Associated to any finite simple graph Γ is the <i>chromatic polynomial </i>PΓ(q) whose complex zeros are called the <i>chromatic zeros </i>of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γ_n}∞_<i>n</i>-0 built recursively using a substitution rule expressed in terms of a generating graph. For each <i>n</i>, let <i>μn</i> denote the probability measure that assigns a Dirac measure to each chromatic zero of Γ_<i>n</i>. Under a mild hypothesis on the generating graph, we prove that the sequence <i>μn</i> converges to some measure <i>μ</i> as <i>n</i> tends to infinity. We call <i>μ</i> the limiting measure of <i>chromatic zeros</i> associated to {Γ_n}∞_<i>n-</i>0. In the case of the Diamond Hierarchical Lattice we prove that the support of <i>μ</i> has Hausdorff dimension two.<div><br></div><div>The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove anew equidistribution theorem that can be used to relate the chromatic zeros of ahierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.<br></div>

<div>Wildlife diseases are an increasingly growing concern for public health managers, conservation biologists, and society at large. These diseases may be zoonotic -- infective wildlife are able to spread pathogens to human populations. Animal or plant species of conservation concern may also be threatened with extinction or extirpation due to the spread of novel pathogens into their native ranges. In this thesis, I develop some mathematical methods for understanding the dynamics of vector-borne diseases in wildlife populations which include several elements of host and vector biology. </div><div><br></div><div>We consider systems where a vector-borne pathogen is transmitted to a host population wherein individuals either die to disease or recover, remaining chronically infective. Both ordinary differential equations (ODE) and individual based (IBM) models of such systems are formulated then applied to a specific system of wildlife disease: avian malaria in Hawaiian honeycreeper populations -- where some species endure disease-induced mortality rates exceeding 90\%. The ODE model predicts that conventional management methods cannot fully stop pathogen transmission.</div><div><br></div><div>Vector dispersal and reproductive biology may also play a large role in the transmission of vector-borne diseases in forested environments. Using an IBM which models dispersal and mosquito reproductive biology, we predict that reducing larval habitat at low elevations is much more effective than at higher elevations. The ODE model is extended to include distinct populations of sensitive and tolerant hosts. We find that the form which interaction between the hosts takes has a significant impact on model predictions.</div>

Neural synchrony in the brain at rest is usually variable and intermittent, thus intervals of predominantly synchronized activity are interrupted by intervals of desynchronized activity. Prior studies suggested that this temporal structure of the weakly synchronous activity might be functionally significant: many short desynchronizations may be functionally different from few long desynchronizations, even if the average synchrony level is the same. In this thesis, we use computational neuroscience methods to investigate the effects of (i) spike-timing dependent plasticity (STDP) and (ii) noise on the temporal patterns of synchronization in a simple model. The model is composed of two conductance-based neurons connected via excitatory unidirectional synapses. In (i) these excitatory synapses are made plastic, in (ii) two different types of noise implementation to model the stochasticity of membrane ion channels is considered. The plasticity results are taken from our recently published article, while the noise results are currently being compiled into a manuscript.<br><br>The dynamics of this network is subjected to the time-series analysis methods used in prior experimental studies. We provide numerical evidence that both STDP and channel noise can alter the synchronized dynamics in the network in several ways. This depends on the time scale that plasticity acts on and the intensity of the noise. However, in general, the action of STDP and noise in the simple network considered here is to promote dynamics with short desynchronizations (i.e. dynamics reminiscent of that observed in experimental studies) over dynamics with longer desynchronizations.

<p>Recent innovations in combinatorial dynamical systems permit them to be studied with algorithmic methods. One such method from topological data analysis, called persistent homology, allows one to summarize the changing homology of a sequence of simplicial complexes. This dissertation explicates three methods for capturing and summarizing changes in combinatorial dynamical systems through the lens of persistent homology. The first places the Conley index in the persistent homology setting. This permits one to capture the persistence of salient features of a combinatorial dynamical system. The second shows how to capture changes in combinatorial dynamical systems at different resolutions by computing the persistence of the Conley-Morse graph. Finally, the third places Conley's notion of continuation in the combinatorial setting and permits the tracking of isolated invariant sets across a sequence of combinatorial dynamical systems. </p>

