Mathematical Sciences Department Theses and Dissertations

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Exact Solutions to the Six-Vertex Model with Domain Wall Boundary Conditions and Uniform Asymptotics of Discrete Orthogonal Polynomials on an Infinite Lattice
(2011-03-09) Liechty, Karl Edmund; Bleher, Pavel, 1947-; Its, Alexander R.; Lempert, Lazlo; Kitchens, Bruce, 1953-
In this dissertation the partition function, $Z_n$, for the six-vertex model with domain wall boundary conditions is solved in the thermodynamic limit in various regions of the phase diagram. In the ferroelectric phase region, we show that $Z_n=CG^nF^{n^2}(1+O(e^{-n^{1-\ep}}))$ for any $\ep>0$, and we give explicit formulae for the numbers $C, G$, and $F$. On the critical line separating the ferroelectric and disordered phase regions, we show that $Z_n=Cn^{1/4}G^{\sqrt{n}}F^{n^2}(1+O(n^{-1/2}))$, and we give explicit formulae for the numbers $G$ and $F$. In this phase region, the value of the constant $C$ is unknown. In the antiferroelectric phase region, we show that $Z_n=C\th_4(n\om)F^{n^2}(1+O(n^{-1}))$, where $\th_4$ is Jacobi's theta function, and explicit formulae are given for the numbers $\om$ and $F$. The value of the constant $C$ is unknown in this phase region. In each case, the proof is based on reformulating $Z_n$ as the eigenvalue partition function for a random matrix ensemble (as observed by Paul Zinn-Justin), and evaluation of large $n$ asymptotics for a corresponding system of orthogonal polynomials. To deal with this problem in the antiferroelectric phase region, we consequently develop an asymptotic analysis, based on a Riemann-Hilbert approach, for orthogonal polynomials on an infinite regular lattice with respect to varying exponential weights. The general method and results of this analysis are given in Chapter 5 of this dissertation.
D-bar and Dirac Type Operators on Classical and Quantum Domains
(2012-08-29) McBride, Matthew Scott; Klimek, Slawomir; Cowen, Carl C.; Ji, Ron; Dadarlat, Marius
I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.
Mathematical Models of Basal Ganglia Dynamics
(2013-07-12) Dovzhenok, Andrey A.; Rubchinsky, Leonid; Kuznetsov, Alexey; Its, Alexander R.; Worth, Robert; Mukhin, Evgeny
Physical and biological phenomena that involve oscillations on multiple time scales attract attention of mathematicians because resulting equations include a small parameter that allows for decomposing a three- or higher-dimensional dynamical system into fast/slow subsystems of lower dimensionality and analyzing them independently using geometric singular perturbation theory and other techniques. However, in most life sciences applications observed dynamics is extremely complex, no small parameter exists and this approach fails. Nevertheless, it is still desirable to gain insight into behavior of these mathematical models using the only viable alternative – ad hoc computational analysis. Current dissertation is devoted to this latter approach. Neural networks in the region of the brain called basal ganglia (BG) are capable of producing rich activity patterns. For example, burst firing, i.e. a train of action potentials followed by a period of quiescence in neurons of the subthalamic nucleus (STN) in BG was shown to be related to involuntary shaking of limbs in Parkinson’s disease called tremor. The origin of tremor remains unknown; however, a few hypotheses of tremor-generation were proposed recently. The first project of this dissertation examines the BG-thalamo-cortical loop hypothesis for tremor generation by building physiologically-relevant mathematical model of tremor-related circuits with negative delayed feedback. The dynamics of the model is explored under variation of connection strength and delay parameters in the feedback loop using computational methods and data analysis techniques. The model is shown to qualitatively reproduce the transition from irregular physiological activity to pathological synchronous dynamics with varying parameters that are affected in Parkinson’s disease. Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. Besides tremor-related bursting activity BG structures in Parkinson’s disease also show increased synchronized activity in the beta-band (10-30Hz) that ultimately causes other parkinsonian symptoms like slowness of movement, rigidity etc. Suppression of excessively synchronous beta-band oscillatory activity is believed to suppress hypokinetic motor symptoms in Parkinson’s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson’s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronized dynamics used to study the delayed feedback stimulation. The second project of this dissertation explores the action of delayed feedback stimulation on partially synchronous oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces the fine temporal structure of the synchronous dynamics observed experimentally. Modeling results suggest that delayed feedback DBS in Parkinson’s disease may boost rather than suppresses synchronization and is therefore unlikely to be clinically successful. Single neuron dynamics may also have important physiological meaning. For instance, bistability – coexistence of two stable solutions observed experimentally in many neurons is thought to be involved in some short-term memory tasks. Bistability that occurs at the depolarization block, i.e. a silent depolarized state a neuron enters with excessive excitatory input was proposed to play a role in improving robustness of oscillations in pacemaker-type neurons. The third project of this dissertation studies what parameters control bistability at the depolarization block in the three-dimensional conductance-based neuronal model by comparing the reduced dopaminergic neuron model to the Hodgkin-Huxley model of the squid giant axon. Bifurcation analysis and parameter variations revealed that bistability is mainly characterized by the inactivation of the Na+ current, while the activation characteristics of the Na+ and the delayed rectifier K+ currents do not account for the difference in bistability in the two models.
