Homogeneous Domains Research Articles

Numerical Analysis of the Effect of Subgrid Variability in a Physically Based Hydrological Model on Runoff, Soil Moisture, and Slope Stability

AbstractIn coarse resolution hydrological modeling, we face the problem of subgrid variability, the effects of which are difficult to express and are often hidden in the parameterization and calibration. We present a numerical experiment with the physically based hydrological model ParFlow‐CLM with which we quantify the effect of subgrid heterogeneities in headwater catchments within the cell size typically used for regional hydrological applications. We simulate homogeneous domains and domains with subgrid heterogeneities in topography or soil thickness for two climates and soil types. The presence of side slope is the main error source, leading to large underestimation of runoff, and marginally also of evapotranspiration. The spatial distribution of soil saturation in the presence of subgrid variability in topography also leads to underestimation of landslide risk. Soil thickness is the second influential subgrid property, affecting soil moisture distribution and surface runoff formation. Results are consistent for the climates and the soil types considered. The topographic wetness index approach is tested as a way to downscale soil moisture simulations within the domain. Although this method is successful in reproducing some spatial variability and patterns, it fails when the coarse grid mean soil saturation is inaccurate or subgrid topography does not represent subsurface flow paths accurately. We conclude that ignoring subgrid variability in topography and soil thickness in coarse‐scale hydrological models may lead locally to underestimation of runoff and slope instability. Users of such models should be aware of these biases and consider ways to include subgrid effects in coarse‐scale hydrological predictions.

Homogeneous almost complex manifolds and their compact quotients

This paper investigates the (non)existence of compact quotients of the homogeneous almost-complex strongly-pseudoconvex manifolds discovered and classified by Gaussier–Sukhov [Wong–Rosay theorem in almost complex manifolds, http:www.arXiv.org:math.CV/0307335 ; On the geometry of model almost complex manifolds with boundary, Math. Z. 254(3) (2006) 567–589] and Lee [Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J. 54(1) (2006) 179–205; Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math. 623 (2008) 123–160].

Elemental and Nd isotopic compositions of zoned titanite in mafic microgranular enclaves of the Early Cretaceous Sanguliu granitic pluton in the North China Craton: Insights into magma mixing process

Widespread Mesozoic granitic plutons in the North China Craton (NCC) are products of remelting of the crust associated with large-scale Mesozoic destruction of the NCC. Mafic microgranular enclaves (MMEs) are common in these plutons and are attributed to magma mixing. However, the involved magmatic processes of the mixing are poorly known. In this paper, we reported complex zoning patterns, in situ U–Pb ages, and elemental and Nd isotopic compositions of titanite grains in the MMEs and host monzogranite of the Sanguliu pluton in the NCC. Titanite grains in the MMEs and monzogranite have similar U–Pb ages of ca.130 Ma, and display different internal textures and compositions. They can thus be divided into four types, i.e., types 1 and 2 from the host monzogranite, and types 3 and 4 from the MMEs. Type 1 is euhedral in shape, whereas type 2 has a euhedral inner domain and a thin deuteric rim with a sharp contact between them. Types 3 and 4 are angular grains. Type 3 shows a typical core-mantle-rim texture with a resorbed core, a mantle and an irregular rim, whereas type 4 contains a homogeneous inner domain and a discontinuous rim. Most titanite grains of the four types display oscillatory, fir-tree and sector zoning and have crystallization temperatures ranging from 680 to 750 °C based on the Zr-in-titanite thermometry, indicating a magmatic origin. The core of type 3 titanite has the highest εNd(t) of −11.3 to −12.5 among others, close to that of the mafic dykes (εNd(t) = −7.8 to −11.6) that intrude the Sanguliu pluton. The core can be interpreted as crystallized from the mixed magmas with more mafic components that were derived from metasomatized subcontinental lithospheric mantle (SCLM) beneath the NCC. The mantle of type 3 and the inner domain of type 4 titanite have εNd(t) (−13.0 to −15.3) nearly identical to that of the MMEs (−14.3 to −15.4), but distinctly higher than that of types 1 and 2 titanite (−15.9 to −18.3). The rims of types 3 and 4 titanite have εNd(t) (−16.1 to −18.3), identical to that of type 1 and the inner domain of type 2 titanite, however, they have REE concentrations and Th/U and Nb/Ta similar to the rim of type 2 titanite, clearly indicative of crystallization from evolved, hydrated, granitic magmas. A three-stage growth model is thus proposed to explain the core-mantle-rim texture of the titanite in the MMEs by a magma mixing process. The core likely crystallized from the mixed magma with more mafic components. It was then partially or totally resorbed by mixed, dioritic magma due to chemical disequilibrium, which was followed by the re-precipitation from the dioritic magma, forming the mantle. The rim is an overgrowth from the evolved, hydrated, granitic magma. This study demonstrates that the complex zoning patterns and compositions of titanite in the MMEs can be used to investigate magmatic processes including magma mixing.

