An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing $n\times n$ matrix functions $D(·)$ defined on some open interval $?=(a\text{,}b)$ of $\mathbb{R}$ and their limit values $D(a)$ and $D(b)$ at the endpoints a and b which are, in general, selfadjoint relations in ${\mathbb{C}}^n$. Certain space decompositions induced by the matrix function $D(·)$ are made explicit by means of the limit values $D(a)$ and $D(b)$. They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

Boundary regularity criteria for the 6D steady Navier–Stokes and MHD equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier–Stokes and MHD equations are Hölder continuous near boundary provided that either $r^{?3}{\int }_{B_r^{+}}|u(x){|}^{3}dx$ or $r^{?2}{\int }_{B_r^{+}}|\mathrm{?}u(x){|}^{2}dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong–Strain [5].     

Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field

In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in $H^2({\mathbb{R}}^3)$ are obtained by introducing a new linearized system with respect to $(n^{\gamma }?{\stackrel{?}{n}}^{\gamma },n?\stackrel{?}{n},P?\stackrel{?}{P},u,H)$ for constants $\stackrel{?}{n}\ge 0$ and $\stackrel{?}{P}>0$, and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of $(n?\stackrel{?}{n},n^{\gamma }?{\stackrel{?}{n}}^{\gamma })$ in $H^2({\mathbb{R}}^3)$ norm.     

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in $L^1({\mathbb{R}}^2)\cap L_{\sigma }^2({\mathbb{R}}^2)$. Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ${6u(t)6}_2=o(t^{?1})$ as $t\to \infty$ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow $u(t)$ lies in the Hardy space $H^1({\mathbb{R}}^2)$ for $t>0$, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ${6u(t)6}_2=o(t^{?\frac{3}{2}})$ as $t\to \infty$ with cyclic symmetry introduced by Brandolese [2].     

Neumann problem for a quasilinear elliptic equation in a varying domain

We investigate the Neumann problem for a nonlinear elliptic operator of Leray–Lions type in ${\Omega }^{(s)}=\Omega \backslash F^{(s)}$, $s=1,2,\dots$, where Ω is a domain in ${\mathbf{R}}^n$$(n?3)$, $F^{(s)}$ is a closed set located in the neighborhood of a $(n?1)$-dimensional manifold Γ lying inside Ω. We study the asymptotic behavior of $u^{(s)}$ as $s\to \infty$, when the set $F^{(s)}$ tends to Γ. To cite this article: M. Sango, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Non-localization of eigenfunctions for Sturm–Liouville operators and applications

In this article, we investigate a non-localization property of the eigenfunctions of Sturm–Liouville operators $A_a=?{?}_{xx}+a(?)\mathrm{Id}$ with Dirichlet boundary conditions, where $a(?)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(?)$ on a measurable subset ω of $(0,L)$, where $e(?)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all subsets ω having a prescribed measure and all $L^{\infty }$ potential functions $a(?)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.     

The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations

We consider asymptotically autonomous semilinear parabolic equations

$u_t+Au=f(t,u).$

Suppose that $f(t,.)\to f^{\pm }$ as $t\to \pm \infty$, where the semiflows induced by

(*)$u_t+Au=f^{\pm }(u)$

are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation g with $g(t)\to 0$ as $\left|t\right|\to \infty$, every solution of

$u_t+Au=f(t,u)+g(t)$

is a connection between equilibria $e^{\pm }$ of (*) with $m(e^{?})\ge m(e^{+})$. Moreover, if the Morse indices satisfy $m(e^{?})=m(e^{+})$, then u is isolated by linearization.     

Elliptic operators with unbounded diffusion,drift and potential terms

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\mathrm{\Delta }+b|x{|}^{\alpha ?1}\frac{x}{|x|}?\mathrm{?}?c|x{|}^{\beta }$ with domain $D(A_p)=\{u\in W^{2,p}({\mathbb{R}}^N)|Au\in L^p({\mathbb{R}}^N)\}$ generates a strongly continuous analytic semigroup $T(?)$ provided that $\alpha >2,\beta >\alpha ?2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [4] and in [16]. Moreover we show that $T(?)$ is consistent, immediately compact and ultracontractive.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let $G^{\sigma }$ be a weighted oriented graph with skew adjacency matrix $S(G^{\sigma })$. Then $G^{\sigma }$ is usually referred as the weighted oriented graph associated to $S(G^{\sigma })$. Denote by $?(G^{\sigma }\text{;}\lambda )$ the characteristic polynomial of the weighted oriented graph $G^{\sigma }$, which is defined as$?(G^{\sigma }\text{;}\lambda )=\mathit{det}(\lambda I_n-S(G^{\sigma }))=\sum _{i=0}^n{a}_i(G^{\sigma }){\lambda }^{n-i}\text{.}$In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.