Arbitrary Constants Research Articles

Heisenberg soft hair on Robinson-Trautman spacetimes

We study 4 dimensional (4d) gravitational waves (GWs) with compact wavefronts, generalizing Robinson-Trautman (RT) solutions in Einstein gravity with an arbitrary cosmological constant. We construct the most general solution of the GWs in the presence of a causal, timelike, or null boundary when the usual tensor modes are turned off. Our solution space besides the shape and topology of the wavefront which is a generic compact, smooth, and orientable 2d surface Σ, is specified by a vector over Σ satisfying the conformal Killing equation and two scalars that are arbitrary functions over the causal boundary, the boundary modes (soft hair). We work out the symplectic form over the solution space using covariant phase space formalism and analyze the boundary symmetries and charges. The algebra of surface charges is a Heisenberg algebra. Only the overall size of the compact wavefront and not the details of its shape appears in the boundary symplectic form and is canonical conjugate to the overall mass of the GW. Hence, the information about the shape of the wavefront can’t be probed by the boundary observer. We construct a boundary energy-momentum tensor and a boundary current, whose conservation yields the RT equation for both asymptotically AdS and flat spacetimes. The latter provides a hydrodynamic description for our RT solutions.

Results of the analytical solution of the problem of radial vibrations of disks of variable thickness

An analytical solution is obtained for the problem of radial vibrations of disks of variable thickness. A disk is considered that is rigidly fixed along the inner circular contour (ρ=0.2) and free on the outer contour (ρ=1). The thickness of the disk varies according to the law H=H0(ρν+μ+Cρν-μ)2, where H0,C,μ are arbitrary constants; ν is the Poisson's ratio. The exact solution of the problem is known only for H=const and H=1/ρ3. However, these solutions are not sufficient to study the vibrations of disks of other configurations. The proposed law of thickness variation H(ρ) allows us to obtain exact solutions to the problem at any value of the constant coefficients H0, C, μ, ν. By varying the values of these coefficients within a single given function, it is possible to set the disk profile of the desired appearance. The methods used to obtain these solutions are based on appropriate mathematical transformations of the original equation. The problem of disk oscillations is solved for four variants of thickness change. The natural frequencies for the first three forms of vibration are calculated. Comparison of the natural frequencies found for the three cases of the disk profile gently sloping indicates an increase in their values with an increase in the bending of the disk thickness. Based on the obtained eigenfunctions, the stresses were calculated and the nature of their distribution along the radial coordinate of the disk was determined. The strength of the disks under resonant radial vibrations was evaluated using a special criterion. It is found that the most limiting, i.e., destructive principal stress σ1=σr at the first (main) form of vibration should be chosen from the ratio σr≈0.79 [σ-1], where [σ-1] is the endurance limit of the disk material under uniform loading. The results obtained can be used to predict the stress-strain state of disks of variable profile under their radial vibrations

New Solutions of Benney-Luke Equation Using The (G’/G,1/G) Method

The Benney-Luke equation has contributed to studying the propagation of the water wave surfaces. This paper illustrates the (G’/G,1/G)-method to obtain the solutions of the Benney-Luke equation and an extension of the Benney-Luke equation. The new types of solutions are also constructed to gather the performance and visualization in three dimensions for observing the behaviors. The solutions are found in the expressions of hyperbolic functions giving the general performance by selecting arbitrary constants.

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

Far-Field Estimation Method of Antennas Above Conductive Plane Using a Partially Spherical Near-Field

For evaluating the practical performance of land mobile communication antennas via near-field measurement, it is essential to measure the electromagnetic field, including the housing in which the antennas are mounted. When measuring a heavy housing, scattered waves from the supporting turntable for the antenna under test (AUT) disturb the electromagnetic field. A technique for eliminating the influence of scattered waves is required to evaluate the abnormal AUT radiation performance. This paper proposes a method for estimating the far-field when the antenna exists above an infinite conductive plane, using partially spherical near-field measurements that include the turntable. The proposed method estimates the far-field by combining the equivalent source reconstruction and the far-field estimation with the reflection coefficient method. In the proposed method, the equivalent source distribution surrounding an AUT is reconstructed from a partially spherical near-field to remove the effects of the turntable. Through numerical calculations and experiments, this paper shows that the proposed method can estimate the far-field pattern above an infinite conductive plane, with arbitrary electric constants from a partially spherical near-field. The far-field estimation error is less than -20 dB when the upper hemispherical near-field can be obtained.

