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1. Improved quantitative unique continuation for complex-valued drift equations in the plane

   https://doi.org/10.1515/forum-2022-0114  

   Davey, Blair; Kenig, Carlos; Wang, Jenn-Nan (June 2022, Forum Mathematicum)

   Abstract In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane.We consider equations of the form Δ ⁢ u + W ⋅ ∇ ⁡ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} ,where W = W 1 + i ⁢ W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued.Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity,we prove new global unique continuation estimates.This improvement is accomplished by reducing our equations to vector-valued Beltrami systems.Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme. 

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2. Uniform congruence counting for Schottky semigroups in SL2(𝐙)

   https://doi.org/10.1515/crelle-2016-0072  

   Magee, Michael; Oh, Hee; Winter, Dale (August 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})} ,and for {q\in\mathbf{N}} , let {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroupof level q . Let δ denote the Hausdorff dimension of the limit set of Γ.We prove the following uniform congruence counting theoremwith respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R :for all positive integer q with no small prime factors, \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(%\mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon}) as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q .Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})} ,which arises in the study of Zaremba’s conjecture on continued fractions. 

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3. Range Closest-Pair Search in Higher Dimensions

   https://doi.org/10.1007/978-3-030-24766-9_20  

   Chan, Timothy M.; Rahul, Saladi; Xue, Jie (January 2019, Proc. Algorithms and Data Structures Symposium (WADS))

   Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S∩Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on R^2. In this paper, we study RCP search in higher dimensions. We give the first nontrivial RCP data structures for orthogonal, simplex, halfspace, and ball queries in R^d for any constant d. Furthermore, we prove a conditional lower bound for orthogonal RCP search for d≥3. 

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4. Primitivity rank for random elements in free groups

   https://doi.org/10.1515/jgth-2021-0150  

   Kapovich, Ilya (March 2022, Journal of Group Theory)

   Abstract For a free group F r F_{r} of finite rank r ≥ 2 r\geq 2 and a non-trivial element w ∈ F r w\in F_{r} , the primitivity rank π ⁢ ( w ) \pi(w) is the smallest rank of a subgroup H ≤ F r H\leq F_{r} such that w ∈ H w\in H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ \pi(w)=\infty ).The set of all subgroups of F r F_{r} of rank π ⁢ ( w ) \pi(w) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) \operatorname{Crit}(w) .These notions were introduced by Puder (2014).We prove that there exists an exponentially generic subset V ⊆ F r V\subseteq F_{r} such that, for every w ∈ V w\in V , we have π ⁢ ( w ) = r \pi(w)=r and Crit ⁡ ( w ) = { F r } \operatorname{Crit}(w)=\{F_{r}\} . 

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5. Energy Harvesting Performance of a Flapping Foil with Leading Edge Motion

   Prier, M. (April 2019, Dissertation)

   F or c e d at a f or a fl a p pi n g f oil e n er g y h ar v e st er wit h a cti v e l e a di n g e d g e m oti o n o p er ati n g i n t h e l o w r e d u c e d fr e q u e n c y r a n g e i s c oll e ct e d t o d et er mi n e h o w l e a di n g e d g e m oti o n aff e ct s e n er g y h ar v e sti n g p erf or m a n c e. T h e f oil pi v ot s a b o ut t h e mi dc h or d a n d o p er at e s i n t h e l o w r e d u c e d fr e q u e n c y r a n g e of 𝑓𝑓 𝑓𝑓 / 𝑈𝑈 ∞ = 0. 0 6 , 0. 0 8, a n d 0. 1 0 wit h 𝑅𝑅 𝑅𝑅 = 2 0 ,0 0 0 − 3 0 ,0 0 0 , wit h a pit c hi n g a m plit u d e of 𝜃𝜃 0 = 7 0 ∘ , a n d a h e a vi n g a m plit u d e of ℎ 0 = 0. 5 𝑓𝑓 . It i s f o u n d t h at l e a di n g e d g e m oti o n s t h at r e d u c e t h e eff e cti v e a n gl e of att a c k e arl y t h e str o k e w or k t o b ot h i n cr e a s e t h e lift f or c e s a s w ell a s s hift t h e p e a k lift f or c e l at er i n t h e fl a p pi n g str o k e. L e a di n g e d g e m oti o n s i n w hi c h t h e eff e cti v e a n gl e of att a c k i s i n cr e a s e d e arl y i n t h e str o k e s h o w d e cr e a s e d p erf or m a n c e. I n a d diti o n a di s cr et e v ort e x m o d el wit h v ort e x s h e d di n g at t h e l e a di n g e d g e i s i m pl e m e nt f or t h e m oti o n s st u di e d; it i s f o u n d t h at t h e m e c h a ni s m f or s h e d di n g at t h e l e a di n g e d g e i s n ot a d e q u at e f or t hi s p ar a m et er r a n g e a n d t h e m o d el c o n si st e ntl y o v er pr e di ct s t h e a er o d y n a mi c f or c e s. 

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