Department: Mathematics ![[clear]](close-x.png "Remove this limiter")
19 matches in the database.
These are records: 1 - 19.
1.
Agarwala, Edward K.
Food For Thought: When Information Optimization Fails to Optimize Utility.
Degree: MS, Mathematics, 2009, Case Western Reserve University
► Information maximization criteria have been used to account for the physiology of…
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▼ Information maximization criteria have been used to account for the physiology of sensory systems as diverse as receptive fields in the primary visual and auditory cortices, and olfaction. We investigated a model of an organism searching for food by taking successive samples from an environment in which food particles diffuse stochastically from a slowly and randomly moving source. In the limit of large food concentrations we reduced our high dimensional model system to a Markov chain on a small number of equivalence classes. In this system we made rigorous quantitative comparisons of different search strategies based on (i) maximizing the searcher's information about the food source's location, (ii) maximizing the likelihood of landing on the source, and (iii) hybrid strategies combining aspects of (i) and (ii). In terms of long-term expected food benefit we found that each strategy was superior to the others depending on the source's rate of movement.
Advisors/Committee Members: Thomas, Dr. Peter.
Subjects: Biology; Mathematics
Keywords: Information theory; behavior; maximum likelihood; foraging; sensory systems
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2.
Caglar, Umut.
Floating Bodies.
Degree: MS, Mathematics, 2010, Case Western Reserve University
► We introduce various concepts of floating bodies, namely Ulam's floating body, Dupin's…
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▼ We introduce various concepts of floating bodies, namely Ulam's floating body, Dupin's floating body, and the convex floating body and we give examples for the different concepts. In particular, we investigate if counterexamples to certain long standing open problems for the one type of bodies can be used for the other. One of the problems is Ulam's problem asking whether spheres are the only solids that can float in equilibrium in every direction. The other one is the homothety problem which asks if a body must be an ellipsoid if it is homothetic to its floating body. While there are partial solutions to these problems, complete solutions still need to be found.
Advisors/Committee Members: Werner, Elisabeth.
Subjects: Mathematics
Keywords: convex floating bodies; Ulam's problem; Homothety conjecture
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3.
Chou, Yonggang.
Some results on bilinear control systems with rank-one inputs.
Degree: PhD, Mathematics, 1995, Case Western Reserve University
► We shall be treating bilinear single-input control systems in n-space, dot x…
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▼ We shall be treating bilinear single-input control systems in n-space, dot x = (A + uB)x, u: R1→[-1,1],with the special property that the 'control' matrix B has rank 1, i.e., B = bc* for nonzero n-vectors b, c. Let cal At cdot p denote the set attainable from an initial point p at time t ge 0 by admissible controls u(.). It is known that the 'extremal' controls are directly available: a measurable control u: (0, t) → (-1, 1) steers p to the boundary of the attainable set cal At cdot p if, and only if, u takes on the extreme values ±1 only, is piecewise constant, and has at most n - 1 switches (under mild generic conditions on the data A, b, c, and for small t > 0; this is a strong version of the bang-bang principle, see (9)). Naturally, every time-optimal control is extremal; however, usually the converse fails, so that not all extremal controls are time-optimal (unless the initial point p is locally controllable; see Fig. 2). Chapter 2 solves the 'recognition problem', this suggests: to determine completely, in terms of the system data A, b, c and the initial point p, which extremal controls are indeed time-optimal. In the second part (Chapter 3), we assume that the initial point p is local ly controllable. We introduce and study the terminal manifolds of these attainable sets cal At cdot p for small times t, and then construct the optimal feedback equation corresponding to the system. Finally (Chapter 4) we apply the theory of discontinuous differential equations to study the so-called measurement-stability of this optimal feedback equation, and we will prove that the equation is measurement-stable
Advisors/Committee Members: Hajek, Otomar.
Subjects: Mathematics
Keywords: Bilinear control systems; Rank-one inputs
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4.
Chuechote, Suparat.
