In classical electrodynamics, accelerated charges emit electromagnetic radiation. The associated energy loss of the particle can be modeled as a radiation reaction force, proportional to the change in acceleration. Including this force in the usual dynamical equation for a charged particle yields the Abraham-Lorentz-Dirac (ALD) equation. For point particles, solutions to the ALD equation are problematic. First, they require additional initial conditions beyond those needed in regular Newtonian dynamics, but more concerning are the presence of unphysical run-away solutions and acausal trajectories. These problems arise in the point-like limit of the theory, so it is possible that quantum mechanics is the more appropriate framework with which to address the problem. In order to explore the similarities and differences of the classical and quantum frameworks, we compute classical trajectories of charged particles in the presence of a Coulomb field, including radiation reaction. We compare the associated classical cross-section with the quantum field-theoretic cross-section for Coulombic scattering, including radiative corrections. With these calculations, we will be able to assess when the field theoretic formulation of single-photon emission can approximate the classical radiation reaction.

Biochar is used as a soil amendment and is created from plant biomass through pyrolysis. Biochar is thought to help plants grow by increasing the fertility of soil as well as impacting water retention in soils. This led to the question: Does biochar increase the water retention of soils found in the Lower Yakima Valley, allowing for a reduction in the amount of water that is needed to grow bell pepper plants in this water-stressed region? This research delves into the effectiveness of biochar by testing water level usage in real world plants grown in two concentrations of biochar (0.5% and 1%) applied to soils trial plots located in the Heritage University Research Farm. A random generator was used to create a split-plot design with three concentrations of biochar (0% or control, 0.5%, 1%) each with three replicates. The spilt-plot contained two different watering schemes. A total of 576 greenhouse grown Capsicum annum plants were transplanted to 18 plots, each plot containing 32 plants. To test the efficiency of biochar in soil water retention, soil moisture data, plant height, dry weight of the plants, and yield abundance were measured. Results indicate that the presence of biochar in soil can lead to increases in biomass of up to 100% in certain conditions. However, there were irrigation engineering complications that led the inability to reliably control the irrigation rates in the two split plots (originally planned for 100% in section 1, 80% reduced flow in section 2). The soil moisture data were inconsistent and highly variable reflecting potential instrument calibration issues, multiple instances of damage to the drip tape leading to water leaks, and persistent problems with the timing mechanisms for watering lines. This experiment led to significant improvement our methods for the following bell pepper research for the following year.

Here we explore applications of multivariable calculus for studying three dimensional wave media in our environment. We employ regression based methods to derive analytic formulae for real wave. Using multivariable optimization methods, we derive the maxima, minima and saddle points of three dimensional functions. We use advanced data visualization methods to study the divergence and curl and illustrate how these can be used to study ocean waves- including their vorticity and circulation. The project provides hands on exploration of real world environmental science problems with advanced data visualization and shows how divergence and curl can be used to measure circulation and vorticity parameters of real wave media involving ocean waves. Real world manifestations of scalar and vector fields in our environment are also presented.

Monte Carlo simulations employ random probability distribution statistics to estimate areas and volumes. Here, we employ Monte Carlo simulations to estimate the numerical value of π. We inscribe a circle in a square board and throw N darts using random values for both x and y. The probability that the dart lies within the circle = area of circle/area of square. This relationship allows us to estimate π. We wrote EXCEL/JAVA code for this research. The accuracy of estimated π is improved as the number of darts N --> ∞. This research allows us to combine mathematics, computer programming and data visualization to estimate π. Important applications of Monte Carlo simulations to find areas and volumes of complex objects including rivers, landscapes and organisms which cannot be represented by analytic functions will be discussed.

Every non-negative integer can be written using what is known as the factorial base representation. We define this notion and explore certain combinatorial structures arising from the arithmetic of these representations. In particular, we investigate the sum-of-digits function, carry sequences, and a partial order referred to as digital dominance. Finally, we describe an analog of a classical theorem due to Kummer that relates the combinatorial objects of interest by constructing a variety of new integer sequences. Many of the combinatorial objects we describe are of interest in their own right and provide interesting avenues for future discoveries. For instance, the carry sequences we describe correspond to a new family of integer partitions that are analogous to hyper-binary partitions, which have been extensively studied. Moreover, the digital dominance order most likely contains interesting combinatorial information about the integers as represented in the factorial base. Finally, our analog of Kummer's theorem has helped to explain how the classical version of Kummer's theorem can be altered to fit with almost any positional numeration system.

Conventional farming techniques involve large quantities of chemical fertilizers, which often leach into bodies of water causing eutrophication. The influx of excess nutrients from fertilizer results in rapid increase of aquatic algal populations followed by dissolved oxygen depletion. This process creates regions of low oxygen that negatively impact the water quality of many major lakes and coastal regions. To address this problem, the sensitivity of plant growth-promoting bacteria (PGPB) to excess fertilizer was investigated. We hypothesize that the addition of high amounts of fertilizer will result in smaller, less productive plants and diminished rhizosphere colonization of the PGPB, Bacillus subtilis. A modified Kirby-Bauer disk diffusion test was performed as an initial assessment of the effect of varied fertilizer concentrations on B. subtilis. Roots of romaine lettuce seedlings were then inoculated with B. subtilis and grown in soil treated with the same range of fertilizer concentrations. The effects of each treatment on plant growth were determined using total leaf area, quantification of the rhizosphere colonization by B. subtilis, and carbon assimilation measured with a LI-6800 Portable Photosynthesis System. These results are an important step towards establishing guidelines for appropriate application of agricultural fertilizer in order to mitigate the frequency and severity of eutrophication events in aquatic systems.