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Full-Text Articles in Physical Sciences and Mathematics
An Exposition Of Kasteleyn's Solution Of The Dimer Model, Eric Stucky
An Exposition Of Kasteleyn's Solution Of The Dimer Model, Eric Stucky
HMC Senior Theses
In 1961, P. W. Kasteleyn provided a baffling-looking solution to an apparently simple tiling problem: how many ways are there to tile a rectangular region with dominos? We examine his proof, simplifying and clarifying it into this nearly self-contained work.
A Combinatorial Exploration Of Elliptic Curves, Matthew Lam
A Combinatorial Exploration Of Elliptic Curves, Matthew Lam
HMC Senior Theses
At the intersection of algebraic geometry, number theory, and combinatorics, an interesting problem is counting points on an algebraic curve over a finite field. When specialized to the case of elliptic curves, this question leads to a surprising connection with a particular family of graphs. In this document, we present some of the underlying theory and then summarize recent results concerning the aforementioned relationship between elliptic curves and graphs. A few results are additionally further elucidated by theory that was omitted in their original presentation.
A Plausibly Deniable Encryption Scheme For Personal Data Storage, Andrew Brockmann
A Plausibly Deniable Encryption Scheme For Personal Data Storage, Andrew Brockmann
HMC Senior Theses
Even if an encryption algorithm is mathematically strong, humans inevitably make for a weak link in most security protocols. A sufficiently threatening adversary will typically be able to force people to reveal their encrypted data. Methods of deniable encryption seek to mend this vulnerability by allowing for decryption to alternate data which is plausible but not sensitive. Existing schemes which allow for deniable encryption are best suited for use by parties who wish to communicate with one another. They are not, however, ideal for personal data storage. This paper develops a plausibly-deniable encryption system for use with personal data storage, …
Chromatic Polynomials And Orbital Chromatic Polynomials And Their Roots, Jazmin Ortiz
Chromatic Polynomials And Orbital Chromatic Polynomials And Their Roots, Jazmin Ortiz
HMC Senior Theses
The chromatic polynomial of a graph, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph. We can then find the orbital chromatic polynomial of a graph and a group of automorphisms of the graph, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph when acted upon by the group. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture …