Physical Sciences and Mathematics Commons^™

Reasoning & Proof In The Hs Common Core, Laurie O. Cavey

Laurie O. Cavey

No abstract provided.

Cardinal Invariants And The Borel Tukey Order, Samuel Coskey

Samuel Coskey

Many proofs of inequalities between cardinal characteristics of the continuum are combinatorial in nature. These arguments can be carried out in any model of set theory, even a model of CH where the inequalities themselves are trivial. Thus, such arguments appear to establish a stronger relationship than a mere inequality. The Borel Tukey order was introduced by Blass in a 1996 article to address just this. Specifically, he observed that the combinatorial information linking two cardinal characteristics is often captured by a pair of Borel maps called a Borel Tukey morphism. The existence of a Borel Tukey morphisms between …

How Do Mathematicians Make Sense Of Definitions?, Laurie O. Cavey, Margaret T. Kinzel, Thomas A. Kinzel, Kathleen L. Rohrig, Sharon B. Walen

Laurie O. Cavey

It seems clear that students’ activity while working with definitions differs from that of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians’ and students’ work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). As part of an ongoing study, we chose to look closely at how mathematicians make sense of definitions in hopes of informing the ways in which we interpret students’ activity and support their understanding of definitions. We conducted interviews with mathematicians in an attempt to reveal their process when making sense of definitions. A striking …

Proportional Reasoning 101, Laurie O. Cavey

Laurie O. Cavey

No abstract provided.

A Note On Mustata's Computation Of Multiplier Ideals Of Hyperplane Arrangements, Zach Teitler

Zach Teitler

In 2006, M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give a simpler proof using a log resolution and generalize to non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini–Procesi in 1995, we also simplify the result. Indeed, Mustat¸˘a’s result expresses the multiplier ideal as an intersection, and our result uses (generally) fewer terms in the intersection.

Fast Multilevel Evaluation Of 1-D Piecewise Smooth Radial Basis Function Expansions, Oren E. Livne, Grady B. Wright

Grady Wright

No abstract provided.

An Informal Introduction To Computing With Chern Classes, Zach Teitler

Zach Teitler

Some time ago, Dr. Teitler began work on a short expository set of notes on Chern classes in algebraic geometry, particularly in the context of enumerative problems. The notes are not polished. Some day he hopes to finish them; in the meantime, here is a draft, in PDF format (21 pages).

Radial Basis Function Interpolation: Numerical And Analytical Developments, Grady Wright

Grady Wright

The Radial Basis Function (RBF) method is one of the primary tools for interpolating multidimensional scattered data. The methods' ability to handle arbitrarily scattered data, to easily generalize to several space dimensions, and to provide spectral accuracy have made it particularly popular in several different types of applications. Some of the more recent of these applications include cartography, neural networks, medical imaging, and the numerical solution of partial differential equations (PDEs). In this thesis we study three issues with the RBF method that have received very little attention in the literature.

First, we focus on the behavior of RBF interpolants …