On one extremal property of Korovkin's means

We point out that$$\inf\limits_{L \in L_n} \sup\limits_{\substack{f \in C_{2\pi}\\f \ne const}} \frac{\max \| f(x) - L(f, x) \|}{\omega^*_2(f, \pi/n + 1)} = \frac{1}{2}$$where $C_{2\pi}$ is the space of periodic continuous functions on real domain, $L_n$ is the set of linear operators that map $C_{2\pi}$ to the set of trigonometric polynomials of order no greater than $n$ ($n = 0,1,\ldots$), $\omega_2(f, t) = \sup\limits_{x, |h| \leqslant t} |f(x-h) - 2f(x) + f(x+h)|$, $\omega^*_2(f, t)$ is the concave hull of the function $\omega_2(f, t)$. In this equality, the infimum is attained for Korovkin's means.

Dual-phase flow in two-layer porous media: a radial composite Laplace domain approximation

The main purpose of this work is to present an interpretation method for injectivity test in a two-layer reservoir that can be extended to a multilayer approach, based on new analytical solutions to the well pressure response. The developed formulation uses a radially composite reservoir approach and considers that the water front propagation may be approximated by a piston-like flow displacement. The reservoir is assumed to be laterally infinite and properties such as permeability and porosity may be different in each layer. The solutions were developed in the Laplace domain and then inverted to real domain using the Stehfest Algorithm. The proposed formulation was then validated by comparison with a numerical flow simulator. Results showed a good agreement between the numerical simulator and the analytical model. Also, a sensitivity study was done by comparing the results of different scenarios varying oil viscosities and injection flow rate to assess how these properties affect the pressure and pressure derivative profiles.

ON THE LIMITS OF SOME p-ADIC SCHNEIDER CONTINUED FRACTIONS

In the present paper, we first generalize some convergence results for continued fractions given in real domain and p-adic domain. However, we prove the transcendence of a p-adic number given by it's Schneider continued fractions, such that the sequence of partial quotients is a Thue-Morse sequence.

Piezomagnetic fields associated with a dislocation source in a layered elastic medium

Summary The piezomagnetic effect is defined as a change in magnetisation with applied stress. Changes in the geomagnetic field caused by the piezomagnetic effect, referred to as the piezomagnetic field, have been theoretically estimated and compared by previous studies to interpret observed variations in the geomagnetic field. However, the piezomagnetic field estimated in previous studies may not provide an accurate estimation because they ignored spatial variations in elasticity, leading to only a rough approximation of the properties of Earth's crust. In this paper, a semi-analytical procedure for calculating the piezomagnetic field arising from a point dislocation source embedded in a layered elastic medium is derived. Following a well-established method of the vector surface harmonic expansion, all of the governing equations written in partial differential equations in a real domain, together with the linear constitutive law of the piezomagnetic effect, are converted to a set of ordinary differential equations in a wavenumber domain. Equations in the wavenumber domain are solved analytically, and each component of the piezomagnetic field in the real domain is obtained after applying the Hankel transform. By using the derived procedure, the piezomagnetic and displacement fields due to a finite fault with strike-slip, dip-slip, and tensile-opening mechanisms are estimated for media with layered elasticity structures. Results for a finite fault are obtained by integrating the point source solution over the fault plane. The results of the numerical analysis allow the effect of heterogeneities in rigidity on the piezomagnetic effect to be examined and implications for the findings of previous investigations to be drawn. In cases where the moment-release at the dislocation source is fixed, the effect of the rigidity differences between upper and lower layers on the piezomagnetic field is minor even in the case where the Curie point depth is near the source of dislocation. This result is in contrast to a previous study that assumed the Mogi model and suggested that heterogeneities in the horizontal direction may be of importance when combined with layered rigidity structures. A contrast is seen between the piezomagnetic and displacement fields corresponding to models with layered rigidity structures: the piezomagnetic field is roughly proportional to the moment-release on a source fault, whereas the displacement field is proportional to slip or opening of the fault. Provided that the rigidity of the crust increases with increasing depth, the calculated piezomagnetic field is likely to have been underestimated in many earlier studies, which assumed uniform rigidity and a geodetically inverted size of slip.

On inequalities for $L_p$-norms of fractional derivatives on the real domain

We obtain new inequalities for fractional Marchaud derivatives of functions defined on the whole real domain, in integral metric ($1 \leqslant p < \infty$); for $p = 1$ we establish the sharpness of obtained inequalities.