Department: Mathematics ![[clear]](close-x.png "Remove this limiter")
8 matches in the database.
These are records: 1 - 8.
1.
Bhattacharjee, Papiya.
Minimal Prime Element Space of an Algebraic Frame.
Degree: PhD, Mathematics, 2009, Bowling Green State University
► The following dissertation investigates algebraic frames. Formally speaking, a frame is a…
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▼ The following dissertation investigates algebraic frames. Formally speaking, a frame is a complete lattice which satisfies a strengthened distributive law where finite infima distribute over arbitrary suprema. In particular, we are interested in focussing on a certain space associated with an algebraic frame: the space of minimal prime elements. In the first half of the dissertation we will investigate different interesting properties of these topological spaces in terms of the algebraic properties of the frame. In one of our main results we state internal conditions of an algebraic frame which will ensure its minimal prime element space is compact.In Chapter 5 we will describe the radical of an algebraic frame. This is a generalization in context to the frame of radical ideals of a commutative ring with identity. We will demonstrate that the radical of an algebraic frame is an algebraic frame. The last part of the dissertation focuses on extensions of algebraic frames. We will generalize the notions of rigid extension, r-extension and r*-extension which are known in the theory of lattice-ordered groups. Our main result will characterize rigid extensions in several ways. We will answer the following question: “Which type of extensions between two algebraic frames will ensure a homeomorphism between their corresponding minimal prime element spaces?” This question had been looked at and answered for lattice-ordered groups by Conrad and Martinez in [4] and later by McGovern in [17]. We will also provide an important example from the theory of rings of continuous functions. In this example, we will construct an extension of algebraic frames which will demonstrate that an r*-extension and an r-extension are two different concepts. In the end we will provide several open questions which may lead to future study.
Advisors/Committee Members: McGovern, Warren Wm.
Subjects: Mathematics
Keywords: Algebraic Frame; Compact; Pseudocomplement; Minimal Prime; Hull-Kernel Topology; Inverse Topology; Rigid Extension; r-extension; r*-extension
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2.
Carr, Benjamin.
On Flips of Unitary Buildings.
Degree: PhD, Mathematics, 2010, Bowling Green State University
► In the Gorenstein-Lyons-Solomon revision of the proof of the classification theorem for…
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▼ In the Gorenstein-Lyons-Solomon revision of the proof of the classification theorem for finite simple groups, one of the techniques in the identification step involves identifying a particular finite simple group (the minimal counterexample to the theorem) with a known finite simple group. Phan's theorems and the Curtis-Tits theorem provide some techniques for accomplishing this identification. Both these theorems can be phrased in terms of groups acting flag transitively on a simply connected geometry. One method of producing geometries suitable for proving Phan-type theorems is to study geometries induced by flips on twin buildings. The purpose of this work is to classify flips of the building associated to the geometry of totally isotropic subspaces of a non-degenerate unitary spaces over a finite field of odd characteristic. A secondary goal is to study some properties of geometries related to these flips. We prove that there are up to similarity only four flips, and that for sufficiently large unitary spaces the resulting geometries are simply connected. We then appeal to Tits' Lemma to prove Phan-type theorems for certain flag-transitive automorphism groups of these geometries.
Advisors/Committee Members: Blok, Dr. Rieuwert.
Subjects: Mathematics
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3.
Deters, Ian Nathaniel.
On The Cyclicity And Synthesis Of Diagonal Operators On The Space Of Functions Analytic On A Disk.
Degree: PhD, Mathematics, 2009, Bowling Green State University
► A diagonal operator on the space of functions holomorphic on a disk…
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▼ A diagonal operator on the space of functions holomorphic on a disk of finite radius is a continuous linear operator having the monomials as eigenvectors. In this dissertation, necessary and sufficient conditions are given for a diagonal operator to be cyclic. Necessary and sufficient conditions are also given for a cyclic diagonal operator to admit spectral synthesis, that is, to have as closed invariant subspaces only the closed linear span of sets of eigenvectors. In particular, it is shown that a cyclic diagonal operator admits synthesis if and only if one vector, not depending on the operator, is cyclic. It is also shown that this is equivalent to existence of sequences of polynomials which seperate and have minimum growth on the eigenvalues of the operator.
Advisors/Committee Members: Seubert, Steven.
