Open Access. Powered by Scholars. Published by Universities.®
- Discipline
-
- Number Theory (6)
- Analysis (2)
- Computer Sciences (2)
- Geometry and Topology (2)
- Theory and Algorithms (2)
-
- Applied Mathematics (1)
- Arts and Humanities (1)
- Astrophysics and Astronomy (1)
- Discrete Mathematics and Combinatorics (1)
- Music (1)
- Music Theory (1)
- Numerical Analysis and Scientific Computing (1)
- Other Applied Mathematics (1)
- Other Astrophysics and Astronomy (1)
- Other Mathematics (1)
- Stars, Interstellar Medium and the Galaxy (1)
- Keyword
-
- Group Theory (2)
- Tverberg (2)
- Abstract Algebra (1)
- Abstraction (1)
- Action-angle variables (1)
-
- Algebra (1)
- Algebraic topology (1)
- Associativity (1)
- Basis (1)
- Binary Operation (1)
- Borda Count (1)
- Borsuk-Ulam (1)
- CSP (1)
- Cayley graph (1)
- Chinese Remainder Theorem (1)
- Conjugacy (1)
- Connectivity (1)
- Conserved quantities (1)
- Constraint satisfaction problem (1)
- Convex Geometry (1)
- Decision Problems (1)
- Determinant (1)
- Direct imaging (1)
- Divisibility (1)
- Elliptic Curves (1)
- Exoplanets (1)
- Factorization (1)
- Finite Fourier Analysis (1)
- Flow variables (1)
- Kepler problem (1)
Articles 1 - 15 of 15
Full-Text Articles in Algebra
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Senior Projects Spring 2023
Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be …
Hidden Symmetries Of The Kepler Problem, Julia Kathryn Sheffler
Hidden Symmetries Of The Kepler Problem, Julia Kathryn Sheffler
Senior Projects Spring 2022
The orbits of planets can be described by solving Kepler’s problem which considers the motion due to by gravity (or any inverse square force law). The solutions to Kepler’s problem, for energies less then 0, are ellipses, with a few conserved quantities: energy, angular momentum and the Laplace-Runge-Lenz (LRL) vector. Each conserved quantity corresponds to symmetries of the system via N ̈other’s theorem. Energy conservation relates to time translations and angular momentum to three dimensional rotations. The symmetry related to the LRL vector is more difficult to visualize since it lives in phase space rather than configuration space. To understand …
Tverberg Type Partitions: Sub-Regular And Elliptical Polygons, Tobias Golz Timofeyev
Tverberg Type Partitions: Sub-Regular And Elliptical Polygons, Tobias Golz Timofeyev
Senior Projects Spring 2021
Tverberg's theorem states that given a set S of T(r,d)=(r-1)(d+1)+1 points in Rd, there exists a partition of S into r subsets whose convex hulls intersect. A feature of Tverberg's theorem is that T(r,d) is tight, so in this senior project we investigate Tverberg-type results when |S|. We found that in R2, given a set S of T(r,2)-2=3r-4 points, and assuming r=r1 r2, there exists a partition of S into r sets such that when grouped into r1 collections of r2 sets, the convex hulls of each collection overlap, and we …
N-Cycle Splines Over Sexy Rings, Jacob Tilden Cummings
N-Cycle Splines Over Sexy Rings, Jacob Tilden Cummings
Senior Projects Spring 2020
In this project we abstract the work of previous bard students by introducing the concept of splines over non-integers, non-euclidean domains, and even non-PIDs. We focus on n-cycles for some natural number n. We show that the concept of flow up class bases exist in PID splines the same way they do in integer splines, remarking the complications and intricacies that arise when abstracting from the integers to PIDs. We also start from scratch by finding a flow up class basis for n-cycle splines over the real numbers adjoin two indeterminates, denoted R[x,y] which necessitate more original techniques.
Connectedness In Cayley Graphs And P/Np Dichotomy For Quay Algebras, Thuy Trang Nguyen
Connectedness In Cayley Graphs And P/Np Dichotomy For Quay Algebras, Thuy Trang Nguyen
Senior Projects Spring 2020
This senior thesis attempts to determine the extent to which the P/NP dichotomy of finite algebras (as proven by Bulatov, et.al in 2017) can be cast in terms of connectedness in Cayley graphs. This research is motivated by Prof. Robert McGrail's work ``CSPs and Connectedness: P/NP-Complete Dichotomy for Idempotent, Right Quasigroups" published in 2014 in which he demonstrates the strong correspondence between tractability and total path-connectivity in Cayley graphs for right, idempotent quasigroups. In particular, we will introduce the notion of total V-connectedness and show how it could be potentially used to phrase the dichotomy in terms of connectivity for …
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 …
An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, Rachel K. Nalecz
An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, Rachel K. Nalecz
Senior Projects Spring 2018
We describe an implementation of the solution to the conjugacy problem in Thompson's group V as presented by James Belk and Francesco Matucci in 2013. Thompson's group V is an infinite finitely presented group whose elements are complete binary prefix replacement maps. From these we can construct closed abstract strand diagrams, which are certain directed graphs with a rotation system and an associated cohomology class. The algorithm checks for conjugacy by constructing and comparing these graphs together with their cohomology classes. We provide a complete outline of our solution algorithm, as well as a description of the data structures which …
Determinantal Conditions On Integer Splines, Kathryn Elizabeth Blaine
Determinantal Conditions On Integer Splines, Kathryn Elizabeth Blaine
Senior Projects Fall 2018
In this project, we work with integer splines on graphs with positive integer edge labels. We focus on graphs that are (m, n)-cycles for some natural numbers m, n, specifically the diamond graph, which consists of two triangles joined at an edge. We extend previous research on integer splines over the diamond graph. In particular, we prove that a set of splines on the diamond graph forms a basis if and only if it satisfies a certain determinantal criterion.
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Senior Projects Spring 2016
Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group's irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs.
Lose Big, Win Big, Sum Big: An Exploration Of Ranked Voting Systems, Erin Else Stuckenbruck
Lose Big, Win Big, Sum Big: An Exploration Of Ranked Voting Systems, Erin Else Stuckenbruck
Senior Projects Spring 2016
Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.
Associativity Of Binary Operations On The Real Numbers, Samuel Joseph Audino
Associativity Of Binary Operations On The Real Numbers, Samuel Joseph Audino
Senior Projects Spring 2016
It is known that there is an agreed upon convention of how to go about evaluating expressions in the real numbers. We colloquially call this PEMDAS, which is short for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is also called the Order of Operations, since it is the order in which we execute the operators of a given expression. When we remove this convention and begin to execute the operators in every possible order, we begin to see that this allows for many different values based on the order in which the operations are executed. We will investigate this question …
Basis Criteria For N-Cycle Integer Splines, Ester Gjoni
Basis Criteria For N-Cycle Integer Splines, Ester Gjoni
Senior Projects Spring 2015
In this project we work with integer splines on graphs with positive integer edge labels. We focus on graphs that are n-cycles for some natural number n. We find an explicit condition for when a set of splines can form a module basis for n-cycle splines. In general, a set of splines forms a Z-module basis if and only if their determinant is equal to the product of the edge labels divided by the greatest common divisor of those edge labels.
Filtering Irreducible Clifford Supermodules, Julia C. Bennett
Filtering Irreducible Clifford Supermodules, Julia C. Bennett
Senior Projects Spring 2011
A Clifford algebra is an associative algebra that generalizes the sequence R, C, H, etc. Filtrations are increasing chains of subspaces that respect the structure of the object they are filtering. In this paper, we filter ideals in Clifford algebras. These filtrations must also satisfy a “Clifford condition”, making them compatible with the algebra structure. We define a notion of equivalence between these filtered ideals and proceed to analyze the space of equivalence classes. We focus our attention on a specific class of filtrations, which we call principal filtrations. Principal filtrations are described by a single element in complex projective …