Amplification in a Stochastic Two Dimensional Model of Eukaryotic Gradient Sensing.
Degree: MS, Mathematics, 2010, Case Western Reserve University
► Chemotaxis is the directed migration of cells guided by chemical gradients. Chemotaxis…
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▼ Chemotaxis is the directed migration of cells guided by chemical gradients. Chemotaxis combinesseveral biological mechanisms, the first of which is gradient sensing. The accuracy with which a cell can determine the direction of an external chemical gradient is limited by fluctuations arising from the discrete nature of second messenger release and diffusion processes within the small volume of a living cell. We implement a stochastic version a Balanced Inactivation gradient sensing model introduced by Levine et al. 2006 in a two dimensional geometry. We develop a fixed timestep approach in which the probabilities of individual molecules making chemical transitions is handled as a system of multinomial random variables. With this numerical platform we investigate the relationship between the amplification of the gradient signal, nonlinear saturation at large gradients, and fundamental limits on the accuracy of the gradient sensing mechanism.
Advisors/Committee Members: Thomas, Peter.
Keywords: Ampliï¬cation; STOCHASTIC; Markov transition; GRADIENT
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5.
Daras, Tryfon Ioannis.
Some large and moderate deviations results for exchangeable sequences.
Degree: PhD, Mathematics, 1995, Case Western Reserve University
► Let (Ω,cal A,P) be a probability space and assume that the probability…
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▼ Let (Ω,cal A,P) be a probability space and assume that the probability measure P has the following representation: P(·) = intΘ P(θ,·)dm(θ) where (Θ,cal M,m) is a probability space (with Θ a first countable topological space) and P(·,·) a probability transition function on Θ×cal A. aLet also, Yjspj=1infty be a sequence of random variables defined on (Ω,cal A,P) and taking values in an arbitrary measurable space (S,cal S) and assume that for each θinΘ, Yjspj=1infty is an i.i.d. sequence under P(θ,·). Let the empirical measures of Yjspj=1infty be defined by: Ln=1over nsumspj=1nδYj with δx the Dirac measure at the point x, and let νn=cal LP(Ln). a Define: Mn=nover bn(Ln-μ) with μ=P o (Y1)-1 and nspn=1infty a positive real sequence such that: bnover n1over2uildreln→inftyoverlongrightarrowinfty, bnover nuildreln→inftyoverlongrightarrow 0eqno(*) and let ildeνn=cal LP(Mn). In chapter 2 of this dissertation, we study Large Deviations for the sequence of probability measures νnspn=1infty. In chapter 3, a Moderate Deviations result with normalizing constants spn2over nspn=1infty, for the sequence ildeνspn=1infty is proved aNow, let (S,cal S) (Rd,cal Bd,dge1 and define eqalign S0 and = 0cr Sj and= sumspi=1jYi, j=1,2,3,··· and denote by sn(t),t in [0,1] the polygonal line in Rd determined by the points (jover n, Sjover xn), j=0,1,2,···, n (trajectories of Yjspj=1infty), with xnspn=1infty a positive real sequence and let μn=cal LP=(sn(·)). In chapter 4, we study Large Deviations for the sequence of probability measures μnspn=1infty, when xn = n. Finally, in chapter 5 we prove a Moderate Deviations result with normalizing constants xspn2over nspn=1infty, for the sequence μnspn=1infty when xn=bn and bn is as in (*)
Advisors/Committee Members: De Acosta, Alejandro.
Subjects: Mathematics
Keywords: Deviations for exchangeable sequences
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6.
Feng, Qingfu.
On Pre-image Topological Pressures.
