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Full-Text Articles in Logic and Foundations of Mathematics
Primality Proving Based On Eisenstein Integers, Miaoqing Jia
Primality Proving Based On Eisenstein Integers, Miaoqing Jia
Honors Theses
According to the Berrizbeitia theorem, a highly efficient method for certifying the primality of an integer N ≡ 1 (mod 3) can be created based on pseudocubes in the ordinary integers Z. In 2010, Williams and Wooding moved this method into the Eisenstein integers Z[ω] and defined a new term, Eisenstein pseudocubes. By using a precomputed table of Eisenstein pseudocubes, they created a new algorithm in this context to prove primality of integers N ≡ 1 (mod 3) in a shorter period of time. We will look at the Eisenstein pseudocubes and analyze how this new algorithm works with the …
Reading Between The Lines: Verifying Mathematical Language, Tristan Johnson
Reading Between The Lines: Verifying Mathematical Language, Tristan Johnson
Honors Theses
A great deal of work has been done on automatically generating automated proofs of formal statements. However, these systems tend to focus on logic-oriented statements and tactics as well as generating proofs in formal language. This project examines proofs written in natural language under a more general scope of mathematics. Furthermore, rather than attempting to generate natural language proofs for the purpose of solving problems, we automatically verify human-written proofs in natural language. To accomplish this, elements of discourse parsing, semantic interpretation, and application of an automated theorem prover are implemented.
Cantor's Infinity, Michael Warrener
Cantor's Infinity, Michael Warrener
Honors Theses
At the heart of mathematics is the quest to find patterns and order in some set of similar structures, whether these be shapes, functions, or even numbers themselves. In the late 1800’s, there was a strong focus in the mathematical community on the study of real numbers and sequences of real numbers. Mathematicians quickly realized, however, that in order to do any meaningful investigation into the properties of sequences of real numbers, they needed a better definition of real numbers than the loose intuitions that had been sucient for the generations prior. This led Georg Cantor (March 3, 1845 - …
The Calculus Of Variations, Erin Whitney
The Calculus Of Variations, Erin Whitney
Honors Theses
The Calculus of Variations is a highly applicable and advancing field. My thesis has only scraped the top of the applications and theoretical work that is possible within this branch of mathematics. To summarize, we began by exploring a general problem common to this field, finding the geodesic be-tween two given points. We then went on to define and explore terms and concepts needed to further delve into the subject matter. In Chapter 2, we examined a special set of smooth functions, inspired by the Calabi extremal metric, and used some general theory of convex functions in order to de-termine …
An Introduction To The P-Adic Numbers, Charles I. Harrington
An Introduction To The P-Adic Numbers, Charles I. Harrington
Honors Theses
One way to construct the real numbers involves creating equivalence classes of Cauchy sequences of rational numbers with respect to the usual absolute value. But, with a different absolute value we construct a completely different set of numbers called the p-adic numbers, and denoted Qp.
A General Look At Posets Rings And Lattices, Courtney A. Phillips
A General Look At Posets Rings And Lattices, Courtney A. Phillips
Honors Theses
A lattice is a type of structure that aims to organize certain relationships that exist between members of a set. This thesis seeks to define lattices, and demonstrate the different types. It will give examples of lattices, as well as various ways to describe and classify them.