Uniform approximation by universal series on arbitrary sets

By considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

Estimates for the green function and existence of positive solutions for higher-order elliptic equations

We establish a3G-theorem for the iterated Green function of(−∆)pm, (p≥1,m≥1), on the unit ballBofℝn(n≥1)with boundary conditions(∂/∂ν)j(−∆)kmu=0on∂B, for0≤k≤p−1and0≤j≤m−1. We exploit this result to study a class of potentials𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems(−∆)pmu=h(‧,u), inD(in the sense of distributions),lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for0≤k≤p−1, whereD=BorB\{0}andhis a Borel measurable function onD×(0,∞)satisfying some appropriate conditions related to𝒥m,n(p).

CANONICAL BEHAVIOR OF BOREL FUNCTIONS ON SUPERPERFECT RECTANGLES

We describe a list of canonical functions from (ωω)2 to ℝ such that every Borel measurable function from (ωω)2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.

On planar sets with prescribed packing dimensions of line sections

We prove that for an arbitrary Borel measurable function f on the space of all planar lines there exists a set which intersects almost every line [lscr ] in a set of packing dimension f([lscr ]).

The identifiability of mixtures of distributions

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

The identifiability of mixtures of distributions

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).