Arbitrary Function Of Time Research Articles

Cotton gravity is not predictive

It is well known that the theory of Cotton gravity proposed by Harada is trivially solved by all isotropic and homogeneous cosmologies. We show that this under-determination is more general. More precisely, the degree of arbitrariness in the solutions increases with the degree of symmetry. The inverse problem of finding the matter source generating a given spacetime geometry is similarly indeterminate in the case of highly symmetric configurations. We give two simple examples. The first is that of static spherically symmetric solutions, which depend on an arbitrary function of the radial coordinate. The second is that of anisotropic cosmologies, which depend on an arbitrary function of time.

Pt4: Phantom-4D, an open-source software tool for creating time-evolving 3D phantoms

pt4, an open-source software tool to describe time-evolving phantoms is presented. pt4 allows users to create detailed time-evolving phantoms for testing novel 4D-CT reconstruction algorithms. Ground-truth volumes and simulated X-ray projections can be produced at arbitrary time-points during time-evolution and with customisable pixel dimensions, noise models, and X-ray source trajectories. Phantoms are built up from 3D primitives whose parameters and attenuation can be made arbitrary functions of time. This feature permits both complex continuous and discontinuous time-evolution necessary for thorough testing of 4D-CT reconstruction algorithms. Various phantoms built using pt4 are also presented to demonstrate the versatility of pt4 phantom description. pt4 is written in C++ and is highly parallelised leading to a performant implementation which is feasible to use for up to thousand-of-voxel volume pixel dimensions.

A Normalization Condition for the Probability Current in Some Remarkable Cases

Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest a way to determine this function: imposing a suitable normalization condition. The application of our proposal to the case of the harmonic oscillator is discussed.

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

Painlevé analysis, infinite dimensional symmetry group and symmetry reductions for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finite-dimensional subalgebra of physical transformations is studied. The commutation relations of the subalgebra, which have been established by selecting the Laurent polynomials in t, are calculated. This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics. Meanwhile, the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.

Unsteady incompressible laminar boundary layer: time and space variable molecular viscosity

The subject of this work is two approaches to describe the laminar unsteady flow of an incompressible fluid in the boundary layer. In the first approach, the viscosity of the fluid and the acceleration with which the plane is set in motion are considered constant. In essence, this is Rayleigh's problem. The solution obtained on the basis of these assumptions asymptotically converges to the well-known self-similar Stokes solution. It is important that the solutions of Stokes and Rayleigh asymptotically at large values of time correspond to the disappearance of shear stresses between the liquid and moving plane after acceleration. A paradox emerges the equations derived by Stokes to describe internal friction indicate the absence of the same friction between a moving body and fluid. Since research using the calculus of variation methods revealed that the molecular viscosity inside the stationary boundary layer should depend on the distance to the moving surface, the corresponding non-steady problem was considered. As a result, as before for the steady case, solutions describing both non-gradient and gradient flows of incompressible fluid in the boundary layer are obtained. The asymptotic analysis of the transition to steady flow testifies the consistency of these solutions. For the case of non-gradient flow, a comparison of the classical solution with the solution corresponding to the extreme fluid flow rate carried by the moving surface is made. It is shown that according to the solution obtained on the basis of the calculus of variation approach, the shear stress on the surface does not disappear anywhere after the motion becomes steady but, as expected, acquires a constant value. The research methods are purely theoretical and the results are analyzed by comparison with available theoretical and experimental data and compliance with the fundamental laws of physics, in particular the law of conservation of energy. These methods are based on the construction of analytical mathematical models, which are differential equations in partial derivatives supplemented with appropriate physical initial and boundary conditions. In addition, Euler's differential equations for the extreme of functional theory are used (in this paper, this is the extreme of fluid flow rate across the cross-section of the boundary layer). When solving these equations, the well-known Fourier method of variable separation is used. Arbitrary functions of time arising during partial integration (by one of the variables – the spatial coordinate) are determined from the conditions of asymptotic convergence of the solutions of non-steady problems to the corresponding solutions of steady problems. Conclusions. The presented results are of fundamental importance for understanding the physics of the flow around aircraft parts, as they indicate the contradiction of the existing idea of the reversibility of direct and inverse problems: the motion of a body in a still fluid and the flow of a fluid around an immobile body.

