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.     

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 $N\ge 3$.     

Equitable two-colorings of uniform hypergraphs

An equitable two-coloring of a hypergraph $H=(V,E)$ is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with the property B problem Radhakrishnan and Srinivasan proved that if $H$ is a $k$-uniform hypergraph with maximum vertex degree $\Delta \left(H\right)$ satisfying $\Delta \left(H\right)⩽c\frac{2^{k-1}}{\sqrt{klnk}}$ for some absolute constant $c>0$, then $H$ is 2-colorable. By using the Lovász Local Lemma for negatively correlated events and the random recoloring method we prove that if $H$ either is a simple hypergraph or has a lot of vertices, then under the same condition on the maximum vertex degree it has an equitable coloring with two colors. We also obtain a general result for equitable colorings of partial Steiner systems.     

Some results on Vizing’s conjecture and related problems

The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs $G$ and $H$, $\gamma \left(G\square H\right)\ge \gamma \left(G\right)\gamma \left(H\right)$. We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every $k\ge 1$ that ${\gamma }_{rk}\left(G\square H\right)\ge \frac{k}{k+1}\gamma \left(G\right)\gamma \left(H\right)$, which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two.     

Improving the Clark–Suen bound on the domination number of the Cartesian product of graphs

A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, $\gamma (G\square H)\ge \gamma \left(G\right)\gamma \left(H\right)∕2$, where $\gamma$ stands for the domination number, and $G\square H$ is the Cartesian product of graphs $G$ and $H$. In this note, we improve this bound by employing the 2-packing number $\rho \left(G\right)$ of a graph $G$ into the formula, asserting that $\gamma (G\square H)\ge (2\gamma \left(G\right)?\rho \left(G\right))\gamma \left(H\right)∕3$. The resulting bound is better than that of Clark and Suen whenever $G$ is a graph with $\rho \left(G\right)<\gamma \left(G\right)∕2$, and in the case $G$ has diameter 2 reads as $\gamma (G\square H)\ge (2\gamma \left(G\right)?1)\gamma \left(H\right)∕3$.     

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.     

Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering

We take advantage of recent (see Graf et al., 2008; Pages and Wilbertz, 2012) and new results on optimal quantization theory to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE) and nonlinear filtering problems. For both problems, a first improvement relies on a Pythagoras like Theorem for quantized conditional expectation. While allowing for some locally Lipschitz continuous conditional densities in nonlinear filtering, the analysis of the error brings into play a new robustness result about optimal quantizers, the so-called distortion mismatch property: the $L^s$-mean quantization error induced by $L^r$-optimal quantizers of size $N$ converges at the same rate $N^{?\frac{1}{d}}$ for every $s\in (0,r+d)$.     

On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.