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1. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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2. 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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3. On the edit distance function of the random graph

   https://doi.org/10.1017/S0963548321000353  

   Martin, Ryan R.; Riasanovsky, Alex W. (March 2022, Combinatorics, Probability and Computing)

   Abstract Given a hereditary property of graphs $$\mathcal{H}$$ and a $$p\in [0,1]$$ , the edit distance function $$\textrm{ed}_{\mathcal{H}}(p)$$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $$\mathcal{H}$$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $$\mathcal{H}$$ and the $$\mathcal{H}$$ -chromatic number of a random graph. Let $$\mathcal{H}$$ be the property of forbidding an Erdős–Rényi random graph $$F\sim \mathbb{G}(n_0,p_0)$$ , and let $$\varphi$$ represent the golden ratio. In this paper, we show that if $$p_0\in [1-1/\varphi,1/\varphi]$$ , then a.a.s. as $$n_0\to\infty$$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $$p\in [1/3,2/3]$$ for any $$p_0\in (0,1)$$ . A primary tool in the proof is the categorization of p -core coloured regularity graphs in the range $$p\in[1-1/\varphi,1/\varphi]$$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques. 

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4. Examining the Rotation of the Planet-hosting M Dwarf GJ 3942

   https://doi.org/10.3847/2515-5172/ad95e9  

   Fonseca, Andrew; Dodson-Robinson, Sarah (November 2024, Research Notes of the AAS)

   Abstract Based on radial velocities, EXORAP photometry, and activity indicators, the HArps-n red Dwarf Exoplanet Survey (HADES) team reported a 16.3 days rotation period for the M dwarf GJ 3942. However, an estimate of the magnitude-squared coherence between the HADES RV and Hαtime series has significant peaks at frequencies 1/16 day⁻¹and 1/32 day⁻¹. We turn to TESS photometry to test the hypothesis that the true rotation period is 32 days with 16 days harmonic. Although the average TESS periodogram has peaks at harmonics of 1/16 day⁻¹, the harmonic sequence is not fully resolved according to the Rayleigh criterion. The TESS observations suggest a 1/16 day⁻¹rotation frequency and a 1/32 day⁻¹subharmonic, though resolution makes the TESS rotation detection ambiguous. 

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5. On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

   https://doi.org/10.1007/s10955-020-02648-3  

   Huang, Hui; Liu, Jian-Guo; Pickl, Peter (December 2020, Journal of Statistical Physics)

   null (Ed.)

   Abstract We rigorously justify the mean-field limit of an N -particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N -particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ( $$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ( $$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions. 

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