Newtonian Potential Research Articles

Type-2 Fuzzy Sets and Newton’s Fuzzy Potential in an Algorithm of Classification Objects of a Conceptual Space

This paper deals with Gärdenfors’ theory of conceptual spaces. Let {mathcal {S}} be a conceptual space consisting of 2-type fuzzy sets equipped with several kinds of metrics. Let a finite set of prototypes tilde{P}_1,ldots ,tilde{P}_nin mathcal {S} be given. Our main result is the construction of a classification algorithm. That is, given an element {tilde{A}}in mathcal {S}, our algorithm classifies it into the conceptual field determined by one of the given prototypes tilde{P}_i. The construction of our algorithm uses some physical analogies and the Newton potential plays a significant role here. Importantly, the resulting conceptual fields are not convex in the Euclidean sense, which we believe is a reasonable departure from the assumptions of Gardenfors’ original definition of the conceptual space. A partitioning algorithm of the space mathcal {S} is also considered in the paper. In the application section, we test our classification algorithm on real data and obtain very satisfactory results. Moreover, the example we consider is another argument against requiring convexity of conceptual fields.

On an alternative approach for mixed boundary value problems for the Laplace equation

Abstract In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , when the boundary surface S = ∂ ⁡ Ω {S=\partial\Omega} is divided into two disjoint parts S D {S_{D}} and S N {S_{N}} where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 {L_{2}} -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ ( Ω ) {W^{1}_{2}(\Omega)} . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p {L_{p}} -based Besov spaces, which under appropriate boundary data implies C α {C^{\alpha}} -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ {\overline{\Omega}} with α = 1 2 - ε {\alpha=\frac{1}{2}-\varepsilon} , where ε > 0 {\varepsilon>0} is an arbitrarily small number.

How Inflationary Gravitons Affect the Force of Gravity

We employ an unregulated computation of the graviton self-energy from gravitons on the de Sitter background to infer the renormalized result. This is used to quantum-correct the linearized Einstein equation. We solve this equation for the potentials that represent the gravitational response to a static, point mass. We find large spatial and temporal logarithmic corrections to the Newtonian potential and to the gravitational shift. Although suppressed by a minuscule loop-counting parameter, these corrections cause perturbation theory to break down at large distances and late times. Another interesting fact is that gravitons induce up to three large logarithms, whereas a loop of massless, minimally coupled scalars produces only a single large logarithm. This is in line with corrections to the graviton mode function: a loop of gravitons induces two large logarithms, whereas a scalar loop gives none.

On the traction problem for steady elastic oscillations equations: the double layer potential ansatz

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.

Numerical exploration of the motion of variable mass test particle on the perturbed circular restricted three-body configuration

This paper represents analytical and numerical investigations of the motion properties of a test particle having a variable mass according to Jeans’ law and subject to the influence of a heterogeneous primary having N layers of different densities of a part and a second body producing a modified Newton potential on the other hand. We also assume that the whole system is perturbed by Coriolis force and centrifugal force. The equations of motion that we derive, are used to determine the locations of equilibrium points, Poincaré surfaces of section, basins of attraction as well as to study questions related to the stability of equilibrium points.

Triangular Equilibria in R3BP under the Consideration of Yukawa Correction to Newtonian Potential

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of α and λ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where α ∈ − 1 , 1 is the coupling constant of Yukawa force to gravitational force, and λ ∈ 0 , ∞ is the range of Yukawa force. It is observed that as λ ⟶ ∞ , the mean-motion of the primaries n ⟶ 1 + α 1 / 2 and as λ ⟶ 0 , n ⟶ 1 . Further, it is observed that the mean-motion is unity, i.e., n = 1 for α = 0 , n > 1 if α > 0 and n < 1 when α < 0 . The triangular equilibria are not affected by α and λ and remain the same as in the classical case of restricted three-body problem. But, α and λ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter μ c = μ 0 + f α , λ , where μ 0 = 0.0385209 ⋯ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that μ c = μ 0 either for α = 0 or λ = 0.618034 , and the critical mass parameter μ c possesses maximum ( μ c max ) and minimum ( μ c min ) values in the intervals − 1 < α < 0 and 0 < α < 1 , respectively, for λ = 1 / 3 .

