Terms Of Elliptic Integrals Research Articles

Two-dimensional disordered lattice networks with substrate

The paper deals with the development of a theory for describing two-dimensional (2D) random lattice networks of resistors with a particular topology. We consider a 2D anisotropic random lattice where each node of the network is connected to a reference node (substrate) through a given random resistor. This topology is of great interest both for theoretical and practical applications. Moreover, the theory is able to take into account the similar, but more interesting problem with a capacitive coupling with the substrate. The analytical results allow us to obtain the average behaviour of such networks, i.e. the electrical characterisation of the corresponding physical systems. This effective medium theory is developed starting from the properties of the lattice Green's function of the network and from an ad hoc mean field procedure. An accurate analytical study of the related lattice Green's functions has been conducted obtaining closed-form results expressed in terms of elliptic integrals. All the theoretical results have been verified by means of numerical Monte-Carlo simulations obtaining a remarkably good agreement between numerical and theoretical values, both in resistive and capacitive systems.

Boundary-Integral method for calculating poloidal axisymmetric AC magnetic fields

This paper presents a boundary-integral equation (BIE) method for the calculation of poloidal axisymmetric magnetic fields applicable in a wide range of ac frequencies. The method is based on the vector potential formulation, and it uses the Green's functions of Laplace and Helmholtz equations for the exterior and interior of conductors, respectively. The paper focuses on a calculation of axisymmetric Green's function for the Helmholtz equation which is both simpler and more accurate than previous approaches. We use three different techniques for calculation of Green's function, depending on the parameter range. For low and high ac frequencies, we use a power series expansion in terms of elliptical integrals and an asymptotic series in terms of modified Bessel functions of second kind, respectively. For the intermediate frequency range, we use Gauss-Chebyshev-Lobatto quadratures. We verify the accuracy of the method by comparing its results with the analytical solution for a sphere in a uniform external ac field. We illustrate the application of the method by analyzing a composite model inductor containing an external secondary circuit.

Disordered lattice networks: general theory and simulations: Research Articles

In this work we develop a theory for describing random networks of resistors of the most general topology. This approach generalizes and unifies several statistical theories available in literature. We consider an n-dimensional anisotropic random lattice where each node of the network is connected to a reference node through a given random resistor. This topology includes many structures of great interest both for theoretical and practical applications. For example, the one-dimensional systems correspond to random ladder networks, two-dimensional structures model films deposited on substrates and three-dimensional lattices describe random heterogeneous materials. Moreover, the theory is able to take into account the anisotropic percolation problem for two- and three-dimensional structures. The analytical results allow us to obtain the average behaviour of such networks, i.e. the electrical characterization of the corresponding physical systems. This effective medium theory is developed starting from the properties of the lattice Green's function of the network and from an ad hoc mean field procedure. An accurate analytical study of the related lattice Green's functions has been conducted obtaining many closed form results expressed in terms of elliptic integrals. All the theoretical results have been verified by means of numerical Monte-Carlo simulations obtaining a remarkably good agreement between numerical and theoretical values. Copyright © 2005 John Wiley & Sons, Ltd.

A comparative study of rectal dose histograms in prostate brachytherapy: Some analytic and numerical results

A cumulative dose histogram is the graph of an integral function integrated over a domain VCR3 and is dubbed the dose-volume histogram (DVH), the dose-surface histogram (DSH) or the dose-wall histogram (DWH), depending on the dimension and structure of the region V. This paper presents a comparative study of the three rectal dose histograms for sixty patients as well as for a cylindrical model of the rectum; in particular, the DSH and DVH for the cylindrical model with one point source are computed analytically in terms of elliptic integrals. The difference among the three relative dose histograms, averaged over the sixty patients, is less than 5%, whereas that between DVH and DWH for various wall-thickness can be as large as 3-12 cm3 in the range 60-100 Gy. The paper also contains an error analysis using two simple models of the rectum, for which the true DSH and DWH can be computed via numerical integration, to evaluate the effect of digitization. The digitized computation agrees quite well with the pre-digitization numerical integration, within 1% or 0.2 cm3, because of the low dose-gradient effect near the rectum in prostate brachytherapy.