有界等質領域上のholomorphic multiplier 表現について

有界等質領域上のholomorphic multiplier 表現について

Strain relaxation dynamics of multiferroic orthorhombic manganites

Resonant ultrasound spectroscopy has been used to characterise strain coupling and relaxation behavior associated with magnetic/magnetoelectric phase transitions in GdMnO3, TbMnO3 and TbMn0.98Fe0.02O3 through their influence on elastic/anelastic properties. Acoustic attenuation ahead of the paramagnetic to colinear-sinusoidal incommensurate antiferromagnetic transition at ∼41 K correlates with anomalies in dielectric properties and is interpreted in terms of Debye-like freezing processes. A loss peak at ∼150 K is related to a steep increase in electrical conductivity with a polaron mechanism. The activation energy, E a, of ≳0.04 eV from a loss peak at ∼80 K is consistent with the existence of a well-defined temperature interval in which the paramagnetic structure is stabilised by local, dynamic correlations of electric and magnetic polarisation that couple with strain and have relaxation times in the vicinity of ∼10−6 s. Comparison with previously published data for Sm0.6Y0.4MnO3 confirms that this pattern may be typical for multiferroic orthorhombic RMnO3 perovskites (R = Gd, Tb, Dy). A frequency-dependent loss peak near 10 K observed for TbMnO3 and TbMn0.98Fe0.02O3, but not for GdMnO3, yielded E a ⩾ ∼0.002 eV and is interpreted as freezing of some magnetoelastic component of the cycloid structure. Small anomalies in elastic properties associated with the incommensurate and cycloidal magnetic transitions confirm results from thermal expansion data that the magnetic order parameters have weak but significant coupling with strain. Even at strain magnitudes of ∼0.1–1‰, polaron-like strain effects are clearly important in defining the development and evolution of magnetoelectric properties in these materials. Strains associated with the cubic–orthorhombic transition due to the combined Jahn–Teller/octahedral tilting transition in the vicinity of 1500 K are 2–3 orders of magnitude greater. It is inevitable that ferroelastic twin walls due to this transition would have significantly different magnetoelectric properties from homogeneous domains due to magnetoelastic coupling with steep strain gradients.

Rigid polyurethane foams reinforced with ultrafine powder from automotive shredder residue

AbstractWith the booming of the abandoned cars, automobile shredding residual (ASR) increase sharply in recent years. ASRs contain complex components that are difficult to separate and recycle, causing serious environmental impacts. Herein, we report a new strategy for ASR recycling to prepare high value‐added rigid polyurethane foams (RPUFs) with excellent mechanical and thermal properties. Ultrafine ASR powder with homogeneous domain size and activated functional groups, added to the precursor of RPUFs as reinforcing fillers, was fabricated through solid‐state shear milling (S3M) technology. As a nucleating agent, ASR particles can participate in the foaming procedure of RPUFs, which decreased the cell size to ~0.2 mm. Owing to the strong interaction between the ASR and polyurethane matrix, the thermal degradation temperature increased to 246.7°C, 42.4°C higher than the neat one. Inspiringly, the ASR particles can act as reinforcing elements in the cell walls, which can not only increase the compressive strength and modulus to 1.79 and 0.1 MPa respectively, 81% and 71% higher than the neat sample, but also enhance the long‐term fatigue resistance of the foam. This strategy achieves the recycling and utilization of ASR, and expands the application scope of RPUFs as well.

Compactness of the [formula omitted]-Neumann problem on domains with bounded intrinsic geometry

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the ∂¯-Neumann operator on (0,q)-forms is equivalent to the boundary not containing any q-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.