Nonsymmetric wormholes and localized big rip singularities in Einstein-Weyl gravity

The inclusion of the Weyl squared term in the gravitational action is one of the most simple, yet non trivial modifications to General Relativity at high energies. Nevertheless the study of the spherically-symmetric vacuum solutions of this theory has received much attention only in recent times. A new type of asymptotically flat wormhole which does not match symmetrically at a finite radius with another sheet of the spacetime is presented. The outer spacetime is characterized by a newtonian potential with a Yukawa correction, and has gravitational properties that can be arbitrarily close to the ones of a Schwarzschild black hole. The internal spacetime instead possesses a singularity at $r=\infty$ with the topology of a 2-dimensional sphere. The expansion scalar of geodesics reaching this singularity diverges in a finite amount of proper time, with a striking resemblance with the future singularity of the Big Rip cosmological scenario. In terms of the external Yukawa hair and mass $M$, these new wormholes fill a large region of the two-dimensional parameter space of physical solutions with $M>0$. On the contrary black holes, both of Schwarzschild and non-Schwarzschild nature, are confined on a line. We argue that this type of wormholes are ideal black hole mimickers.

Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

We study the two-phase Stokes flow driven by surface tension for two fluids of different viscosities, separated by an asymptotically flat interface representable as graph of a differentiable function. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single and double layer potential, spectral results on the corresponding integral operators, and abstract results on nonlinear parabolic evolution equations.

Integration of the modified double layer potential of the vector boundary element method for eddy current problems

The boundary element method for the eddy current problem (BEM-ECP) was proposed in a number of papers and is applicable to important tasks such as the problem of inductive heating and transmission of electromagnetic energy. BEM-ECP requires the construction of a system of linear algebraic equations in which the matrix is inherently dense and is constructed out of element matrices. For the process of the element matrix computation, two cases are normally considered: far-field interaction and near-field interaction, because the construction of element matrices requires integration of a singular function. In this article, we suggest a transform that allows computing the matrix components of the near-singular interaction part while implementing only the single and double layer potentials. The previously suggested modified double layer potential (MDLP) can be integrated by means of this transform, which simplifies the program implementation of BEM-ECP significantly. Solving model problems, we analyse the drawbacks of the previously suggested approach. This analysis includes the proof of the MDLP singularity that makes the integration of this potential a rather difficult task without the help of our transform. The previously suggested approach does not work well with surfaces that are not smooth. Our approach does consider such cases, which is its main advantage. We demonstrate this on the model problems with known analytical solutions.

Asymptotic analysis of subwavelength halide perovskite resonators

Halide perovskites are promising materials with many significant applications in photovoltaics and optoelectronics. Their highly dispersive permittivity relation leads to a non-linear relationship between the frequency and the wavenumber. This, in turn, means the resonance of the system is described by a highly non-linear eigenvalue problem, which is mathematically challenging to understand. In this paper, we use integral methods to quantify the resonant properties of halide perovskite nano-particles. We prove that, for arbitrarily small particles, the subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. We also characterize the hybridized subwavelength resonant frequencies of a dimer of two halide perovskite particles. Finally, we examine the specific case of spherical resonators and demonstrate that our new results are consistent with previous works.

The challenge of sixfold integrals: the closed-form evaluation of Newton potentials between two cubes

The challenge of explicitly evaluating, in elementary closed form, the weakly singular sixfold integrals for potentials and forces between two cubes has been taken up at various places in the mathematics and physics literature. It created some strikingly specific results, with an aura of arbitrariness, and a single intricate general procedure due to Hackbusch. Those scattered instances were mostly addressing the problem head-on by successive integration, while keeping track of a thicket of primitives generated at intermediate stages. In this paper, we present a substantially easier and shorter approach, based on a Laplace transform of the kernel. We clearly exhibit the structure of the results as obtained by an explicit algorithm, just computing with rational polynomials. The method extends, up to the evaluation of single integrals, to higher dimensions. Among other examples, we easily reproduce Fornberg’s startling closed-form solution of Trefethen’s two-cubes problem and Waldvogel’s symmetric formula for the Newton potential of a rectangular cuboid.

