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.     

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.     

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 $N\ge 3$.     

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 $\text{Aut}(\stackrel{￣}{T})=\mathrm{PSL}(2,\mathbb{C})$.     

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.     

A new type of approximation for the radical quintic functional equation in non-Archimedean (2,β)-Banach spaces

Let $\mathbb{R}$ be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean $(2,\beta )$-normed spaces $(X,{6?,?6}_{?,\beta })$ and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equation

$f\left(\sqrt[5]{x^5+y^5}\right)=f(x)+f(y),x,y\in \mathbb{R}.$

Also, under some weak natural assumptions on the function $\gamma :\mathbb{R}\times \mathbb{R}\times X\to [0,\infty )$, we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when $f:\mathbb{R}\to X$ satisfy the following radical quintic inequality

${6f\left(\sqrt[5]{x^5+y^5}\right)?f(x)?f(y),z6}_{?,\beta }\le \gamma (x,y,z),x,y\in \mathbb{R}?\{0\},z\in X,$

with $x\ne ?y$.     

Global Calderón–Zygmund estimates for a class of nonlinear elliptic equations with Neumann data

In this paper we obtain the following global Calderón–Zygmund estimates

$B\left(\left|\mathbf{f}\right|\right)\in L^q(\mathrm{\Omega })?B\left(\left|\mathrm{?}u\right|\right)\in L^q(\mathrm{\Omega })\text{for any}q\ge 1$

in a convex domain Ω of the weak solution for a class of quasilinear elliptic equations with the Neumann data

$\begin{array}{ccc}\hfill \text{div}\left(a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u\right)& \hfill =\hfill & \text{div}\left(a\left(\left|\mathbf{f}\right|\right)\mathbf{f}\right)\text{in}\mathrm{\Omega },\hfill \\ \hfill a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u?\nu & \hfill =\hfill & a\left(\left|\mathbf{f}\right|\right)\mathbf{f}?\nu \text{on}?\mathrm{\Omega },\hfill \end{array}$

where $B(t)={\int }_0^t\tau a(\tau )d\tau$ for $t\ge 0$. Moreover, we remark that

$B(t)=t^p\log ?(1+t)$

satisfies the given conditions in this work.     

A counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.