Structural and optical properties of MOCVD AllnN epilayers

7] M.-Y. Ryu, C.Q. Chen, E. Kuokstis, J.W. Yang, G. Simin, M. Asif Khan, Appl. Phys. Lett. 80 (2002) 3730. [8] D. Xu, Y. Wang, H. Yang, L. Zheng, J. Li, L. Duan, R. Wu, Sci. China (a) 42 (1999) 517. [9] H. Hirayama, A. Kinoshita, A. Hirata, Y. Aoyagi, Phys. Stat. Sol. (a) 188 (2001) 83. [10] Y. Chen, T. Takeuchi, H. Amano, I. Akasaki, N. Yamada, Y. Kaneko, S.Y. Wang, Appl. Phys. Lett. 72 (1998) 710. [11] Ig-Hyeon Kim, Hyeong-Soo Park, Yong-Jo Park, Taeil Kim, Appl. Phys. Lett. 73 (1998) 1634. [12] K. Watanabe, J.R. Yang, S.Y. Huang, K. Inoke, J.T. Hsu, R.C. Tu, T. Yamazaki, N. Nakanishi, M. Shiojiri, Appl. Phys. Lett. 82 (2003) 718

Ricci flow on compact K\"ahler manifolds of positive bisectional curvature

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of K\"ahler-Einstein metrics on $M^n$, but instead on the first author's Harnack inequality for the K\"ahler-Ricc flow, and a very recent local injectivity radius estimate of Perelman for the Ricci flow.Comment: 4 page

Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).

Correct order on some certain weighted representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+ka_2$. In 2016, Qu proved that $$\liminf_{n\rightarrow\infty}r_{1,k}(A,n)=\infty$$ providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu's result and obtained that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{\log n}>0.$$ In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that $$\liminf_{n\rightarrow\infty}\frac{r_{1,k}(A,n)}{n}>0.$$ Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.