Subjects: Mathematics
Keywords: cyclic vectors; spectral synthesis; invariant subspaces; diagonal operators
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4.
Drees, Kevin Michael.
Cp(X,ℤ).
Degree: PhD, Mathematics, 2009, Bowling Green State University
► We examine the ring of continuous integer-valued continuous functions on a topological…
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▼ We examine the ring of continuous integer-valued continuous functions on a topological space X, denoted C(X,ℤ), endowed with the topology of pointwise convergence, denoted Cp(X,ℤ). We first deal with the basic properties of the ring C(X,ℤ) and the space Cp(X,ℤ). We find that the concept of a zero-dimensional space plays an important role in our studies. In fact, we find that one need only assume that the domain space is zero-dimensional; this is similar to assume the space to be Tychonoff when studying C(X), where C(X) is the ring of real-valued continuous functions. We also find the space Cp(X,ℤ) is itself a zero-dimensional space. Next, we consider some specific topological properties of the space Cp(X,ℤ) that can be characterized by the topological properties of X. We show that if Cp(X,ℤ) is topologically isomorphic to Cp(Y,ℤ), then the spaces X and Y are homeomorphic to each other, this is much like a the theorem by Nagata from 1949. We show that if X is a zero-dimensional space, then there is a zero-dimensional space Y such that X is embedded in Cp(Y,ℤ). Thus every zero-dimensional space can be viewed as a collection of integer-valued continuous functions. We consider and prove the collection of all linear combinations of characteristic functions on clopen (open and closed) subsets is a dense subspace of Cp(X,ℤ). We then consider when the space Cp(X,ℤ) are Gδ- and Fς-subsets of the collection of all functions from X to ℤ (a Gδ-subset is a countable intersection of open subsets and a Fς-subset is a countable union of closed subsets). We make classifications for when Cp(X,ℤ) is a discrete space, metrizable space, Frechet-Urysohn space, sequential space, and k-space. We end with some results on cardinal invariants and the relationships between the tightness and Lindelöf numbers of related spaces.
Advisors/Committee Members: McGovern, Warren Wm.
Subjects: Mathematics
Keywords: pointwise topology; rings of continuous functions; zero-dimensional; weight; character; metrizable space; Frechet-Urysohn space
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5.
Henthorn, Melanie Lea.
Examples of Diagonal Operators That Fail Spectral Synthesis on Spaces of Analytic Functions.
Degree: PhD, Mathematics, 2011, Bowling Green State University
► The following dissertation is a discussion regarding a specific class of operators…
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▼ The following dissertation is a discussion regarding a specific class of operators acting on spaces of functions analytic in a region. Both the space of functions analytic in the entire complex plane and the space of functions analytic in the unit disk are complete metrizable topological vector spaces. Then a diagonal operator D on either of these spaces is defined to be a continuous linear map, sending the space into itself, that has the monomials as its eigenvectors. If we let {λn} be the eigenvalues corresponding to these eigenvectors, and if we write an function f in terms of a power series f(z)=∑∞n=0anzn, then we observe D(∑∞n=0anzn)=∑∞n=0anλnzn. A closed subspace M is invariant for D if Df is in M for all f in M. The study of invariant subspaces is a popular topic in modern operator theory. We observe that the closed linear span of the orbit is the smallest closed invariant subspace for D containing f. If every invariant subspace for a diagonal operator D on a space of analytic functions can be expressed as a closed linear span of some subset of the eigenvectors of D, we say that D admits spectral synthesis. Until recently, it was not known whether or not every diagonal operator on the space of functions analytic on a disk admitted spectral synthesis. The dissertation of K. Overmoyer gave the first examples of nonsynthetic operators on this space. The present dissertation provides two new classes of operators which also fail synthesis on the space of functions analytic on the unit disk. It also provides the first known class of operators which fail spectral synthesis on the space of entire functions.
Advisors/Committee Members: Wade, J. Gordon.
Subjects: Mathematics
Keywords: spectral synthesis; diagonal operators; spaces of analytic functions; nonsynthetic operators
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6.
Overmoyer, Kate.
Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators.
