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1.
Hurth, Tobias.
Limit theorems for a one-dimensional system with random switchings.
Degree: MS, Mathematics, 2010, Georgia Tech
URL: http://hdl.handle.net/1853/37201 
► We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a…
(more)
▼ We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute its unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero and infinity and derive analogues of classical probabilistic results such as the central limit theorem and large deviations principle.
Advisors/Committee Members: Bakhtin, Yuri (Committee Member), Bunimovich, Leonid (Committee Member), Koltchinskii, Vladimir (Committee Member).
Subjects/Keywords: One-dimensional random dynamical system; Large deviations principle; Central limit theorem; Invariant density; Driving vector fields; One force - one solution principle; Random switchings; Limit theorems (Probability theory); Random dynamical systems
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APA (6th Edition):
Hurth, T. (2010). Limit theorems for a one-dimensional system with random switchings. (Masters Thesis). Georgia Tech. Retrieved from http://hdl.handle.net/1853/37201
Chicago Manual of Style (16th Edition):
Hurth, Tobias. “Limit theorems for a one-dimensional system with random switchings.” 2010. Masters Thesis, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/37201.
MLA Handbook (7th Edition):
Hurth, Tobias. “Limit theorems for a one-dimensional system with random switchings.” 2010. Web. 03 Mar 2021.
Vancouver:
Hurth T. Limit theorems for a one-dimensional system with random switchings. [Internet] [Masters thesis]. Georgia Tech; 2010. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/37201.
Council of Science Editors:
Hurth T. Limit theorems for a one-dimensional system with random switchings. [Masters Thesis]. Georgia Tech; 2010. Available from: http://hdl.handle.net/1853/37201
Georgia Tech
2.
Amato, Alberto.
Leading points concepts in turbulent premixed combustion modeling.
Degree: PhD, Mechanical Engineering, 2014, Georgia Tech
URL: http://hdl.handle.net/1853/52247
► The propagation of premixed flames in turbulent flows is a problem of wide physical and technological interest, with a significant literature on their propagation speed…
(more)
▼ The propagation of premixed flames in turbulent flows is a problem of wide physical and technological interest, with a significant literature on their propagation speed and front topology. While certain scalings and parametric dependencies are well understood, a variety of problems remain. One major challenge, and focus of this thesis, is to model the influence of fuel/oxidizer composition on turbulent burning rates.
Classical explanations for augmentation of turbulent burning rates by turbulent velocity fluctuations rely on global arguments - i.e., the turbulent burning velocity increase is directly proportional to the increase in flame surface area and mean local burning rate along the flame. However, the development of such global approaches is complicated by the abundance of phenomena influencing the propagation of turbulent premixed flames. Emphasizing key governing processes and cutting-off interesting but marginal phenomena appears to be necessary to make further progress in understanding the subject.
An alternative approach to understand turbulent augmentation of burning rates is based upon so-called "leading points", which are intrinsically local properties of the turbulent flame. Leading points concepts suggest that the key physical mechanism controlling turbulent burning velocities of premixed flames is the velocity of the points on the flame that propagate farthest out into the reactants. It is postulated that modifications in the overall turbulent combustion speed depend solely on modifications of the burning rate at the leading points since an increase (decrease) in the average propagation speed of these points causes more (less) flame area to be produced behind them. In this framework, modeling of turbulent burning rates can be thought as consisting of two sub-problems: the modeling of (1) burning rates at the leading points and of (2) the dynamics/statistics of the leading points in the turbulent flame. The main objective of this thesis is to critically address both aspects, providing validation and development of the physical description put forward by leading point concepts.
To address the first sub-problem, a comparison between numerical simulations of one-dimensional laminar flames in different geometrical configurations and statistics from a database of direct numerical simulations (DNS) is detailed. In this thesis, it is shown that the leading portions of the turbulent flame front display a structure that on average can be reproduced reasonably well by results obtained from model geometries with the same curvature. However, the comparison between model laminar flame computations and highly curved flamelets is complicated by the presence of negative (i.e., compressive) strain rates, due to gas expansion. For the highest turbulent intensity investigated, local consumption speeds, curvatures, strain rates and flame thicknesses approach the maximum values obtained by the laminar model geometries, while other cases display substantially lower values.