Solutions with pure radiation and gyratons in 3D massive gravity theories

We find exact solutions of topologically massive gravity (TMG) and new massive gravity (NMG) in 2 + 1 dimensions (3D) with an arbitrary cosmological constant, pure radiation, and gyratons, i.e. with possibly non-zero T uu and T ux in canonical coordinates. Since any ‘reasonable’ geometry in 3D (i.e. admitting a null geodesic congruence) is either expanding Robinson–Trautman ( Θ≠0 ) or Kundt ( Θ=0 ), we focus on these two classes. Assuming expansions Θ=1/r (‘GR-like’ Robinson–Trautman) or Θ=0 (general Kundt), we systematically integrate the field equations of TMG and NMG and identify new classes of exact solutions. The case of NMG contains an additional assumption of g ux being quadratic in r, which is automatically enforced in TMG as well as in 3D GR. In each case, we reduce the field equations as much as possible and identify new classes of solutions. We also discuss various special subclasses and study some explicit solutions.

An anisotropic cosmological model with Bianchi type III universe in f(G) gravity

The paper is devoted to exploring the exact solution of the modified Einstein's field in the setting of spatially homogeneous and anisotropic Bianchi type III spacetime in modified f(G) gravity, where G is Gauss-Bonnet invariant. Here, by employing the hyperbolic hybrid scale factor a = emt (sinh(t))n in which m,n are positive constants, with the power-law f (G)=βGm+1 model where β,m are arbitrary constants, we computed the physical and geometrical properties of the cosmological parameters of the model. The results of the model parameters are well satisfied with recent cosmological observational data. To get the exact solution of the field equation of the Bianchi type III model in the presence of an anisotropic dark fluid, we consider the relation in which the shear scalar (σ) is proportional to the expansion scalar (θ), resulting in C = An. Furthermore, the energy conditions for the power-law f(G) model are graphically examined, and it turns out that the null energy condition (NEC), weak energy condition (WEC), and dominant energy condition (DEC) are well satisfied except for the strong energy condition (SEC). The violation of the strong energy conditions (SEC) indicates the f(G) model supports the universe's current expansion with negative pressure, having a quintessence model in the present and Λ cold dark matter (CDM) model in the future.

Generalized ghost dark energy in [formula omitted] gravity

In this manuscript, we use the correspondence principle to construct the generalized ghost dark energy f(Q) model, where Q is the non-metricity. For this purpose, we use the Friedman–Robertson–Walker universe model with power-law scale factor. The energy density in this model linearly depends on the Hubble parameter, with a sub-leading term of H2, resulting in an energy density ρD=ξH+ζH2, where ξ and ζ are arbitrary constants. We reconstruct interacting fluid scenario of this dark energy with dark matter. The reformulated f(Q) gravity model shows how different epochs of the cosmic history can be reproduced by this modified gravity model. The physical characteristics of the model are discussed through the equation of state parameter with the planes of ωD−ωD′ and r−s. We also explore the stability criteria for the interacting model using the squared speed of sound. The equation of state parameter indicates phantom phase of the universe and the squared speed of sound represents stability of the interacting model. The (ωD−ωD′)-plane indicates freezing region, while the (r−s)-plane corresponds to the Chaplygin gas. We conclude that our findings coincide with the latest observational data.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Solution of linear PDEs obtained using Laplace transform

We would like to find the solution of PDEs obtained using Laplace transform. The solution of linear PDEs using Laplace transform has the advantage that an arbitrary constant is specifically displayed as an initial value. Of course, even if the Laplace transform used as a tool is replaced with many other Laplace-type transforms, the obtained result still holds. In addition, the solution of example 2 is obtained using ChatGPT and compared with the exact solution.

Improved limited view ultrasound tomography via machine learning.

