The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

Weak solutions of semilinear elliptic equation involving Dirac mass

In this paper, we study the elliptic problem with Dirac mass

(1)$\{\begin{array}{c}?\mathrm{\Delta }u=V{u}^p+k{\delta }_0\text{in}{\mathbb{R}}^N,\hfill \\ \underset{|x|\to +\infty }{\lim }?u(x)=0,\hfill \end{array}$

where $N>2$, $p>0$, $k>0$, ${\delta }_0$ is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in ${\mathbb{R}}^N?\{0\}$, with non-empty support and satisfying

$0\le V(x)\le \frac{{\sigma }_1}{|x{|}^{a_0}(1+|x{|}^{a_{\infty }?a_0})},$

with $a_0<N$, $a_0<a_{\infty }$ and ${\sigma }_1>0$. We obtain two positive solutions of (1) with additional conditions for parameters on $a_{\infty },a_0$, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.     

Proof of a conjecture on monomial graphs

Let e be a positive integer, p be an odd prime, $q=p^e$, and ${\mathbb{F}}_q$ be the finite field of q elements. Let $f,g\in {\mathbb{F}}_q[X,Y]$. The graph $G_q(f,g)$ is a bipartite graph with vertex partitions $P={\mathbb{F}}_q^3$ and $L={\mathbb{F}}_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l]=[l_1,l_2,l_3]\in L$ if and only if $p_2+l_2=f(p_1,l_1)$ and $p_3+l_3=g(p_1,l_1)$. If $f=XY$ and $g=X{Y}^2$, the graph $G_q(XY,X{Y}^2)$ contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs $G_q(f,g)$ containing no cycles of length less than eight and not isomorphic to the graph $G_q(XY,X{Y}^2)$, even without requiring them to be edge-transitive. So far, no such graphs $G_q(f,g)$ have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture.     

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

The chemotaxis system

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right),\hfill \\ v_t=\mathrm{\Delta }v?uv,\hfill \end{array}(?)$

is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)?{\mathbb{R}}^n$, where $R>0$ and $n\ge 2$.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data $(u_0,v_0)$ satisfying $u_0\ge 0$ and $v_0>0$, the present paper establishes the existence of a globally defined pair $(u,v)$ of radially symmetric functions which are continuous in $(\overline{\mathrm{\Omega }}?\{0\})\times [0,\infty )$ and smooth in $(\overline{\mathrm{\Omega }}?\{0\})\times (0,\infty )$, and which solve the corresponding initial-boundary value problem for (?) with $(u(?,0),v(?,0))=(u_0,v_0)$ in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data.     

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.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

Extension operators and twisted sums of c0 and C(K) spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space X containing a non-complemented copy Y of $c_0$ such that the quotient space $X/Y$ is isomorphic to $C(K)$?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube $2^{{\omega }_1}$ or K is a separable scattered compact space of height 3 and weight ${\omega }_1$, then every twisted sum of $c_0$ and $C(K)$ is trivial.We also construct nontrivial twisted sums of $c_0$ and $C(K)$ for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces $K?L$ which do not admit an extension operator $C(K)\to C(L)$.     

Pruned discrete random samples

Let $X_i,i\in \mathbb{N}$, be independent and identically distributed random variables with values in ${\mathbb{N}}_0$. We transform (‘prune’) the sequence $\{X_1,\dots ,X_n\},n\in \mathbb{N}$, of discrete random samples into a sequence $\{0,1,2,\dots ,Y_n\},n\in \mathbb{N}$, of contiguous random sets by replacing $X_{n+1}$ with $Y_n+1$ if $X_{n+1}>Y_n$. We consider the asymptotic behaviour of $Y_n$ as $n\to \infty$. Applications include path growth in digital search trees and the number of tables in Pitmanʼs Chinese restaurant process if the latter is conditioned on its limit value.     

Regularity results for the minimum time function with Hörmander vector fields

In a bounded domain of ${\mathbb{R}}^n$ with boundary given by a smooth $(n?1)$-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\dots ,X_N\}$ subject to Hörmander's bracket generating condition. We investigate the regularity of the viscosity solution T of such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of $\{X_1,\dots ,X_N\}$ is a symplectic manifold. We apply our results to several examples.     

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring $k[X,Y,Z]$ over a field k of characteristic zero extend to the polynomial ring $R[X,Y,Z]$ over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on $k[X_1,X_2,X_3,X_4]$ of rank at most three is a polynomial ring over k.     

Two remarks on liftings of maps with values into S1

Given a map $u\in L_{\mathrm{loc}}^1(\Omega ,S^1)$ with some regularity: ${|u|}_X=R<\infty$, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u={\mathrm{e}}^{\mathrm{i}\phi }$) with the same regularity and with an optimal control ${|\phi |}_X?g(R)$. Two cases are treated here:(i) ${|?|}_X$ is a $W^{s,p}(0,1)$-seminorm, with $0<s<1<p$ and $sp>1$. We find a lifting φ such that ${|\phi |}_{W^{s,p}(I)}?C(R+R^{1/s})$ and we show that the exponent $1/s$ cannot be improved.(ii) ${|?|}_X$ is the $\mathit{BV}(\Omega )$-seminorm where $\Omega ?{\mathbf{R}}^d$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^1$, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $\phi \in \mathit{BV}(\Omega )$ satisfying ${|\phi |}_{\mathit{BV}}?2R$. To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).