To address the second sub-problem, the…
Advisors/Committee Members: Lieuwen, Timothy C. (advisor), Seitzman, Jerry M. (committee member), Genzale, Caroline (committee member), Yeung, P.K. (committee member), Bakhtin, Yuri (committee member).
Subjects/Keywords: Premixed flames; Turbulent combustion; Leading points; Flame stretch; G-equation
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APA (6th Edition):
Amato, A. (2014). Leading points concepts in turbulent premixed combustion modeling. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/52247
Chicago Manual of Style (16th Edition):
Amato, Alberto. “Leading points concepts in turbulent premixed combustion modeling.” 2014. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/52247.
MLA Handbook (7th Edition):
Amato, Alberto. “Leading points concepts in turbulent premixed combustion modeling.” 2014. Web. 03 Mar 2021.
Vancouver:
Amato A. Leading points concepts in turbulent premixed combustion modeling. [Internet] [Doctoral dissertation]. Georgia Tech; 2014. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/52247.
Council of Science Editors:
Amato A. Leading points concepts in turbulent premixed combustion modeling. [Doctoral Dissertation]. Georgia Tech; 2014. Available from: http://hdl.handle.net/1853/52247
Georgia Tech
3.
Einav, Amit.
Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.
Degree: PhD, Mathematics, 2011, Georgia Tech
URL: http://hdl.handle.net/1853/42788
► The presented work deals with two distinct problems in the field of Mathematical Physics. The first part is dedicated to an 'almost' solution of Villani's…
(more)
▼ The presented work deals with two distinct problems in the field of Mathematical Physics.
The first part is dedicated to an 'almost' solution of Villani's conjecture, a known
conjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzmann equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equilibrate is proportional to the number of particles in the system. Our main result in this part is a proof, up to an epsilon, of that conjecture, showing that for all practical purposes we can consider it to be true.
The second part of the presentation is based on a joint work with Prof. Michael Loss and is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to intersection of hyperplanes. The newly found inequality is sharp and the functions that attain equality in it are completely classified.
Advisors/Committee Members: Loss, Michael (Committee Chair), Bakhtin, Yuri (Committee Member), Carlen, Eric (Committee Member), Harrell, Evans (Committee Member), Heil, Christopher (Committee Member).
Subjects/Keywords: Fractional laplacian; Villani's conjecture; Entropy production; Kac's model; Trace inequality; Mathematical physics; Statistical mechanics; Transport theory; Particle methods (Numerical analysis); Inequalities (Mathematics)
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APA (6th Edition):
Einav, A. (2011). Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/42788
Chicago Manual of Style (16th Edition):
Einav, Amit. “Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.” 2011. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/42788.
MLA Handbook (7th Edition):
Einav, Amit. “Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian.” 2011. Web. 03 Mar 2021.
Vancouver:
Einav A. Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. [Internet] [Doctoral dissertation]. Georgia Tech; 2011. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/42788.
Council of Science Editors:
Einav A. Two problems in mathematical physics: Villani's conjecture and trace inequality for the fractional Laplacian. [Doctoral Dissertation]. Georgia Tech; 2011. Available from: http://hdl.handle.net/1853/42788
4.
Yurchenko, Aleksey.
Some problems in the theory of open dynamical systems and deterministic walks in random environments.
Degree: PhD, Mathematics, 2008, Georgia Tech
URL: http://hdl.handle.net/1853/26549
► The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a…
(more)
▼ The first part of this work deals with open dynamical systems. A natural question of how the survival probability
depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found
that this dependency could be very essential. The main results are related to the holes with equal sizes
(measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period.
Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results
are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory
where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole
is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that
specific features of the dynamics may play a role comparable to the size of the hole.
In the second part we consider some classes of cellular automata called Deterministic Walks in Random
Environments on Z
1. At first we deal with the system with constant rigidity and Markovian distribution
of scatterers on Z
1. It is shown that these systems have essentially the same properties as DWRE on
Z
1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with
non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws
for the dynamics of perturbations propagating in such environments with aging are obtained.
Advisors/Committee Members: Bunimovich, Leonid (Committee Chair), Bakhtin, Yuri (Committee Member), Cvitanovic, Predrag (Committee Member), Houdre, Christian (Committee Member), Weiss, Howard (Committee Member).
