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# Theses and Dissertations - Department of Mathematics

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### Browsing Theses and Dissertations - Department of Mathematics by Author "Borie, Richard B."

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Item The Corona Theorem for the multiplier algebras on weighted Dirichlet spaces(University of Alabama Libraries, 2009) Kidane, Berhanu Tekle; Trent, Tavan T.; University of Alabama TuscaloosaShow more In this dissertation we give a proof of "The Corona Theorem for Infinitely Many Functions for the Multiplier Algebras on Weighted Dirichlet Spaces", and we obtain explicit estimates on the size of the solution. We denote the open unit disc of the complex plane by D, and for α in (0, 1) we denote by Dα the Weighted Dirichlet Spaces of all holomorphic functions on D, and byShow more Item Groups whose non-permutable subgroups satisfy certain conditions(University of Alabama Libraries, 2012) Karatas, Zekeriya Yalcin; Dixon, Martyn R.; University of Alabama TuscaloosaShow more In this dissertation, we determine the structure of groups whose non-permutable subgroups satisfy certain conditions. In Chapter 1, we give the definitions and well-known results that we will use during the dissertation. In Chapter 2, we express our main result, which states that an infinite rank $mathfrak{X}$-group with all proper subgroups permutable or of finite rank has all subgroups permutable. Before proving our main result in Chapter 4, we establish some preliminary results in Chapter 3 which are used in proving the main result. In Final Chapter, we study the class of locally graded groups with all subgroups permutable or nilpotent of bounded class $c$. We prove that such groups are soluble of derived length bounded by a number depending on $c$. This chapter contains preliminary investigations into the problem of the structure of groups with all subgroups permutable or soluble.Show more Item Groups with condition on non-permutable subgroups(University of Alabama Libraries, 2015) Chataut, Laxmi Kant; Dixon, Martyn R.; University of Alabama TuscaloosaShow more In this dissertation, we study the structure of groups satisfying the weak minimal condition and weak maximal condition on non-permutable subgroups. In Chapter 1, we discuss some definitions and well-known results that we will be using during the dissertation. In Chapter 2, we establish some preliminary results which will be useful during the proof of the main results. In Chapter 3, we express our main results, one of which states that a locally finite group satisfying the weak minimal condition on non-permutable subgroups is either Chernikov or quasihamiltonian. We also prove that, a generalized radical group satisfying the weak minimal condition on non-permutable subgroups is either Chernikov or is soluble-by-finite of finite rank. In the Final Chapter, we will discuss the class of groups satisfying the weak maximal condition on non-permutable subgroups.Show more Item Polydegree properties of polynomial automorphisms(University of Alabama Libraries, 2016) Perry, Kaitlyn Anne; Lewis, Drew; University of Alabama TuscaloosaShow more The group of automorphisms of the affine plane has the structure of an amalgamated free product of the triangular and affine subgroups. This leads us to the polydegree: the unique sequence of degrees of the triangular automorphisms in the amalgamated free product decomposition of the automorphism. This group is also endowed with the structure of an infinite dimensional algebraic variety. The interaction between these two structures is not well understood. We use the Valuation Criterion, due to Furter, to study the interaction between these structures; in particular, it allows us to see if an automorphism in G is also in the closure of $G_d$, where d is the polydegree sequence. In this paper, we will discuss a method that gives us new results concerning a class of automorphisms with a polydegree of length one being contained in the closure (in the Zariski topology) of a set of automorphisms with a polydegree of length 2.Show more