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1. Bicycling geodesics are Kirchhoff rods

   https://doi.org/10.1088/1361-6544/acd613  

   Bor, Gil; Jackman, Connor; Tabachnikov, Serge (May 2023, Nonlinearity)

   Abstract A bicycle path is a pair of trajectories in ${\mathbb{R}}^n$, the ‘front’ and ‘back’ tracks, traced out by the endpoints of a moving line segment of fixed length (the ‘bicycle frame’) and tangent to the back track. Bicycle geodesics are bicycle paths whose front track’s length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n⩾3$each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily ofKirchhoff rods, a class of curves introduced in 1859 by Kirchhoff, generalizing the planar elastic curves of Bernoulli and Euler. 

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2. Fractional polarization of extragalactic sources in the 500 deg2 SPTpol survey

   https://doi.org/10.1093/mnras/stz2905  

   Gupta, N; Reichardt, C L; Ade, P A; Anderson, A J; Archipley, M; Austermann, J E; Avva, J S; Beall, J A; Bender, A N; Benson, B A; et al (December 2019, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT We study the polarization properties of extragalactic sources at 95 and 150 GHz in the SPTpol 500 deg2 survey. We estimate the polarized power by stacking maps at known source positions, and correct for noise bias by subtracting the mean polarized power at random positions in the maps. We show that the method is unbiased using a set of simulated maps with similar noise properties to the real SPTpol maps. We find a flux-weighted mean-squared polarization fraction 〈p2〉 = [8.9 ± 1.1] × 10−4 at 95 GHz and [6.9 ± 1.1] × 10−4 at 150 GHz for the full sample. This is consistent with the values obtained for a subsample of active galactic nuclei. For dusty sources, we find 95 per cent upper limits of 〈p2〉95 < 16.9 × 10−3 and 〈p2〉150 < 2.6 × 10−3. We find no evidence that the polarization fraction depends on the source flux or observing frequency. The 1σ upper limit on measured mean-squared polarization fraction at 150 GHz implies that extragalactic foregrounds will be subdominant to the CMB E and B mode polarization power spectra out to at least ℓ ≲ 5700 (ℓ ≲ 4700) and ℓ ≲ 5300 (ℓ ≲ 3600), respectively, at 95 (150) GHz. 

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3. Assessing Scale Dependence on Local Sea Level Retrievals from Laser Altimetry Data over Sea Ice

   https://doi.org/10.3390/rs12223732  

   Tian, Liuxi; Xie, Hongjie; Ackley, Stephen F.; Mestas-Nuñez, Alberto M. (November 2020, Remote Sensing)

   null (Ed.)

   The measurement of sea ice elevation above sea level or the “freeboard” depends upon an accurate retrieval of the local sea level. The local sea level has been previously retrieved from altimetry data alone by the lowest elevation method, where the percentage of the lowest elevations over a particular segment length scale was used. Here, we provide an evaluation of the scale dependence on these local sea level retrievals using data from NASA Operation IceBridge (OIB) which took place in the Ross Sea in 2013. This is a unique dataset of laser altimeter measurements over five tracks from the Airborne Topographic Mapper (ATM), with coincidently high-spatial resolution images from the Digital Mapping System (DMS), that allows for an independent sea level validation. The local sea level is first calculated by using the mean elevation of ATM L1B data over leads identified by using the corresponding DMS imagery. The resulting local sea level reference is then used as ground truth to validate the local sea levels retrieved from ATM L2 by using nine different percentages of the lowest elevation (0.1%, 0.5%, 1%, 1.5%, 2%, 2.5%, 3%, 3.5%, and 4%) at seven different segment length scales (1, 5, 10, 15, 20, 25, and 50 km) for each of the five ATM tracks. The closeness to the 1:1 line, R2, and root mean square error (RMSE) is used to quantify the accuracy of the retrievals. It is found that all linear least square fits are statistically significant (p < 0.05) using an F test at every scale for all tested data. In general, the sea level retrievals are farther away from the 1:1 line when the segment length scale increases from 1 or 5 to 50 km. We find that the retrieval accuracy is affected more by the segment length scale than the percentage scale. Based on our results, most retrievals underestimate the local sea level; the longer the segment length (from 1 to 50 km) used, especially at small percentage scales, the larger the error tends to be. The best local sea level based on a higher R2 and smaller RMSE for all the tracks combined is retrieved by using 0.1–2% of the lowest elevations at the 1–5 km segment lengths. 

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4. Algebraic area enumeration for open lattice walks

   https://doi.org/10.1088/1751-8121/aca573  

   Ouvry, Stéphane; Polychronakos, Alexios P (December 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We calculate the number of open walks of fixed length and algebraic area on a square planar lattice by an extension of the operator method used for the enumeration of closed walks. The open walk area is defined by closing the walks with a straight line across their endpoints and can assume half-integer values in lattice cell units. We also derive the length and area counting of walks with endpoints on specific straight lines and outline an approach for dealing with walks with fully fixed endpoints. 

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5. The Traverse Ridge Paleoseismic Site and Ruptures Crossing the Boundary Between the Provo and Salt Lake City Segments of the Wasatch Fault Zone, Utah, United States

   https://doi.org/10.3389/feart.2021.607018  

   Toké, Nathan A.; Phillips, Joseph; Langevin, Christopher; Kleber, Emily; DuRoss, Christopher B.; Hiscock, Adam I.; McDonald, Greg N.; Wells, Jack D.; Carlson, J. Kade; Horns, Daniel M. (March 2021, Frontiers in Earth Science)

   How structural segment boundaries modulate earthquake behavior is an important scientific and societal question, especially for the Wasatch fault zone (WFZ) where urban areas lie along multiple fault segments. The extent to which segment boundaries arrest ruptures, host moderate magnitude earthquakes, or transmit ruptures to adjacent fault segments is critical for understanding seismic hazard. To help address this outstanding issue, we conducted a paleoseismic investigation at the Traverse Ridge paleoseismic site (TR site) along the ∼7-km-long Fort Canyon segment boundary, which links the Provo (59 km) and Salt Lake City (40 km) segments of the WFZ. At the TR site, we logged two trenches which were cut across sub-parallel traces of the fault, separated by ∼175 m. Evidence from these exposures leads us to infer that at least 3 to 4 earthquakes have ruptured across the segment boundary in the Holocene. Radiocarbon dating of soil material developed below and above fault scarp colluvial packages and within a filled fissure constrains the age of the events. The most recent event ruptured the southern fault trace between 0.2 and 0.4 ka, the penultimate event ruptured the northern fault trace between 0.6 and 3.4 ka, and two prior events occurred between 1.4 and 6.2 ka (on the southern fault trace) and 7.2 and 8.1 ka (northern fault trace). Colluvial wedge heights of these events ranged from 0.7 to 1.2 m, indicating the segment boundary experiences surface ruptures with more than 1 m of vertical displacement. Given these estimates, we infer that these events were greater than Mw 6.7, with rupture extending across the entire segment boundary and portions of one or both adjacent fault segments. The Holocene recurrence of events at the TR site is lower than the closest paleoseismic sites at the adjacent fault segment endpoints. The contrasts in recurrence rates observed within 15 km of the Fort Canyon fault segment boundary may be explained conceptually by a leaky segment boundary model which permits spillover events, ruptures centered on the segment boundary, and segmented ruptures. The TR site demonstrates the utility of paleoseismology within segment boundaries which, through corroboration of displacement data, can demonstrate rupture connectivity between fault segments and test the validity of rupture models. 

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