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1. The Taylor coefficients of the Jacobi theta constant θ3

   https://doi.org/10.1007/s11139-018-0109-5  

   Romik, Dan (January 2019, The Ramanujan Journal)

   We study the Taylor expansion around the point x=1 of a classical modular form, the Jacobi theta constant θ3. This leads naturally to a new sequence (d(n)) = 1, 1, -1, 51, 849, -26199, ... of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of θ3. We prove several results about the numbers d(n) and conjecture that they satisfy the congruence d(n)≡(−1)^(n-1) (mod 5) and other similar congruence relations. 

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2. On the Degree of Polynomials Computing Square Roots Mod p

   https://doi.org/10.4230/LIPIcs.CCC.2024.25  

   Kedlaya, Kiran S; Kopparty, Swastik (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   For an odd prime p, we say f(X) ∈ F_p[X] computes square roots in F_p if, for all nonzero perfect squares a ∈ F_p, we have f(a)² = a. When p ≡ 3 mod 4, it is well known that f(X) = X^{(p+1)/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p ≡ 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p. We show that for all p ≡ 1 mod 4, the degree of a polynomial computing square roots has degree at least p/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p/3. In the other direction, Agou, Deliglése, and Nicolas [Agou et al., 2003] showed that for infinitely many p ≡ 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p/8. 

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3. X-Ray Spectroscopy in the Microcalorimeter Era 4: Optical Depth Effects on the Soft X-Rays Studied with Cloudy

   https://doi.org/10.3847/1538-4357/ac7eb9  

   Chakraborty, Priyanka; Ferland, Gary J.; Chatzikos, Marios; Fabian, Andrew C.; Bianchi, Stefano; Guzmán, Francisco; Su, Yuanyuan (August 2022, The Astrophysical Journal)

   Abstract In this paper, we discuss atomic processes modifying the soft X-ray spectra from optical depth effects like photoelectric absorption and electron scattering suppressing the soft X-ray lines. We also show the enhancement in soft X-ray line intensities in a photoionized environment via continuum pumping. We quantify the suppression/enhancement by introducing a “line modification factor ( f mod ).” If 0 ≤ f mod ≤ 1, the line is suppressed, which could be the case in both collisionally ionized and photoionized systems. If f mod ≥ 1, the line is enhanced, which occurs in photoionized systems. Hybrid astrophysical sources are also very common, where the environment is partly photoionized and partly collisionally ionized. Such a system is V1223 Sgr, an Intermediate Polar binary. We show the application of our theory by fitting the first-order Chandra Medium Energy Grating (MEG) spectrum of V1223 Sgr with a combination of Cloudy -simulated additive cooling-flow and photoionized models. In particular, we account for the excess flux for O vii , O viii , Ne ix , Ne x , and Mg xi lines in the spectrum found in a recent study, which could not be explained with an absorbed cooling-flow model. 

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4. Determination of ring-level dynamic modulus of elasticity in loblolly pine from measurements of ultrasonic velocity and specific gravity

   https://doi.org/10.1093/forestry/cpac063  

   Dahlen, Joseph; Auty, David; Eberhardt, Thomas L.; Schimleck, Laurence; Pokhrel, Nawa Raj; Achim, ed., Alexis (February 2023, Forestry)

   Abstract Wood stiffness (modulus of elasticity, MOE) is an important property for conifer wood, with the variability in MOE largely being a function of both the specific gravity (SG) (wood density) and the angle of the microfibrils within the S2 layer of longitudinal tracheids. Rapid analysis techniques can be used together to quantify MOE; while SG can be determined with relative ease, this is not the case for microfibril angle, requiring expensive X-ray diffraction equipment. An alternative to microfibril angle is to measure longitudinal acoustic velocity. The objective of this study was to measure and then model the within tree variation in dynamic MOE (MOEdyn) by developing the methodology to measure ultrasonic velocity (USV) in radial samples from pith to bark using ultrasonic frequencies (>20 kHz). A total of 419 pith-to-bark radial strips, collected from multiple height levels in 92 loblolly pine (Pinus taeda) trees, were processed to obtain matching SG (2mm longitudinal) and USV (8.2-mm longitudinal) samples. Ring-by-ring SG was measured using X-ray densitometry and time-of-flight USV was measured at a 10-mm radial resolution from pith to bark. A subset of samples was sent to SilviScan to determine microfibril angle using X-ray diffraction. The relationship between microfibril angle and USV was strong (R2 = 0.91, RMSE = 2.6°). Nonlinear mixed-effects models were then developed to predict radial variation in SG, USV and MOEdyn. Fixed effects for the models, which included cambial age and height of disk within tree, had pseudo R2 values of 0.67 for SG (RMSE = 0.051), 0.71 for USV (RMSE = 316 m/s) and 0.69 for MOEdyn (RMSE = 1.9 GPa). When combined with SG measurements from X-ray densitometry, USV measurements from pith to bark are a powerful tool for assessing variability in wood stiffness. 

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5. Essential dimension via prismatic cohomology

   https://doi.org/10.1215/00127094-2023-0071  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (October 2024, Duke Mathematical Journal)

   NA (Ed.)

   For X a smooth, proper complex variety we show that for p≫0, the restriction of the mod p cohomology Hi(X,Fp) to any Zariski open has dimension at least h0,iX . The proof uses the prismatic cohomology of Bhatt and Scholze. We use this result to obtain lower bounds on the p-essential dimension of covers of complex varieties. For example, we prove the p-incompressibility of the mod p homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large p. By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain X, there exist p-congruence covers that are p-incompressible. 

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