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Full-Text Articles in Mathematics
Fixed Points Of Functions Below The Line Y = X, Grace Fryling, Harrison Rouse
Fixed Points Of Functions Below The Line Y = X, Grace Fryling, Harrison Rouse
Undergraduate Mathematics Day: Past Content
This paper concerns fixed points of functions whose graphs lie on or below the line y = x. Using the Monotone Convergence Theorem, we show that positive fixed points of such functions are “attracting on the right” so long as we include a couple of further assumptions about these functions near their fixed points. As an illustrative example, we confirm that this is the case for the function y = x sin x; the positive fixed points of this function “attract on the right” and “repel on the left.” Further, we generalize by showing that differentiability is in fact not …
Finding An Effective Shape Parameter Strategy To Obtain The Optimal Shape Parameter Of The Oscillatory Radial Basis Function Collocation In 3d, Quinnlan Aiken, Annika Murray, Ar Lamichhane
Finding An Effective Shape Parameter Strategy To Obtain The Optimal Shape Parameter Of The Oscillatory Radial Basis Function Collocation In 3d, Quinnlan Aiken, Annika Murray, Ar Lamichhane
Undergraduate Mathematics Day: Past Content
Recent research into using the Method of Approximate Particular Solutions to numerically solve partial differential equations, has shown promising results. High levels of accuracy can be obtained when implementing this method, however the success of this collocation method is dependent on a shape parameter that is found in nearly all radial basis functions. If the shape parameter is not appropriately chosen, then it can provide an unacceptable result. Two shape parameter strategies are considered, a random variable shape parameter strategy and a leave-one-out cross validation strategy. The main objective of this work is to assess the viability of using these …
Using Circle Packings To Approximate Harmonic Measure Distribution Functions, Ella Wilson
Using Circle Packings To Approximate Harmonic Measure Distribution Functions, Ella Wilson
Undergraduate Mathematics Day: Past Content
Harmonic measure distribution functions, h-functions, encode information about the geometry of domains in the plane. Specifically, given a domain and a basepoint in the domain, for a fixed radius, r, the value h(r) is the probability that a Brownian particle first exits the domain within distance r of the basepoint. There are many domains for which we can compute h-functions, such as the disk and the inside and outside of a wedge. However, exact computation is often difficult or impossible for more complicated domains, so we need methods to approximate these h-functions. In this paper, we develop two methods for …
Efficient Conformal Binary Classification Under Nearest Neighbor, Maxwell Lovig
Efficient Conformal Binary Classification Under Nearest Neighbor, Maxwell Lovig
Undergraduate Mathematics Day: Past Content
There are many types of statistical inferences that can be used today: Frequentist, Bayesian, Fiducial, and others. However, Vovk introduced a new version of statistical inference known as Conformal Predictions. Conformal Predictions were designed to reduce the assumptions of standard prediction methods. Instead of assuming all observations are drawn independently and identically distributed, we instead assume exchangeability. Meaning, all N! possible orderings of our N observations are equally likely. This is more applicable to fields such as machine learning where assumptions may not be easily satisfied. In the case of binary classification, Vovk provided the nearest neighbors (NN) measure which …