Virus transport from drywells under constant head conditions: A modeling study

Many arid and semi-arid regions of the world face challenges in maintaining the water quantity and quality needs of growing populations. A drywell is an engineered vadose zone infiltration device widely used for stormwater capture and managed aquifer recharge. To our knowledge, no prior studies have quantitatively examined virus transport from a drywell, especially in the presence of subsurface heterogeneity. Axisymmetric numerical experiments were conducted to systematically study virus fate from a drywell for various virus removal and subsurface heterogeneity scenarios under steady-state flow conditions from a constant head reservoir. Subsurface domains were homogeneous or had stochastic heterogeneity with selected standard deviation (σ) of lognormal distribution in saturated hydraulic conductivity and horizontal (X) and vertical (Z) correlation lengths. Low levels of virus concentration tailing can occur even at a separation distance of 22 m from the bottom of the drywell, and 6-log10 virus removal was not achieved when a small detachment rate (kd1=1 × 10⁻⁵ min⁻¹) is present in a homogeneous domain. Improved virus removal was achieved at a depth of 22 m in the presence of horizontal lenses (e.g., X=10 m, Z=0.1 m, σ=1) that enhanced the lateral movement and distribution of the virus. In contrast, faster downward movement of the virus with an early arrival time at a depth of 22 m occurred when considering a vertical correlation in permeability (X=1 m, Z=2 m, σ=1). Therefore, the general assumption of a 1.5–12 m separation distance to protect water quality may not be adequate in some instances, and site-specific microbial risk assessment is essential to minimize risk. Microbial water quality can potentially be improved by using an in situ soil treatment with iron oxides to increase irreversible attachment and solid-phase inactivation.

Correction to: Infinite Dimensional Holomorphic Homogeneous Regular Domains

We correct an error in Lemma 2.3 in our paper with the above title, published in J. Geom. Anal. 30: 223–247.

Application of Finite Element Residual Scheme to Steady Unconfined Flow Subject to Rainfall.(Dept.C)

A two- dimensional finite element method is used to study the problem of steady-state unconfined seepage due to rainfall through porous media. The method is based on the finite element residnal scheme. Homogeneous and nonhomogeneous soil domains are considered and solved numerically. Numerical results are compared with that obtained from the analytical approximations to be in the same order of magnitude. To facilitate its p ractical use, the scheme has been programmed for computer solution using FORTRAN-IV language.

Feature extraction via prototype margin distance maximizing criterion for subspace learning

Locality sensitive discriminant analysis (LSDA) is an effective feature extraction algorithm, which captures the local structure and label information to build margin for classification. However, the margin of LSDA is easily affected by outliers. To overcome the limitations of LSDA, we propose a new margin-based feature extraction algorithm called prototype margin distance maximizing criterion (PMDMC). Specifically, for each class, we first find its either heterogeneous nearest marginal samples to build the between-class marginal patch. While one side does not characterize the margin. Therefore, for between-class marginal patch, we turn back to find its corresponding within-class marginal patch in the original class. With the between-class marginal patch and within-class marginal patch, we build a novel margin. Finally, we use the Fisher-like criterion to find the optimal feature subspace. In this feature subspace, the margins between different classes are enlarged by maximizing the distance between two heterogeneous marginal patches, and simultaneously minimizing the distance between the prototypes (class means) and their corresponding within-class marginal patches. The obtained margins are more discriminative than before. What's more, under this margin design, the outliers which are usually lying in margins can be pulled close into their respective homogeneous domains. From the comparative experiments performed on the ORL, Yale, and AR face databases, we prove the effectiveness of the proposed PMDMC.

Simulation for non-homogeneous transport equation by Nyström method

In this work we solve numerically the one-dimensional transport equation with semi-reflective boundary conditions and non-homogeneous domain. The proposed methodology consists of applying the Nyström method in order to discretize the integral formulation of this problem which is an equation involving weakly singular integral operators. For this purpose, analytical and computational techniques were applied to deal with the singularities. The Nyström method is an integral method which approximates the integral operator by a numerical quadrature and turns the integral equation into a finite dimensional linear system. This formulation allows us to use any function to describe both scattering cross section and total cross section. The algorithm is implemented in C language with the use of routines of GNU scientific library and computational techniques for code optimization. The scalar flux was calculated for two numerical quadrature, namely Gauss-Legendre quadrature and Boole's rule. The numerical results were determined for transport problem with homogeneous and non-homogeneous domains. In order to validate the proposed method-ology, our numerical results were compared with those from the literature and presented with several correct significant digits.

