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1. Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

   https://doi.org/10.3847/1538-4357/acfda9  

   Webb, G. M. ; Xu, Y. ; Biermann, P. L. ; Al-Nussirat, S. ; Mostafavi, P. ; Li, G. ; Barghouty, A. F. ; Zank, G. P. ( November 2023 , The Astrophysical Journal)

   A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution functionf₀(r,p) at cylindrical radiusrfrom the jet axis (assumed to lie along thez-axis) is given by convolving the particle momentum spectrum$f_0(\infty ,p\prime )$with the Green’s function$G(r,p;p\prime )$, which describes the monoenergetic spectrum solution in which$f_0\to \delta (p-p\prime )$asr→ ∞ . Previous work by Webb et al. studied only the Green’s function solution for$G(r,p;p\prime )$. In this paper, we explore for the first time, solutions for more general and realistic forms for$f_0(\infty ,p\prime )$. The flow velocityu=u(r)e_zis along the axis of the jet (thez-axis).uis independent ofz, andu(r) is a monotonic decreasing function ofr. The scattering time$\tau {(r,p)={\tau }_0(p/p_0)}^{\alpha }$in the shear flow region 0 <r<r₂, and$\tau {(r,p)={\tau }_0(p/p_0)}^{\alpha }{(r/r_2)}^s$, wheres> 0 in the regionr>r₂is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r,p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution${\psi }_p(r,p;p\prime )$that particles observed at (r,p) originated fromr→ ∞ with momentum$p\prime$. The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.

    

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2. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin ; Owen, Art ( March 2023 , Mathematics of Computation)

   Let$f$be analytic on$[0,1]$with$|f^{(k)}(1/2)|\leqslant A\alpha ^kk!$for some constants$A$and$\alpha >2$and all$k\geqslant 1$. We show that the median estimate of$\mu =\int _0^1f(x)\,\mathrm {d} x$under random linear scrambling with$n=2^m$points converges at the rate$O(n^{-c\log (n)})$for any$c> 3\log (2)/\pi ^2\approx 0.21$. We also get a super-polynomial convergence rate for the sample median of$2k-1$random linearly scrambled estimates, when$k/m$is bounded away from zero. When$f$has a$p$’th derivative that satisfies a$\lambda$-Hölder condition then the median of means has error$O( n^{-(p+\lambda )+\epsilon })$for any$\epsilon >0$, if$k\to \infty$as$m\to \infty$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.

    

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3. Bounds on eigenfunctions of quantum cat maps

   https://doi.org/10.1088/1402-4896/ad0331  

   Kim, Elena ; Koirala, Robert ( October 2023 , Physica Scripta)

   We studyℓ^∞norms ofℓ²-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000,Communications in Mathematical Physics,211, 659–686)) we show that there exists a sequence of eigenfunctionsuwith$\parallel u{\parallel }_{\infty }\gtrsim {\left(\log N\right)}^{-1/2}$. For general eigenfunctions we show the upper bound$\parallel u{\parallel }_{\infty }\lesssim {\left(\log N\right)}^{-1/2}$. Here the semiclassical parameter is$h={\left(2\pi N\right)}^{-1}$. Our upper bound is analogous to the one proved by Bérard in P Bérard (1977,Mathematische Zeitschrift,155, 249-276) for compact Riemannian manifolds without conjugate points.

    

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4. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak ; Augat, Meric ; Jury, Michael ; Sargent, Meredith ( November 2023 , Proceedings of the American Mathematical Society)

   Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial$f$in$d$freely noncommuting arguments, find a free polynomial$p_n$, of degree at most$n$, to minimize$c_n ≔\|p_nf-1\|^2$. (Here the norm is the$\ell ^2$norm on coefficients.) We show that$c_n\to 0$if and only if$f$is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the$d$-shift.

    

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5. What Does the Geometry of the Hβ BLR Depend On?

   https://doi.org/10.3847/1538-4357/accc84  

   Villafaña, Lizvette ; Williams, Peter R. ; Treu, Tommaso ; Brewer, Brendon J. ; Barth, Aaron J. ; U, Vivian ; Bennert, Vardha N. ; Guo, Hengxiao ; Bentz, Misty C. ; Canalizo, Gabriela ; et al ( May 2023 , The Astrophysical Journal)

   We combine our dynamical modeling black-hole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured cross-correlation time lags and line widths to recover individual scale factors,f, used in traditional reverberation-mapping analyses. We extend our sample by including prior results from Code for AGN Reverberation and Modeling of Emission Lines (caramel) studies that have utilized our methods. Aiming to improve the precision of black-hole mass estimates, as well as uncover any regularities in the behavior of the broad-line region (BLR), we search for correlations betweenfand other AGN/BLR parameters. We find (i) evidence for a correlation between the virial coefficient${\log }_{10}(f_{\mathrm{mean},\sigma })$and black-hole mass, (ii) marginal evidence for a similar correlation between${\log }_{10}(f_{\mathrm{rms},\sigma })$and black-hole mass, (iii) marginal evidence for an anticorrelation of BLR disk thickness with${\log }_{10}(f_{\mathrm{mean},\mathrm{FWHM}})$and${\log }_{10}(f_{\mathrm{rms},\mathrm{FWHM}})$, and (iv) marginal evidence for an anticorrelation of inclination angle with${\log }_{10}(f_{\mathrm{mean},\mathrm{FWHM}})$,${\log }_{10}(f_{\mathrm{rms},\sigma })$, and${\log }_{10}(f_{\mathrm{mean},\sigma })$. Last, we find marginal evidence for a correlation between line-profile shape, when using the root-mean-square spectrum,${\log }_{10}{(\mathrm{FWHM}/\sigma )}_{\mathrm{rms}}$, and the virial coefficient,${\log }_{10}(f_{\mathrm{rms},\sigma })$, and investigate how BLR properties might be related to line-profile shape usingcaramelmodels.

    

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