On the dynamics of a higher order rational difference equations

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation

$y_{n+1}=a{y}_n+b{y}_{n?t}+c{y}_{n?l}+\frac{d{y}_{n?k}+e{y}_{n?s}}{\alpha y_{n?k}+\beta y_{n?s}},n=0,1,\dots ,$

with positive parameters and non-negative initial conditions. Numerical examples to the difference equation are given to explain our results.     

On the Brezis–Nirenberg problem on S3, and a conjecture of Bandle–Benguria

We consider the following Brezis–Nirenberg problem on ${\mathbf{S}}^3$

$?{\mathrm{\Delta }}_{{\mathbf{S}}^3}u=\lambda u+u^5\text{in}D,u>0\text{in}D\text{and}u=0\text{on }?D,$

where D is a geodesic ball on ${\mathbf{S}}^3$ with geodesic radius ${\theta }_1$, and ${\mathrm{\Delta }}_{{\mathbf{S}}^3}$ is the Laplace–Beltrami operator on ${\mathbf{S}}^3$. We prove that for any $\lambda <?\frac{3}{4}$ and for every ${\theta }_1<\pi$ with $\pi ?{\theta }_1$ sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

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.     

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 w-integral closure of integral domains

Let D be an integral domain with quotient field K, $\overline{D}$ the integral closure of D, X an indeterminate over D, and $N_v=\{f\in D[X]|{(A_f)}_v=D\}$. Let w be the 1-operation on D defined by $I_w=\{x\in K|\text{there is a finitely generated ideal}A\text{such that}{A}^{\mathrm{-1}}=D\text{and}xA\subseteq I\}$, and let $D^w=\{u\in K|u{I}_w\subseteq I_w\text{for some nonzero finitely generated ideal}I\text{of}D\}$. Then $D^w$, called the w-integral closure of D, is an integrally closed overring of D. In this paper, we show that $D^w=\overline{D}{[X]}_{N_v}\cap K$ and $D^w{[X]}_{N_v}=\overline{D}{[X]}_{N_v}$. Using this result, we give several w-integral closure analogs of the integral closure. We also study the w-integral closure of UMT-domains and strong Mori domains.     

Weighted bilinear Hardy inequalities

We characterize the weights w, $w_1$, $w_2$ such that the weighted bilinear Hardy inequality$\left(\underset{a}{\overset{b}{\int }}\right(\underset{a}{\overset{x}{\int }}f{)}^q(\underset{a}{\overset{x}{\int }}g{)}^{q}w(x)dx{)}^{\frac{1}{q}}?C(\underset{a}{\overset{b}{\int }}{f}^{p_1}{w}_1{)}^{\frac{1}{p_1}}(\underset{a}{\overset{b}{\int }}{g}^{p_2}{w}_2{)}^{\frac{1}{p_2}}$ holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, $p_1$ and $p_2$ with $1<q,p_1,p_2<\infty$. We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.     

Existence of weak solutions to stochastic evolution inclusions

We prove the existence of a weak mild solution (or mild solution-measure) to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space $\mathrm{d}{X}_t\in A{X}_t\mathrm{d}t+F(t\text{,}{X}_t)\mathrm{d}t+G(t\text{,}{X}_t)\mathrm{d}{W}_t$ where W is a cylindrical Wiener process, A is a linear operator which generates a $C_0$-semigroup, F and G are multifunctions with convex compact values satisfying a linear growth condition and a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures. In the case when F and G are single-valued, we obtain the existence of a strong solution. To cite this article: A. Jakubowski et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur les nombres de Fibonacci de la forme qkyp

We study the equation $F_n=q^k{y}^p$ where q is a prime number and k is a positive integer. We solve it for all $q?1(\mathrm{mod}4)$ and get partial results when $q\equiv 1(\mathrm{mod}4)$. In particular, we answer Ribenboim's question about $F_n=2^k{y}^p$. To cite this article: Y. Bugeaud et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

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.