Solving an abstract nonlinear eigenvalue problem by the inverse iteration method

Let $\left( X,\left\Vert \cdot\right\Vert_{X}\right)$ and $\left( Y,\left\Vert \cdot\right\Vert_{Y}\right)$ be Banach spaces over $\mathbb{R},$ with $X$ uniformly convex and compactly embedded into $Y.$ The inverse iteration method is applied to solve the abstract eigenvalue problem $A(w)=\lambda\left\Vert w\right\Vert_{Y}^{p-q}B(w),$ where the maps $A:X\rightarrow X^{\star}$ and $B:Y\rightarrow Y^{\star}$ are homogeneous of degrees $p-1$ and $q-1,$ respectively.Comment: This paper generalizes, for an abstract setting, another previously posted in ArXiv (see arXiv:1510.03941v2), which has not been submitted to or published in any other sourc

Variations on the Theme of Life

Variations on the Theme of Life is an interactive media piece, an iOS application that generates a unique musical experience using the mathematics of cellular automata and Conway\u27s Game of Life

The Future of Emotional Harm

Why should tort law treat claims for emotional harm as a second-class citizen? Judicial skepticism about these claims is long entrenched, justified by an amalgam of perceived problems ranging from proof difficulties for causation and the need to constrain fraudulent claims, to the ubiquity of the injury, and a concern about open-ended liability. To address this jumble of justifications, the law has developed a series of duty limitations to curb the claims and preclude them from reaching the jury for individualized analysis. The limited duty approach to emotional harm is maintained by the latest iteration of the Restatement (Third) of Torts. This Article argues that many of the justifications for curtailing this tort have been discredited by scientific developments. In particular, the rapid advances in neuroscience give greater insight into the changes that occur in the brain from emotional harm. Limited duty tests should no longer be used as proxies for validity or justified by the presumed untrustworthiness of the claim. Instead, validity evidence for emotional harm claims—like evidence of physical harm—should be entrusted to juries. This approach will reassert the jury’s role as the traditional factfinder, promote corrective justice and deterrence values, and lead to greater equity for negligent infliction of emotional distress (NIED) claimants. The traditional limitations on tort recovery, including the rules of evidence and causation, are more than adequate to avoid opening the floodgates to emotional distress claims

The Future of Emotional Harm

Why should tort law treat claims for emotional harm as a second-class citizen? Judicial skepticism about these claims is long entrenched, justified by an amalgam of perceived problems ranging from proof difficulties for causation and the need to constrain fraudulent claims, to the ubiquity of the injury, and a concern about open-ended liability. To address this jumble of justifications, the law has developed a series of duty limitations to curb the claims and preclude them from reaching the jury for individualized analysis. The limited duty approach to emotional harm is maintained by the latest iteration of the Restatement (Third) of Torts. This Article argues that many of the justifications for curtailing this tort have been discredited by scientific developments. In particular, the rapid advances in neuroscience give greater insight into the changes that occur in the brain from emotional harm. Limited duty tests should no longer be used as proxies for validity or justified by the presumed untrustworthiness of the claim. Instead, validity evidence for emotional harm claims—like evidence of physical harm—should be entrusted to juries. This approach will reassert the jury’s role as the traditional factfinder, promote corrective justice and deterrence values, and lead to greater equity for negligent infliction of emotional distress (NIED) claimants. The traditional limitations on tort recovery, including the rules of evidence and causation, are more than adequate to avoid opening the floodgates to emotional distress claims

A comparison of two phosphorus soil tests as inputs to a pasture growth model : a thesis presented in partial fulfilment of the requirements for the degree of Master of Agricultural Science in Soil Science at Massey University

