Terms Of Quadratures Research Articles

Existence and uniqueness of solution to the system of integral equations in the planar Earth, Sun and satellite system

This manuscript delves into the exploration of the existence and uniqueness of the nth level approximation within the context of the inertial restricted three-body problem. In this scenario, two massive celestial bodies, namely Earth and the Sun, are held stationary along a straight line, while a less massive object serves as an artificial satellite. Within the manuscript, we have uncovered solutions expressed in terms of quadratures, infinite series, and transcendental functions. Our investigation employs a system of integral equations to address the challenge posed by these two immobile centers. The process initiates with the derivation of the equations of motion for the Earth, Sun, and satellite system, all considered within the inertial coordinate system. Subsequently, we formulate the nth level approximation and present its solution for the linear integral equations system. We also meticulously determine the conditions necessary for the solution to converge. Additionally, we engage in an in-depth discussion regarding the existence of such a solution. Moreover, the manuscript firmly establishes the uniqueness of this solution, assuring its singularity. Furthermore, we undertake a rigorous analysis to quantify the error associated with the nth level approximated solution.

Modified phase-generated carrier demodulation of fiber-optic interferometric ultrasound sensors

We propose and demonstrate a modified phase-generated carrier (PGC) demodulation scheme optimized for detection of ultrasound using interferometric sensors with sinusoidal fringes. The sensor used in demonstration is made from a pair of weak fiber Bragg-gratings at the ends of a coiled fiber that form a low-finesse Fabry-Perot interferometer. The phase of the laser source is modulated using an electro-optic phase modulator to generate the carrier signal and obtain 2 quadrature (the sine and cosine) terms at the first and the second order carrier frequencies. The signal of interest (ultrasound) has much higher frequency than the environmental perturbation but a very small amplitude that causes only small phase shift. Using small-signal approximation, for each of the 2 quadrature terms, we separate the contributions from the environmental perturbations (quasi-DC component) and from the ultrasound (AC component). The AC components that contain the information of the ultrasound signal are then further amplified with a large gain. The signal of interest is constructed by simple algebraic operations on the 2 quasi-DC components and the 2 amplified AC components involving multiplying and summing. This work provides a simple and robust demodulation method with potentially high sensitivity for fiber-optic interferometric ultrasound sensors.

Slowly rotating black holes in quasitopological gravity

While cubic Quasi-topological gravity is unique, there is a family of quartic Quasi-topological gravities in five dimensions. These theories are defined by leading to a first order equation on spherically symmetric spacetimes, resembling the structure of the equations of Lovelock theories in higher-dimensions, and are also ghost free around AdS. Here we construct slowly rotating black holes in these theories, and show that the equations for the off-diagonal components of the metric in the cubic theory are automatically of second order, while imposing this as a restriction on the quartic theories allows to partially remove the degeneracy of these theories, leading to a three-parameter family of Lagrangians of order four in the Riemann tensor. This shows that the parallel with Lovelock theory observed on spherical symmetry, extends to the realm of slowly rotating solutions. In the quartic case, the equations for the slowly rotating black hole are obtained from a consistent, reduced action principle. These functions admit a simple integration in terms of quadratures. Finally, in order to go beyond the slowly rotating regime, we explore the consistency of the Kerr-Schild ansatz in cubic Quasi-topological gravity. Requiring the spacetime to asymptotically match with the rotating black hole in GR, for equal oblateness parameters, the Kerr-Schild deformation of an AdS vacuum, does not lead to a solution for generic values of the coupling. This result suggests that in order to have solutions with finite rotation in Quasi-topological gravity, one must go beyond the Kerr-Schild ansatz.

On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole

It is well known that the equations governing the evolution of scalar, electromagnetic, and gravitational perturbations of the background geometry of a Schwarzschild black hole can be reduced to a single master equation. We use Kovacic’s algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of this master equation. We prove that the algebraically special Liouvillian solutions χ and χ∫dr *χ2, initially found by Chandrasekhar in the gravitational case, are the only Liouvillian solutions to the master equation. We show that the Liouvillian solution χ∫dr *χ2 is a product of elementary functions, one of them being a polynomial solution P to an associated confluent Heun equation. P admits a finite expansion both in terms of truncated confluent hypergeometric functions of the first kind, and also in terms of associated Laguerre polynomials. Remarkably both expansions entail not constant coefficients but appropriate function coefficients instead. We highlight the relation of these results with inspiring new developments. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher.