Degree: PhD, Mathematics, 2005, Case Western Reserve University
► Following the notions of pre-image topological entropies defined and studied in the…
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▼ Following the notions of pre-image topological entropies defined and studied in the recent papers [3] by Cheng and Newhouse, [6] by Diebig, Diebig and Nitecki, [7] by Hurley, [8] by Langevin and Przytycki, [11] by Nitecki, and [12] by Nitecki and Przytycki, we define the notions of so-called pre-image topological pressures. It is shown that those pre-image topological pressures have almost the same basic properties, such as the Power Rule, the Sub-product Rule and the Topological Invariance as the classic topological pressure does (in [17]). It is shown that one of those Pre-iamge topological pressures has some sort of Variational Principle, which is also an extension of the analogous results for the topological pressure in [17] and for the pre-image entropy in [3]. We do not know if there is such a Variational Principle for the other two pre-image topological pressures (and even for the corresponding pre-image entropies). The techniques used here are traditional, and can be found in [3] and [17].
Advisors/Committee Members: Hurley, Michael.
Subjects: Education, Mathematics
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7.
Hahn, Philip James.
Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions.
Degree: PhD, Mathematics, 2004, Case Western Reserve University
► Propagation of functional or pathological ionic disturbances in biological systems plays an…
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▼ Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. Potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. Examined here is propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author for this dissertation. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice. Three models are studied using the method developed. First, the Fitzhugh-Nagumo equation is used to illustrate the method. Second, the continuous Nelkin-Yaari model, describing spreading excitation in brain tissue, is examined. Third, a novel model of non-synaptic pulse propagation in hippocampal slices is developed and analyzed. Investigation of this new model shows that potassium wave behavior in the CA1 region can be explained using descriptions of only two phenomena: action potential spike dynamics in response to elevated potassium and simple sink functions that allow for the formation of a wave backside and refractory time.
Advisors/Committee Members: Alexander, James.
Subjects: Mathematics
Keywords: Diffusion; excursion; Potassium
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8.
Hu, Yiming.
Topics on the stochastic Burgers’ equation.
Degree: PhD, Mathematics, 1994, Case Western Reserve University
► In this dissertation, we study some properties of statistical solutions of the…
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▼ In this dissertation, we study some properties of statistical solutions of the Burgers’ equation which including: (a) A survey of some of known results, including the Kraichnan’s theory of Burgers’ turbulence, the mapping closure method, and connections with the weakly asymmetric, simple exclusion interacting particle system; (b) Extremal rearrangement properties; (c) A Maximum principle for statistical solutions of Burgers’ equation; (d) Limit behavior for quadratic forms of moving averages related to the study of Hopf-Cole solutions.
Advisors/Committee Members: Woyczynski, Wojbor A.
Keywords: Topics stochastic Burgers equation
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9.
Kouri, Drew P.
A Nonlinear Response Model for Single Nucleotide Polymorphism Detection Assays.
Degree: MS, Mathematics, 2008, Case Western Reserve University
► Malaria is a significant cause of mortality in the tropical regions of…
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▼ Malaria is a significant cause of mortality in the tropical regions of the world, such as Papua New Guinea (PNG). Efforts to combat malaria are impeded by the development of drug resistant mutants. We generated a model describing the chemical and molecular properties of the ligase detection reaction (LDR) fluorescent microsphere assay (FMA) for single nucleotide polymorphism (SNP) detection and employed numerical optimization techniques to determine the parameters of this model. First, we implemented the Levenberg-Marquardt Nonlinear Least Squares (NLLS) algorithm and estimated the model parameters from simulated data representing a control dilution/mixing experiment. Second, we used these parameters as well as parameters estimated from experimental data to generate possible distributions of parasite concentration. These distributions can, in principle, inform us of how drug resistance is distributed in a given sample and throughout PNG communities in general. Furthermore, they allow us to evaluate a drug's effectiveness in that community.
Advisors/Committee Members: Thomas, Peter.
Subjects: Mathematics
Keywords: xc; Trust Region; del; µ; LDR
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10.
Kuceyeski, Amy Frances.
Efficient Computational and Statistical Models of Hepatic Metabolism.