Glasshouse and field studies were carried out to investigate relationships between plant growth and extractable soil phosphorus and between fertilizer phosphorus and extractable soil phosphorus respectively. The purpose of the studies was to provide information with which to quantify the parameters of a simple model designed to predict relative pasture yield as a function of soil and fertilizer phosphorus. The relationship between yield and water-extractable soil P differed markedly between two soils of different P retention properties in glasshouse studies using both intact cores and conventional pots. To illustrate this difference, the levels of water-extractable P (0-8 cm depth) in intact cores required for 90% of maximum yield were 12.7 and 2.6 μg/g soil in the soils of lower and higher P retention respectively. In contrast, the relationship between yield and Olsen (bicarbonate-extractable) P was much less soil type dependent. The corresponding levels of Olsen P in intact soil cores required for 90% of maximum yield were 17.7 and 17.8 ug/g soil respectively. For modelling purposes, the Olsen procedure was therefore considered to provide a more suitable index of plant available soil P from which to predict pasture production on soils differing in P retention. The proportion of yield variation accounted for by differences in extractable soil P was 25% or less in initial harvests from the intact cores, 50-75% in later harvests from the intact cores and 89-97% in the pot experiments. The results of the intact core experiments, however, were considered to be more directly applicable to the field situation than were the results of the pot experiments. Seasonal changes in extractable soil P in Tokomaru silt loam included an increase during the dry season to reach a peak in late autumn followed by a decline in winter. The magnitude of these changes with respect to Olsen P was approximately 2.5 and 5 μg/g soil in the 0-8 cm and 0-4 cm depths respectively. A subsequent decline in extractable soil P during the spring and second summer was attributed largely to plant uptake of soil P and its loss in discarded clippings. The application of superphosphate increased extractable soil P in proportion to the rate applied. The increases per unit of applied fertilizer P, in both absolute terms and relative to an initial (time-zero) increase, were greater in a soil of low P retention (Tokomaru) than in a soil of high P retention (Ramiha). Water-extractable P (0-8 cm depth) was increased on average by 2.3 and 0.2 μg/g in the Tokomaru and Ramiha soils respectively six months after the application of 40 kg P/ha as super-phosphate. The corresponding average increases in Olsen P (2.7 and 1.1 μg/g) were greater, and differed less between the soils, than the increases in water-extractable P. Thus, neither soil P extraction procedure was independent of soil type in terms of the effects of applied fertilizer P. For modelling purposes the effects of applied fertilizer would need to be assessed in a wider range of soils. The level of water-extractable P in stored, air-dry soils was found to undergo short-term fluctuations, apparently due to changes in the conditions of extraction such as variations in the pH of distilled water. Longer-term increases of 25-100% in the level of water-extractable P of stored soils also occurred. No reason for the latter changes was apparent

Examination of Molinism

What is the driving force behind salvation? Is it God’s sovereign will, enacting His efficacious grace upon the heart of man? Or is it the free will of man himself, choosing to accept the grace that has been extended to him? This is the age-old question behind the argument of sovereignty versus free will. Luis de Molina, a sixteenth century Jesuit theologian, believed that God, through His omniscience and omnipotence, can predestine an individual for salvation while keeping the free will of that individual intact. This system, known as Molinism, stands on three main principles: a wholly libertarian account of man’s free will, the conviction that the grace the Lord extends to complete salvific acts is not in itself intrinsically efficacious, and the assumption of the truth of the concept of Scientia media, or Middle Knowledge

Computing Dynamic Heterogeneous-Agent Economies: Tracking the Distribution

Theoretical formulations of dynamic heterogeneous-agent economiestypically include a distribution as an aggregate state variable. This paperintroduces a method for computing equilibrium of these models by including a distribution directly as a state variable if it is finite-dimensional or a fine approximation of it if infinite-dimensional. The method accurately computes equilibrium in an extreme calibration of Huffman's (1987) overlapping-generations economy where quasi-aggregation, the accurate forecasting of prices using a small state space, fails to obtain. The method also accurately solves for equilibrium in a version of Krusell and Smith's (1998) economy wherein quasi-aggregation obtains but households face occasionally binding constraints. The method is demonstrated to be not only accurate but also feasible with equilibria for both economies being computed in under ten minutes in Matlab. Feasibility is achieved by using Smolyak's (1963) sparse-grid interpolation algorithm to limit the necessary number of gridpoints by many orders of magnitude relative to linear interpolation. Accuracy is achieved by using Smolyak's algorithm, which relies on smoothness, only for representing the distribution and not for other state variables such as individual asset holdings.Numerical Solutions, Heterogeneous Agents, Projection Methods