The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z

This paper answers a question asked by Ed Pegg Jr. in 2001: "What is the maximal value of min(A,B,C)/ gcd(A,B,C) for A^x + B^y = C^z with A,B,C >= 1; x,y,z >= 3?" Equations of this form are analyzed, showing how they map to exponential Diophantine equations with coprime bases. A search algorithm is provided to find the largest min/gcd value within a given equation range. The algorithm precalculates a multi-gigabyte lookup table of power residue information that is used to eliminate over 99% of inputs with a single array lookup and without any further calculations. On inputs that pass this test, the algorithm then performs further power residue tests, avoiding modular powering by using lookups into precalculated tables, and avoiding division by using multiplicative inverses. This algorithm is used to show the largest min/gcd value for all equations with C^z <= 2^100.Comment: Body: 16 pages, Appendices: 11 pages, 5 tables, 1 figur

A comparison of the speed and quality of manuscript and cursive handwriting at the college level.

Thesis (M.Ed.)--Boston Universit

Developing, Implementing and Evaluating Policies to Support Fisheries Co-management

The aim of this document is to bring together a number of the lessons relating to the development, implementation and evaluation of policies to support co-management that have emerged from projects undertaken through the DFID Fisheries Management Science Programme (FMSP) and elsewhere. It is beyond the scope of this document to provide a comprehensive analysis or guide. It seeks to highlight some experiences and some areas that need to be considered by policy makers when attempting to develop sustainably co-managed fisheries. This document is targeted to fisheries policy makers, and decision-makers concerned with the fisheries sector

Step Scaling with off-shell renormalisation

We make use of twisted boundary conditions for off-shell Rome-Southampton renormalisation. This allows to define the vertex function precisely, at a fixed physical momentum that need not be one of the Fourier modes of a simulation. This definition includes choosing the orientation with respect to lattice axes, and so lattice artefacts including ${\cal O}(4)$ breaking then have a valid Symanzik expansion. Excellent statistical precision is afforded by volume plane-wave sources, enabling both a theoretically and statistically clean continuum limit to be taken. Thereafter all $p^2$ dependence can be unambiguously identified as continuum anomolous running. The use of non-exceptional momenta has been found to greatly reduce the dependence of non-perturbative vertex functions on both mass and $p^2$. We illustrate how this can be developed into a practical scheme for step-scaling the RI/SMOM approach with initial results. The size of the possible steps is continuous, rather than discrete allowing one to take arbitrarily small steps, and the scheme inherits from RI/MOM the property that it is easy to implement for general operators.Comment: 35 page

SOX10 commonly stains scar in Mohs sections

Sox10 immunostaining is used for the diagnosis and margin evaluation of melanocytic lesions. Sox10 was initially thought not to stain fibrohistiocytic processes. Consequently, it was believed to reliably distinguish desmoplastic melanoma from scar. However, recent data from formalin sections suggest Sox10 is less specific than previously thought. In this report, we demonstrate that Sox10-stained Mohs sections commonly show strong, fractional staining of scar. When using Sox10 with frozen section immunohistochemistry, Mohs practitioners should recognize the potential of this marker to stain scar to avoid overdiagnosis of desmoplastic melanoma

A Native Hymenopteran Predator of \u3ci\u3eAgonopterix Alstroemeriana\u3c/i\u3e (Lepidoptera: Oecophoridae) in East-Central Illinois

Agonopterix alstroemeriana is a European oecophorid moth that defoliates poison hemlock (Conium maculatum), a noxious Eurasian weed extensively naturalized throughout temperate Australia, New Zealand, North America, and South America. Throughout western North America, and increasingly in the Midwest and Northeast, A. alstroemeriana has been utilized in poison hemlock eradication programs. We report, for the first time, predation on A. alstroemeriana by Euodynerus foraminatus (Hymenoptera: Eumenidae), a native solitary wasp that paralyzes these and other lepidopteran larvae and uses them to provision its nests. The presence of an effective predator may reduce the impact of A. alstroemeriana in biological control programs

Head-mounted spatial instruments II: Synthetic reality or impossible dream

A spatial instrument is defined as a spatial display which has been either geometrically or symbolically enhanced to enable a user to accomplish a particular task. Research conducted over the past several years on 3-D spatial instruments has shown that perspective displays, even when viewed from the correct viewpoint, are subject to systematic viewer biases. These biases interfere with correct spatial judgements of the presented pictorial information. The design of spatial instruments may not only require the introduction of compensatory distortions to remove the naturally occurring biases but also may significantly benefit from the introduction of artificial distortions which enhance performance. However, these image manipulations can cause a loss of visual-vestibular coordination and induce motion sickness. Consequently, the design of head-mounted spatial instruments will require an understanding of the tolerable limits of visual-vestibular discord