Subjects/Keywords: Open dynamical systems; Escape rate; Autocorrelation function; Dynamical systems; Holes; Dynamics; Chaotic behavior in systems
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APA ·
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Export
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APA (6th Edition):
Yurchenko, A. (2008). Some problems in the theory of open dynamical systems and deterministic walks in random environments. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/26549
Chicago Manual of Style (16th Edition):
Yurchenko, Aleksey. “Some problems in the theory of open dynamical systems and deterministic walks in random environments.” 2008. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/26549.
MLA Handbook (7th Edition):
Yurchenko, Aleksey. “Some problems in the theory of open dynamical systems and deterministic walks in random environments.” 2008. Web. 03 Mar 2021.
Vancouver:
Yurchenko A. Some problems in the theory of open dynamical systems and deterministic walks in random environments. [Internet] [Doctoral dissertation]. Georgia Tech; 2008. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/26549.
Council of Science Editors:
Yurchenko A. Some problems in the theory of open dynamical systems and deterministic walks in random environments. [Doctoral Dissertation]. Georgia Tech; 2008. Available from: http://hdl.handle.net/1853/26549
5.
Webb, Benjamin Zachary.
Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems.
Degree: PhD, Mathematics, 2011, Georgia Tech
URL: http://hdl.handle.net/1853/39521
► This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex…
(more)
▼ This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex systems. This includes the spectral properties of such systems and in particular their influence on the systems dynamics. In the second part of this dissertation, or Chapter IV, we consider a new class of one-dimensional dynamical systems or functions with an eventual negative Schwarzian derivative motivated by some maps arising in neuroscience. To aid in understanding the interplay between the graph structure of a network and its dynamics we first introduce the concept of an isospectral graph reduction in Chapter I. Mathematically, an isospectral graph transformation is a graph operation (equivalently matrix operation) that modifies the structure of a graph while preserving the eigenvalues of the graphs weighted adjacency matrix. Because of their properties such reductions can be used to study graphs (networks) modulo any specific graph structure e.g. cycles of length n, cliques of size k, nodes of minimal/maximal degree, centrality, betweenness, etc. The theory of isospectral graph reductions has also lead to improvements in the general theory of eigenvalue approximation. Specifically, such reductions can be used to improved the classical eigenvalue estimates of Gershgorin, Brauer, Brualdi, and Varga for a complex valued matrix. The details of these specific results are found in Chapter II. The theory of isospectral graph transformations is then used in Chapter III to study time-delayed dynamical systems and develop the notion of a dynamical network expansion and reduction which can be used to determine whether a network of interacting dynamical systems has a unique global attractor. In Chapter IV we consider one-dimensional dynamical systems of an interval. In the study of such systems it is often assumed that the functions involved have a negative Schwarzian derivative. Here we consider a generalization of this condition. Specifically, we consider the functions which have some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. This includes both systems with regular as well as chaotic dynamic properties.
Advisors/Committee Members: Bunimovich, Leonid (Committee Chair), Bakhtin, Yuri (Committee Member), Dieci, Luca (Committee Member), Randall, Dana (Committee Member), Weiss, Howie (Committee Member).
Subjects/Keywords: Schwarzian derivative; Global stability; Dynamical networks; Spectral equivalence; Graph transformations; Complex matrices; Attractors (Mathematics); Eigenvalues
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APA ·
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MLA ·
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CSE |
Export
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APA (6th Edition):
Webb, B. Z. (2011). Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/39521
Chicago Manual of Style (16th Edition):
Webb, Benjamin Zachary. “Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems.” 2011. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/39521.
MLA Handbook (7th Edition):
Webb, Benjamin Zachary. “Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems.” 2011. Web. 03 Mar 2021.
Vancouver:
Webb BZ. Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems. [Internet] [Doctoral dissertation]. Georgia Tech; 2011. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/39521.
Council of Science Editors:
Webb BZ. Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems. [Doctoral Dissertation]. Georgia Tech; 2011. Available from: http://hdl.handle.net/1853/39521
Georgia Tech
6.
Hurth, Tobias.
Invariant densities for dynamical systems with random switching.