The Dynamics of Twisted Tent Maps
(2013-07-12) Chamblee, Stephen Joseph; Misiurewicz, Michał, 1948-; Roeder, Roland; Geller, William; Eremenko, Alexandre; Mukhin, Evgeny
This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.
Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models
(2013-11-06) Bothner, Thomas Joachim; Its, Alexander R.; Bleher, Pavel, 1947-; Tarasov, Vitaly; Eremenko, Alexandre; Mukhin, Evgeny
We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.
Commutants of composition operators on the Hardy space of the disk
(2013-11-06) Carter, James Michael; Cowen, Carl C.; Klimek, Slawomir; Perez, Rodrigo A.; Chin, Raymond; Bell, Steven R.; Mukhin, Evgeny
The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.
Superstable manifolds of invariant circles
(2013-12-10) Kaschner, Scott R.; Roeder, Roland; Bleher, Pavel, 1947-; Misiurewicz, Michał, 1948-; Buzzard, Gregory; Mukhin, Evgeny
Let f:X\rightarrow X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of \mathbb P^1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z\rightarrow z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by resenting two examples with a < b for which W^s_loc(S) is not real analytic in the neighborhood of any point.
Restrictions to Invariant Subspaces of Composition Operators on the Hardy Space of the Disk
(2014-01-29) Thompson, Derek Allen; Cowen, Carl C.; Ji, Ronghui ; Klimek, Slawomir; Bell, Steven R.; Mukhin, Evgeny
Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.
Locally compact property A groups
(2014-05) Harsy Ramsay, Amanda R.; Ji, Ronghui
In 1970, Serge Novikov made a statement which is now called, "The Novikov Conjecture" and is considered to be one of the major open problems in topology. This statement was motivated by the endeavor to understand manifolds of arbitrary dimensions by relating the surgery map with the homology of the fundamental group of the manifold, which becomes diffi cult for manifolds of dimension greater than two. The Novikov Conjecture is interesting because it comes up in problems in many different branches of mathematics like algebra, analysis, K-theory, differential geometry, operator algebras and representation theory. Yu later proved the Novikov Conjecture holds for all closed manifolds with discrete fundamental groups that are coarsely embeddable into a Hilbert space. The class of groups that are uniformly embeddable into Hilbert Spaces includes groups of Property A which were introduced by Yu. In fact, Property A is generally a property of metric spaces and is stable under quasi-isometry. In this thesis, a new version of Yu's Property A in the case of locally compact groups is introduced. This new notion of Property A coincides with Yu's Property A in the case of discrete groups, but is different in the case of general locally compact groups. In particular, Gromov's locally compact hyperbolic groups is of Property A.
Multiscale mathematical modeling of ocular blood flow and oxygenation and their relevance to glaucoma
(2016-06-14) Carichino, Lucia; Guidoboni, Giovanna; Harris, Alon; Arciero, Julia Concetta
Glaucoma is a multifactorial ocular disease progressively leading to irreversible blindness. There is clear evidence of correlations between alterations in ocular hemodynamics and glaucoma; however, the mechanisms giving rise to these correlations are still elusive. The objective of this thesis is to develop mathematical models and methods to help elucidate these mechanisms. First, we develop a mathematical model that describes the deformation of ocular structures and ocular blood flow using a reduced-order fluid-structure interaction model. This model is used to investigate the relevance of mechanical and vascular factors in glaucoma. As a first step in expanding this model to higher dimensions, we propose a novel energy-based technique for coupling partial and ordinary differential equations in blood flow, using operator splitting. Next, we combine clinical data and model predictions to propose possible explanations for the increase in venous oxygen saturation in advanced glaucoma patients. We develop a computer-aided manipulation process of color Doppler images to extract novel waveform parameters to distinguish between healthy and glaucomatous individuals. The results obtained in this work suggest that: 1) the increase in resistance of the retinal microcirculation contributes to the influence of intraocular pressure on retinal hemodynamics; 2) the influence of cerebrospinal fluid pressure on retinal hemodynamics is mediated by associated changes in blood pressure; 3) the increase in venous oxygen saturation levels observed among advanced glaucoma patients depends on the value of the patients’ intraocular pressure; 4) the normalized distance between the ascending and descending limb of the ophthalmic artery velocity profile is significantly higher in glaucoma patients than in healthy individuals.

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