<p dir="ltr">The original definition of Nash Equilibrium applied to normal form games, but the notion has now been extended to various other forms of games including leader-follower games (Stackelberg games), extensive form games, stochastic games, games of incomplete information, cooperative games, and so on. We focus on general-sum stochastic Stackelberg games in this work. An example where such games would be natural to consider is in security games where a defender wishes to protect some targets through deployment of limited resources and an attacker wishes to strategically attack the targets to benefit themselves. The hierarchical order of play arises naturally since the defender typically acts first and deploys a strategy, while the attacker observes the strategy ofthe defender before attacking. Another example where this framework fits is in testing during epidemics, where the leader (the government) sets testing policies and the follower (the citizens) decide at every time step whether to get tested. The government wishes to minimize the number of infected people in the population while the follower wishes to minimize the cost of getting sick and testing. This thesis presents a learning algorithm for players to converge to their stationary policies in a general sum stochastic sequential Stackelberg game. The algorithm is a two time scale implicit policy gradient algorithm that provably converges to stationary points of the optimization problems of the two players. Our analysis allows us to move beyond the assumptions of zero-sum or static Stackelberg games made in the existing literature for learning algorithms to converge.</p><p dir="ltr"><br></p>

<p>In the real world, uncertainty is a common challenging problem faced by individuals, organizations, and firms. Decision quality is highly impacted by uncertainty because decision makers lack complete information and have to leverage the loss and gain in many possible outcomes or scenarios. This study explores dynamic decision making (with known distributions) and decision learning (with unknown distributions but some samples) in not-for-profit operations and supply chain management. We first study dynamic staffing for paid workers and volunteers with uncertain supply in a nonprofit operation where the optimal policy is too complex to compute and implement. Then, we consider dynamic inventory control and pricing under both supply and demand uncertainties where unmet demand is lost leading to a challenging non-concave dynamic problem. Furthermore, we explore decision learning from limited data of focal system and available data of related but different systems by transfer learning, cross learning, and co-learning utilizing the similarities among related systems.</p>

<p dir="ltr">The increasing availability of spatio-temporal information in the form of detailed time-resolved images sampled at very high framing rates has resulted in a search for mathematical techniques capable of extracting and relaying the pertinent underlying physics governing complex flows. Analysis relying on the usage of a solitary spectral, correlation, or modal decomposition techniques to identify dynamically significant information from large datasets may give an incomplete description of these phenomena. Moreover, fully resolved spatio-temporal measurements of these complex flow fields are needed for a complete and accurate description across a wide spectrum of length and time scales. The primary goals of this dissertation are address these challenges in two key aspects: (1) to improve and demonstrate the novel application of complementary data analysis and modal decomposition techniques to quantify the dynamics of flow systems that exhibit intricate patterns and behaviors in both space and time, and (2) to make advancements in achieving and characterizing high-resolution and high-speed quantitative measurements of turbulent mixing fields.</p><p dir="ltr">In the first goal, two canonical flow fields are considered, including an acoustically excited co-axial jet and a bluff-body stabilized flame. The local susceptibility of a nonreacting, cryogenic, coaxial-jet, rocket injector to transverse acoustics is characterized by applying dynamical systems theory in conjunction with complementary wavelet-based spectral decomposition to high-speed backlit images of flow field. The local coupling of the jet with external acoustics is studied as a function of the relative momentum flux ratio between the outer and inner jets, giving a quantitative description of the dynamical response of each jet to external acoustics as a function of the downstream distance from the nozzle.</p><p dir="ltr">Bluff bodies are a common feature in the design of propulsion systems owing to their ability to act as flame holders. The reacting wake behind the bluff body consists of a recirculation bubble laden with hot-products and wrapped between separated shear layers. The wake region of a bluff body is systematically investigated utilizing a technique known as robust dynamic mode decomposition (DMD) to discern the onset of the thermoacoustic instability mode, which is highly detrimental to aerospace propulsion systems. The approach enables quantification of the spatial distribution and behavior of coherent structures observed from different flows as a function of the equivalence ratio.</p><p dir="ltr">As modal decomposition techniques employ a certain degree of averaging in time, a novel space-and-time local filtering technique utilizing the well-defined characteristics of wavelets is introduced with a goal of temporally resolving the spatial evolution of irregular flow instabilities associated with specific frequencies. This provides insight into the existence of transient sub-modal characteristics representing intermittencies within seemingly stable modes. The flow fields obtained from the same two canonical flows are interrogated to demonstrate the utility of the technique. It has been shown that temporally resolved flow features obtained from wavelet filtering satisfactorily track the same modal featured derived from DMD, but reveal sub-modal spatial distortions or local non-stationarity of specific modal frequencies on a frame-by-frame basis.</p><p dir="ltr">Finally, to improve the ability to study the dynamical behavior of complex flows across the full range of spatio-temporal scales present, advancements are reported in the spatial and temporal quantitative measurement of the scalar quantities in turbulent mixing fields utilizing 100 kHz planar laser-induced fluorescence (PLIF) and Rayleigh scattering imaging of acetone. The imaging system provided a resolution of 55 µm with a field-of-view mapping of 18.5 µm/pixel on the camera sensor, which is three times better spatial resolution than the previous reported work to-date for similar flow fields that were investigated at 1/10^th the current measurement rate. The power spectra of instantaneous mixture fraction fluctuations adhere to Kolmogorov's well-established -5/3 law, showing that the technique captures a significant range of dissipation scales. This observation underscores the ability to study mixing dynamics throughout the turbulent by fully resolving scalar fluctuations up to 30 kHz. This enhanced spatio-temporal resolution allows for a more detailed investigation of the dynamical behavior of turbulent flows with complex modal interactions down to the smallest diffusion limited mixing scales.</p>