Degree: PhD, Mathematics, 2009, Case Western Reserve University
► Computational models provide a useful tool for experimentalists in understanding the processes…
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▼ Computational models provide a useful tool for experimentalists in understanding the processes occurring in a biological system that may otherwise be impossible toobserve directly. The pivotal role of the liver in metabolic regulation makes it a challenging organ to model and simulate. Computational models that can adequately describe hepatic metabolism further the understanding of the functions within the organ. This thesis designs, identifies and analyzes three computational models of hepatic metabolism which account for the complexity of liver biochemistry, hepatic heterogeneity and perfused organ states. These models are governed by systems of ordinary or partial differential equations that depend on a large number of parameters that need to be identified. The classical deterministic parameter estimation problem is recast in the form of Bayesian statistical inference, allowing the integration of a priori belief and data from several experiments. In this approach, the unknowns are modeled as random variables and their values are probability densities. Effcient Markov Chain Monte Carlo techniques are designed and adapted to draw samples effectively from the parameter densities. Setting deterministic models inside a statistical framework makes it possible to study the correlations of different pathways with the time courses of metabolites. This methodology is applied to quantify the sensitivity of various hepatic pathways related to glucose production to redox state under varying conditions, providing insight into the regulation of hepatic gluconeogenesis. The Bayesian framework that we utilize allows us to incorporate into our parameter estimation process information available prior to considering the data. We show that the choices made in the encoding of this a priori information may affect both the parameter estimation and the corresponding model predictions by introducing three priors for a particular model and scrutinizing their effects.
Advisors/Committee Members: Calvetti, Dr. Daniela.
Subjects: Biochemistry; Biomedical research; Mathematics
Keywords: Computational models of hepatic metabolism; Bayesian inference; MCMC sampling; Prior comparison; Correlation Study; Gluconeogenesis
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11.
Li, Zhuobin.
SCHISTOSOMIASIS TRANSMISSION AND CONTROL IN A DISTRIBUTED HETEROGENEOUS HUMAN-SNAIL ENVIRONMENT IN COASTAL KENYA.
Degree: PhD, Mathematics, 2008, Case Western Reserve University
► Schistosomiasis, a disease caused by parasitic flukes, infects 207 million people worldwide,…
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▼ Schistosomiasis, a disease caused by parasitic flukes, infects 207 million people worldwide, especially in South America, the Middle East and Southeast Asia. The schistosome parasite undergoes a complex life cycle that involves two hosts: a definitive human/animal host and an intermediate snail host. Based on transmission estimates from field studies, there are strong seasonal variations in snail populations, as well as agedependent infection patterns and heterogeneities among different human and snail clusters that strongly affect transmission risk. Optimal control strategies for timing and resource allocation need to be defined using non-linear models that can account for these important environmental and behavioral patterns. Here we outline a mathematical model of heterogeneous schistosome transmission for distributed human/snail population clusters, age-dependent water contact behavior, and seasonal changes in environment and use this model to define optimal strategies for parasite control.
Advisors/Committee Members: Gurarie, David.
Keywords: Schistosomiasis; mathematical modeling; two-stage population model; distributed environment; heterogeneity; control strategy
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12.
Munch, James Joseph.
Blind Image Deconvolution with Conditionally Gaussian Hypermodels.
Degree: MS, Mathematics, 2011, Case Western Reserve University
► This paper presents an alternating, iterative implementation of the Bayesian Framework as…
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▼ This paper presents an alternating, iterative implementation of the Bayesian Framework as a means for solving blind deconvolution problems. Specically, a blind deblurring problem is reviewed as proof of this concept. In this paper it will be shown that, given proper a priori beliefs about a system, along with a blurred image and an initial estimate for the kernel, the important properties of both the original signal and kernel, which are used to generate the blurred image, are recovered by this approach.
Advisors/Committee Members: Calvetti, Daniela.
Subjects: Mathematics
Keywords: Deconvolution; Blind Deconvolution; Bayesian; Hypermodels; Imaging
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13.
Sabbaghan, Masoud.
Non-coalescent minimal distal flows.
