<p>In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp. </p>

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<p>The main functions of the kidney take place in the nephrons. For their proper operation, nephrons need to be supplied with a stable blood flow that remains constant despite fluctuations of arterial pressure. Such stability is provided by the afferent arterioles, which are unique vessels in the kidney capable of adjusting diameter. By doing so, afferent arterioles regulate blood delivery downstream, where the nephrons are located. The afferent arterioles respond to signals initiated by two mechanisms: the myogenic response which operates to absorb pressure perturbations within the vasculature, and tubuloglomerular feedback which operates to stabilize salt reabsorption.</p><p>In this thesis, a mathematical model of the renal nephrovasculature that represents both mechanisms in a dynamical context is developed. For this purpose, de- tailed representations of the myogenic mechanism of vascular smooth muscles and the tubular processes are developed and combined in a single comprehensive model. The resulting model is formulated with a large number of ordinary differential equations that represent the intracellular processes of arteriolar smooth muscles, coupled with a number of partial differential equations, mainly of the advection-diffusion-reaction type, that represent blood flow, glomerular filtration and the tubular processes. Due to its unique activation characteristics, the myogenic response is formulated with a set of delay differential equations.</p><p>The model is utilized to assess a verity of physiological phenomena: the conduction of vasomotor responses along the afferent arteriole, autoregulation under physiologic as well as pathophysiologic conditions, and renal oxygenation. A first attempt to model the impact of diabetes mellitus on renal hemodynamics is also made. Further, an application with clinical significance is presented. Namely, renal oxygenation is estimated under conditions that simulate those observed during cardiopulmonary surgery. Results indicate the development of renal hypoxia, which suggests an important pathway for the development of acute kidney injury.</p> / Dissertation

A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs.

<p> Chapter 1 of this thesis describes the historical development and application of size-structured models and their role in modeling the population dynamics of biological systems. Chapter 2 presents a general model for the interaction of a size-structured population with its environment. The vital rates of the individuals are assumed to depend on a number of variables including the total population and the environment. A nonstandard finite difference approximation is developed for the general model and the convergence of the scheme to the unique weak solution of the nonlinear system of partial differential equations coupled with ordinary differential equations is established. Potential applications are presented in various fields ranging from shared resource dynamics to invasive species. Chapter 3 presents a first order finite difference scheme for phytoplankton dynamics subject to a vector of environmental variables in a well-mixed reactor. The biophysical problem yields a nonlinear partial differential equation coupled with a general system of nonlinear ordinary differential equations which represent the time-varying environmental conditions in the water column. Convergence results are established for the coupled system using the finite difference method. Numerical trials are presented to illustrate the performance of the model against a real dataset. Taking into account the combined effects of aggregate growth and coagulation, the model is used to investigate the role of the active fraction of the cell aggregate on algal bloom dynamics and nutrient consumption rates. The model is well-suited to complement microcosm studies and can accommodate the effects of a wide range of environmental conditions. Chapter 4 summarizes the main results, research and practical challenges associated with the application of general size-structured models, and suggestions for future work.</p>

Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work along with previous work by others seeks to establish a base case in answering the question of whether a given finite mutation type cluster algebra exhibits a maximal green sequence. In this paper we will provide a triangulation for orientable surfaces of genus n with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.

In this dissertation, a new method is developed to study BVPs of the modified Helmholtz and Helmholtz equations in a semi-infinite strip subject to the Poincare type, impedance and higher order boundary conditions. The main machinery used here is the theory of Riemann Hilbert problems, the residue theory of complex variables and the theory of integral transforms. A special kind of interconnected Laplace transforms are introduced whose parameters are related through branch of a multi-valued function. In the chapter 1 a brief review of the unified transform method used to solve BVPs of linear and non-linear integrable PDEs in convex polygons is given. Then unified transform method is applied to the BVP of the modified Helmholtz equation in a semi-infinite strip subject to the Poincare type and impedance boundary conditions. In the case of BVP of the modified Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions, two scalar RHPs are derived, then the closed form solutions of the given BVP are derived. The difficulty in application of the unified transform method to BVP of the Helmholtz equation in a semi infinite strip is discussed later on. The chapter 2 contains application of the finite integral transform (FIT) method to study the BVP for the Helmholtz equation in a semi-infinite strip subject to the Poincare type and impedance boundary conditions. In the case of the impedance boundary conditions, a series representation of the solution of the BVP for the Helmholtz equation in a semi-infinite strip is derived. The Burniston-Siewert method to find integral representations of a certain transcendental equation is presented. The roots of this equation are required for both methods, the FIT method and the RHP based method. To implement the Burniston-Siewert method, we solve a scalar RHP on several segments of the real axis. In chapter 3, we have applied the new method to study the Poincare type and impedance BVPs for the Helmholtz equation in a semi-infinite strip. In the case of the Poincare type boundary conditions an order two vector RHP is derived. In general, it is not possible to find closed form solution of an order two vector RHP. In the case of the impedance boundary conditions two scalar RHPs are derived whose closed form solutions are found. Then the series representation for solution of the BVP of the Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions, is recovered using the inverse transform operator, and the residue theory of complex variables. The numerical results are presented for various values of the parameters involved. It is observed that the FIT method and the new method generate exactly the same solution of the BVP of the Helmholtz equation in a semi-infinite strip subject to the impedance boundary conditions. In chapter 4, we have applied the new method to study the acoustic scattering from a semi-infinite strip subject to higher order boundary conditions. Two scalar RHPs are derived whose closed form solutions are found. A unique solution of the problem is obtained.

Highly connected matroids are consistently useful in the analysis of matroid structure. Round matroids, in particular, were instrumental in the proof of Rota's conjecture. Chapter 2 concerns a class of matroids with similar properties to those of round matroids. We provide many useful characterizations of these matroids, and determine explicitly their regular members. Tutte proved that a 3-connected matroid with every element in a 3-element circuit and a 3-element cocircuit is either a whirl or the cycle matroid of a wheel. This result led to the proof of the 3-connected splitter theorem. More recently, Miller proved that matroids of sufficient size having every pair of elements in a 4-element circuit and a 4-element cocircuit are spikes. This observation simplifies the proof of Rota's conjecture for GF(4). In Chapters 3 and 4, we investigate matroids having similar restrictions on their small circuits and cocircuits. The main result of each of these chapters is a complete characterization of the matroids therein.

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