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Full-Text Articles in Algebra
Polygonal Analogues To The Topological Tverberg And Van Kampen-Flores Theorems, Leah Leiner
Polygonal Analogues To The Topological Tverberg And Van Kampen-Flores Theorems, Leah Leiner
Senior Projects Spring 2019
Tverberg’s theorem states that any set of (q-1)(d+1)+1 points in d-dimensional Euclidean space can be partitioned into q subsets whose convex hulls intersect. This is topologically equivalent to saying any continuous map from a (q-1)(d+1)-dimensional simplex to d-dimensional Euclidean space has q disjoint faces whose images intersect, given that q is a prime power. These continuous functions have a Fourier decomposition, which admits a Tverberg partition when all of the Fourier coefficients, except the constant coefficient, are zero. We have been working with continuous functions where all of the Fourier coefficients except the constant and one other coefficient are zero. …
The Conditional Probability That An Elliptic Curve Has A Rational Subgroup Of Order 5 Or 7, Meagan Kenney
The Conditional Probability That An Elliptic Curve Has A Rational Subgroup Of Order 5 Or 7, Meagan Kenney
Senior Projects Spring 2019
Let E be an elliptic curve over the rationals. There are two different ways in which the set of rational points on E can be said to be divisible by a prime p. We will call one of these types of divisibility local and the other global. Global divisibility will imply local divisibility; however, the converse is not guaranteed. In this project we focus on the cases where p=5 and p=7 to determine the probability that E has global divisibility by p, given that E has local divisibility by p.
Factorization Lengths In Numerical Monoids, Maya Samantha Schwartz
Factorization Lengths In Numerical Monoids, Maya Samantha Schwartz
Senior Projects Spring 2019
A numerical monoid M generated by the natural numbers n_1, ..., n_k is a subset of {0, 1, 2, ...} whose elements are non-negative linear combinations of the generators n_1, ..., n_k. The set of factorizations of an element in M is the set of all the different ways to write that element as a linear combination of the generators. The length of a factorization of an element is the sum of the coefficients of that factorization. Since an element in a monoid can be written in different ways in terms of the generators, its set of factorization lengths may …