Vol 51, No 5 (2016)

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = Δ_E² + q to be separated in the space L² (M) on a complete Riemannian manifold (M,g) with metric g, where Δ_E 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 Δ_E²u ∈ L² (M) and qu ∈ L² (M).

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.

The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter ß for which these operators are bounded on mixed norm spaces L(p, q, ß) on the unit ball in Cⁿ.

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 B_n(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

\(\sum\nolimits_{n = 0}^\infty {{c_n}\sigma \left( {{Z^n}} \right)} ,where\sigma \left( {{Z^n}} \right) = {S_n}\left( z \right) = \frac{{{B_{n + 1}}\left( {z + 1} \right) - {B_{n + 1}}\left( 1 \right)}}{{n + 1}}\)