This text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The Atiyah-Singer index theorem and its applications are developed (without complete proofs) via the heat equation method. Zeta functions for Laplacians and analytic torsion are also treated, and the recently uncovered relation between index theory and analytic torsion is laid out. The text is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning. There are over 100 exercises with hints.

set of signals (usually time series) at each of a collection of pixels (in two dimensions) or voxels (in three dimensions). Building from such data, various forms of higher-level data representations are employed in neuroimaging. Traditionally, two-and three-dimensional images have, naturally, been the norm, but increasingly in recent years there has emerged a substantial interest in network-based representations.1.1. Motivation. Let G = (V, E) denote a graph, based on d = |V | vertices. In this setting, the vertices v ∈ V correspond to regions of interest (ROIs) in the brain, often pre-defined through considerations of the underlying neurobiology (e.g., the putamen or the cuneus). Edges {u, v} ∈ E between vertices u and v are used to denote a measure of association between the corresponding ROIs. Depending on the imaging modality used, the notion of 'association' may vary. For example, in diffusion tensor imaging (DTI), associations are taken to be representative of structural connectivity between brain regions. On the other hand, in functional magnetic resonance imaging (fMRI), associations are instead thought to represent functional connectivity, in the sense that the two regions of the brain participate together in the achievement of some higher-order function, often in the context of performing some task (e.g., counting from 1 to 10).With neuroimaging now a standard tool in clinical neuroscience, and with the advent of several major neuroscience research initiatives -perhaps most prominent being the recently announced Brain Research Accelerated by Innovative Neurotechnologies (BRAIN) initiative -we are quickly moving towards a time in which we will have available databases composed of large collections of secondary data in the form of network-based data objects. Faced with databases in which networks are a fundamental unit of data, it will be necessary to have in place the statistical tools to answer such questions as, "What is the 'average' of a collection of networks?" and "Do these networks differ, on average, from a given nominal network?," as well as "Do two collections of networks differ on average?" and "What factors (e.g., age, gender, etc.) appear to contribute to differences in networks?", or finally, say, "Has there been a change in the networks for a given subpopulation from yesterday to today?" In order to answer these and similar questions, we require network-based analogues of classical tools for statistical estimation and hypothesis testing.While these classical tools are among the most fundamental and ubiquitous in use in practice, their extension to network-based datasets, however, is not immediate and, in fact, can be expected to be highly non-trivial. The main challenge in such an extension is due to the simple fact that networks 2 are not Euclidean objects (for which classical methods were developed)rather, they are combinatorial objects, defined simply through their sets of vertices and edges. Nevertheless, our work here in this paper demonstrates that networks can be associated with ce...

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Abstract. Rocket observations of spatially localized bursts of intense lower hybrid wave radiation have been made for decades. As the ability of rocket observations to measure small-scale structure increases, scientists are better able to understand the physics responsible for the observations. However, rocket measurements are limited in their ability to fully diagnose the phenomenon they seek to measure. Therefore a laboratory experiment was undertaken at the Large Plasma Device (LAPD) at the University of California at Los Angeles to study the interaction of lower hybrid waves with a fieldaligned density depletion. The laboratory parameters were chosen to mimic the physics in the ionosphere. Laboratory experiments allow one not only to study the effect of different interaction parameters, like wave frequency or striation size, but also to fully diagnose the wave fields in three spatial dimensions and time. These experiments have shown that large-amplitude wave fields are localized to the areas of largest density gradient. The field pattern is independent of striation size and shape. The experimental results relate strongly to space-based observations. The laboratory results, therefore, can act as a guidepost to assist the understanding of the more complex and difficult-to-understand plasmas encountered in the ionosphere. IntroductionThe topic of wave interactions with plasma nonuniformities is a rich field which is now under investigation in the laboratory. The natural evolution of laboratory experiments to study these interactions should hold interest for those in the space physics community. Space, ionospheric, and magnetospheric plasmas are nonuniform in some way. A well-designed laboratory experiment has the potential to provide measurements in detail far greater than can currently be obtained by in situ measurements. Such detail can provide new insight into the physical mechanisms involved and can help direct the development of theories to explain the space-based observations. The observation of lower hybrid wave interactions with a field-aligned density depletion by sounding rockets in the ionosphere provided the inspiration for a series of laboratory experiments at the Large Plasma Device (LAPD) at the University of California at Los Angeles (UCLA). These experiments were designed so that their results would not only provide information about the wave/plasma nonuniformity interaction in general but also provide information about the ionospheric observations. Lower hybrid wave radiation has been observed in the ionosphere for decades [e. were not able to make any statement about the relationship of these bursts, which they called "spikelets," to changes in the plasma density. In fact, while they believed that these spikelets could be stationary structures, they could not discount the possibility that the structures moved at the ion drift velocity. They were, however, able to demonstrate a correlation between the spikelets and transverse ion acceleration. Ions accelerated at the rocket altitudes (450-650 ...

As water availability becomes a growing challenge in various regions throughout the world, desalination and wastewater reclamation through technologies such as reverse osmosis (RO) are becoming more important. Nevertheless, many open questions remain regarding the internal structure of thin-film composite RO membranes. In this work, fully aromatic polyamide films that serve as the active layer of state-of-the-art water filtration membranes were investigated using high-angle annular dark-field scanning transmission electron microscopy tomography. Reconstructions of the 3D morphology reveal intricate aspects of the complex microstructure not visible from 2D projections. We find that internal voids of the active layer of compressed commercial membranes account for less than 0.2% of the total polymer volume, contrary to previously reported values that are two orders of magnitude higher. Measurements of the local variation in polyamide density from electron tomography reveal that the polymer density is highest at the permeable surface for the two membranes tested and establish the significance of surface area on RO membrane transport properties. The same type of analyses could provide explanations for different flux variations with surface area for other types of membranes where the density is distributed differently. Thus, 3D reconstructions and quantitative analyses will be crucial to characterize the complex morphology of polymeric membranes used in next-generation water-purification membranes.