We study the Born-Oppenheimer approximation for a symmetric linear triatomic molecule in two space dimensions. We compute energy levels up to errors of order ε⁵, uniformly for three bounded vibrational quantum numbers n₁, n₂, and n₃; and nuclear angular momentum quantum numbers â ≤ kε^-3/4 for k > 0. Here the small parameter ε is the fourth root of the ratio of the electron mass to an average nuclear mass. / Master of Science

Channel morphology in alluvial rivers results from the interactions among the flow of water and sediment, the grain size distribution of the material in transport, and the characteristics of the materials making up the channel boundary. Many modern river management problems depend upon the ability to predict channel behavior in response to changes in the delivery of sediment. Sediment budgets provide a framework for explicitly evaluating the links between sediment delivery to and export from a river, and changes in storage. In the work presented here I have developed sediment budgets at three different spatial and temporal scales in an effort to gain insight to channel response to a change in sediment supply. In Chapter 2, I present a bed load budget for the Snake River in Grand Teton National Park (GTNP), Wyoming. The analysis was designed to evaluate the effects of 50 years of flow regulation on net sediment flux and, thus, sediment storage for the Snake River below Jackson Lake Dam. In Chapter 3 I present a sediment mass balance constructed for a single flood on an aggrading 4-km reach of the middle Provo River, Utah. Sediment accumulation in the Provo River had driven significant point bar growth, and the sediment budget was designed to explicitly link patterns in sediment flux with morphologic change. In Chapter 4, I present the results from a physical experiment designed to further evaluate the effect of changing sediment supply on point bar morphology in a single meander bend. The experiment was conducted in a field-scale flume, the Outdoor StreamLab (OSL), at the University of Minnesota. In each of the cases I present here, the channel was subject to sediment accumulation due to either an increase in sediment supply (Provo River and OSL) or a decrease in transport capacity (Snake River). The analyses provide insight into processes governing channel response to changes in sediment supply and highlight the inherent challenges and uncertainties associated with sediment budgets, regardless of the scale of the analysis.

<p>Have you thought of why you get tired or why you get hungry? Something in your body keeps track of time. It is almost like you have a clock that tells you all those things.</p><p>And indeed, in the suparachiasmatic region of our hypothalamus reside cells which each act like an oscillator, and together form a coherent circadian rhythm to help our body keep track of time. In fact, such circadian clocks are not limited to mammals but can be found in many organisms including single-cell, reptiles and birds. The study of such rhythms constitutes a field of biology, chronobiology, and forms the background for my research and this thesis.</p><p>Pioneers of chronobiology, Pittendrigh and Aschoff, studied biological clocks from an input-output view, across a range of organisms by observing and analyzing their overt activity in response to stimulus such as light. Their study was made without recourse to knowledge of the biological underpinnings of the circadian pacemaker. The advent of the new biology has now made it possible to "break open the box" and identify biological feedback systems comprised of gene transcription and protein translation as the core mechanism of a biological clock.</p><p>My research has focused on a simple transcription-translation clock model which nevertheless possesses many of the features of a circadian pacemaker including its entrainability by light. This model consists of two nonlinear coupled and delayed differential equations. Light pulses can reset the phase of this clock, whereas constant light of different intensity can speed it up or slow it down. This latter property is a signature property of circadian clocks and is referred to in chronobiology as "Aschoff's rule". The discussion in this thesis focus on develop a connection and also a understanding of how constant light effect this clock model.</p>

Have you thought of why you get tired or why you get hungry? Something in your body keeps track of time. It is almost like you have a clock that tells you all those things. And indeed, in the suparachiasmatic region of our hypothalamus reside cells which each act like an oscillator, and together form a coherent circadian rhythm to help our body keep track of time. In fact, such circadian clocks are not limited to mammals but can be found in many organisms including single-cell, reptiles and birds. The study of such rhythms constitutes a field of biology, chronobiology, and forms the background for my research and this thesis. Pioneers of chronobiology, Pittendrigh and Aschoff, studied biological clocks from an input-output view, across a range of organisms by observing and analyzing their overt activity in response to stimulus such as light. Their study was made without recourse to knowledge of the biological underpinnings of the circadian pacemaker. The advent of the new biology has now made it possible to "break open the box" and identify biological feedback systems comprised of gene transcription and protein translation as the core mechanism of a biological clock. My research has focused on a simple transcription-translation clock model which nevertheless possesses many of the features of a circadian pacemaker including its entrainability by light. This model consists of two nonlinear coupled and delayed differential equations. Light pulses can reset the phase of this clock, whereas constant light of different intensity can speed it up or slow it down. This latter property is a signature property of circadian clocks and is referred to in chronobiology as "Aschoff's rule". The discussion in this thesis focus on develop a connection and also a understanding of how constant light effect this clock model.

