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1. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    

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2. CLASS Data Pipeline and Maps for 40 GHz Observations through 2022

   https://doi.org/10.3847/1538-4357/acf293  

   Li 李, Yunyang 云炀 ; Eimer, Joseph R. ; Osumi, Keisuke ; Appel, John W. ; Brewer, Michael K. ; Ali, Aamir ; Bennett, Charles L. ; Bruno, Sarah Marie ; Bustos, Ricardo ; Chuss, David T. ; et al ( October 2023 , The Astrophysical Journal)

   The Cosmology Large Angular Scale Surveyor (CLASS) is a telescope array that observes the cosmic microwave background over 75% of the sky from the Atacama Desert, Chile, at frequency bands centered near 40, 90, 150, and 220 GHz. This paper describes the CLASS data pipeline and maps for 40 GHz observations conducted from 2016 August to 2022 May. We demonstrate how well the CLASS survey strategy, with rapid (∼10 Hz) front-end modulation, recovers the large-scale Galactic polarization signal from the ground: the mapping transfer function recovers ∼67% (85%) ofEEandBB(VV) power atℓ= 20 and ∼35% (47%) atℓ= 10. We present linear and circular polarization maps over 75% of the sky. Simulations based on the data imply the maps have a white noise level of$110\mu \mathrm{K}\mathrm{arcmin}$and correlated noise component rising at low-ℓasℓ^−2.4. The transfer-function-corrected low-ℓcomponent is comparable to the white noise at the angular knee frequencies ofℓ≈ 18 (linear polarization) andℓ≈ 12 (circular polarization). Finally, we present simulations of the level at which expected sources of systematic error bias the measurements, finding subpercent bias for the Λ cold dark matterEEpower spectra. Bias fromE-to-Bleakage due to the data reduction pipeline and polarization angle uncertainty approaches the expected level for anr= 0.01BBpower spectrum. Improvements to the instrument calibration and the data pipeline will decrease this bias.

    

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3. Magnetic Misalignment of Interstellar Dust Filaments

   https://doi.org/10.3847/1538-4357/acb0c4  

   Cukierman, Ari J. ; Clark, S. E. ; Halal, George ( April 2023 , The Astrophysical Journal)

   We present evidence for scale-independent misalignment of interstellar dust filaments and magnetic fields. We estimate the misalignment by comparing millimeter-wave dust-polarization measurements from Planck with filamentary structures identified in neutral-hydrogen (Hi) measurements from Hi4PI. We find that the misalignment angle displays a scale independence (harmonic coherence) for features larger than the Hi4PI beamwidth (16.′2). We additionally find a spatial coherence on angular scales of$\mathit{}(1°)$. We present several misalignment estimators formed from the auto- and cross-spectra of dust-polarization and Hi-based maps, and we also introduce a map-space estimator. Applied to large regions of the high-Galactic-latitude sky, we find a global misalignment angle of ∼2°, which is robust to a variety of masking choices. By dividing the sky into small regions, we show that the misalignment angle correlates with the parity-violatingTBcross-spectrum measured in the Planck dust maps. The misalignment paradigm also predicts a dustEBsignal, which is of relevance in the search for cosmic birefringence but as yet undetected; the measurements ofEBare noisier than those ofTB, and our correlations ofEBwith misalignment angle are found to be weaker and less robust to masking choices. We also introduce an Hi-based dust-polarization template constructed from the Hessian matrix of the Hiintensity, which is found to correlate more strongly than previous templates with Planck dustBmodes.

    

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4. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra ; Inamdar, Tanmay ; Quanrud, Kent ; Varadarajan, Kasturi ; Zhang, Zhao ( June 2022 , Journal of Combinatorial Optimization)

   We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$$w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$$f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$$k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$$S\subseteq N$such that$$f_i(S) \ge k_i$$$f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$$1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$$r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$$O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$$k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$$(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$$O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$$f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.

    

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5. Introducing isodynamic points for binary forms and their ratios

   https://doi.org/10.1007/s40627-022-00112-4  

   Hägg, Christian ; Shapiro, Boris ; Shapiro, Michael ( January 2023 , Complex Analysis and its Synergies)

   The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group$${\mathcal {M}}$$$M$on the set of triangles, Kimberling (Encyclopedia of Triangle Centers,http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below theisodynamicmap associating to a univariate polynomial of degree$$d\ge 3$$$d\ge 3$with at most double roots a polynomial of degree (at most)$$2d-4$$$2d-4$such that this map commutes with the action of the Möbius group$${\mathcal {M}}$$$M$on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called theisodynamic pointsof the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

    

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