ALMA Thermal Observations of Europa

We present four daytime thermal images of Europa taken with the Atacama Large Millimeter Array. Together, these images comprise the first spatially resolved thermal dataset with complete coverage of Europa's surface. The resulting brightness temperatures correspond to a frequency of 233 GHz (1.3 mm) and a typical linear resolution of roughly 200 km. At this resolution, the images capture spatially localized thermal variations on the scale of geologic and compositional units. We use a global thermal model of Europa to simulate the ALMA observations in order to investigate the thermal structure visible in the data. Comparisons between the data and model images suggest that the large-scale daytime thermal structure on Europa largely results from bolometric albedo variations across the surface. Using bolometric albedos extrapolated from Voyager measurements, a homogenous model reproduces these patterns well, but localized discrepancies exist. These discrepancies can be largely explained by spatial inhomogeneity of the surface thermal properties. Thus, we use the four ALMA images to create maps of the surface thermal inertia and emissivity at our ALMA wavelength. From these maps, we identify a region of either particularly high thermal inertia or low emissivity near 90 degrees West and 23 degrees North, which appears anomalously cold in two of our images.Comment: 9 pages, 3 figures, accepted for publication in the Astronomical Journa

Forced, Non-Linear Vibration of Integral Shroud Turbine Blades

This thesis examines the forced, non-linear vibration of integral shroud turbine blades. The shroud of this type of blade is integral with the foil and root. During turbine operation small gaps are generally present between adjacent shrouds. If the amplitude of blade vibration is sufficient, adjacent shrouds will contact. This contact creates a dynamic non-linearity. A complete row of blades is considered in the analysis. The Ritz averaging method is used to develop an approximate displacement solution. Frequency response curves for a representative turbine blade are presented in the usual format

Positivity and nonstandard graded Betti numbers

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo-Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that, given a simplicial projective toric variety and a module M over its coordinate ring, the multigraded Betti numbers of M are contained in a particular polytope when M satisfies an appropriate positivity condition.Comment: 12 page

Linear syzygies of curves in weighted projective space

We develop analogues of Green's $N_p$-conditions for subvarieties of weighted projective space, and we prove that such $N_p$-conditions are satisfied for high degree embeddings of curves in weighted projective space. A key technical result links positivity with low degree (virtual) syzygies in wide generality, including cases where normal generation fails.Comment: 29 page

Minimal free resolutions of differential modules

We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of differential modules. Our main result in this direction explains a sense in which the minimal free resolution of a differential module is a deformation of the minimal free resolution of its homology; this leads to structural results that mirror classical theorems about minimal free resolutions of modules.Comment: 20 page

Tate resolutions on toric varieties

We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier- Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson's resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties.Comment: 31 pages. To appear in the Journal of the European Mathematical Society (JEMS