Degree: PhD, Mathematics, 2011, Bowling Green State University
► A diagonal operator acting on the space H(B(0,R)) of functions analytic on…
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▼ A diagonal operator acting on the space H(B(0,R)) of functions analytic on the disk B(0,R) where 0 < R ≤ ∞ is defined to be any continuous linear map on H(B(0,R)) having the monomials z n as eigenvectors. In this dissertation, examples of diagonal operators D acting on the spaces H(B(0,R)) where 0
Subjects: Mathematics
Keywords: invariant subspaces; diagonal operators; spectral synthesis; cyclicity
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7.
Shapla, Tanweer J.
INFERENCE OF ATTRIBUTABLE RISK FOR MULTIPLE EXPOSURE LEVELS UNDER CROSS-SECTIONAL SAMPLING DESIGN.
Degree: PhD, Mathematics, 2006, Bowling Green State University
► Attributable risk (AR) plays an important role in assessing the relationship between…
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▼ Attributable risk (AR) plays an important role in assessing the relationship between the risk factor and the disease in public health and biomedical sciences. This research is intended to develop point and interval estimation procedures for the inference of the attributable risk when the data set is obtained by means of a cross-sectional sampling design. In this thesis, we develop a novel approach for estimating the variance of the Maximum Likelihood Estimate of AR for a dichotomous risk factor by using the Delta method. The new method is computationally much easier than the existing method using the Fisher Information Matrix. This method has also been extended for a risk factor with multiple exposure levels without and with confounders. The performance of the new method has been justified with real life examples and by the Monte Carlo simulation. The simulation shows that the confidence interval estimator performs very well in terms of the coverage probability and the average length of the interval estimated. For small sample case where large sample approximation theory can not be applied, we develop inference procedure for a dichotomous risk factor using exact test regarding positive association between the risk factor and disease outcome which has never been considered before for attributable risk. This procedure has been extended for a risk factor with multiple exposure levels. The attributable risk has also been studied for intermediate base-level which is useful for detecting the significance of a particular level of risk factor with multiple exposure levels. This technique can be used to amalgamate some of the insignificant exposure levels and hence reduce the exposure levels of the undertaken risk factor. Statistical properties of attributable risk have been explored under certain conditions on the cell probabilities. The behavior of the test of positive dependence using the test statistics based on the estimate of AR and logarithm of the odds ratio, log OR has been studied. It has been shown that in some subsets of the alternative, the test using the test statistic based on the estimate of AR is better than the test using the test statistic based on the estimate of log OR, and in some other subsets, the conclusion is in converse direction. In an exact test for small sample, it has been shown that the two statistics based on the estimates of AR and log OR are equivalent.
Advisors/Committee Members: Nguyen, Truc.
Subjects: Statistics
Keywords: p11; ARk; p12; p.1; AR; ATTRIBUTABLE RISK; p01
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8.
Shifflet, Daniel R.
Optimally Clean Rings.
Degree: PhD, Mathematics, 2011, Bowling Green State University
► In 1999 Nicholson introduced the definition that an element of a ring…
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▼ In 1999 Nicholson introduced the definition that an element of a ring is called strongly clean if it can be written as the sum of a unit and an idempotent that commute. Similarly, a ring is called strongly clean if each of its elements is strongly clean. While many popular collections of rings have been shown to possess this characteristic, there are some that do not. Perhaps most surprising is the fact that there are still large collections that have yet to be classified. One such example in this final group is the set of formal power series rings. We know not all these structures are strongly clean, but some are. Which ones? To answer this question we investigated conditions on a ring that imply the extension to a formal power series ring would still be strongly clean. Using Peirce Decompositions and Corners, we were able to isolate the structure needed. Simply stated as a surjective group homomorphism or the solvability of a ring commutator, it is shown that our definition of optimally clean is sufficient to satisfy the extension in question. Further, we were able to to verify the equivalence of strongly and optimally clean within the context of formal power series rings. Extending on this success, we then investigated similar conditions for an extension to the skew power series ring to be strongly clean. This led to our analogous definition of skew optimally clean and proof of its sufficiency for this result. Additionally, we provide examples verifying our conditions to be distinct from previously established definitions. Finally, we verify our new conditions to include all other classes that have been shown sufficient to extend to a strongly clean formal power series ring, making this the most general result to date. Unfortunately, there are not yet enough examples to allow us to determine whether or not they are also necessary properties.
Advisors/Committee Members: McGovern, Dr. Warren Wm.
Subjects: Mathematics
Keywords: power series ring; strongly clean ring
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