Showing papers on "Singularity published in 1981"

Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses

Abstract: In this paper, the problem of sensitivity, reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability, of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n \times n matrices of H^{\infty} frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility, of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity, to output disturbances can be reduced to an optimal value approaching the singularity, measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitivity resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.

A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves

Abstract: Any graph-manifold can be obtained by plumbing according to some plumbing graph I\ A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity. In this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3-manifolds M(TX) and A/fTj) ob- tained by plumbing according to two graphs Tx and T2 are homeomorphic (3-mani- folds are oriented 3-manifolds throughout this paper, and homeomorphisms of 3-manifolds are orientation preserving). We then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one-parameter families of complex curves. These results are described below. Since the class of 3-manifolds obtainable by plumbing is precisely the class of graph-manifolds, which were classified, with minor exceptions, by Waldhausen (24), a calculus for plumbing is in some sense implicit in Waldhausen's work. Moreover, it has been known for some time that the calculus can be put in a form like the one given here, but the details have never appeared in the literature. A related calculus for plumbing trees has been worked by Bonahon and Siebenmann (1) in order to classify their "algebraic knots". We describe the calculus in greater generality than is needed for the present applications, since this involves minimal extra work, and the calculus is needed elsewhere ((4), (14), and (15)). In particular, in an appendix we describe two generalizations of it. The calculus consists of a collection of moves one can do to a plumbing graph T without altering the plumbed manifold M(T). To see these moves are sufficient, we describe how they can be used to reduce any graph to a normal form which is

Scattering of electromagnetic waves from random media with strong permittivity fluctuations

Abstract: By taking into account the singularity of the dyadic Green's function in the renormalization method, a theory is derived for vector electromagnetic wave propagation in a random medium with large permittivity fluctuations and with anisotropic correlation function. The strong fluctuation theory is then applied to a discrete scatterer problem in which the permittivity can assume only two values. The results are found to be consistent with those derived from discrete scatterer theory for all values of dielectric constants of the scatterers.

Intermittency in nonlinear dynamics and singularities at complex times

Abstract: High-pass filtering of turbulent velocity signals is known to produce intermittent bursts. This is, as shown, a general property of dynamical systems governed by nonlinear equations with band-limited random forces or intrinsic stochasticity. It is shown that singularities for complex times determine the very-high-frequency behavior of the solution and show up in the high-pass filtered signal as bursts centered at the real part of the singularity and with overall amplitude decreasing exponentially with the imaginary part. Near a singularity, nonlinear interactions, however weak they may be on the real axis, acquire unbounded strength. Investigations of singularities by nonperturbative methods is thus essential for quantitative analysis of high-frequency or high-wave-number properties. In contrast to results based on two-point closures, the high-frequency dissipation-range spectrum is actually not universal with respect to the low-frequency forcing. Unlimited intermittency is demonstrated, i.e., the flatness of the high-pass filtered solution grows indefinitely with filter frequency. This gives strong support to a conjecture of Kraichnan [Phys. Fluids 10, 2080 (1967)] about intermittency in the dissipation range of turbulent flows. The analysis is carried out in great detail for the nonlinear Langevin equation $m\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{v}=\ensuremath{-}\ensuremath{\gamma}v\ensuremath{-}{v}^{3}+f(t)$. Lorenz's three mode system and Burgers's model are also discussed. Conjectures are made about Navier-Stokes turbulence which can be checked experimentally and numerically.

On the singular behavior at the vertex of a bi-material wedge

Abstract: Information on the singular behavior at the vertex of a bi-material wedge is the objective of this paper. A summary of the necessary conditions, which depend heavily on the associated eigenvalue equation, for stress singularities of O(r -λ 1n r) as r→0 or O(r -λ) as r→0 is stated. The eigenvalue equations arising from a wide range of boundary and interface conditions are then provided. Bi-material wedge problems that have been subjected to singularity analyses of some generality in the literature are briefly reviewed.

Symmetry and bifurcations of momentum mappings

Abstract: The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.

Numerical Differentiation of Analytic Functions

Abstract: It is well known that the classical difference formulas for evaluating high derivatives of a real function f(ζ) are very ill-conditioned. However, if the function f(ζ) is analytic and can be evaluated for complex values of ζ, the problem can be shown to be perfectly well-conditioned. An algorithm that performs this evaluation for an arbitrary analytic function f(~) is described. A short FORTRAN program for generating up to 50 leading derivatives is to be found in the algorithm section of this issue. To use this program, no knowledge is required either of the method or of the analytical nature (e.g., position of nearest singularity, its type) of the function.

