# Mathematics

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Item Encryption using the rubik's cube.(Wheaton College (MA), 2019) Feng, WeiqiRubik's Cubes, estimated to be the world's besting-selling toy, is a fascinating puzzle that I believe most people have attempted to solve. However, as anyone who has tried knows, Rubik's Cubes are hard to solve. Cryptography shares the essence with playing Rubik's Cubes; to fix the scrambled pattern. In this thesis, we discuss the advantages and challenges of using the Rubik's Cube as the basis of an encryption system.We also test the Rubik's Cube encryption under the formal definition of security while seek additional opportunities provided by using 4 by 4 by 4 and larger cubes. Finally we construct a key-exchange protocol based on the Rubik's Cube using a structure similar to Diffie-Hellman key exchange.Item Examining different measures used to detect gerrymandering.(Wheaton College (MA), 2019) Helmreich, RaeGerrymandering is the act of changing political boundaries so as to favor one political party or class in an election. Much effort has been made in recent years to mathematically detect gerrymandering in order to prevent this manipulation. Markov Chain Monte Carlo (MCMC) methods have been used to sample the distribution of legal and reasonable redistrictings of a particular area, which allows comparison of specific districtings with the space of reasonable legal districtings. Depending on the state, districting plans in America are often required to be {esc}(3z{esc}(Bcompact{esc}(3y{esc}(B, though this term is loosely defined by the law. Fortunately, the MCMC method can take into account the compactness of districts, but of course this requires a strict definition of compactness. Unfortunately, different groups have used a multitude of measures of compactness. An open question in this field is what relationship, if any, exists between diffferent measures of compactness. Two of the methods that have been used are the Isoperimetric Score and the Cut Edge Score of a districting plan. Isoperimetric Score looks at the geography of a districting plan, while Cut Edge Score examines the relational structure of a districting plan. Looking at these two seemingly different measures of compactness, I have found that prioritizing Cut Edge Score with MCMC methods may result in more compact districting plans, both in terms of Isoperimetric Score and Cut Edge Score, more easily than when taking into account Isoperimetric Score. Moreover, there is consistently a strong relationship between the two scores. In ad- dition, I looked at the Population Balance Score of districting plans, which measures how close a districting plan is to having an equal population in each district. Look- ing at this score, I have found that in certain cases prioritizing Cut Edge Score in MCMC methods results in better Population Balance Scores than when prioritizing Isoperimetric Score.Item Hidden markov models for music classification.(Wheaton College (MA), 2019) Liu, XinruItem Kings in generalized tournaments.(Wheaton College (MA)., 2018) Zhang, ChengThis thesis explores how to find and construct kings in three generalizations of tourna- ment: semi-complete digraphs, oriented graphs and quasi-transitive oriented graphs.In Chapter 3 and Chapter 4, we present a way to interpret semi-complete digraphs and oriented graphs as tournaments with “ties” (we call the “ties” in semi-complete digraphs “double ties”, and the “ties” in oriented graphs “ties”). In Chapter 3, we prove there exists an (n, k) semi-complete digraph if and only if n ≥ k ≥ 1, and all the (n, k) semi-complete digraphs that exist can be constructed with at most 1 double tie. In Chapter 4, we prove there exists an (n, k) oriented graph for all n ≥ k ≥ 0 except (1, 0), (2, 2), (3, 2), and (4, 4) oriented graphs, and we prove that all the (n, k) oriented graphs that exist can be constructed with at most 1 tie.The main focus of this thesis is quasi-transitive oriented graph, which is discussed in Chapter 5. We show an interesting fact that all the quasi-transitive oriented graphs can be condensed into tournaments by “tie component condensations”. Then, we show that the tie component condensation on a quasi-transitive oriented graph is a most efficient condensation to tournament in all the condensations to tournaments defined on all the oriented graph with the same tie structure. Finally we prove that the kings in a quasi- transitive oriented graph Q are related to the kings in the “underlying tournament of Q” (result of Q after tie component condensation). This result gives us a way to understand the properties of kings in quasi-transitive oriented graphs using the properties of king in tournaments.Item Perfect colorings of a design with 2-dimensional euclidean crystallographic symmetry group.(Wheaton College (MA)., 2018) Xu, XiThis thesis introduces perfect colorings and the systematic way of generating perfect colorings. We study the perfect colorings for designs whose symmetry groups are 2-dimensional Euclidean crystallographic groups. Two-dimensional Euclidean crys- tallographic groups are discrete subgroups of the isometry group of the Euclidean plane. We classify crystallographic groups by their ranks. Crystallographic groups of rank 0, rank 1, and rank 2 are explained with details and examples. Theorems that related to the number of perfect colorings for designs with crystallographic symmetry groups are constructed and proved.