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Full-Text Articles in Other Mathematics
Emergence And Complexity In Music, Zoe Tucker
Emergence And Complexity In Music, Zoe Tucker
HMC Senior Theses
How can we apply mathematical notions of complexity and emergence to music, and how can these mathematical ideas then inspire new musical works? Using Steve Reich's Clapping Music as a starting point, we look for emergent patterns in music by considering cases where a piece's complexity is significantly different from the total complexity of each of the individual parts. Definitions of complexity inspired by information theory, data compression, and musical practice are considered. We also consider the number of distinct musical pieces that could be composed in the same manner as Clapping Music. Finally, we present a new musical …
Sudoku Variants On The Torus, Kira A. Wyld
Sudoku Variants On The Torus, Kira A. Wyld
HMC Senior Theses
This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.
The Document Similarity Network: A Novel Technique For Visualizing Relationships In Text Corpora, Dylan Baker
The Document Similarity Network: A Novel Technique For Visualizing Relationships In Text Corpora, Dylan Baker
HMC Senior Theses
With the abundance of written information available online, it is useful to be able to automatically synthesize and extract meaningful information from text corpora. We present a unique method for visualizing relationships between documents in a text corpus. By using Latent Dirichlet Allocation to extract topics from the corpus, we create a graph whose nodes represent individual documents and whose edge weights indicate the distance between topic distributions in documents. These edge lengths are then scaled using multidimensional scaling techniques, such that more similar documents are clustered together. Applying this method to several datasets, we demonstrate that these graphs are …
Combinatorial Polynomial Hirsch Conjecture, Sam Miller
Combinatorial Polynomial Hirsch Conjecture, Sam Miller
HMC Senior Theses
The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …