We discuss the finite-fuel, singular stochastic control problem of optimally tracking the standard Brownian motion x+W (·) started at x ∈ IR, by an adapted process ξ(·) = ξ+(·)−ξ−(·) of bounded total variation ξ̌(t) = ξ+(t) + ξ−(t) ≤ y, ∀ 0 ≤ t < ∞, so as to minimize the total expected discounted cost

Every characteristic function φ can be written in the following way: φ(ξ) = 1 h(ξ) + 1 , where h(ξ) = { 1/φ(ξ)− 1 if φ(ξ) 6= 0 ∞ if φ(ξ) = 0. This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form φa(ξ) = a h(ξ)+a , where φ(ξ) is a fixed characteris...

In this article a group-theoretical aspect of the method of dimensional reduction is presented. Then, on the base of symmetry analysis of an anisotropic space geometrical description of dimensional reduction of equations for vector massless field is given. Formula for calculating components of the energymomentum tensor from the variables of the field factor-equations is derived. The covariant e...

For each real number ξ, let λ̂2(ξ) denote the supremum of all real numbers λ such that, for each sufficiently large X , the inequalities |x0| ≤ X , |x0ξ − x1| ≤ X and |x0ξ − x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero, and let ω̂2(ξ) denote the supremum of all real numbers ω such that, for each sufficiently large X , the dual inequalities |x0 + x1ξ + x2ξ| ≤ X, |x1| ≤ X and |x2...

This paper studies the homotopy invariant cat(X, ξ) introduced in [5]. Given a finite cell-complex X, we study the function ξ → cat(X, ξ) where ξ varies in the cohomology space H 1 (X; R). Note that cat(X, ξ) turns into the classical Lusternik-Schnirelmann category cat(X) in the case ξ = 0. Interest in cat(X, ξ) is based on its applications in dynamics where it enters estimates of complexity of...

Pin structures on vector bundles are the natural generalization of Spin structures to the case of nonoriented bundles. Spin(n) is the central Z/2Z extension (or double cover) of SO(n) and Pin−(n) and Pin(n) are two different central extensions of O(n), although they are topologically the same. The obstruction to putting a Spin structure on a bundle ξ (= R → E → B) is w2(ξ) H(B;Z/2Z); for Pin it...

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 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 deal with a Carleson measure type problem for the tent spaces ATpq(α) in unit disc of complex plane. They consist analytic functions Tpq(α) introduced by Coifman, Meyer and Stein. Well known like Bergman arise as special case this family. Let s,t,p,q∈(0,∞) α>0. find necessary sufficient conditions on positive Borel μ order to exist constant C such that∫T(∫Γ(ξ)|f(z)|tdμ(z))s/t|dξ|≤C‖f‖Tpq(α)s...