Degree: PhD, Mathematics, 2014, Georgia Tech
URL: http://hdl.handle.net/1853/52274
► We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a…
(more)
▼ We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a two-component Markov process whose first component is a stochastic process on a finite-dimensional smooth manifold and whose second component is a stochastic process on a finite collection of smooth vector fields that are defined on the manifold. We identified sufficient conditions for uniqueness and absolute continuity of the invariant measure associated to this Markov process. These conditions consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where the manifold is the real line or a subset of the real line, we studied regularity properties of the invariant densities of absolutely continuous invariant measures. We showed that invariant densities are smooth away from critical points of the vector fields. Assuming in addition that the vector fields are analytic, we derived the asymptotically dominant term for invariant densities at critical points.
Advisors/Committee Members: Bakhtin, Yuri (advisor), Bunimovich, Leonid (committee member), de la Llave, Rafael (committee member), Koltchinskii, Vladimir (committee member), Popescu, Ionel (committee member), Mattingly, Jonathan C. (committee member).
Subjects/Keywords: Randomly switched ODEs; Piecewise deterministic Markov processes; Invariant densities; Hypoellipticity
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APA ·
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APA (6th Edition):
Hurth, T. (2014). Invariant densities for dynamical systems with random switching. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/52274
Chicago Manual of Style (16th Edition):
Hurth, Tobias. “Invariant densities for dynamical systems with random switching.” 2014. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/52274.
MLA Handbook (7th Edition):
Hurth, Tobias. “Invariant densities for dynamical systems with random switching.” 2014. Web. 03 Mar 2021.
Vancouver:
Hurth T. Invariant densities for dynamical systems with random switching. [Internet] [Doctoral dissertation]. Georgia Tech; 2014. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/52274.
Council of Science Editors:
Hurth T. Invariant densities for dynamical systems with random switching. [Doctoral Dissertation]. Georgia Tech; 2014. Available from: http://hdl.handle.net/1853/52274
Georgia Tech
7.
Dever, John William.
Local space and time scaling exponents for diffusion on compact metric spaces.
Degree: PhD, Mathematics, 2018, Georgia Tech
URL: http://hdl.handle.net/1853/60250
► We provide a new definition of a local walk dimension beta that depends only on the metric and not on the existence of a particular…
(more)
▼ We provide a new definition of a local walk dimension beta that depends only on the metric and not on the existence of a particular regular Dirichlet form or heat kernel asymptotics. Moreover, we study the local Hausdorff dimension alpha and prove that any variable Ahlfors regular measure of variable dimension Q is strongly equivalent to the local Haudorff measure with Q equal to alpha, generalizing the constant dimensional case. Additionally, we provide constructions of several variable dimensional spaces, including a new example of a variable dimensional Sierpinski carpet. We show also that there exist natural examples where alpha and beta both vary continuously. We prove that beta is greater than or equal to two provided the space is doubling. We use the local exponent beta in time-scale renormalization of discrete time random walks, that are approximate at a given scale in the sense that the expected jump size is the order of the space scale. In analogy with the variable Ahlfors regularity space scaling condition involving alpha, we consider the condition that the expected time to leave a ball scales like the radius of the ball to the power beta of the center. Under this local time scaling condition along with the local space scaling condition of Ahlfors regularity, we then study the Gamma and Mosco convergence of the resulting continuous time approximate walks as the space scale goes to zero. We prove that a non-trivial Dirichlet form with Dirichlet boundary conditions on a ball exists as a Mosco limit of approximate forms. One of the novel ideas in this construction is the use of exit time functions, analogous to the torsion functions of Riemannian geometry, as test functions to ensure the resulting domain contains enough functions. We also prove tightness of the associated continuous time processes.
Advisors/Committee Members: Bellissard, Jean (advisor), Harrell, Evans (advisor), Loss, Michael (committee member), Tao, Molei (committee member), Bakhtin, Yuri (committee member), Cvitanović, Predrag (committee member), Teplyaev, Alexander (committee member).
Subjects/Keywords: Local walk dimension; Variable Ahlfors regularity; Local dimension; Metric geometry; Variable exponent; Random walks on fractal graphs; Mean exit time; Gamma convergence; Mosco convergence; Weak convergence; Diffusion on fractals
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APA (6th Edition):
Dever, J. W. (2018). Local space and time scaling exponents for diffusion on compact metric spaces. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/60250
Chicago Manual of Style (16th Edition):
Dever, John William. “Local space and time scaling exponents for diffusion on compact metric spaces.” 2018. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/60250.
