Bi-Sobolev mappings and Kp-distortions in the plane

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms $f:\mathbb{D}\stackrel{onto}{?}\mathbb{D}$, $f\in W_{\mathrm{loc}}^{1,1}(\mathbb{D};\mathbb{D})$, where $\mathbb{D}?{\mathbb{R}}^2$ is the unit disk, such that there exists a constant ${\mathbb{K}}_p\ge 0$ for which the following distortion inequality

$|Df(x){|}^p\le {\mathbb{K}}_p|J_f(x){|}^{p?1}\text{a.e.}x\in \mathbb{D}$

is satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11]. Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with ${\mathbb{K}}_p={\mathbb{K}}_{p,f}(x)$ not necessarily constant, we identify the homeomorphism f whose p-distortion function ${\mathbb{K}}_{p,f}(x)$ is minimal in $L^1$ norm.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

Double scattering channels for 1D NLS in the energy space and its generalization to higher dimensions

We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.     

Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal–Lacey theorem asserts that the Banach space $C(K)$ of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ${?}_2$. More recently, Ka?kol and Saxon [20] proved that the space $C_p(K)$ endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence $C_p(\beta \mathbb{N})$ lacks infinite-dimensional separable quotient algebras. This motivates the following question: (?) Does$C_p(K)$admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does $C_p(\beta \mathbb{N})$ admit SQ? Our main theorem implies that $C_p(K)$ has SQ for any compact space K containing a copy of $\beta \mathbb{N}$. Consequently, this result reduces problem (?) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of $\beta \mathbb{N}$). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ?), that for some Efimov space K the space $C_p(K)$ has SQ. Some applications of the main result are provided.     

The spherical p-harmonic eigenvalue problem in non-smooth domains

We prove the existence of p-harmonic functions under the form $u(r,\sigma )=r^{?\beta }\omega (\sigma )$ in any cone $C_S$ generated by a spherical domain S and vanishing on $?C_S$. We prove the uniqueness of the exponent β and of the normalized function ω under a Lipschitz condition on S.     

Witt kernels of quadratic forms for purely inseparable multiquadratic extensions in characteristic 2

The aim of this paper is to give a complete answer to the question of hyperbolicity of nonsingular quadratic forms over purely inseparable multiquadratic extensions in characteristic . This completes partial computations of Mammone and Moresi.

     

Automorphisms and holomorphic differentials in characteristic p

Let G be a group of automorphisms of a function field F of genus g_F over an algebraically closed field K. The space $\text{Ω}{}_{\mathrm{F}}$ of holomorphic differentials of F is a g_F^? dimensional K-space. In response to a query of Hecke, Chevalley and Weil (Abh. Math. Sem. Univ. Hamburg, 10 (1934), 358–361) completely determined the structure of $\text{Ω}{}_{\mathrm{F}}$ as representation space for G in the classical case. They carried out the proof for the special case in which F is unramified over the fixed field of G. The case of cyclic ramified extensions had been previously considered by Hurwitz (Math. Ann., 41 (1893), 37–45). Weil (Abh. Math. Sem. Univ. Hamburg, 11 (1935), 110–115) gave a proof in the general case. The treatment in the last two papers is analytical. In characteristic p, the problem is open. If G is cyclic and F is unramified over the fixed field E of G, Tamagawa (Proc. Japan Acad., 27 (1951), 548–551) proved that the representation is the direct sum of one identity representation of degree 1 and g_E ? 1 regular representations. The principal object of this paper is an extension of Tamagawa's result to arbitrary cyclic extensions of p-power degree. The number of times an indecomposable representation of given degree occurs in the representation of G on $\text{Ω}{}_{\mathrm{F}}$ is explicitly determined in terms of g_E and the Witt vector characterizing the extension $\text{F}\text{E}$. The paper also contains a purely algebraic proof of the result of Chevalley and Weil for arbitrary cyclic extensions of degree relatively prime to p. Using character theory, it can be extended to arbitrary groups of order relatively prime to the characteristic.     

The PBW filtration and convex polytopes in type B

We study the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type ${\mathtt{B}}_n$. We prove in various cases, including all multiples of the adjoint representation and all irreducible finite-dimensional representations for ${\mathtt{B}}_3$, that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of examples for favourable modules and graded combinatorial character formulas.     

