consistency found in order, frequency, or location of transformation topics within textbooks by publisher or grade level. The structure of transformation lessons in three series (Prentice Hall, Glencoe, and UCSMP) was similar, with transformation lesson content at a simplified level and student low level of cognitive demand in transformation tasks. The types of exercises found predominately focused on students applying content studied in the narrative of lessons. The typical problems and issues experienced by students when working with transformations, as identified in the literature, received little support or attention in the lessons. The types of tasks that seem to embody the ideals in the process standards, such as working a problem backwards, were found on few occurrences across all textbooks examined. The level of cognitive demand required for student exercises predominately occurred in the Lower-Level, and Lower-Middle categories. Research indicates approximately the last fourth of textbook pages are not likely to be studied during a school year; hence topics located in the final fourth of textbook pages might not provide students the opportunity to experience geometric transformations in that year. This was found to be the case in some of the textbooks examined, therefore students might not have the opportunity to study geometric transformations during some middle grades, as was the case for the Glencoe (6, 7), and the UCSMP (6) textbooks, or possibly during their entire middle grades career as was found with the Prentice Hall (6, 7, Prealgebra) textbook series.

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