Prior literature on online communities proposes an important function for social roles. Theoretical models argue that specific roles are critical for community success, and claim that individual roles shift in predictable ways over the lifetime of the community. However these models are currently hard to assess, as they do not provide systematic definitions of roles, nor empirical work evaluating the impact of roles on community success. Further they do not specify interrelations between different roles. This thesis addresses these questions in the context of enterprise online communities. Using gold-standard systematically-defined social roles, we explore role behaviors and their impacts using both quantitative and qualitative measures of community success. We find evidence contradicting prior theoretical claims about role shifts and the division of labor between roles. We also examine language styles within communities, finding a complex relationship in the types of support language that engender success. To develop new models of relations between roles, we utilize graphical methods to discover important community subgroups centering around leadership. These subgroups are important predictors of community success, significantly extending the explanatory power of existing network theories. Furthermore, subgroups serve as sources of significant topic-setting language. This work elucidates the workings of online communities at a user level, increasing theoretical understanding and suggesting how new community tools might be developed that promote community success.

Signals describing the energy levels of quantum mechanical systems are, by definition, sparse

in the energy domain. Processing these signals via sparsity promoting methods is thus

reasonable and, as this dissertation argues, valuable.

Quantum mechanical energy levels are determined experimentally through NMR spec-

troscopy where noise, peak blurring, and long experiment times impede progress. We show

how l1-penalized optimization can lead to improved signal quality and reduce data acquisi-

tion time in NMR spectroscopy.

Quantum mechanical signal processing is central to MRI reconstruction. MRI data ac-

quisition and reconstruction is highly time-consuming and expensive. We provide a fast

converging algorithm based on minimizing a combination total variation and framelet norms

which produces high-quality images from undersampled MRI data.

In the field of numerical analysis, all the eigenvalues of a Hermitian matrix may be com-

puted by simulating a fictitious quantum dynamical system with Hamiltonian corresponding

to the matrix of interest and then determining the energy levels of this fictitious system. By

determining the energy levels with l1-penalized optimization we show that the number of

simulation steps can be significantly reduced.

Quantum mechanical systems have spatial components as well. When the spatial domain is partitioned according to the location of potential wells, one often finds low-energy

wavefunctions tend to localize within the confines of each partition. For a given partition,

the energy levels of its corresponding localized wavefunctions often make up only a small

fraction of the complete range of energy levels. For situations where only a few eigenpairs

are sought we introduce a "projection-correction" method allowing us to efficiently compute

only the low-energy eigenpairs which localize within a given spatial partition. In contrast to

standard methods for eigenvalue computation which specify only a part of the spectrum, our

method also allows one to isolate regions of space where prior information on eigenfunction

locality is known.