MLA Handbook (7th Edition):
Dever, John William. “Local space and time scaling exponents for diffusion on compact metric spaces.” 2018. Web. 03 Mar 2021.
Vancouver:
Dever JW. Local space and time scaling exponents for diffusion on compact metric spaces. [Internet] [Doctoral dissertation]. Georgia Tech; 2018. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/60250.
Council of Science Editors:
Dever JW. Local space and time scaling exponents for diffusion on compact metric spaces. [Doctoral Dissertation]. Georgia Tech; 2018. Available from: http://hdl.handle.net/1853/60250
8.
Ma, Jinyong.
Topics in sequence analysis.
Degree: PhD, Mathematics, 2012, Georgia Tech
URL: http://hdl.handle.net/1853/45908
► This thesis studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young…
(more)
▼ This thesis studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young diagrams, associated with a random word of size n whose letters are independently drawn from an alphabet of size m=m(n). When the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. Since the length of the top row of the Young diagrams is the length of the longest (weakly) increasing subsequence of the random word, the corresponding large deviations follow. When the letters are drawn with non-uniform probability, a control of both highest probabilities will ensure that the length of the top row of the diagrams satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, are obtained for both models. In the second part, we investigate the order of the r-th, 1<= r < +∞, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, the r-th central moment is shown to be of order n
r/2. In particular, when r=2, we get the order of the variance of the longest common subsequence.
Advisors/Committee Members: Houdre, Christian (Committee Chair), Bakhtin, Yuri (Committee Member), Foley, Robert (Committee Member), Koltchinskii, Vladimir (Committee Member), Matzinger, Heinrich (Committee Member), Popescu, Ionel (Committee Member).
Subjects/Keywords: Longest common subsequence; Young diagrams; Large deviations; Sequential analysis; Representations of groups
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APA ·
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APA (6th Edition):
Ma, J. (2012). Topics in sequence analysis. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/45908
Chicago Manual of Style (16th Edition):
Ma, Jinyong. “Topics in sequence analysis.” 2012. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/45908.
MLA Handbook (7th Edition):
Ma, Jinyong. “Topics in sequence analysis.” 2012. Web. 03 Mar 2021.
Vancouver:
Ma J. Topics in sequence analysis. [Internet] [Doctoral dissertation]. Georgia Tech; 2012. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/45908.
Council of Science Editors:
Ma J. Topics in sequence analysis. [Doctoral Dissertation]. Georgia Tech; 2012. Available from: http://hdl.handle.net/1853/45908
9.
Minsker, Stanislav.
Non-asymptotic bounds for prediction problems and density estimation.
Degree: PhD, Mathematics, 2012, Georgia Tech
URL: http://hdl.handle.net/1853/44808
► This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the…
(more)
▼ This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning.
Given a random couple (X,Y) with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. Prediction rule is constructed from a sequence of observations sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for the performance of active learning methods. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem.
The second part of this thesis is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P. Given a collection of functions H, the goal of dictionary learning is to construct a prediction rule for Y given by a linear combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that the proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance.
Advisors/Committee Members: Koltchinskii, Vladimir (Committee Chair), Bakhtin, Yuri (Committee Member), Balcan, Maria-Florina (Committee Member), Houdre, Christian (Committee Member), Romberg, Justin (Committee Member).
Subjects/Keywords: Active learning; Sparse recovery; Oracle inequality; Confidence bands; Infinite dictionary; Estimation theory Asymptotic theory; Estimation theory; Distribution (Probability theory); Prediction theory; Active learning; Algorithms; Mathematical optimization; Chebyshev approximation
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APA ·
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MLA ·
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APA (6th Edition):
Minsker, S. (2012). Non-asymptotic bounds for prediction problems and density estimation. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/44808
Chicago Manual of Style (16th Edition):
Minsker, Stanislav. “Non-asymptotic bounds for prediction problems and density estimation.” 2012. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/44808.
