Tangent Lines

In [1] Leibniz published the first treatment of the subject of calculus. An English translation can be found in [2]; he says that to find a tangent is to draw a right line, which joins two points of the curve having an infinitely small difference, or the side of an infinite angled polygon produced, which is equivalent to the curve for us. Today, according to [3], a straight line is said to be a tangent line of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f\u27(c) where f\u27 is the derivative of f. At first glance, it would appear that this second definition is simply a more precise version of the first; indeed, the cited Wikipedia article states this sentiment explicitly. In this paper we examine cases where Wikipedia\u27s definition is more strict than Leibniz\u27s original one, and present two attempts at formulating a more general, but still precise, definition

Elliptic Curves

Elliptic curves have found widespread use in number theory and applications thereof, such as cryptography. In this paper we will first examine the basic theory of elliptic curves and then look specifically at how they can be used to construct cryptographic systems more efficient than their counterparts, and how they can be used to generate proofs for or against primality

Investigating the cyclization of enediyne analogs using density functional theory

Enediynes are organic molecules that readily undergo a thermal rearrangement, now commonly known as the Bergman cyclization, to a cyclic para diradical form. Interest in this rearrangement was renewed when it was found to be crucial to the mechanism of cytotoxicity in a variety of natural products containing the enediyne structural moiety. Cyclization of these molecules leads to DNA strand scission and ultimately cell death. Recent efforts by medicinal chemists to discover therapeutically relevant enediyne derivatives have been complemented by computational approaches, which seek to compute energies and energetic barriers to cyclization that can accurately predict the behavior of these molecules in vivo. Here we demonstrate this approach for cis-hex-3-ene-1,5-diyne and two of its analogs using density functional theory, discuss the validity of its predictions, and investigate the effect of basis set on the description of these molecules’ reactivity.Pharmac

Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps

Redesigning Choice Architecture for the College Selection and Application Process for High School Juniors

Choice architecture, a term coined in the 2008 book Nudge: Improving Decisions About Health, Wealth, and Happiness by behavioral scientist Richard H. Thaler and professor of law Cass R. Sunstein, is the belief that no choice can be neutral because the way in which a choice is presented influences the decision-maker. Designers, perfectly positioned to structure choices in both print and digital format, should equally embrace this role and handle it with great responsibility. Rather than use the power to structure choice architecture to bolster consumerism, we should instead harness the opportunity to help individuals make choices that improve their health, finances, relationships, and prospects - thereby creating change that could scale to families, communities, states, and even entire nations. This project is an exploration of the ways a designer can structure the choice architecture for the first major life decision many young folks face - which college to attend. Considered within the context of the current West Virginia college landscape, and using qualitative and quantitative data gathered from a group representing the target audience, the project provides a model for leading students through the complex process of evaluating options and making a decision that is best suited to their situation, without encroaching on their freedom of choice

Population Structure of Tursiops truncatus in North Inlet, South Carolina and Use of Inlet as a Nursery Ground

Abundance, distribution and movement patterns of bottlenose dolphin populations are becoming more frequently studied along the Atlantic Coast from Virginia to Florida. One aspect of dolphin populations that has not been a focus of study is the use of nursery grounds by females with calves. Nursery grounds for bottlenose dolphin populations have been identified in Sarasota Bay, Florida and near Beaufort, North Carolina. This study analyzed data collected from September 1997 to June 2006 on focal follows and photo identification to determine if females were utilizing North Inlet, South Carolina as a nursery ground. There was no significant difference in the number of calves + young-of-year (YOY) sighted between seasons, indicating that females did not use the Inlet more during any one season to rear their young. Two resident females gave birth from mid-May to early June and two from the end of June to the beginning of July. We could not conclude an accurate time of birth for the fifth mom in the study

Solutions Network Formulation Report. Visible/Infrared Imager/Radiometer Suite and Landsat Data Continuity Mission Simulated Data Products for the Great Lakes Basin Ecological Team

The proposed solution would simulate VIIRS and LDCM sensor data for use in the USGS/USFWS GLBET DST. The VIIRS sensor possesses a spectral range that provides water-penetrating bands that could be used to assess water clarity on a regional spatial scale. The LDCM sensor possesses suitable spectral bands in a range of wavelengths that could be used to map water quality at finer spatial scales relative to VIIRS. Water quality, alongshore sediment transport and pollutant discharge tracking into the Great Lakes system are targeted as the primary products to be developed. A principal benefit of water quality monitoring via satellite imagery is its economy compared to field-data collection methods. Additionally, higher resolution satellite imagery provides a baseline dataset(s) against which later imagery can be overlaid in GIS-based DST programs. Further, information derived from higher resolution satellite imagery can be used to address public concerns and to confirm environmental compliance. The candidate solution supports the Public Health, Coastal Management, and Water Management National Applications

Sports in the time of COVID-19

Linfield coaches and staff repeated these steps more than 900 times in the past year, administering COVID-19 tests to student-athletes. They did all these tests – in addition to temperature checks, rigorous cleanings and more – in the hopes of bringing back athletic competition to Linfield