Mathematics electronic theses and dissertations (MU)The electronic theses and dissertations of the Department of Mathematics.https://hdl.handle.net/10355/53812019-03-24T04:02:37Z2019-03-24T04:02:37ZThe absolute functional calculus for sectorial operatorsKucherenko, Tamarahttps://hdl.handle.net/10355/41552017-08-09T20:47:09Z2005-01-01T00:00:00ZThe 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/10355/468872018-12-12T18:05:41Z2015-01-01T00:00:00ZAge-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 DelbrÃ¼ck 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/10355/55932018-11-14T21:15:45Z2008-01-01T00:00:00ZAlgebraic 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.; Includes bibliographical references.; Thesis (Ph. D.) University of Missouri-Columbia 2008.; Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.
2008-01-01T00:00:00ZAlmost everywhere convergence for modified Bochner Riesz means at the critical index for [rho] [greater than or equal to] 2Annoni, Marco, 1981-https://hdl.handle.net/10355/82832018-12-10T18:49:21Z2010-01-01T00:00:00ZAlmost everywhere convergence for modified Bochner Riesz means at the critical index for [rho] [greater than or equal to] 2
Annoni, Marco, 1981-
The Fourier transform is a mathematical operation that can be used with its inverse to rewrite a function as a sum of waves. It has been a useful mathematical tool for many applied sciences. Sometimes Fourier inversion is not possible in the classic sense and needs to be generalized. This is often done in a standard way, after choosing a summability method. A famous and much studied one is the method of the Bochner-Riesz means. We use techniques and results of harmonic analysis (Plancharel-type inequalities, partitions of the euclidean space, an analytic continuation argument, maximal operators, duality, potentials etc.) to investigate the method of the Bochner-Riesz means modified by A. Seeger. We prove that the Fourier inversion with respect to the modified Bochner-Riesz means holds pointwise almost everywhere for a certain class of functions. First of all, this result refines a Theorem of A. Carbery, J. Rubio de Francia and L. Vega. Secondly, it highlights the connection between the choice of the method one can use to invert the Fourier transform and the class of functions on which the method will work. Finally, it also shows how to generalize certain techniques to a scenario where we lack certain algebraic properties.
Title from PDF of title page (University of Missouri--Columbia, viewed on May 24, 2010).; The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file.; Dissertation advisor: Dr. Loukas Grafakos.; Vita.; Includes bibliographical references.; Ph. D. University of Missouri--Columbia 2010.; Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.
2010-01-01T00:00:00Z