MLA Handbook (7th Edition):
Minsker, Stanislav. “Non-asymptotic bounds for prediction problems and density estimation.” 2012. Web. 03 Mar 2021.
Vancouver:
Minsker S. Non-asymptotic bounds for prediction problems and density estimation. [Internet] [Doctoral dissertation]. Georgia Tech; 2012. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/44808.
Council of Science Editors:
Minsker S. Non-asymptotic bounds for prediction problems and density estimation. [Doctoral Dissertation]. Georgia Tech; 2012. Available from: http://hdl.handle.net/1853/44808
10.
Gong, Ruoting.
Small-time asymptotics and expansions of option prices under Levy-based models.
Degree: PhD, Mathematics, 2012, Georgia Tech
URL: http://hdl.handle.net/1853/44798
► This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several…
(more)
▼ This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component.
An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming
smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a
consequence of our tail expansions, the polynomial expansion in t of the transition
density is also obtained under mild conditions.
The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel
second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as
well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities
are also addressed.
Advisors/Committee Members: Houdre, Christian (Committee Chair), Bakhtin, Yuri (Committee Member), Dai, Jiangang (Committee Member), Koltchinskii, Vladimir (Committee Member), Popescu, Ionel (Committee Member), Swiech, Andrzej (Committee Member).
Subjects/Keywords: CGMY model; Stochastic volatility model; Implied volatility; Small time asymptotics; Levy process; Asymptotic expansions; Lévy processes; Options (Finance)
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APA (6th Edition):
Gong, R. (2012). Small-time asymptotics and expansions of option prices under Levy-based models. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/44798
Chicago Manual of Style (16th Edition):
Gong, Ruoting. “Small-time asymptotics and expansions of option prices under Levy-based models.” 2012. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/44798.
MLA Handbook (7th Edition):
Gong, Ruoting. “Small-time asymptotics and expansions of option prices under Levy-based models.” 2012. Web. 03 Mar 2021.
Vancouver:
Gong R. Small-time asymptotics and expansions of option prices under Levy-based models. [Internet] [Doctoral dissertation]. Georgia Tech; 2012. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/44798.
Council of Science Editors:
Gong R. Small-time asymptotics and expansions of option prices under Levy-based models. [Doctoral Dissertation]. Georgia Tech; 2012. Available from: http://hdl.handle.net/1853/44798
11.
Hoffmeyer, Allen Kyle.
Small-time asymptotics of call prices and implied volatilities for exponential Lévy models.
Degree: PhD, Mathematics, 2015, Georgia Tech
URL: http://hdl.handle.net/1853/53506
► We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to asset-price…
(more)
▼ We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to asset-price models whose log returns structure is a Lévy process. We consider two main problems. First, we consider very general Lévy models that are in the domain of attraction of a stable random variable. Under some relatively minor assumptions, we give first-order at-the-money call-price and implied volatility asymptotics. In the case where our Lévy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t¹/ᵃℓ(t) where ℓ is a slowly varying function and α ∈ (1,2). We also give an example of a Lévy model which exhibits this new type of behavior where ℓ is not asymptotically constant. In the case of a Lévy process with Brownian component, we find that the order of convergence of the call price is √t. Second, we investigate the CGMY process whose call-price asymptotics are known to third order. Previously, measure transformation and technical estimation methods were the only tools available for proving the order of convergence. We give a new method that relies on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using only the characteristic function of the Lévy process. While this method does not provide a less technical approach, it is novel and is promising for obtaining second-order call-price asymptotics for at-the-money options for a more general class of Lévy processes.
Advisors/Committee Members: Houdre, Christian (advisor), Bakhtin, Yuri (committee member), Koltchinskii, Vladimir (committee member), Peng, Liang (committee member), Figueroa-Lopez, Jose E. (committee member).
Subjects/Keywords: CGMY process; Levy process; Small-time asymptotics; Asymptotic expansions; Regular variation; Options pricing; Finance
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APA ·
Chicago ·
MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Hoffmeyer, A. K. (2015). Small-time asymptotics of call prices and implied volatilities for exponential Lévy models. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/53506
Chicago Manual of Style (16th Edition):
Hoffmeyer, Allen Kyle. “Small-time asymptotics of call prices and implied volatilities for exponential Lévy models.” 2015. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/53506.
