Analytical Continuation within the Freundlich Adsorption Model. Comment on Edet, U.A.; Ifelebuegu, A.O. Kinetics, Isotherms, and Thermodynamic Modeling of the Adsorption of Phosphates from Model Wastewater Using Recycled Brick Waste. Processes 2020, 8, 665

Using experimental data for the adsorption of phosphates out of wastewater on waste recycled bricks, published independently in MDPI Processes before (2020), this message re-visits the mathematical theory of the Freundlich adsorption model. It demonstrates how experimental data are to be deeper treated to model the saturation regime and to bridge a chasm between those areas where the data fit the Freundlich power function and where a saturation of surface adsorption centers occurs.

Correction of thermographic images based on the minimization method of Tikhonov functional

The paper considers the method of correction of thermographic images (thermograms) obtained by recording in the infrared range of radiation from the surface of the object under study using a thermal imager. A thermogram with a certain degree of reliability transmits an image of the heat-generating structure inside the body. In this paper, the mathematical correction of images on a thermogram is performed based on an analytical continuation of the stationary temperature distribution as a harmonic function from the surface of the object under study towards the heat sources. The continuation is carried out by solving an ill-posed mixed problem for the Laplace equation in a cylindrical region of rectangular cross-section. To construct a stable solution to the problem, the principle of the minimum of the Tikhonov smoothing functional we used.

Possibilities of trend analysis in the preprocessing of geophysical fields on a model example

A model example shows the negative consequences of the trend analysis for the solution of the problem of interpretation of geophysical data. As an alternative, an analytical downward continuation of the measurement data from the earth's surface is proposed. Keywords: preprocessing, modeling, computational procedure, polynomial trend, principal component analysis, upward recalculation of the field, analytical continuation.

Применение модифицированного метода дискретных источников и метода диаграммных уравнений к решению задачи дифракции волн на теле с шероховатой границей

A two-dimensional problem of diffraction on a cylindrical body with a rough boundary is considered. In this work, two aspects of the problem of diffraction on the body with irregular boundary are investigated. First, the problem of diffraction by bodies with random perturbations of the boundary is considered. As an example, diffraction by a rough circular cylinder is considered. The results of calculating the averaged scattering diagram obtained using the modified method of discrete sources are compared with the results obtained using the method of small perturbations. The second goal of this work is to clarify the degree of influence of the small perturbations of the scatterer boundary on the geometry of the set of singularities of the analytical continuation of the diffraction field. As an example, the problem of diffraction by the cylindrical body with a cross-section in the form of a rough multi-leaf, specified in polar and elliptical coordinates, is considered.

Analytical continuation of two-dimensional wave fields

Wave fields obeying the two-dimensional Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing ideal boundary conditions. This is for example the case for the classical canonical problems of diffraction by a half-line or a segment. In the present work, it is shown that such wave fields admit an analytical continuation into the domain of two complex coordinates. The branch sets of such continuation are given and studied in detail. For a generic scattering problem, it is shown that the set of all branches of the multi-valued analytical continuation of the field has a finite basis. Each basis function is expressed explicitly as a Green’s integral along so-called double-eight contours. The finite basis property is important in the context of coordinate equations, introduced and used by the authors previously, as illustrated in this article for the particular case of diffraction by a segment.

Direct calculation of mutual information of distant regions

We consider the (Rényi) mutual information, $$ {I}^{(n)}\left(A,B\right)={S}_A^{(n)}+{S}_B^{(n)}-{S}_{A\cup B}^{(n)} $$ I n A B = S A n + S B n − S A ∪ B n , of distant compact spatial regions A and B in the vacuum state of a free scalar field. The distance r between A and B is much greater than their sizes RA,B. It is known that $$ {I}^{(n)}\left(A,B\right)\sim {C}_{AB}^{(n)}{\left\langle 0\left|\phi (r)\phi (0)0\right|\right\rangle}^2 $$ I n A B ∼ C AB n 0 ϕ r ϕ 0 0 2 . We obtain the direct expression of $$ {C}_{AB}^{(n)} $$ C AB n for arbitrary regions A and B. We perform the analytical continuation of n and obtain the mutual information. The direct expression is useful for the numerical computation. By using the direct expression, we can compute directly I(A, B) without computing SA, SB and SA∪B respectively, so it reduces significantly the amount of computation.