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We explore lattice point counting and the method of stationary phase through the lens of questions about the number of lattice points on and near surfaces with vanishing curvature. Our focus is on spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic Heisenberg norms for α ≥ 2. We accomplish this through a transformative process that takes a number theory question about counting lattice points and translates it into that of an analytical estimation of measure. This process relies on truncating and scaling the n-dimensional integer lattice to produce a fractal-like set. By introducing a measure on this resulting set and using elementary Fourier analysis, the counting problem is transformed into one of bounding an energy integral.This process uses principles of fractal geometry and oscillatory integrals. Primary challenges that arise are the presence of vanishing curvature and uneven dilations. Following a discussion and formal estimate of the curvature of the Heisenberg spheres, we utilize the method of stationary phase to compute a bound on the Fourier transform of their surface measures.Our work is inspired by that of Iosevich and Taylor (2011) and Garg, Nevo, and Taylor (2015). We present an extension of the main result in the former to surfaces with vanishing curvature. Furthermore, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.Additionally, we present a mini-course on the basics of stationary phase—a quick-start guide to stationary phase in practice. This includes a discussion of the formulation of oscillatory integrals and their solutions with a focus on the impact of geometric properties (e.g. curvature) on the estimates for the decay of the Fourier transform. It further serves as a supplement to [Shakarchi and Stein, Functional Analysis: Chapter 8, Sections 2 and 3], providing detailed expositions of the proofs of fundamental results.We conclude with a discussion of potential avenues for improvement and interesting problems that could be considered utilizing our methods.

Overactive Bladder (OAB) syndrome affects nearly 33 million adults in the US alone and results in severe health-related quality-of-life issues. Detrusor Overactivity (DO) is thought to be an important underlying cause of OAB and bladder cystometry – a test to measure bladder pressures during filling and voiding phases, is the clinical standard for diagnosis of DO. However, cystometry has not evolved for decades, and analysis of cystometric data is still largely subjective, dependent on physician interpretation and, mostly focused on overt contractile events and voiding behavior. Patient reported outcome measures, which are subjective and non-quantitative, have instead filled this diagnostic gap, resulting in challenges for both clinicians and patients. This thesis addresses the limitations of current bladder diagnostics by 1) developing a scalable workflow to measure and quantify cystometric signals during filling and 2) applying the workflow and measurement techniques across various clinical and preclinical in vitro and in vivo models and studies to improve utility of cystometric outcomes and increase reliability and accuracy of clinically diagnosing DO.As a first step, a scalable workflow was developed from an initial set of clinical cystometry data sets to quantify bladder pressure signals during filling, with the goal of rapid near-term translation to routine clinical application. This workflow was used to analyze the acquired signals from preclinical studies and existing clinical data sets, which demonstrate its scalability and utility across different model systems. Initially, in vitro porcine tissue models were considered for measurement of contractile activity of bladder tissue strips. Numerous studies were conducted to mimic physiological and pathological conditions during bladder filling, by tissue stretching and addition of carbachol and their impacts on the contractile activities of the strips were measured and quantified. To understand bladder physiology in a clinical setting, retrospective clinical studies were conducted where clinical cystometry data sets were analyzed with a focus on filling-related bladder pressure signals. Further, the analysis was extended to DO and non-DO patients where the success of the workflow was demonstrated in not just differentiating the two groups but being able to do so with half the duration of available signals, thus showing potential to reduce cystometry duration by 50% as well as eliminating the need to initiate voiding. To bridge the gap between in vitro and clinical studies, preclinical in vivo large animal sheep studies were conducted. Leveraging the physiological and anatomical similarities of sheep and humans, multiple sets of filling cystometry data were captured from sheep in both awake behaving and anesthetized states and the data was analyzed using the workflow. Results showed similarities between in vivo signals, and those of clinical non-DO patients as well as in vitro tissue studies. Further, these studies demonstrated that sheep are viable preclinical models to mimic bladder physiology and pathologies, and test different therapeutic efficacies. This thesis demonstrates the utility of measuring and quantifying filling-related signals of the bladder in diagnosis of bladder dysfunction. Further, parallels between preclinical and clinical models of bladder dysfunction are shown. The concepts described here reflect the need for quantification of clinical data and point to a much-needed shift in clinical diagnosis of bladder dysfunction from subjective interpretation to objective quantified outcomes.

Abstract: Kernel methods are often used for nonlinear regression and classification in machine learning because they are computationally cheap and accurate. Fourier basis and wavelet basis are the bases that can efficiently approximate the kernel functions. In previous research, Bayesian approximate kernel regression with Fourier transform has been proposed. With the proposed method, we use the analytic properties of the reproducing kernel Hilbert space (RKHS) to define a linear vector space that captures nonlinear structures. We map the data into a low-dimensional randomized feature space using Fourier transform and convert kernel function into operations of a linear machine. A Bayesian approximate kernel regression model is then formulated with the application of a generalized kernel model and the Bayesian method. We replace Fourier transform with wavelet transform in randomized feature space to approximate kernel functions. We formulate a new Bayesian approximate kernel model with wavelet transform and use the Gibbs sampler to compute the parameters of the model. We then make a comparison between the performance of Fourier based and wavelet-based Bayesian approximate kernels solving both regression and classification problems.