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We study connections between the Wp1$W^1_p$‐differentiability and the Lp$L_p$‐differentiability of Sobolev functions. We prove that Wp1$W^1_p$‐differentiability implies the Lp$L_p$‐differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the Wp1$W^1_p$‐differentiability of Sobolev functions capp$\operatorname{cap}_p$‐almost everywhere. [ABSTRACT FROM AUTHOR]

Let G be a tree on n vertices and let L = D − A denote the Laplacian matrix on G. The second-smallest eigenvalue λ 2 (G) > 0 , also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that 'behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs. [ABSTRACT FROM AUTHOR]

The combination of aquaculture industry with floating offshore wind turbines has the potential to generate significant economic advantages for both industries. To investigate this potential, the present study focuses on analyzing the heave, and pitch dynamic responses of a Spar-type floating offshore wind turbine that incorporates an aquaculture feeding system. A series of water tank model tests, together with numerical calculations, were conducted using a 1/56 scale model of a 2 MW, displacement 3500 tons, floating Spar-type wind turbine. The feeding system was placed inside the Spar and slightly above the waterline by adjusting the configuration of the total weight. The weight of the feeding system in the experiments is 100 tons, capable of sustaining 300 tons of fish for an entire week, and the realistic applications have been expanded using the numerical calculation. For this reason, the present study serves a good case study for general understanding, because the integration of the feeding system inevitably raises the center of gravity of the structure and potentially affects its overall stability. The experiments revealed no discernible increase in the heave motion. Moreover, the pitch motion theoretically increased, but occasionally decreased in the experiments with the overall inclination angles being less than 1.2 degrees during the experiments. As a result, the present study supports the practice of integrating a Spar-type wind turbine with feeding systems. Future research should continue to comprehensively examine, both experimentally and numerically, the motion responses of the wind turbine and aquaculture facilities with varying configurations. [ABSTRACT FROM AUTHOR]

Abstract.This article extends the potential theory to countable transient jump processes based on the works of Kemeny, Snell, and Knapp (1976) and Syski (1979). First, it presents the construction of excessive measures. Using excessive measures, this article defines elementary concepts in potential theory. Thus most conclusions for Markov chains can be generalized to jump processes. One of the most important results is the variational formulas for the energy of equilibrium potentials, which holds for asymmetric processes. [ABSTRACT FROM AUTHOR]

The ultra-high-speed aerodynamically alleviated marine vehicle (AAMV) is a high-performance vessel that combines a hydrodynamic configuration and an aerodynamic wing to reduce wave-making resistance during the high-speed planing phase. The forces of the AAMV exhibit strong nonlinear and water–air coupling characteristics, resulting in particularly complex motion characteristics. This paper presents a longitudinal and lateral stability model of the AAMV, which considers the effects of aerodynamic alleviation. Additionally, a numerical model of wind and wave turbulence forces is established, which considers viscous correction based on the potential theory. Finally, the effect of wind and wave turbulence forces on the motion stability of the AAMV under regular and irregular waves is analyzed by numerical solution. The simulation results demonstrate the influence of these disturbance forces on the stability of the AAMV under different sea states. The motion parameters of the AAMV exhibit a pronounced response to changes in sea state level. The aerodynamically alleviated effect is enhanced as speed increases, and the influence of winds and waves on the AAMV is greatly weakened, reducing the possibility of instability. During the cruising phase under class V sea state, the pitch, roll, and heave response are 0.210°, 0.0229°, and 0.0734 m, respectively. This effect can effectively improve the motion stability of the AAMV in winds and waves. [ABSTRACT FROM AUTHOR]

We analyze the recently introduced family of preconditioners in [M. M. Rana et al., SIAM J. Sci. Comput., 43 (2021), pp. S475-S495] for the stage equations of implicit Runge--Kutta methods for s-stage methods. We simplify the formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general s-stage method and use these to obtain convergence rate estimates for preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis is based on understanding the inherent matrix structure of these problems and exploiting it to qualitatively predict and explain the main observed features of the GMRES convergence behavior, using tools from approximation and potential theory based on Schwarz--Christoffel maps for curves and close, connected domains in the complex plane. We illustrate our analysis with numerical experiments showing very close correspondence of the estimates and the observed behavior, suggesting the analysis reliably captures the essence of these preconditioners. [ABSTRACT FROM AUTHOR]

