Invariant Manifolds in the Hamiltonian-Hopf Bifurcation

Abstract We study the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter ν. The eigenvalues of the linearized system are pure imaginary for ν \u3c 0 and complex with nonzero real part for ν \u3e 0. (These are the same basic assumptions as found in the Hamiltonian-Hopf bifurcation theorem of the authors.) For ν \u3e 0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν \u3c 0 there are no longer stable and unstable manifolds attached to the equilibrium. We study the evolution of these manifolds as the parameter is varied. If the sign of a certain term in the normal form is positive then for small positive ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. This happens at the Lagrange equilibrium point L4 of the restricted three-body problem at the Routh critical value µ1. On the other hand if the sign of this term in the normal form is negative then for ν = 0 the stable and unstable manifolds persists and then as ν decreases from zero they detach from the equilibrium to follow a hyperbolic periodic solution. Joint work with Dieter Schmidt

Cost Efficient Tillage and Rotation Options for Mitigating GHG Emissions from Agriculture in Eastern Canada

The economic efficiency of cropping options to mitigate GHG emissions with agriculture in Eastern Canada was analyzed. Data on yield response to tillage (moldboard plow and chisel plow) and six corn based rotations were obtained from a 20-year field experiment in Ontario. Budgets were constructed for each cropping system while GHG emissions were measured for soil carbon and were estimated for nitrous oxide according to IPCC methodology. Complex crop rotations with legumes, such as corn-corn-soybeans-wheat with red clover underseeded, have higher net returns and substantially (more than 1 Mg ha1 year1) lower GHG emissions than continuous corn. Reduced tillage reduces GHG emissions due to lower input use but no sequestration effect could be found in the soil from tillage. Rotation had a much bigger effect on the mitigation potential of GHG emissions than tillage. However, opportunity costs of more than $200 per Mg CO2 eq ha1 year1 indicate the limits to increase the mitigation potential beyond the level of the economic best cropping system.Environmental Economics and Policy,

The Interest Rate Roller Coaster: Factors that Influence Interest Rate Fluctuations

Millions of people are affected by interest rates each day. Whether obtaining a car loan, a mortgage loan, or any other type of loan, the price one pays for this loan is ultimately the interest rate charged. Interest rates are also a factor for investors. The return (income) one receives from investing in bonds, stocks, and mutual funds will be determined by the interest rate or yield of these securities. In turn, interest rates will affect people\u27s decisions in choosing the right opportunity (borrowing vs. investing) that will minimize costs or maximize returns. What many people do not realize, however, is how changes in the stock market or the money supply can affect the interest rate they will be charged on a loan, or receive on a particular security. This essay will examine these and other factors that influence interest rate fluctuations. Although much research has been done in this area, my goal is to provide a current assessment of factors that affect interest rates. In addition, this essay will reinforce previous conclusions and offer insight into recent fluctuations

The Amplitude and Spectral Index of the Large Angular Scale Anisotropy in the Cosmic Microwave Background Radiation

In many cosmological models, the large angular scale anisotropy in the cosmic microwave background is parameterized by a spectral index, $n$, and a quadrupolar amplitude, $Q$. For a Peebles-Harrison-Zel'dovich spectrum, $n=1$. Using data from the Far Infra-Red Survey (FIRS) and a new statistical measure, a contour plot of the likelihood for cosmological models for which $-1 < n < 3$ and $0 \le Q \le 50~\mu{\rm K}$ is obtained. We find that the likelihood is maximum at $(n, Q) = (1.0, 19 uK)$. For constant $n$ the likelihood falls to half its maximum at $Q \approx 14 uK$ and $25 uK$ and for constant $Q$ the likelihood falls to half its maximum at $n \approx 0.5$ and $1.4$. Regardless of $Q$, the likelihood is always less than half its maximum for $n < -0.4$ and for $n > 2.2$, as it is for $Q 44$.Comment: Ten Page

Scale invariance and critical gravitational collapse

We examine ways to write the Choptuik critical solution as the evolution of scale invariant variables. It is shown that a system of scale invariant variables proposed by one of the authors does not evolve periodically in the Choptuik critical solution. We find a different system, based on maximal slicing. This system does evolve periodically, and may generalize to the case of axisymmetry or of no symmetry at all.Comment: 7 pages, 3 figures, Revtex, discussion modified to clarify presentatio

Changing Groundwater Levels in the Sandstone Aquifers of Northern Illinois and Southern Wisconsin: Impacts on Available Water Supply

