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1. Families of Well Approximable Measures

   https://doi.org/10.2478/udt-2021-0003  

   Fairchild, Samantha; Goering, Max; Weiß, Christian (June 2021, Uniform distribution theory)

   Abstract We provide an algorithm to approximate a finitely supported discrete measureμby a measureν_Ncorresponding to a set ofNpoints so that the total variation betweenμandν_Nhas an upper bound. As a consequence ifμis a (finite or infinitely supported) discrete probability measure on [0, 1]^dwith a sufficient decay rate on the weights of each point, thenμcan be approximated byν_Nwith total variation, and hence star-discrepancy, bounded above by (logN)N⁻¹. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measureμ, there exist finite sets whose star-discrepancy with respect toμis at most ${\left(log\hspace{0.17em}N\right)}^{d-\frac{1}{2}}{N}^{-1}${\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}}. Moreover, we close a gap in the literature for discrepancy in the cased=1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure. 

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2. Algebraic stability of meromorphic maps descended from Thurston’s pullback maps

   https://doi.org/10.1090/tran/8221  

   Ramadas, Rohini (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n n . If ϕ \phi has a fully ramified periodic point p ∞ p_{\infty } and satisfies certain additional conditions, then, by work of Koch, ϕ \phi induces a meromorphic self-map R ϕ R_{\phi } on the moduli space M 0 , n \mathcal {M}_{0,n} ; R ϕ R_{\phi } descends from Thurston’s pullback map on Teichmüller space. Here, we relate the dynamics of R ϕ R_{\phi } on M 0 , n \mathcal {M}_{0,n} to the dynamics of ϕ \phi on S 2 S^2 . Let ℓ \ell be the length of the periodic cycle in which the fully ramified point p ∞ p_{\infty } lies; we show that R ϕ R_{\phi } is algebraically stable on the heavy-light Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n-\ell ) light points. 

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3. K → μ+μ− beyond the standard model

   https://doi.org/10.1007/JHEP03(2022)048  

   Dery, Avital; Ghosh, Mitrajyoti (March 2022, Journal of High Energy Physics)

   A bstract We analyze the New Physics sensitivity of a recently proposed method to measure the CP-violating $$ \mathcal{B} $$ B ( K S → μ + μ − ) ℓ =0 decay rate using K S − K L interference. We present our findings both in a model-independent EFT approach as well as within several simple NP scenarios. We discuss the relation with associated observables, most notably $$ \mathcal{B} $$ B ( K L → π 0 $$ \nu \overline{\nu} $$ ν ν ¯ ). We find that simple NP models can significantly enhance $$ \mathcal{B} $$ B ( K S → μ + μ − ) ℓ =0 , making this mode a very promising probe of physics beyond the standard model in the kaon sector. 

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4. Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

   https://doi.org/10.1515/acv-2021-0053  

   Akman, Murat; Hofmann, Steve; Martell, José María; Toro, Tatiana (May 2022, Advances in Calculus of Variations)

   Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u )   and   L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-%\operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion[M. Akman, S. Hofmann, J. M. Martell and T. Toro,Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition,preprint 2021, https://arxiv.org/abs/1901.08261v3 ]is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaininga square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work ofDahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above. 

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5. Weather at the core: defining and categorizing geomagnetic excursions and reversals

   https://doi.org/10.1093/gji/ggae415  

   Constable, Catherine; Morzfeld, Matthias (November 2024, Geophysical Journal International)

   SUMMARY Paleomagnetic records provide us with information about the extreme geomagnetic events known as excursions and reversals, but the sparsity of available data limits detailed knowledge of the process and timing. To date there are no agreed on criteria for categorizing such events in terms of severity or longevity. In an analogy to categorizing storms in weather systems, we invoke the magnitude of the global (modified) paleosecular variation index $$P_{i_D}$$ to define the severity of the magnetic field state, ranging in level from 0 to 3, and defined by instantaneous values of $$P_{i_D}$$ with level 0 being normal ($$P_{i_D}\lt 0.5$$) to extreme ($$P_{i_D}\ge 15$$). We denote the time of entry to an excursional (or reversal) event by when $$P_{i_D}$$ first exceeds 0.5, and evaluate its duration by the time at which $$P_{i_D}$$ first returns below its median value, termed the end of event threshold. We categorize each excursional event according to the peak level of $$P_{i_D}$$ during the entire event, with a range from Category-1 (Cat-1) to Cat-3. We explore an extended numerical dynamo simulation containing more than 1200 events and find that Cat-1 events are the most frequent (72 per cent), but only rarely lead to actual field reversals where the axial dipole, $$g_1^0$$, has reversed sign at the end of the event. Cat-2 account for about 20 per cent of events, with 34 per cent of those leading to actual reversals, while Cat-3 events arise about 8 per cent of the time but are more likely to produce reversals (43 per cent). Higher category events take as much as 10 times longer than Cat-1 events. Two paleomagnetic field models separately cover the Laschamp excursion and Matuyama–Brunhes (M-B) reversal which are Cat-2 events with respective durations of 3.6 and 27.4 kyr. It seems likely that Cat-2 may be an underestimate for M-B due to limitations in the paleomagnetic records. Our overall results suggest no distinction between excursions and reversals other than a reversal having the ending polarity state opposite to that at the start. 

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