Large engineering structures like ships and other floating structures require reliable and complex computations to ensure their structural integrity over their lifetime. This work presents an analysis of the structural integrity of an aged single-bottom tanker during its towing from Croatia to a scrapheap in Turkey. The ship is subjected to wave loading (corresponding to the global response) and towing force (corresponding to the local response). Computations were performed by a general hydro-structure tool that combines a 3D finite element (FE) structural model and a 3D boundary element (BE) model based on the Rankine panel theory, according to the guidelines of the relevant classification society. Time domain simulations were performed to take account of Froude-Krylov nonlinearities for a wave determined within the Equivalent Design Wave (EDW) approach. After the calculation of wave loading and its transfer to the FE model of a ship structure, the stresses were calculated and compared with the allowable values. The local strength assessment of the ship forecastle deck structure was also performed, considering the prescribed towing force as an imposed load. Both the procedure and the used computational tools are general and can be applied to any kind of ship or other floating structure. [ABSTRACT FROM AUTHOR]

Based on a potential theoretical approach, the subsurface stress field is calculated for an elastic half-space which is subject to normal and uniaxial tangential surface tractions that—in the case of elastic decoupling—correspond to rigid normal and tangential translations of a circular surface domain. The stress fields are obtained explicitly and in closed form as the imaginary parts of compact complex-valued expressions. The stress state in the surface and on the central axis are considered in detail. As, within specific approximations that have been discussed at length in the literature, any tangential contact problem with friction can be understood as a certain incremental series of such rigid translations, the solutions presented here can serve as the basis of very fast superposition algorithms for the analysis of subsurface stress fields in general tangential contact problems with friction. This idea is demonstrated by means of the frictional tangential contact between an elastic half-space and a rigid cylindrical flat punch with rounded corners. [ABSTRACT FROM AUTHOR]

A method for computing the Riesz a-capacity, 0 < α ≤ 2, of a general set K ⊂C ℝd is given. The method is based on simulations of isotropic ct-stable motion paths in d-dimensions. The familiar walk-on-spheres method, often utilized for simulating Brownian motion, is modified to a novel walk-in-and-out-of-balls method adapted for modeling the stable path process on the exterior of regions "probed" by this type of generalized random walk. It accounts for the propensity of this class of random walk to jump through boundaries because of the path discontinuity. This method allows for the computationally efficient simulation of hitting locations of stable paths launched from the exterior of probed sets. Reliable methods of computing capacity from these locations are given, along with non-standard confidence intervals. Illustrative calculations are performed for representative types of sets K, where both a and d are varied. [ABSTRACT FROM AUTHOR]

The vertical center of gravity (VCG) has a significant impact on the roll motion response of a surface ship, particularly oil tankers based on the oil level in the tanker after discharging oil at several stations or positional changes, such as changes in the superstructure and deck structure. This study examined the motion response of the Korea very large crude carrier 2 (KVLCC2) at various VCGs, especially roll motion when the VCG changed. The potential theory in the Ansys AQWA program was used as a numerical simulation method to calculate the motion response. On the other hand, the calculations obtained through potential theory overestimated the roll amplitudes during resonance and lacked precision. Therefore, roll damping is a necessary parameter that accounts for the viscosity effect by performing an experimental roll decay. The roll decay test estimated the roll damping coefficients for various VCGs using Froude’s method. The motion response of the ship in regular waves was evaluated for various VCGs using the estimated roll-damping coefficients. In addition, the reliability of the numerical simulation in motion response was verified with those of the experiment method reported elsewhere. The simulation results showed that the responses of the surge, sway, heave, pitch, and yaw motion were not affected by changing the VCG, but the natural frequency and magnitude of the peak value of the roll motion response varied with the VCG.

The Westergaard half-space problem has been solved using the potential theory in this work. It is a classical theme in elasticity theory that seeks to find the displacements and stresses in the half-space caused by known boundary loads. It has important applications in analyzing stresses and displacement fields in soil due to applied points and distributed loads on the boundary caused by structures placed on the soil. It is governed by stress–strain, strain-displacement, and equilibrium equations. Horizontal inextensibility is assumed in developing the problem, simplifying the displacement formulation to a three-dimensional (3D) Laplace equation. The potential theory is applied to find the vertical displacement. Stress–displacement equations obtained from the simultaneous use of the kinematic and stress-strain equations are used to obtain the stress fields. The specific problem of point load at the origin was considered and solved. The equilibrium of internal vertical stresses and the external vertical load is used to find the integration constant. Hence, vertical displacements were found. The stress fields were found from the stress–displacement equations. The expressions for the vertical displacements and stresses were found to be exact within the framework of the theory used, as they satisfied all the governing equations of the problem. However, the solutions become unbounded at the origin due to the singularity of the vertical displacement and stresses. The obtained solutions are identical to previously obtained solutions.