The ground state of long-range Schrödinger equations and static q q ¯ $$ q\overline{q} $$ potential

Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d Schr\"odinger operators $-\frac{d^{2}}{dx^{2}}+\lambda\,V(x)$ with long-range potential $V(x)$. In particular, we consider even potentials obeying $\int_{\mathbb R}dx\, V(x)<0$ with large $x$ asymptotics $V\sim -a/x^{2}-b/x^{3}+\cdots$. The associated Schr\"odinger operator is known to admit a bound state for $\lambda\to 0^{+}$, but the binding energy is rigorously non-analytic at $\lambda=0$. Its asymptotic expansion starts at order $\mathcal O(\lambda)$, but contains higher corrections $\lambda^{n}\,\log^{m}\lambda$ with all $0\le m\le n-1$ and standard Rayleigh-Schr\"odinger perturbation theory fails order by order in $\lambda$. We discuss various analytical tools to tame this problem and provide the general expansion of the binding energy at $\mathcal O(\lambda^{3})$ in terms of quadratures. The method is tested on a soluble potential that is fully under control, and on various non-soluble cases as well. A supersymmetric case, arising in the study of the quark-antiquark potential in $\mathcal N=6$ ABJ(M) theory, is also exploited to provide a further non-trivial consistency check. Our analytical results confirm at third order a remarkable exponentiation of the leading infrared logarithms, first noticed in $\mathcal N=4$ SYM where it may be proved by Renormalization Group arguments. We prove this interesting feature at all orders at the level of the Schr\"odinger equation for general potentials in the considered class.

Development of Zeldovich’s approach for cosmological distances measurement in the Friedmann Universe

We present our development of Zeldovich's ideas for the measurement of the cosmological angular diameter distance (ADD) in the Friedmann Universe. We derive the general differential equation for the ADD measurement which is valid for an open, spatially-flat and closed universe, and for any stress energy tensor. We solve the mentioned equations in terms of quadratures in a form suitable for further numerical investigations for the present universe filled by radiation, (baryonic and dark) matter and dark energy. We perform the numerical investigation in the absence of radiation, and show the strong dependence ADD on the filling of the cone of light rays (CLR). The difference of the empty and totally filled CLR may reach 600-700 Mps. for the redshift $f\simeq 3$.

Kinetic models for historical processes of fast invasion and aggression.

In the last few decades many investigations have been devoted to theoretical models in new areas concerning description of different biological, sociological, and historical processes. In the present paper we suggest a model of the Nazi Germany invasion of Poland, France, and the USSR based on kinetic theory. We simulate this process with the Cauchy boundary problem for two-element kinetic equations. The solution of the problem is given in the form of a traveling wave. The propagation velocity of a front line depends on the quotient between initial forces concentrations. Moreover it is obtained that the general solution of the model can be expressed in terms of quadratures and elementary functions. Finally it is shown that the front-line velocities agree with the historical data.

Vector Definitions for Control of Single-Phase and Unbalanced Three-Phase Systems

The d- q transformation applied to three-phase balanced systems allows for convenient definitions of vectors, which are consistent with the usual notion of phasors in sinusoidal steady state. If harmonics are neglected, then the dynamic models of balanced power-electronic systems in the d- q variables are also time-invariant, which facilitates vector control design. Alternative definitions are required for single-phase systems and unbalanced three-phase systems, around which similar controllers can then be built. This paper presents a unified theoretical analysis of these definitions. Generalizations of the time-delay-based definition are presented, which are also shown to be amenable to vector control. The dynamic equations for zero-sequence components reveal that the strategy to avoid cross-coupling between in-phase and quadrature terms in typical vector-control schemes is applicable to the control of zero-sequence vector components as well.