Light bending and gravitational lensing in energy-momentum-squared gravity

In the present work, we derive the motion of light in the weak-field limit of energy-momentum-squared gravity (EMSG). To do so, we introduce the post-Newtonian expansion of this modified theory of gravity. It is shown that, in addition to the Newtonian potential, a new EMSG potential affects the trajectory of photons. As a result, in this theory, photons do not behave as predicted by general relativity (GR). To evaluate the EMSG theory by the Solar System tests, we study light deflection and Shapiro time delay. Regarding the results obtained in Bertotti et al. [Nature (London) 425, 374 (2003)] and Shapiro et al. [Phys. Rev. Lett. 92, 121101 (2004)], we restrict the free parameter of the theory and show that it lies within the range $\ensuremath{-}4.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}<{f}_{0}^{\ensuremath{'}}<8.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}$. This interval is in agreement with those derived in Nazari et al. [Phys. Rev. D 105, 044014 (2022)] and Akarsu et al. [Phys. Rev. D 97, 124017 (2018)]. This consistency manifests that this theory passes these Solar System tests with flying colors. Interestingly, it turns out that the magnitude of the EMSG correction strongly depends on the density of the deflector. So, we investigate the possible effects of EMSG on images of a light source microlensed by a compact dense object such as neutron stars. It is estimated that the EMSG correction to the position of lensed images could be as large as (1--0.1) microarcseconds, which may be detected by future high-resolution missions. Moreover, the total magnification and the shape of light curves are obtained in the EMSG theory. It is revealed that except for a small deviation, the overall behavior of the EMSG light curves is similar to that in GR. We also show that as long as the light source and the dense lens are aligned, the EMSG correction is effective, and the combined light of the lensed images is different from the GR case. This issue makes it possible to observe signatures of this theory in the microlensing regime.

Nonrelativistic Boson stars as N-body quantum systems

In this work, we show that the structural configuration of a collection of generic nonrelativistic bosons forming a gravitationally bound Bose–Einstein condensate can be interpreted as a nonrelativistic boson star. With the approach followed in this work, we can analyze with a concise and straightforward procedure the equilibrium properties of nonrelativistic boson stars viewed as a Bose–Einstein condensate. The system’s behavior is obtained by analyzing its fundamental constituent properties, i.e. the associated single-particle properties. Additionally, we show that by expressing the corresponding Newtonian gravitational potential, under certain circumstances, as a harmonic oscillator potential one, we can describe the conditions in which the nonrelativistic boson star can form equilibrium configurations. In order to analyze the boson star’s structural configuration, we employ four different ansätzes commonly used in the literature. These ansätzes allow us to compare the cloud’s structural properties of the boson star, which leads us to obtain several gravitational equilibrium configurations from compact objects matching the size of typical stars to gigantic systems comparable to the size of galaxy cluster dark matter halos. Finally, we show that these ansätzes predict, qualitatively speaking, the same structural and gravitational equilibrium configurations for different values of the parameters involved.

Truncated Newton Kernel Ridge Regression for Prediction of Porosity in Additive Manufactured SS316L

Despite the many benefits of additive manufacturing, the final quality of the fabricated parts remains a barrier to the wide adoption of this technique in industry. Predicting the quality of parts using advanced machine learning techniques may improve the repeatability of results and make additive manufacturing accessible to different fields. This study aims to integrate data extracted from various sources and use them to obtain accurate predictions of relative density with respect to the governing process parameters. Process parameters such as laser power, scan speed, hatch distance, and layer thickness are used to predict the relative density of 316L stainless steel specimens fabricated by selective laser melting. An extensive dataset is created by systematically combining experimental results from prior studies with the results of the current work. Analysis of the collected dataset shows that the laser power and scan speed significantly impact the relative density. This study compares ridge regression, kernel ridge regression, and support vector regression using the data collected for SS316L. Computational results indicate that kernel ridge regression performs better than both ridge regression and support vector regression based on the coefficient of determination and mean square error.