Based on a potential theoretical approach, the subsurface stress field is calculated for an elastic half-space which is subject to normal and uniaxial tangential surface tractions that—in the case of elastic decoupling—correspond to rigid normal and tangential translations of a circular surface domain. The stress fields are obtained explicitly and in closed form as the imaginary parts of compact complex-valued expressions. The stress state in the surface and on the central axis are considered in detail. As, within specific approximations that have been discussed at length in the literature, any tangential contact problem with friction can be understood as a certain incremental series of such rigid translations, the solutions presented here can serve as the basis of very fast superposition algorithms for the analysis of subsurface stress fields in general tangential contact problems with friction. This idea is demonstrated by means of the frictional tangential contact between an elastic half-space and a rigid cylindrical flat punch with rounded corners.

We prove that for recurrent, reversible graphs, the following conditions are equivalent: (a) existence and uniqueness of the potential kernel, (b) existence and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack inequality, and (d) one-endedness of the wired uniform spanning tree. In particular this gives a proof of the anchored (and in fact also elliptic) Harnack inequality on the UIPT. This also complements and strengthens some results of Benjamini et al. (Ann Probab 29(1):1–65, 2001). Furthermore, we make progress towards a conjecture of Aldous and Lyons by proving that these conditions are fulfilled for strictly subdiffusive recurrent unimodular graphs. Finally, we discuss the behaviour of the random walk conditioned to never return to the origin, which is well defined as a consequence of our results. [ABSTRACT FROM AUTHOR]

It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in the d‐dimensional ball, thus generalizing the existing literature. [ABSTRACT FROM AUTHOR]

The classical Fatou theorem identifies bounded harmonic functions on the unit disk with bounded measurable functions on the boundary circle. We extend this theorem to bounded harmonic maps. [ABSTRACT FROM AUTHOR]

The combination of aquaculture industry with floating offshore wind turbines has the potential to generate significant economic advantages for both industries. To investigate this potential, the present study focuses on analyzing the heave, and pitch dynamic responses of a Spar-type floating offshore wind turbine that incorporates an aquaculture feeding system. A series of water tank model tests, together with numerical calculations, were conducted using a 1/56 scale model of a 2 MW, displacement 3500 tons, floating Spar-type wind turbine. The feeding system was placed inside the Spar and slightly above the waterline by adjusting the configuration of the total weight. The weight of the feeding system in the experiments is 100 tons, capable of sustaining 300 tons of fish for an entire week, and the realistic applications have been expanded using the numerical calculation. For this reason, the present study serves a good case study for general understanding, because the integration of the feeding system inevitably raises the center of gravity of the structure and potentially affects its overall stability. The experiments revealed no discernible increase in the heave motion. Moreover, the pitch motion theoretically increased, but occasionally decreased in the experiments with the overall inclination angles being less than 1.2 degrees during the experiments. As a result, the present study supports the practice of integrating a Spar-type wind turbine with feeding systems. Future research should continue to comprehensively examine, both experimentally and numerically, the motion responses of the wind turbine and aquaculture facilities with varying configurations.

The ultra-high-speed aerodynamically alleviated marine vehicle (AAMV) is a high-performance vessel that combines a hydrodynamic configuration and an aerodynamic wing to reduce wave-making resistance during the high-speed planing phase. The forces of the AAMV exhibit strong nonlinear and water–air coupling characteristics, resulting in particularly complex motion characteristics. This paper presents a longitudinal and lateral stability model of the AAMV, which considers the effects of aerodynamic alleviation. Additionally, a numerical model of wind and wave turbulence forces is established, which considers viscous correction based on the potential theory. Finally, the effect of wind and wave turbulence forces on the motion stability of the AAMV under regular and irregular waves is analyzed by numerical solution. The simulation results demonstrate the influence of these disturbance forces on the stability of the AAMV under different sea states. The motion parameters of the AAMV exhibit a pronounced response to changes in sea state level. The aerodynamically alleviated effect is enhanced as speed increases, and the influence of winds and waves on the AAMV is greatly weakened, reducing the possibility of instability. During the cruising phase under class V sea state, the pitch, roll, and heave response are 0.210°, 0.0229°, and 0.0734 m, respectively. This effect can effectively improve the motion stability of the AAMV in winds and waves.

