Let $G$ be a locally compact group (usually a reductive algebraic group over an algebraic number field $F$). The main aim of the theory of Automorphic Forms is to understand the right regular representation of the group $G$ on the space $L^{2}(\Gamma \ G)$ for certain \emph{nice} closed subgroups $\Gamma$. Usually, $\Gamma$ is taken to be a lattice or even an arithmetic subgroup. In the case of uniform lattices, the space $L^{2}(\Gamma \ G)$ decomposes into a direct sum of irreducible unitary representations of the group $G$ with each such representation $\pi$ occurring with a \emph{finite} multiplicity $m(\Gamma, \pi)$. It seems quite difficult to obtain an explicit formula for this multiplicity; however, the limiting behaviour of these numbers in case of certain \emph{nice} sequences of subgroups $(\Gamma_{n})_{n}$ seems more tractable. We study this problem in the global set-up where $G$ is the group of adelic points of a reductive group defined over the field of rational numbers and the relevant subgroups are the maximal compact open subgroups of $G$. As is natural and traditional, we use the Arthur trace formula to analyse the multiplicities. In particular, we expand the geometric side to obtain the information about the spectral side---which is made up from the multiplicities $m(\Gamma, \pi)$. The geometric side has a contributions from various conjugacy classes, most notably from the unipotent conjugacy class. It is this \emph{unipotent} contribution that is the subject of Part III of this thesis. We estimate the contribution in terms of level of the maximal compact open subgroup and make conclusions about the limiting behaviour. Part IV is then concerned with the spectral side of the trace formula where we show (under certain conditions) that the trace of the discrete part of the regular representation is the only term that survives in the limit.

Queueing theory has often been applied to study communication and service queueing systems such as call centers, hospital emergency departments and ride-sharing platforms. Unfortunately, it is complicated to analyze queueing systems. That is largely because the arrival and service processes that mainly determine a queueing system are uncertain and must be represented as stochastic processes that are difficult to analyze. In response, service providers might be able to partially capture the main characteristics of systems given partial data information and limited domain knowledge. An effective engineering response is to develop tractable approximations to approximate queueing characteristics of interest that depend on critical partial information. In this thesis, we contribute to developing high-quality approximations by studying tight bounds for the transient and the steady-state mean waiting time given partial information. We focus on single-server queues and multi-server queues with the unlimited waiting room, the first-come-first-served service discipline, and independent sequences of independent and identically distributed sequences of interarrival times and service times. We assume some partial information is known, e.g., the first two moments of inter-arrival and service time distributions. For the single-server GI/GI/1 model, we first study the tight upper bounds for the mean and higher moments of the steady-state waiting time given the first two moments of the inter-arrival time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. For the tight upper bound of the transient mean waiting time, we formulate the problem as a non-convex non-linear program, derive the gradient of the transient mean waiting time over distributions with finite support, and apply classical non-linear programming theory to characterize stationary points. We then develop and apply a stochastic variant of the conditional gradient algorithm to find a stationary point for any given service-time distribution. We also establish necessary conditions and sufficient conditions for stationary points to be three-point distributions or special two-point distributions. Our studies indicate that the tight upper bound for the steady-state mean waiting time is attained asymptotically by two-point distributions as the upper mass point of the service-time distribution increases and the probability decreases, while one mass of the inter-arrival time distribution is fixed at 0. We then develop effective numerical and simulation algorithms to compute the tight upper bound. The algorithms are aided by reductions of the special queues with extremal inter-arrival time and extremal service-time distributions to D/GI/1 and GI/D/1 models. Combining these reductions yields an overall representation in terms of a D/RS(D)/1 discrete-time model involving a geometric random sum of deterministic random variables, where the two deterministic random variables have different values, so that the extremal waiting times need not have a lattice distribution. We finally evaluate the tight upper bound to show that it offers a significant improvement over established bounds. In order to understand queueing performance given only partial information, we propose determining intervals of likely performance measures given that limited information. We illustrate this approach for the steady-state waiting time distribution in the GI/GI/K queue given the first two moments of the inter-arrival time and service time distributions plus additional information about these underlying distributions, including support bounds, higher moments, and Laplace transform values. As a theoretical basis, we apply the theory of Tchebycheff systems to determine extremal models (yielding tight upper and lower bounds) on the asymptotic decay rate of the steady-state waiting-time tail probability, as in the Kingman-Lundberg bound and large deviations asymptotics. We then can use these extremal models to indicate likely intervals of other performance measures. We illustrate by constructing such intervals of likely mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, yielding practical levels of accuracy.

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In many environmental and climate applications, extreme data are spatial by nature, and hence statistics of spatial extremes is currently an important and active area of research dedicated to developing innovative and flexible statistical models that determine the location, intensity, and magnitude of extreme events. In particular, the development of flexible sub-asymptotic models is in trend due to their flexibility in modeling spatial high threshold exceedances in larger spatial dimensions and with little or no effects on the choice of threshold, which is complicated with classical extreme value processes, such as Pareto processes. In this thesis, we develop new flexible sub-asymptotic extreme value models for modeling spatial and spatio-temporal extremes that are combined with carefully designed gradient-based Markov chain Monte Carlo (MCMC) sampling schemes and that can be exploited to address important scientific questions related to risk assessment in a wide range of environmental applications. The methodological developments are centered around two distinct themes, namely (i) sub-asymptotic Bayesian models for extremes; and (ii) flexible marked point process models with sub-asymptotic marks. In the first part, we develop several types of new flexible models for light-tailed and heavy-tailed data, which extend a hierarchical representation of the classical generalized Pareto (GP) limit for threshold exceedances. Spatial dependence is modeled through latent processes. We study the theoretical properties of our new methodology and demonstrate it by simulation and applications to precipitation extremes in both Germany and Spain. In the second part, we construct new marked point process models, where interest mostly lies in the extremes of the mark distribution. Our proposed joint models exploit intrinsic CAR priors to capture the spatial effects in landslide counts and sizes, while the mark distribution is assumed to take various parametric forms. We demonstrate that having a sub-asymptotic distribution for landslide sizes provides extra flexibility to accurately capture small to large and especially extreme, devastating landslides.

In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graph-theoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phase-lock oscillations.

In dense hot star winds, the infrared and radio continua are dominated by free‐free opacity and recombination emission line spectra. In the case of a spherically symmetric outflow that is isothermal and expanding at constant radial speed, the radiative transfer for the continuum emission from a dense wind is analytic. Even the emission profile shape for a recombination line can be derived. Key to these derivations is that the opacity scales with only the square of the density. These results are well‐known. Here an extension of the derivation is developed that also allows for line blends and the inclusion of an additional power‐law dependence beyond just the density dependence. The additional power‐law is promoted as a representation of a radius dependent clumping factor. It is shown that differences in the line widths and equivalent widths of the emission lines depend on the steepness of the clumping power‐law. Assuming relative level populations in LTE in the upper levels of He II, an illustrative application of the model to Spitzer/IRS spectral data of the carbon‐rich star WR 90 is given (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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