A Sustainable Closed-Loop Supply Chains Inventory Model Considering Optimal Number of Remanufacturing Times

The mathematical modeling of reverse logistics inventory systems ignores the fact that returned items may arrive out of sequence, i.e., with different number of remanufacturing times. Moreover, such modeling assumes that the retuned items may retain the same quality upon recovery regardless of how many times they have been previously remanufactured. This paper develops a new mathematical expression of the percentage of retuned items that can be remanufactured a finite number of times. The proposed expression is modeled as a function of the expected number of times an item can be remanufactured in its lifecycle and the number of times an item can be technologically (or optimally) remanufactured based on its quality upon recovery. The model developed in this paper considers joint production and remanufacturing options. The return rate is a varying demand-dependent rate, which is a decision variable with demand, product deterioration, manufacturing, and remanufacturing rates being arbitrary functions of time. The model considers the initial inventory of returned items in the mathematical formulation, which enables decision-makers to adjust all functions and input parameters for subsequent cycles. Illustrative examples indicate that dependent purchasing price of recovery items and the incorporation of remanufacturing investment cost significantly impact the optimal remanufacturing policy.

On the constant roll complex scalar field inflationary models

In this paper we wish to point out the possibility of using a complex scalar field in a constant roll inflationary model, as needed for observational viability. We extend the idea of real field inflaton with constant rate of roll to a complex field, showing the feasibility of solving Einstein Klein-Gordon equations constrained by an appropriate form of constant roll definition. As compared to the well known (two-parametric class of) real field models, there is one more degree of flexibility in constant roll inflationary solutions which is represented by an arbitrary function of time, γ(t). We work with an arbitrary but constant function γ (where γ = 0 refers to the corresponding real field model) and find new inflationary class of potentials. In this class of models, the behavior of real and complex field models are similar in some aspects, for example the solutions with large constant roll parameter are not stable and should be considered as early time transients. These field solutions relax at late time on a dual attractor trajectory. However, complex fields phase space trajectories reach this stable regime after real fields. We performed the stability analysis on γ function space solutions and found that dynamically stable trajectories in phase space are stable under γ variations. We extended this study by considering multifield models of constant roll inflation with non-canonical kinetic terms. By enlarging the size of field space, we showed that a multifield constant roll model is dynamically a single field effective theory. If field space is parametrized by N non-canonical fields, there will be N free parameters in the potential that can be attributed to the interaction between the fields.

Hydrodynamic Permeability in Axisymmetric Flows of Viscous Fluids through an Annular Domains with Porous Layer

Mass, energy, and momentum transfer processes between fluid-saturated porous media and the adjacent free flow occur in many natural and technical systems. The flow dynamics in the porous region and the adjacent free flow is strongly controlled by the mechanisms at the common interface and conditions on the outer surface of the free-flow. The present paper considers unsteady axisymmetric flows of viscous fluids through an annular domain with a porous layer covering a cylindrical solid core. Fluid flow in the domain filled with porous material and in transparent domain is described by Brinkman model and Navier Stokes equations, respectively. Analytical solutions for the dimensionless velocity fields in the Laplace domain are determined using Bessel functions, Laplace transform, and the appropriate interface and boundary conditions. The inversion of the Laplace transforms is done with the help of a numerical algorithm. In addition, the hydrodynamic permeability is determined. The dependence of the dimensionless velocity fields and of hydrodynamic permeability on characteristic parameters of the porous layer is numerically and graphically discussed. Since the velocity on the outer surface is given by an arbitrary function of time, the results in this paper could be used to study various filtration problems.

