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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. Matrix models and deformations of JT gravity

   https://doi.org/10.1098/rspa.2020.0582  

   Witten, Edward (December 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   Recently, it has been found that Jackiw-Teitelboim (JT) gravity, which is a two-dimensional theory with bulk action − 1 / 2 ∫ d 2 x g ϕ ( R + 2 ) , is dual to a matrix model, that is, a random ensemble of quantum systems rather than a specific quantum mechanical system. In this article, we argue that a deformation of JT gravity with bulk action − 1 / 2 ∫ d 2 x g ( ϕ R + W ( ϕ ) ) is likewise dual to a matrix model. With a specific procedure for defining the path integral of the theory, we determine the density of eigenvalues of the dual matrix model. There is a simple answer if W (0) = 0, and otherwise a rather complicated answer. 

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3. Four-manifolds of pinched sectional curvature

   https://doi.org/10.1016/j.aim.2018.07.003  

   Cao, Xiaodong; Tran, Hung (January 2022, Pacific journal of mathematics)

   In this paper, we study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 5 6 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere, or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature. 

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4. A Highly Efficient Dual-band Harmonic-tuned GaN RF Synchronous Rectifier with Integrated Coupler and Phase Shifter

   https://doi.org/10.1109/MWSYM.2019.8700900  

   Hoque, Md Aminul; Ali, Sheikh Nijam; Mokri, Mohammad Ali; Gopal, Srinivasan; Chahardori, Mohammad; Heo, Deukhyoun (June 2019, 2019 IEEE MTT-S International Microwave Symposium (IMS))

   This paper presents a dual-band RF rectifying circuit for wireless power transmission at 1.17 GHz and 2.4 GHz. A dual-band harmonic-tuned inverse-class F/class-F mode power amplifier using a 10 W GaN device has been utilized to implement the proposed rectifier with an on-board coupler and phase shifter. The matching circuit is precisely designed so that the circuit operates in inverse class F and class F mode in the lower and upper frequency bands using dual-band harmonic tuning, respectively. Measurement results show that the rectifier circuit has 78% and 76% efficiencies at 1.17 GHz and 2.4 GHz frequency bands, respectively. To the best of the authors' knowledge, this rectifier is the first demonstration of a dual-band harmonic-tuned synchronous rectifier using a GaN HEMT device with an integrated coupler and phase-shifter for a watt-level RF input power. 

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5. Convergence of graph Laplacian with kNN self-tuned kernels

   https://doi.org/10.1093/imaiai/iaab019  

   Cheng, Xiuyuan; Wu, Hau-Tieng (September 2021, Information and Inference: A Journal of the IMA)

   Abstract Kernelized Gram matrix $$W$$ constructed from data points $$\{x_i\}_{i=1}^N$$ as $$W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $$\sigma $$, and a common practice called self-tuned kernel adaptively sets a $$\sigma _i$$ at each point $$x_i$$ by the $$k$$-nearest neighbor (kNN) distance. When $$x_i$$s are sampled from a $$d$$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $$L_N$$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $$W^{(\alpha )}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $$, where $$\hat{\rho }$$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $$\alpha $$. When $$\alpha = 1$$, the limiting operator is the weighted manifold Laplacian $$\varDelta _p$$. Specifically, we prove the point-wise convergence of $$L_N f $$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $$\hat{\rho }$$ which bounds the relative estimation error $$|\hat{\rho } - \bar{\rho }|/\bar{\rho }$$ uniformly with high probability, where $$\bar{\rho } = p^{-1/d}$$ and $$p$$ is the data density function. Our theoretical results reveal the advantage of the self-tuned kernel over the fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $$d$$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data. 

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