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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)

   Abstract 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 $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }$in the shear flow region 0 <r<r₂, and $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }{\left(r/r_2\right)}^s$, 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. Deep Einstein@Home All-sky Search for Continuous Gravitational Waves in LIGO O3 Public Data

   https://doi.org/10.3847/1538-4357/acdad4  

   Steltner, B; Papa, M A; Eggenstein, H-B; Prix, R; Bensch, M; Allen, B; Machenschalk, B (July 2023, The Astrophysical Journal)

   Abstract We present the results of an all-sky search for continuous gravitational waves in the public LIGO O3 data. The search covers signal frequencies 20.0 Hz ≤f≤ 800.0 Hz and a spin-down range down to −2.6 × 10⁻⁹Hz s⁻¹, motivated by detectability studies on synthetic populations of Galactic neutron stars. This search is the most sensitive all-sky search to date in this frequency/spin-down region. The initial search was performed using the first half of the public LIGO O3 data (O3a), utilizing graphical processing units provided in equal parts by the volunteers of the Einstein@Home computing project and by the ATLAS cluster. After a hierarchical follow-up in seven stages, 12 candidates remain. Six are discarded at the eighth stage, by using the remaining O3 LIGO data (O3b). The surviving six can be ascribed to continuous-wave fake signals present in the LIGO data for validation purposes. We recover these fake signals with very high accuracy with our last stage search, which coherently combines all O3 data. Based on our results, we set upper limits on the gravitational-wave amplitudeh₀and translate these into upper limits on the neutron star ellipticity and on ther-mode amplitude. The most stringent upper limits are at 203 Hz, withh₀= 8.1 × 10⁻²⁶at the 90% confidence level. Our results exclude isolated neutron stars rotating faster than 5 ms with ellipticities greater than $5\times {10}^{-8}\left(\frac{d}{100\mathrm{pc}}\right)$within a distancedfrom Earth andr-mode amplitudes $\alpha \ge {10}^{-5}\left(\frac{d}{100\mathrm{pc}}\right)$for neutron stars spinning faster than 150 Hz. 

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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)

   Abstract 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. Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

   https://doi.org/10.1515/acv-2024-0005  

   Mengesha, Tadele; Schikorra, Armin; Seesanea, Adisak; Yeepo, Sasikarn (November 2024, Advances in Calculus of Variations)

   Abstract We extend the Calderón–Zygmund theory for nonlocal equations tostrongly coupled system of linear nonlocal equations ${\mathcal{ℒ}}_A^{s}u=f${\mathcal{L}^{s}_{A}u=f}, where the operator ${\mathcal{ℒ}}_A^s${\mathcal{L}^{s}_{A}}is formally given by ${\mathcal{ℒ}}_A^{s}u={\int }_{{ℝ}^n}\frac{A(x,y)}{{\left|x-y\right|}^{n+2s}}\frac{\left(x-y\right)\otimes \left(x-y\right)}{{\left|x-y\right|}^2}\left(u\left(x\right)-u\left(y\right)\right)𝑑y.$\mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-%y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For $0<s<1${0<1}and $A:{ℝ}^n\times {ℝ}^n\to ℝ${A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}}taken to be symmetric and serving asa variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot ,y)${A(\,\cdot\,,y)}is uniformly Holder continuous and ${inf}_{x\in {ℝ}^n}A(x,x)>0${\inf_{x\in\mathbb{R}^{n}}A(x,x)>0}, then for $f\in L_{\mathrm{loc}}^p${f\in L^{p}_{\rm loc}}, for $p\ge 2${p\geq 2}, the solution vector $u\in H_{\mathrm{loc}}^{2s-\delta ,p}${u\in H^{2s-\delta,p}_{\rm loc}}for some $\delta \in (0,s)${\delta\in(0,s)}. 

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5. On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

   https://doi.org/10.1090/tran/9165  

   Dong, Hongjie; Li, Haigang; Xu, Longjuan (September 2024, Transactions of the American Mathematical Society)

   In this paper, we consider higher regularity of a weak solution $(\mathbf{u},p)$to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\mathrm{\delta <\#comment/>}}$in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mathrm{\mu <\#comment/>}}$, where $s$is a positive integer, $\mathrm{\delta <\#comment/>}\in <\#comment/>(0,1)$, and $\mathrm{\mu <\#comment/>}\in <\#comment/>(0,1]$, we show that $D\mathbf{u}$and $p$are piecewise $C^{s,{\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}}$, where ${\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}=min\{\frac{1}{2},\mathrm{\mu <\#comment/>},\mathrm{\delta <\#comment/>}\}$. Our result is new even in the 2D case with piecewise constant coefficients. 

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