We explain a general mathematical method that allows one to calculate the electrostatic potential created by a uniformly charged rectangular plate with arbitrary length and width at an arbitrary point in space. Exact analytical results for the electrostatic potential due to a uniformly charged finite rectangular plate are shown for special cases in order to illustrate the implementation of the formalism. Results of this nature are very important to various problems in physical sciences, applied mathematics, and potential theory. [ABSTRACT FROM AUTHOR]

The Laplace–Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace–Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard 1/r kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace–Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries. [ABSTRACT FROM AUTHOR]

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs. [ABSTRACT FROM AUTHOR]

The floating photovoltaic (FPV) array, which consists of tens or hundreds of rows of floating photovoltaic systems, exhibits great economic and environmental benefits. An FPV array arranged in the ocean will be subjected to the combined action of wind, waves, and currents during the installation, service, and maintenance processes; however, systematic research in this field is still lacking. As a representative of the FPV array, this study on the hydrodynamic characteristics of a single-row FPV system is a fundamental prerequisite for the safe design of large-scale FPV arrays in the near future. In this paper, based on the potential theory, a numerical model of a single-row FPV system with ten floating platforms is established with ANSYS-AQWA software. Following this, the hydrodynamic coefficients of a single floating platform are calculated and evaluated. After that, the dynamic responses of the FPV system under different load combinations are explored. Finally, the influence mechanism of wave parameters on the hydrodynamic characteristics of the FPV system is discussed in detail, including the wavelength and wave height. The results indicate that, within the scope of this study, the floating platforms have good seakeeping and stability. The wave load dominates for all conditions, and the presence of the current will affect the parameters of the wave. In addition, the relative relationship between the wavelength and the length of FPV system will affect their resonance effect, and the increase in wave height will significantly enhance the dynamic response of the FPV system. This study can help to provide references for the safety design of FPV arrays. [ABSTRACT FROM AUTHOR]

A two-dimensional (0xz) numerical model that makes it possible to calculate the initial elevation of the water surface in the source of a tsunami in a basin of variable depth is developed in the potential theory of an incompressible fluid under the approximation of an instantaneous deformation of the sea bottom. The model makes it possible to take into account the contribution of the horizontal component of the bottom deformation and the smoothing effect of the water layer through the use of the σ coordinate. To test the numerical model, we obtained an analytical solution to the problem of the initial elevation in a basin with a flat sloping bottom with a bottom deformation of a triangular shape. The results of the test show that for a spatial step typical for numerical tsunami models, there is close agreement between the numerical and analytical solutions. Using the developed σ model, we calculate the initial elevations of the water surface during the Kuril earthquake on January 13, 2007 and the Great East Japan Tohoku earthquake on March 11, 2011 (along the selected 2D sections). The results obtained are used to test an approximate method for calculating the initial elevation, known as the Kajiura filter, in which the ocean depth is assumed to be constant throughout the area of the source of the tsunami. [ABSTRACT FROM AUTHOR]

The boundary element method (BEM) is used by applying Cauchy's formula to the boundary of the water movement domain under a dam. By approximating the border with a polygon through linear interpolation, the relationships between the complex velocities on each edge of the polygon are analytically deduced. For the case of the flow domain described by a semi- circular closed contours, the numerical values of the velocity are computed and compared with those obtained only analytically. Conclusions on the analytical and numerical context are drawn. [ABSTRACT FROM AUTHOR]

In this paper we characterize the minimizer for a class of nonlocal perturbations of the Coulomb energy. We show that the minimizer is the normalized characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, is smooth off the origin, and is sufficiently small. This result can be seen as the stability of ellipsoids as energy minimizers, since the minimizer of the Coulomb energy is the normalized characteristic function of a ball. [ABSTRACT FROM AUTHOR]

