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We introduce the theory of Latin squares and their partial transversals. Furthermore, we present the history and some applications of Latin squares. We also prove using elementary methods that a 6 x 6 Latin square has a partial transversal of length 5. After developing some of the theory of so-called partial Latin squares we provide a simple proof of a result originally due to Woolbright, which gives a lower bound for the length of the longest partial transversal in an n x n Latin square. Followingly we improve slightly the best known lower bound for the length of the longest partial transversal in an n x n Latin square.

This paper develops several classical algorithms and cryptosystems in cryptography, and develops the theory of elliptic curves to reveal the improvements provided by elliptic curve cryptography. The prerequisites to this paper are an understanding of groups, fields, and some elementary number theory.

In the early 1830s, a young French mathematician named Èvariste Galois laid the foundations of group theory, although he never precisely defined groups. Galois studied groups in the context of sets of arrangements and his ideas were reformulated into a more abstract setting in the twentieth century. This paper provides precise definitions for constructs closely related to Galois's original notion of group theory and explores important group properties in that context, demonstrating that the modern concepts of the group, subgroup, normal subgroup, and solvable group can be expressed in terms of arrangement sets.

In this paper we investigate a general class of solutions to various partial differential equations known as solitons or stable solitary wave solutions. We introduce necessary background by considering general solutions of the classical wave equation and some of its variants, focusing on features of linearity, non-linearity, dissipation and dispersion. The Korteweg-de Vries (KdV) equation is presented as an iconic non-linear dispersive wave equation that admits soliton solutions. How soliton solutions are approximated motivates an introduction to the Padè approximation, which seeks convergence by expressing a solution as a quotient G/F of polynomials of exponentially decaying functions. The Padè approximation motivates a substitution that decouples the KdV equation into a pair of equations on the polynomials G and F. The decoupled version of the KdV equation is then greatly simplified by introducing a bilinear differentiation operator known as Hirota's D-operator. Another substitution allows Hirota's D-operator to express the KdV equation in a single bilinear form. This final form illustrates how the perturbation method can be used to produce exact soliton and multi-soliton solutions. The generation of multi-solition solutions in an almost additive fashion with this method is summarized as a non-linear superposition principle. Connections between Hirota's method, Kac-Moody algebras and quantum field theory are briefly mentioned.

We explain an approach, due originally to Barnes, to tiling problems using some commutative algebra. We investigate in particular the occurence of coloring arguments in tiling problems. The only prerequisites are linear algebra and familiarity with rings and ideals.

Determining when two links are equivalent is one of the central goals of knot theory. This paper describes the Conway polynomial, a link invariant that offers one approach to this problem. When calculating the Conway polynomial of the (n,2) torus knots, we encounter the familiar patterns of Pascal's triangle and the Fibonacci sequence.