Brooks's theorem for measurable colorings

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $\mu$-measurable $d$-coloring with respect to any Borel probability measure $\mu$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.Comment: Minor correction

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Attribution of chemistry-climate model initiative (CCMI) ozone radiative flux bias from satellites

The top-of-atmosphere (TOA) outgoing longwave flux over the 9.6 µm ozone band is a fundamental quantity for understanding chemistry–climate coupling. However, observed TOA fluxes are hard to estimate as they exhibit considerable variability in space and time that depend on the distributions of clouds, ozone (O3), water vapor (H2O), air temperature (Ta), and surface temperature (Ts). Benchmarking present-day fluxes and quantifying the relative influence of their drivers is the first step for estimating climate feedbacks from ozone radiative forcing and predicting radiative forcing evolution. To that end, we constructed observational instantaneous radiative kernels (IRKs) under clear-sky conditions, representing the sensitivities of the TOA flux in the 9.6 µm ozone band to the vertical distribution of geophysical variables, including O3, H2O, Ta, and Ts based upon the Aura Tropospheric Emission Spectrometer (TES) measurements. Applying these kernels to present-day simulations from the Chemistry-Climate Model Initiative (CCMI) project as compared to a 2006 reanalysis assimilating satellite observations, we show that the models have large differences in TOA flux, attributable to different geophysical variables. In particular, model simulations continue to diverge from observations in the tropics, as reported in previous studies of the Atmospheric Chemistry Climate Model Intercomparison Project (ACCMIP) simulations. The principal culprits are tropical middle and upper tropospheric ozone followed by tropical lower tropospheric H2O. Five models out of the eight studied here have TOA flux biases exceeding 100 mW m−2 attributable to tropospheric ozone bias. Another set of five models have flux biases over 50 mW m−2 due to H2O. On the other hand, Ta radiative bias is negligible in all models (no more than 30 mW m−2). We found that the atmospheric component (AM3) of the Geophysical Fluid Dynamics Laboratory (GFDL) general circulation model and Canadian Middle Atmosphere Model (CMAM) have the lowest TOA flux biases globally but are a result of cancellation of opposite biases due to different processes. Overall, the multi-model ensemble mean bias is −133±98  mW m−2, indicating that they are too atmospherically opaque due to trapping too much radiation in the atmosphere by overestimated tropical tropospheric O3 and H2O. Having too much O3 and H2O in the troposphere would have different impacts on the sensitivity of TOA flux to O3 and these competing effects add more uncertainties on the ozone radiative forcing. We find that the inter-model TOA outgoing longwave radiation (OLR) difference is well anti-correlated with their ozone band flux bias. This suggests that there is significant radiative compensation in the calculation of model outgoing longwave radiation

Folner tilings for actions of amenable groups

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.Comment: Minor revisions. Final versio

Local and Remote Mean and Extreme Temperature Response to Regional Aerosol Emissions Reductions

The climatic implications of regional aerosol and precursor emissions reductions implemented to protect human health are poorly understood. We investigate the mean and extreme temperature response to regional changes in aerosol emissions using three coupled chemistryclimate models: NOAA GFDL CM3, NCAR CESM1, and NASA GISS-E2. Our approach contrasts a long present-day control simulation from each model (up to 400 years with perpetual year 2000 or 2005 emissions) with 14 individual aerosol emissions perturbation simulations (160240 years each). We perturb emissions of sulfur dioxide (SO2) and/or carbonaceous aerosol within six world regions and assess the statistical significance of mean and extreme temperature responses relative to internal variability determined by the control simulation and across the models. In all models, the global mean surface temperature response (perturbation minus control) to SO2 and/or carbonaceous aerosol is mostly positive (warming) and statistically significant and ranges from +0.17 K (Europe SO2) to -0.06 K (US BC). The warming response to SO2 reductions is strongest in the US and Europe perturbation simulations, both globally and regionally, with Arctic warming up to 1 K due to a removal of European anthropogenic SO2 emissions alone; however, even emissions from regions remote to the Arctic, such as SO2 from India, significantly warm the Arctic by up to 0.5 K. Arctic warming is the most robust response across each model and several aerosol emissions perturbations. The temperature response in the Northern Hemisphere midlatitudes is most sensitive to emissions perturbations within that region. In the tropics, however, the temperature response to emissions perturbations is roughly the same in magnitude as emissions perturbations either within or outside of the tropics. We find that climate sensitivity to regional aerosol perturbations ranges from 0.5 to 1.0 K (W m(exp -2))(exp -1) depending on the region and aerosol composition and is larger than the climate sensitivity to a doubling of CO2 in two of three models. We update previous estimates of regional temperature potential (RTP), a metric for estimating the regional temperature responses to a regional emissions perturbation that can facilitate assessment of climate impacts with integrated assessment models without requiring computationally demanding coupled climate model simulations. These calculations indicate a robust regional response to aerosol forcing within the Northern Hemisphere midlatitudes, regardless of where the aerosol forcing is located longitudinally. We show that regional aerosol perturbations can significantly increase extreme temperatures on the regional scale. Except in the Arctic in the summer, extreme temperature responses largely mirror mean temperature responses to regional aerosol perturbations through a shift of the temperature distributions and are mostly dominated by local rather than remote aerosol forcing