Degree: PhD, Mathematics, 1993, Case Western Reserve University
► In 1970, W. Parry and P. Walters constructed an example of a…
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▼ In 1970, W. Parry and P. Walters constructed an example of a non-coalescent minimal distal flow, that is, a minimal distal flow with an endomorphism which is not an automorphism (7). In this dissertation we give a general construction of such examples in a systematic way.
Advisors/Committee Members: Wu, Ta-Sun.
Subjects: Mathematics
Keywords: Non-coalescent minimal distal flows
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14.
Sabree, Benjamin David.
A Pedagogical Investigation of the Development of General Relativity Using Differential Forms.
Degree: MS, Mathematics, 2008, Case Western Reserve University
► General relativity is widely applicable to many areas of current physics research.…
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▼ General relativity is widely applicable to many areas of current physics research. Some ‘math first’ treatments of the subject employ differential forms while others do not. This paper advocates those approaches that utilize differential forms by first outlining some of the general mathematical advantages of forms and then by comparing developments of selected topics from the different treatments.
Advisors/Committee Members: Singer, David.
Subjects: Mathematics; Physics
Keywords: general relativity; pedagogy; differential forms
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15.
Seadler, Bradley T.
Signed-Measure Valued Stochastic Partial Differential Equations with Applications in 2D Fluid Dynamics.
Degree: PhD, Mathematics, 2012, Case Western Reserve University
► We note the interesting phenomenon that the Kantorovich-Rubinstein metric is not complete…
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▼ We note the interesting phenomenon that the Kantorovich-Rubinstein metric is not complete on the space of signed measures. Consequently, we introduce a new metric with a useful partial completeness property. With this metric, a general result about the Hahn-Jordan decomposition of solutions of stochastic partial differential equations is shown. These general results are applied to the smoothed Stochastic Navier-Stokes equations. As an application, we derive that the vorticity of the fluid is conserved for a solution of the Stochastic Navier-Stokes equations.
Advisors/Committee Members: Kotelenez, Dr. Peter.
Subjects: Aerospace Engineering; Mathematics; Physics
Keywords: Stochastic; Partial; Differential; Equations; Hahn; Jordan; Fluid; Dynamics; 2D
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16.
Steinberg, Daniel Howard.
Elastic curves in hyperbolic space.
Degree: PhD, Mathematics, 1995, Case Western Reserve University
► An elastic curve is a regular curve γ with fixed endpoints and…
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▼ An elastic curve is a regular curve γ with fixed endpoints and fixed tangent directions at endpoints which is an equilibrium of the functional: Fλ(γ) = ∫γ k2 + λ ds where k2 is the squared curvature of γ and λ is an arbitrary constant. In this thesis we classify the elastic curves in the hyperbolic plane and determine all of the closed solutions. For each rotation index we analyze the structure of critical points among closed curves. In rotation index 0 we get a unique figure eight curve for each positive length. In rotation index 1 it is shown that the only critical points of Fλ are circles for λ > 0. It may follow that curve straightening (following the negative gradient flow induced by F′ in the space of closed regular curves) converges in rotation index 1. In rotation index 2, F′ has no minimum value and so the Palais-Smale condition C) fails (it is known to fail for rotation index greater than 2). There is also a complete description of energy verses length behavior of closed elastic curves in each rotation index.
Advisors/Committee Members: Singer, David A.
Subjects: Mathematics
Keywords: Elastic curves; Hyperbolic space
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17.
Tolmatz, Leonid.
Exact tail asymptotics of a certain Wiener functional.
Degree: PhD, Mathematics, 1992, Case Western Reserve University
The exact tail asymptotics of the distribution of the functional intspo1vert Bsvert ds on the Wiener process Bs: Sge o is obtained.
Advisors/Committee Members: de Acosta, Alejandro.
Subjects: Mathematics
Keywords: Exact tail asymptotics; Wiener functional
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18.
Wang, Xiaoxia.
