A steadystate, semianalytical model of energetic particle acceleration in radiojet shear flows due to cosmicray viscosity obtained by Webb et al. is generalized to take into account more general cosmicray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function
We present probabilistic interpretations of solutions to semilinear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975
 Award ID(s):
 2204615
 NSFPAR ID:
 10469058
 Publisher / Repository:
 IOP Publishing
 Date Published:
 Journal Name:
 Nonlinearity
 Volume:
 36
 Issue:
 11
 ISSN:
 09517715
 Format(s):
 Medium: X Size: p. 61046123
 Size(s):
 p. 61046123
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract f _{0}(r ,p ) at cylindrical radiusr from the jet axis (assumed to lie along thez axis) is given by convolving the particle momentum spectrum with the Green’s function ${f}_{0}(\infty ,p\prime )$ , which describes the monoenergetic spectrum solution in which $G(r,p;p\prime )$ as ${f}_{0}\to \delta (pp\prime )$r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for $G(r,p;p\prime )$ . The flow velocity ${f}_{0}(\infty ,p\prime )$ =u u (r ) _{z}is along the axis of the jet (thee z axis). is independent ofu z , andu (r ) is a monotonic decreasing function ofr . The scattering time in the shear flow region 0 < $\tau {(r,p)={\tau}_{0}(p/{p}_{0})}^{\alpha}$r <r _{2}, and , where $\tau {(r,p)={\tau}_{0}(p/{p}_{0})}^{\alpha}{(r/{r}_{2})}^{s}$s > 0 in the regionr >r _{2}is 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 that particles observed at ( ${\psi}_{p}(r,p;p\prime )$r ,p ) originated fromr → ∞ with momentum . The acceleration of ultrahighenergy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described. $p\prime $ 
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 superpolynomial convergence rate for the sample median of$2k1$ 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 quasiMonte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 
Abstract We study
ℓ ^{∞}norms ofℓ ^{2}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 eigenfunctionsu with . For general eigenfunctions we show the upper bound $\parallel u{\parallel}_{\infty}\gtrsim {\left(\mathrm{log}N\right)}^{1/2}$ . Here the semiclassical parameter is $\parallel u{\parallel}_{\infty}\lesssim {\left(\mathrm{log}N\right)}^{1/2}$ . Our upper bound is analogous to the one proved by Bérard in P Bérard (1977, $h={\left(2\pi N\right)}^{1}$Mathematische Zeitschrift ,155 , 249276) for compact Riemannian manifolds without conjugate points. 
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$c_n ≔\p_nf1\^2$ . (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. 
Abstract We combine our dynamical modeling blackhole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured crosscorrelation time lags and line widths to recover individual scale factors,
f , used in traditional reverberationmapping 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 blackhole mass estimates, as well as uncover any regularities in the behavior of the broadline region (BLR), we search for correlations betweenf and other AGN/BLR parameters. We find (i) evidence for a correlation between the virial coefficient and blackhole mass, (ii) marginal evidence for a similar correlation between ${\mathrm{log}}_{10}({f}_{\mathrm{mean},\sigma})$ and blackhole mass, (iii) marginal evidence for an anticorrelation of BLR disk thickness with ${\mathrm{log}}_{10}({f}_{\mathrm{rms},\sigma})$ and ${\mathrm{log}}_{10}({f}_{\mathrm{mean},\mathrm{FWHM}})$ , and (iv) marginal evidence for an anticorrelation of inclination angle with ${\mathrm{log}}_{10}({f}_{\mathrm{rms},\mathrm{FWHM}})$ , ${\mathrm{log}}_{10}({f}_{\mathrm{mean},\mathrm{FWHM}})$ , and ${\mathrm{log}}_{10}({f}_{\mathrm{rms},\sigma})$ . Last, we find marginal evidence for a correlation between lineprofile shape, when using the rootmeansquare spectrum, ${\mathrm{log}}_{10}({f}_{\mathrm{mean},\sigma})$ , and the virial coefficient, ${\mathrm{log}}_{10}{(\mathrm{FWHM}/\sigma )}_{\mathrm{rms}}$ , and investigate how BLR properties might be related to lineprofile shape using ${\mathrm{log}}_{10}({f}_{\mathrm{rms},\sigma})$caramel models.