Dissertationen zum Thema „Applied Mathematics|Mathematics|Theoretical Mathematics“

A fundamental question in Complexity Science is how numerous dynamic processes coordinate with each other on multiple levels of description to form a complex whole—a multiscale coordinative structure (e.g. a community of interacting people, organs, cells, molecules etc.). This dissertation includes a series of empirical, theoretical and methodological studies of rhythmic coordination between multiple agents to uncover dynamic principles underlying multiscale coordinative structures. First, a new experimental paradigm was developed for studying coordination at multiple levels of description in intermediate-sized (N = 8) ensembles of humans. Based on this paradigm, coordination dynamics in 15 ensembles was examined experimentally, where the diversity of subjects’ movement frequency was manipulated to induce different grouping behavior. Phase coordination between subjects was found to be metastable with inphase and antiphase tendencies. Higher frequency diversity led to segregation between frequency groups, reduced intragroup coordination, and dispersion of dyadic phase relations (i.e. relations at different levels of description). Subsequently, a model was developed, successfully capturing these observations. The model reconciles the Kuramoto and the extended Haken-Kelso-Bunz model (for large- and small-scale coordination respectively) by adding the second-order coupling from the latter to the former. The second order coupling is indispensable in capturing experimental observations and connects behavioral complexity (i.e. multistability) of coordinative structures across scales. Both the experimental and theoretical studies revealed multiagent metastable coordination as a powerful mechanism for generating complex spatiotemporal patterns. Coexistence of multiple phase relations gives rise to many topologically distinct metastable patterns with different degrees of complexity. Finally, a new data-analytic tool was developed to quantify complex metastable patterns based on their topological features. The recurrence of topological features revealed important structures and transitions in high-dimensional dynamic patterns that eluded its non-topological counterparts. Taken together, the work has paved the way for a deeper understanding of multiscale coordinative structures.

GRAY, SCOTT THOMAS. A Distributed ParameterLiver Model of Benzene Transport and Metabolism in Humans and Mice -Developmental, Theoretical, and Numerical Considerations. (Under thedirection of Hien T. Tran.)

In the Clean Air Act of 1970, the U. S. Congress names benzene ahazardous air pollutant and directs certain government agencies toregulate public exposure. Court battles over subsequent regulations haveled to the need for quantitative risk assessment techniques. Models forhuman exposure to various chemicals exist, but most current modelsassume the liver is well-mixed. This assumption does not recognize (mostsignificantly) the spatial distribution of enzymes involved in benzenemetabolism.

The development of a distributed parameter liver model that accountsfor benzene transport and metabolism is presented. The mathematicalmodel consists of a parabolic system of nonlinear partial differentialequations and enables the modeling of convection, diffusion, andreaction within the liver. Unlike the commonly used well-mixed model,this distributed parameter model has the capacity to accommodate spatialvariations in enzyme distribution.

Preliminary investigations on the qualitative behavior of thedistributed liver model have included simulations with orthograde andretrograde bloodflow through mouse liver tissue. Simulation of humanexposure with the partial differential equation and the existingordinary differential equation model are presented and compared.Finally, the dependence of the solution on model parameters is explored.

In this dissertation we study combinatorial Hodge Laplacians on simplicial com-

plexes using tools generalized from spectral graph theory. Specifically, we consider

generalizations of graph Cheeger numbers and graph random walks. The results in

this dissertation can be thought of as the beginnings of a new spectral theory for

simplicial complexes and a new theory of high-dimensional expansion.

We first consider new high-dimensional isoperimetric constants. A new Cheeger-

type inequality is proved, under certain conditions, between an isoperimetric constant

and the smallest eigenvalue of the Laplacian in codimension 0. The proof is similar

to the proof of the Cheeger inequality for graphs. Furthermore, a negative result is

proved, using the new Cheeger-type inequality and special examples, showing that

certain Cheeger-type inequalities cannot hold in codimension 1.

Second, we consider new random walks with killing on the set of oriented sim-

plexes of a certain dimension. We show that there is a systematic way of relating

these walks to combinatorial Laplacians such that a certain notion of mixing time

is bounded by a spectral gap and such that distributions that are stationary in a

certain sense relate to the harmonics of the Laplacian. In addition, we consider the

possibility of using these new random walks for semi-supervised learning. An algo-

rithm is devised which generalizes a classic label-propagation algorithm on graphs to

simplicial complexes. This new algorithm applies to a new semi-supervised learning

problem, one in which the underlying structure to be learned is flow-like.

In the last few decades, data analysis techniques have had to expand to handle large sets of data with complicated structure. This includes identifying low dimensional structure in high dimensional data, analyzing shape and image data, and learning from or classifying large corpora of text documents. Common Bayesian and Machine Learning techniques rely on using the unique geometry of these data types, however departing from Euclidean geometry can result in both theoretical and practical complications. Bayesian nonparametric approaches can be particularly challenging in these areas.

This dissertation proposes a novel approach to these challenges by working with convenient embeddings of the manifold valued parameters of interest, commonly making use of an extrinsic distance or measure on the manifold. Carefully selected extrinsic distances are shown to reduce the computational cost and to increase accuracy of inference. The embeddings are also used to yield straight forward derivations for nonparametric techniques. The methods developed are applied to subspace learning in dimension reduction problems, planar shapes, shape constrained regression, and text analysis.

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.