Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--Oregon State University, 2000. / Typescript (photocopy). Includes bibliographical references (leaves 29-30). Also available on the World Wide Web.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2010. / Directed by Jerry Vaughan; submitted to the Dept. of Mathematics and Statistics. Title from PDF t.p. (viewed Jul. 14, 2010). Non-Latin script record Includes bibliographical references (p. 30).

Thesis (Ph.D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Ozlem Ergun; Committee Member: Dana Randall; Committee Member: Joel Sokol; Committee Member: Shabbir Ahmed; Committee Member: William Cook.

Thesis (Ph. D.)--Oregon State University, 1994. / Typescript (photocopy). Includes bibliographical references (leaves 41-45). Also available on the World Wide Web.

<p> The principal object of this study is to find weak and strong type estimates concerning functions in weighted Lp spaces and their maximal functions. We also apply these results to the study of convolution integrals. </p> / Thesis / Master of Science (MSc)

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main focus of this work is to study the classical Calder\'n-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

This thesis presents new results on a few problems involving maximum flows. The first topic we explore is maximum flow network interdiction. The second topic we explore is reoptimization heuristics for rapidly solving an entire sequence of Maximum Flow Problems. In the Cardinality Maximum Flow Network Interdiction Problem (CMFNIP), an interdictor chooses R arcs to delete from an s-t flow network so as to minimize the maximum flow on the network induced on the undeleted arcs. This is an intensively studied problem that has nontrivial applications in military strategy, intercepting contraband and flood control. CMFNIP is a strongly NP-hard special case of the Maximum Flow Network Interdiction Problem (MFNIP), where each arc has an interdiction cost and the interdictor is constrained by an interdiction budget. Although there are several papers on MFNIP, very few theoretical results have been documented. In this talk, we introduce two exponentially large classes of valid inequalities for CMFNIP and prove that they can be separated in polynomial time. Second, we prove that the integrality gap of the commonly used integer linear programming formulation for CMFNIP is contained in the set Omega(|V| ^(1 e)) where |V| is the number of nodes in the network and e is in the interval (0,1). We prove that this result holds even when the linear programming relaxation is strengthened with our two classes of valid inequalities and we note that this result immediately extends to MFNIP. In the second part of this defense, we explore incremental algorithms for solving an online sequence of Maximum Flow Problems (MFPs). Sequences of MFPs arise in a diverse collection of settings including computational biology, finger biometry, constraint programming and real-time scheduling. To initiate this study, we develop an algorithm for solving a sequence of MFPs when the ith MFP differs from the (i-1)st MFP, for each possible i, in that the underlying networks differ by exactly one arc. Second, we develop maximum flow reoptimization heuristics to rapidly compute a robust minimum capacity s-t cut in light of uncertain arc capacities. Third, we develop heuristics to efficiently compute a maximum expected maximum flow in the context of two-stage stochastic programming. We present computational results illustrating the practical performance of our algorithms.

The study of biological diversity has seen a tremendous growth over the past few decades. Among the commonly used indices capturing both the richness and evenness of a community, the Berger-Parker index, which relates to the maximum proportion of all species, is particularly effective. However, when the number of individuals and species grows without bound this index changes, and it is important to develop statistical tools to measure this change. In this thesis, we introduce two estimators for this maximum: the multinomial maximum and the length of the longest increasing subsequence. In both cases, the limiting distribution of the estimators, as the number of individuals and species simultaneously grows without bound, is obtained. Then, constructing the 95% confidence intervals for the maximum proportion helps improve the comparison of the Berger-Parker index among communities. Finally, we compare the two approaches by examining their associated bias corrected estimators and apply our results to environmental data.