Angular Integral Research Articles

Headphones

A system and method for conducting surveys of vocalizing wildlife by determining the direction species of wildlife that emit vocalizations is disclosed. The system includes a sound receiving unit (12) with a set of three microphones (14, 16, 18) for monitoring the vocalizations. The signals received by the microphones are supplied to an audio signal processing circuit (20) that produces a set of subtractive signals indicating the relative signal strength of the received signals and a digital signal representative of the period of the received vocalizations. The signals produced by the audio signal processing circuit are supplied to a central processing unit (22). The central processing unit calculates the frequency of the vocalization based digital signal representative of the period of the signal and based on a set of these signals determines the species that emitted the vocalizations. A angular detection circuit integral with the audio signal processing circuit determines the angular direction relative to the sound receiving unit of from where the vocalization originated based on the subtractive signals.

Stabilization of Flexible One-Link Arm Position: Stability Domains in the Space of Feedback Gains

A one-link flexible arm with a load at the end is considered. The arm can be rotated around its hub in a horizontal plane. This rotation is actuated by a DC motor. The dynamics of a controlled flexible arm are described by an infinite-dimensional mathematical model, which contains the Bernoulli-Euler beam equation, equations of the actuator motion and boundary con ditions. The control voltage fed to the motor is a linear com bination of arm angular position, integral, rotation speed, arm deformations, tip position, and velocity. The principal goal of the article is to find linear feedback gains such that the desired arm position is asymptotically sta ble. This problem of stability arises when an elastic manipula tor is designed. We have analytically described the boundaries of the domains of asymptotic stability in the space of feedback gains and constructed these domains in different cases. It is shown that the stability domains for a flexible arm belong to and are less than the corresponding domains for a rigid arm. We have shown that under feedback on tip position and veloc ity, the one-link flexible manipulator is unstable if damping is small.

Zeeman studies of the first excited states of palladium and metal free phthalocyanines in Shpol’skii matrix

Phthalocyanines (Pcs) are planar molecules of relatively high symmetry and exhibit magneto-optic effects in many ways. We report the Zeeman effect of the first excited states of palladium (Pd–) and free-base (H2–) Pcs in a 5 K Shpol’skii matrix by direct observation of their fluorescence position shifts. From the experimental data an apparent angular momentum integral, coupling the lowest two Jahn–Teller stabilized crystal field split first excited states, Λ′, was found to be 2.7±0.1 ℏ for PdPc. No conclusive Λ′ was deduced for H2Pc but its value can be estimated to be around the theoretical value of 3 ℏ. The experimental data of H2Pc revealed a complex structure in the fluorescence band examined, which complicated the analysis of Λ′ but helped unravel the biexponential decay problem of the structure. We infer one narrower band in the structure has the previously reported lifetime of τ 6.3 ns, and the other broader band being ∼1.8 cm−1 higher, has τ 3.7 ns. We believe this composite situation also exists for the other tautomeric structure since it also exhibited biexponential decay.

Transient excitation of a straight thin wire segment over an interface between two dielectric half spaces

The transient excitation of a straight thin wire segment parallel to a plane interface between two homogeneous dielectric half spaces is analyzed by the continuous‐time, discretized‐space approach. The analysis is carried out in three steps. First, the subproblem of a single wire embedded in a homogeneous dielectric medium, excited by a voltage source and an incident electric field, is studied with the aid of a time domain integral equation. This equation is discretized in space and transformed to the frequency domain. This procedure results in a system of linear equations of a fixed dimension which is solved by marching on in frequency. Second, the subproblem of a horizontal, pulsed dipole over the interface between two homogeneous dielectric half spaces is considered. The reflected field in the upper medium is computed by spectral techniques. With the aid of a time domain Weyl representation, a fixed, composite Gaussian quadrature rule is derived for the semi‐infinite integral involved. The angular integral is evaluated in closed form. Third, for the complete problem, the reflected field in the upper medium is expressed as a superposition of contributions from dipole sources on the wire axis, and subsequently treated as a secondary incident field in the integral equation for the current on the wire. This integral equation is then solved by combining the techniques developed for the two subproblems. Representative numerical results are presented and discussed.

On inverse problems in dynamics of gravitating systems

Abstract An analog of Szebehely's equation is found for non-stationary potential on the basis of the angular momentum integral.

Orbit determination by four observations

AbstractAn elliptic orbit is determined from two short‐arc pairs of observations at different oppositions by the angular momentum integral. Other methods for initial orbit determination than the classical do exist.

Hyperspherical integration and the triple-cross vertex graphs

We have extended the hyperspherical variables method to the analytical calculation of the angular integral of the box graph. We discuss the applications of our results to the analytical calculation of the QED contribution to the electron g-2 of the set of three-loop triple-cross vertex graphs.

Pulse propagation on a semiinfinite horizontal wire above ground: Near‐zone fields

The near zone electric field distribution due to a transient current pulse propagating on a semiinfinite wire above the ground is investigated. A quasi‐transverse electromagnetic approximation to the modal equation for the propagation wave number is used to obtain the time history of the pulse on the wire. Spectral domain techniques are then used to obtain an expression for the electric field. In the resulting two‐dimensional Sommerfeld integrals, representation of the angular integral in terms of incomplete Lipschitz‐Hankel integrals allows for the efficient computation of the frequency domain near and intermediate zone fields. A numerical inverse Fourier transform is then applied to obtain the transient response. Numerical results, presented in both the time and frequency domains, are used to determine when the previously obtained far zone approximations can be applied. The results in this paper are for the quasi‐transverse electromagnetic component of current; however, similar techniques can be applied to the Earth‐attached and continuous modes.

Numerical computation of 2D sommerfeld integrals— Deccomposition of the angular integral

Spectral domain techniques are frequently used in conjunction with Galerkin's method to obtain the current distribution on planar structures. When this technique is employed, a large percentage of the computation time is spent filling the impedance matrix. Therefore, it is important to develop accurate and efficient numerical techniques for the computation of the impedance elements, which can be written as two-dimensional (2D) Sommerfeld integrals. Once the current distribution has been found, then the near-zone electric field distribution can be obtained by computing another set of 2D Sommerfeld integrals. The computational efficiency of the 2D Sommerfeld integrals can be improved in two ways. The first method, which is discussed in this paper, involves finding a new way to compute the inner angular integral in the polar representation of these integrals. It turns out that the angular integral can be decomposed into a finite number of incomplete Lipschitz-Hankel integrals, which in turn can be calculated using series expansions. Therefore, the angular integral can be computed by summing a series instead of applying a standard numerical integration algorithm. This new technique is found to be more accurate and efficient when piecewise-sinusoidal basis functions are used to analyze a printed strip dipole antenna in a layered medium. The incomplete Lipschitz-Hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi-infinite integral.

Numerical computation of 2D sommerfeld integrals— A novel asymptotic extraction technique

The accurate and efficient computation of the elements in the impedance matrix is a crucial step in the application of Galerkin's method to the analysis of planar structures. As was demonstrated in a previous paper, it is possible to decompose the angular integral, in the polar representation for the 2D Sommerfeld integrals, in terms of incomplete Lipschitz-Hankel integrals (ILHIs) when piecewise sinusoidal basis functions are employed. Since Bessel series expansions can be used to compute these ILHIs, a numerical integration of the inner angular integral is not required. This technique provides an efficient method for the computation of the inner angular integral; however, the outer semi-infinite integral still converges very slowly when a real axis integration is applied. Therefore, it is very difficult to compute the impedance elements accurately and efficiently. In this paper, it is shown that this problem can be overcome by using the ILHI representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi-infinite integral. The usefulness of this asymptotic extraction technique is demonstrated by applying it to the analysis of a printed strip dipole antenna in a layered medium.

A new method for computing the reaction between two rooftop basis functions in a planar structure

Accurate and efficient techniques are developed for computing spectral domain reaction integrals involving rooftop basis functions. Representing the angular integral in terms of special functions not only reduces the two-dimensional integration by one dimension, but it also leads to the development of a novel asymptotic extraction technique.

Deducing hard thermal loops from Ward identities

In hot gauge theories, hard thermal loops are the dominant loop corrections at soft momenta. We show that the hard thermal loops in arbitrary amplitudes are determined, up to an angular integral, by their Ward identities.

On a new form of the transfer equation with applications to multiple scattering of polarized light

The stationary transfer equation for polarized light in a plane-parallel medium is transformed into a new equivalent form. In this Q-form transfer equation the intensity vector of the radiation field is equated to an angular integral of its spatial derivative. Expansion in generalized spherical functions enables us to derive explicit expressions for the integral kernel, which become divergent in the case of conservative scattering. In this case, a definite Q-form is obtained by means of K- and F-integrals, which permit us to reduce the original conservative transfer equation to a nonconservative pseudo-transfer equation. The convenience of the Q-form transfer equation is demonstrated for radiative transfer problems in atmospheres with internal sources proportional to integer powers of the optical depth. Explicit formulae are derived for the powers 0 and 1, which yield the intensity vector of the internal radiation field in terms of the surface Green's function matrices. Moreover, the Q-form transfer equations are employed to derive a set of nonlinear integral relations for Green's function matrices. In particular, a new nonlinear equation is obtained, which relates the Green's function matrix of a finite homogeneous medium to its surface values. Thus, we generalize the classical nonlinear equations that are usually derived by using principles of invariance.