The K-pulse concept

Abstract: Two general classes of inversion techniques may be distinguished: those utilizing angular or spatial frequency analysis over as wide a range of target aspects as practicable, and those involving relatively few aspects but employing as wide a frequency spectrum as possible. The latter technique is discussed, and a special excitation waveform called the "kill-pulse" or K-pulse is proposed. By analogy with lumped and distributed networks, a unique aspect-invariant excitation waveform is postulated for an isolated scatterer. This waveform, of finite and minimal duration, then characterizes the pole spectrum of the scatterer as used in the singularity expansion method (SEM) or for target identification. The derivation of this excitation waveform and its relation to one or more surface waves is illustrated by several examples. Using surface waves initiated at certain points of an object, the complex attenuation of circumnavigating waves is estimated by the geometrical theory of diffraction. Three-parameter characteristic equations are derived which predict pole strings with excellent accuracy for conducting spheres and cylinders and lesser accuracy for prolate spheroids of several axial ratios. For spheres and cylinders, the same characteristic equation also yields good estimates of cavity resonance frequencies.

Critical exponents for the percolation problem and the Yang-Lee edge singularity

Abstract: Gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields. The results have already been reported, give the exponents to third order in epsilon where epsilon =6-d and d is the dimensionality of space. The class of models includes the percolation problem and the Yang-Lee edge singularity, so the authors give explicit results for the exponents to order epsilon 3 in these cases. By resummation methods, based on the symptotic behaviour of the epsilon expansion, they obtain numerical estimates for these exponents for a number of interesting values of d.

Analysis of a circular microstrip disk antenna with a thick dielectric substrate

Abstract: The problem of a circular microstrip disk excited by a probe is solved using rigorous analysis. The disk is assumed to have zero thickness, and the current on the probe is taken to be uniform. Using vector Hankel transforms the problem is formulated in terms of vector dual-integral equations, from which the unknown current can be solved for. Due to the singular nature of the current distribution arising from probe excitation, the direct application of Galerkin's basis function expansion method gives a slowly convergent result. Therefore the singular part of the current is removed since the singularity is known a priori. The unknown current to be solved for is then regular and tenable to Galerkin's method of analysis. It is shown that this analysis agrees with the single-mode approximation when the dielectric substrate layer is thin, and that it deviates from the single-mode approximation when the substrate layer is thick. Excellent agreement of both the computed real and imaginary parts of the input impedance with experimental data is noted. The radiation patterns and the current distributions on the disk are also-presented.

Augmented electric‐ and magnetic‐field integral equations

Abstract: Augmented electric- and magnetic-field integral equations, which preserve the basic simplicity, solution capability, and pure electric- and magnetic-field character of Maue's original integral equations, are introduced to eliminate the spurious resonances from the exterior solution of the original integral equations. The exact dependence of the original and augmented integral equations on the geometry of the principal area (self patch) which excludes the singularity of their kernels is also determined, and alternate forms for the integral equations are provided that avoid integrals dependent upon the geometry of the principal area. Numerical results obtained for scattering from the perfectly conducting cube, sphere, and infinite circular cylinder confirm the theoretically predicted result that the augmented integral equations eliminate the spurious resonances for all perfectly conducting scatterers except the sphere.

Singularities in Classical Mechanical Systems

Abstract: Singularities in the equations of motion of a classical mechanical system usually play a dominant role in the global phase portrait of the system. By a singularity we mean a point or set of points where the system is undefined, as in the case of a collision between two or more of the particles in the n-body problem. Such singularities often lead to a complicated global orbit structure. Not only do certain solutions tend to run off the phase space, but also nearby solutions tend to behave in an erratic or unpredictable manner. Numerical studies of such systems are often inconclusive because of this erratic behavior. And power series or other analytic techniques often yield only a very local description of solutions near the singularity, one which gives no hint of the global complexity of the system.

A semi-analytic solution for nonlinear standing waves in deep water

Abstract: A method of time-dependent conformal mapping is introduced to simplify the power-series solution procedure for time- and space-periodic standing waves in deep water. A solution has been found to 25th order in the wave amplitude. The values of certain coefficients are determined by the requirement that secular terms must be suppressed. Because the series for the wave profile is not always uniformly convergent, Pade approximants are used for summation. For very high waves, the slope of the surface has at least two relative maxima. The singularity structure of the solution is also discussed.

