Terms Of Elementary Functions Research Articles

Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

This paper presents a method for parameterizing new Lorentz spacetime coordinates based on coupled parameters. The role of symmetry in rapidity in special relativity is explored, and invariance is obtained for new spacetime intervals with respect to the Lorentz transformation. Using the Euler–Hamilton equations, an additional angular rapidity and perpendicular rapidity are obtained, and the Hamiltonian and Lagrangian of a relativistic particle are expanded into rapidity spectra. A so-called passage to the limit is introduced that makes it possible to decompose physical quantities into spectra in terms of elementary functions when explicit decomposition is difficult. New rapidity-dependent Lorentz spacetime coordinates are obtained. The descriptions of particle motion using the old and new Lorentz spacetime coordinates as applied to plane laser pulses are compared in terms of the particle kinetic energy. Based on a classical model of particle motion in the field of a plane monochromatic electromagnetic wave and that of a plane laser pulse, rapidity-dependent spectral decompositions into elementary functions are presented, and the Euler–Hamilton equations are derived as rapidity functions in 3+1 dimensions. The new and old Lorentz spacetime coordinates are compared with the Fermi spacetime coordinates. The proper Lorentz groups SO(1,3) with coupled parameters using the old and new Lorentz spacetime coordinates are also compared. As a special case, the application of Lorentz spacetime coordinates to a relativistic hydrodynamic system with coupled parameters in 1+1 dimensions is demonstrated.

The Shocks in Josephson Transmission Line Revisited

We continue our previous studies of the shocks in the lossy Josephson transmission line (JTL). The article consists of two parts. In the first part, we analyze the scattering of the “sound” (small amplitude small wave vector harmonic wave) on the shock wave and calculate the reflection and the transmission coefficients. In the second part, we show that the kinks, which we previously studied only in the lossless JTL, exist also in the lossy JTL and study the similarities and the dissimilarities between the shocks and the kinks there. We find that the nonlinear equation describing the weak kinks and the weak shocks can be integrated (in particular cases) in terms of elementary functions. We also show that the profile of the shock in the lossy JTL demonstrates oscillatory behavior with leading peaks resembling solitary waves if the losses are weak.

Novel class of susceptible–infectious–recovered models involving power-law interactions

It is shown that the ordinary SIR (Susceptible–Infectious–Recovered) epidemic model exhibits features that are common to a class of compartmental models with power-law interactions. Within this class of theoretical models, the standard SIR model emerges as a singular non-integrable model. Various integrable models, whose solutions are defined explicitly or implicitly in terms of elementary functions, are discovered within the same class. A Hamiltonian dynamics with position-depending forces underlies a sub-class of these models. The general class of models is very flexible and capable of describing epidemics characterized by a finite or indefinite lifespan. In the last case, the compartment population distributions evolve in time exhibiting exponential or power-law tails.

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices

The Ablowitz–Ladik equations, hereafter called and , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of in the periodic setting, showing in particular that the background solution of is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of are always singular on curves of spacetime, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution (i) evolves according to into a recurrence of the above exact solutions, in the case of one and two unstable modes, and (ii) evolves according to into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect.

On Geometric Interpretations of Euler’s Substitutions

We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such “irrational integrals” can be expressed in terms of elementary functions by one of three of Euler’s substitutions. It is less well known that the Euler substitutions have a geometric interpretation. In the framework of this interpretation, one can see that the number 3 is not the most suitable. We show that it is natural to introduce a fourth Euler substitution. In his original treatise, Leonhard Euler used two substitutions which are sufficient to cover all cases.

Anisotropic Strange Stars with Nonlinear Equation of State

The modeling of superdense matter is an interesting research area and in the last decades, such models allow explain the behavior of massive objects as neutron stars, quasars, pulsars and white dwarfs. Taking local anisotropy into consideration, in this paper we present a new classes of solutions for the Einstein´s field equations in a spherically symmetric spacetime under the influence of an electric field considering a quadratic equation of state with a particular form the metric potential that depends on an adjustable parameter. The obtained solutions can be written in terms of elementary functions, namely polynomials and algebraic functions. The graphs generated show that physical variables such as metric potentials, radial pressure, energy density, charge density, anisotropy, radial speed sound are consistent with realistic stellar models. The results of this research can be useful in the development and description of new models of compact structures. Keywords: quadratic equation of state, Einstein-Maxwell system, metric potential, adjustable parameter, compact structures

