6.14 a Proving Lemma 6.4.1: $(\I,r,m) \vDash C_S \varphi$ iff $(\I,r',m') \vDash \varphi$ for all points (r′,m′) that are S-reachable from (r, m).

INTRODUCTION 1.Soil moisture patterns are important to predicting water quality of the runoff generated. Much attention in hydrology is paid to soil moisture as a risk indicator of surface runoff generation e.g. . Surface runoff quickly transports water to streams causing pulses of high flow that increase flooding potential, impact wildlife habitat, and transport sediment and dissolved contaminants from the land surface. The ability to predict where and when runoff is likely to be generated guides the planning and implementation of management practices that reduce these negative impacts. Traditionally, the temporal and spatial variability of soil moisture patterns is identified through two approaches. Complex, distributed hydrologic models such as the Soil Moisture Routing model predict patterns fairly well, but require climate and landscape data at fine resolutions to be reliable. In contrast, indices that draw on watershed-level hydrologic drivers such as terrain and soil properties can provide fast, simple estimates of soil moisture patterns. These topographic indices, originally developed by , can be used to quickly estimate patterns in regions where soil moisture spatial variability is driven by topographic changes and shallow soil depths. In the Northeastern United States, topographic indices have been shown to work well to predict soil moisture patterns . Soil moisture measurements can be collected rapidly and easily in the field, however due to signfiicant spatial and temporal variability, achieving a coverage for watersheds is a time-intensive endevor. Remote sensing techniques are in development, but the resolution of their predicitions is too coarse for meaningful guidance in management practices . In this project we are interested in using geostatistics and the topographic index models to extroplate soil moisture measurements for comparison with patterns predicted by a more complicated, uncalibrated hydrologic model. To do this, we use a geostatistical tool for characterizing spatial patterns, the semivariogram model. Semivariograms quantify the change in variance between two points in a field based on their distance. The sill represents the distance at very large distances (large being relative to the data). The distance at which the sill is stable is considered the range. At this distance, spatial correlation no longer exists. The variance in repeated measurements at the same location is given as the nugget of the semivariogram. Kriging is a geostatistical technique that makes use of the semivariogram for interpolating data where it does not exist. Kriging with additional information, such as the output of the topographic index models, can be used to influence the spatial pattern. This is particularly useful where linear distance is not the only important correlation for a spatial pattern. In the case of soil moisture, a location in a stream may be continually saturated to a similar degree as a location far downstream that is also along a high accumalting flowpath. A location much closer in spatial distance to the saturated area may be significantly drier, perhaps because of a steep slope that transmits water away quickly. For this reason, straight kriging with the soil moisture data will present limited spatial results. In some variograms the range will occur at very close distances or spatial correlation will not be apparent. In this project we investigate the locations and dates where good spatial correlation is evident and use topographic index information to improve kriging predictions.

No attractors in Hamiltonian system It’s not true that neighboring initial conditions will converge to the fixed state after long time. As we saw from the Jupiter problem, there are some regions where the initial conditions for chaotic and periodic trajectories lies infinitesimally close but lead to different results. The chaotic initial conditions will not make the system converge to certain state. Therefore, there is no attractor in this 3-body problem. Stable Hamiltonian systems can withstand a minor perturbation _in the process_ and the system will still end up where it’s supposed to. Unstable Hamiltonian system will diverge if perturbed _in the process_ and the system will not come to the original equilibrium regardless of whether the system becomes chaotic or not.

Small satellites can enable a new kind of mission architecture: inspecting larger satellites on orbit in close proximity without mechanical contact. Induction coupling is a new actuation technology that can augment on-orbit servicing by exploiting eddy-current forces and torques. Current technologies for applying forces and torques between two spacecraft share a glaring disadvantage: they require direct contact or propellant. By using the forces between a magnetic field and the electric currents it induces in a target, an induction coupler can control the relative position and orientation between a chaser spacecraft and a target without physical contact. A system utilizing these eddy-current effects places relatively few requirements on the target and chaser compared to other proposed electromagnetic actuation concepts. This paper presents a system overview of a contactless induction coupler, outlines those requirements through the analysis of an inspection mission on the International Space Station, and traces them to flight applications through ongoing experimental work.