Hall Voltage Dependence on Inversion Layer Geometry in the Quantum Hall-Effect Regime

Abstract: A calculation of the Hall voltage is presented within a model of a finite two-dimensional inversion layer. An explicit form for the electric field is obtained and this is found to have a power-law singularity in the corners of the inversion layer. This singularity is most pronounced in the quantum Hall-effect regime where the Hall angle approaches $\frac{\ensuremath{\pi}}{2}$. The error in measuring the Hall voltage in this regime due to the shorting effect of the source and drain is calculated. This is found to be negligible at the level required for a new determination of the fine-structure constant and development of a new resistance standard using inversion-layer measurements in the quantum Hall-effect regime. Limitations of the model and other possible sources of error are briefly discussed.

Crack-tip fields for fast fracture of an elastic-plastic material

Abstract: A n asymptotic analysis of the near-tip fields is given for transient crack propagation in an elastic-plastic material. The material is characterized by J 2 flow theory together with a bilinear effective stress-strain curve. Both plane stress and plane strain conditions have been considered. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variation of the near-tip stresses, all as functions of the crack-tip speed and the ratio of the slopes of the two portions of the bilinear stress-strain relation. It was found that the results are much more sensitive to the elastic-plastic constitutive relation than to the crack speed. This result is important for numerical analyses of dynamic crack propagation problems.

Removal of Goldstein's singularity at separation, in flow past obstacles in wall layers

Abstract: It is shown that, in the flow of a viscous wall layer past a relatively steep obstacle at the wall, the Goldstein (1948) singularity generated in the classical boundary-layer approach to separation is removable in a physically sensible fashion. The removal is effected by means of a sequence of local double structures, the last of which arises just beyond separation owing to the occurrence of a further singularity which is also removable and describes the necessary complete breakaway of the viscous layer from the wall. The novel forms of the local pressure–displacement relations are the key elements allowing the solution to retain physical reality throughout. Beyond the breakaway the reattachment process takes place only at a relatively large distance downstream, before the motion returns to its original uniform shear form. The present flow configuration, the first we know of where Goldstein's singularity proves to be removable, has important applications in both internal and external flows at high Reynolds numbers and these are also discussed.

Singularity subtraction in the numerical solution of integral equations

Abstract: The singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.

Correlation functions of amorphous multiphase systems

Abstract: We give two general integral expressions for the first and second derivatives of the so-called stick probability functions which are commonly used in analyzing x-ray-scattering results. Then it is shown that by letting the length of the stick go to zero, the limit of the second derivative can be expressed in terms of an integral over the singularity lines of the surfaces which separate the different phases of the sample. In this way one has achieved the generalization of the well-known result that the limit of the second derivative is always zero when phase boundaries are smooth.

The physical content of the singularity expansion method

Abstract: The singularity expansion method describes the echo return from pulsed radar signals in terms of a Prony‐series superposition of damped sinusoids. These are obtained as the residues of poles of the scattering amplitude in the complex frequency plane. For the example of radar scattering from a conducting sphere, we show that these poles can be regrouped into infinite subsets whose residues sums represent individual creeping waves, the nth pole in the subseries corresponding to the resonance caused by a standing creeping wave with n+1/2 wavelength spanning the circumference. The corresponding Prony subseries thus appears simply as a mathematical device which synthesizes the physical creeping wave.

On a generalization of Kaden's problem

Abstract: Kaden's problem of the roll-up of an initially planar semi-infinite vortex sheet with a parabolic distribution of circulation is extended to include vortex sheets exhibiting a general power law circulation distribution, resulting in the presence of a power law, and in one case a logarithmic-like, velocity-field singularity. Both semi-infinite and infinite initially plane sheets with this property are considered and the form of their roll-up in the similarity plane, into single and double-branched spirals respectively, is obtained numerically. Estimates of the Betz constant obtained from the solutions are found to be significantly different from values predicted by the Betz approximation.

Molecular wave functions in momentum space

Abstract: Momentum-space calculations exhibit two kinds of advantages over position space: First, the numerical solution of Hartree-Fock equation is feasible without expansion of the wave functions in a particular basis. Equations only exhibit one avoidable singularity even for the multicenter case. Several mathematical techniques are presented, including standard fast Fourier-transform (FFT) techniques and numerical calculation of the involved convolutions. Second, momentum representation contributes in an original way to a better understanding of several physical problems arising in quantum chemistry. The two-body density matrix involving the electronic correlation are examined in both position and momentum space. If an expansion in Gaussian functions is used, momentum space renders feasible the obtainment of a multidimensional fully correlated wave function, starting from the Hartree-Fock solution.