Kerr-fully diving into the abyss: analytic solutions to plunging geodesics in Kerr

We present closed-form solutions for the generic class of plunging geodesics in the extended Kerr spacetime using Boyer–Lindquist coordinates. We also specialise to the case of test particles plunging from the innermost precessing stable circular orbit (ISSO) and unstable spherical orbits. We find these solutions in the form of elementary and Jacobi elliptic functions parameterised by Mino time. In particular, we demonstrate that solutions for the ISSO case can be determined almost entirely in terms of elementary functions, depending only on the spin parameter of the black hole and the radius of the ISSO. Furthermore, we introduce a new equation that characterises the radial inflow from the ISSO to the horizon, taking into account the inclination. For ease of application, our solutions have been implemented in a Mathematica package that is available as part of the KerrGeodesics package in the Black Hole Perturbation Toolkit.

Tight bounds for the median of a gamma distribution.

The median of a standard gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form 2-1/k(A + k): an upper bound with A = e-γ (with γ being the Euler-Mascheroni constant) and a lower bound with [Formula: see text]. These bounds are valid over the entire domain of k > 0, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.

A Note on Some Generalized Hypergeometric Reduction Formulas

Herein, we calculate reduction formulas for some generalized hypergeometric functions m+1Fmz in terms of elementary functions as well as incomplete beta functions. For this purpose, we calculate the n-th order derivative of the function zγBzα,β with respect to z. As corollaries, we obtain reduction formulas of these m+1Fmz functions for argument unity in terms of elementary functions, as well as beta functions.

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index . For Cauchy and Brownian motions (i.e. ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

Shielding effect of ring dislocation dipole on penny-shaped crack

Mechanical behavior of strongly crystalline materials can be significantly influenced by the internal dislocation structure and their iteration with micro crack-like defects. Thus, study on the crack-dislocation interaction problems can help to understand the failure mechanism of crystalline materials. Previous theoretical analysis have been mostly limited to plane problems, where the cracks are idealized as straight slit with infinite or zero out plane dimension and the dislocations slip along an inclined straight path starting from the crack tip. In this paper, we explore a new interaction mechanism under axisymmetric deformation state. An unconventional case of a penny-shaped crack interacts with a ring dislocation dipole is considered. Both the ring dislocation dipole and crack are modelled as axisymmetric edge dislocation loops. Solutions for the dislocation loops are obtained using the GKS based method. The mixed boundary value problem is converted into Abel integral equations, which can be solved analytically. Exact closed form solutions for the stress intensity factors (SIFs) in terms of elementary functions are obtained. Numerical studies are conducted to investigate the shielding effect of the ring dislocation dipole on the crack tip SIFs. Conditions for the dislocation to shield the crack are discussed. The present solutions can also be used as weight functions to study the axisymmetric interaction between penny shaped crack and other types of eigenstrain sources.

INTEGRATION OF A THIRD-ORDER ODE VIA ANALYTICAL AND GEOMETRICAL METHODS

Analytical and geometrical methods are applied to integrate an ordinary differential equation of third order. The main objective is to compare both approaches and show the possibilities that each one of them offers in the integration process of the considered equation, specially when not only Lie point symmetries but also generalized symmetries are involved. The analytical method of order reduction by using a generalized symmetry provides the general solution of the equation but in terms of a primitive that cannot be explicitly evaluated. On the other hand, the application of geometrical tools previously reported in the recent literature leads to two functionally independent first integrals of the equation without any kind of integration. In order to complete the integration of the given third-order equation, a third independent first integral arises by quadrature as the primitive of a closed differential one-form. From these first integrals, the expression of the general solution of the equation can be expressed in parametric form and in terms of elementary functions.

New asymptotic representations of the noncentral t‐distribution

AbstractNew asymptotic approximations of the noncentral t distribution are given a generalization of the Student's t distribution. Using new integral representations, we give new asymptotic expansions not only for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some cases, we accept more than one large parameter. These results are not only in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.

Two-loop superfield effective potential for N=2 , d=3 supersymmetric quantum electrodynamics with higher derivatives

In the $\mathcal{N}=2$, $d=3$ superspace, we consider a higher-derivative generalization of the supersymmetric quantum electrodynamics, where the higher-derivative operator is a polynomial function of the d'Alembertian with arbitrary degree. For this theory, we use the background field quantization in a higher-derivative $R_\xi$ gauge to explicitly calculate the superfield effective potential up to two loops in the K\"{a}llerian approximation. This superfield effective potential is obtained in a closed form and in terms of elementary functions.