Non-homogenous nanofluid model for 3D convective flow in enclosures filled with hydrodynamically and thermally heterogeneous porous media

The non-homogenous two-phase Buongiorno's nanofluids model is applied to investigate the convection process within cubic enclosures filled with homogenous/heterogeneous porous medium. The homogenous/heterogeneous properties are considered in the overall permeability and thermal conductivity of the porous elements. Different cases upon the type of the medium are considered, namely, homogeneous porous domain, heterogeneous porous domain in all directions, heterogeneous cavity with stratification in X-Y plane, heterogeneous cavity with stratification in X-Z plane and heterogeneous cavity with stratification in Y-Z plane. The worked mixture is Al2O3ewater nanofluid and the Brownian motion parameters, the thermophores parameter and the dynamic viscosity are variables. The SIMPLE algorithm together with the finite volume method are extended to the three dimensional case and applied to treat the governing equations. It is remarkable that the heterogeneous case in all directions (η1=η2=η3=1.5) gives the largest values of the average Nusselt coefficient while the homogenous case has the lowest rate of the heat transfer. In all the considered porous cases, the growing in the Rayleigh and Darcy numbers enhances the average Nusselt number.

Social dilemmas in off-lattice populations

Exploring the possible consequences of spatial reciprocity on the evolution of cooperation is an intensively studied research avenue. Related works assumed a certain interaction graph of competing players and studied how particular topologies may influence the dynamical behavior. In this paper we apply a numerically more demanding off-lattice population approach which could be potentially relevant especially in microbiological environments. As expected, results are conceptually similar to those which were obtained for lattice-type interaction graphs, but some spectacular differences can also be revealed. On one hand, in off-lattice populations spatial reciprocity may work more efficiently than for a lattice-based system. On the other hand, competing strategies may separate from each other in the continuous space concept, which gives a chance for cooperators to survive even at relatively high temptation values. Furthermore, the lack of strict neighborhood results in soft borders between competing patches which jeopardizes the long term stability of homogeneous domains. We survey the major social dilemma games based on pair interactions of players and reveal all analogies and differences compared to on-lattice simulations.

Hybrid finite volume-finite element methods for hydro-mechanical analysis in highly heterogeneous porous media

In this study, two classes of advanced finite volume schemes, including Multi-Point Flux Approximation (MPFA) and Dual Discrete Finite Volume (DDFV) method, have been employed in conjunction with the finite element (FE) geomechanics simulator to model the coupled fluid-solid system. Fully saturated porous media with poroelastic behavior, random field permeability and elastic modulus are considered as parameters. The performance of the proposed hydro-mechanical models, including MPFA O-FEM and DDFV-FEM, is examined through different test cases. First, the models are validated and compared with the closed-form solutions in the homogeneous domain. Second, the methods' stability and convergence properties are analyzed in highly heterogeneous porous media with different permeability and elastic modulus realizations. Finally, implementing the models in a large-size realistic problem of a synthetic reservoir accomplishes this comprehensive study. Although both methods agree well with the analytical solutions in homogeneous porous media, the DDFV-FEM yields higher accuracy. In the cases where the random field of permeability varies over several orders of magnitude (4th and more), the MPFA O-FEM suffers from significant oscillations and even instability. The analysis illustrates that MPFA O-FEM exhibits oscillation in the cases with high anisotropy ratio, while the DDFV-FEM demonstrates superior performance for handling anisotropy and heterogeneity.

Quantitative WDS compositional mapping using the electron microprobe

Abstract While much progress has been made in electron-probe microanalysis (EPMA) to improve the accuracy of point analysis, the same level of attention has not always been applied to the quantification of wavelength-dispersive spectrometry (WDS) X-ray intensity maps at the individual pixel level. We demonstrate that the same level of rigor applied in traditional point analysis can also be applied to the quantification of pixels in X-ray intensity maps, along with additional acquisition and quantitative processing procedures to further improve accuracy, precision, and mapping throughput. Accordingly, X-ray map quantification should include pixel-level corrections for WDS detector deadtime, corrections for changes in beam current (beam drift), changes in standard intensities (standard drift), high-accuracy removal of background intensities, quantitative matrix corrections, quantitative correction of spectral interferences, and, if required, time-dependent corrections (for beam and/or contamination sensitive materials). The purpose of quantification at the pixel level is to eliminate misinterpretation of intensity artifacts, inherent in raw X-ray intensity signals, that distort the apparent abundance of an element. Major and minor element X-ray signals can contain significant artifacts due to absorption and fluorescence effects. Trace element X-ray signals can contain significant artifacts where phases with different average atomic numbers produce different X-ray continuum (bremsstrahlung) intensities, or where a spectral interference, even an apparently minor one, can produce a false-positive intensity signal. The methods we propose for rigorous pixel quantification require calibration of X-ray intensities on the instrument using standard reference materials, as we already do for point analysis that is then used to quantify multiple X-ray maps, and thus the relative time overhead associated with such pixel-by-pixel quantification is small. Moreover, the absolute time overhead associated with this method is usually less than that required for quantification using manual calibration curve methods while resulting in significantly better accuracy. Applications to geological, synthetic, or engineering materials are numerous as quantitative maps not only show compositional 2D variation of fine-grained or finely zoned structures but also provide very accurate quantitative analysis, with precision approaching that of a single point analysis, when multiple-pixel averaging in compositionally homogeneous domains is utilized.