Tomographic reconstruction is used extensively in medicine, non-destructive testing and geology. In an ideal situation where measurements are taken at all angles around an object, known as full view configuration, a full reconstruction of the object can be produced. One of the major issues faced in tomographic imaging is when measurements cannot be taken freely around the object under inspection. This may be caused by the size and geometry of the object or difficulty accessing from particular directions. The resulting limited view transducer configuration leads to a large deterioration in image quality, thus it is very beneficial to employ a compensation algorithm. At present, the most effective compensation algorithms require a large amount of computing power or a bespoke case-by case approach, often with numerous arbitrary constants which must be tuned for a specific application. This work proposes a machine learning based approach to perform the limited view compensation. The model is based around an autoencoder architecture. It is trained on an artificial dataset, taking advantage of the ability to generate arbitrary limited view images given a full view input. The approach is evaluated on ten laser-scanned corrosion maps and the results compared to positivity regularisation - a limited view compensation algorithm similar in the speed of execution and generalisation potential. The algorithms are compared for root mean squared error (RMSE) across the image, and maximum absolute error (MAE). Furthermore, they are visually compared for subjective quality. Compared to the conventional algorithm, the ML-based approach improves on the MAE in eight out of the ten cases. The conventional approach performs better on mean RMSE, which is explained by the ML outputting inaccurate background level, which is not a critical ability. Most importantly, the visual inspection of outputs shows the ML approach reconstructs the images better, especially in the case of irregular corrosion patches. Compared to limited view images, the ML method improves both the RMSE and MAE by 41%.

Novel Traveling Wave Solutions of Jaulent-Miodek Equations and Coupled Konno-Oono Systems and Their Dynamics

This research article deals with analytical solutions to two problems. The first is the (1+1)-coupled Jaulent-Miodek system of equations, which is associated with the energy-dependent Schrödinger potential, whereas the second problem, the system of coupled Konno-Oono equations relates to complexity and chaos in electromagnetic fields. Similarity reductions via Lie-symmetry analysis is performed for the systems to derive their analytical solutions. Since Lie symmetry involves arbitrary constants in the infinitesimals, this opens up more possibilities for getting a rich variety of analytical solutions for both real-life problems. The analytical solutions are supplemented graphically to understand them in a better way. Traveling wave profiles are obtained eventually. Solution for CKOEs are different from the earlier research (Kumar and Kumar 2022a; Kumar et al. 2022) as far as the authors are aware.

Integral Equations of Volterra Typewith Two Boundary and One Interior Singular Point

Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.

A self-similar solution for unsteady isothermal and adiabatic flows behind the shock wave in a dusty non-ideal rotating medium under the action of magnetic field

In this paper, the similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric mixture of small solid particles of micro-size and a non-ideal gas with azimuthal magnetic field are obtained. In the mixture, the solid particles are distributed uniformly. For isothermal and adiabatic flows, the Lie group analysis is used to discuss all possible cases of similarity solutions. The local Lie group generator’s arbitrary constants generate a range of solutions with power and exponential law shock paths. A detailed discussion of the similarity solution for isothermal and adiabatic flows with a power-law shock path is presented. The constant density for the ambient medium has been assumed. The effect of changing the Alfven Mach number, non-idealness parameter, and other parameters on shock wave strength and flow variables is explored. These parameters are observed to have a decaying impact on shock waves. The effect of varying the Alfven Mach number, non-idealness parameter and other parameters on shock wave strength and flow variables is investigated. The parameter of non-idealness and Alfven Mach number are noted to have a decaying effect on shock waves.

Pole-skipping of gravitational waves in the backgrounds of four-dimensional massive black holes

Pole-skipping is a property of gravitational waves dictated by their behaviour at horizons of black holes. It stems from the inability to unambiguously impose ingoing boundary conditions at the horizon at an infinite discrete set of Fourier modes. The phenomenon has been best understood, when such a description exists, in terms of dual holographic (AdS/CFT) correlation functions that take the value of ‘0/0’ at these special points. In this work, we investigate details of pole-skipping purely from the point of view of classical gravity in 4d massive black hole geometries with flat, spherical and hyperbolic horizons, and with an arbitrary cosmological constant. We show that pole-skipping points naturally fall into two categories: the algebraically special points and a set of pole-skipping points that is common to the even and odd channels of perturbations. Our analysis utilises and generalises (to arbitrary maximally symmetric horizon topology and cosmological constant) the ‘integrable’ structure of the Darboux transformations, which relate the master field equations that describe the evolution of gravitational perturbations in the two channels. Finally, we provide new insights into a number of special cases: spherical black holes, asymptotically Anti-de Sitter black branes and pole-skipping at the cosmological horizon in de Sitter space.

