A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.     

Reduced limit for semilinear boundary value problems with measure data

We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μ$-\mathrm{\Delta }u+g°u=\mu$ in a smooth bounded domain Ω⊂R^N$\Omega \subset R^N$. Let {_μn}$\{{\mu }_n\}$ and {_νn}$\{{\nu }_n\}$ be sequences of measure in Ω and ∂Ω   respectively. Assume that there exists a solution _un$u_n$ with data (_μn,_νn)$({\mu }_n,{\nu }_n)$, i.e., _un$u_n$ satisfies the equation with μ=_μn$\mu ={\mu }_n$ and has boundary trace _νn${\nu }_n$. Further assume that the sequences of measures converge in a weak sense to μ and ν   respectively while {_un}$\{u_n\}$ converges to u   in L¹(Ω)$L^1(\Omega )$. In general u   is not a solution of the boundary value problem with data (μ,ν)$(\mu ,\nu )$. However there exists a pair of measures (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ such that u   is a solution of the boundary value problem with this data. The pair (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ is called the reduced limit of the sequence {(_μn,_νn)}$\{({\mu }_n,{\nu }_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3].     

On multilinear polynomials in four variables evaluated on matrices

Let K   be an algebraically closed field of characteristic 0 and let _Mn(K)$M_n(K)$, n?3$n?3$, be the matrix ring over K  . We will show that the image of any multilinear polynomial in four variables evaluated on _Mn(K)$M_n(K)$ contains all matrices of trace 0.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Gaussian bounds for higher-order elliptic differential operators with Kato type potentials

Let P(D)$P(D)$ be a nonnegative homogeneous elliptic operator of order 2m   with real constant coefficients on Rⁿ${\mathbb{R}}^n$ and V   be a suitable real measurable function. In this paper, we are mainly devoted to establish the Gaussian upper bound for Schrödinger type semigroup e^−tH$e^{-tH}$ generated by H=P(D)+V$H=P(D)+V$ with Kato type perturbing potential V  , which naturally generalizes the classical result for Schrödinger semigroup e^−t(Δ+V)$e^{-t(\mathrm{\Delta }+V)}$ as V∈_K2(Rⁿ)$V\in K_2({\mathbb{R}}^n)$, the famous Kato potential class. Our proof significantly depends on the analyticity of the free semigroup e^−tP(D)$e^{-tP(D)}$ on L¹(Rⁿ)$L^1({\mathbb{R}}^n)$. As a consequence of the Gaussian upper bound, the L^p$L^p$-spectral independence of H is concluded.     

Relativization makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d)$\mathrm{Res}(d)$, as well as their tree-like versions, Res^?(d)${\mathrm{Res}}^{?}(d)$. The contradictions we use are natural combinatorial principles: the Least number principle  , _LNPn${\mathrm{LNP}}_n$ and an ordered variant thereof, the Induction principle  , _IPn${\mathrm{IP}}_n$.