Exploration of processes governing microbial reductive dechlorination in a heterogeneous aquifer flow cell

Although microbial reductive dechlorination (MRD) has proven to be an effective approach for in situ treatment of chlorinated ethenes, field implementation of this technology is complicated by many factors, including subsurface heterogeneity, electron donor availability, and distribution of microbial populations. This work presents a coupled experimental and mathematical modeling study designed to explore the influence of heterogeneity on MRD and to assess the suitability of microcosm-derived rate parameters for modeling complex heterogeneous systems. A Monod-based model is applied to simulate a bioremediation experiment conducted in a laboratory-scale aquifer cell packed with aquifer material from the Commerce Street Superfund site in Williston, VT. Results reveal that (uncalibrated) model application of microcosm-derived dechlorination and microbial growth rates for transformation of trichloroethene (TCE), cis-1,2-dichloroethene (cis-DCE), and vinyl chloride (VC) reproduced observed aquifer cell concentration levels and trends. Mean relative errors between predicted and measured effluent concentrations were quantified as 6.7%, 27.0%, 41.5%, 32.0% and 21.6% over time for TCE, cis-DCE, VC, ethene and total volatile fatty acids (fermentable electron donor substrate and carbon source), respectively. The time-averaged extent of MRD (i.e., ethene formation) was well-predicted (4% underprediction), with modeled MRD exhibiting increased deviation from measured values under electron donor limiting conditions (maximum discrepancy of 14%). In contrast, simulations employing a homogeneous (uniform flow) domain resulted in underprediction of MRD extent by an average of 13%, with a maximum discrepancy of 45%. Model sensitivity analysis suggested that trace amounts of natural dissolved organic carbon served as an important fermentable substrate, providing up to 69% of the reducing equivalents consumed for MRD under donor-limiting conditions. Aquifer cell port concentration data and model simulations revealed that ethene formation varied spatially within the domain and was associated with regions of longer residence times. These results demonstrate the strong influence of subsurface heterogeneity on the accuracy of MRD predictions, and highlight the importance of subsurface characterization and the incorporation of flow field uncertainty in model applications for successful design and assessment of in situ bioremediation.

On the attraction power of critical state in granular materials

The aim of this paper is to offer a fresh perspective on the classic concept of critical state (CS) in granular materials by suggesting that CS can be defined through the use of a single proportional strain test. In classic conventional testing, CS manifests itself under constant lateral stress and controlled strain in one given direction whenever continuous shearing is applied without change being induced to material volume. However, a comparison between proportional strain tests and biaxial tests simulated with DEM has clearly shown that the CS line (CSL) characterized by stresses, void ratio and fabric indexes can act as an attractor. The mechanical responses and fabric metrics evolve along dilatant proportional strain loading paths according to similar values after the strain level has become large enough to wipe out the material memory in the homogeneous domains considered in this analysis, i.e., the shear band area in dense samples and the whole area in loose samples. This suggests that the micro-structure of a granular material subjected to any dilatant proportional strain loading paths evolves while preserving its ability to withstand shearing without volume change at any time. Therefore, the CS concept can be generalized to a wide class of loading paths which shows that CS acts as a general attractor irrespective of the loading path considered.

Holomorphic function spaces on homogeneous Siegel domains

We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results from the more particular cases of the upper half-plane, Siegel domains of tube type over irreducible cones, or symmetric, irreducible Siegel domains of type II.

О СОВМЕСТНОМ ПРИМЕНЕНИИ КОЛЛОКАЦИОННОГО МЕТОДА ГРАНИЧНЫХ ЭЛЕМЕНТОВ И МЕТОДА ФУРЬЕ ДЛЯ РЕШЕНИЯ ЗАДАЧ ТЕПЛОПРОВОДНОСТИ В КОНЕЧНЫХ ЦИЛИНДРАХ С ГЛАДКИМИ НАПРАВЛЯЮЩИМИ

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .