Browsing by Subject "Fourier analysis"

Now showing 1 - 6 of 6

Combinatorial and probabilistic techniques in harmonic analysis
(2012-05) Lewko, Mark J., 1983-; Vaaler, Jeffrey D.; Beckner, William; Pavlovic, Natasa; Rodriguez-Villegas, Fernando; Zuckerman, David
We prove several theorems in the intersection of harmonic analysis, combinatorics, probability and number theory. In the second section we use combinatorial methods to construct various sets with pathological combinatorial properties. In particular, we answer a question of P. Erdos and V. Sos regarding unions of Sidon sets. In the third section we use incidence bounds and bilinear methods to prove several new endpoint restriction estimates for the Paraboloid over finite fields. In the fourth and fifth sections we study a variational maximal operators associated to orthonormal systems. Here we use probabilistic techniques to construct well-behaved rearrangements and base changes. In the sixth section we apply our variational estimates to a problem in sieve theory. In the seventh section, motivated by applications to sieve theory, we disprove a maximal inequality related to multiplicative characters.
Forward modeling of internal kinematic structures of disk galaxies in MaNGA
(2018-12-07) Leung, Andrew S.; Gebhardt, Karl; Drory, Niv
SDSS-IV MaNGA is a large IFU survey with a goal of observing ~ 104 galaxies by the survey’s end. MaNGA produces spatially resolved kinematic maps that describe the motions of stars and gas in each galaxy, but these maps include the effects of beam smearing. The primary objective of this project is the development, testing and validation of a code designed to accurately extract intrinsic kinematic models from spectroscopic measurements of disk galaxies observed by the MaNGA survey. The code uses a thin disk model including perturbations around circular orbits up to second order to describe bar, spiral and oval features. In this Thesis, we document our implementation of the algorithm and present a small sample of our results to demonstrate the behavior of our code. The code is available online at http://github.com/aleung12/manga/ under the MIT License.
Multi-resolution modeling of the mitral valve : a novel computational pipeline for patient-specific simulations of valve repair
(2019-02-01) Khalighi, Amir Hossein; Sacks, Michael S.; Barr, Ronald E; Bogard, David G; Neptune, Richard R; Gorman, Robert C
The mitral valve (MV) is the left atrio-ventricular heart valve that regulates blood flow direction during the cardiac cycle. Among the four heart valves, MV is the most problematic one, with MV-related pathologies directly afflicting 5% of the population in the industrialized world. Over the past 25 years, computational simulations of the MV based on biomechanical models have gained significant credibility in understanding valve function and improving surgical treatments. However, MV models with proven predictive power have yet to be developed on a patient-specific basis from clinical imaging data. The main challenge is that ultrasound, which is the prevailing imaging modality in the clinic, struggles to capture the full MV shape and its fine-scale geometric details. Thus, computational modeling of the MV for clinical applications first requires overcoming the obstacle that complete MV models cannot be developed directly from clinical images. In this Ph.D. project, we tackled this challenge through a detailed anatomical analysis of the MV constituents to better understand the comprising components of the MV apparatus and their impact on the MV modeling. This knowledge was then used to systematically identify the key characteristic of predictive MV modeling, build patient-specific models, and perform simulations of the MV repair. Remarkably, we established a framework to build faithful computational models of the MV for predictive surgical simulations based only on the information that can be acquired in the clinic and prior to the operation.
Peak thermal conductivity measurements of bulk boron arsenide crystals and individual carbon nanotubes
(2021-08-13) Zhou, Yuanyuan, active 21st century; Shi, Li, Ph. D.; Wang, Yaguo; Bahadur, Vaibhav; Tutuc, Emanuel
High-thermal conductivity materials are useful for thermal management applications and fundamental studies of phonon transport. Conventional criteria suggests high thermal conductivity only exists in strongly bonded simple crystal structures of light elements, such as diamond, graphite, graphene, and carbon nanotubes (CNTs). In comparison, recent theories and experiments have shown zincblende boron Arsenide (BAs) as the first known semiconductor with a room-temperature thermal conductivity close to 1000 W m⁻¹ K⁻¹. The unusual high thermal conductivity is achieved via an unconventional route based on isotopically pure heavy atom and a large mass ratio of constituent atoms, the latter of which results in a large energy gap between the acoustic and option phonon polarizations and bunching of the acoustic phonon dispersions. These features in the phonon band structure limit three-phonon scattering and scattering by isotopic impurities. Past measurements of several ultrahigh thermal conductivity materials, including BAs bulk crystals, were not able to obtain the peak thermal conductivity, which is expected to appear at a low temperature and contains insight into the competition between extrinsic phonon-defect and phonon-boundary scattering with intrinsic phonon-phonon processes. Meanwhile, past thermal conductivity measurements of CNTs are subjected to errors caused by contact thermal resistance. The observed peak temperatures are much higher than those reported for bulk graphite. The results suggest that extrinsic phonon scattering mechanisms dominate intrinsic phonon-phonon scattering that is predicted to give rise to non-diffusive phonon transport phenomena including hydrodynamic, ballistic, and quantized phonon transport regimes. Here we report a peak thermal conductivity measurement method based on differential Wheatstone bridge measurements of the small temperature drop between two thin film resistance thermometers patterned directly on a bulk sample. With the use of a mesoscale silicon bar sample as the calibration standard, this method is able to obtain results that agree with past measurements of large bulk silicon crystals at high temperatures and first principles calculation results that accounts for additional phonon-boundary scattering in the sample. The agreement demonstrates the accuracy of this measurement method for peak thermal conductivity measurements of high-thermal conductivity materials. This method was employed to measure the peak thermal conductivity of several BAs crystals. In addition, a multi-probe thermal transport measurement method was used to determine both the contact thermal resistance and the intrinsic thermal conductance of different segments of the same individual multi-walled CNT samples simultaneously and directly. The differential thin film resistance thermometry method is expected to address the need of accurate peak thermal conductivity measurement methods and find use in the ongoing search of high-thermal conductivity materials for thermal management. The obtained peak thermal conductivity measurements of BAs can help to advance the understanding of phonon scatterings by phonons, boundaries, and defects in ultrahigh thermal conductivity materials. The thermal transport measurement of CNTs validates the multi-probe method for probing intrinsic thermal conductivity of nanostructures, and can provide an essential tool for further studying hydrodynamic, ballistic, and quantized phonon transport phenomena in high-quality CNTs and other low-dimensional structures.
Receptive Field Inference with Localized Priors
(Public Library of Science, 2011-10-27) Park, Mijung; Pillow, Jonathan W.
The linear receptive field describes a mapping from sensory stimuli to a one-dimensional variable governing a neuron's spike response. However, traditional receptive field estimators such as the spike-triggered average converge slowly and often require large amounts of data. Bayesian methods seek to overcome this problem by biasing estimates towards solutions that are more likely a priori, typically those with small, smooth, or sparse coefficients. Here we introduce a novel Bayesian receptive field estimator designed to incorporate locality, a powerful form of prior information about receptive field structure. The key to our approach is a hierarchical receptive field model that flexibly adapts to localized structure in both spacetime and spatiotemporal frequency, using an inference method known as empirical Bayes. We refer to our method as automatic locality determination (ALD), and show that it can accurately recover various types of smooth, sparse, and localized receptive fields. We apply ALD to neural data from retinal ganglion cells and V1 simple cells, and find it achieves error rates several times lower than standard estimators. Thus, estimates of comparable accuracy can be achieved with substantially less data. Finally, we introduce a computationally efficient Markov Chain Monte Carlo (MCMC) algorithm for fully Bayesian inference under the ALD prior, yielding accurate Bayesian confidence intervals for small or noisy datasets.
Some inequalities in Fourier analysis and applications
(2014-05) Kelly, Michael Scott; Vaaler, Jeffrey D.
We prove several inequalities involving the Fourier transform of functions which are compactly supported. The constraint that the functions have compact support is a simplifying feature which is desirable in applications, but there is a trade-off in control of other relevant quantities-- such as the mass of the function. With applications in mind, we prove inequalities which quantify these types of trade-offs.