Study of transient analysis of a conductive beam carrying an electrical current subjected to magnetic field with elastically restrained ends

The present study considers the nonlinear vibration behavior of a beam with general boundary conditions that carry an electrical current in the magnetic field. This paper discusses the magnetic couple, the transverse magnetic force, the electrical current, and the damper. By contrast, the magnetic field is selected as an arbitrary function of time. Under certain hypotheses, Hamilton's principle is used along with Maxwell's equations to derive the governing equation. An elastically restrained beam carrying an electrical current is also solved using Galerkin's method under a magnetic field. Thus, the effect of the rotational and the translational support flexibilities, the magnetic field, and other parameters are evaluated. For a more detailed investigation, some numerical examples are investigated to present the simplicity and efficiency of this formulation. Based on the numerical results, it is clear that the natural frequency of the ferromagnetic beam is sensitive to the angle and magnetic field. By increasing magnetic field intensity, the magnitude of the natural frequency of the beam increases. But with the increase of the angle, the frequency value decreases. Therefore, at larger angles, the impact of the intensity of the magnetic field will be less. Also, it is determined from the results that the beam deflection in various magnetic fields indicates a significant effect of the boundary conditions, not only on the dynamic response of a damped beam but also on the rate of damping of the response. The dynamic response under the magnetic field is decreased when the beam experiences a stiffer constant in its support. The results are shown that the effect of stiffening for the transitional support is more significant than that of the rotational support. Also, the influence of the boundary constraints becomes smaller when the magnetic field becomes smaller.

Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory

A general approach is presented that offers exact analytical solutions for the time-evolution of quantum spin systems during parametric waveforms of arbitrary functions of time. The proposed method utilizes the path-sum method that relies on the algebraic and combinatorial properties of walks on graphs. A full mathematical treatment of the proposed formalism is presented, accompanied by an implementation in Matlab. Using computation of the spin dynamics of monopartite, bipartite, and tripartite quantum spin systems under chirped pulses as exemplar parametric waveforms, it is demonstrated that the proposed method consistently outperforms conventional numerical methods, including ODE integrators and piecewise-constant propagator approximations.

Motion of Particles around Time Conformal Dilaton Black Holes

In this paper, the geodesic motion of neutral and test particles around the time conformal (TC) Dilaton black hole (BH) is investigated using the eϵg(t) as the time conformal factor in which g(t) is an arbitrary function of time and ϵ is a perturbation parameter. The function g(t) leads to (ta) by utilizing the well-known approximate Noether symmetry (ANS). Furthermore, we discuss the effect of magnetic fields and find the location of stable and unstable orbits w. r. t time, graphically. After that, in the presence and absence of a magnetic field, we interrogate the crucial physical parameters such as effective potential (Ueff), effective force (Feff) and escape velocity (ν⊥). We find the unstable and stable regions of particles for different values of angular momentum (Lz) and magnetic field (B) near the TC Dilaton BH. Moreover, the effects of the Dilaton parameter (μ) on neutral and charged particles are also discussed, which provide some new features. The important results in this study could estimate the powerful relativistic jets originating from the BH.

Flexible parametric survival analysis with multiple timescales: Estimation and implementation using stmt

In this article, we describe methodology that allows for multiple timescales using flexible parametric survival models without the need for time splitting. When one fits flexible parametric survival models on the log-hazard scale, numerical integration is required in the log likelihood to fit the model. The use of numerical integration allows incorporation of arbitrary functions of time into the model and hence lends itself to the inclusion of multiple timescales in an appealing way. We describe and exemplify these methods and show how to use the command stmt , which implements these methods, alongside its postestimation commands.

Maxwell’s Equations in Homogeneous Spaces for Admissible Electromagnetic Fields

Maxwell’s vacuum equations are integrated for admissible electromagnetic fields in homogeneous spaces. Admissible electromagnetic fields are those for which the space group generates an algebra of symmetry operators (integrals of motion) that is isomorphic to the algebra of group operators. Two frames associated with the group of motions are used to obtain systems of ordinary differential equations to which Maxwell’s equations reduce. The solutions are obtained in quadratures. The potentials of the admissible electromagnetic fields and the metrics of the spaces contained in the obtained solutions depend on six arbitrary time functions, so it is possible to use them to integrate field equations in the theory of gravity.