<p>Biochemical reaction networks with a sufficiently large number of molecules may be represented as systems of differential equations. Many networks receive inputs that fluctuate continuously in time. These networks may never settle down to a static equilibrium and are of great interest both mathematically and biologically. Biological systems receive inputs that vary on multiple time scales. Hormonal and neural inputs vary on a scale of seconds or minutes; inputs from meals and circadian rhythms vary on a scale of hours or days; and long term environmental changes (such as diet, disease, and pollution) vary on a scale of years. In this thesis, we consider the limiting behavior of networks in which the input is on a different time scale compared to the reaction kinetics within the network.</p> <p>We prove analytic results of how the variance of reaction rates within a system compares to the variance of the input when the input is on a different time scale than the reaction kinetics within the network. We consider the behavior of simple chains, single species complex networks, reversible chains, and certain classes of non-linear systems with time-scaled stochastic input, as the input speeds up and slows down. In all cases, as the input fluctuates more and more quickly, the variance of species within the system approaches to zero. As the input fluctuates more and more slowly, the variance of the species approaches the variance of the input, up to a normalization factor.</p> / Dissertation

The ground state electron wavefunction of some molecules has a non-zero angular momentum about the internuclear axis. Molecular rotational momentum can couple with this angular momentum, splitting the energy degeneracy of the two directions of motion about the internuclear axis. Performing a Born-Oppenheimer approximation of such a system will break the relevant energy degeneracy at eighth order. This degeneracy breaking is known as L-doubling. / Master of Science

Proposed designs for Very Large Floating Structures motivate us to understand water-wave propagation through arrays of hundreds, or possibly thousands, of floating structures. The water-wave problems we study are each formulated under the usual conditions of linear wave theory. We study the frequency-domain problem of water-wave propagation through a periodically arranged array of structures, which are solved using a variety of methods. In the first instance we solve the problem for a periodically arranged infinite array using the method of matched asymptotic expansions for both shallow and deep water; the structures are assumed to be small relative to the wavelength and the array periodicity, and may be fixed or float freely. We then solve the same infinite array problem using a numerical approach, namely the Rayleigh-Ritz method, for fixed cylinders in water of finite depth and deep water. No limiting assumptions on the size of the structures relative to other length scales need to be made using this method. Whilst we aren t afforded the luxury of explicit approximations to the solutions, we are able to compute diagrams that can be used to aid an investigation into negative refraction. Finally we solve the water-wave problem for a so-called strip array (that is, an array that extends to infinity in one horizontal direction, but is finite in the other), which allows us to consider the transmission and reflection properties of a water-wave incident on the structures. The problem is solved using the method of multiple scales, under the assumption that the evolution of waves in a horizontal direction occurs on a slower scale than the other time scales that are present, and the method of matched asymptotic expansions using the same assumptions as for the infinite array case.

In this thesis, we use asymptotic methods to solve problems of wave propagation through infinite and finite (only consider those that are finite in one direction) arrays of scatterers. Both two- and three-dimensional arrays are considered. We always assume the scatterer size is much smaller than both the wavelength and array periodicity. Therefore a small parameter is involved and then the method of matched asymptotic expansions is applicable. When the array is infinite, the elastic wave scattering in doubly-periodic arrays of cavity cylinders and acoustic wave scattering in triply-periodic arrays of arbitrary shape rigid scatterers are considered. In both cases, eigenvalue problems are obtained to give perturbed dispersion approximations explicitly. With the help of the computer-algebra package Mathematica, examples of explicit approximations to the dispersion relation for perturbed waves are given. In the case of finite arrays, we consider the multiple resonant wave scattering problems for both acoustic and elastic waves. We use the methods of multiple scales and matched asymptotic expansions to obtain envelope equations for infinite arrays and then apply them to a strip of doubly or triply periodic arrays of scatterers. Numerical results are given to compare the transmission wave intensity for different shape scatterers for acoustic case. For elastic case, where the strip is an elastic medium with arrays of cavity cylinders bounded by acoustic media on both sides, we first give numerical results when there is one dilatational and one shear wave in the array and then compare the transmission coefficients when one dilatational wave is resonated in the array for normal incidence. Key words: matched asymptotic expansions, multiple scales, acoustic scattering, elastic scattering, periodic structures, dispersion relation.

Sandy beaches are often highly allochthonous, depending on external subsidies of carbon and nutrients. Despite this, sandy beach macrofaunal assemblages have received little attention regarding their response to enhanced primary productivity generated from coastal upwelling. This thesis investigates the influence of upwelling on macrofaunal assemblages over a variety of spatial and temporal scales. Spatially, four regions were examined across two biogeographic provinces to remove temperature as a confounding factor, and limit biogeography-specific effects. A nested hierarchical design enabled both large and small scales to be examined and generalities about upwelling effects across multiple areas to be considered. Sampling was conducted in two seasons, and over two years, to test the persistence of any effects. Biogeography and region had the strongest influences on macrofaunal biodiversity. Upwelling influenced macrofaunal assemblages in every region when analyses were conducted at the species level. However, the particular effect, positive or negative, differed among regions depending on local factors, and between the response variables, abundance and biomass. Coarser scales of taxonomy, feeding guild and developmental mode were investigated; however, the influence of upwelling generally became weaker and more varied, and occasionally disappeared. Seasonality was greater on the South Coast but was still important in some analyses on the West Coast. At the small-scale, variation within-beaches was lower than between beaches, assemblage structure remained stable over time, and consistent zonation was not present. The influence of temperature on filtration rate and oxygen consumption of Donax serra was investigated to test a driving mechanism for assemblage responses to upwelling. Feeding ability was significantly reduced at colder temperatures indicating an important factor which may be involved in determining assemblage structure. These results suggest that alterations to upwelling regimes predicted under climate change scenarios will impact sandy beach macrofauna, however the specific outcome will depend on multiple contextual factors.

The work presents recent contributions in the field of computational neuroscience and sketches possible research perspectives.