The admissibility of M11 over number fields

A group G is $\mathbb{Q}$-admissible if there exists a G-crossed product division algebra over $\mathbb{Q}$. The $\mathbb{Q}$-admissibility conjecture asserts that every group with metacyclic Sylow subgroups is $\mathbb{Q}$-admissible. We prove that the Mathieu group $M_{11}$ is $\mathbb{Q}$-admissible, in contrast to any other sporadic group.     

Extensions to the boundary of Riemann maps on varying domains in the complex plane

We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.     

An Arakelov inequality in characteristic p and upper bound of p-rank zero locus

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus g?1 over characteristic p with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of p-rank zero in a semi-stable family over characteristic p with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. The parallel results for smooth families of Abelian varieties over k with W₂-lifting assumption are also obtained.     

Deformations of Bloch groups and Aomoto dilogarithms in characteristic p

In this paper, we study the Bloch group B₂(F₂[ε]) over the ring of dual numbers of the algebraic closure of the field with p elements, for a prime p?5. We show that a slight modification of Kontsevich?s -logarithm defines a function on B₂(F₂[ε]). Using this function and the characteristic p version of the additive dilogarithm function that we previously defined, we determine the structure of the infinitesimal part of B₂(F₂[ε]) completely. This enables us to define invariants on the group of deformations of Aomoto dilogarithms and determine its structure. This final result might be viewed as the analog of Hilbert?s third problem in characteristic p.     

Decay estimates for oscillatory integrals with polynomial phase in terms of p and p or in terms of p and p

We obtain estimates for certain oscillatory integrals with polynomial (degree n) phase, p(t). These estimates are stated in terms of differences between the roots, real or complex, of p⁽ⁿ⁻³⁾(t) and p⁽ⁿ⁻²⁾(t) or between p⁽ⁿ⁻²⁾(t) and p⁽ⁿ⁻¹⁾(t). The sharpness of these results is also explored. This result is a partial generalization of the results found in [J. Math. Anal. Appl. 280 (2003) 424].     

The Soergel category and the redotted Webster algebra

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.     

Local existence,lower mass bounds,and a new continuation criterion for the Landau equation

We consider the spatially inhomogeneous Landau equation with soft potentials. First, we establish the short-time existence of solutions, assuming the initial data has sufficient decay in the velocity variable and regularity (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass throughout the domain. The resulting lower bounds are sub-Gaussian, which we show is optimal. The proof of mass-spreading is based on a stochastic process, and makes essential use of nonlocality. By combining this theorem with prior results, we derive two important applications: $C^{\infty }$-smoothing, even for initial data with vacuum regions, and a continuation criterion (the solution can be extended as long as the mass and energy densities stay bounded from above). This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.     

The enhanced Sanov theorem and propagation of chaos

We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.     

On the g-circulant solutions to the matrix equation Am = λJ, II

Let g and n be positive integers and let $\text{k =}\text{n(g, n)}\text{(g}{}^{\mathrm{m}}\text{, n)}$. If θ(x) is a multiple of Σ_{i = 0}^{k ? 1}xⁱ, then the g-circulant whose Hall polynomial is equal to θ(x) satisfies the matrix equation in the title. If the g-circulant whose Hall polynomial is equal to Σ_{i = 0}^{h ? 1}xⁱ satisfies the matrix equation in the title, then h is a multiple of k.     

Control,observation and polynomial decay for a coupled heat-wave system

This Note is devoted to study the control, observation and polynomial decay of a linearized 1-d model for fluid–structure interaction, where a wave and a heat equation evolve in two bounded intervals, with natural transmission conditions at the point of interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. The controllability and observability of the system through the wave component are derived from sidewise energy estimate and Carleman inequalities. As for the control and observation through the heat component, we need to develop first a careful spectral high frequency analysis for the underlying semigroup, which yields a new Ingahm-type inequality. It is shown that the controllable/observable subspace for both cases are quite different. Also, we obtain a sharp polynomial decay rate for the energy of smooth solutions. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).