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- All Subjects: Equations, Quartic
- Creators: Kaliszewski, Steven
Description
The theory of frames for Hilbert spaces has become foundational in the study of wavelet analysis and has far-reaching applications in signal and image-processing. Originally, frames were first introduced in the early 1950's within the context of nonharmonic Fourier analysis by Duffin and Schaeffer. It was then in 2000, when M. Frank and D. R. Larson extended the concept of frames to the setting of Hilbert C*-modules, it was in that same paper where they asked for which C*-algebras does every Hilbert C*-module admit a frame. Since then there have been a few direct answers to this question, one being that every Hilbert A-module over a C*-algebra, A, that has faithful representation into the C*-algebra of compact operators admits a frame. Another direct answer by Hanfeng Li given in 2010, is that any C*-algebra, A, such that every Hilbert C*-module admits a frame is necessarily finite dimensional. In this thesis we give an overview of the general theory of frames for Hilbert C*-modules and results answering the frame admittance property. We begin by giving an overview of the existing classical theory of frames in Hilbert spaces as well as some of the preliminary theory of Hilbert C*-modules such as Morita equivalence and certain tensor product constructions of C*-algebras. We then show how some results of frames can be extended to the case of standard frames in countably generated Hilbert C*-modules over unital C*-algebras, namely the frame decomposition property and existence of the frame transform operator. We conclude by going through some proofs/constructions that answer the question of frame admittance for certain Hilbert C*-modules.
ContributorsJaime, Arturo (Author) / Kaliszewski, Steven (Thesis director) / Spielberg, Jack (Committee member) / Aguilar, Konrad (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
This dissertation will cover two topics. For the first, let $K$ be a number field. A $K$-derived polynomial $f(x) \in K[x]$ is a polynomial that
factors into linear factors over $K$, as do all of its derivatives. Such a polynomial
is said to be {\it proper} if
its roots are distinct. An unresolved question in the literature is
whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples
are known of proper $K$-derived quartics for a quadratic number field $K$, although other
than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field
is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$
over which there exist proper $K$-derived quartics. The search finds examples for
$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\
For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.
factors into linear factors over $K$, as do all of its derivatives. Such a polynomial
is said to be {\it proper} if
its roots are distinct. An unresolved question in the literature is
whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples
are known of proper $K$-derived quartics for a quadratic number field $K$, although other
than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field
is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$
over which there exist proper $K$-derived quartics. The search finds examples for
$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\
For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.
ContributorsCarrillo, Benjamin (Author) / Jones, John (Thesis advisor) / Bremner, Andrew (Thesis advisor) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2019