This thesis is to develop a novel consensus algorithm or protocol for multi-agent systems in the event of communication link failure over the network. The structure or topology of the system is modeled by an algebraic graph theory, and defined as a discrete time-invariant system with a second-order dynamics. The communication link failure is governed by a Bernoulli process. Lyapunov-based methodologies and Linear Matrix Inequality (LMI) techniques are then applied to find an appropriate controller gain by satisfying the sufficient conditions of the error dynamics. Therefore, the controller with the calculated gain is guaranteed to drive the system to reach a consensus. Finally, simulation and experiment studies are carried out by using two Mobile Robot Pioneer 3-DXs and one Pioneer 3-AT as a team to verify the proposed work.

abstract: In this work, I present a Bayesian inference computational framework for the analysis of widefield microscopy data that addresses three challenges: (1) counting and localizing stationary fluorescent molecules; (2) inferring a spatially-dependent effective fluorescence profile that describes the spatially-varying rate at which fluorescent molecules emit subsequently-detected photons (due to different illumination intensities or different local environments); and (3) inferring the camera gain. My general theoretical framework utilizes the Bayesian nonparametric Gaussian and beta-Bernoulli processes with a Markov chain Monte Carlo sampling scheme, which I further specify and implement for Total Internal Reflection Fluorescence (TIRF) microscopy data, benchmarking the method on synthetic data. These three frameworks are self-contained, and can be used concurrently so that the fluorescence profile and emitter locations are both considered unknown and, under some conditions, learned simultaneously. The framework I present is flexible and may be adapted to accommodate the inference of other parameters, such as emission photophysical kinetics and the trajectories of moving molecules. My TIRF-specific implementation may find use in the study of structures on cell membranes, or in studying local sample properties that affect fluorescent molecule photon emission rates. / Dissertation/Thesis / Masters Thesis Applied Mathematics 2019

Nonparametric Bayesian inference has widespread applications in statistics and machine learning. In this thesis, we examine the most popular priors used in Bayesian non-parametric inference. The Dirichlet process and its extensions are priors on an infinite-dimensional space. Originally introduced by Ferguson (1983), its conjugacy property allows a tractable posterior inference which has lately given rise to a significant developments in applications related to machine learning. Another yet widespread prior used in nonparametric Bayesian inference is the Beta process and its extensions. It has originally been introduced by Hjort (1990) for applications in survival analysis. It is a prior on the space of cumulative hazard functions and it has recently been widely used as a prior on an infinite dimensional space for latent feature models. Our contribution in this thesis is to collect many diverse groups of nonparametric Bayesian tools and explore algorithms to sample from them. We also explore machinery behind the theory to apply and expose some distinguished features of these procedures. These tools can be used by practitioners in many applications.

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Climate change is currently a central problem in ecology, with far-reaching effects on species that may be diffcult to quantify. Ectothermic species which rely on environmental cues to complete successive stages of their life history are especially sensitive to temperature changes and so are good indicators of the impacts of climate change on ecosystems. Based on data collected in growth experiments for the alpine butterfly Parnassius smintheus (Rocky Mountain Apollo), a novel mathematical model is presented to study developmental rate in larval insects. The movement of an individual through larval instars is treated as a discrete-time four-outcome Bernoulli process, where class transition and death are assigned temperature-dependent probabilities. Transition and mortality probabilities are estimated using maximum likelihood estimation techniques. This adult emergence model is then integrated into a reproductive success model, and multi-year implications of climate change on the population dynamics of P. smintheus are explored. / Applied Mathematics