High-RI resist polymers for 193 nm immersion lithography

A critical aim within the field of 193 nm immersion lithography is the development of high refractive index immersion fluids and resists. Increases in the refractive index (RI) of the immersion fluid will result in increases in the numerical aperture and depth of focus. Increasing the RI of resist polymers will improve exposure latitude for the process. A challenge for increasing the RI of resist polymers is to do so without detrimentally affecting other properties of the polymer such as transparency, line edge roughness, adhesion and plasma etch resistance. It is well known in the literature that introducing sulfur, bromine or aromatic groups into a polymer structure will increase its RI. However, due to the relatively strong absorption of phenyl groups at 193 nm these groups have to be avoided. Furthermore, the use of bromine poses problems associated with contamination of the silicon wafer. Hence, in this study, a systematic approach has been used to increase the sulfur content of 193 nm type resist polymers, by synthesis of sulfur-containing monomers and by performing bulk modifications of the polymer. The effect of sulfur content on the RI at 193 nm was then investigated. A broad study of the relationship between molecular structure and RI dispersion from 250-180 nm has also been undertaken, and conclusions drawn using QSPR methodologies. Finally, the effect of sulfur content on other lithography parameters, such as transparency, adhesion and plasma etch resistance, was also evaluated

Stratospheric ozone chemistry feedbacks are not critical for the determination of climate sensitivity in CESM1(WACCM)

The Community Earth System Model‐Whole Atmosphere Community Climate Model (CESM1‐WACCM) is used to assess the importance of including chemistry feedbacks in determining the equilibrium climate sensitivity (ECS). Two 4×CO2 model experiments were conducted: one with interactive chemistry and one with chemical constituents other than CO2 held fixed at their preindustrial values. The ECS determined from these two experiments agrees to within 0.01 K. Similarly, the net feedback parameter agrees to within 0.01 W m−2 K−1. This agreement occurs in spite of large changes in stratospheric ozone found in the simulation with interactive chemistry: a 30% decrease in the tropical lower stratosphere and a 40% increase in the upper stratosphere, broadly consistent with other published estimates. Off‐line radiative transfer calculations show that ozone changes alone account for the difference in radiative forcing. We conclude that at least for determining global climate sensitivity metrics, the exclusion of chemistry feedbacks is not a critical source of error in CESM

Novel high-index resists for 193nm immersion lithography and beyond

A preliminary Quantitative Structure Property Relationship (QSPR) model for predicting the refractive index of small molecules and polymers at 193 nm is presented. Although at this stage the model is only semiquantitative we have found it useful for screening databases of commercially-available compounds for high refractive index targets to include in our program of synthesis of high refractive index resist polymers. These resists are targeted for use in 2nd and 3rd generation 193 nm immersion lithography. Using this methodology a range of targets were identified and synthesized via free radical polymerization. Novel resist polymers were also synthesized via Michael addition polymerization. Preliminary dose to clear experiments identified a number of promising candidates for incorporation into high refractive index resist materials. Furthermore, we have demonstrated imaging of a high index resist using water-based 193 nm immersion lithography