2019-2020

The topic of this presentation is statistical hypothesis testing when the populations from which the data are drawn are known only with a given probability distribution. Some important areas of application for which such a situation arises will be noted briefly. The specific cases to be considered are testing a one-sided hypothesis involving two populations. An illustrative small data set, involving six observations, is used to demonstrate relevant approaches and calculations for such testing. Both a frequentist approach and a Bayesian approach will be discussed. In both approaches, use is made of all possible data configurations along with their corresponding probabilities. Various measures of goodness are put forth for each of the two approaches. A simulation approach will be described for larger data sets.

Models that predict the simultaneous movement of liquid water, vapor and heat in the shallow subsurface has many practical interests, given the critical role these processes play in the global water and energy balances. Given the limitation in computational capabilities, important details describing the interaction between heat and water near the subsurface, including the soil and its spatial heterogeneity, are big challenges to be accounted for in current global climate models.

In this talk, I will present a mathematical homogenization procedure that applies the two-scale formulation and asymptotic analysis into elliptic operators, and in particular, into the non-linear coupled system that describes the interaction between water and heat near the soil surface. The resulting 3D-upscaled (homogenized) model uses parameters such as air temperature, solar radiation, among others, as boundary conditions, making it more realistic and ready for coupling with the atmosphere. The use of such upscaled model allows a more accurate prediction of evaporation and water's budget that can be used into large-scale climate modeling efforts to better quantify the change of the climate, its forecast and its impact.

Within the talk, I will point to problems that I would like to pursue further research and collaboration with colleagues from the department. Since multiple scale has a wide range of applications, I hope to be seek possibility for research and interaction in other areas, besides porous media.

In recent years, researchers have been granted an unprecedented window into the inner workings of such cellular processes as the circadian rhythm, the cell cycle, apoptosis (cell death), and metabolism. These biochemical systems, however, can have dozens of components interacting over hundreds of reaction channels, which presents several significant challenges for mathematical modelers: (1) the high dimension of the state space, (2) the non-linearities of the interaction kinetics, and (3) the typically unknown rate parameters.

In this talk, I will introduce the background for both the ordinary differential equation (deterministic) and continuous-time Markov chain (stochastic) modeling frameworks for biochemical reaction systems. I will briefly introduce some historical work from chemical reaction network theory which cuts through the complexity of these systems by connecting the topological structure of the interaction network to the dynamical system's behavior. Recent results on steady states, discrete extinction events, and computational methods will be presented.

I would like to introduce a numerical method called nonstandard finite difference (NSFD) method that when applied to differential equations will preserve many nice properties of the solutions to the equations. First, I will talk about the exact finite difference methods and the reason behind the concept of NSFD methods. Then I will present many population models as examples, including predator-prey model, Lotka-Volterra competition model, SI and SIS epidemic models, to illustrate the powerful and useful NSFD methods that transform differential equations to difference equations. Possible future research directions will be presented as well.

Information asymmetry reflects the risk and uncertainty faced by investors and is a measure of a firm's transparency. High information asymmetry could increase the cost of external financing, which in turn impede a firm's leverage (debt-asset ratio) adjustment. This paper studies the adjustment speed toward the target leverage in the presence of information asymmetry using micro-level data from China. In contrast to the previous studies, we allow heterogeneity in the adjustment speed coefficient by modeling it as a nonparametric function of information asymmetry and other firm characteristics. This refinement not only allows for more flexibility in the model, but it also facilitates further exploration into the differences and determinants of firms' financing behavior. We uniquely build the firm-level measure of information asymmetry into the traditional partial leverage adjustment framework. Based on our firm-level measure of the adjustment speed, our paper explores why the leverage adjustment speed matters by examining its association with the corporate performance indicators. We find that China's firms do have leverage targets and they slowly adjust toward these targets. We also find that the adjustment speed decreases with an increase in information asymmetry. Overall, firms which converge toward their targets faster perform better in value, profitability, investment and costs.

The Space-time Discontinuous Galerkin (ST-DG) method is an excellent method to discretize problems on deforming domains. This method uses DG to discretize both in the spatial and temporal directions, allowing for an arbitrarily high order approximation in space and time. Furthermore, this method automatically satisfies the geometric conservation law which is essential for accurate solutions on time-dependent domains. We present a higher-order accurate Hybridizable or Embedded Discontinuous Galerkin (DG-H or DG-E) method for incompressible flows. These discretizations are energy stable and guarantee a pointwise divergence-free velocity field on simplicial meshes. Numerical results will be presented to illustrate the method.

Quadratic growth condition has been used extensively in recent developments of algorithms to guarantee fast convergence and stable performance. However, checking this property is a difficult task, usually proceeded case-by-case. In this presentation, unified second-order necessary and sufficient conditions for quadratic growth condition are provided. Our main application is to stable signal and image recovery, at which we study the theoretical guarantees of l^1 and l^{1,2} analysis regularization such as discrete total variation and fussed Lasso when solving sparse linear inverse problems.

Foodborne infection is a major public health concern. According to WHO, every year 10% of people suffer from foodborne illnesses worldwide. A dose-response relationship measures an infection risk of a pathogen and it is widely used in quantitative microbial risk assessment (QMRA). While many descriptive models (e.g. exponential and log-normal) can demonstrate the dose-response relationship of an outbreak they suffer from the lack of a mechanistic basis. To account for the within-host interaction of a pathogen and to measure its infection potential, we develop several mathematical models. These models mimic the gastrointestinal pathway of a food pathogen and demonstrate the critical steps for the pathogen and host when exposure may turn into an infection. Due to the frequent outbreaks of Listeria monocytogenes and its severe impact on public health and economy we consider this bacterium as the model pathogen. These models can be used as complementary tools to assess the pathogen survival in a host and to build the dose-response relationships.

The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics. My talk will be a tour of the history of the principle based on the book with the same title that I wrote in collaboration with Anthony Bloch, and published by Cambridge University Press. The book examines the principle and its fundamental role in science. including – with varying levels of mathematical sophistication - explanations from historical sources, discussion of classic papers, and original worked examples. The result is a story that is understandable to those with a modest mathematical background, as well as to researchers and students in physics and the history of physics.

In the broadest terms, Ecological Stoichiometry is the study of the intertwined relationships between the balance of energy, chemical elements, organisms and their interactions within ecosystems. Applying this to mathematical population models lets us explore how nutrient dynamics influence the interaction between organisms and their environments. In this talk, we’ll review some of the history behind one of the most famous population models of all time, the Lotka-Volterra equations and construct a recently developed version that incorporates ideas from Ecological Stoichiometry. In addition, we’ll explore some of the broader implications these ideas have in cancer modeling, plant-virus interactions and how rising atmospheric C02 levels may interact with human nutrition.