Flow of a plastically anisotropic layer between parallel planes in a region with a sink

A self-similar solution to the problem of flow of a plastically anisotropic layer between parallel planes in a region with a sink is constructed. The problem can be reduced to solving an unstudied Riccati equation with a small parameter. The perturbation method is used to find two different solutions expressed in terms of quadratures.

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.

Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.

Two non-holonomic integrable problems tracing back to Chaplygin

The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange’s gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.

Traveling wave solutions of the nonlinear dispersive Klein–Gordon equations

In this paper, the traveling wave solutions to the nonlinear dispersive Klein–Gordon equation is obtained. There are five forms of nonlineraity that are studied in this paper. While in each of the cases, the results are in terms of quadratures, the numerical simulation of the solutions are given for completeness.

Обобщение преобразования Чаплыгина и явное интегрирование шарового подвеса

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.

Model of film coating with weakly curved surface

We use the perturbation method to construct a solution of the plane problem of elasticity for a film-foundation composite where the film surface is weakly curved. In the case where the film surface has a periodic shape, the problem solution in each approximation is represented in terms of Fourier series with coefficients expressed in terms of quadrature. In the first approximation, we obtain the stresses on the film surface and on the interphase surface in terms of the surface curvature, the film average thickness, and the film-to-foundation Young’s modulus ratio.

The diffusion coefficient of nonlinear Brownian motion

Nonlinear Brownian motion (BM) refers to cases where the damping constant and possibly also the noise intensity in the Langevin equation depend on the velocity of the particle. Such velocity dependence is encountered in cases where Stokes' linear friction law does not apply, for relativistic Brownian particles, and for models of active motion of biological objects. For an arbitrary velocity dependence of damping and noise intensity, the diffusion coefficient can be given in terms of quadratures. We evaluate and discuss this quadrature formula for the three different cases. For a nonlinear friction (being larger at high speed than expected from Stokes' friction) and for the relativistic BM we obtain in general diffusion coefficients that are smaller than those for linear BM. The diminished diffusion in these equilibrium systems has different physical reasons. For the nonequilibrium model of active motion we demonstrate that diffusion can be minimized at a finite noise intensity.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ and μ. We consider several special cases of this problem: (a) for μ known a priori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known a priori, μ is determined by a single deformation field up to a constant. This includes as a special case that for which the term λ∇ ⋅ u ≡ 0. (c) Finally, if neither λ nor μ is known a priori, but Poisson's ratio ν is known, then μ and λ are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for μ based on the compressible elasticity equations is unstable in the limit λ → ∞. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.

New numerical integrators based on solvability and splitting

When Lie-group integrators such as those based on the Magnus expansion are applied to linear systems of ODEs, it is necessary to evaluate matrix exponentials. This leads to a reduction in their computational efficiency when the dimension of the matrix is very large. For quadratic Lie groups it is possible to approximate the matrix exponential by a rational function and still preserve the Lie-group structure, but this is no longer true in the important case of the special linear group. In this paper we propose a new class of integration algorithms especially designed to avoid this problem. They are based on expressing the solution as a product of upper and lower triangular matrices obtained explicitly in terms of quadratures. We analyse the main features of the procedure and its feasibility as a practical numerical method.

A sphere rolling on the inside surface of a cone

We analyse the motion of a sphere that rolls without slipping on the inside of a conical surface having its axis in the direction of the constant gravitational field of the Earth. This non-holonomic system admits a solution in terms of quadratures. We exhibit that for each set of conditions defining the motion there is only one circular orbit which is stable. We show that its solutions can be found using an analogy with central force problems. We also discuss the case of motion with no gravitational field, that is, of motion on a freely falling cone.

A note on point source diffraction by a wedge

The object of this paper is to give new expressions for the wave field produced when a time harmonic point source is diffracted by a wedge with Dirichlet or Neumann boundary conditions on its faces. The representation of the total field is expressed in terms of quadratures of elementary functions, rather than Bessel functions, which is usual in the literature. An analogous expression is given for the three‐dimensional free‐space Green′s function.