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1. SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

   https://doi.org/10.1017/s1446788718000198  

   LIU, WENCAI (August 2019, Journal of the Australian Mathematical Society)

   In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence $${\mathcal{D}}$$ , we obtain a sharp criterion such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ for a certain one-parameter family of $$\unicode[STIX]{x1D713}$$ . Also, under a minor condition on pseudo-absolute-value sequences $${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$$ , we obtain a sharp criterion on a general sequence $$\unicode[STIX]{x1D713}(n)$$ such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ . 

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2. COUNTING CONJUGACY CLASSES IN

   https://doi.org/10.1017/S0004972718000047  

   HULL, MICHAEL; KAPOVICH, ILYA (June 2018, Bulletin of the Australian Mathematical Society)

   We show that if a finitely generated group $$G$$ has a nonelementary WPD action on a hyperbolic metric space $$X$$ , then the number of $$G$$ -conjugacy classes of $$X$$ -loxodromic elements of $$G$$ coming from a ball of radius $$R$$ in the Cayley graph of $$G$$ grows exponentially in $$R$$ . As an application we prove that for $$N\geq 3$$ the number of distinct $$\text{Out}(F_{N})$$ -conjugacy classes of fully irreducible elements $$\unicode[STIX]{x1D719}$$ from an $$R$$ -ball in the Cayley graph of $$\text{Out}(F_{N})$$ with $$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$$ of the order of $$R$$ grows exponentially in $$R$$ . 

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3. Analysis of passive flexion in propelling a plunging plate using a torsion spring model

   https://doi.org/10.1017/jfm.2018.736  

   Arora, N.; Kang, C.-K.; Shyy, W.; Gupta, A. (December 2018, Journal of Fluid Mechanics)

   We mimic a flapping wing through a fluid–structure interaction (FSI) framework based upon a generalized lumped-torsional flexibility model. The developed fluid and structural solvers together determine the aerodynamic forces, wing deformation and self-propelled motion. A phenomenological solution to the linear single-spring structural dynamics equation is established to help offer insight and validate the computations under the limit of small deformation. The cruising velocity and power requirements are evaluated by varying the flapping Reynolds number ( $$20\leqslant Re_{f}\leqslant 100$$ ), stiffness (represented by frequency ratio, $$1\lesssim \unicode[STIX]{x1D714}^{\ast }\leqslant 10$$ ) and the ratio of aerodynamic to structural inertia forces (represented by a dimensionless parameter $$\unicode[STIX]{x1D713}$$ ( $$0.1\leqslant \unicode[STIX]{x1D713}\leqslant 3$$ )). For structural inertia dominated flows ( $$\unicode[STIX]{x1D713}\ll 1$$ ), pitching and plunging are shown to always remain in phase ( $$\unicode[STIX]{x1D719}\approx 0$$ ) with the maximum wing deformation occurring at the end of the stroke. When aerodynamics dominates ( $$\unicode[STIX]{x1D713}>1$$ ), a large phase difference is induced ( $$\unicode[STIX]{x1D719}\approx \unicode[STIX]{x03C0}/2$$ ) and the maximum deformation occurs at mid-stroke. Lattice Boltzmann simulations show that there is an optimal $$\unicode[STIX]{x1D714}^{\ast }$$ at which cruising velocity is maximized and the location of optimum shifts away from unit frequency ratio ( $$\unicode[STIX]{x1D714}^{\ast }=1$$ ) as $$\unicode[STIX]{x1D713}$$ increases. Furthermore, aerodynamics administered deformations exhibit better performance than those governed by structural inertia, quantified in terms of distance travelled per unit work input. Closer examination reveals that although maximum thrust transpires at unit frequency ratio, it is not transformed into the highest cruising velocity. Rather, the maximum velocity occurs at the condition when the relative tip displacement $${\approx}\,0.3$$ . 

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4. A COMPACTNESS PRINCIPLE FOR MAXIMISING SMOOTH FUNCTIONS OVER TOROIDAL GEODESICS

   https://doi.org/10.1017/S0004972718001636  

   STEINERBERGER, STEFAN (August 2019, Bulletin of the Australian Mathematical Society)

   Let $$f\in C^{2}(\mathbb{T}^{2})$$ have mean value 0 and consider $$\begin{eqnarray}\sup _{\unicode[STIX]{x1D6FE}\,\text{closed geodesic}}\frac{1}{|\unicode[STIX]{x1D6FE}|}\biggl|\int _{\unicode[STIX]{x1D6FE}}f\,d{\mathcal{H}}^{1}\biggr|,\end{eqnarray}$$ where $$\unicode[STIX]{x1D6FE}$$ ranges over all closed geodesics $$\unicode[STIX]{x1D6FE}:\mathbb{S}^{1}\rightarrow \mathbb{T}^{2}$$ and $$|\unicode[STIX]{x1D6FE}|$$ denotes its length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $$\unicode[STIX]{x1D6FE}$$ attaining the supremum in terms of the smoothness of the function: for all $$s\geq 2$$ , $$\begin{eqnarray}|\unicode[STIX]{x1D6FE}|^{s}{\lesssim}_{s}\biggl(\max _{|\unicode[STIX]{x1D6FC}|=s}\Vert \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}f\Vert _{L^{1}(\mathbb{T}^{2})}\biggr)\Vert \unicode[STIX]{x1D6FB}f\Vert _{L^{2}}\Vert f\Vert _{L^{2}}^{-2}.\end{eqnarray}$$ 

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5. On the computability of rotation sets and their entropies

   https://doi.org/10.1017/etds.2018.45  

   BURR, MICHAEL A.; SCHMOLL, MARTIN; WOLF, CHRISTIAN (February 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ . 

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