In 2014-15, the Illinois State Water Survey conducted their largest synoptic measurement of water levels (i.e., heads) in Cambrian-Ordovician sandstone wells since 1980. The study covered 33 counties in the northern half of Illinois where demands for water are satisfied, in part, by sandstone aquifers. The Wisconsin Geological and Natural History Survey also measured sandstone wells in 10 counties in southern Wisconsin. These observations were used to generate head contours of the sandstone aquifers. These contours provide insight into the direction and magnitude of groundwater flow. They also can be compared with historic measurements, providing insight into the impact of changing groundwater withdrawals through time. In predevelopment conditions, heads in the Cambrian-Ordovician sandstone aquifers were near or above land surface. Due to pumping from the sandstone aquifers, heads have decreased over time; this decrease is referred to as drawdown. In 2014, drawdown in northeastern Illinois was typically over 300 ft and exceeded 800 ft in the Joliet region. Three factors drove this large drawdown. First, demands for water from sandstone aquifers are much greater in northeastern Illinois than in the rest of the study region. Second, the sandstone aquifers are overlain by aquitards, which are low permeable materials that limit vertical infiltration of water. Third, the Sandwich Fault Zone limits water flowing into the sandstone aquifers of northeastern Illinois from the south. Heads near the center of the cone of depression continue to have a decreasing trend. The more severe drawdown in northeastern Illinois has resulted in local areas where heads have fallen below the top of the sandstone, known as desaturation. Desaturation of a sandstone aquifer can create a number of water quality and quantity concerns. The uppermost sandstone, the St. Peter, was observed to be partially desaturated in portions of Will, Kane, and Kendall Counties under non-pumping conditions. Other areas in these counties are at risk of desaturation under pumping conditions or with the installation of additional wells connecting the St. Peter to deeper, more heavily stressed sandstones. Simulations from a groundwater flow model indicate that the risk of desaturation will increase with increased future withdrawals. Despite the relatively small demand for water throughout much of central Illinois, heads have been declining since predevelopment, likely due to the shale overlying the sandstone. This shale serves as an aquitard, minimizing vertical infiltration of groundwater to the sandstone. Sustained drawdown in this region could potentially induce flow from the southern half of the state, where water in the sandstone is highly saline and not suitable as a drinking water supply. Drawdown in northwestern Illinois was also typically small (<100 ft), primarily due to two factors: 1) low demands from the sandstone aquifers and 2) the absence of shale aquitards. The notable exception is in Winnebago County, near Rockford, where demands are historically high and drawdown was on the order of 100-200 ft. While the quantity of water in the aquifer is not a concern in this region, large withdrawals could result in reductions of natural groundwater discharge to surface waters, impacting stream ecosystems under low flow conditions. Drawdown since predevelopment was over 300 ft in southeastern Wisconsin, with the greatest drawdown in Waukesha County of over 400 ft. Recent trends indicate heads in the Waukesha area are recovering, although they are still well below predevelopment levels. [This report is also associated with the fact sheets: Changing Groundwater Levels in the Cambrian-Ordovician Sandstone Aquifers of Northern Illinois, 1980-2014, Groundwater Availability in Northeastern Illinois from Deep Sandstone Aquifers, and Sources of Water for Communities in Northeastern Illinois.published or submitted for publicationis peer reviewedOpe

Materials Characterization of Additively Manufactured Components for Rocket Propulsion

To advance Additive Manufacturing (AM) technologies for production of rocket propulsion components the NASA Glenn Research Center (GRC) is applying state of the art characterization techniques to interrogate microstructure and mechanical properties of AM materials and components at various steps in their processing. The materials being investigated for upper stage rocket engines include titanium, copper, and nickel alloys. Additive manufacturing processes include laser powder bed, electron beam powder bed, and electron beam wire fed processes. Various post build thermal treatments, including Hot Isostatic Pressure (HIP), have been studied to understand their influence on microstructure, mechanical properties, and build density. Micro-computed tomography, electron microscopy, and mechanical testing in relevant temperature environments has been performed to develop relationships between build quality, microstructure, and mechanical performance at temperature. A summary of GRC's Additive Manufacturing roles and experimental findings will be presented

Humour in Nietzsche's style

Nietzsche's writing style is designed to elicit affective responses in his readers. Humour is one of the most common means by which he attempts to engage his readers' affects. In this article, I explain how and why Nietzsche uses humour to achieve his philosophical ends. The article has three parts. In part 1, I reject interpretations of Nietzsche's humour on which he engages in self‐parody in order to mitigate the charge of decadence or dogmatism by undermining his own philosophical authority. In part 2, I look at how Nietzsche uses humour and laughter as a critical tool in his polemic against traditional morality. I argue that one important way in which Nietzsche uses humour is as a vehicle for enhancing the effectiveness of his ad hominem arguments. In part 3, I show how Nietzsche exploits humour's social dimension in order to find and cultivate what he sees as the right kinds of readers for his works