An efficient boundary-type meshfree method for analysis of two-dimensional laser heating problems

The purpose of this study is to formulate a truly and effective boundary-type meshfree method for analysis of 2D transient heat conduction problems including moving laser heat sources. Being based on the method of fundamental solutions (MFS), the present work formulates new time-dependent formulations to solve the problem directly in the time-domain. The motion path and the intensity of the laser heat source over a moving area can be arbitrary functions of time and space. In the proposed formulations, the particular solution associated to the heat source function is evaluated using a simple approach without using any internal points or cells. By solving two example problems, the efficiency and accuracy of the proposed method are demonstrated. The analyses show that the presented boundary-type meshfree method is relatively more efficient than other domain-type techniques. From the analyses, it is also observed that very accurate results can be obtained by the proposed method employing large time-step size and small number of collocation points.

Dynamics of particle near time conformal slowly rotating Kerr black hole

In this work, we have developed the time conformal slowly rotating Kerr black hole using the approximate Noether symmetry approach. It is found that the slowly rotating Kerr black hole admits time conformal factor eεf(t), where ε is very small perturbation and f(t) is an arbitrary function of time. We have obtained the numerical solutions of geodesics for lowly rotating Kerr black hole and time conformal slowly rotating Kerr black hole. We have discussed the neutral and charged particle dynamics moving around time conformal slowly rotating Kerr black hole in the absence/presence of magnetic field. Moreover, we have analyzed graphically the behavior of effective force, effective potential and escape velocity of particles in the absence/presence of magnetic field near the event horizon. We have examined the conditions under which particle can escape from the surrounding of black hole after colliding with another particle. We have found the unstable orbits of particles near event horizon due to the effect of magnetic field and angular momentum.

Tomographic Description of a Quantum Wave Packet in an Accelerated Frame.

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

ЭЛЕКТРОИМПУЛЬСНЫЙ НАГРЕВ РЕАКЦИОННОСПОСОБНЫХ СОСТАВОВ ТОНКОСТЕННЫМИ ПРОВОДНИКАМИ-ТЭНАМИ В РЕЖИМАХ МНОГООЧАГОВОГО ЗАЖИГАНИЯ МЕТАТЕЛЬНЫХ ЗАРЯДОВ

The problem of electric pulse heating of a medium by the heater made of metal foil with an ideal thermal contact with the medium is solved. The solution is obtained in the form of rapidly converging series for ohmic heating power, which is given as an arbitrary function of time. Using the obtained solution, a dimensionless criterion is introduced for quasi-homogeneous heating of the heater up to a melting point with intensive heat transfer to environment. It is shown that with a fixed heating duration, the criterion value depends weakly on the type of a heating power function, which allows ignoring parameters of the external electrical circuit of the device while choosing the heater thickness. The definition of a "thin" heater is introduced, which admits of bulk droplet destruction during heating of a reactive medium. In the "thin" heater approximation, quadrature solutions to the thermal problem are obtained under heat exchange according to Newton's law and ideal thermal contact, when the heater power is an arbitrary function of time. Dimensionless similarity parameters of the simulated processes are identified in all solutions, which include thermal and physical characteristics of the heater and medium, as well as the thickness of the heater and the time of its heating up to the melting point. The applicability of the "thin" heater model for calculating the thermal state of a high-energy pyrotechnic coating is shown. As an example, the permissible values of the thickness of the magnesium foil heater are determined, which ensure its uniform heating up to the melting point in 1 and 5 milliseconds under the ideal thermal contact with ignited coating. The obtained quadrature solutions to the thermal problems and the calculated results are applicable in the design of compact electric pulse devices for contact multi-point ignition of various reactive compositions with efficient consumption of electric energy.

Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs

Dynamical PDEs that have a spatial divergence form possess conservation laws that involve an arbitrary function of time. In one spatial dimension, such conservation laws are shown to describe the presence of an x -independent source/sink; in two and more spatial dimensions, they are shown to produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint on initial/boundary data for which the dynamical PDE will be well posed. Several examples of nonlinear PDEs from applied mathematics and integrable system theory are used to illustrate these results.

EQUATIONS OF PLANETARY SYSTEMS MOTION

The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.