Department of Mathematicshttp://hdl.handle.net/2104/47732015-05-23T07:38:55Z2015-05-23T07:38:55ZQuadratic Lyapunov theory for dynamic linear switched systems.Eisenbarth, Geoffrey B.http://hdl.handle.net/2104/88992015-02-18T09:05:17Z2014-01-28T00:00:00ZQuadratic Lyapunov theory for dynamic linear switched systems.
Eisenbarth, Geoffrey B.
In this work, a special class of time-varying linear systems in the arbitrary time scale setting is examined by considering the qualitative properties of their solutions. Building on the work of J. DaCunha and A. Ramos, Lyapunov's Second (or Direct) Method is utilized to determine when the solutions to a given switched system are asymptotically stable.
Three major classes of switched systems are analyzed which exhibit a convenient containment scheme so as to recover early results as special cases of later, more general results. The stability of switched systems under both arbitrary and particular switching is considered, in addition to design parameters of the time scale domain which also imply stability. A new approach to Lyapunov theory for time scales is then considered for switched systems which do not necessarily belong to any class of systems, contrasting and generalizing previous results. Finally, extensions of the contained theory are considered and a nontrivial generalization of a major result by D. Liberzon and A. Agrachev is investigated and conjectured.
2014-01-28T00:00:00ZExistence and uniqueness of solutions of boundary value problems by matching solutions.Liu, Xueyan, 1978.http://hdl.handle.net/2104/88412015-02-18T09:04:45Z2013-09-24T00:00:00ZExistence and uniqueness of solutions of boundary value problems by matching solutions.
Liu, Xueyan, 1978.
In this dissertation, we investigate the existence and uniqueness of boundary value problems for the third and nth order differential equations by matching solutions. Essentially, we consider the interval [a, c] of a BVP as the union of the two intervals [a, b] and [b, c], analyze the solutions of the BVP on each, and then match the proper ones to be the unique solution on the whole domain. In the process of matching solutions, boundary value problems with different boundaries, especially at the matching point b, would be quite different for the requirements of conditions on the nonlinear term. We denote the missing derivatives in the boundary conditions at the matching point b by k₁ and k₂. We show how y(ᵏ²)(b) varies with respect to y(ᵏ¹)(b), where y is a solution of the BVP on [a, b] or [b, c]. Under certain conditions on the nonlinear term, we can get a monotone relation between y(ᵏ²)(b) and y(ᵏ¹)(b), on [a, b] and [b, c], respectively. If the monotone relations are different on [a, b] and [b, c], then we can finally get a unique value for y(ᵏ¹)(b) where the k₂nd derivative of two solutions on [a, b] and [b, c] are equal and we can join the two solutions together to obtain the unique solution of our original BVP. If the relations are the same, then we will arrive at the situation that the k₂nd order derivatives of two solutions at b on [a, b] and [b, c] are decreasing with respect to the k₁st derivatives at b at different rates, and by analyzing the relations more in detail, we can finally get a unique value for the k₁st derivative of solutions of BVP's on [a, b] and [b, c], which are matched to be a unique solution of the BVP on [a, c]. In our arguments, we use the Mean Value Theorem and the Rolle's Theorem many times. As the simplest models, third order BVP's are considered first. Then, in the following chapters, nth order problems are studied. Lastly, we provide an example and some ideas for our future work.
2013-09-24T00:00:00ZA combinatorial property of Bernstein-Gelfand-Gelfand resolutions of unitary highest weight modules.Hartsock, Gail.http://hdl.handle.net/2104/88342015-02-18T09:04:38Z2013-09-24T00:00:00ZA combinatorial property of Bernstein-Gelfand-Gelfand resolutions of unitary highest weight modules.
Hartsock, Gail.
It follows from a formula by Kostant that the difference between the highest weights of consecutive parabolic Verma modules in the Bernstein-Gelfand-Gelfand-Lepowsky resolution of the trivial representation is a single root. We show that an analogous property holds for all unitary representations of simply laced type. Specifically, the difference between consecutive highest weights is a sum of positive noncompact roots all with multiplicity one.
2013-09-24T00:00:00ZOn a ring associated to F[x].Aceves, Kelly Fouts.http://hdl.handle.net/2104/88072015-02-18T09:04:11Z2013-09-24T00:00:00ZOn a ring associated to F[x].
Aceves, Kelly Fouts.
For a ﬁeld F and the polynomial ring F [x] in a single indeterminate, we deﬁne
Ḟ [x] = {α ∈ End_F(F [x]) : α(ƒ) ∈ ƒF [x] for all ƒ ∈ F [x]}. Then Ḟ [x] is naturally isomorphic to F [x] if and only if F is inﬁnite. If F is ﬁnite, then Ḟ [x] has cardinality continuum. We study the ring Ḟ[x] for ﬁnite ﬁelds F. For the case that F is ﬁnite, we discuss many properties and the structure of Ḟ [x].
2013-09-24T00:00:00Z