FLRW Cosmology with Hybrid Scale Factor in f(R, Lm) Gravity

In this paper, we aim to describe the cosmic late-time acceleration of the Universe in f(R,Lm) gravity framework proposed by Harko (2010) with the help of an equation of state for strange quark matter. To achieve this, we adopt a specific form of f(R,Lm) gravity as f(R,Lm) =R/2}+ Lnm, where n is arbitrary constants. Here we utilize a hybrid scale factor to resolve the modified field equations in the context of f(R,Lm) gravity for an isotropic and homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) metric in presence of strange quark matter (SQM). Also, we analyze the dynamics of energy density, pressure and the state finder parameters and explained the distinctions between our model and the current dark energy models in the presence of SQM. We observed a transition from an accelerating to a decelerating phase in the Universe, followed by a return to an accelerating phase at late times. Also, we analyzed the state finder diagnostic as well equation of state parameter and found that the model exhibited quintessence-like behavior. The conclusion drawn from our investigation was that the proposed f(R, Lm) cosmological model aligns well with recent observational studies and effectively describes the cosmic acceleration observed during late times.

Square-root parametrization of dark energy in f(Q) cosmology

This paper is a parametrization of the equation of state (EoS) parameter of dark energy (DE), which is parameterized using square-root (SR) form i.e. , where ω 0 and ω 1 are free constants. This parametrization is examined in the context of the recently suggested f(Q) gravity theory as an alternative to general relativity (GR), in which gravitational effects are attributed to the non-metricity scalar Q with the functional form f(Q) = Q + α Q n , where α and n are arbitrary constants. We derive observational constraints on model parameters using the Hubble dataset with 31 data points and the Supernovae (SNe) dataset from the Pantheon samples compilation dataset with 1048 data points. For the current model, the evolution of the deceleration parameter, density parameter, EoS for DE, and Om(z) diagnostic have all been investigated. It has been shown that the deceleration parameter favors the current accelerated expansion phase. It has also been shown that the EoS parameter for DE has a quintessence nature at this time.

Cosmological models with variable parameters

The present study deals with the investigation of the Friedmann-Lemaitre-Robertson-Walker models (often FLRW-models) with time varying G and ∧ in the framework of General theory of Relativity. The Einstein field equations have been solved by considering the time-varying deceleration parameter q(t) and Scale factor , where m and β are arbitrary constants. We have analysed the value of m, which will generate a transition for universe from early decelerating phase to recent acceleration phase. The physical and graphic behaviour have also been planned to study in this communication.

Propagation of magnetogasdynamics spherical shock wave in a gravitating gas with radiation heat flux

This article presents a mathematical model for characterizing the dynamic behavior of spherical shock waves in a self-gravitating, radiating ideal gas with the presence of an azimuthal magnetic field, emphasizing adiabatic conditions. The analysis assumes that the environment just ahead of the shock wave is stationary, and it accounts for variations in density, magnetic field, and fluid velocity within the disturbed medium just behind the shock front. Furthermore, the impact of thermal radiation within the context of an optically thin limit is incorporated into the energy equation of the governing system. Employing the Lie invariance method, the set of partial differential equations governing the flow within this medium is transformed into a system of nonlinear ordinary differential equations through the use of similarity variables. Four distinct cases of similarity solutions are derived by selecting different values for the arbitrary constants associated with the generators. Among these four cases, only two yield similarity solutions, one assuming a power-law shock path and the other an exponential-law shock path. In the case of a power-law shock path, the resulting set of nonlinear ordinary differential equations is numerically solved using the 4th-order Runge-Kutta method in MATLAB software. The article thoroughly discusses the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), ϕ (ambient density exponent), and G0 (gravitational parameter), on the flow properties. The findings are presented graphically to provide a comprehensive understanding of the effects of these parameters.

A study on the optimal incident angle of sub-wavelength grating polarization beam splitter

One problem faced by the sub-wavelength grating polarization beam splitter is the lack of a generally-used formula able to conduct accurate calculation of the incident angle. With that background, by the design method of optical film polarization beam splitter, this work realizes an precise calculation under the condition of arbitrary grating constants by transforming the splitter into an optical-film-structure model. According to tests, the incident angles of grating calculated by the method proposed here are consistent with those obtained from the FDTD simulation, illustrating the reliability of applying the polarization beam splitting design method of optical thin films to polarization beam splitting gratings.