Disordered lattice networks: general theory and simulations

In this work we develop a theory for describing random networks of resistors of the most general topology. This approach generalizes and unifies several statistical theories available in literature. We consider an n-dimensional anisotropic random lattice where each node of the network is connected to a reference node through a given random resistor. This topology includes many structures of great interest both for theoretical and practical applications. For example, the one-dimensional systems correspond to random ladder networks, two-dimensional structures model films deposited on substrates and three-dimensional lattices describe random heterogeneous materials. Moreover, the theory is able to take into account the anisotropic percolation problem for two- and three-dimensional structures. The analytical results allow us to obtain the average behaviour of such networks, i.e. the electrical characterization of the corresponding physical systems. This effective medium theory is developed starting from the properties of the lattice Green's function of the network and from an ad hoc mean field procedure. An accurate analytical study of the related lattice Green's functions has been conducted obtaining many closed form results expressed in terms of elliptic integrals. All the theoretical results have been verified by means of numerical Monte-Carlo simulations obtaining a remarkably good agreement between numerical and theoretical values. Copyright © 2005 John Wiley & Sons, Ltd.

A Lie Symmetry Connection between Jacobi’s Modular Differential Equation and Schwarzian Differential Equation

In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi’s modular differential equation. A six-dimensional Lie symmetry algebra is obtained and its symmetry generators are found to be given in terms of elliptic integrals. As a consequence the transformation between Jacobi’s modular differential equation and the well-known Schwarzian differential equation is derived.

Slow nonlinear oscillations in a circular well

The period $T$ of the Helmholtz mode in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir is determined in terms of elliptic integrals. It is shown that $T$ decreases monotonically with increasing amplitude $A$ and is within 1% (10%) of the linear limit $T_0$ for $A{/}h_0\,{<}\,0.4(1.0)$, where $h_0$ is the ambient depth.

New Geometrical Applications of the Elliptic Integrals: The Mylar Balloon

An explicit parameterization in terms of elliptic integrals (functions) for the Mylar balloon is found which then is used to calculate various geometric quantities as well as to study all kinds of geodesics on this surface.

The Two Fixed Centers: An Exceptional Integrable System

It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures, a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space. We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation number’) as a function of the two integrals of motion.

Electroencephalography in ellipsoidal geometry

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.

Solid angle subtended by a cylindrical detector at a point source in terms of elliptic integrals

The solid angle subtended by a right circular cylinder at a point source located at an arbitrary position generally consists of a sum of two terms: that defined by the cylindrical surface (Ω cyl ) and the other by either of the end circles (Ω circ ) . We derive an expression for Ω cyl in terms of elliptic integrals of the first and third kinds and give similar expressions for Ω circ using integrals of the first and second kinds. These latter can be used alternatively to an expression also in terms of elliptic integrals, due to Philip A. Macklin and included as a footnote in Masket (Rev. Sci. Instrum. 28 (3) (1957) 191). The solid angle subtended by the whole cylinder when the source is located at an arbitrary location can then be calculated using elliptic integrals.

Shapes of Submerged Funicular Arches

In this paper, the analytical forms of the shapes of submerged funicular arches are expressed in terms of elliptic integrals. The initial radius of curvature at the apex is related to the water depth at the apex and the axial compressive force. The shape of the submerged funicular arches depends on the ratio between this initial radius of curvature and the water depth at the apex. Using the analytical expressions, the maximum span and height of the submerged funicular arches can be determined explicitly. Besides, the analytical expressions are useful in determining the design parameters for the submerged funicular arches accurately.

Approximation of Light‐Ray Deflection Angle and Gravitational Lenses in the Schwarzschild Metric. I. Derivation and Quasar Lens

We calculate a second-order approximation for the light-ray deflection in a Schwarzschild metric. The approximation is based on Darwin's exact solution for the deflection angle, which is written in terms of elliptic integrals. Even if the derived equation is almost as simple as the usual Einstein deflection angle, it follows the Darwin solution more faithfully. This enables relatively simple calculations of gravitational lensing in axially symmetric conditions with a small value of the impact parameter b. We demonstrate the effects of second-order accuracy by modeling the lensed quasar QSO B0957+561A, B. Our simple toy model leads to a 4% difference in mass estimate for the deflector, when comparing the Einstein solution with the approximation. The naive point mass model is certainly inadequate when modeling a realistic quasar lens, but it demonstrates clearly the effects of the second-order accuracy and could, nevertheless, be used for some supermassive black hole models.

