Mathematicshttp://hdl.handle.net/1903/22612018-08-17T12:12:03Z2018-08-17T12:12:03ZThe Uncertainty Principle in Harmonic Analysis and Bourgain's TheoremPowell, Alexander M.http://hdl.handle.net/1903/210872018-07-17T15:41:11Z2003-01-01T00:00:00ZThe Uncertainty Principle in Harmonic Analysis and Bourgain's Theorem
Powell, Alexander M.
We investigate the uncertainty principle in harmonic analysis and how it constrains the uniform localization properties of orthonormal bases. Our main result generalizes a theorem of Bourgain to construct orthonormal bases which are uniformly well-localized in time and frequency with respect to certain generalized variances. In a related result, we calculate generalized variances of orthonormalized Gabor systems. We also answer some interesting cases of a question of H. S. Shapiro on the distribution of time and frequency means and variances for orthonormal bases.
2003-01-01T00:00:00ZA latent variable modeling framework for analyzing neural population activityWhiteway, Matthewhttp://hdl.handle.net/1903/209962018-07-17T06:18:41Z2018-01-01T00:00:00ZA latent variable modeling framework for analyzing neural population activity
Whiteway, Matthew
Neuroscience is entering the age of big data, due to technological advances in
electrical and optical recording techniques. Where historically neuroscientists have
only been able to record activity from single neurons at a time, recent advances
allow the measurement of activity from multiple neurons simultaneously. In fact, this
advancement follows a Moore’s Law-style trend, where the number of simultaneously
recorded neurons more than doubles every seven years, and it is now common to see
simultaneous recordings from hundreds and even thousands of neurons.
The consequences of this data revolution for our understanding of brain struc-
ture and function cannot be understated. Not only is there opportunity to address
old questions in new ways, but more importantly these experimental techniques will
allow neuroscientists to address new questions entirely. However, addressing these
questions successfully requires the development of a wide range of new data anal-
ysis tools. Many of these tools will draw on recent advances in machine learning
and statistics, and in particular there has been a push to develop methods that can
accurately model the statistical structure of high-dimensional neural activity.
In this dissertation I develop a latent variable modeling framework for analyz-
ing such high-dimensional neural data. First, I demonstrate how this framework can
be used in an unsupervised fashion as an exploratory tool for large datasets. Next, I
extend this framework to incorporate nonlinearities in two distinct ways, and show
that the resulting models far outperform standard linear models at capturing the
structure of neural activity. Finally, I use this framework to develop a new algorithm
for decoding neural activity, and use this as a tool to address questions about how
information is represented in populations of neurons.
2018-01-01T00:00:00ZConsistency of Spectral Clustering with Functional Magnetic Resonance Image DataMoon, Jessiehttp://hdl.handle.net/1903/209912018-07-17T06:17:45Z2018-01-01T00:00:00ZConsistency of Spectral Clustering with Functional Magnetic Resonance Image Data
Moon, Jessie
Functional magnetic resonance imaging (fMRI) is a non-invasive technique for studying brain activity. It uses the amount of blood flowing through a brain, referred to as the blood oxygenation level dependent (BOLD) signal. However analyzing the fMRI signals is challenging because of its complicated spatio-temporal correlation structure and its massive amount of data.
There are several brain atlases available but researchers observe that fMRI signals are not coherent even within the same area in a brain atlas. Therefore providing parcellation of a brain, especially based on its functional connectivity, is necessary to understand brain activities.
One of the techniques that are used for a brain parcellation is spectral clustering. It is a well-used technique in many areas of studies, such as physics and engineering. However, its asymptotic behavior, whether spectral clustering will produce consistent clustering as samples grow large, is not fully clarified. In addition, there has previously been no available mathematical justification of the large-sample properties of spectral clustering when the data are dependent.
Von Luxburg et al. (2008) showed the consistency of eigenfunctions of spectral clustering under the assumption that data are independent and identically distributed. Because fMRI signals are spatially dependent, applying her results to fMRI data analysis is not appropriate. In this thesis, we extend von Luxburg's work to 3-dimensional spatially dependent data satisfying strong mixing conditions, which will be the case for fMRI data.
We applied the spectral clustering algorithm to simulated data to see how the algorithm can be affected by perturbation in a similarity matrix. There are two simulated data experiments. The first type of simulated data is similar to the stochastic block model, and the second is sampled independently from a Gaussian random field distribution with correlation.
We applied spectral clustering to various regions of interest (ROIs) both for a single subject and for multiple subjects. We also provided methods to analyze data from multiple subjects using spectral clustering and compared these methods using several criteria.
2018-01-01T00:00:00ZHydrodynamic Limits of the Boltzmann EquationLu, Chuntinghttp://hdl.handle.net/1903/209512018-07-17T06:09:45Z2018-01-01T00:00:00ZHydrodynamic Limits of the Boltzmann Equation
Lu, Chunting
This dissertation studies two problems that are related to the question of how solutions of the Boltzmann equation behave in various fluid dynamic regimes. The Boltzmann equation models so-called rarefied gases of identical particles, for which all but binary collisions between particles can be neglected. When the mean free path of gas particles is small comparing to the macroscopic length scale, one can derive fluid equations from the Boltzmann equations.
The first problem is to establish the acoustic limit for a family of appropriately scaled DiPerna-Lions solutions with finite zeroth to second moments over $\RD$. Every initial data with finite zeroth to second moments has a unique nonhomogeneous global Maxwellian associated with it by matching values of conserved quantities. The fluid fluctuations converge to a unique limit governed by the solution of an acoustic system with variable coefficients. This differs from the acoustic system with constant coefficient obtained by scaling the Boltzmann equation around a homogeneous Maxwellian (cf. Bardos-Golse-Levermore (2000), Golse-Levermore (2002)). Moreover, unlike the regimes around the homogeneous Maxwellian, there is no higher order Navier-Stokes correction in the regime around the nonhomogeneous Maxwellian.
The second problem is the approximation of solutions to the linearized Boltzmann equation by solutions of the linearized compressible Navier-Stokes system and by solutions of the weakly dissipative linearized compressible Navier-Stokes system over a periodic domain. We show that if the initial data of the linearized Boltzmann equation is smooth enough and lies within the fluid regime, then fluid moments of its solutions are close to the associated linearized compressible Navier-Stokes system in $L^2(\TD)$ uniformly for $t> 0$. We also show that solutions of the weakly dissipative linearized compressible Navier-Stokes systems approximate solutions of the linearized compressible Navier-Stokes system uniformly for $t > 0$ in $L^2(\TD)$. Therefore, we justified weakly dissipative linearized compressible Navier-Stokes approximation to the linearized Boltzmann equation. Our work differs from that of Ellis and Pinsky \cite{ellis1975} in that (1) we consider a periodic domain instead of $\RD$, and (2) the collision kernels we consider include those arising from inverse power potentials, as well as the hard sphere case considered in Ellis and Pinsky (1975).
2018-01-01T00:00:00Z