In this paper, we calculate electrical resistance scaling exponents for analogues of "carpet" and "sponge" fractals in higher-dimensional embedding spaces. The calculation idealizes the voids that define such fractals as spherical, and exploits the elementary theory of dielectrics. Possible applications include models of elementary particles with "extra" dimensions, and new large-dimension methods for the theory of fractals. [ABSTRACT FROM AUTHOR]

In this paper we give a new, and shorter, proof of Huber’s theorem [Theorem 13 in Huber (Comment Math Helve 32:13–72, 1958)] which affirms that for a connected open Riemann surface endowed with a complete conformal Riemannian metric, if the negative part of its Gaussian curvature has finite mass, then the Riemann surface is homeomorphic to the interior of a compact surface with boundary, and thus it has finite topological type. We will also show that such Riemann surface is parabolic [Theorem 15 in Huber (Comment Math Helve 32:13–72, 1958)]. [ABSTRACT FROM AUTHOR]

A submarine is optimized to operate below the water surface because it spends most of its time in a submerged condition. However, the performance in free surface conditions is also important because it is unavoidable for port departure and arrival. Generally, potential flow theory is used for seakeeping analysis of a surface ship and is known for excellent numerical accuracy. In the case of a submarine, the accuracy of potential theory is high underwater but is low in free surface conditions because of the nonlinearity near the free surface area. In this study, the seakeeping performance of a Canadian Victoria Class submarine in regular waves was investigated to improve the numerical accuracy in free surface conditions by using computational fluid dynamics (CFD). The results were compared to those of model tests. In addition, the potential theory software Hydrostar developed by Bureau Veritas was also used for seakeeping performance to compare with CFD results. From the calculation results, it was found that the seakeeping analysis by using CFD gives good results compared with those of potential theory. In conclusion, seakeeping analysis based on CFD can be a good solution for estimating the seakeeping performance of submarines in free surface conditions.

The geometry of a compressor leading edge has an important effect on the aerodynamic performance at an off-designed incidence angle. The current geometric design methods of the leading edge are usually developed based on the flow characteristics at the designed incidence angle. However, few research focuses on the quantitative rules of the leading edge flow characteristics at the off-designed incidence angle in a compressor cascade. This situation restricts the further optimization and development of the leading edge geometry design method. In this paper, starting from the research of a potential cascade theory, the singularity point, where the surface velocity approaches infinity in the leading edge region, is eliminated by applying the characteristic that the ratio of the velocity increasement on the incidence angle in the plate cascade and the isolated plate flow is finite. Secondly, the equivalent pitch lengths based on 1/cos(β) and VI caused by a diffuser deceleration in the cascade passage were employed to correct the effect of the stagger angle. Finally, by introducing the isolated flow around the thick airfoil and considering the influence of the camber line geometry, a model of the variation of the surface velocity near the leading edge under the off-designed incidence angle, named the velocity increment on incidence angle, is derived from any compressor cascade. Hence, the relation between the off-designed incidence angle and the designed incidence angle of the surface velocity in a cascade blade is established, and it depends only on the geometrical parameters. Through a verification using numerical calculations and experimental measurement, the explicit formula for the velocity increment on incidence angle proposed in this paper has high precision near the leading edge. [ABSTRACT FROM AUTHOR]

The three-dimensional traction problem for steady elastic oscillations equations is studied. Representability of its solution by means of a double layer potential is considered instead of the more usual simple layer potential. [ABSTRACT FROM AUTHOR]

Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing active sound intensity, and uncorrelated sound waves arriving isotropically from all directions. Are there ideal source layouts to synthesize a maximum diffuse sound field within? As methods, we employ numeric simulations and undertake a series of considerations based on uncorrelated source layouts at a finite radius. Statistically expected active sound intensity and sound energy density are insightful and highlight the relation of active sound intensity to potential theory. Correspondingly, both Gauß’ divergence and Newton’s spherical shell theorem apply, and they provide valuable insights. In a circular layout, uncorrelated elementary point-source fields decaying by 1/√r ideally compose an extended sound field of vanishing active sound intensity; in spherical layouts this is the case with a 1/r decay. None of the layouts synthesizes a perfectly constant sound energy density inside. Theory and simulation offer a broad basis for understanding the synthesis of diffuse sound fields with uncorrelated sources in the free sound field.

Let $ D $ be a bounded domain in $ {{\mathbb R}}^3 $ with a closed, smooth, connected boundary $ S $, $ N $ be the outer unit normal to $ S $, $ k > 0 $ be a constant, $ u_{N^{\pm}} $ are the limiting values of the normal derivative of $ u $ on $ S $ from $ D $, respectively $ D': = {{\mathbb R}}^3\setminus \bar{D} $; $ g(x, y) = \frac{e^{ik|x-y|}}{4\pi |x-y|} $, $ w: = w(x, \mu): = \int_S g_{N}(x, s)\mu(s)ds $ be the double-layer potential, $ u: = u(x, \sigma): = \int_S g(x, s)\sigma(s)ds $ be the single-layer potential. In this paper it is proved that for every $ w $ there is a unique $ u $, such that $ w = u $ in $ D $ and vice versa. This result is new, although the potential theory has more than 150 years of history. Necessary and sufficient conditions are given for the existence of $ u $ and the relation $ w = u $ in $ D' $, given $ w $ in $ D' $, and for the existence of $ w $ and the relation $ w = u $ in $ D' $, given $ u $ in $ D' $.

We compute the equilibrium measure in dimension d = s + 4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s = 0 ( d = 4 ), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d = s + 3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler–Lagrange variational characterization, the Funk–Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions. [ABSTRACT FROM AUTHOR]

Rudolph Clausius is well known as a pioneer of the mechanical theory of heat (1857) and as the creator of the concept of entropy (1865). Oftentimes, he is also called the discoverer of the second law of thermodynamics although some argue that this law was already established by Sadi Carnot in 1824 (while still based on the caloric theory). But beyond any doubt, it was Clausius who gave in 1850 the first mathematically correct formulation of the first law (in its differential form that is still valid today, dQ = dU + pdV) and a particularly stringent exposition of both the necessity and independence of the two laws, indeed a logical masterpiece. This paper focuses on his concept of mathematical physics for the development of theoretical physics, contributions that have changed physics well beyond the field of thermodynamics. [ABSTRACT FROM AUTHOR]

The present contribution aims at presenting a semi-analytical solution methodology of the linear hydrodynamic diffraction and radiation problems induced by two coaxial moonpool-type floaters subjected to incident waves. The flow field around the floaters is subdivided into ring-shaped fluid regions, in each of which axisymmetric eigenfunction-expansions of the velocity potential are made. The applied theoretical formulation is complemented by numerical-panel methodologies, using pulsating singularities distributed over the floaters' wetted surface. Extensive numerical results in terms of exciting forces and hydrodynamic coefficients are given by applying frequency-domain techniques. The effect of the water trapped between the solids' vertical walls on the floaters' hydrodynamics is investigated and discussed. The presented analysis led to some remarkable trapping phenomena in the enclosed water areas, broadening the resonance-frequency bandwidth of the floaters when compared to a single moonpool body. [ABSTRACT FROM AUTHOR]

Following Bojarski and Vekua, we have studied the Dirichlet problem lim z → ζ Re f z = φ ζ as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ(ζ) in bounded domains D of the complex plane ℂ, where f satisfies the degenerate Beltrami equation f z ¯ = μ z f z , μ z < 1 , a.e. in D. Assuming that D is an arbitrary simply connected domain, we have established, in terms of μ, the BMO and FMO criteria, as well as a number of other integral criteria, on the existence and representation of regular discrete open solutions to the stated above problem. We have also proven similar theorems on the existence of multivalued solutions to the problem with single-valued real parts in an arbitrary bounded domain D with no boundary component degenerated to a single point. Finally, we have given a similar solvability and representation results concerning the Dirichlet problem in such domains for the degenerate A-harmonic equation associated with the Beltrami equation. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the geometric based relationship between the Katugampola fractional calculus and a Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system. Using the potential-theoretic approach with some classical analytical tools, we have derived some kinds of fractal dimensions of the graph of the Katugampola fractional integral of this fractal function. It has been shown that there is a linear relationship between the order of the Katugampola fractional integral and the fractal dimension of the graph of this generalized Weierstrass function. Numerical results have also been provided to corroborate such linear connection. • A Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system has been investigated. • The Hausdorff, Packing and Box dimensions of the graph of the Katugampola fractional integral of the Weierstrass-type function have been derived. • The geometric based linear connection between the Katugampola fractional calculus and the generalized Weierstrass function has been established. • Numerical simulations have also been carried out to corroborate such linearity with required figures and tables provided. [ABSTRACT FROM AUTHOR]

