We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, ...

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

The evolution of probability distribution functions (PDFs) of continuous density, velocity and velocity derivatives ( deformation tensor) fields in the theory of cosmological gravitational instability are considered. We show that in the Newtonian theory the dynamical equations cannot be reduced to the closed set of Lagrangian equations. Since continuous fields from galaxy surveys need sufficien...

Local network community detection is the task of finding a single community of nodes concentrated around few given seed nodes in a localized way. Conductance is a popular objective function used in many algorithms for local community detection. This paper studies a continuous relaxation of conductance. We show that continuous optimization of this objective still leads to discrete communities. W...

On the half line [0,∞) we study first order differential operators of the form B 1 i d dx +Q(x), where B := ( B1 0 0 −B2 ) , B1, B2 ∈ M(n,C) are self–adjoint positive definite matrices and Q : R+ → M(2n,C), R+ := [0,∞), is a continuous self–adjoint off–diagonal matrix function. We determine the self–adjoint boundary conditions for these operators. We prove that for each such boundary value prob...

On the half line [0,∞) we study first order differential operators of the form B 1 i d dx +Q(x), where B := ( B1 0 0 −B2 ) , B1, B2 ∈ M(n,C) are self–adjoint positive definite matrices and Q : R+ → M(2n,C), R+ := [0,∞), is a continuous self–adjoint off–diagonal matrix function. We determine the self–adjoint boundary conditions for these operators. We prove that for each such boundary value prob...

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix Σ of an elliptically symmetric t distribution on R with degrees of freedom ν > 1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U . Her...

and Applied Analysis 3 We now define two maps S1 and S2 : Ω → C t0,∞ , R as follows: S1x t ⎧⎨ ⎩ a t x t − τ , t ≥ t1, S1x t1 , t0 ≤ t ≤ t1, S2x t ⎪⎨ ⎪⎩ − 1 n − 1 ! ∫∞ t s − t n−1p s f x s − σ ds, t ≥ t1, S2x t1 v t − v t1 , t0 ≤ t ≤ t1. 2.5 We will show that for any x, y ∈ Ω we have S1x S2y ∈ Ω. For every x, y ∈ Ω and t ≥ t1 we obtain S1x t ( S2y ) t ≤ a t v t − τ − 1 n − 1 ! ∫∞ t s − t n−1p s ...

Abstract. Let (Ω,B, P ) be a probability space, A ⊂ B a sub-σ-field, and μ a regular conditional distribution for P given A. Necessary and sufficient conditions for μ(ω)(A) to be 0-1, for all A ∈ A and ω ∈ A0, where A0 ∈ A and P (A0) = 1, are given. Such conditions apply, in particular, when A is a tail sub-σ-field. Let H(ω) denote the A-atom including the point ω ∈ Ω. Necessary and sufficient ...