MLA Handbook (7th Edition):
Hoffmeyer, Allen Kyle. “Small-time asymptotics of call prices and implied volatilities for exponential Lévy models.” 2015. Web. 03 Mar 2021.
Vancouver:
Hoffmeyer AK. Small-time asymptotics of call prices and implied volatilities for exponential Lévy models. [Internet] [Doctoral dissertation]. Georgia Tech; 2015. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/53506.
Council of Science Editors:
Hoffmeyer AK. Small-time asymptotics of call prices and implied volatilities for exponential Lévy models. [Doctoral Dissertation]. Georgia Tech; 2015. Available from: http://hdl.handle.net/1853/53506
12.
Li, Yao.
Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks.
Degree: PhD, Mathematics, 2012, Georgia Tech
URL: http://hdl.handle.net/1853/49013
► The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency…
(more)
▼ The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first chapter is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.
Advisors/Committee Members: Yi, Yengfei (advisor), Zhou, Hao-min (committee member), Kemp, Melissa Lambeth (committee member), Chow, Shui-Nee (committee member), Bakhtin, Yuri (committee member).
Subjects/Keywords: Stochastic dynamical system; Fokker-Planck equation; Systematic measure; Systems biology; Bioinformatics; Fokker-Planck equation Numerical solutions; Stochastic differential equations
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APA ·
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APA (6th Edition):
Li, Y. (2012). Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/49013
Chicago Manual of Style (16th Edition):
Li, Yao. “Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks.” 2012. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/49013.
MLA Handbook (7th Edition):
Li, Yao. “Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks.” 2012. Web. 03 Mar 2021.
Vancouver:
Li Y. Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks. [Internet] [Doctoral dissertation]. Georgia Tech; 2012. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/49013.
Council of Science Editors:
Li Y. Stochastic perturbation theory and its application to complex biological networks – a quantification of systematic features of biological networks. [Doctoral Dissertation]. Georgia Tech; 2012. Available from: http://hdl.handle.net/1853/49013
13.
Almada Monter, Sergio Angel.
Scaling limit for the diffusion exit problem.
Degree: PhD, Mathematics, 2011, Georgia Tech
URL: http://hdl.handle.net/1853/39518
► A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from…
(more)
▼ A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory.
Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks.
The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.
Advisors/Committee Members: Bakhtin, Yuri (Committee Chair), Bunimovich, Leonid (Committee Member), Cvitanovic, Pedrag (Committee Member), Houdre, Christian (Committee Member), Koltchinskii, Vladimir (Committee Member), Swiech, Andrzej (Committee Member).
Subjects/Keywords: Stochastic calculus; Small noise; Stochastic dynamics; Probability; Dynamical systems; Stochastic differential equations; Stochastic analysis; Dynamics
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APA ·
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MLA ·
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APA (6th Edition):
Almada Monter, S. A. (2011). Scaling limit for the diffusion exit problem. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/39518
Chicago Manual of Style (16th Edition):
Almada Monter, Sergio Angel. “Scaling limit for the diffusion exit problem.” 2011. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/39518.
MLA Handbook (7th Edition):
Almada Monter, Sergio Angel. “Scaling limit for the diffusion exit problem.” 2011. Web. 03 Mar 2021.
Vancouver:
Almada Monter SA. Scaling limit for the diffusion exit problem. [Internet] [Doctoral dissertation]. Georgia Tech; 2011. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/39518.
Council of Science Editors:
Almada Monter SA. Scaling limit for the diffusion exit problem. [Doctoral Dissertation]. Georgia Tech; 2011. Available from: http://hdl.handle.net/1853/39518
Georgia Tech
14.
Pearson, John Clifford.
The noncommutative geometry of ultrametric cantor sets.
Degree: PhD, Mathematics, 2008, Georgia Tech
URL: http://hdl.handle.net/1853/24657
► An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a…
(more)
▼ An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.
Advisors/Committee Members: Bellissard, Jean (Committee Chair), Baker, Matt (Committee Member), Bakhtin, Yuri (Committee Member), Garoufalidis, Stavros (Committee Member), Putnam, Ian (Committee Member).