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.

Landau theory of second-order phase transitions in ferroelectric films

A detailed discussion is given of the predictions of the Landau-Devonshire theory for the behavior of thin ferroelectric films. It is assumed that the phase transition in the bulk is second order, and the theory is extended by the inclusion in the free-energy density of a term proportional to $|\ensuremath{\nabla}P{|}^{2}$ together with a boundary condition involving an extrapolation length \ensuremath{\delta}. Expressions for the polarization profile $P(z)$ are given in terms of Jacobi elliptic functions; some of these are in a clearer and simpler form than has been available previously. Interesting analytic results for the entropy in terms of elliptic integrals and elliptic functions are also presented. Graphical illustrations of the main results are included. Careful consideration is given to the nature of the phase transition, and it is concluded that it is second order in all cases.

Analytical Approach to Large Deformation Problems of Frame Structures. In Case of a Square Frame with Rigid Joints.

In elements used as flexible linking devices and structures, the main characteristic is a fairly large deformation without exceeding the elastic limit of the material. This property is of both analytical and technological interests. Previous studies of large deformation have been generally concerned with a single member (e.g. a cantilever beam, a simply supported beam, etc.). However, there are very few large deformation studies of assembled members such as frames. This paper deals with a square frame with rigid joints, loaded diagonally in either tension or compression by a pair of opposite forces. Analytical solutions for large deformation are obtained in terms of elliptic integrals, and are compared with the experimental data. The agreement is found to be fairly close.

Analytic solution for two bubbles in a hele-shaw cell

A four-parameter family of exact solutions is reported for two unequal bubbles moving steadily in a Hele-Shaw cell when surface tension is neglected. The solutions are given in closed form in terms of elliptic integrals and are found to be in good agreement with shapes observed in the experiments by Ikeda and Maxworthy [Phys. Rev. A 41, 4367 (1990)]. An analytic solution for a finger with a bubble is also obtained as a special case of the general two-bubble solution. The relevance of these exact solutions for a selection theory for the cases of a finger with a bubble and a pair of bubbles is also briefly discussed.

Large deformation analysis of a square frame with rigid joints

In elements used as flexible linking devices and structures, the main characteristic is a fairly large deformation without exceeding the elastic limit of the material. This property is of both analytical and technological interests. Previous studies of large deformation have been generally concerned with a single member (e.g. a cantilever beam, a simply supported beam, etc.). However, there are very few large deformation studies of assembled members such as frames. This paper deals with a square frame with rigid joints, loaded diagonally in either tension or compression by a pair of opposite forces. Analytical solutions for large deformation are obtained in terms of elliptic integrals, and are compared with the experimental data. The agreement is found to be fairly close.

Analytical solutions for the Newtonian gravitational field induced by matter within axisymmetric boundaries

An analytical method originally applied to the problem of the actuator disc in fluid mechanics has been applied to the closely analogous problem of constructing the classical Newtonian potential and attractions. The method can treat axisymmetric problems and also non-axisymmetric cases where matter is confined within axisymmetric boundaries. The potential and attractions for the generalized thin finite disc can be given in closed form in terms of elliptic integrals and elementary functions. For the general case of matter within an axisymmetric boundary, the potentials and attractions can be evaluated as one-dimensional integrals of albeit complex analytical expressions. These expressions represent the fields induced by matter in an extended region as a distribution of gravitating discs. For certain special cases, such as matter bounded by a circular cylinder and also for matter distributed in a spherical region, closed-form solutions can be given that appear to be new. Some non-axisymmetric results are also given for the thin disc of infinite radial extent.

Stability of a pipe through which a string is pulled

Stability of an elastic pipe through which a string is pulled with the constant velocity is studied by the Liapunov– Schmidt method. It is assumed that imperfections in shape (small initial deformation) and loading (distributed load along the axis of the pipe) are present. Stability boundary is obtained from the eigenvalues of the linearized equations. It is shown that the bifurcation is super-critical. The conditions guaranteeing that imperfections introduced here form a universal unfolding are stated. The post-critical shape of perfect pipe is determined by numerical integration of the corresponding system of equations. For this system of equations we also found two new first integrals. For the case of Bernoulli–Euler model of the pipe the post-critical shape is expressed in terms of elliptic integrals.