Sub-elliptic boundary value problems in flag domains

Abstract A flag domain in ℝ 3 {\mathbb{R}^{3}} is a subset of ℝ 3 {\mathbb{R}^{3}} of the form { ( x , y , t ) : y < A ⁢ ( x ) } {\{(x,y,t):y<A(x)\}} , where A : ℝ → ℝ {A\colon\mathbb{R}\to\mathbb{R}} is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 {\bigtriangleup^{\flat}=X^{2}+Y^{2}} in flag domains Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , with L 2 {L^{2}} -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 {L^{2}} -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.

Jeans instability in an expanding universe with dissipation

Jeans instability is analyzed in an expanding universe within the framework of BGK model of the Boltzmann equation and Poisson equations. The background is characterized by a comoving Maxwellian distribution function and a spacetime Newtonian gravitational potential which satisfy the BGK model of the Boltzmann–Poisson equations without the necessity to invoke “Jeans swindle.” The perturbations of the distribution function and Newtonian gravitational potentials from their background states are represented by plane waves of small amplitudes and a differential equation for the density contrast is determined. The density contrast differential equation was solved numerically and it is shown: (i) Jeans instability is characterized by perturbation wavelengths larger than Jeans wavelength where the density contrast grows with time. The growth of the density contrast is less accentuated for the case where the particle collisions are considered due to an energy dissipation; (ii) for perturbation wavelengths smaller than Jeans wavelength the density contrast has an oscillatory behavior in time and the oscillations for the case where the collisions are taken into account fade away in time due to the energy dissipation.

A fast regularisation of a Newtonian vortex equation

We consider equations of the form u_t = \nabla \cdot ( \gamma(u) \nabla \operatorname{N}(u)) , where \operatorname{N} is the Newtonian potential (inverse of the Laplacian) posed in the whole space \mathbb{R}^d , and \gamma(u) is the mobility. For linear mobility, \gamma(u)=u , the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex of constant intensity in space u=c_1t^{-1} supported in a ball that spreads in time like c_2t^{1/d} , thus showing a discontinuous leading front. In this paper we propose the model with sublinear mobility \gamma(u)=u^\alpha , with 0<\alpha<1 , and prove that non-negative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularisation of the previous one. In particular, we find that the equivalent of the previous vortex is an explicit self-similar solution decaying in time like u=O(t^{-1/\alpha}) with a space tail with size u=O(|x|^{- d/(1-\alpha)}) . We restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burgers’ equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions. We also construct numerical finite-difference convergent schemes.

Fate of observers in circular motion

In Newtonian physics or in general relativity, energy dissipation causes observers moving along circular orbits to slowly spiral towards the source of the gravitational field. We show that the loss of energy has the same effect in any theory of gravity respecting the weak equivalence principle, by exhibiting an intimate relation between the energy of a massive test particle and the stability of its orbit. Ultimately, massive particles either plunge, or are driven towards minima of the generalized Newtonian potential, where they become static. In addition, we construct a toy metric which displays an unbound innermost stable circular orbit, allowing particles that reach this orbit to be expelled away.

Constraining modified gravity with quantum optomechanics

We derive the best possible bounds that can be placed on Yukawa- and chameleon-like modifications to the Newtonian gravitational potential with a cavity optomechanical quantum sensor. By modelling the effects on an oscillating source-sphere on the optomechanical system from first-principles, we derive the fundamental sensitivity with which these modifications can be detected in the absence of environmental noise. In particular, we take into account the large size of the optomechanical probe compared with the range of the fifth forces that we wish to probe and quantify the resulting screening effect when both the source and probe are spherical. Our results show that optomechanical systems in high vacuum could, in principle, further constrain the parameters of chameleon-like modifications to Newtonian gravity.

Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.