For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ, x) ∈ T1 × R+ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ with the following properties: 1. Γ is the closure of the graph of a function x = φ(θ). It attracts Lebesgue-a.e. starting point in T1 × R+. The set {θ : φ(θ) 6= 0} is meager but has full 1-dimensional Lebesgue mea...

Let S be a subset of R with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of...

The advection equation is used to model the transport of material whose mass is conserved. If the vector field governing the advection is divergence-free, then the advection equation reduces to the usual transport equation which is used to model linear diffusion processes such as transport of neutrons, scattering of light and propagation of -rays in a scattering medium [1]. The uncontrolled adv...

The Idea for this article was given by a problem in real analysis. We wanted to determine the one-dimensional Lebesgue-measure of the set f(C)9 where C stands for the classical triadic Cantor set and/is the Cantor-function, which is also known as "devil's staircase." We could see immediately that to determine the above measure we needed to know which dyadic rationals were contained in C. We soo...

The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σadditive measure a ...

Let D be a bounded domain in C with C boundary. Many holomorphic function spaces over D have been introduced in last half century, such as Hardy, Bergman, Besov and Sobolev spaces. Properties (boundary behaviour ect.) of functions in those function spaces have been received a great deal of studies. Readers can see parts of them from the following books [26] [29], [66], [71], [75], [76] and refe...

Let H be a Hilbert space of analytic functions on the open unit disc D such that the operator Mζ of multiplication with the identity function ζ defines a contraction operator. In terms of the reproducing kernel for H we will characterize the largest set ∆(H) ⊆ ∂D such that for each f, g ∈ H, g 6= 0 the meromorphic function f/g has nontangential limits a.e. on ∆(H). We will see that the question...

An innnite system of stochastic diierential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V , and V is shown to be the unique solution of a nonlinear stochastic partial diierential equation (SPDE). Conditions are given under which the weighted empirical measure has an L 2-density with respect ...

Let μ be a measure with compact support, with orthonormal polynomials {pn}, and associated reproducing kernels {Kn}. We show that bulk universality holds in measure in {ξ : μ (ξ) > 0}. More precisely, given ε, r > 0, the linear Lebesgue measure of the set of ξ with μ (ξ) > 0 and for which

The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σadditive measure a ...