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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Mathematics
The invariant subspace problem asks if every bounded linear operator on a Banach space has a nontrivial closed invariant subspace. Per Enflo has shown this is false in general, however it is known that every compact operator has an invariant subspace. The purpose of this project is to explore introductory results in functional analysis. Specifically we are interested in understanding compact operators and the proof that all compact operators on a Hilbert space have an invariant subspace. In the process of doing this we build up many examples and theorems relating to operators on a Hilbert or Banach space. Continuing …
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Mathematics
We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and briefly suggest how symmetry and representations are involved in quantum theory.
Analysis Of Roms Estimated Posterior Error Utilizing 4dvar Data Assimilation, Joseph Patrick Horton
Analysis Of Roms Estimated Posterior Error Utilizing 4dvar Data Assimilation, Joseph Patrick Horton
Mathematics
The appropriateness of the approximate error calculated by the Regional Ocean Modeling System (ROMS) is analyzed using Four-Dimensional Data Assimilation (4DVAR) performed on a numerical model of the San Luis Obispo Bay. An effective method of sampling data to minimize the actual error associated with the assimilated numerical model is explored by using different data sampling methods. An idealized state of the SLO bay region ("Real Run") is created to be used as the real ocean, then a numerical model of this region is created approximating this Real Run; this is known as the "Simulated State". By taking samples from …
Completeness Of Ordered Fields, James Forsythe Hall
Completeness Of Ordered Fields, James Forsythe Hall
Mathematics
The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other areas of mathematics, such as nonstandard analysis. Continuing, we study the completeness of non-Archimedean fields, and provide several examples of such fields with varying degrees of properties, using nonstandard analysis to produce some relatively "nice" (in particular, they are Cantor complete) final examples. As a small detour, …
Exact Solutions For Wind-Driven Coastal Upwelling And Downwelling Over Sloping Bathymetry, Dana Lynn Duke, Paul Derek Sinz
Exact Solutions For Wind-Driven Coastal Upwelling And Downwelling Over Sloping Bathymetry, Dana Lynn Duke, Paul Derek Sinz
Mathematics
The dynamics of wind-driven coastal upwelling and downwelling are studied using a simplified dynamical model. Exact solutions are examined as a function of time and over a family of sloping bathymetries. Assumptions in the two-dimensional model include a frictionless ocean interior below the surface Ekman layer, and no alongshore dependence of the variables; however, dependence in the cross-shore and vertical directions is retained. Additionally, density and alongshore momentum are advected by the cross-shore velocity in order to maintain thermal wind. The time-dependent initial-value problem is solved with constant initial stratification and no initial alongshore flow. An alongshore pressure gradient is …
Knowing When To Say When: An Expanded Description Of Stopping Problems And Their Solutions, Eric C. Bauer
Knowing When To Say When: An Expanded Description Of Stopping Problems And Their Solutions, Eric C. Bauer
Mathematics
No abstract provided.