Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

On the stability of flat complex vector bundles over parallelizable manifolds

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G/\mathrm{\Gamma }$, where G is a complex connected Lie group and Γ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_{\rho }$ associated with any irreducible representation $\rho :\mathrm{\Gamma }?\text{GL}(r,\mathbb{C})$. More precisely, we prove that $E_{\rho }$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where E is a stable vector bundle. All the rational Chern classes of E vanish, in particular, its degree is zero.We deduce a stability result for flat holomorphic vector bundles $E_{\rho }$ of rank 2 over $G/\mathrm{\Gamma }$. If an irreducible representation $\rho :\mathrm{\Gamma }?\text{GL}(2,\mathbb{C})$ satisfies the condition that the induced homomorphism $\mathrm{\Gamma }?\mathrm{PGL}(2,\mathbb{C})$ does not extend to a homomorphism from G, then $E_{\rho }$ is proved to be stable.     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

Further results on logarithmic terraces

Let p be an odd prime, and let x be a primitive root of p. Suppose that we write the elements of ${\mathbb{Z}}_{p-1}$ as $1,2,\dots ,p-1,$ and that, wherever we evaluate $x^l(\mathrm{mod}p)$, we always write it as one of $1,2,\dots ,p-1.$ Let $?=(l_1,\dots ,l_{p-1})$ be a terrace for ${\mathbb{Z}}_{p-1}$. Then $?$ is said to be a logarithmic terrace if $\mathbf{e}=(e_1,\dots ,e_{p-1})$, defined by $e_i\equiv x^{l_i}(\mathrm{mod}p)$, is also a terrace for ${\mathbb{Z}}_{p-1}$. We study properties of logarithmic terraces, in particular investigating terraces which are simultaneously logarithmic for two different primitive roots.     

The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small [5]. It has been conjectured by Nagylaki and Lou [2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in [6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.     

The full automorphism group of T‾

Let $\stackrel{￣}{G}$ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over $\mathbb{C}$. Let $\stackrel{￣}{T}?\stackrel{￣}{G}$ be the closure of a maximal torus $T?G$. We prove that the group of all automorphisms of the variety $\stackrel{￣}{T}$ is the semi-direct product $N_G(T)?D$, where $N_G(T)$ is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if $G\ne \mathrm{PSL}(2,\mathbb{C})$. Note that if $G=\mathrm{PSL}(2,\mathbb{C})$, then $\stackrel{￣}{T}=\mathbb{C}{\mathbb{P}}^1$ and so in this case $\text{Aut}(\stackrel{￣}{T})=\mathrm{PSL}(2,\mathbb{C})$.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.