Approximate Solution Using Elementary Functions of Mixed Problem with Boundary Conditions of the Second Kind for One-Dimensional Wave Equation

The paper considers a mixed problem with boundary conditions of the second kind for a one-dimensional wave equation. The solution to this problem is written in integral form using the Green’s function. For practical use, this solution is of little use, since, firstly, the Green’s function is a trigonometric series and, therefore, its calculation presents certain difficulties, secondly, it is necessary to calculate approximately the five integrals with the Green’s function included in the solution of the problem, and, thirdly, it is extremely difficult to estimate the error of the approximate calculation of the solution. In this work, these difficulties are overcome, namely, simple expression for the Green’s function is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piecewise linear, piecewise quadratic and piecewise cubic functions, and, finally, a simple and efficient estimate of the approximation error is obtained. The error estimate is linear in the grid steps of the problem and uniform in the spatial variable at any fixed point in time. Thus, an approximate solution of the problem with an arbitrarily small error is effectively expressed in terms of elementary functions. An example of solving the problem by the proposed method is given, and graphs of the exact and approximate solutions are plotted.

On the Su–Schrieffer–Heeger model of electron transport: Low‐temperature optical conductivity by the Mellin transform

AbstractWe describe the low‐temperature optical conductivity as a function of frequency for a quantum‐mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heeger tight‐binding Hamiltonian for noninteracting spinless electrons on a one‐dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo‐type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three‐dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.

Exact solution of the Einstein field equations for a spherical shell of fluid matter

We determine the exact solution of the Einstein field equations for the case of a spherically symmetric shell of liquid matter, characterized by an energy density which is constant with the Schwarzschild radial coordinate r between two values $$r_{1}$$ and $$r_{2}$$ . The solution is given in three regions, one being the well-known analytical Schwarzschild solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined mostly analytically but partially numerically, within the matter region. The solutions for the temporal coefficient of the metric and for the pressure within this region are given in terms of a non-elementary but fairly straightforward real integral. For some values of the parameters this integral can be written in terms of elementary functions. We show that in this solution there is a singularity at the origin, and give the parameters of that singularity in terms of the geometrical and physical parameters of the shell. This does not correspond to an infinite concentration of matter, but in fact to zero energy density at the center. It does, however, imply that the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This gravitational field is repulsive with respect to the origin, and thus has the effect of stabilizing the geometrical configuration of the matter, since any particle of the matter that wanders out into either one of the vacuum regions tends to be brought back to the bulk of the matter by the gravitational field.

Projectile motion with quadratic drag

Two-dimensional coupled nonlinear equations of projectile motion with air resistance in the form of quadratic drag are often treated as inseparable and solvable only numerically. However, when they are recast in terms of the angle between the projectile velocity and the horizontal, they become completely uncoupled and possess analytic solutions for projectile velocities as a function of that angle. The equations relating the time and position coordinates to this angle are not integrable in terms of elementary functions but are easy to integrate numerically. Additionally, energy equations explicitly including dissipation terms can be developed as integrals of the equations of motion. One-dimensional numerical integrations can be treated in a pedagogically straightforward way using numerical analysis software or even within a spreadsheet, making this topic accessible to undergraduates. We present this approach with sample numerical results for velocity components, trajectories, and energy-balance of a baseball-sized projectile.

On the Krýlov–Bogoliúbov-Mitropólsky and Multiple Scales Methods for Analyzing a Time Delay Duffing–Helmholtz Oscillator

This study is divided into two important axes; for the first one, a new symmetric analytical (approximate) solution to the Duffing–Helmholtz oscillatory equation in terms of elementary functions is derived. The obtained solution is compared with the numerical solution using 4th Range–Kutta (RK4) approach and with the exact analytical solution that is obtained using elliptic functions. As for the second axis, we consider the time-delayed version for the same oscillator taking the impact of both forcing and damping terms into consideration. Some analytical approximations for the time delayed Duffing–Helmholtz oscillator are derived using two different perturbation techniques, known as Krylov–Bogoliubov–Mitropolsky method (KBMM) and the multiple scales method (MSM). Moreover, these perturbed approximations are analyzed numerically and compared with the RK4 approximations.

Uniform Convergent Expansions of the Error Function in Terms of Elementary Functions

We derive a new analytic representation of the error function {text {erf}},, z in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector vert arg (pm z)vert <pi /4. The expansion is accompanied by realistic error bounds.