 Award ID(s):
 1800251
 NSFPAR ID:
 10315033
 Date Published:
 Journal Name:
 Dolomites Research Notes on Approximation
 Volume:
 14
 ISSN:
 20356803
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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null (Ed.)Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientationpreserving branched covering whose postcritical 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 selfmap 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 heavylight Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n\ell ) light points.more » « less

A bstract We analyze the New Physics sensitivity of a recently proposed method to measure the CPviolating $$ \mathcal{B} $$ B ( K S → μ + μ − ) ℓ =0 decay rate using K S − K L interference. We present our findings both in a modelindependent 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.more » « less

Abstract Let $\gamma (t)=(P_{1}(t),\ldots ,P_{n}(t))$ where $P_{i}$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma (t)\}$ in sets of positive density $\epsilon $ in $[0,1]^{n}$ with a gap estimate $t\geq \delta (\epsilon )$ when $P_{i}$’s are arbitrary, and in $[0,N]^{n}$ with a gap estimate $t\geq \delta (\epsilon )N^{n}$ when $P_{i}$’s are of distinct degrees where $\delta (\epsilon )=\exp \left (\exp \left (c\epsilon ^{4}\right )\right )$ and $c$ only depends on $\gamma $. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain’s reduction are primarily utilised. For the first result, dimensionreducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the cornertype Roth theorem previously proven by the first author and Guo.

Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1sided nontangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scaleinvariant/quantitative versions of openness and pathconnectedness. Let us assume also that Ω satisfies the socalled capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two realvalued (nonnecessarily 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 1sided 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 nontangential estimates for solutions of operators as before. We establish that bounded weak nullsolutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak nullsolution, the associated square function can be controlled by the nontangential 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.more » « less

Abstract Scattering of high energy particles from nucleons probes their structure, as was done in the experiments that established the nonzero size of the proton using electron beams 1 . The use of charged leptons as scattering probes enables measuring the distribution of electric charges, which is encoded in the vector form factors of the nucleon 2 . Scattering weakly interacting neutrinos gives the opportunity to measure both vector and axial vector form factors of the nucleon, providing an additional, complementary probe of their structure. The nucleon transition axial form factor, F A , can be measured from neutrino scattering from free nucleons, ν μ n → μ − p and $${\bar{\nu }}_{\mu }p\to {\mu }^{+}n$$ ν ¯ μ p → μ + n , as a function of the negative fourmomentum transfer squared ( Q 2 ). Up to now, F A ( Q 2 ) has been extracted from the bound nucleons in neutrino–deuterium scattering 3–9 , which requires uncertain nuclear corrections 10 . Here we report the first highstatistics measurement, to our knowledge, of the $${\bar{\nu }}_{\mu }\,p\to {\mu }^{+}n$$ ν ¯ μ p → μ + n crosssection from the hydrogen atom, using the plastic scintillator target of the MINERvA 11 experiment, extracting F A from free proton targets and measuring the nucleon axial charge radius, r A , to be 0.73 ± 0.17 fm. The antineutrino–hydrogen scattering presented here can access the axial form factor without the need for nuclear theory corrections, and enables direct comparisons with the increasingly precise lattice quantum chromodynamics computations 12–15 . Finally, the tools developed for this analysis and the result presented are substantial advancements in our capabilities to understand the nucleon structure in the weak sector, and also help the current and future neutrino oscillation experiments 16–20 to better constrain neutrino interaction models.more » « less