Mathematics Theses and Dissertationshttp://hdl.handle.net/1903/27932016-05-29T11:01:02Z2016-05-29T11:01:02ZBayesian Estimation of the Inbreeding Coefficient for Single Nucleotide Polymorphism Using Complex Survey DataXue, Zhenyihttp://hdl.handle.net/1903/173072016-03-29T04:16:42Z2015-01-01T00:00:00ZBayesian Estimation of the Inbreeding Coefficient for Single Nucleotide Polymorphism Using Complex Survey Data
Xue, Zhenyi
In genome-wide association studies (GWAS), single nucleotide polymorphism (SNP) is often used as a genetic marker to study gene-disease association. Some large scale health sample surveys have recently started collecting genetic data. There is now growing interest in developing statistical procedures using genetic survey data. This calls for innovative statistical methods that incorporate both genetic and statistical sampling.
Under simple random sampling, the traditional estimator of the inbreeding coefficient is given by 1 - (number of observed heterozygotes) / (number of expected heterozygotes). Genetic data quality control reports published by the National Health and Nutrition Examination Survey (NHANES) and the Health and Retirement Study (HRS) use this simple estimator, which serves as a reasonable quality control tool to identify problems such as genotyping error. There is, however, a need to improve on this estimator by considering different features of the complex survey design. The main goal of this dissertation is to fill in this important research gap. First, a design-based estimator and its associated jackknife standard error estimator are proposed. Secondly, a hierarchical Bayesian methodology is developed using the effective sample size and genotype count. Lastly, a Bayesian pseudo-empirical likelihood estimator is proposed using the expected number of heterozygotes in the estimating equation as a constraint when maximizing the pseudo-empirical likelihood. One of the advantages of the proposed Bayesian methodology is that the prior distribution can be used to restrict the parameter space induced by the general inbreeding model.
The proposed estimators are evaluated using Monte Carlo simulation studies. Moreover, the proposed estimates of the inbreeding coefficients of SNPs from APOC1 and BDNF genes are compared using the data from the 2006 Health and Retirement Study.
2015-01-01T00:00:00ZANALYSIS OF STEADY-STATE AND DYNAMICAL RADIALLY-SYMMETRIC PROBLEMS OF NONLINEAR VISCOELASTICITYStepanov, Alexeyhttp://hdl.handle.net/1903/172782016-03-29T04:10:34Z2015-01-01T00:00:00ZANALYSIS OF STEADY-STATE AND DYNAMICAL RADIALLY-SYMMETRIC PROBLEMS OF NONLINEAR VISCOELASTICITY
Stepanov, Alexey
This thesis treats radially symmetric steady states and radially symmetric motions of nonlinearly elastic and viscoelastic plates and shells subject to dead-load and hydrostatic pressures on their boundaries and with the plate subject to centrifugal force. The plates and shells are described by specializations of the exact (nonlinear) equations of three-dimensional continuum mechanics. The treatment in every case is very general and encompasses large classes of constitutive functions (characterizing the material response).
We first treat the radially symmetric steady states of plates and shells and the radially symmetric steady rotations of plates. We show that the existence, multiplicity, and qualitative behavior of solutions for problems accounting for the live loads due to hydrostatic pressure and centrifugal force depend critically on the material properties of the bodies, physically reasonable refined descriptions of which are given and examined here with great care, and on the nature of boundary conditions.The treatment here, giving new and sharp results, employs several different mathematical tools, ranging from phase-plane analysis to the mathematically more sophisticated direct methods of the Calculus of Variations, fixed-point theorems, and global continuation methods, each of which has different strengths and weaknesses for handling intrinsic difficulties in the mechanics.
We then treat the initial-boundary-value problems for the radially symmetric motions of annular plates and spherical shells that consist of a nonlinearly viscoelastic material of strain-rate type. We discuss a range of physically natural constitutive equations. We first show that when the material is strong in a suitable sense relative to externally applied loads, solutions exist for all time, depend continuously on the data, and consequently are unique. We study the role of the constitutive restrictions and that of the regularity of the data in ensuring the preclusion of a total compression and of an infinite extension for finite time. We then show that when the material is not sufficiently strong then under certain conditions on the (hydrostatic) pressure terms there are globally defined unbounded solutions and there are solutions that blow up in finite time.
