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We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Letn ≥ 2. Suppose a subset Ω ofn‐dimensional Euclidean spacesatisfies −Ω = Ωcand Ω + v = Ωc(up to measure zero sets) for every standard basis vector. For anyand for anyq ≥ 1, letand let. For anyx ∈ ∂Ω, letN(x) denote the exterior normal vector atxsuch that ‖N(x)‖2 = 1. Let. Our main result shows thatBhas the smallest Gaussian surface area among all such subsets Ω, less a small error:In particular,Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10−9would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.
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Natural and Experimental Constraints on a Flow Law for Dislocation‐Dominated Creep in Wet Quartz
Abstract We present a flow law for dislocation‐dominated creep in wet quartz derived from compiled experimental and field‐based rheological data. By integrating the field‐based data, including independently calculated strain rates, deformation temperatures, pressures, and differential stresses, we add constraints for dislocation‐dominated creep at conditions unattainable in quartz deformation experiments. A Markov Chain Monte Carlo (MCMC) statistical analysis computes internally consistent parameters for the generalized flow law: = Aσne−(Q+VP)/RT. From this initial analysis, we identify differenteffectivestress exponents for quartz deformed at confining pressures above and below ∼700 MPa. To minimize the possible effect of confining pressure, compiled data are separated into “low‐pressure” (<560 MPa) and “high‐pressure” (700–1,600 MPa) groups and reanalyzed using the MCMC approach. The “low‐pressure” data set, which is most applicable at midcrustal to lower‐crustal confining pressures, yields the following parameters: log(A) = −9.30 ± 0.66 MPa−n−r s−1;n = 3.5 ± 0.2;r = 0.49 ± 0.13;Q = 118 ± 5 kJ mol−1; andV = 2.59 ± 2.45 cm3 mol−1. The “high‐pressure” data set produces a different set of parameters: log(A) = −7.90 ± 0.34 MPa−n−r s−1;n = 2.0 ± 0.1;r = 0.49 ± 0.13;Q = 77 ± 8 kJ mol−1; andV = 2.59 ± 2.45 cm3 mol−1. Predicted quartz rheology is compared to other flow laws for dislocation creep; the calibrations presented in this study predict faster strain rates under geological conditions by more than 1 order of magnitude. The change innat high confining pressure may result from an increase in the activity of grain size sensitive creep.
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- Award ID(s):
- 1650173
- PAR ID:
- 10361488
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Geophysical Research: Solid Earth
- Volume:
- 126
- Issue:
- 5
- ISSN:
- 2169-9313
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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