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1. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo ; de la Llave, Rafael ( March 2022 , Nonlinearity)

   The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation${\mathcal{H}}_{\epsilon }(p,q,I,\phi ,t)=h(I)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i(q_i)\right)+\epsilon H_1(p,q,I,\phi ,t),$where$(p,q)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$,$(I,\phi )\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}.

    

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2. Volumetric Rates of Luminous Red Novae and Intermediate-luminosity Red Transients with the Zwicky Transient Facility

   https://doi.org/10.3847/1538-4357/acc2b9  

   Karambelkar, Viraj R. ; Kasliwal, Mansi M. ; Blagorodnova, Nadejda ; Sollerman, Jesper ; Aloisi, Robert ; Anand, Shreya G. ; Andreoni, Igor ; Brink, Thomas G. ; Bruch, Rachel ; Cook, David ; et al ( May 2023 , The Astrophysical Journal)

   Luminous red novae (LRNe) are transients characterized by low luminosities and expansion velocities, and they are associated with mergers or common-envelope ejections in stellar binaries. Intermediate-luminosity red transients (ILRTs) are an observationally similar class with unknown origins, but they are generally believed to be either electron-capture supernovae in super-asymptotic giant branch stars or outbursts in dusty luminous blue variables (LBVs). In this paper, we present a systematic sample of eight LRNe and eight ILRTs detected as part of the Census of the Local Universe (CLU) experiment on the Zwicky Transient Facility (ZTF). The CLU experiment spectroscopically classifies ZTF transients associated with nearby (<150 Mpc) galaxies, achieving 80% completeness form_r< 20 mag. Using the ZTF-CLU sample, we derive the first systematic LRNe volumetric rate of${7.8}_{-3.7}^{+6.5}\times {10}^{-5}$Mpc⁻³yr⁻¹in the luminosity range −16 ≤M_r≤ −11 mag. We find that, in this luminosity range, the LRN rate scales as$\mathit{dN}/\mathit{dL}\propto L^{-2.5\pm 0.3}$—significantly steeper than the previously derived scaling ofL^−1.4±0.3for lower-luminosity LRNe (M_V≥ −10 mag). The steeper power law for LRNe at high luminosities is consistent with the massive merger rates predicted by binary population synthesis models. We find that the rates of the brightest LRNe (M_r≤ −13 mag) are consistent with a significant fraction of them being progenitors of double compact objects that merge within a Hubble time. For ILRTs, we derive a volumetric rate of${2.6}_{-1.4}^{+1.8}\times {10}^{-6}$Mpc⁻³yr⁻¹forM_r≤ −13.5 mag, which scales as$\mathit{dN}/\mathit{dL}\propto L^{-2.5\pm 0.5}$. This rate is ∼1%–5% of the local core-collapse supernova rate and is consistent with theoretical ECSN rate estimates.

    

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3. Asymptotics of noncolliding q-exchangeable random walks

   https://doi.org/10.1088/1751-8121/acedda  

   Petrov, Leonid ; Tikhonov, Mikhail ( August 2023 , Journal of Physics A: Mathematical and Theoretical)

   We consider a process of noncollidingq-exchangeable random walks on$\mathbb{Z}$making steps 0 (‘straight’) and −1 (‘down’). A single random walk is calledq-exchangeable if under an elementary transposition of the neighboring steps$(\text{down},\text{straight})\to (\text{straight},\text{down})$the probability of the trajectory is multiplied by a parameter$q\in (0,1)$. Our process ofmnoncollidingq-exchangeable random walks is obtained from the independentq-exchangeable walks via the Doob’sh-transform for a nonnegative eigenfunctionh(expressed via theq-Vandermonde product) with the eigenvalue less than 1. The system ofmwalks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is theq-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncollidingq-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel ofq-distributed random lozenge tilings of sawtooth polygons. In the limit as$m\to \mathrm{\infty }$,$q=e^{-\gamma /m}$withγ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

    

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4. The Pan-STARRS1 z > 5.6 Quasar Survey. III. The z ≈ 6 Quasar Luminosity Function

   https://doi.org/10.3847/1538-4357/aca7ca  

   Schindler, Jan-Torge ; Bañados, Eduardo ; Connor, Thomas ; Decarli, Roberto ; Fan, Xiaohui ; Farina, Emanuele Paolo ; Mazzucchelli, Chiara ; Nanni, Riccardo ; Rix, Hans-Walter ; Stern, Daniel ; et al ( January 2023 , The Astrophysical Journal)

   We present thez≈ 6 type-1 quasar luminosity function (QLF), based on the Pan-STARRS1 (PS1) quasar survey. The PS1 sample includes 125 quasars atz≈ 5.7–6.2, with −28 ≲M₁₄₅₀≲ −25. With the addition of 48 fainter quasars from the SHELLQs survey, we evaluate thez≈ 6 QLF over −28 ≲M₁₄₅₀≲ −22. Adopting a double power law with an exponential evolution of the quasar density (Φ(z) ∝ 10^k(z−6);k= −0.7), we use a maximum likelihood method to model our data. We find a break magnitude of$M^{*}=-{26.38}_{-0.60}^{+0.79}\mathrm{mag}$, a faint-end slope of$\alpha =-{1.70}_{-0.19}^{+0.29}$, and a steep bright-end slope of$\beta =-{3.84}_{-1.21}^{+0.63}$. Based on our new QLF model, we determine the quasar comoving spatial density atz≈ 6 to be$n(M_{1450}<-26)={1.16}_{-0.12}^{+0.13}{\mathrm{cGpc}}^{-3}$. In comparison with the literature, we find the quasar density to evolve with a constant value ofk≈ −0.7, fromz≈ 7 toz≈ 4. Additionally, we derive an ionizing emissivity of${ϵ}_{912}(z=6)={7.23}_{-1.02}^{+1.65}\times {10}^{22}\mathrm{erg}{\mathrm{s}}^{-1}{\mathrm{Hz}}^{-1}{\mathrm{cMpc}}^{-3}$, based on the QLF measurement. Given standard assumptions, and the recent measurement of the mean free path by Becker et al. atz≈ 6, we calculate an Hiphotoionizing rate of Γ_{H I}(z= 6) ≈ 6 × 10⁻¹⁶s⁻¹, strongly disfavoring a dominant role of quasars in hydrogen reionization.

    

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5. Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales

   https://doi.org/10.1088/1361-6544/acf17c  

   Han, Yuxi ; Jang, Jiwoong ( August 2023 , Nonlinearity)

   We study the rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales, where the Hamiltonian$H=H(x,y,p):{\mathbb{R}}^n\times {\mathbb{T}}^n\times {\mathbb{R}}^n\to \mathbb{R}$depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is$O(t\sqrt{ϵ})$by optimal control formulas, scale separations and curve cutting techniques. We also show the rate$O(\frac{\sqrt{ϵ}}{\lambda })$of homogenization for the static problem based on the same idea. Additionally, we provide examples that illustrate the rate of convergence for the Cauchy problem is optimal for$0<t<\sqrt{ϵ}$and$t\sim \sqrt{ϵ}$.

    

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