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1. The CHARA Array Interferometric Program on the Multiplicity of Classical Be Stars: New Detections and Orbits of Stripped Subdwarf Companions

   https://doi.org/10.3847/1538-4357/ad13ec  

   Klement, Robert ; Rivinius, Thomas ; Gies, Douglas R. ; Baade, Dietrich ; Mérand, Antoine ; Monnier, John D. ; Schaefer, Gail H. ; Lanthermann, Cyprien ; Anugu, Narsireddy ; Kraus, Stefan ; et al ( February 2024 , The Astrophysical Journal)

   Rapid rotation and nonradial pulsations enable Be stars to build decretion disks, where the characteristic line emission forms. A major but unconstrained fraction of Be stars owe their rapid rotation to mass and angular momentum transfer in a binary. The faint, stripped companions can be helium-burning subdwarf OB-type stars (sdOBs), white dwarfs (WDs), or neutron stars. We present optical/near-infrared Center for High Angular Resolution Astronomy (CHARA) interferometry of 37 Be stars selected for spectroscopic indications of low-mass companions. From multiepochH- and/orK-band interferometry plus radial velocities and parallaxes collected elsewhere, we constructed 3D orbits and derived flux ratios and absolute dynamical masses of both components for six objects, quadrupling the number of anchor points for evolutionary models. In addition, a new wider companion was identified for the known Be + sdO binary 59 Cyg, while auxiliary Very Large Telescope Interferometer/GRAVITY spectrointerferometry confirmed circumstellar matter around the sdO companion to HR 2142. On the other hand, we failed to detect any companion to the six Be stars withγCas–like X-ray emission, with sdOB and main-sequence companions of the expected spectroscopic mass being ruled out for the X-ray-prototypical starsγCas andπAqr, leaving elusive WDs as the most likely companions, as well as a likely explanation of the X-rays. No low-mass main-sequence close companions were identified for the other stars.

    

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2. Spectroscopic follow-up of black hole and neutron star candidates in ellipsoidal variables from Gaia DR3

   https://doi.org/10.1093/mnras/stad2130  

   Nagarajan, Pranav ; El-Badry, Kareem ; Rodriguez, Antonio C. ; van Roestel, Jan ; Roulston, Benjamin ( July 2023 , Monthly Notices of the Royal Astronomical Society)

   We present multi-epoch spectroscopic follow-up of a sample of ellipsoidal variables selected from Gaia Data Release 3 (DR3) as candidates for hosting quiescent black holes (BHs) and neutron stars (NSs). Our targets were identified as BH/NS candidates because their optical light curves – when interpreted with models that attribute variability to tidal distortion of a star by a companion that contributes negligible light – suggest that the companions are compact objects. From the likely BH/NS candidates identified in recent work accompanying Gaia DR3, we select 14 of the most promising targets for follow-up. We obtained spectra for each object at 2–10 epochs, strategically observing near conjunction to best constrain the radial velocity semi-amplitude. From the measured semi-amplitudes of the radial velocity curves, we derive minimum companion masses of $M_{2,\, \rm min} \le 0.5 \, {\rm M}_{\odot }$ in all cases. Assuming random inclinations, the typical inferred companion mass is $M_2 \sim 0.15\, {\rm M}_{\odot }$. This makes it unlikely that any of these systems contain a BH or NS, and we consider alternative explanations for the observed variability. We can best reproduce the observed light curves and radial velocities with models for unequal-mass contact binaries with star-spots. Some of the objects in our sample may also be detached main-sequence binaries, or even single stars with pulsations or star-spot variability masquerading as ellipsoidal variation. We provide recommendations for future spectroscopic efforts to further characterize this sample and more generally to search for compact object companions in close binaries.

    

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3. Cohomology of line bundles on the incidence correspondence

   https://doi.org/10.1090/btran/173  

   Gao, Zhao ; Raicu, Claudiu ( January 2024 , Transactions of the American Mathematical Society, Series B)

   For a finite dimensional vector space$V$of dimension$n$, we consider the incidence correspondence (or partial flag variety)$X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }$, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on$X$in characteristic$p>0$. If$n=3$then$X$is the full flag variety of$V$, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic$0$, the cohomology groups are described for all$V$by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When$n=3$, we recover the recursive description of characters from recent work of Linyuan Liu, while for general$n$we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

    

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4. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn ; Wake, Preston ( October 2022 , Proceedings of the American Mathematical Society, Series B)

   We show that for primes$N, p \geq 5$with$N \equiv -1 \bmod p$, the class number of$\mathbb {Q}(N^{1/p})$is divisible by$p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv -1 \bmod p$, there is always a cusp form of weight$2$and level$\Gamma _0(N^2)$whose$\ell$th Fourier coefficient is congruent to$\ell + 1$modulo a prime above$p$, for all primes$\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$extension of$\mathbb {Q}(N^{1/p})$.

    

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5. Tight Bounds for ℓ 1 Oblivious Subspace Embeddings

   https://doi.org/10.1145/3477537  

   Wang, Ruosong ; Woodruff, David P. ( January 2022 , ACM Transactions on Algorithms)

   An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications. 

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