MATHEMATICAL MODELS OF SCHISTOSOMIASIS TRANSMISSION, MORBIDITY AND CONTROL WITH APPLICATIONS TO ENDEMIC COMMUNITIES IN COASTAL KENYA.
Degree: PhD, Mathematics, 2012, Case Western Reserve University
► Schistosomiasis is a tropical parasitic disease caused by blood-dwelling fluke worms of…
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▼ Schistosomiasis is a tropical parasitic disease caused by blood-dwelling fluke worms of the genus Schistosoma. More than 200 million people worldwide, especially in sub-Saharan Africa, the Middle East and Southeast Asia, suffer from this disease. The disease is characterized by focal epidemiology and over-dispersed distribution pattern, with higher infection rates in children than in adults. Beside acute symptoms, chronic schisto-infection results in indirect morbidity such as cognitive and physical impairment in children. While some advances have been made in the control of the disease through population-based chemotherapy more efforts are required to achieve the goal of disease elimination with limited resources. World Health Organization has proposed several schistosomiasis control guidelines and programs by, and their outcomes need to be assessed. Mathematical models can be used for prediction and control analysis. Here we develop several of them that focus on specific parts and processes of schisto-infection and disease. They are i) dynamics model of snail population with rainfall input; ii)heterogeneous transmission for distributed human/snail population systems, and over-dispersed worm burden in host populations; iii) the effect of chronic schisto-infection on childhood growth and development. The models were calibrated using epidemiological and demographic data from a coastal region in eastern Kenya. We used these models to simulate and predict the effect of different control strategies, to do their cost-benefit analysis and find the optimal regimens. Our models are based on nonlinear differential equations, and exploit diverse mathematical tools and techniques for analysis. For numeric simulations, calibration and symbolic algebra we used Wolfram Mathematica 7.
Advisors/Committee Members: Gurarie, David.
Subjects: Mathematics
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19.
Ye, Deping.
Topics in Convex Geometry and Phenomena in High Dimension.
Degree: PhD, Mathematics, 2009, Case Western Reserve University
► This dissertation deals with topics in convex geometry and phenomena in high…
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▼ This dissertation deals with topics in convex geometry and phenomena in high dimension. Convex geometry studies geometry of convex bodies of fixed dimension, while phenomena in high dimension aims to understand the structure of large dimensional objects. Our objects of study in convex geometry are related to Lp affine surface areas and mixed p-affine surface areas. In particular, we prove new Lp affine isoperimetric inequalities for all p in [−∞,1). For p ≥ 1, such inequalities had been established earlier. We generalize these inequalities to the analogous inequalities for mixed p-affine surface area. We also prove new Alexandrov-Fenchel type inequalities for mixed p-affine surface area. To generalize the Lp affine surface and mixed p-affine surface area to all p, we study the asymptotic behavior of the volume of the illumination surface bodies and the polar bodies of the surface bodies. Properties of the illumination surface bodies are established, for instance, we show that the illumination surface bodies are star-convex, but not necessarily convex. As applications of our asymptotic results, we give geometric interpretations of functionals associated with convex bodies, such as Lp affine surface area, surface area, and mixed p-affine surface area. Moreover, we establish, for all p ≠ −n, a duality formula which shows that Lp affine surface area of a convex body K equals Ln2/p affine surface area of the polar body K°. In phenomena in high dimension, we will study the relative sizes of various sets of quantum states. We obtain two sided estimates for the Bures volume of an arbitrary subset of the set of N × N density matrices in terms of the Hilbert-Schmidt volume of that subset. For general subsets, our results are essentially optimal (for large N). As applications, we derive, in particular, nontrivial lower and upper bounds for the Bures volume of sets of separable states and for sets of states with positive partial transpose.
Advisors/Committee Members: Werner, Elisabeth.
Subjects: Mathematics
Keywords: affine isoperimetric inequality; Lp affine surface area; Lp Brunn- Minkowski theory; mixed $p$-affine surface area; Bures metric; Bures volume; separable states; positive partial transpose
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