Distributional Hessian and divdiv complexes on triangulation and cohomology

In this paper, we construct discrete versions of some Bernstein-Gelfand-Gelfand (BGG) complexes, i.e., the Hessian and the divdiv complexes, on triangulations in 2D and 3D. The sequences consist of finite elements with local polynomial shape functions and various types of Dirac measure on subsimplices. The construction generalizes Whitney forms (canonical conforming finite elements) for the de Rham complex and Regge calculus/finite elements for the elasticity (Riemannian deformation) complex from discrete topological and Discrete Exterior Calculus perspectives. We show that the cohomology of the resulting complexes is isomorphic to the continuous versions, and thus isomorphic to the de~Rham cohomology with coefficients.Comment: keywords: Bernstein-Gelfand-Gelfand sequences, cohomology, finite element exterior calculus, discrete exterior calculus, Regge calculu

Nonstandard finite element de Rham complexes on cubical meshes

We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.Comment: 31 page

Comparisons of serum miRNA expression profiles in patients with diabetic retinopathy and type 2 diabetes mellitus

OBJECTIVES: The aim of this study was to compare the expression levels of serum miRNAs in diabetic retinopathy and type 2 diabetes mellitus. METHODS: Serum miRNA expression profiles from diabetic retinopathy cases (type 2 diabetes mellitus patients with diabetic retinopathy) and type 2 diabetes mellitus controls (type 2 diabetes mellitus patients without diabetic retinopathy) were examined by miRNA-specific microarray analysis. Quantitative real-time polymerase chain reaction was used to validate the significantly differentially expressed serum miRNAs from the microarray analysis of 45 diabetic retinopathy cases and 45 age-, sex-, body mass index- and duration-of-diabetes-matched type 2 diabetes mellitus controls. The relative changes in serum miRNA expression levels were analyzed using the 2-ΔΔCt method. RESULTS: A total of 5 diabetic retinopathy cases and 5 type 2 diabetes mellitus controls were included in the miRNA-specific microarray analysis. The serum levels of miR-3939 and miR-1910-3p differed significantly between the two groups in the screening stage; however, quantitative real-time polymerase chain reaction did not reveal significant differences in miRNA expression for 45 diabetic retinopathy cases and their matched type 2 diabetes mellitus controls. CONCLUSION: Our findings indicate that miR-3939 and miR-1910-3p may not play important roles in the development of diabetic retinopathy; however, studies with a larger sample size are needed to confirm our findings

Electrically empowered microcomb laser

Optical frequency comb underpins a wide range of applications from communication, metrology, to sensing. Its development on a chip-scale platform -- so called soliton microcomb -- provides a promising path towards system miniaturization and functionality integration via photonic integrated circuit (PIC) technology. Although extensively explored in recent years, challenges remain in key aspects of microcomb such as complex soliton initialization, high threshold, low power efficiency, and limited comb reconfigurability. Here we present an on-chip laser that directly outputs microcomb and resolves all these challenges, with a distinctive mechanism created from synergetic interaction among resonant electro-optic effect, optical Kerr effect, and optical gain inside the laser cavity. Realized with integration between a III-V gain chip and a thin-film lithium niobate (TFLN) PIC, the laser is able to directly emit mode-locked microcomb on demand with robust turnkey operation inherently built in, with individual comb linewidth down to 600 Hz, whole-comb frequency tuning rate exceeding $\rm 2.4\times10^{17}$ Hz/s, and 100% utilization of optical power fully contributing to comb generation. The demonstrated approach unifies architecture and operation simplicity, high-speed reconfigurability, and multifunctional capability enabled by TFLN PIC, opening up a great avenue towards on-demand generation of mode-locked microcomb that is expected to have profound impact on broad applications