Near-field scattering by passive and active small spherical particles

Near-field scattering functions of small spheres are discussed and presented as a function of the distance from the scatterer. Although the total scattered energy defined by the angular integral of the scattering function does not depend on the distance from the particle, the scattering at any given angle in the near field does. In particular, apparent negative scattered energy flux values along the radius vector in given angular ranges are emphasized. The appearance of negative values was reported previously; however, they are shown here to reach any high normalized value, relative to the total scattering, depending mainly on the scatterer size, the distance from the scatterer, and the complex index of refraction. The negative values that were treated previously merely as numerical results are explained in terms of the Ricatti–Bessel-function asymptotic expansion. It is also shown that, excluding the scattering region near 90 deg, the physical significance of these high negative values in the near field is usually negligible relative to the overall field when the coherent interference with the incident wave is taken into account. In the far field the negative values disappear.

A comparison of programming paradigms for the parallel computation of racah coefficients: An application of transputers to computational atomic physics

Computational atomic physics continues to place considerable demands on processor power. In recent years this need has been met by supercomputers such as the CRAY-1, the CRAY-XMP and the CRAY-2. In this paper we consider the calculation of Racah coefficients. In a typical atomic scattering or atomic structure computation tens of thousands of Racah coefficients are needed in the construction of angular integrals. These computations exhibit fine-grain parallelism and cannot be vectorised. Consequently, angular integral computations cannot exploit the architecture of the CRAY machines. In this paper we consider the ability of transputer networks to meet this computational need. Two programming paradigms are described for the parallel computation of a set of over fifteen thousand Racah coefficients first tabulated by Simon et al [1] in 1954. Timings for the two paradigms, computed on a network of seventeen T414 transputers, are presented and discussed.

Swirling recirculating flow an a liquid-metal column generated by a rotating magnetic field

In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting.One-dimensional stirring, where the swirl is independent of z and θ, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow.The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments.The analysis predicts that the swirl velocity scales on {B(σ/ρω)½}ωR, has a maximum at the bottom of the driven region, and penetrates an axial distance of the order ℝR away from the forced region. (For turbulent flow the Reynolds number ℝ must be based on an effective eddy viscosity.) All these features were reproduced experimentally.

Path integral on S2: the Rosen-Morse oscillator

Guided by the group chain SO(3) contains/implies O(2), the authors construct an angular path integral for the symmetric Rosen-Morse oscillator on S2. By explicit path integration, they obtain the normalised energy eigenfunction as well as the exact energy spectrum.

Bounded motion for a modified three-body problem

The energy and the angular momentum integral of motion for the planar three point problem cannot assure bounded motion. In this paper it is shown that if one of the points is replaced by a homogeneous sphere then bounded motion can be found. The zero velocity surfaces for this modified problem are found and their evolution is described.

A statistical theory of orientational ordering at the free surface of a liquid of ellipsoidal molecules

The surface tension γ of a system of ellipsoidal molecules is evaluated approximately using a generalized Fowler-Kirkwood-Buff model. The pair potential is modelled as u(r/σ) where r is the centre of mass distance and σ is an angle dependent range parameter determined by the shape anisotropy of the molecule. It is shown that if the pair correlation function g scales as g(r/σ) γ can be mapped onto the value for a system of spheres, multiplied by an angular integral which takes into account approximately both the anisotropic molecular shape as well as possible orientational order. It is shown that γ is lowered when the orientational order parameter Q ≠ 0, implying that isotropic molecular liquids may be partially ordered near the free surface. The surface favours parallel ordering of rod-like particles in a direction in the plane of the surface. For plate-like particles, ordering with the plates in the plane of the surface is favoured. In both cases the anchoring energies increase sharply with the shape anisotropy of the molecule. These results are qualitatively consistent with the surface properties of non-polar nematic liquid crystals.

Improbability of Collisions in Newtonian Gravitational Systems of Specified Angular Momentum

The angular momentum integral of the equations of motion of the n-body problem of celestial mechanics defines a family of lower dimensional analytic manifolds in the phase space. It is shown that if n is greater than two, then the intersection of the set of initial conditions which lead to collision with each manifold is of measure zero and of Baire first category in the manifold. Thus, the collision initiating set satisfying an arbitrary value of angular momentum is small in both a measure theoretic and topological sense. It follows as a corollary that the converse of Sundman's theorem of total collapse is not true when n is greater than two. Introduction. Consider a Newtonian gravitational system of n point masses. Associated with such a system is a vector constant c, the angular momentum of the system. Let c and n > 3 be arbitrarily chosen and consider only those systems of n point masses associated with this choice of c. In this work it is shown that collisions of point masses in these gravitational systems are improbable and that the set of initial conditions which lead to collisions in these systems is small in both a measure theoretic and topological sense. It follows as a corollary that the converse of Sundman's theorem of total collapse is not true when n > 2. The equations of motion of a Newtonian gravitational system, written as a system of first order differential equations of degree 6n, are