Department: Arts and Sciences : Mathematics ![[clear]](close-x.png "Remove this limiter")
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1.
ENDELMAN, ROBIN CAROL.
Degenerations of Elliptic Solutions to the Quantum Yang-Baxter Equation.
Degree: PhD, Arts and Sciences : Mathematics, 2002, University of Cincinnati
► In this dissertation, we study the connection between well-known elliptic solutions of…
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▼ In this dissertation, we study the connection between well-known elliptic solutions of the Yang-Baxter equation and trigonometric degenerations, and construct a new family of rational solutions, which generalize to higher dimensions the Jordanian R-matrices. We introduce a framework for studying solutions of the Yang-Baxter equation for operators on function spaces, which generalizes Shibukawa's R-operators. We extend the standard results on twisting of R-matrices to our R-operators, and to a spectral parameter-dependent twist which gives rise to a two-parameter variation of the Yang-Baxter equation. Applying such a twist to the Shibukawa-Ueno R-operators, we give a unified description of Belavin's R-matrices, the Cremmer-Gervais' R-matrices, and the generalized Jordanian R-matrices. We give new finite dimensional representations of these operators in each of the elliptic and trigonometric cases, from which we obtain the trigonometric and rational degenerations.
Advisors/Committee Members: Hodges, Dr. Timothy J.
Subjects: Mathematics
Keywords: Yang-Baxter Equation; R-Matrix
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2.
Gonchigdanzan, Khurelbaatar.
ALMOST SURE CENTRAL LIMIT THEOREMS.
Degree: PhD, Arts and Sciences : Mathematics, 2001, University of Cincinnati
► The almost sure central limit theorem (ASCLT) has been discovered by two…
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▼ The almost sure central limit theorem (ASCLT) has been discovered by two works by Brosamler (1988) and Schatte (1988) and extensively studied in the past decade. In the dissertation we investigate ASCLT and its extensions to weakly dependent random variables. Its strong approximation is also considered for both independent and dependent random variables. The goal is to prove ASCLT, 'logarithmic' limit theorems and related invariance principles for weakly dependent random variables.
Advisors/Committee Members: Peligrad, Magda.
Subjects: Mathematics; Statistics
Keywords: LIMIT THEOREMS; DEPENDEND VARIABLES; ALMOST SURE CENTRAL LIMIT THEOREMS; LOGARITHMIC AVERAGE; G-MIXING, STRONG MIXING, ASSOCIATED VARIABLES
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3.
Stancescu, Daniel O.
Bootstrap Methods for the Estimation of the Variance of Partial Sums.
Degree: PhD, Arts and Sciences : Mathematics, 2001, University of Cincinnati
► Given a stationary sequence of random variables with long range dependence, we…
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▼ Given a stationary sequence of random variables with long range dependence, we introduce an estimator for the variance of the partial sums, based on a resampling procedure, called the circular block bootstrap method. The properties, the consistency and the Central Limit Theorem for this estimator are studied in this paper, and some simulation results are also presented.
Advisors/Committee Members: Peligrad, Magda.
Subjects: Statistics; Mathematics
Keywords: BOOTSTRAP; VARIANCE OF PARTIAL SUMS; LONG RANGE DEPENDENCE; MIXING SEQUENCES; ASSOCIATED SEQUENCES
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4.
Wang, Guojun.
Some Bayesian Methods in the Estimation of Parameters in the Measurement Error Models and Crossover Trial.
Degree: PhD, Arts and Sciences : Mathematics, 2004, University of Cincinnati
► In this dissertation, we use Bayesian methods to estimate parameters in measurement…
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▼ In this dissertation, we use Bayesian methods to estimate parameters in measurement error models and in the two-period crossover trial. The reference prior approach is used to estimate parameters in the measurement error models, including simple normal structural models, Berkson models, structural models with replicates, and the hybrid models. Reference priors are derived. Jeffreys prior is obtained as a special case of reference priors. The posterior properties are studied. Simulation-based comparisons are made between the reference prior approach and the maximum likelihood method. A fractional Bayes factor (FBF) approach is used to estimate the treatment effect in the two-period crossover trial. The reference priors and the FBF are derived. The FBF is used to combine the carryover-effect model and the no-carryover-effect model. Markov chain Monte Carlo simulation is used to implement the Bayesian analysis.
Advisors/Committee Members: Sivaganesan, Dr. Siva.
Subjects: Statistics; Mathematics
Keywords: Measurement Error Model; Structural Model; Reference Prior; Jeffreys Prior; Crossover Trial; Fractional Bayes Factor(Fbf); Markov Chain Monte Carlo Simulation
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5.
Yi, Zhuobiao.
Identification of General Source Terms in Parabolic Equations.
Degree: PhD, Arts and Sciences : Mathematics, 2002, University of Cincinnati
► In this dissertation we propose stable numerical solutions for several ill-posed problems.…
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▼ In this dissertation we propose stable numerical solutions for several ill-posed problems. The problems mainly investigated include general source terms identification in the one-dimensional and two-dimensional inverse heat conduction problem (IHCP),and the simultaneous identification of the temperature and the general source term in the one-dimensional and two-dimensional ICHP. Each numerical solution consists of a regularization procedure, here based on the mollification method, and a numerical marching scheme for the solution of the stabilized problem. The numerical algorithms are presented with stability and error analysis, numerical implementation and a set of numerical results.
Advisors/Committee Members: Murio, Dr. Diego.
Subjects: Mathematics
Keywords: IHCP; general source tem; ill-posed; inverse problem; mollification
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6.
Yu, Weiming.
Identification of Coefficients in Reaction-Diffusion Equations.
Degree: PhD, Arts and Sciences : Mathematics, 2004, University of Cincinnati
► In this dissertation, a “local” approach for the numerical identification of space…
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▼ In this dissertation, a “local” approach for the numerical identification of space dependent coefficients in equation/systems of nonlinear parabolic equations – that approach steady state in a finite time – is introduced. Stability and error analysis results, together with some numerical examples of interest are also presented.
Advisors/Committee Members: Murio, Dr. Diego.
Subjects: Mathematics
Keywords: Reaction diffusion equation; mollification; identification
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