The practical importance of these results is that for each problem involving live loads they furnish thresholds in material response delimiting materials for which solutions are ill behaved. A mathematical or numerical study limited to a particular class of materials may dangerously indicate well-behaved solutions when there are other realistic materials for which solutions are ill behaved. Moreover this work furnishes so-called trivial solutions for the subsequent study (not given here) of bifurcation of stable equilibrium configurations from these trivial solutions.
2015-01-01T00:00:00ZQuasiperiodicity and ChaosDas, Suddhasattwahttp://hdl.handle.net/1903/172772016-03-29T04:10:16Z2015-01-01T00:00:00ZQuasiperiodicity and Chaos
Das, Suddhasattwa
In this work, we investigate a property called ``multi-chaos'' is which a chaotic set has densely many hyperbolic periodic points of unstable dimension $k$ embedded in it, for at least 2 different values of $k$. We construct a family of maps on the torus having this property. They serve as a paradigm for multi-chaos occurring in higher dimensional systems. One of the factors that leads to this strong form of chaos is the occurrence of a quasiperiodic orbit transverse to an expanding sub-bundle of the tangent bundle. Hence, a key step towards identifying multi-chaos numerically is finding quasiperiodic orbits in high dimensional systems. To analyze quasiperiodic orbits, we develop a method of weighted ergodic averages and prove that these averages have super-polynomial convergence to the Birkhoff average. We also show how this accelerated convergence of the ergodic averages over quasiperiodic trajectories enable us to compute the rotation number, Fourier series and Lyapunov exponents of quasiperiodic orbits with a high degree of precision ($\approx 10^{-30}$).
2015-01-01T00:00:00ZAlgorithms and Generalizations for the Lovasz Local LemmaHarris, Davidhttp://hdl.handle.net/1903/172762016-03-29T04:10:04Z2015-01-01T00:00:00ZAlgorithms and Generalizations for the Lovasz Local Lemma
Harris, David
The Lovasz Local Lemma (LLL) is a cornerstone principle of the probabilistic method for combinatorics. This shows that one can avoid a large of set of “bad-events” (forbidden configurations of variables), provided the local conditions are satisfied. The original probabilistic formulation of this principle did not give efficient algorithms. A breakthrough result of Moser & Tardos led to an framework based on resampling
variables which turns nearly all applications of the LLL into efficient algorithms. We extend and generalize the algorithm of Moser & Tardos in a variety of ways.
We show tighter bounds on the complexity of the Moser-Tardos algorithm, particularly its parallel form. We also give a new, faster parallel algorithm for the LLL.
We show that in some cases, the Moser-Tardos algorithm can converge even thoughthe LLL itself does not apply; we give a new criterion (comparable to the LLL) for determining when this occurs. This leads to improved bounds for k-SAT and hypergraph coloring among other applications.
We describe an extension of the Moser-Tardos algorithm based on partial resampling, and use this to obtain better bounds for problems involving sums of independent random variables, such as column-sparse packing and packet-routing. We describe a variant of the partial resampling algorithm specialized to approximating column-sparse covering integer programs, a generalization of set-cover. We also give hardness reductions and integrality gaps, showing that our partial resampling based algorithm obtains nearly optimal approximation factors.
We give a variant of the Moser-Tardos algorithm for random permutations, one of the few cases of the LLL not covered by the original algorithm of Moser & Tardos. We use this to develop the first constructive algorithms for Latin transversals and hypergraph packing, including parallel algorithms.
We analyze the distribution of variables induced by the Moser-Tardos algorithm. We show it has a random-like structure, which can be used to accelerate the Moser-Tardos algorithm itself as well as to cover problems such as MAX k-SAT in which we only partially avoid bad-events.
2015-01-01T00:00:00Z