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##### Riemann problem in the weighted spaces L 1(ρ)

In the unit disc bounded by the circle T = {z, |z| = 1} we consider the Riemann boundary value problem in the weighted space L1(ρ), where $$\rho \left( t \right) = {\prod\nolimits_{k = 1}^m {\left| {t - {t_k}} \right|} ^{{\alpha _k}}}$$ , tk ∈ T, k = 1, 2,..., m, and αk, k = 1, 2,..., m are real numbers. The question of interest is to determine an analytic outside the circle T function ϕ(z), ϕ(∞) = 0 to satisfy $${\lim _{r \to 1 - 0}}||{\Phi ^ + }\left( {rt} \right) - a\left( t \right){\Phi ^ - }\left( {{r^{ - 1}}t} \right) - f\left( t \right)|{|_{{L^1}\left( {{\rho _r}} \right)}} = 0$$ , where f ∈ L1(ρ), a(t) ∈ Cδ(T), δ>0, and ρr are some continuations of function ρ inside the circle. The normal solvability of this problem is established.

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•  •  ##### Analytic summability of real and complex functions

Gamma-type functions satisfying the functional equation f(x+1) = g(x)f(x) and limit summability of real and complex functions were introduced by Webster (1997) and Hooshmand (2001). However, some important special functions are not limit summable, and so other types of such summability are needed. In this paper, by using Bernoulli numbers and polynomials Bn(z), we define the notions of analytic summability and analytic summand function of complex or real functions, and prove several criteria for analytic summability of holomorphic functions on an open domain D. As consequences of our results, we give some criteria for absolute convergence of the functional series $$\sum\nolimits_{n = 0}^\infty {{c_n}\sigma \left( {{Z^n}} \right)} ,where\sigma \left( {{Z^n}} \right) = {S_n}\left( z \right) = \frac{{{B_{n + 1}}\left( {z + 1} \right) - {B_{n + 1}}\left( 1 \right)}}{{n + 1}}$$ . Finally, we state some open problems.

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•  •  ##### On the robustness to small trends of parameter estimation for continuous-time stationary models with memory

The paper deals with a question of robustness of inferences, carried out on a continuoustime stationary process contaminated by a small trend, to this departure from stationarity.We show that a smooth periodogram approach to parameter estimation is highly robust to the presence of a small trend in themodel. The obtained result is a continuous version of that of Hede and Dai (Journal of Time Series Analysis, 17, 141-150, 1996) for discrete time processes. Open Access
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•  •  ##### Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = ΔE2 + q to be separated in the space L2 (M) on a complete Riemannian manifold (M,g) with metric g, where ΔE is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L2 (M) if for all u ∈ L2 (M) such that Au ∈ L2 (M) we have ΔE2u ∈ L2 (M) and qu ∈ L2 (M).

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