Entropy maximization and temperature First of all, Ω₁(E₁)δE₁ and Ω₂(E₂)δE₂ are the phase space volume for these two subsystems. Since most of the volume in phase space is near the surface, we can instead write Ω₁(E₁) and Ω₂(E₂) to denote the two volumes in phase space. Second, it’s clear that the probability density of subsystem 1 having energy E₁ is proportional to the ’allowed’ volume product of the two subsystems. \rho_1(E_1) \propto \Omega_1(E_1)\Omega_2(E_2) where E₂ = E − E₁. And finally, we impose the normalization condition: \int \rho_1(E_1)dE_1 = A \int \Omega_1(E_1)\Omega_2(E-E_1)dE_1 = A \Omega(E) = 1 A is the proportional constant and equal to 1/Ω(E). Therefore, \rho_1(E_1) = {\Omega(E)} If we maximize the probability at E₁: {dE_1} = 0 {\Omega_1}{dE_1} - {\Omega_2}{dE_2} = 0 By plugging the definition of entropy S = kBlog(Ω), {k_B}{dE_1} - {k_B}{dE_2} = 0 {dE_1} + {dE_1} = {dE_1}(S(E_1)+S(E_2)) = 0 ,which implies the maximization of the sum of entropies of both subsystems. For (6), we see that if the probability density ρ(E₁) is maximized, {T_1} = {T_2}

TO DO - identify data set - complete outline - book room? - come up with quiz questions to identify suitable students

The steady increase in white-tailed deer populations and largely unnoticed invasion of earthworms are changing some of the important underlying processes that support North American forest understory communities . Amidst a rapidly-changing set of ecological drivers, earthworm invasion and increased browse pressure from white tailed deer are two of the most important stressors to forest plant populations . Northern forests in North America have developed in the absence of earthworms, at least as far back as the last glaciation . Human activity has facilitated the introduction and dispersion of earthworms to even remote forested areas through logging roads, dumping of fishing bait, and the relocation of fill or horticultural materials . Whereas uninvaded forest soils often build up a thick leaf-fermenting-humus (LFH) layer through slow, localized decomposition of organic material, earthworm-dominated forests are characterized by bare mineral soil, and can completely lack a stratified soil profile . Many native plants require the unique habitat that the forest floor provides for protection of slowly germinating seedlings, nutrient and water retention, access to mycorrhizal symbionts and temperature buffering . Drivers of change in understory plant communities are also occurring above-ground. In much of North eastern North America, browse pressure from rapidly growing white-tailed deer (_Odocoileus virginianus_) populations is dramatically altering forest understory communities, restricting regeneration and causing extinction of native plants at the local level . Long-term datasets show that overall species diversity decreases with high deer browse pressure, and selective browsing disproportionately affects palatable species, while releasing unpalatable plants competition . Deer browse impacts are compounded in slow-growing perennials, as they preferentially browse the largest individual plants that maintain growth rates of the plant population . To look at the combined and individual effects of deer and earthworms, I set up experimental plantings of a large number of native forest understory species (n=15) in spring 2012. Because earthworm and deer impacts on seedlings are the net effect of a myriad of factors, I selected species with many different traits that could conceivably respond to these factors. Thus far I have been recording seedling survival, and have been surprised to find earthworms decrease seedling survival in 11 of 15 species, and had no effect on 4 species. This is surprising, because field surveys have found several plant species included in my study are associated with earthworm-invaded environments . However, these snapshots of plant communities do not provide information on the impacts of deer and earthworms on plant population growth rates. As my seedlings become reproductive, I can build a stage-structured model to determine how deer and earthworms impact native plant population growth, both individually and together. I expected that, 1) most plants growing in earthworm-invaded areas will be slower to progress through the vegetative stages 2) plant species that appear to benefit from earthworm invasion (sedges, etc.) will have higher seed set (theta will be large) 3) unfenced plots will see more regressions to smaller stages due to deer browse 4) deer and earthworms will have a synergistic effect on decreasing population growth.

