Mathematics electronic theses and dissertations (MU)The electronic theses and dissertations of the Department of Mathematics.https://hdl.handle.net/10355/53812020-02-25T18:32:19Z2020-02-25T18:32:19ZAbsolute continuity of parabolic measure and the initial-dirichlet problemGrenschaw, Alyssahttps://hdl.handle.net/10355/699462020-02-25T15:57:07Z2019-01-01T00:00:00ZAbsolute continuity of parabolic measure and the initial-dirichlet problem
Grenschaw, Alyssa
This thesis is devoted to the study of parabolic measure corresponding to a divergence form parabolic operator. We first extend to the parabolic setting a number of basic results that are well known in the elliptic case. Then following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set [omega] R(n+1), assuming as a background hypothesis only that the essential boundary of [omega] satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in [Lp], for some p< [infinity]. Finally, we prove that for the heat equation, BMO-solvability implies scale invariant quantitative absolute continuity of caloric measure with respect to surface measure, in an open set [omega] with time-backwards ADR boundary. Moreover, the same results apply to the parabolic measure associated to a uniformly parabolic divergence form operator (L), with estimates depending only on dimension, the ADR constants, and parabolicity, provided that the continuous Dirichlet problem is solvable for (L) in [omega]. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.
2019-01-01T00:00:00ZThe absolute functional calculus for sectorial operatorsKucherenko, Tamarahttps://hdl.handle.net/10355/41552019-08-15T17:23:52Z2005-01-01T00:00:00ZThe absolute functional calculus for sectorial operators
Kucherenko, Tamara
We introduce the absolute functional calculus for sectorial operators. This notion is stronger than the common holomorphic functional calculus. We are able to improve a key theorem related to the maximal regularity problem and hence demonstrate the power and usefulness of our new concept. In trying to characterize spaces where sectorial operators have absolute calculus, we find that certain real interpolation spaces play a central role. We are then extending various known results in this setting. The idea of unifying theorems about sectorial operators on real interpolation spaces permeates our work and opens paths for future research on this subject.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.; Title from title screen of research.pdf file viewed on (July 18, 2006); Includes bibliographical references.; Vita.; Thesis (Ph. D.) University of Missouri-Columbia 2005.; Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.
2005-01-01T00:00:00ZAge-dependent branching processes and applications to the Luria-Delbrck experimentOveys, Hesamhttps://hdl.handle.net/10355/468872020-01-30T18:48:44Z2015-01-01T00:00:00ZAge-dependent branching processes and applications to the Luria-Delbrck experiment
Oveys, Hesam
Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbruck that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution has spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by D. E. Lea, C. A. Coulson, and J. B. S. Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.
2015-01-01T00:00:00ZAlgebraic resolution of formal ideals along a valuationEl Hitti, Samar, 1979-https://hdl.handle.net/10355/55932020-02-20T18:45:05Z2008-01-01T00:00:00ZAlgebraic resolution of formal ideals along a valuation
El Hitti, Samar, 1979-
Let X be a possibly singular complete algebraic variety, defined over a field [kappa] of characteristic zero. X is nonsingular at [rho] [element of] X if OX,[rho] is a regular local ring. The problem of resolution of singularities is to show that there exists a nonsingular complete variety X, which birationally dominates X. Resolution of singularities (in characteristic zero) was proven by Hironaka in 1964. The valuation theoretic analogue to resolution of singularities is local uniformization. Let [logical or] be a valuation of the function field of X, [logical or] dominates a unique point [rho], on any complete variety [upsilon] , which birationally dominates X. The problem of local uniformization is to show that, given a valuation [logical or] of the function field of X, there exists a complete variety [upsilon] , which birationally dominates X such that the center of [logical or] on [upsilon], is a regular local ring. Zariski proved local uniformization (in characteristic zero) in 1944. His proof gives a very detailed analysis of rank 1 valuations, and produces a resolution which reflects invariants of the valuation. We extend Zariski's methods to higher rank to give a proof of local uniformization which reflects important properties of the valuation. We simultaneously resolve the centers of all the composite valuations, and resolve certain formal ideals associated to the valuation.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.; Title from title screen of research.pdf file (viewed on June 4, 2009); Vita.; Thesis (Ph. D.) University of Missouri-Columbia 2008.
2008-01-01T00:00:00Z