Conformal and CR mappings on Carnot groups

We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component

Focal Points of Introducing China Dream Into Classroom, Into Textbooks and Into University Students’ Mind

Concerning the work of introducing China Dream into class, into textbooks and into university students’ mind, three focal points need to be highly stressed, namely, laying the solid ideological foundation of practicing China Dream, enriching and expanding the primary coverage of practicing China Dream, and then clarifying the conjunction between students’ voluntary practicing China Dream and their self-fulfillment

Global well-posedness for the 2D incompressible heat conducting Navier-Stokes equations with temperature-dependent coefficients and vacuum

We consider the initial boundary problem of 2D non-homogeneous incompressible heat conducting Navier-Stokes equations with vacuum, where the viscosity and heat conductivity depend on temperature in a power law of Chapman-Enskog. We derive the global existence of strong solution to the initial-boundary value problem, which is not trivial, especially for the nonisentropic system with vacuum. Significantly, our existence result holds for the cases that the viscosity and heat conductivity depend on $\theta$ with possibly different power laws (i.e., $\mu=\theta^{\alpha}, \kappa=\theta^{\beta}$ with constants $\alpha,\beta\geq0$) with smallness assumptions only on $\alpha$ and the measure of initial vacuum domain. In particular, the initial data can be arbitrarily large. Moreover, it is obtained that both velocity and temperature decay exponentially as time tends to infinity.Comment: 28 pages. Any comments are welcom

Optimal lifting of Levi-degenerate hypersurfaces and applications to the Cauchy--Szeg\"o projection

We consider a family of Levi-degenerate finite type hypersurfaces in $\mathbb C^2$, where in general there is no group structure. We lift these domains to stratified Lie groups via a constructive proof, which optimizes the well-known lifting procedure to free Lie groups of general manifolds defined by Rothschild and Stein. This yields an explicit version of the Taylor expansion with respect to the horizontal vector fields induced by the sub-Riemannian structure on these hypersurfaces. Hence, as an application, we establish the Schatten class estimates for the commutator of the Cauchy--Szeg\"{o} projection with respect to a suitable quasi-metric defined on the hypersurface

The Kohn-Laplacian and Cauchy-Szeg\"{o} projection on Model Domains

We study the Kohn-Laplacian and its fundamental solution on some model domains in $\mathbb C^{n+1}$, and further discuss the explicit kernel of the Cauchy-Szeg\"o projections on these model domains using the real analysis method. We further show that these Cauchy-Szeg\"o kernels are Calder\'on-Zygmund kernels under the suitable quasi-metric

Type IIs restriction based combinatory modulation technique for metabolic pathway optimization

Additional file 1: Table S1. Oligonucleotides used in this study

Combinatory optimization of chromosomal integrated mevalonate pathway for β-carotene production in Escherichia coli

Additional file 1: Table S1. Primers used in this work. Table S2. Modulating genes of mvaS-mvaA-mavD1 operon for improving β-carotene production. Table S3. Modulating genes of Hmg1-erg12 operon for improving β-carotene production. Table S4. Sequences of representative artificial regulatory parts. Table S5. Plasmids used in this work. Table S6. Escherichia coli strains used in this work. Table S7. Calculated strength of mvaS and Hmg1 RBS, RBS sequence and relative β-carotene yield of strains from Re-modulation libraries. Figure S1. Two-step recombination method for inserting Hmg1-erg12 operon in E. coli chromosome. Figure S2. Two-step recombination method for modulating gene expression in E. coli chromosome by different artificial regulatory parts