Subjects/Keywords: Fractal geometry; Noncommutative geometry; Cantor set; Riemannian manifolds; Noncommutative differential geometry; Geometry; Cantor sets
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MLA ·
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Export
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APA (6th Edition):
Pearson, J. C. (2008). The noncommutative geometry of ultrametric cantor sets. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/24657
Chicago Manual of Style (16th Edition):
Pearson, John Clifford. “The noncommutative geometry of ultrametric cantor sets.” 2008. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/24657.
MLA Handbook (7th Edition):
Pearson, John Clifford. “The noncommutative geometry of ultrametric cantor sets.” 2008. Web. 03 Mar 2021.
Vancouver:
Pearson JC. The noncommutative geometry of ultrametric cantor sets. [Internet] [Doctoral dissertation]. Georgia Tech; 2008. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/24657.
Council of Science Editors:
Pearson JC. The noncommutative geometry of ultrametric cantor sets. [Doctoral Dissertation]. Georgia Tech; 2008. Available from: http://hdl.handle.net/1853/24657
Georgia Tech
15.
Palmer, Ian Christian.
Riemannian geometry of compact metric spaces.
Degree: PhD, Mathematics, 2010, Georgia Tech
URL: http://hdl.handle.net/1853/34744
► A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a…
(more)
▼ A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ – in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the
space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
Advisors/Committee Members: Bellissard, Jean (Committee Chair), Bakhtin, Yuri (Committee Member), Belegradek, Igor (Committee Member), Cvitanovic, Predrag (Committee Member), Gangbo, Wilfrid (Committee Member).
Subjects/Keywords: Noncommutative Geometry; Metric Spaces; Geometry, Riemannian; Metric spaces; Hausdorff measures
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APA (6th Edition):
Palmer, I. C. (2010). Riemannian geometry of compact metric spaces. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/34744
Chicago Manual of Style (16th Edition):
Palmer, Ian Christian. “Riemannian geometry of compact metric spaces.” 2010. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/34744.
MLA Handbook (7th Edition):
Palmer, Ian Christian. “Riemannian geometry of compact metric spaces.” 2010. Web. 03 Mar 2021.
Vancouver:
Palmer IC. Riemannian geometry of compact metric spaces. [Internet] [Doctoral dissertation]. Georgia Tech; 2010. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/34744.
Council of Science Editors:
Palmer IC. Riemannian geometry of compact metric spaces. [Doctoral Dissertation]. Georgia Tech; 2010. Available from: http://hdl.handle.net/1853/34744
Georgia Tech
16.
Litherland, Trevis J.
On the limiting shape of random young tableaux for Markovian words.
Degree: PhD, Mathematics, 2008, Georgia Tech
URL: http://hdl.handle.net/1853/26607
► The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite,…
(more)
▼ The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.
Advisors/Committee Members: Houdre, Christian (Committee Chair), Bakhtin, Yuri (Committee Member), Foley, Robert (Committee Member), Koltchinskii, Vladimir (Committee Member), Lifshitz, Mikhail (Committee Member), Matzinger, Heinrich (Committee Member), Popescu, Ionel (Committee Member).
Subjects/Keywords: Cyclic matrices; Brownian functional; Young tableaux; Markovian words; Invariance principle; Random variables; Probabilities; Asymptotic distribution (Probability theory)
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APA ·
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MLA ·
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Export
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APA (6th Edition):
Litherland, T. J. (2008). On the limiting shape of random young tableaux for Markovian words. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/26607
Chicago Manual of Style (16th Edition):
Litherland, Trevis J. “On the limiting shape of random young tableaux for Markovian words.” 2008. Doctoral Dissertation, Georgia Tech. Accessed March 03, 2021.
http://hdl.handle.net/1853/26607.
MLA Handbook (7th Edition):
Litherland, Trevis J. “On the limiting shape of random young tableaux for Markovian words.” 2008. Web. 03 Mar 2021.
Vancouver:
Litherland TJ. On the limiting shape of random young tableaux for Markovian words. [Internet] [Doctoral dissertation]. Georgia Tech; 2008. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/1853/26607.
Council of Science Editors:
Litherland TJ. On the limiting shape of random young tableaux for Markovian words. [Doctoral Dissertation]. Georgia Tech; 2008. Available from: http://hdl.handle.net/1853/26607
. 
Open Access Theses and Dissertations
