If P is an integer polynomial denote the degree of P by ∂(P) and let H(P) be the maximum of the absolute value of the coefficients of P. Define Λ(P)=2[superscript ∂(P)]H(P) and for a fixed prime p let C[subscript p] denote the completion of the algebraic closure of the p-adic...
Geometric Problems become increasingly intractable and difficult to visualize as the number of dimensions increases beyond three. Inductions from lower dimensional spaces are possible yet often awkward. This thesis shows how elementary linear algebra, vector calculus, and combinatorics offer improved methods for calculating the dihedral angles of n-simplicies and proving...
We show that Pappus Curves, introduced by R. Schwartz to study his dynamical system in the real projective plane generated by iterated applications of the classical Pappus Theorem, are algebraic exactly in the linear case. Our approach is to use properties of projective curves such as singular points, genus, number...
The notion of a normal number and the Normal Number Theorem date back over 100 years. Émile Borel first stated his Normal Number Theorem in 1909. Despite their seemingly basic nature, normal numbers are still engaging many mathematicians to this day. In this paper, we provide a reinterpretation of the...
Finding new examples of compact simply connected spaces admitting a Riemannian metric of positive sectional curvature is a fundamental problem in differential geometry. Likewise, studying topological properties of families of manifolds is very interesting to
topologists. The Eschenburg spaces combine both of those interests: they are positively curved Riemannian manifolds...
We use the theory of continued fractions over function fields in the setting of hyperelliptic curves of equation y²=f(x), with deg(f)=2g+2. By introducing a new sequence of polynomials defined in terms of the partial quotients of the continued fraction expansion of y, we are able to bound the sum of...
In real networks, identifying dense regions is of great importance. For example, in a network that represents academic collaboration, authors within the densest component of the graph tend to be the most prolific. Dense subgraphs often identify communities in social networks. And dense subgraphs can be used to discover regulatory...
Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL₂(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of...
The goal of this paper is to classify linear operators with octonionic coefficients and octonionic variables. While building up to the octonions we also classify linear operators over the quaternions and show how to relate the linear operators over the quaternions and octonions to matrices. We also construct a basis...