INTRODUCTION AND PREVIOUS WORK The electronic properties of a material are primarelly dictated by three distinct effects. 1. The ion potentials, 2. the electron-electron interaction, and 3. externally applied fields A full solution accounting for all of these interaction would yield a many body problem that would typically be unsolvable. For this reason it is necessary to make approximations to obtain meaningful results. The most common of this approximations would be to ignore electron-electron interactions and assume a stationary periodic potential. This near-free electron problem leads to the Band Structure Theory that is often solved with assistance of Bloch Theorem . This solution leads to a very simple classification of metals, and insulators based on their band structure and Fermi Energy. This method, although often successful, it fails by omitting electron-electron interactions and lattice distortions. A clear example of this is NiO; by its half filed orbitals one would predict this material to be a metal- however it has insulating properties due to electron-electron interactions. The presence of two opposite mechanism driving that drives to either metallic or insulating behavior can lead to some pretty interesting results. As an example of this consider a N hydrogen atoms distributed at a constant density. The system is described by two non-commuting operators; the Kinetic energy, and the ion potentials. The kinetic energy of the electrons will trend to spread the the electrons, and the ion potential (driven by the coulomb force) will trend to localize the electrons near the protons. The electrons therefore have to phases; bound, or unbound. At low densities, the attractive potential can act over long distances which localizes the electrons. This is an insulating state. As we increase the density we pass a critical density after which the attractive potential is screened by other particles in the systems for the electrons to be bound. This leads to a metallic state at high densities. This change in properties can be characterized as quantum phase transition. In general a quantum phase transition can be induce by varying the relative strength of two non-commuting operators. It is therefore possible to externally alter the relative strength of these mechanisms by varying an external parameter i.e. pressure . This metal-insulator transition as quantum phase transition is shown in figure 1. It is worth noting that proper transition is only attained at zero temperature. This is because at higher temperature there is always some conduction, at which point it becomes a discussion of how good of an insulator it is.

Abstract will go here in final report.

     In this project, the effects of electron transport due to the Charge Density Wave (CDW) transition will be investigated. Following a review of how CDWs are formed, a review of several publications that discuss possible applications of CDW-based devices, and practical value of CDWs for electo-optical applications will be discussed. Specifically, pristine and Ca-intercalated bilayer graphene will be considered for photodetector and bolometer applications. Using transport formalism developed in class, figures of merit for such devices using CDWs will be discussed quantitatively.Theory and Identification of CDW     Charge Density Waves occur due to a reconfiguration of both the electronic and lattice structures as a result of the interactions of the electrons with both themselves, and the lattice. Given that a certain amount of coupling between electrons and phonons exists, the phonon dispersion will undergo a strong change at lower temperatures. This causes the phonon mode to form a standing wave, (that is to say, a certain nonzero wavevector of energy zero.) Consequently, the electron density, which is coupled to the phonon density, also undergoes oscillations. The CDW state occurs when such a reconfiguration is energetically preferable.     Even though an exact, overlapping model for CDW occurence from any mechanism, and of any dimensionality is yet elusive, a simple 1D model exists that illustrates the CDW occurence at \(T=0K\). Following the derivation proposed in the work by Rossnagel [1], which closely follows the mean-field theory approach taken by Gruner [2], an electronic susceptibility can be extracted.

Topological insulators are materials that exhibit a unique quantum state of matter. This state is often referred to as the quantum spin hall state and has not been observed in any other type of material. This state of matter has very unique transport properties and in the last 10 years there has been much speculation about potential applications to take advantage of these properties. In this paper, the potential and applicability of a resonance tunneling diode constructed of topological insulator material is investigated. Conventional resonance tunneling diode properties are briefly reviewed and then used as comparison to a theoretical topological insulator resonance tunneling diode.