The floating photovoltaic (FPV) array, which consists of tens or hundreds of rows of floating photovoltaic systems, exhibits great economic and environmental benefits. An FPV array arranged in the ocean will be subjected to the combined action of wind, waves, and currents during the installation, service, and maintenance processes; however, systematic research in this field is still lacking. As a representative of the FPV array, this study on the hydrodynamic characteristics of a single-row FPV system is a fundamental prerequisite for the safe design of large-scale FPV arrays in the near future. In this paper, based on the potential theory, a numerical model of a single-row FPV system with ten floating platforms is established with ANSYS-AQWA software. Following this, the hydrodynamic coefficients of a single floating platform are calculated and evaluated. After that, the dynamic responses of the FPV system under different load combinations are explored. Finally, the influence mechanism of wave parameters on the hydrodynamic characteristics of the FPV system is discussed in detail, including the wavelength and wave height. The results indicate that, within the scope of this study, the floating platforms have good seakeeping and stability. The wave load dominates for all conditions, and the presence of the current will affect the parameters of the wave. In addition, the relative relationship between the wavelength and the length of FPV system will affect their resonance effect, and the increase in wave height will significantly enhance the dynamic response of the FPV system. This study can help to provide references for the safety design of FPV arrays.

The increasing adoption of power electronic devices may lead to large disturbance in DC power grids. Traditionally, the Brayton-Moser mixed potential theory is utilized to address large-signal stability (LSS) analysis. However, this theory proposes sufficient criteria only for large-signal global stability, which provides general but fuzzy information about the stability of DC microgrids. To solve this issue, we propose for the first time a comprehensive hyperlocal large-signal stability analysis (HL-LSS) that is adaptable to large-scale complex power grids. The novel proposed analysis studies the complex situation in nonlinear DC grids with multiple equilibrium points, including stability analysis of each equilibrium point and convergence analysis of state trajectories. The region of attraction (ROA) of microgrids is also estimated.

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'. [ABSTRACT FROM AUTHOR]

We give a geometric description of the parabolic bifurcation locus in the space Rat d of all rational functions on P 1 of degree d > 1, generalising the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions. [ABSTRACT FROM AUTHOR]

In this paper, we characterize explicitly the minimizer of HT ht in the case of a I physical confinement i , that is, we assume the confining potential F to be the indicator function of an elliptic domain of HT ht . Proof Assume first that HT ht , the unit vector along the HT ht -axis. This condition is proved by showing that the Fourier transform of HT ht is a positive distribution on test functions vanishing at zero. Thus, HT ht , and hence HT ht . [Extracted from the article]

We develop a numerical method for solving three-dimensional problems of fluid filtration and absorption in a piecewise homogeneous medium by means of boundary integral equations. This method is applied to a simulation of the lymph flow in a lymph node. The lymph node is considered as a piecewise homogeneous domain containing porous media. The lymph flow is described by Darcy's law. Taking into account the lymph absorption, we propose an integral representation for the velocity and pressure fields, where the lymph absorption imitates the lymph outflow from a lymph node through a system of capillaries. The original problem is reduced to a system of boundary integral equations, and a numerical algorithm for solving this system is provided. We simulate the lymph velocity and pressure as well as the total lymph flux. The method is verified by comparison with experimental data. [ABSTRACT FROM AUTHOR]

We study properties of A -harmonic and A -superharmonic functions involving an operator having generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. In particular, Harnack's Principle and Minimum Principle are provided for A -superharmonic functions. [ABSTRACT FROM AUTHOR]

Let D be a bounded domain in ℝ³ with a closed, smooth, connected boundary S, N be the outer unit normal to S, k > 0 be a constant, ... are the limiting values of the normal derivative of u on S from D, respectively D' := ℝ³ \ D; g(x, y) = ..., w :=(x, μ) := ∫x gN(x, s)μ(s)ds be the double-layer potential, u := u(x, σ) := ∫s g(x, s)σ(s)ds be the single-layer potential. In this paper it is proved that for every w there is a unique u, such that w = u in D and vice versa. This result is new, although the potential theory has more than 150 years of history. Necessary and sufficient conditions are given for the existence of u and the relation w = u in D', given w in D', and for the existence of w and the relation w = u in D', given u in D'. [ABSTRACT FROM AUTHOR]