Finite Element Methods For Interface Problems On Local Anisotropic Fitting Mixed Meshes

A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new finite element method is proposed for second order elliptic interface problems based on the resulting mesh. Optimal approximation capabilities on anisotropic elements are proved in both the $H^1$ and $L^2$ norms. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results

Scheme to Achieve Silicon Topological Photonics

We derive in the present work topological photonic states purely based on silicon, a conventional dielectric material, by deforming a honeycomb lattice of silicon cylinders into a triangular lattice of cylinder hexagons. The photonic topology is associated with a pseudo time reversal (TR) symmetry constituted by the TR symmetry respected in general by the Maxwell equations and the $C_6$ crystal symmetry upon design, which renders the Kramers doubling in the present photonic system with the role of pseudo spin played by the circular polarization of magnetic field in the transverse magnetic mode. We solve Maxwell equations, and demonstrate new photonic topology by revealing pseudo spin-resolved Berry curvatures of photonic bands and helical edge states characterized by Poynting vectors.Comment: 5 pages, 5 figure

Topological Properties of Electrons in Honeycomb Lattice with Kekul\'{e} Hopping Textures

Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the $\mathbb{Z}_2$ index associated with a pseudo time-reversal symmetry emerging from the $C_6$ symmetry of the Kekul\'{e} hopping texture, where the angular momentum of orbitals accommodated on the hexagonal "artificial atoms" behaves as the pseudospin. The size of topological gap is proportional to the hopping-integral difference, which can be larger than typical spin-orbit couplings by orders of magnitude and potentially renders topological electronic transports available at high temperatures.Comment: 7 pages, 7 figure

The Weyl Integration Model for KAK decomposition of Reductive Lie Group

The Weyl integration model presented by An and Wang can be effectively used to reduce the integration over $G$-space. In this paper, we construct an especial Weyl integration model for KAK decomposition of Reductive Lie Group and obtain an integration formula which implies that the integration of $L^1$-integrable function over reductive Lie group $G$ can be carried out by first integrating over each conjugacy class and then integrating over the set of conjugacy classes.Comment: 10 page

Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms

We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem

A 28/37/39GHz Multiband Linear Doherty Power Amplifier in Silicon for 5G Applications

This paper presents the first multiband mm-wave linear Doherty PA in silicon for broadband 5G applications. We introduce a new transformer-based on-chip Doherty power combiner, which can reduce the impedance transformation ratio in power back-off (PBO) and thus improve the bandwidth and power-combining efficiency. We also devise a "driver-PA co-design" method, which creates power-dependent uneven feeding in the Doherty PA and enhances the Doherty operation without any hardware overhead or bandwidth compromise. For the proof of concept, we implement a 28/37/39-GHz PA fully integrated in a standard 130-nm SiGe BiCMOS process, which occupies 1.8mm2. The PA achieves a 52% -3-dB small-signal S21 bandwidth and a 40% -1-dB large-signal saturated output power (Psat) bandwidth. At 28/37/39GHz, the PA achieves +16.8/+17.1/+17-dBm Psat, +15.2/+15.5/+15.4-dBm P1dB, and superior 1.72/1.92/1.62 times efficiency enhancement over class-B operation at 5.9/6/6.7-dB PBO. Moreover, the PA demonstrates multi-Gb/s data rates with excellent efficiency and linearity for 64QAM in all the three 5G bands. This PA advances the state of the art for Doherty, wideband, and 5G silicon PAs in mm-wave bands. It supports drop-in upgrade for current PAs in existing mm-wave systems and opens doors to compact system solutions for future multiband 5G massive MIMO and phased-array platforms

An Algorithm for Deciding the Summability of Bivariate Rational Functions

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$ such that $f(x,y)=\Delta_x g(x,y) + \Delta_y h(x,y)$. Recently, Chen and Singer presented a method for deciding whether a rational function is summable. To implement their method in the sense of algorithms, we need to solve two problems. The first is to determine the shift equivalence of two bivariate polynomials. We solve this problem by presenting an algorithm for computing the dispersion sets of any two bivariate polynomials. The second is to solve a univariate difference equation in an algebraically closed field. By considering the irreducible factorization of the denominator of $f(x,y)$ in a general field, we present a new criterion which requires only finding a rational solution of a bivariate difference equation. This goal can be achieved by deriving a universal denominator of the rational solutions and a degree bound on the numerator. Combining these two algorithms, we can decide the summability of a bivariate rational function.Comment: 18 page

Convergence rates in the law of large numbers under sublinear expectations

In this note, we study convergence rates in the law of large numbers for independent and identically distributed random variables under sublinear expectations. We obtain a strong $L^p$-convergence version and a strongly quasi sure convergence version of the law of large numbers.Comment: 12 page

Matrix Linear Discriminant Analysis

We propose a novel linear discriminant analysis approach for the classification of high-dimensional matrix-valued data that commonly arises from imaging studies. Motivated by the equivalence of the conventional linear discriminant analysis and the ordinary least squares, we consider an efficient nuclear norm penalized regression that encourages a low-rank structure. Theoretical properties including a non-asymptotic risk bound and a rank consistency result are established. Simulation studies and an application to electroencephalography data show the superior performance of the proposed method over the existing approaches

Quantum Anomalous Hall Effect in a Perovskite and Inverse-Perovskite Sandwich Structure

Based on first-principles calculations, we propose a sandwich structure composed of a G-type anti-ferromagnetic (AFM) Mott insulator LaCrO$_3$ grown along the [001] direction with one atomic layer replaced by an inverse-perovskite material Sr$_3$PbO. We show that the system is in a topologically nontrivial phase characterized by simultaneous nonzero charge and spin Chern numbers, which can support a spin-polarized and dissipationless edge current in a finite system. Since these two materials are stable in bulk and match each other with only small lattice distortions, the composite material is expected easy to synthesize.Comment: 4 pages, 4 figure