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1. Rationality of twists of the Siegel modular variety of genus 2 and level 3

   https://doi.org/10.1090/proc/15854  

   Calegari, Frank; Chidambaram, Shiva (February 2022, Proceedings of the American Mathematical Society)

   Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 

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2. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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3. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

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4. The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

   https://doi.org/10.1007/s11242-024-02070-3  

   Arbabi, Sepehr; Sahimi, Muhammad (March 2024, Transport in Porous Media)

   Abstract Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient$${\varvec{\nabla }}P$$ $\nabla P$and the fluid velocityv. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude$$\omega _z$$ ${\omega }_z$of the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$ ${\omega }_z$, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocityv, given by,$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$ $-\nabla P=(\mu /K_e)v+{\beta }_n\rho {\left|v\right|}^{2}v$, provides accurate representation of the numerical data, where$$\mu$$ $\mu$and$$\rho$$ $\rho$are the fluid’s viscosity and density,$$K_e$$ $K_e$is the effective Darcy permeability in the linear regime, and$$\beta _n$$ ${\beta }_n$is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation. 

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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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