An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

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 Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

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$\Omega =B_1$.     

p-Laplacian boundary value problems on exterior regions

This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?R^N$U?{\mathbb{R}}^N$ with dimension N?3$N?3$. Existence-uniqueness results when p∈(1,N)$p\in (1,N)$ are provided in a space E^1,p(U)$E^{1,p}(U)$ of functions that contains W^1,p(U)$W^{1,p}(U)$. Functions in E^1,p(U)$E^{1,p}(U)$ are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an L^p$L^p$-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

A converse of Loewner–Heinz inequality and applications to operator means

Let f(t)$f(t)$ be an operator monotone function. Then A?B$A?B$ implies f(A)?f(B)$f(A)?f(B)$, but the converse implication is not true. Let A?B$A?B$ be the geometric mean of A,B?0$A,B?0$. If A?B$A?B$, then B⁻¹?A?I$B^{-1}?A?I$; the converse implication is not true either. We will show that if f(λB+I)⁻¹?f(λA+I)?I$f{(\lambda B+I)}^{-1}?f(\lambda A+I)?I$ for all sufficiently small λ>0$\lambda >0$, then f(λA+I)?f(λB+I)$f(\lambda A+I)?f(\lambda B+I)$ and A?B$A?B$. Moreover, we extend it to multi-variable matrices means.     

Dimension spectra of random subfractals of self-similar fractals

The dimension of a point x   in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}$\{x\}$) is the algorithmic information density of x  . Roughly speaking, this is the least real number dim(x)$\dim (x)$ such that r×dim(x)$r\times \dim (x)$ bits suffice to specify x   on a general-purpose computer with arbitrarily high precision 2^−r$2^{-r}$. The dimension spectrum of a set X   in Euclidean space is the subset of [0,n]$[0,n]$ consisting of the dimensions of all points in X.     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.