<p dir="ltr">There is a growing trend towards the development of parsimonious surrogate models for studying physical phenomena. While they typically offer less accuracy, these models bypass the computational costs of numerical methods, usually by multiple orders of magnitude, allowing statistical applications such as sensitivity analysis, stochastic treatments, parametric problems, and uncertainty quantification. Researchers have explored generalized surrogate frameworks leveraging Gaussian processes, various basis function expansions, support vector machines, and neural networks. Dynamical fields, represented through time-dependent partial differential equation, pose a particular hardship for existing frameworks due to their high dimensional representation, and possibly multi-scale solutions.</p><p dir="ltr">In this work, we present a novel architecture for solving time-dependent partial differential equations using co-ordinate neural networks and time-marching updates through hyper-networks. We show that it provides a temporally meshed and spatially mesh-free solution which are causally coherent as justified through a theoretical treatment of Lie groups. We showcase results on some benchmark problems in computational physics while discussing their performance against similar physics-informed approaches like physics-informed DeepOnets and Physics informed neural networks.</p>

Growing concerns over global warming and increasing fuel costs have pushed the traditional fuel-based centralized electrical grid to the forefront of mounting public pressure. These concerns will only intensify in the future, owing to the growth in electricity demand. Such growths require increased generation of electricity to meet the demand, and this means more carbon footprint from the electrical grid. To meet the growing demand economically by using clean sources of energy, the electrical grid needs significant structural and operational changes to cope with various challenges. Microgrids (µGs) can be an answer to the structural requirement of the electrical grid. µGs integrate renewables and serve local needs, thereby, reducing line losses and improving resiliency. However, stochastic nature of electricity harvest from renewables makes its integration into the grid challenging. The time varying and intermittent<br>nature of renewables and consumer demand can be mitigated by the use of storages and dynamic load scheduling. Automated dynamic load scheduling constitutes the operational changes that could enable us to achieve energy efficiency in the grid.<br>The current research works on automated load scheduling primarily focuses on scheduling residential and commercial building loads, while the current research on manufacturing scheduling is based on static approaches with very scarce literature on job shop scheduling. However, residential, commercial and, industrial sector, each contribute to about one-third of the total electricity consumption. A few research<br>works have been done focusing on dynamic scheduling in manufacturing facilities for energy efficiency. In a smart grid scenario, consumers are coupled through electricity<br>pool and storage. Thus, this research investigates the problem of integrating production line loads with building loads for optimal scheduling to reduce the total electricity<br>cost in a µG.<br>This research focuses on integrating the different types of loads from different types of consumers using automated dynamic load scheduling framework for sequential decision making. After building a deterministic model to be used as a benchmark, dynamic load scheduling models are constructed. Firstly, an intelligent algorithm is developed for load scheduling from a consumer’s perspective. Secondly, load scheduling model is developed based on central grid controller’s perspective. And finally, a reinforcement learning model is developed for improved load scheduling by sharing<br>among multiple µGs. The performance of the algorithms is compared against different well-known individualistic strategies, static strategies and, optimal benchmark<br>solutions. The proposed dynamic load scheduling framework is model free with minimum assumptions and it outperforms the different well-known heuristics and static strategies while obtains solutions comparable to the optimal benchmark solution.<br>The future electrical grid is envisioned to be an interconnected network of µGs. In addition to the automated load scheduling in a µG, coordination among µGs by<br>demand and capacity sharing can also be used to mitigate stochastic nature of supply and demand in an electrical grid. In this research, demand and resource sharing<br>among µGs is proposed to leverage the interaction between the different µGs for developing load scheduling policy based on reinforcement learning. <br>

<p>Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.</p>