Open Access. Powered by Scholars. Published by Universities.®
Science and Mathematics Education
Department of Mathematics: Dissertations, Theses, and Student Research
Articles 1 - 2 of 2
Full-Text Articles in Education
Numerical Integration Of Linear And Nonlinear Wave Equations, Laura Lynch
Numerical Integration Of Linear And Nonlinear Wave Equations, Laura Lynch
Department of Mathematics: Dissertations, Theses, and Student Research
We begin our study with an analysis of various numerical methods and boundary conditions on the well-known and well-studied advection and wave equations, in particular we look at the FTCS, Lax, Lax-Wendroff, Leapfrog, and Iterated Crank Nicholson methods with periodic, outgoing, and Dirichlet boundary conditions. We will then extend our study to the nonlinear equation gtt = gxx – gt2/g, introduced by Khoklov and Novikov. The nonlinearities are similar to those seen in General Relativity, and thus our analysis establishes the effects of numerical integration and boundary condition choices on the long-term stability …
Factorability In The Ring Z[√–5], Laura Lynch
Factorability In The Ring Z[√–5], Laura Lynch
Department of Mathematics: Dissertations, Theses, and Student Research
The fundamental theorem of arithmetic says that any integer greater than 2 can be written uniquely as a product of primes. For the ring Z[√–5], although unique factorization holds for ideals, unique factorization fails for elements. We investigate both elements and ideals of Z[√–5]. For elements, we examine irreducibility (the analog of primality) in Z[√–5] and look at how often and how badly unique fac- torization fails. For ideals, we examine irreducibility again and a proof for unique factorization.